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Content.
Balkan Journal of Mechanical Transmissions, Volume 2 (2012), Issue 1, pp. 1, ISSN 2069–5497.
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Balkan Journal of
Mechanical Transmissions
Volume 2 (2012), Issue 1, pp. 1
ISSN 2069–5497
CONTENT
CHAPTER. Editorial
DOBRE, G. Editorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
CHAPTER. Contributed papers
ALEXE, A., DOBRE, G. A new vision on the information management solutions of the product
development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
IVANOV, K. Design of toothed continuously variable transmission in the form of gear variator . . . . . . . 11
IVANOV, K. Dynamics of gear differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
KAMBLE, S., CHAVAN, U., JOSHI, S. Valve jump minimization using n-harmonic cam profile . . . . . .27
KUZMANOVIC, S., RACKOV, M., RAFA, K. Benefits of the application of cycloidal backlash gear
reducers with more eccentric shafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
MARTIKKA, H. Gearbox modification to obtain continuous controllability . . . . . . . . . . . . . . . . . . . . . . 39
MIHAILIDIS, A., GATSIOS, S. Simulation of gear contact on a two-disk test rig . . . . . . . . . . . . . . . . . 45
NENOV, P., VARBANOV, V., ANGELOVA, E., KALOYANOV, B. Design and optimisation of
cylindrical gear drives based on IS0 6336 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
RADULESCU, M. An extension of the electromechanical analogy in the domain of hydrostatic
transmissions. Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
VEJA, A. D. Contributions concerning the roller gearing tooth profile generation . . . . . . . . . . . . . . . . . . 71
ROmanian
Association of
MEchanical
Transmissions
(ROAMET)
Balkan Association of
Power Transmissions
(BAPT)
2
Balkan Journal of Mechanical Transmissions
Volume 2 (2012), Issue 1, pp. 2 ISSN 2069–5497
EDITORIAL George DOBRE
Publisher & Editor-in-Chief
The Balkan Journal of Mechanical Transmissions (BJMT) became a visible presence in the world of technical publications. This has come as a result of the co-ordinated actions of two representative European associations: the Balkan Association of Power Transmissions (BAPT) and the Romanian Association of Mechanical Transmissions (ROAMET). BJMT is available online and it is a free publication. No similar publication currently exists in Europe dedicated to mechanical transmissions, and as such BJMT attracted a number of specialists in one of the areas of main mechanical engineering, namely power and particularly mechanical transmissions.
The main scope of the journal is to disseminate a high number of papers of high scientific and practical quality in its thematic area of recognized social-economic benefits. This scope is correlated with other aims targetted by the Editorial Board of the journal:
1. to achieve a recognition through indexation in Thomson ISI and other abstracting services;
2. following also from this previous reason, to always have a number of high quality submissions awaiting publication.
The present issue is published after the successful conclusion of the 4th International Conference on Power Transmissions, June 20-23, Sinaia, Romania. The organizational effort for the PT 12 conference started two years ago, along with the publication of the website, see www.pt12.pub.ro, and elaboration of the call for papers. The website has been regularly updated to give all present and current information. Some activities have been carried out online (sending of papers, registration forms). The first major solution adopted by the Local Organizing Committee was to publish the proceedings in English in the Mechanism and Machine Science Bookseries under the auspices of the SPRINGER Publishing House. This book will be
indexed in e-resources such as:
SpringerLink Contemporary;
ScienceDirect;
Scopus;
Web of Science;
Cambridge Scientific Abstracts (CSA);
Ei Compendex (CPX);
Google and Google Scholar, etc.
The Local Organizing Committee of PT 12 conference received more than 80 papers, from which about 65 papers have been retained for presentation and 58 have been presented. There are 106 authors from 16 different countries, as follows: Bulgaria – 13 authors; China – 1; Croatia – 2; Germany – 2; Greece – 3; India - 3; Italy – 2; Japan – 9; Kazakhstan – 1; Netherlands – 2; Republic of Macedonia – 2; Republic of Moldova – 3; Romania – 39; Russia – 1; Serbia – 18; USA – 3.
Thus the real success of PT 12 conference is a veritable impetus to continue under the best auspices the work at the Balkan Journal of Mechanical Transmissions.
We remind our readers that the Balkan Journal of Mechanical Transmissions is a biannual electronic publication that is planned to be issued in the last month of each semester, i.e. June and December.
CORRESPONDENCE
George DOBRE, Prof. Dr. University POLITEHNICA Faculty of Mechanical Engineering and Mechatronics 313 Splaiul Independentei 060042 Bucharest, Romania [email protected]
ROmanian Association of MEchanical
Transmissions (ROAMET)
Balkan Association of Power Transmissions
(BAPT)
3
Balkan Journal of Mechanical Transmissions
(BJMT)
Volume 2 (2012), Issue 1, pp. 3-10
ISSN 2069–5497
A NEW VISION ON THE INFORMATION MANAGEMENT SOLUTIONS OF THE PRODUCT DEVELOPMENT
Adrian ALEXE, George DOBRE
ABSTRACT. This paper will highlight the criteria needed for an information management solution to offer customers a scalable enterprise grade solution. This collaborative approach is designed to bring together people, processes, data and systems to deliver products to market faster. The idea of a PLM solution is not something new but through PLM you can bring a set of new business processes that coordinate and synchronize the activities and deliverables starting from the administration and controlling the information that helps companies to develop the right product for the right market at the right time till the product is obsolete.
KEYWORDS. Product Lifecycle Management, Information Management Solutions, Engineering Design, Requirements Management, Synchronous Product Development
1. INTRODUCTION
Leading companies that are delivering complex products in different markets must overcome many challenges to manage products throughout their enterprise. The most difficult challenges are integrating complex systems engineering processes, promoting collaboration among multiple engineering disciplines, and enabling the sharing of intellectual property among globally dispersed teams.
Companies searching the way to manage complex products efficiently can leverage a comprehensive product lifecycle management (PLM) solution to:
• reuse intellectual property as to reduce time to market;
• increase innovation;
• improve overall traceability of the requirements;
The literature in area of PLM is very reach; some references are given in paper, including the theoretical approaches (Blackburn at al., 1996; Leffingwell, 1997; Grady, 1999; Hofmann and Lehner, 2001; Rainey, 2005) and results of companies (DASSAULT Systèmes, IBM, SIEMENS, PTC, etc.).
In today market only a CAD solution is not enough as to bring innovative products at the right time in the market. Now it is very important to integrate products and processes among the company and this can be done only with the help of an information management solution that can leverage the knowledge in the extended enterprise.
Information management solutions for product development processes have evolved significantly since the beginning due to the Internet evolution (web 2.0) and have, as objective, the improvement of productivity (decreasing costs and product cycles increasing in the same time the quality of the product).
The problem that remains is how to implement these solutions in different engineering fields and what are the new methodologies that can be used to integrate complex engineering processes and to coordinate and synchronize these activities.
Lots of companies are studying the PLM problem in general, but they are not providing information about implementation methodology and best practices that must be followed. They are not specialists in engineering as to provide companies a set of best practices that will suite companies that are producing specific products and are having dedicated procedures.
For sure it is not visible in literature the best in class functionalities that top companies are using for successful product development. This is because best practices are corporate knowledge and their main advantage on the market. For sure one can identify the software tools that performing companies are using for design, manufacturing and now for managing the engineering data.
Two of the mostly used tools by top companies are requirements management and synchronous product development (SPD). Companies must avoid the inefficiencies in managing products that result in costly product development headaches ranging from rework to customer dissatisfaction and rising development costs.
Requirements management allows organizations to capture the “voice of the customer” and translate that information into new products. Two essential areas of focus that enable companies to best leverage requirements management are process integration and traceability. Process integration eliminates communication barriers by creating a collaborative environment for sharing ideas, requirements and data throughout the product lifecycle. Increased innovation, in turn, fosters ideas that improve products and new
ROmanian
Association of
MEchanical
Transmissions
(ROAMET)
Balkan Association of
Power Transmissions
(BAPT)
4
product introductions. In addition, continuous traceability - from capturing the customer needs to product definition - makes a significant difference in project cycle times and cost reductions.
With SPD one can coordinate and synchronize the activities of global teams that are working in many engineering domains. SPD will allow top companies to take a systems-based approach that requires simultaneous and connected development that occur between software, electronics and mechanical functions at the earliest design stages and continue throughout the product development cycle.
This article analyzes the new idea of bringing together the voice of the customer (the first step of requirements management) with SPD in an information management solution that will allow companies to invest more in the concept phase. By investing more time in the concept phase the application of these new tools will allow companies to diminish the errors, speed up the production, increase the functionalities of the product and secondly one can come in market with state of the art products.
The paper presents a view over requirements management and SPD providing some application aspects regarding the combination of these two tools in a PLM system at companies are discussed. Final conclusions are emphasized.
2. REQUIREMENTS MANAGEMENT
2.1. Introduction
Requirements management is the consistent, prioritized, and monitored approach for administering and controlling the information that helps an enterprise develop the „right product for the right market at the right time”. The challenge for most enterprises is not capturing customer requirements (needs). The true challenge is the need for continuous communication, change management and traceability
(enforcement) of customer requirements throughout the development cycle.
As it is mentioned on IBM website in a recent industry study, organizations surveyed incurred a cost of as much as 60 percent on time and budget when they used poor requirements management practices. Organizations with poor business analysis capabilities had three times as many project failures as successes. With proper requirements management and definition, organizations can reduce project overruns by as much as 20 percent by limiting the number of inaccurate, incomplete, and omitted requirements.
2.2. Product development costs can decrease with effective requirement management
As mentioned by Leffingwell (1997), requirements definition in the early stages of a product is a major factor in rising development costs. Requirements errors specifically can account for 70% to 85% of rework costs.
For example, product rework can:
• represent about 40% of a development organization’s total spend — with a significant effort focused on correcting requirements defects as found Grady, Robert B. (1999);
• consume 30% to 50% of total product development costs as shown by Boehm and Papaccio (1988), while requirements errors specifically account for 70% to 85% of rework costs as showed Leffingwell (1997).
Fig. 1 illustrates, correcting requirements errors after a product is released can cost over 100 times more than anticipated (Grady, 1999). Best-in-class companies have learned to make the requirements definition visible early in the product development lifecycle.
Requirements Design Prototype/Code Test/Validate Production
Relative cost to correct a defect
Development phase
Fig 1. Relative cost to correct a requirement defect depending on when it is discovered
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In a typical product development lifecycle, eliciting, authoring, analyzing, and managing requirements represents about 10% of a project’s resources. As demonstrate Hofmann and Lehner (2001) the most successful projects spend roughly 28% of their resources on requirements. This research also shows that early requirements management improves time-to-market and helps keep costs on target.
2.2. Effective requirements management challenges
Today, most standalone requirements management solutions fall short after the initial requirements capture and analysis. Many companies use groupware applications and manual processes to manage product requirements. These manual processes are slow and prone to error. Groupware solutions create “information silos” which are not integrated with downstream processes directly. Project teams must spend costly manual person hours to ensure traceability, driving up product overhead costs.
Requirements management vastly improves decision support by linking business functions and key areas of product development, providing:
• a global collaboration via a common platform that fosters innovation by bridging the gap across all disciplines to share requirements, design and product launch data;
• effective traceability allowing direct links to product line planning, systems design (features/options), designed product definitions (Engineering Bill of Material - BOM releases), and use-case testing (prototype and test);
• automated governance (rules) to enforce validation at each development state (e.g., system engineering, design, prototype, and test), ensuring that all requirements are met.
2.3. Downstream process integration and traceability
Integration of requirements with downstream processes removes “information silos”. This collaborative environment leverages shared ideas and data throughout the product development cycle. A common, shared view of customer needs drives innovation, whether an evolutionary change in existing products or a revolutionary new product. Collaboration also supports continuous improvement of products and business processes. Typically today 61% of a company’s profits are generated from new innovative products (i.e. new ideas, radical concepts) and 86% from product extensions of current product lines.
3. SYNCHRONOUS PRODUCT DEVELOPMENT
3.1. Introduction
Synchronous Product Development (SPD) is a set of business processes that coordinate and synchronize the activities and deliverables of global product development teams, working in multiple engineering disciplines, in any phase of the product lifecycle (as showed by ENOVIA Matrix One (2006) in Fig. 2). SPD serves to bridge together the various design disciplines, and at the same time, ensure that designs reflect the needs of customers and the capabilities of the various manufacturing sites.
A major impediment to system development is ensuring that design activities are properly aligned with requirements. This is even more complex when
the product’s capabilities require early design collaboration between multiple design disciplines. Engineers and designers in each discipline perform fantastic feats to squeeze maximum functionality out of their designs. However, efforts are often compromised when they are confronted with limitations imposed upon their designs by the other
engineering disciplines trying to fulfill competing product needs.
Consider this scenario: A new interactive product that is being developed requires significant software modifications in order to meet new marketing requirements. The software modifications (enhancements) result in unacceptable response
Fig. 2. Synchronous product development (about ENOVIA Matrix One (2006))
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times, so a new processor is required to provide greater computational speed and capacity. But the new processor puts out more heat and thus requires changes be made to the product package to improve heat dissipation. Because identification of the need for a new processor and an upgraded package occur late in the design cycle, product launch has to be delayed. Which design team made a mistake? Did they all make mistakes? Which team needs to re-work its product design and slip far behind schedule on this project and perhaps on other projects as well? And who will be responsible for the extra costs incurred for
re-developing a solution? Situations like this often give rise to software, electronics and mechanical design teams viewing and treating one another as adversaries rather than as teammates working together toward a common goal.
Scenarios such as the one outlined above make systems design a real challenge. In addition, product development teams must also cope with late-stage design changes caused by shifting customer needs and website-specific manufacturing capabilities.
Fig. 3. New car product development lifecycle without SPD (about ENOVIA Matrix One, 2006)
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As seen in Fig. 3, there is little benefit from designing an incredible new product if it does not satisfy customer requirements or cannot be manufactured at a reasonable cost, if at all. Yet, many companies have disjointed processes in place for keeping design teams abreast of shifting product requirements and manufacturing- based design parameters. Design teams must focus on design, not on tracking down requirements of the upstream and downstream teams that should be providing product requirements data to the designers. This is where having a strategy of Synchronous Product Development combined with requirements management pays off. With these two strategies in place, designers will work from a single source of product information that is continuously updated by upstream and downstream functions. Moreover changes are automatically pushed to the various design teams as soon as they occur, which enables designers to focus on their core work and adjust designs before they are locked. Fig. 4 depicts an organization that has effectively implemented SPD processes to streamline all dimensions of product development through information-sharing between design realms and their upstream and downstream stakeholders of product management and manufacturing. As the Fig. 4 illustrates, processes are in place to automatically furnish the design teams with all the information they require to do their job right – the first time.
Synchronous Product Development is achieved when companies establish and adhere to the three crucial components of product development. These are:
• a product planning framework;
• one unified product design;
• aligned product views between design and manufacturing.
3.2. A product planning framework
A product planning framework, often referred to as “system design,” is a top-down approach to decomposing products into planned product options and features across all design disciplines to ensure consistent and interoperable development processes. Most companies’ software, electronics and mechanical design disciplines utilize independent design logic and processes. But systems-based design provides one unified set of processes to ensure that all design disciplines are operating in unison. A product planning framework enables and mandates that all design disciplines take the following critical steps when designing components parts to ensure a systems approach:
• capture requirements;
• plan product options and features to account for multiple design disciplines;
• track design iterations;
• validate that requirements have been satisfied;
• design for part re-use and modularity;
This type of requirements management and product planning inherent in system design has proven successful in software development environments and Aerospace & Defense product development environments – both of which have traditionally had a highly specialized need to continuously capture and validate requirements data. But today, as companies in all industries need to develop increasingly complex products to meet shorter market opportunities, they must establish a product planning framework – prior to defining their products.
3.3. One unified product design
With a solid understanding of a product’s requirements product development teams can begin to design the product itself. This “product design” occurs in two stages: “informal” and “formal.”
Informal product design occurs when designers are free to explore any solution to satisfy a product’s requirements. In this stage, designers must have the ability to “white board” their ideas and collaborate on them in an environment that is unencumbered by design rules. Informal product design should also enable designers to view and mark-up one another’s work within any design tool and across multiple design tools. Such a free flowing, ad hoc environment enables designers to be as creative as possible, enabling true innovation to take place.
“Formal” product design brings order to the creative environment. In the formal product design stage, rules are applied to design processes to ensure that all of a product’s component parts are accounted for and seen by all stakeholders – in a unified environment. The rules mandate that design teams take the following steps:
• consolidate part design content into a single definition of the engineering BOM;
• adhere to one cross-functional engineering change process;
• qualify and manage purchased parts;
Taking these steps, designers are able to follow a path to product development success that is predictable, consistent and repeatable.
3.4. Aligned product views between design and manufacturing
Many product companies have adopted a Design Anywhere, Build Anywhere strategy, and as a result, have manufacturing sites spread around the world. Very often, manufacturing plants require product design changes to suit their “local” needs and/or capabilities. Not all plants can work from the same Manufacturing BOM because of variables such as local supplier constraints, manufacturing capabilities and/or local market requirements and regulations. To ensure that each site manufactures products to specification (as design teams intended),
Fig. 4. New car product development lifecycle with SPD (about ENOVIA Matrix One, 2006)
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manufacturing teams must be able to collaborate with design teams on unreleased Engineering BOM is to detect and correct problems before release to manufacturing. Aligning design and manufacturing views of the product can be achieved by combining Engineering BOM management with Manufacturing BOM management. This step is easy to achieve if companies leverage their formal product definition environment, the previous stage in Synchronous Product Development. By bringing Manufacturing BOM management into the formal product definition stage within engineering, design and manufacturing teams can apply all the rules and processes of the formal product definition environment to the Manufacturing BOM. The result: streamlined product launch and fewer costly product recalls.
Many product companies already leverage elements of Synchronous Product Development and should not look to throw away processes that already prove effective for launching new products. Doing so would disrupt design teams too much and interfere with ongoing competitive practices. However, bringing planning and order to existing product development activities will provide greater levels of speed and consistency in driving new products to market. Thus, companies should look to improve their current product development activities by identifying areas for improvement and applying SPD best practices and solutions to them in a staged approach. This approach provides several key benefits mentioned below.
• Product development teams can continue to work in an environment that is comfortable and well-known to them.
• Return of investment (ROI) of SPD investments can be proven at each stage of implementation prior to investment in subsequent stages.
• Companies can target the areas of greatest need first to gain competitive advantage and then obtain executive commitment for future stages.
4. SOME APPLICATION ASPECTS
4.1. Introduction
As this article described previously it has been proved successful in Aerospace & Defense product development environment but this new solutions and work methodologies can be applied to any company that is delivering complex products.
In this article one will concentrate the application of requirements management and SPD in the industrial equipment segment that includes a variety of companies who are designing and manufacturing diverse set of products:
• machine tools;
• heavy machinery, such as mining and construction equipment;
• farm machinery;
• packaging machinery
• air conditioning refrigeration and heating
machinery.
Companies in this segment cover a wide range of industries and operating models. In general, all of these industries have a need for information management solutions in the product definition lifecycle and in managing the design supply chain.
Although diverse in function, this group of products has many things in common regarding product design, manufacturing and management of the supply chain; all areas that require PLM enablement.
A short definition of these common processes in a company and more exactly in an industrial equipment company that require an information management system is given below.
4.2. Design activities in industrial equipment industry
Companies from the industrial equipment industry are delivering complex products that include many subassemblies with mechanical, electronic, and software components that require close tolerances and interactions. These products are purchased by industrial customers and are often made-to-order. These products are usually major capital investments for customers, with prices ranging from hundreds of thousands to millions of dollars.
The pressure on this type of companies is to innovate as well as hold to high quality and performance standards at competitive costs. Design reuse is essential in leveraging the intellectual assets and increasing the pace of innovation at a typical best in class companies, the ability of PLM to help people locate and quickly share types of design information (e.g., 3D models, design context, parameters, etc.) is critical.
Usually companies that are designing industrial equipment products are dividing the general assembly in multiple subassemblies and in multiple parts especially per engineering domains: software, electrical, mechanical, etc. These sets of subassemblies and parts are allocated to different teams and different peoples. Under the delivery pressure they are getting all the parts together and building the new products. On the last 100 meters one will observe that they have issues with the final assembly due to the lack of collaboration between teams.
4.2. Design chain management
Due to the fact that a product from the industrial equipment industry is composed by millions of parts, it is impossible for a company to develop all of them internally. In this case they will work with a supply chain and they have to share product information also with customers and suppliers. The lack of a PLM solution will mean lots of time spent by sending the details by emails and FPT (File Protocol Transfer).
One will not detail here the problems that will appear due to the lack of communication, misunderstandings and exceeded delivery time.
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An information management solution enables this sharing and collaboration throughout design and manufacturing processes.
4.3. Manufacturing
Manufacturing these products involves very complex bills of materials plus a combination of in house and contracted manufacturing of parts and assemblies. Without an information management system companies will not be able to streamline their processes and will not be able to share design and product structure information both as drawings, and as 3D geometry.
The manufacturing processes should be firstly simulated by the technical design team and in the end sent to the shop floor or to the suppliers. It’s a must for the design team to have all the details (size, controller, maximum speed) of the machine that will manufacture the part as to simulate the most efficient cycle time.
4.4. Requirements management solution combined with SPD for an industrial equipment
company
It is known (UNESCO) that there is a lack of engineers in the world. This means that all the engineering functions (design, manufacturing, maintenance, etc) will be affected in the future.
From our experience in applying the requirements management solution combined with SDP for Romanian companies that are working in industrial equipment industry we have clear results that if they are investing more time in the research and development (R&D) they will decrease the errors and implicit the number of engineers in the shop floor.
Our suggestion for this investment in R&D means to create three distinct engineering offices. One will take care of design and new functionalities, one for engineering and the last for fabrication technology and manufacturing preparation.
The first office will take care of understanding the “voice of the customer” (usually their customers that are demanding different types of machines like CNC (computer numerical control) machine tools, comprising horizontal boring and milling machines, bridge type milling machines, vertical lathes, automatic roll grinding machines, vertical rotary surface grinders) when they are thinking new solutions. This team will be free to explore any solution. In this way this team will be as creative as possible, enabling innovation to take place.
With an SPD solution, the second team that will practically design and simulate the new technologies that are proposed by the first team will understand exactly the role and the functionality of that assembly, machinery, etc. They will apply the design processes to be sure that all of a product’s component parts are accounted. Another role of this team is to define and create the engineering BOM and to validate the parts that must be bought.
The third team can came closer to these first two
teams as to see how they can build that machinery and what technologies they can use. By getting together these teams with than one can decrease the time sent between engineering, can reduce the number of errors and also will decrease the needed time for manufacturing because one will know exactly from the early beginning what is the best solution and what is best cycle time for manufacturing.
The implementations of this kind of products on the base of requirements management and SPD are taking lot of time, usually 2 to 3 years, due to the series of procedures for complex product. But when the work methodology is stable the advantage will be seen quickly due to the dynamic design environment that will provide there ability to rapidly and iteratively modify their product and process designs.
These types of companies are usually called engineering-centric companies and their competitive advantage is product design. These companies are viewed as innovators that are bringing world – class products to the market. People generally buy from these companies because of the innovative and unique products they sell.
CONCLUSIONS
The purpose of this work was to see the impact of new approaches applied to companies from the industrial equipment.
This article emphasizes a new working methodology of combining these two solutions, and applying them on a real company.
After the implementation on a real company one has seen that even the time of implementing a requirements management solutions combined with SPD is very long it brings a series of competitive advantage that are more valuable for their long time strategy. In today global market the key of success is innovation. As one can see from the above, the synchronous product development and requirements management approaches fits extraordinary for engineering centric companies but not only. For sure these approaches can be applied to manufacturing centric companies but the entire methodology must be re-engineered.
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CORRESPONDENCE
Adrian ALEXE, PhD Student
University POLITEHNICA
Faculty of Mechanical Engineering and Mechatronic
313 Splaiul Independentei
060042 Bucharest, Romania
George DOBRE, Prof. Dr.
University POLITEHNICA
Faculty of Mechanical Engineering and Mechatronic
313 Splaiul Independentei
060042 Bucharest, Romania
11
Balkan Journal of Mechanical Transmissions
(BJMT)
Volume 2 (2012), Issue 1, pp. 11-20
ISSN 2069–5497
DESIGN OF TOOTHED CONTINUOUSLY VARIABLE TRANSMISSION IN THE FORM OF GEAR VARIATOR
Konstantin IVANOV
ABSTRACT. Continuously variable transmission contains the closed gear differential and the controlled device. Recently the idea of construction continuously variable transmission in the form of the mechanism with two degrees of freedom and one input is patented by inventors. However in these patents there is no theoretical basis of interrelation of parameters of drive in the conditions of uncertainty of motion. It does impossible definition of transmission parameters on the set conditions of motion. Earlier it is proved that in such drive there is an additional constraint. The purpose of the present paper is to analyze the additional constraint and to prove presence of kinematic and power definability of transmission. It is proved that closed gear differential in itself carry out a function of a controlling device. The gear differential with the closed contour containing toothed wheels creates additional constraint and supports a transition regime of motion from a start-up to bi-mobile regime with independent stepless regulating. The found regularities allow design gear continuously variable transmission with constant engagement of toothed wheels on the given operational parameters of motion.
KEYWORDS. Gear transmission, closed contour, equilibrium, stepless regulating, design
NOMENCLATURE
Symbol Description CVT Continuously variable transmission
H Carrier
F Force R Reaction M Moment s Displacement v Speed φ Instant angle of turn ω Angular speed a, b, c, d, l, e
Sizes of links
u Transfer ratio
1. INTRODUCTION
Force adaptation in mechanics is a mechanical system ability to be self-maintained adapting to an output variable load.
At the present the transmission mechanisms with the variable transfer ratio (gear boxes) are used in machines with variable technological resistance.
The transmission mechanism is the mechanical system which provides transformation of parameters of input power into required parameters of power on the output link.
The main purpose of the transmission mechanism is adaptation to variable technological loading. Adaptation provides motion of the tool with a speed inversely proportional a loading of the tool.
The connecting gears (gear boxes) with the variable transfer ratio provide a step speed control which has essential defects. The progressive transmissions (CVT – continuously variable transmission) provide a
smooth speed control of motion. Creation of a progressive transmission with the help of the kinematic chain with two degrees of freedom which has the closed automatically adjusting device is possible. Such transmission has one entrance and meets the requirements of theoretical mechanics about equality of number of degrees of freedom with number of generalized coordinates on Targ (1970). According to this condition in the theory of mechanisms and machines for construction of planar mechanisms the principle of Assur is used on Levitsky (1979). By this principle the number of initial (or input) links of a mechanism should be equal to number of degrees of freedom and an output link is the link of structural group with zero mobility (Assur group). However when in the kinematic chain with two degrees of freedom the one input link takes place the closing device imposes additional constraint and transforms the kinematic chain into the mechanism.
In the patent of Ivanov «Transmission with automatically regulating speed» (1996) for the first time a mechanical system with two degrees of freedom with the closing device in the form of the closed mechanical contour is presented. The closed mechanical contour with a self braking gear imposes additional constraint on movement of links.
Transmission with automatically regulating speed contains gear differential and the closing gear executed in the form of the self-braking gear. Transmission possesses effect of power adaptation. The output shaft turns with a speed inversely to drag torque on it.
The closed mechanical contour in the form of centrifugal regulator imposing additional constraint on motion of input and output links of electromechanical
ROmanian
Association of
MEchanical
Transmissions
(ROAMET)
Balkan Association of
Power Transmissions
(BAPT)
12
system with two degrees of freedom is used in the patent of Ivanov, Dmitrieva, Kulanbaev (1986).
Recently the idea of construction of stepless adjustable transmission in the form of gear differential with two degrees of freedom combined with the closed automatically adjusting device in the form of hydro mechanical transmission is patented by Crockett (1990) and Volkov (2004). According to the new idea, the differential with two degrees of freedom contains
the input carrier 1H , the output carrier 2H and
structural group with zero mobility in the form of closed contour with toothed wheels 1 - 2 - 3-6 - 5 - 4 placed between carriers (Fig. 1). The closing automatically adjusting device in the form of the hydro converter imposes the constraint on links of
differential. The input pump P of the hydro converter
is connected to a toothed wheel 1 and the turbine T -
with the carrier 1H . The mechanism designed by this
principle has definability of motion and gets the major technical property - an opportunity to change the transfer ratio depending on an output moment of resistance independently and continuously.
Fig. 1. Stepless adaptive transmission in the form of closed differential with the closing hydro converter
(about Crockett, 1990, or Volkov, 2004)
However the closed mechanical contour in itself imposes a constraint on motion of links and has ability of self-regulation; also the gear differential containing the closed contour with toothed wheels is in itself capable to impose additional constraint on motion of links (Ivanov (1995).
This property is used in a basis of the patent of Harries (1991) on automatically adjusting transfer in the form of the gear differential mechanism. Transfer is executed as differential (without the hydro converter) and contains only the input carrier, the output carrier and the closed contour with toothed wheels placed between carriers. In the operating conditions the self-regulation occurs by the closed contour with toothed wheels. The contour imposes additional constraint on motion of links. For start-up when transfer passes from one-mobile condition in
two-mobile condition the braking of one of wheels is used.
Various variants of gear adaptive transfer designs in the form of gear differential with the closed contour which uses at start-up self-braking are presented in Ivanov's patents (2002, 2004). Ivanov's patents (2010, 2011) on a way of motion transfer with the help of the gear differential mechanism with the closed contour provide an use of inertial properties of the mechanism at start-up. Self-braking and inertial properties provide automatic act of transfer on all regimes of motion.
Closed contour is used for motion transfer in all patented designs of stepless adjustable transfer. However the description of interacting of transfer parameters is executed without theoretically accurate substantiation of an opportunity of motion transfer in the kinematic chain with two degrees of freedom through the closed contour at presence only one entrance. Existing practical development as patents for inventions are, as a rule, the intuitive result which is not containing the theoretical description of the phenomenon of force adaptation. Absence of a theoretical basis of the force adaptation phenomenon makes impossible creation of theoretically substantiated designs of adaptive transmissions.
Earlier in the papers of Ivanov (1995, 2004, 2007) presented on three world congresses under the theory of mechanisms and machines the opportunity of motion transfer in the kinematic chain with two degrees of freedom with the help of the closed contour has been investigated on the basis of the kinematic and force parameters interconnection. The analytical description of motion transfer has been executed on the basis of a principle of virtual works. It has been shown that the closed contour imposes additional constraint on motion of links that results in definability of motion of the kinematic chain with two degrees of freedom at presence only one entrance. The animation model of the gear differential mechanism presented on a site <http://www.madbass.narod.ru/> is developed on the basis of received theoretical dependences. It shows the work of mechanism with variable transfer ratio. However in these works the theoretical substantiation of an opportunity of motion transfer in the kinematic chain with two degrees of freedom and one entrance to an output link is absent.
For the first time the basic theoretical regularity of force adaptation effect on the basis of use of the closed contour in the finished form are published in work of Ivanov (2010 a).
The most exact content of theoretical regularities of force adaptation effect is stated by Ivanov’s (2010 b) article “Theoretical fundamental of toothed continuously variable transmission”.
Effect of force adaptation in mechanics was described by Ivanov (2011). The carried out discovery of the effect allows providing the variable transfer ratio only due to the closed contour using without application of devices of the control varying structure of mechanism. Mechanical properties of the closed contour allow
A
G
F
E
A
D
B
C
1
4
6
H1
2
P T
3
5
H2
13
providing the required transfer ratio independently, smart and automatically.
2. DATA ON A PRIORITY
In the existing scientific literature on mechanics and theory of mechanisms and machines the research of parameters interconnection of the kinematic chain with two degrees of freedom is executed only for a chain having two input links. At present there are no publications on research of parameters interconnection of the kinematic chain with two degrees of freedom having only one entrance except the author’s publications.
Existing practical developments as invention patents are, as a rule, the intuitive result which is not containing the theoretical description of the force adaptation phenomenon.
In the patent of Ivanov (1996) for the first time a mechanical system with two degrees of freedom with the closing device in the form of the closed mechanical contour with a self-braking gear is presented; this contour imposes additional constraint on movement of links.
Transmission with automatically regulating speed contains gear differential and the closing gear executed in the form of the self-braking gear. Transmission possesses effect of power adaptation. The output shaft turns with a speed inversely to drag torque on it.
The closed mechanical contour in the form of centrifugal regulator imposing additional constraint on motion of input and output links of electromechanical system with two degrees of freedom is used in the patent of Ivanov and Dmitrieva (1986).
For the first time the force adaptation effect has been proved theoretically in the paper of Ivanov (1995).
For the first time the basic theoretical regularities of force adaptation effect on the basis of use of the closed contour in the consecutive form are published in work of Ivanov (2010 a).
The most full and correct content of theoretical regularities of the force adaptation effect which set up on the theorem about the closed contour is stated for the first time in work of Ivanov (2010 b).
3. ESSENCE OF FORCE ADAPTATION EFFECT DISCOVERY
The force adaptation effect in mechanics is mechanical system ability to independent accommodating to variable loading on output link by way of changing output speed of motion when input power is constant.
The mechanical system having force adaptation effect represents the kinematic chain with two degrees of freedom which contains an input link, an output link and placed between these links the mobile four-link closed contour in the form of structural group with zero mobility.
The basic new way of motion and forces transfer
corresponds to the circumscribed structure of the kinematic chain. Motion and forces transfer from an input link occurs through the intermediate closed contour on an output link with one degree of freedom.
The closed contour has zero mobility. But at relative motion of links the contour creates additional differential constraint between forces and displacements, imposes this constraint on motion of kinematic chain and provides definability of motion of the kinematic chain with two degrees of freedom at presence only one input.
The closed contour creates additional constraint only due to accomplishment of the following demand: all internal forces of a contour should be expressed through external forces. For this purpose external forces of a contour should be applied on non-adjacent links of a contour in its external kinematic pairs connecting a contour with input and output links.
The closed contour has surprising property to create the additional differential constraint providing definability of motion and force adaptation effect. This property of the closed contour is defined by the theorem about closed contour.
Implementation of force adaptation effect can be executed by the kinematic chains with two degrees of freedom of various types.
4. IMPLEMENTATION OF FORCE ADAPTATION EFFECT BY A LEVER KINEMATIC CHAIN
The lever kinematic chain implementing force adaptation effect (Fig. 2) contains the frame 0, one input link 1, closed four-link contour 2-3-4-5 as 4 class structural Assur group and output link 6. Kinematic
pairs B and K connect the contour with input link1 and with output link 6. These pairs are external pairs
of a contour. Kinematic pairs GDEC ,,, are internal
pairs of a contour.
Fig. 2. The lever kinematic chain implementing
force adaptation effect
The input motive force 1F is transferred from input link
1 to a point B . The output resistance force 6R is
transferred from the output link 6 into a point K .
Points applications B and K of external contour
forces 1F and 6R have external displacements
KB ss , . Points GDEC ,,, of the application of internal
forces of a contour (reactions)
45423532 ,,, RRRR have internal displacements
GDEC ssss ,,, . At known external displacements
Ba
2
G
6D
sK
sB
Kb
d
c
R6
C
F1
A
E
0 1
3
4
5
L
14
points B and K contour internal displacements of
points GDEC ssss ,,, are unequivocally determined.
As the contour links 2 and 5 on which active forces are applied, are not contiguous for each of these links
internal forces 45423532 ,,, RRRR can be expressed
on conditions of a statics through active external
forces 1F and 6R .
Theorem. Mobile four-link the closed contour in composition of the kinematic chain with two degrees of freedom imposes a constraint on motion of links if the active forces transmitted on a contour, are applied to non-adjacent links of a contour.
Let's note that the closed four-link contour represents the fourth class structural Assur group with zero mobility.
At the demonstration of the theorem we shall accept the following assumptions.
1. If in the closed contour there is no relative motion of links in its hinges the static friction takes place. The static friction torque instanced to one links of a contour, balances a moment, created by the external active forces applied on a contour. At presence of equilibrium of external forces the contour goes as a single whole.
2. At relative motion of links of structural group the friction in kinematic pairs is absent, they are ideal and stationary.
For the demonstration of the theorem we shall make for non-adjacent links 2 and 5 the equilibrium conditions by a principle of virtual works accepting the valid displacements for possible. For a link 2:
DCB sRsRsF 42321 += . (1)
For a link 5:
GEK sRsRsR 45356 += . (2)
Let's combine the equations (1) and (2). We shall receive the equation of interconnection of parameters external and internal forces:
++=+ DCKB sRsRsRsF 423261 GE sRsR 4535 + . (3)
The right side of an equation (3) represents the sum of works of internal forces of a contour. All internal forces are determined through known external forces. All internal displacements are determined through external displacements. Hence, work of internal forces on possible internal displacements is determined. Constraints in kinematic pairs are ideal and stationary. Work of external forces cannot pass into work of internal forces. Hence, work of internal forces on possible internal displacements is equal to zero:
045354232 =+++ GEDC sRsRsRsR . (4)
The left side of an equation (3) represents the sum of works of external forces of a contour. At accomplishment of a condition (4) we shall receive from the equation (3) the equilibrium condition for external forces according to a principle of virtual works:
061 =+ KB sRsF . (5)
Thus, the mobile four-link closed contour imposes a constraint on motion of links in the form of equation (5) if active forces are applied to non-adjacent links of a contour, as was to be shown.
For definability of parameters the equation (5) should contain one unknown parameter.
In really acting designs the task of engine parameters
(an input motive force 1F and input displacement Bs )
and output resistance force 6R is natural.
Therefore it is necessary to accept in the equation (5)
as unknown parameter the output displacement Ks .
From the formula (5) in view of a mark of work signs it
follows 61 / RsFs BK = . Or in view of a time:
61 / RvFv BK = ; (6)
Here: Bv is the input speed; Kv - output speed.
The equation (6) provides definability of motion of all mechanism as at known speeds (or displacements)
points B and K speeds (displacement) of other
points GDEC ,,, are determined univocal.
The equation (6) is the equation of differential constraint of the closed contour.
The kinematic chain with two degrees of freedom and with one entrance, containing the closed contour (Fig. 2) gets additional differential constraint and turns to the mechanism with one degree of freedom, providing adaptation of parameters to the motion conditions. The formula (6) characterizes force adaptation effect of the kinematic chain with the closed contour: at
constant parameters of an input power BvF ,1 the
output speed Kv is in inverse proportional
dependence on variable force of resistance 6R .
Let's consider the simplified alternative of the lever mechanism with parallel displacements of points (Fig. 3) for their evident representation in the form of a picture of displacements.
The planar kinematic chain with two degrees of freedom has the frame 0, one input link 1, the mobile closed contour with mobile links 2, 3, 4, 5 and one
output link 6. Kinematic pairs LA, are translational.
Kinematic pairs KGEDCB ,,,,, are rotational.
Sizes of links:
,,, bKMaKLlBDBC ====
2/)(,2 baelba −==+ .
On an input link 1 the external active force (input
motive force) 1F acts. On an output link 6 the external
active output force of resistance 6R acts.
Motion of the kinematic chain is defined by the plot of
instant linear displacements )6,4,3,1( =isi of points
KGEDCB ,,,,,
(of the corresponding links i )
15
which presented in a picture of displacements and correspond to analytical conditions of interconnection
of displacements.
S is the center of final twirl ( the instant center of
speeds) of link 2 (or link 5). Input displacement is
displacement 1ssB = of a point B of a link 1. Output
displacement is displacement 6ssK = of a point K
of a link 6.
Forces and displacements are parallel to axisOx . By
that 43 , ssssss GDEC ==== .
At equality of input and output forces in hinges of a contour the friction of space occurs. 5 moment of frictional force of space of structural group
fM instanced to a link, balances a moment, created
by external forces. Equilibrium of external forces
eFMRF f 161 , == occurs. The structural group goes
as a single whole.
Under condition of 16 FR > the relative motion of links
of a contour occurs. The friction in kinematic pairs is absent, they are ideal and stationary. Let's express internal forces through known external forces. For a
link 2 it is definable reactions 4232 , RR through
external force 1F :
2/14232 FRR == .
For a link 5 it is definable reactions 4535 , RR through
external force 6R :
laRRlbRR 2/,2/ 645635 == .
On links 3 and 4 reactions act:
;; 42243223 RRRR −=−=
.; 45543553 RRRR −=−=
Under act not equal on magnitude of reactions links 3 and 4 appear unbalanced. The forces acting on links 3 and 4:
lbRFRRP 2/2/ 6153233 −=−= ;
laRFRRP 2/2/ 6124544 −=−= .
So, on conditions of statics equilibrium for each of links 3 and 4 is not observed.
However according to the equations (1) - (5) all kinematic chain is in equilibrium by a principle of
virtual works.
Let's receive from the equation (4) in view of marks of works
0445335442332 =++−− sRsRsRsR .
Or:
4245435323 )()( sRRsRR −=− .
Then the equilibrium equation for internal forces of a contour will become:
4433 sPsP = . (7)
The equation (7) shows presence of equilibrium of internal forces on intermediate links 3 and 4 on the move at absence of static equilibrium on each separate intermediate link.
Equilibrium of internal forces results in equilibrium of external forces at the equation (5). The equation (5) is expressing the differential constraint which the contour imposes on relative motion of links. The equation (5) allows determining of output displacement of the kinematic chain at the given parameters of input power and the given force of resistance.
Further on known displacements 61, ss it is possible to
determine displacements of all points of the kinematic chain which owing to definability of motion becomes the mechanism.
Displacements of points of links according to a picture of displacements are connected by formulas
esslss /)(/)( 6113 −=− ;
essbss /)(/)( 6146 −=− .
From here we shall receive displacements links 3 and 4:
elssss /)( 6113 −+= ; (8)
ebssss /)( 6164 −−= . (9)
Thus, the equation (5) provides definability of motion
of all mechanism as at known displacements 61, ss
points B and K displacements 43, ss of other points
GDEC ,,, are determined univocal.
F1
3 P3
E
A 1
N
s3
S
v6
D,G G
6
S
D
s6
s1
K
l
l
b
a
e R6
2
C
B
v1
0A
4
5
L
s6
C, E
K
s4
s1 B
P4
O
x
y
Fig. 3. The kinematic chain with parallel displacements points and a picture of its displacements
16
The equation (5) is of differential constraint of the closed contour. The kinematic chain with two degrees of freedom and one input, containing the closed contour (Fig. 3), gets additional differential constraint and turns to the mechanism with one degree of freedom, providing adaptivity parameters to conditions of motion. The formula (6) characterizes force adaptation effect of the kinematic chain with the closed contour: at constant parameters of an input
power BvF ,
1 the delivery speed Kv is in return
proportional dependence on variable force of
resistance 6R .
It is necessary to note, that according to the theorem of the closed contour only the closed contour has property to impose additional constraint on motion of kinematic links with two degrees of freedom. If, for example, to place in the kinematic chain on Fig. 3
instead of the closed contour between points B and
K structural group BNK (it is shown by a dot line) this structural group will not provide additional
constraint. The received kinematic chain ABNKL remain the kinematic chain with two degrees of
freedom as in group BNK internal reaction in the
hinge N cannot be expressed through known
external forces 1F for this group and 6R in points
B and K .
Let's consider further the kinematic chain with toothed wheels.
5. IMPLEMENTATION OF FORCE ADAPTATION EFFECT BY THE KINEMATIC CHAIN IN THE FORM OF WHEELWORK WITH THE CLOSED CONTOUR
The wheelwork implementing the force adaptation effect represents the closed differential mechanism (Fig. 4).
Fig. 4. Wheelwork with the closed contour
implementing the force adaptation effect
It contains the frame 0, one input carrier 1H , a closed
four-link contour with toothed wheels 1-2-3-6-5-4 and
the output carrier 2H . Solar wheels 1, 4 are united in
the block of wheels 1-4. Ring wheels 3, 6 are united in
the block of wheels 3-6.
The input motive force 1F is transferred from input link
1H to a point B . Output force of resistance 6R is
transferred from output link 2H to a point K .
Application points B and K of a contour have
external displacements KB ss , . Application points
GDEC ,,, of the of internal forces of a contour
(reactions) 45126532 ,,, RRRR have internal
displacements GDEC ssss ,,, .
If the external displacements of points B and K are known then a contour internal displacements of points
GDEC ssss ,,, is univocal determined.
As the contour links 2 and 5 on which active forces are applied are not contiguous then for each of these
links the internal forces 45126532 ,,, RRRR can be
expressed by static conditions through active forces
1HF and 2HR .
Theorem. Mobile four-link closed contour imposes a constraint on motion of links if the active forces transmitted on a contour are applied on non-adjacent links of a contour.
To prove this theorem we shall describe the equilibrium conditions by a principle of virtual works for non-adjacent links 2 and 5. We shall accept the valid displacements as virtual. For a contour links 2 and 5 we shall express reactions
in kinematic pairs EGCD ,,, through the external
forces 21, HH RF are applied in points B and K:
13212 5.0 HFRR == ; (10)
26545 5.0 HRRR == . (11)
Here:
111 / HHH rMF = ;
11212 / rMR = ;
33232 / rMR = ;
44545 / rMR = ;
222 / HHH rMR = .
The following quantities intervene: 21, HH MM -
moments on input and output carriers; 21, HH rr -
radiuses of input and output carriers; 3212, MM -
moments created on the satellite by 2 reactions
3212 , RR on the link of toothed wheels 1 and 3;
6545, MM - moments created on the satellite by 5
reactions 6545 , RR on the link of toothed wheels 4
and 6, )6...2,1( =iri - radiuses of wheels.
Opposite directed moments received from the equations (10) and (11) are transferred to intermediate links 1-4 and 3-6 from satellites 2 and 5:
11121 /5.0 HH rrMM = ;
13123 /5.0 HH rrMM = ;
24254 /5.0 HH rrMM = ;
A A
G
K
E
A
D
С
B
1 4
6
H 2 H1
2
3
5
0
17
./5.0 26256 HH rrMM =
Let's present for each satellite an equilibrium equation by a principle of virtual works. We shall receive for satellites 2 and 5:
BHCD sFsRsR 13212 =+ ; (12)
KHEG sRsRsR 26545 =+ . (13)
Here KGEDCB ssssss ,,,,, are the displacements of
the points KGEDCB ,,,,, .
Let’s express displacements of points through instant angles of turn of links and radiuses:
113311 ,, HHBCD rsrsrs ϕϕϕ === ;
226644 ,, HHKEG rsrsrs ϕϕϕ === ,
in which 264131 ,,,,,HH
ϕϕϕϕϕϕ are the instant
angles of turn of toothed wheels and carriers.
With the account 6341 , ϕϕϕϕ == and a time t
( ttttHHHH/,/,/,/ 22113311 ϕωϕωϕωϕω ==== )
we shall receive:
11332112 HHMMM ωωω =+ , (14)
22365145 HHMMM ωωω =+ . (15)
As satellites 2 and 5 are link of the mechanism as a whole, we shall combine the made expressions. We shall receive a condition of interacting of parameters of the mechanism as a whole:
=+++ 365145332112 ωωωω MMMM
2211 HHHH MM ωω + . (16)
In the left side of equation (16) the sum of powers (corresponding to the sum of works) internal forces of a contour takes place.
In the observed mechanism all internal forces are determined through known external forces, all internal displacements are determined through external displacements. Hence the work (or power) of internal forces on virtual internal displacements is determined. Constraints in kinematic pairs are ideal and stationary. Work of external forces cannot pass in work of internal forces. Hence the work (or power) of internal forces on virtual internal displacements is equal to zero:
0365145332112 =+++ ωωωω MMMM . (17)
The right side of an equation (16) represents the sum of powers (corresponding to the sum of works) external forces of a contour. At accomplishment of a condition (17) we shall receive from the equation (16) equilibrium condition for external forces according to a principle of virtual works
02211 =+ HHHH MM ωω . (18)
The equation (18) expresses additional constraint which is imposed by a contour on motion of links. Thus the mobile four-link closed contour imposes a constraint on motion of links if the active forces transmitted on a contour are applied on non-adjacent links of a contour as was to be shown.
Additional constraint (18) is differential. Additional
constraint provides: 1. transformation of the kinematic chain with two
degrees of freedom in the mechanism with one degree of freedom, that is definability of motion under act of forces;
2. force adaptation effect to output loading at the
given parameters of an input power 11, HHM ω and
the given output moment of resistance 2HM .
According to the formula (18) in view of marks of powers we shall receive
2112 / HHHH MM ωω = . (19)
That is at a constant input power output angular speed is in return proportional dependence on a
variable output moment of resistance 2HM .
We shall receive from the equation (17):
0)()( 3562315421 =+++ ωω MMMM ,
with the account:
23322112 , MMMM −=−= ;
5445 MM −= ;
5665 MM −= .
Driving moments 21M and 23M transmitted on the
link 1 and 3 from the input satellite 2 are positive.
Moments of resistance 54M and 56M transmitted on
the link 4 and 6 from the output satellite 5 are negative. In view of signs of the moments we shall receive:
0)()( 3562315421 =−+− ωω MMMM . (20)
The equation (20) is of works (powers) on intermediate links 1-4 and 3-6, it means the presence of equilibrium on intermediate links 1-4 and 3-6 simultaneously. In the mobile closed contour basic new situation occurs: the equilibrium in statics separately on each intermediate link is absent but equilibrium of intermediate links simultaneously on the motion of all contour takes place. In the closed contour circulation of energy occurs.
The equation (20) contains positive and negative members and characterizes equilibrium of powers on intermediate links of a contour.
As for the observed chain we have:
2154 MM > ;
5623 MM > ,
then from the equation (20) we shall receive:
0)()( 3562312154 =−+−− ωω MMMM . (21)
From here:
3562312154 )()( ωω MMMM −=− . (22)
The equation (22) reflects unknown earlier an analytical form of energy circulation inside a contour in a time of its motion.
Angular speeds 1ω and 3ω of intermediate links 1-4
18
and 3-6 are determined through known angular
speeds of input both output carriers 21, HH ωω and
transfer ratios at the stopped carriers.
Transfer ratios of links of transfer we shall determine
through numbers of wheels 6...,2,1, =izi .
The interconnection of angular speeds of the mechanism is determined by formulas
)1(13
13
11 H
H
H u=−
−
ωω
ωω; (23)
)2(46
23
21 H
H
H u=−
−
ωω
ωω, (24)
where:
13)1(
13/ zzu H
−= ;
46)2(
46/ zzu
H−= .
From (23):
113)1(
131 )( HHHu ωωωω +−= . (25)
From (24):
223)2(
461 )( HHHu ωωωω +−= . (26)
Let's subtract (26) of (25). We shall receive
−+− 113)1(
13)( HH
Hu ωωω 0)( 223)2(
46 =−− HHHu ωωω .
From here:
=++−− 2)2(
461)1(
133)2(
46)1(
13)( H
HH
HHHuuuu ωωω
12 HH ωω −= ,
from which the following angular speed results:
)2(46
)1(13
)1(131
)2(462
3
)1()1(
HH
HH
HH
uu
uu
−
−−−=
ωωω . (27)
Formulas (19), (27), (25) define a sequence of acts by determination of angular speeds of links.
It is necessary to note that at start-up the kinematic chain will be propelled in a condition with one degree of freedom at absence of mobility inside a contour with presence of a static friction in kinematic pairs of a contour before accomplishment of a condition
12 HH MM > .
Thus all kinematic and force parameters are determined and all mechanism has the kinematic and static definability.
6. NUMERICAL CHECK OF THE FOUND
REGULARITIES FOR THE WHEELWORK
For the wheelwork (Fig. 3) the following quantities are
given numerically:
• angular speed for element 1:
11 100 −= sHω ;
• moments on elements 1 and 2:
NmMH 1001 = ; NmMH 2002 = ;
• numbers of teeth of wheels,:
;160,6040,90,30,30 654321 ====== zzzzzz
• module:
mmm 8= ;
• radiuses of toothed wheels:
mmmzr 1202/3082/11 =⋅== ;
mmr 1202 = ; mmr 3603 = ; mmr 1604 = ;
mmr 2405 = ; mmr 6406 = ;
• radiuses of input and output carriers,
mmrrrH 2402/)360120(2/)( 311 =+=+= ;
mmrrrH 4002/)( 642 =+= ;
• transfer ratio of wheels 1 and 3 at the motionless
carrier 1H :
330/90/ 13)1(
13−=−=−= zzu
H;
• transfer ratio of wheels 4 and 6 at the motionless
carrier 2H :
440/160/ 46)2(
46−=−=−= zzu
H.
To determine the angular speeds 312 ,, ωωωH , the
solution explained below could be used.
1. From (19):
=⋅== 200/100100/ 2112 HHHH MM ωω 150 −s .
2. From (27):
=−
−−−=
)2(46
)1(13
)1(131
)2(462
3
)1()1(
HH
HH
HH
uu
uu ωωω
133.3395.1
)5.11(100)91(50 −=
+−
+−+= s .
3. From (25):
=+−= 113)1(
131 )( HHH
u ωωωω
1200100)10033.33)(5.1( −=+−−= s .
4. We compute the moments on toothed wheels using the formulas received from the equations (10) and (11):
== 11121 /5.0 HH rrMM ;40200/1601005.0 Nm=⋅⋅
== 13123 /5.0 HH rrMM ;60200/2401005.0 Nm=⋅⋅
== 24254 /5.0 HH rrMM ;20200/402005.0 Nm=⋅⋅
26256 /5.0 HH rrMM = Nm180200/3602005.0 =⋅⋅= .
5. The check of equilibrium of circulating energy on the equation (22) is the main demonstration of reliability of the received results. Here
19
54212356 , MMMM >> . Multiplying the equation
(22) on "– 1", we shall receive:
3235615421 )()( ωω MMMM −=− .
After substitution of numerical values we shall receive accomplishment of equilibrium of powers:
33.33)60180(200)2040( ⋅−=⋅− ,
resulting the identity:
40004000 ≡ .
Check shows presence of equilibrium of positive power on the block of wheels 1 - 4 and negative power on the block of wheels 3 - 6. Thus equilibrium is carried out. The force and kinematic definability of the mechanism takes place what confirms reliability of presence of force adaptation effect.
7. PRACTICAL IMPLEMENTATION
Practical implementation of discovery is presented by the following developments.
1. Patents of Russian Federation and Kazakhstan (2002, 2004, 2010).
2. The developed design documentation on a row of concrete designs of toothed continuously variable transmissions. The assembly drawing of continuously variable transmission of the drive of the conveyor is presented on Fig. 5.
Fig. 5. The assembly drawing of continuously variable
transmission of the drive of the conveyor (Patents of
Russian Federation and Kazakhstan (2002, 2004,
2010)
3. The computer animation model of toothed continuously variable transmission (Fig. 6) presented on the website: http://www.madbass.narod.ru/. The animation model corresponds to the design of the adaptive wheelwork presented on Fig. 4. It allows seeing a change of motion of links at change of external loading.
4. An acting dummy of toothed continuously variable transmission. The photo of acting dummy of toothed continuously variable transmission presented on Fig. 7. The dummy confirms presence of force adaptation effect in a wheelwork with the closed contour.
Fig. 6. Animation model of toothed continuously
variable transmission
Fig. 7. Photo of an acting dummy of toothed
continuously variable transmission
8. CONCLUSIONS
The developed continuously variable transmission allows to provide the variable transfer ratio only due to use of the closed contour and its properties without application of control devices of the varying structure of the mechanism. Mechanical properties of the closed contour allow to provide the required transfer ratio independently, stepless and automatically.
Creation of toothed continuously variable transmission as the toothed closed differential mechanism with two degrees of freedom is theoretically proved.
Force adaptation effect in the mechanics consisting in a volume, that the kinematic chain with two degrees of freedom containing an input link, an output link and the closed mobile contour placed between them at a constant input power provides the motion of an output link with a speed inversely proportional to loading on it. It is proved that the mobile closed mechanical contour of transmission creates additional constraint and provides the transition regime of motion translating the mechanism of transmission from an one-mobile condition at start-up in two mobile condition of operating regime of motion. It is proved that in operating regime of motion the equilibrium by a principle of the virtual works providing stepless regulating of transfer ratio takes place.
20
The equations of interconnection of force, kinematic and geometric parameters of transmission in kinematics and dynamic are developed. These equations are tested by numerical instances, confirmed with skilled acting samples and the developed computer animation model. The assembly drawing of the toothed continuously variable transmission is presented.
The found regularities allow synthesizing the toothed continuously variable transmission on the given operational parameters of motion, to execute the kinematic and dynamic analysis of transmission and to determine design data of transfer.
Toothed continuously variable transmission as the toothed closed differential mechanism with constant cogging wheels is the elementary transmission of such type and has the reliability corresponding to reliability of a wheelwork. The specified properties allow using transmission both in easy local drives of manipulators and in heavy drives of transport machines including in motor - wheels.
The toothed closed differential mechanism of transmission has force adaptation effect to variable technological loading. Force adaptation allows creating the easy and hardly loaded adaptive drives of machines with the variable transfer ratio dependent on technological resistance (velocipede, motorcycle, automobile, drilling rig, bulldozer, lorry etc.).
REFERENCES
CROCKETT, S. J. (1990). Shiftless continuously-aligning transmission. Patent of USA 4,932,928, Cl. F16H 47/08, U.S. Cl. 475/51; 475/47. 9 p.
HARRIES, J. (1991). Power transmission system comprising two sets of epicyclic gears. Patent of Great Britain GB2238090 (A), 11 p.
IVANOV, K. S. (1995). The Question of the Synthesis of Mechanical Automatic Variable Speed Drives. Proceedings of the Ninth World Congress on the Theory of Machines and Mechanisms, Vol.1, Politechnico di Milano, Italy, August 29/Sept 2, 1995. pp. 580 - 584.
IVANOV, K. S. (1996). Transmission with automatically regulating speed. The preliminary patent of Republic Kazakhstan. No. 3208, 12 p.
IVANOV, K. S. (2004). Discovery of the Force Adaptation Effect. Proceedings of the 11th World Congress in Mechanism and Machine Science. Tianjin, China. pp. 581 - 585.
IVANOV, K. S. (2007). Gear Automatic Adaptive Variator with Constant Engagement of Gears. Proceedings of the 12th World Congress in Mechanism and Machine Science. Besancon. France. Vol. 2. pp. 182 - 188.
IVANOV, K. S. (2010 a). The simplest automatic transfer box. World Congress on Engineering, London - WCE 2010, UK. pp. 1179 – 1184.
IVANOV, K. S. (2010 b). Theoretical fundamental of toothed continuously variable transmission. Theory of mechanisms and machines. Periodical science-methodical journal. N2 (16), Saint-Petersburg State Polytechnic University, pp. 36 – 48.
IVANOV, K. S. (2011). Effect of force adaptation in mechanics. Journal of Mechanics Engineering and Automation. Vol. 1, No. 3. Libertyville, USA, pp. 163 – 180.
IVANOV, K. S., DMITRIEVA, N. A., KULANBAEV, A. M. (1986). Electromechanical automatically adjusting installation. Patent of the USSR, No. 1216489. 6 p.
IVANOV, K. S., KOSS, I. M. (2002). Automatic gearbox. Preliminary patent of Republic Kazakhstan. No. 12236, 8 p.
IVANOV, K. S., VOROGUSHIN, V. A., SHISHKIN, P. A. (2004). Adaptive gearing (variants). Preliminary patent of Republic Kazakhstan, No. 14477, 8 p.
IVANOV, K. S., YAROSLAVCEVA, E. K. (2010 a). Way of automatic and continuous change of a twisting moment and speed of twirl of the output shaft depending on a tractive resistance and the device for its realization. Patent of Russia, 2398989, 10 p.
IVANOV, K. S., YAROSLAVCEVA, E. K. (2010 b). Device of the power transmission with continuously variable transfer ratio (variants). Preliminary patent of Republic Kazakhstan, No. 028847, 12 p.
LEVITSKY, N. I. (1979). Theory of mechanisms and machines. Moscow. Science. 576 p.
TARG, S. M. (1970). Theoretical mechanics. Moscow. Science. 480 p.
VOLKOV, I. V. (2004). Way of automatic and continuous change of a twisting moment and speed of twirl of the output shaft depending on attractive resistance and the device for its realization. The description of the invention to the patent of Russia, No. 2234626, 14 p.
CORRESPONDENCE
Konstantin IVANOV, Prof. D.Sc. Eng. Almaty University of Power Engineering and Telecommunications Faculty of Information Technology Baytursynov Street 126 050013 Almaty, Kazakhstan [email protected]
21
Balkan Journal of Mechanical Transmissions
(BJMT)
Volume 2 (2012), Issue 1, pp. 21-26
ISSN 2069–5497
DYNAMICS OF GEAR DIFFERENTIAL
Konstantin IVANOV
ABSTRACT. Dynamics of the gear differential with two degrees of freedom working in starting mode from start place, in a transitive mode and in a mode of steady movement with arbitrarily assigned values of input moment and the output moments of two output links are presented in this work. The essentially new statement of a problem is presented in the present work: in a kinematic chain with two degrees of freedom one generalized co-ordinate takes place only (instead of two), it is necessary to determine movement of two output links. The internal moment of friction is used as the power factor which provides a workability and definability of movements of the mechanism with two power outputs. Dynamics of gear differential consists of determination of links start acceleration, functions of links angular velocities depend on time in transient mode and links angular velocities in mode of steady motion. It is proved that the gear differential goes in a mode of the established movements adapting to external loads due to output angular speeds when power of one input link is constant and output resistant moments of two output links are arbitrary.
KEYWORDS. Gear differential, dynamics, equations of motion, adaptation
NOMENCLATURE
Symbol Description T Kinetic energy H
Carrier
F Force Q Generalized force M Moment R Reaction ε Angular acceleration φ Angle of turn ω Angular speed J Moment of inertia r Radius of link z Number of teeth u Transfer ratio t Time
1. INTRODUCTION
The gear differential is used often for decomposition of an input power flow on two output power flows. For example, in the automobile with the help of gear differential the input power flow of engine is separated on two output power flows of two automobiles wheels with loadings on them corresponding to conditions of movement.
The one input gear differential is considered not able to work at arbitrary input moments of resistance of two output links because condition of equilibrium is break. For example, it is not able to start because one of output link will be not mobile.
However practice of using of automobile differential testifies to a start possibility and the linear motion of the automobile at different loadings on wheels. For example, one of wheels is slip or wheels move with different angular speeds.
Levitsky (1979) offers the differential equations for
researching of differential. However it is not considered the question of work ability of mechanism at random different values loadings on two output links of mechanism.
Now the following approach to research of movement of gear differential as kinematic chain with two degrees of freedom is used: the kinematic chain has two input links, movement research in dynamics consists in definition of accelerations of two input links, and work of motive forces is not equal to work of forces of resistance (Levitsky, 1979).
However in real practice the automobile gear differential has only one input link connected with a propeller shaft and two output links connected with car wheels. Movement of differential links at uniform car movement on turn is uniform; work of motive forces is equal to work of forces of resistance. Non-uniform movement of differential occurs only at non-uniform movement of the car that is in a transitive mode. Such statement of a research problem of gear differential movement was not considered earlier.
For research of gear differential movement in real practice the essentially new statement of a problem is presented in the present work: in a kinematic chain with two degrees of freedom one generalized co-ordinate takes place only (instead of two), it is necessary to determine movement of two output links. This problem is solved by the account of a friction moment in the differential when an add constraint between wheels is absent.
The idea of present work consists in setting of equilibrium of mechanism with two degrees of freedom with taking in account the internal moment of friction which really exists. The internal moment of friction plays a role of the power factor which provides for workability and definability of motion of mechanism
ROmanian
Association of
MEchanical
Transmissions
(ROAMET)
Balkan Association of
Power Transmissions
(BAPT)
22
with two degrees of freedom at start-up and at established motion.
The purpose of the present work is to make the dynamic analysis of gear differential with account the internal moment of friction. The account of friction must set the possibilities of work ability of mechanism at start, reveal the regularity of correlation of parameters of mechanism in transition mode and possibilities of transition of mechanism in regime of steady motion.
For goal achievement it is used the known regularities of correlation of parameters of mechanism in dynamics (Levitsky, 1979), condition of equilibrium in static and also information about regularity of correlation of kinematics and power parameters of mechanism with two degrees of freedom considered in works of Ivanov (2007, 2010 a, 2010 b and 2011).
2. BASIC EQUATIONS OF DYNAMICS
The gear differential with only one input is presented
in Fig. 1. It contains input carrier H , satellite 2 and
central wheels 1 and 3. Input power from carrier H is delivered at output wheels 1 and 3.
Fig. 1. The gear differential mechanism with one input
The basic equations of dynamics of the gear differential shown on Fig. 1 look like Lagrange second-order equations:
111
QTT
dt
d=
∂
∂−
∂
∂
ϕω; (1)
HHH
QTT
dt
d=
∂
∂−
∂
∂
ϕω. (2)
Here: 1ϕ , Hϕ are angles of turn of links 1 and H , that
are accepted for the generalized coordinates; Hωω ,1
– angular speeds of links 1 and H ; T – kinetic energy
of the mechanism; HQQ ,1 – generalized forces which
we will name resulted moments.
Kinetic energy of the mechanism with balanced links looks like:
+++= 222
222
211(
2
1HH
RKmKJJT ωωω
)233
2 ωω JJ HH ++ . (3)
Here: HJJJJ ,,, 321 are the moments of inertia of links
1, 2, 3 and H relative to the axes, which are going
through the centers of weights of links; K – number of
satellites; 2m – mass of one satellite; HR – radius of
a trajectory of center of satellite.
Angular speed 3ω can be expressed in angular
speeds 1ω and Hω (the generalized speeds) trough
the equation:
HHH uu ωωω )1(
31)(
313 += . (4)
Here )1(
3)(
31,
HH
uu are the transfer ratios:
)(31
)1(3
3
1)(31
1;H
HH
uuz
zu −=−= ,
1z and 3z being the numbers of teeth of wheels 1
and 3.
Angular speed 2ω is found from a similar equation:
HHH uu ωωω )1(
21)(
212 += . (5)
Here:
)(21
)1(2
2
1)(21
1,H
HH
uuz
zu −=−= ,
2z being the number of teeth of wheel 2.
We get after substitution values of 2ω and 3ω in
expression of kinetic energy (3) and grouping members:
211
2111 22 HHHHH JJJT ωωωω ++= ,
in which:
2)(313
2)(212111 ][][
HHuJuKJJJ ++= ;
)1(3
)(313
)1(2
)(2121 H
HH
HH uuJuuKJJ += ;
.][][ 2)1(33
22
2)1(22 HHHHHH uJJRKmuKJJ +++=
Coefficients HJJ 111, and HHJ are the inertial
coefficients. All coefficients in the considered
mechanism do not depend on angles 1ϕ and Hϕ .
From here:
.0;01
=∂
=∂∂
H
TT
ϕϕ
The differentiation of kinetic energy expression on
23
angular speeds 1
ω и H
ω gives:
HHJJT
ωωω 1111
1
+=∂∂
;
.11 HHHHH
JJT
ωωω
+=∂∂
Differentiating these expressions on time we receive:
HHJJT
dt
dεε
ω 11111
+=
∂
∂;
HHHHH
JJT
dt
dεε
ω+=
∂
∂11 .
Here 1ε and Hε are the angular acceleration of links
1 and H .
Finally the equations of movement of the mechanism looks like: 11111 QJJ HH =+ εε ;
.11 HHHHH QJJ =+ εε (6)
The resulted moments of forces 1Q и
HQ also found
from a condition of equality of elementary works (powers) of these forces to work (power) of all external forces applied to links of the mechanism. Considering that external forces act only on links 1, 3
and H as pairs with moments HMMM ,, 31 we
receive:
.331111 ωωωωω MMMQQ HHHH ++=+
Or after substitute of 3ω value:
=+ HHQQ ωω11
HHH
HH uMuMMM ωωωω )1(331
)(31311 +++= .
From here: )1(
33)(
31311 ;HHH
HuMMQuMMQ +=+= .
For solution of system (6) we present it in view of determining relative to angular accelerations:
2111
111
HHH
HHHH
JJJ
JQJQ
−
−=ε ;
21111
1111
HH
HHH
JJJ
JQJQ
−
−=ε
(7)
In the system (7) we put the values of angular accelerations:
dt
d
dt
d HH
ωε
ωε == ,1
1 .
Dividing and integrating, we receive:
12111
111 Ct
JJJ
JQJQ
HHH
HHHH +−
−=ω ;
H
HH
HHH Ct
JJJ
JQJQ+
−
−=
21111
1111ω .
(8)
We determine constants of integration 1C and HC
from starting conditions of movement that correspond to start mode of the mechanism.
3. RESEARCH OF START MODE OF MECHANISM
Let us consider a start mode of the differential mechanism having only one input (Fig. 1). An input
link of the mechanism is carrier H .
Start of the mechanism is impossible on conditions of
balance of static at different output moments 31, MM
on output links 1 and 3 because the link which transmits a bigger reaction on satellite 2 appears stopped. For example, the link 1 will be stopped at
3212 RR > . Start of mechanism with movement of all
links is able only at realization of next conditions: 3212 RR = ;
32122 RRRH += (9)
It is no reason of turning of satellite 2 relative carrier
H at the same time. All mechanism will be motioning like mechanism with one degree of freedom with
angular speed Hω of carrier H relative to motions of
links.
Expression (3) of kinetic energy of such mechanism with the account of:
Hωωωω === 321 ,
will become in next view: ( ++= 21
2
2
1KJJT Hω
)32
2 JJRKm HH +++ .
(10)
Let us designate:
)( 32
2210 JJRKmKJJJ HH ++++= .
Then:
0JT
HH
ωω
=∂∂
;
0)( JT
dt
dH
H
εω
=∂∂
;
.0=∂∂
H
T
ϕ
Taking in account:
3010 MMMQ HH ++= ,
the differential equation of movement of the mechanism will be at start:
.30100 MMMJ HH ++=ε (11)
Here 3010, MM are the moments of resistance at start.
We are getting dependence of input angular speed on
time at start with the help of substitution dt
d HH
ωε = in
the equation (11), dividing of variables and integrating
with account of start conditions ( 0,0 == Ht ω ):
24
.
0
3010 tJ
MMMHH
++=ω (12)
It is starting transition mode of movement after start
and reaching some start angular speed 0Hω after the
expiration of time 0t by all mechanism. The
mechanism with one degree of freedom converses to a mechanism with two degrees of freedom. This conversion goes in result of changing of output moments of resistance in way that appears a reason
of movement of satellite 2 relative to carrier H .
This reason consists in infringement of conditions (9) because there is an acting addition moment of friction
TM in mechanism, having a place in rotary pair
connecting satellite with carrier and is expressed in next correlations of reactions which are transferred to the satellite and correspond to the working moments: 3212 RR ≠ ;
32122 RRRH +≠ . (13)
Equilibrium conditions of the satellite are kept an
account of the moment of friction TM .
The equations of movement of the mechanism in a transition period look like (8).
We find constants of integration 1C and HC from the
start conditions: at 0tt = , there is 01 HHCC ω== . We
get an expression of angular speeds of links in function of time after substitution of these values in (8):
02111
111 H
HHH
HHHH tJJJ
JQJQωω +
−
−= ;
021111
1111H
HH
HHH t
JJJ
JQJQωω +
−
−= .
(14)
Let us designate:
2111
111
HHH
HHHH
JJJ
JQJQk
−
−= ;
.21111
1111
HH
HHH
JJJ
JQJQk
−
−=
Then from (14) we get
.011 Htk ωω += (15)
.0HHH tk ωω += (16)
4. RESEARCH OF A TRANSITIVE MODE
At a transition mode the mechanism goes as system with two degrees of freedom at presence only one input. The transitive mode of motion begins after the mode of getaway (start-up) ending and lasts up to achievement of a mode of the steady (established) motion at which work (or power) of driving forces is equal to work (or power) of resistance forces. Mechanism moves as system with two degrees of freedom having only one input during transition mode of motion. The condition of reaching of steady motion mode corresponds to principle of possible
displacements and has next form
023311 =−−− HTHH MMMM ωωωω . (17)
Here HH ωωω −= 22 – angular speed of the satellite 2
relative to carrier H .
Let us substitute in (17) value H2
ω and values of
angular speeds 23
,ωω from expressions (4), (5) and
after transformations we get:
1)1(2
)1(33
)(21
)(3131 ωω
HTHH
HT
H
HuMuMM
uMuMM
−−
++= . (18)
Let us designate:
.)1(
2)1(
33
)(21
)(3131
1
HTHH
HT
H
HuMuMM
uMuMMu
−−
++= (19)
Then: .11ωω HH u= (20)
The equations (15), (16) and (20) connect among
themselves angular speeds 1,ωωH and time t . We
determine these parameters solving the specified system of three equations. We substitute in the equation (20) values of angular speeds from the
equations (15), (16). We get t - time of transition from
start mode in a mode of steady motion mode after transformations:
.1
011
1H
HH
H
kku
ut ω
−+
= (21)
Further, it is possible to determine angular speeds of
links 1 and H in steady motion mode under formulas (15), (16). Angular speeds of links 3 and 2 are determined under formulas (4) and (5).
Thus, the differential mechanism passes in a mode of the steady (established) motion at arbitrary set external moments on its links.
5. RESEARCH OF A STEADY MOTION MODE
Parameters of steady motion mode (angular speeds of links and time of transition from start mode in a mode of steady motion) depend on the set moments of forces on links and from the set moment of friction. Transition of the mechanism in a mode of steady motion at absence of acceleration of links testifies that the mechanism will be in a condition of balance. Thus as it has been proved in works of Ivanov (2007, 2010 and 2011) conditions of balance of static are carried out for the satellite 2 at arbitrarily assigned reactions transmitted on it. It happens at the account of some
conditional reaction 02R which acts on satellite from
side of support in instantaneous center of speeds of satellite 2. The value of this reaction is determined from a condition of balance of static of the
satellite∑ = 0F . The condition of balance of static of
satellite ∑ = 0M is carried out at the same time with
account of this conditional reaction.
Thus differential is going to steady motion mode at
25
arbitrarily assigned of force moments of differential.
If it happens changing of output moments of
resistance 31, MM in steady motion mode with
constant parameters HHM ω, of input power then
output angular speeds 31,ωω get the corresponding
values from the equations (20) and (4). This rule determines the effect of force adaptation proved in works of Ivanov (2007, 2010 and 2011). There is adaptation to external loadings due to the forced corresponded change of output speeds of motion at constant input power in the mechanism with two degrees of freedom.
Balance of the mechanism with two degrees of freedom is provided at the account of the internal
moment of friction TM really having a place. The
internal moment of friction plays a role of force regulation factor providing workability and definiteness of movement of the differential mechanism with two outputs at the variable output moments of resistance.
It is necessary to note that at absence of the moment
of friction ( 0=TM ) the mechanism degenerates in
system with one degree of freedom because one of the output links transmitting on the satellite the bigger reaction appears stopped.
The required size of the moment of friction can be determined having presented the formula (18) in the following kind:
.)(
21)(
3131
)1(2
)1(331
HT
HHTHH
H uMuMM
uMuMM
++
−−=
ωω
(22)
At 3212 RR > , angular speed 1ω can exist in the limits:
.0 1 Hωω <<
We have at 01 =ω :
0)1(2
)1(33 =−− HTHH uMuMM .
We have at 01 >ω :
THH MuMM >− )1(33 ,
or:
.)1(33 HHT MuMM −>
As we have at 01 =ω :
0)1(33 =− HH MuM ,
:hen: .0min >TM (23)
We have at Hωω =1 :
=−− )1(2
)1(33 HTHH uMuMM
.)(21
)(3131
HT
H uMuMM ++=
We have at Hωω <1 :
31 MMMM HT −−>
or: .31max HT MMMM −+< (24)
Thus the moment of friction TM should be in the
following limits: .0 31 HT MMMM −+<< (25)
Limits of change of angular speed 3ω :
.)1(33 HHH u ωωω << (26)
The efficiency is:
.)( 2
HH
HTHH
M
MM
ωωωω
η−−
= (27)
At:
Hωωω == min22 ,
we have:
.1max ==ηη
At:
01 =ω ,
we have from (5):
.2
21max22 H
z
zzωωω
+==
We receive from (27):
.12
1min
z
z
M
M
H
T−=η (28)
After the substitution of:
max2min2 ,ωω ,
we receive limits of change of efficiency from (27):
.112
1 <<− ηz
z
M
M
H
T (29)
6. BASIC DISTINCTIVE FEATURES OF THE MECHANISM WITH TWO DEGREES OF FREEDOM
The mechanism with two degrees of freedom has the following distinctive features.
1. The mechanism should have the real internal moment of friction transmitted on the satellite from the side of carrier and being on the satellite as the driving moment overcoming forces of resistance as corresponding reactions.
2. The satellite has the instant center of speeds which are not conterminous to the central axis of
the mechanism (otherwise Hωω =2
, relative
angular speed 022 HH =−= ωωω
and the
mechanism passes in a condition with one degree of freedom).
3. In the instant center of speeds of the satellite on it conditional reaction from the frame acts. This
reaction 3212202 RRRR H ++−= corresponds to
a condition of balance of the satellite, ∑ = 0F .
4. The internal moment of friction, TM , should be
determined from a condition of balance of the
26
satellite, 0=∑M , in view of conditional reaction,
02R . The determination of TM without taking into
account this force will result the mechanism in a condition with one degree of freedom. For
example, the value of the moment TM which is
received from a condition of equality to zero of the
sum of moments concerning point B (without
taking into account reaction 02R ) after
substitution in expression (18) will lead to:
0=Hω .
5. Numerical values of some parameters are determined by system of inequalities.
7. CONCLUSIONS
The gear differential with two degrees of freedom having only one input is the efficient mechanism if in it is the internal moment of friction in connection of the satellite with carrier.
Start of gear differential happens in condition with one degree of freedom, when all links of mechanism move with the same angular speed. Start mode going with appearance of accelerations.
Work of mechanism as system with two degree of freedom begins under the action of output moments of resistance, which change symmetric allocation of reactions on satellite. Balance of motion becomes possible with starting to operate the internal moment of friction, which appears at relative movement of satellite in relation to carrier, caused by asymmetric loading on the satellite. Work (capacity) of moment of friction is stabilizing factor resulting the mechanism in steady motion mode. In result, mechanism moves in a transitive mode of movement with different angular speeds of movement of output wheels.
The transitive mode of movement translates the mechanism in steady motion mode in which the effect of force adaptation of output links takes place at constant input power. The effect of force adaptation consists in independent change by the mechanism of speeds of rotation of output wheels depending on loading on them.
The internal moment of friction in the mechanism, chosen in required limits, plays a role of the stabilizing and regulating factor providing workability of the mechanism in different modes of motion and achievement of effect of force adaptation.
REFERENCES
IVANOV, K. S. (2011). Effect of force adaptation in mechanics. Journal of Mechanics Engineering and Automation. Vol. 1, No. 3. Libertyville, USA, pp. 163 – 180.
IVANOV, K. S. (2007). Gear Automatic Adaptive Variator with Constant Engagement of Gears. Proceedings of the 12th World Congress in Mechanism and Machine Science. Besancon. France. Vol. 2. pp. 182 - 188.
IVANOV, K. S. (2010 a). The simplest automatic
transfer box. World Congress on Engineering, London - WCE 2010, UK. pp. 1179 – 1184.
IVANOV, K. S. (2010 b). Theoretical fundamental of toothed continuously variable transmission. Theory of mechanisms and machines. Periodical science-methodical journal. N2 (16), Saint-Petersburg State Polytechnic University, pp. 36 – 48.
LEVITSKY, N. I. (1979). Theory of mechanisms and machines. Moscow. Science. 576 p.
CORRESPONDENCE
Konstantin IVANOV, Prof. D.Sc. Eng. Almaty University of Power Engineering and Telecommunications Faculty of Information Technology Baytursynov Street 126 050013 Almaty, Kazakhstan [email protected]
27
Balkan Journal of Mechanical Transmissions
(BJMT)
Volume 2 (2012), Issue 1, pp. 27-32
ISSN 2069–5497
VALVE JUMP MINIMIZATION USING N-HARMONIC CAM PROFILE
Sandip Kamble, Umesh Chavan, Satishchandra Joshi *
ABSTRACT. Valve jump is a transient condition that occurs in high-speed engines. With jump, the cam and the follower separate owing to excessively unbalanced forces exceeding the spring force during the period of negative acceleration. This is undesirable since the constraint and control of follower motion is not maintained. Valve jump results in short life of the cam flank surface, high noise, vibrations, and poor action. Present work focuses on the development of N-harmonic curve for valve acceleration improvement. Mathematical model of N-harmonic cam profile for high speed diesel engine is constructed. Simulation and experimental analysis shows N-harmonic cam profile has improved dynamic performance compared to polydyne cam.
KEYWORDS: Valve jump, polydyne cam, N-harmonic cam, cam-tappet, contact load
1. INTRODUCTION
Overhead valve train (OHV) is a widespread mechanism for internal combustion engine. In most of internal combustion engine valve cam has polydyne cam profile, however, in this type of valve train a serious problem known as valve jump is likely to occur at high speed. One of the major trends in engine development is increasing operating speed. With increasing engine speeds, the importance of cam-valve train design and its dynamic behavior become more critical. When the valve mechanism runs at a relatively low speed, the dynamic effect is not significant and kinematic analysis is enough to predict valve motion. But as the running speed becomes high, the dynamic characteristics of valve system take on a more important role as mentioned by Bagci and Kurnool (1997), Dresner and Barkan (1995), Horeni (1992) and Takashi et al.(2002).
Objective of this paper is to propose N-harmonic profile for valve cam to minimize jump at high speed. Jump model of N-harmonic cam profile is constructed. Simulation and experimental analysis is carried out to analyse the jump characteristics of polydyne and N- harmonic cam.
2. DYNAMIC ANALYSIS OF VALVE JUMP
The critical design point is where the inertia forces of follower maximum and tends to eliminate contact between the cam and follower. This point occurs in the vicinity of the maximum negative acceleration as shown by Chavan and Joshi (2011). Jump will occur when the negative inertia force of the system exceeds the available spring force. During negative acceleration period of valve opening event, the inertia forces tend to oppose the spring force. If at any instance, the load offered by valve spring is not adequate, then due to surplus inertia force, valve train mass lose contact of cam profile. This phenomenon is called valve jump and it results in unbalanced forces exceeding the spring force during the period of negative acceleration. Jump affects the life of cam
flank surface, high noise and vibrations.
2.1. Valve Jump Model
The valve train system shown in Fig. 1(a) is modeled as a two degree-of-freedom as described by Chen (1982) and Rothbart (2004), lumped-mass system taking into account the mass distribution, the elastic flexibility of all links, the Hertzian contact stiffness, the backlash in joints, and the resulting dynamic model is illustrated in Fig. 1(b). The system has the spring that closes the cam joint at the end effectors (valve) with most of follower train elasticity between it and cam. All the masses in the follower train are lumped into two effective masses m1 and m2. The cam acts against mass less cam shoe that connects to m1 through spring k1 and damper c1. Spring k2 and damper c2 connect to m2 to ground and represent the physical valve spring used to close the joint between cam and follower. The cam lift profile is the input to the dynamic model. The equations for condition of contact between
cam and follower mass m1 )( sz = can be written as
mentioned by Norton, (2002)
xm
kk
m
ccz
m
czz
m
kzx
i
2
21
2
21
2
2
2
2 )(+
−+
−′+−+′′=&& .
(1)
xkxczmFc 111 ++′′= & . (2)
If Fc is positive, then the value of x is valid. If Fc is negative, then separation can be occurred. Equations are solved simultaneously with Fc set to zero at time step, maintaining the initial velocity and displacement conditions. As the solution continues for additional time steps, s and z are compared at each step. When z ≤ s, the cam and follower have regained contact.
2.2 Construction of N- harmonic cam profile
In N-harmonic cam, lift expression is function of sine and cosine values. The lift expression is then differentiated to get velocity and acceleration. In this
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method, values at certain points on lift curve and acceleration curve are defined, which forms the boundary conditions for the equations. These equations are then solved to get the required lift curve.
a) Physical Model
b) Simplified two-mass model
Fig. 1. Mathematical model of Valve train
For N-harmonic cam, the lift expression of cam follower can be expressed as follow mentioned by Gao and Feng. (2004):
∑∑==
+=N
ii
N
ii iBiAh
00
sincos)( ααα . (3)
where: Ai and Bi are undetermined coefficients; α - the cam angle.
The velocity expression of cam follower is:
∑∑==
+−=N
ii
N
ii iiBiiA
d
dh
00
cossin ααα
. (4)
The acceleration expression of cam follower is:
∑∑==
−−=N
ii
N
ii iBiiAi
d
hd
0
2
0
2
2
2
sincos ααα
. (5)
Fig. 2. Theoretical cam lift
In the above equations, coefficients Ai and Bi can’t be selected at random. Firstly, they must subject to the following conditions composed of some equations about undetermined coefficients Ai and Bi.
Condition 1: In Fig. 2, when α = 0, i.e. in the connecting point between opening ramp and base circle, the lift of cam follower is zero, velocity and acceleration are also zero. They are expressed as follows:
∑∑∑===
===N
ii
N
ii
N
ii AiBA
0
2
10
0,0,0 . (6)
Condition 2: In Fig. 2, when α = αR, i.e. in the connecting point between event and ramp, the lift and velocity of cam follower should be two constants hR
and vR and expressed as:
R
N
iRi
N
iRi hiBiA =+∑∑
== 10
sincos αα ; (7)
R
N
iRi
N
iRi viiBiiA =+− ∑∑
== 10
cossin αα . (8)
Condition 3: In Fig. 2, when α = αB, the lift of cam follower runs to maximum lift hmax and the velocity of cam follower is zero. They are expressed as:
max10
sincos hiBiiAN
iBi
N
iBi =+∑∑
==
αα ; (9)
0cossin
10
=+− ∑∑==
N
iBi
N
iBi iiBiiA αα . (10)
Condition 4: In Fig. 2, when α = αE, i.e. in the connecting point between closing event and ramp, the lift and velocity of cam follower should be two constants hE and vE. They are expressed as:
E
N
iEi
N
iEi hiBiA =+∑∑
== 10
sincos αα ; (11)
E
N
iEi
N
iEi viiBiiA =+− ∑∑
== 10
cossin αα . (12)
Condition 5: In Fig. 2, when α = αF, i.e. in the connecting point between closing ramp and base circle, the lift of cam follower is zero, velocity and acceleration are also zero. It can be expressed as follows:
0sincos
10
=+∑∑==
N
iFi
N
iFi iBiA αα ; (13)
0cossin
10
=+− ∑∑==
N
iFi
N
iFi iiBiiA αα ; (14)
29
0sincos
1
2
0
2 =−− ∑∑==
N
iFi
N
iFi iBiiAi αα . (15)
Condition 6: The negative acceleration curve must pass through the points which are given before, i.e. when α = α1, α2, α3, α4, α5, α6, α7 the acceleration should equal to the given constants –A1, –A2, –A3, –A4, –A5, –A6, –A7 as shown in Fig. 3 correspondingly, so the following equations can be written:
k
N
iki
N
iki AiBiiAi =−∑∑
== 1
2
0
2 sincos αα . (16)
where 7...2,1=k .
Fig. 3. Acceleration constraints
Condition 7: The fullness factor ζm should satisfy following equation:
+− )(0 REA αα
=
−−++∑
=
N
i
RiRiEiEi
i
iB
i
iA
i
iB
i
iA
0
cossincossin αααα
= ))(())([( maxmax EEERRBm hhhh −−+−− ααααζ .
(17)
where the fullness factor ζm is used to evaluate the charging performance of a valve train cam. It is defined as:
∫ −
=)(
)(
max REm
y
dy
αααα
ζ . (18)
where yα and ymax denote valve lift and maximum valve lift respectively.
The equations 6H18 are arranged in a matrix form and solved using MATLAB:
][]][[ YXK = . (19)
After getting the coefficients i.e. A0 to A10 and B1 to B10, these coefficients are fed in lift expression to get the values of lift from opening ramp to closing ramp event.
3. EXPERIMENTAL ANALYSIS
The polydyne and N-harmonic cam (Fig. 4) are manufactured and tested for jump speed on test bench (Fig. 5). Since measuring the cam tappet contact loads on an actual operating engine is
extremely difficult, therefore indirect measurement was applied to the actual engine tests, in which the valve jump occurrence engine speed was detected by hearing the thumping sound. When jump occurs, the follower pounds on the cam surface giving a good thumping sound. The machine is a motorized unit consisting of a cam shaft driven by a D.C. Motor. At the free end of cam shaft a cam can be easily mounted. The arrangement of speed regulation is provided.
Fig. 4. Manufactured cams
Fig .5. Photograph of test bench
4. RESULTS AND DISCUSSIONS
In present investigation, MATLAB simulink tool is used as employed by Chavan and Joshi (2010) to solve equations of N-harmonic cam.
The valve motion is simulated using polydyne and N-harmonic profile as valve cam. The lift is calculated from opening ramp to closing ramp event. Valve lift, velocity and acceleration curves are shown in Fig. 6, Fig. 7 and Fig. 8.
Fig. 6. Valve lift
b) Polydine a) N-harmonic
30
Fig. 7. Valve velocity
Fig. 8. Valve acceleration
The result of dynamic calculation shown in Table 1, N-harmonic cam has lower velocity and acceleration. Area under displacement curve of N-harmonic cam is more as compared to polydyne cam. Thus, N-harmonic cam has larger fullness factor.
Table 1. Results of dynamic computation at 1050 rpm
Quantity Polydyne
cam
N-Harmonic
cam
Maximum lift of valve (mm)
6.140 6.169
Fullness Factor (ξ) 0.708 0.900
Area under displacement curve (mm
2)
9.107 11.582
The cam contact force varies at each point on the periphery of cam profile and is higher at nose. Cam contact force at nose for different speeds is simulated. Jump occurs when cam tappet contact load becomes zero as shown in Fig. 9a.
Simulation and experimental result shows that when cam speed increases beyond 750 rpm, the polydyne cam can’t operate normally and jump occurs. However, N-harmonic cam still works well beyond 750 rpm keeping the contact between cam and follower as shown in Fig. 9. In case of N-harmonic cam, cam
tappet contact load increases with increase in cam speed and reverse is the case for polydyne cam. Thus, acceleration constraints provided in N-harmonic cam profile development provides enough flexibility in cam design. Inertia forces increase with the square of the engine speed. The valve spring must have enough force to counteract the inertia forces on the nose. Intersection points (jump speed) of inertia force and spring force curve are found by simulating the forces as shown in Fig. 10 and 11.
a) Polydyne cam
b) N- Harmonic cam
Fig. 9. Variation of contact force at different speed
5. CONCLUSIONS
N-harmonic cam has smooth operation, large fullness factor, and it provides greater flexibility in cam design. It works without jump at relatively higher speed. N-harmonic cams are more suitable for high speed engines as compared to polydyne cams.
REFERENCES
BAGCI, C., KURNOOL, S. (1997). Exact response analysis and dynamic design of cam-follower systems using Laplace transforms, Journal of Mechanical Design, Volume 119, pp. 359-369.
CHAVAN, U. S., JOSHI, S. V. (2011). Synthesis of cam profile using classical splines and the effect of knot locations on the acceleration, jump and interface force of cam follower system, IME (Part C) Journal of Mechanical Engineering Science, Volume 225, Issue 12, pp. 3019-3029.
31
Fig. 10. Variation of inertia force at different polydyne cam speed
Fig.10. Variation of inertia force at different polydyne cam speed
Fig. 11. Variation of inertia force at different N-Harmonic cam speed
CHAVAN, U. S., JOSHI, S. V. (2010). Synthesis, design and analysis of a novel variable lift cam follower system, Int. Journal of Design Engineering,Volume 3, pp. 392- 411.
CHEN, F.Y. (1982). Mechanics and Design of Cam Mechanisms, Pergamon, New York.
DRESNER, T. L., BARKAN, P. (1995). New methods for the dynamic analysis of flexible single input and multi input cam-follower systems, Journal of Mechanical Design, Volume 117, pp. 150-155.
GAO, W., FENG, J. (2004). A Design Approach of Asymmetrical Cam Profile and Its Effect on Performance of High-Speed Automotive Engine, SAE paper 2004-01-0610.
HORENI, B. (1992). Double-mass model of an elastic cam mechanism, Mechanism and Machine Theory, Volume 27, pp. 443-449.
NORTON, R. L. (2002). Cam Design and Manufacturing Handbook, Industrial press, New York.
ROTHBART, H. A. (2004) Cam Design Handbook, New Jersey.
TAKASHI, I., AKIRA, S., BENCHENG, S., MASAHIKO, S., TADAO, O., and MASAHIRO, A. (2002).
Prediction of the dynamic characteristics in valve train design of a diesel engine, SAE-paper 2002-32-1839.
CORRESPONDENCE
Sandip KAMBLE Assistant Professor, Mechanical Engineering, Sharad Institute of Technology, Kolhapur, India. [email protected]
Umesh CHAVAN Associate Professor, Mechanical Engineering.Vishwakarma Institute of Technology, Pune, Maharashtra, India, 411037 [email protected]
Satishchandra JOSHI Prof. Dr., Principal, D.Y. Patil College of Engineering and Professor, Mechanical Engineering, Vishwakarma Institute of Technology, Pune, Maharashtra, India, 411037, [email protected]
32
Tel+91-020-9975586453, 91-020-9850081589, Fax 91-020-24280926, * Corresponding author
33
Balkan Journal of Mechanical Transmissions
(BJMT)
Volume 2 (2012), Issue 1, pp. 33-38
ISSN 2069–5497
BENEFITS OF THE APPLICATION OF CYCLOIDAL BACKLASH GEAR REDUCERS WITH MORE ECCENTRIC SHAFTS
Sinisa KUZMANOVIC, Milan RACKOV, Klara RAFA
ABSTRACT. Backlash gear reducers belong to mechanisms that are now increasingly used in mechanical
engineering, first of all in machine tools for accurate positioning of tools while processing, in measuring machines, scanners and radars, where they are used for accurate transmission of motion, in robotics, where used for accurate positioning of nippers and other working organs, and especially in military field of mechanical engineering (with artillery tools and rocket launchers), where there are used to capture the exact elevation angle and azimuthal angle, aviation, etc. As is known, backlash gear reducers include all reducers with arc gap of less than 10 arc minutes, although for more accurate positioning (1 m to 1 km) arc gap is only slightly greater than 3 arc minutes. This paper analyzes universal cycloidal backlash gear reducers with more eccentric shafts and it was concluded that an increased number of shafts can significantly raise backlash and self – locking, that is for this kind of reducers extremely important.
KEYWORDS. cycloidal, backlash, reducer, more, eccentric shaft
1. INTRODUCTION
Backlash gear reducers are mechanisms designed for transmission of mechanical energy and motion, from driving to working machine, while they are used to reduce number of revolutions and to increase torque, and in some cases, they are used to change direction of rotation and to change the position of the driving machine shaft axis. Backlash gear reducers today have greater application in mechanical engineering, due to the fact that driving machines (engines) are usually driven by electric motors which usually work with a significantly higher number of revolutions then working machines, so reducers are used to adjust the number of revolutions and torque of driving machine to requirements of working machines (KUZMANOVIĆ, S. 2009). Backlash gear reducers are transmitters of motion.
2. THE BASIC CLASSIFICATION OF CYCLOIDAL BACKLASH GEAR REDUCERS
Cycloidal backlash gear reducers are usually categorized by the number of used records on cam:
• reducers with a single cam (overhead panel),
• reducers with twin cam plate and
• reducers with three plates.
Depending on the number of eccentric shafts, cycloidal reducers are divided into:
• gear with a single eccentric shaft,
• gear with two eccentric shafts and
• gear with three eccentric shafts.
Depending on the size of the arc gap ∆ϕz, there are:
• backlash gear reducers, with the angular gap ∆ϕz ≤ 10′ per gear pair,
• low backlash gear reducers, with the angular gap of ∆ϕz ≤ 1° per gear pair and
• industrial gear reducers, with the so-called normal gap with ∆ϕz > 1° (Ondrive Precision Manufacturing).
Here only cycloidal backlash gear reducers will be considered.
3. DEFINITION OF BACKLASH REDUCERS
For the proper functioning of the gear reducers, it is necessary that there is a certain gap between cogs, i.e. that the width of the space between cogs is slightly larger than the width of the cog of coupled gears. The clearance is necessary in order to avoid interference, wear and overheating of cogs, to enable proper lubrication, to compensate deviations in production and to enable thermal expansion. Thereby, it should be noted that there is a gap in the roller bearings, which can also affect the increase of the clearance between cogs.
In the classical reducers the value of that gap is not so important, while in backlash gear reducers it is extremely important and seeks to be reduced to the lowest possible level (Ognjanović, 2000).
Small gap between cogs, in backlash gear reducers, is needed to ensure high positioning accuracy, where the positioning accuracy means the difference between the expected and actual position of the output shaft. As the gear has circular shape, the gap appears as angular size, i.e. as the arc gap.
Arc gap ∆ϕz is the angle of rotation of the output shaft,
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if the input shaft is fixed, i.e. it is stationary, and its value is expressed in arc minutes. In case that the value of this angle is less than 10 arc minutes the reducer is considered as backlash reducer.
It should be noted that the gap in gear pair is just a part of idle motion, which describes a condition in which, based on the mechanism input, does not get any response at the output.
Idle motion ∆ϕo of gear reducers is the output shaft rotation angle of the gear reducer under load with a torque T, at full cycle of changes in torque and at fixed input shaft (Fig. 2). It is the sum of the arc gap and the elastic torsion deformation of the shaft.
The rigidity of cogs, especially the torsion rigidity of the shaft, strongly influences the size of the idle motion of backlash gear reducers and therefore it seeks to increase the rigidity by adopting larger modules, shorter shafts and larger shaft diameters.
The graphic representation of values of the idle motion is given in Fig.1. Although, at present, there is no standard that defines the way of measuring the load, it is usually measured on the output shaft since its value is then much smaller (because of the larger gear diameter) and thereby certainly more interesting for display. The unit for measuring the arc gap, as already stated, is arc minute.
Fig. 1. Graphic representation of idle motion changes with changing loads (NEUGART)
Theoretically speaking, there is no torque required to register gap, i.e. it does not depend on the load. However, in reality, it's not the case, because in order to overcome the internal friction in the reducer and to eliminate the gap between the coupled elements, a certain torque is required. With increasing torque components deform elastically, leading to an increase of the idle motion, which is registered on the output shaft as the rotation angle which depends on the load, and its size is a measure of the rigidity of the reducer.
In reality, changing the load looks a little different (Fig.2). After a certain amount of torque required eliminating all gaps in the system, it is practiced to amend the complete cycle of gear load change (from zero to nominal load in one direction, then reducing the load and changing the torque in the other direction, to nominal value). In this way the "hysteresis loop" of the reducer is generated that enables to determine not only the actual gap, but the torsion
rigidity of the reducer and the idle at any load.
Because of the elasticity of the shaft and gear cogs, the idle motion of the backlash gear reducer is not in the form of straight line, but in the shape of the curve, covering a slightly wider field (Fig. 2). The initial values of arc gaps appear at the torque whose value is typically only 2% of the nominal value of torque.
The actual value of the arc gap for precision servo systems with a small arc gap typically ranges from 2 to 8 arc minutes (measured at the output).
Torsion rigidity (or its reciprocal value - the elasticity) is the size of the elastic angular deformation, which occurs on the output shaft under load, due to elastic deformation of gears and shafts. The rigidity and elasticity were determined on the basis of the measured deformation and the load of the reducer (Fig. 2). Rigidity shows what torque is required for unit value of deformation. A common unit for the rigidity: Nm /arc min. Elasticity indicates what deformation torque creates. A common unit for elasticity is arc min /Nm.
Most of the idle motion occurs due to the elasticity of the cycloidal reducer components. For high accuracy positioning rigidity effects can not be ignored.
Fig. 2. Graphic representation of the real change of the idle motion of backlash gear reducers
(NEUGART)
The absence of standards in this area leads to some confusion in the interpretation of the values given by some manufacturers. For example: Is the given value the maximum or the average value of the gap? Is the given value measured at the load in a first (positive) or in the second (negative) direction, or is it the total value? Do you use a torque on the output shaft, to ensure full seating of gears and bearings, etc?
Repeatability is an indicator of the degree of accuracy in which the positioning can be repeated. Gap and idle motion have a negligible role here. The biggest impact has the quality and accuracy of gear or cam profile making, as well as the positioning accuracy of the measuring device on the servomotor. The type of the applied coupling also has a large influence on the
Idle motion, i.e. the angle of rotation at load direction change
Torsional rigidity
Arc gap ∆φZ
Clearance backlash
Rotation angle of the output shaft
-T, Nm T, Nm 50% 100%
∆φ
∆T
φ
-φ, arc min
2% of Tmax 2% of Tmax
35
rigidity of the system and dynamic response, so it is necessary to consider its properties when choosing it, because the market offers couplings in different variants, with different rigidity and damping capabilities.
4. CYCLOIDAL REDUCERS
4.1. Introduction
Despite their high prices, cycloidal reducers face (encounters) great application in mechanical engineering because of the extremely small gap, a long period of holding such a small gap (due to the large number of cogs in mesh), because of the possibility of transferring high torque and high gear ratios (Kuzmanović and Vereš, 2006; Kuzmanović and Rackov, 2009). They are available in three basic variants with one, two and three eccentric shafts.
4.2. Cycloidal gear reducers with one (a single) eccentric shaft
Cycloidal gear reducers with one eccentric shaft (Fig. 3) in spite of high accuracy, have a slightly lower accuracy from reducers with two and three eccentric shafts.
Fig. 3. Characteristic solution of the cycloidal backlash gear reducer with a single eccentric shaft
(SUMIMOTO DRIVE)
The working principle of this reducers is pretty simple. The input shaft typically has two eccentric, with one cam plate on both of them. Cams are identical and two are used to repeal the radial forces that occur at the contact which enables quieter operation. This ensures a longer and larger contact of the coupled cogs and increases rigidity and enables backlash easier. The rotation of the eccentric shaft causes the coupling of cam plates with rollers which are placed along the rim of the circle, which causes cam plate rolling by rollers. This complex movement of cams is taking over by rollers, which are placed in specially drilled holes in the cam plates, which are an integral
part of the output shaft. So, rolling the cams by rollers causes the rotation of the output shaft (Fig.4).
Fig. 4. Schematic representation of the working principle of the cycloidal backlash gear reducer with a
single eccentric (Sumimoto Drive)
Reducers are used with one, two and three cam plates. The increase of the number of records causes the increase of the accuracy and load and, unfortunately, the price of the gear. For less demanding drives cycloidal backlash gear reducers with just a single cam (overhead panel) are used.
4.3. Cycloidal reducers with two eccentric shafts
Cycloidal reducers with two eccentric shafts (Figure 5) are slightly different from the classical cycloidal reducers, but are able to provide greater accuracy and less arc gap. Cycloidal reducers have very compact design.
Fig. 5. Assembly drawing of the cycloidal backlash gear reducer with two eccentric. The rotation direction
of the output shaft is the same as the input rotation (NABTESCO)
The working principle is very similar to cycloidal reducers with one eccentric, except that the application of two eccentric shafts allows to transmit more torque, and achieves a greater accuracy (Fig. 6 a and b). Namely, the torque from the input shaft through the cylindrical backlash gears is transferred to the two eccentric shafts (Fig. 6 a), from which the movement is transferred to the cam plate (Fig. 6 b), which performs complex rolling by rollers, which are circular placed along the edge of the housing. This complex movement is taken through the same two eccentric shafts, which are molded into the two front panels that are connected via two lunate parts which together represent the output shaft (Fig. 7).
Bearing of output
Cycloid disc
Eccentric high speed shaft
Ring gear housing Output flange
Case
Spur gear
Output flange
Input gear
Shaft
Middle flange Motor flange (option)
Crankshaft Input spline set (option)
Hold
RV gear
Main bearing
Oil seal
Pin
Oil seal
36
a)
b)
Fig. 6. Representation of the way of starting the eccentric shafts (1) and the way of moving cams (2) in the cycloidal backlash gear reducer with two eccentric
(NABTESCO)
Fig. 7. Schematic representation of the working principle of the cycloidal backlash gear reducer with
two eccentric shafts (NABTESCO)
4.4. Cycloidal reducers with three eccentric shafts
Cycloidal reducers with three eccentric shafts (Fig. 8) achieve even greater backlash and capacity.
Cycloidal reducers with three eccentric work on a similar principle as cycloidal reducers with two eccentric shafts. From the input shaft through a planetary backlash gear reducer (Fig.9) all three eccentric shafts are driven simultaneously. The number of shafts ie. gears in conjunction has very
positive effect on reducing arc gap.
Fig. 8. A characteristic solution of the special cycloidal
backlash gear reducer (NABTESCO)
Fig. 9. The schematic representation of the driving
principle of eccentric shafts in the cycliodal backlash gear reducer with three eccentric (NABTESCO)
Eccentric shafts move cam plates (Fig. 10) and roll them on rollers which are placed along the edge of the housing.
Fig. 10. The schema of the rolling of cams on rollers
in cycloidal backlash gear reducer with three eccentric shafts (NABTESCO)
This complex movement which is now much more accurate due to three supports (eccentric shafts) is transmitted through the same three eccentric shafts which are supported in the two front plate molded in the housing of the reducer and interconnected with three outlets in a trapezoid shape (Fig. 11).
Crankshaft
Input gear
Spur gear
Crankshaft
Rotation
Spur gear
Eccentric region
Rotation
Crank movement
Needle bearing
RV gear
Crankshaft rotation angle 0°
Crankshaft (connected to the spur gear)
Case
RV gear
Pin Output shaft
Rotation angle 0°
Rotation angle 360°
Spur gear
Input gear
Crankshaft
Came plate
Crankshaft (connected to the spur gear)
Case
Pin Shaft
Crankshaft rotation angle 0°
Rotation angle 0°
Rotation angle 360°
37
0
1
2
3
4
5
6
7
a) View
b) Lateral projection
Fig. 11. Assembly drawing of the cycloidal backlash gear reducer with three eccentric shafts (NABTESCO)
5. CONCLUSIONS
Backlash reducers are just one of the components of a propulsion (drive) system which includes an electric motor with brake, a counter of the number of revolutions (rev counter) and a processor also. Because of demands for accurate positioning, backlash reducers today have greater application in mechanical engineering. The demand for a compact structure, in order to use the available space more rational and to reduce weight of the whole system, caused the increased use of cycloidal reducers today. Small clearances between cogs and high torsional rigidity of shafts provide precision, exact position, repeatability and high reliability of such reducers. In places requiring high rigidity of the system, especially in places where there is a "contra-torque", which is often the case in mechanical engineering, cycloidal reducers are used because they are very rigid and self - locking, so, due to the high torque on the output, there will be no movement of the taken position of the output shaft.
Certainly, great positioning accuracy, which ranges below 3 arc minutes, especially contributes to their implementation. The main characteristic of the cycloidal reducers is a long life (ie, a long period of backlash retention) as a result of the large number of
cogs in cut and a large number of cam plates (today usually three). Cycloidal reducers despite the extremely high prices, especially those with three eccentric shafts, are of great use in the most responsible systems, where beside the high precision of position taking, a high degree of repeatability is also required. On less responsible places other types of backlash reducers are used. Fig. 12. The schematic representation of the driving principle of eccentric shafts in the cycliodal backlash
gear reducer with two and three eccentric shafts (NABTESCO)
After examining the catalogues it is evident that arc gap values are moving within limits of 6 arc minutes for cycloidal reducers with two eccentric shafts, while in cycloidal reducers with three eccentric shafts have arc gap about 1 arc minute. Based on this, it can be concluded that the application of cycloidal reducers with three eccentric shafts is justified, especially, because they have greater surface contact, and are able to keep backlash long.
ACKNOWLEDGEMENTS
This paper was financially supported by the Ministry of science and technological development of the Republic of Serbia, within project „Development of a New Generation of High Energy Efficiency Wind Generators”, project number 35005.
REFERENCES
KUZMANOVIĆ, S. (2009). Universal Helical Gear Reducers, University of Novi Sad, Faculty of Technical Sciences, Novi Sad, ISBN 978-86-7892-202-2.
KUZMANOVIĆ, S., RACKOV, M. (2009). Development tendencies of universal gear reducers. In: Proceedings of the 3rd International Conference Power Transmissions 09, Kallithea, Greece, 1-2. October 2009, pp. 145-148.
KUZMANOVIĆ, S., VEREŠ, M. (2006). New development trends of universal gear reducers, časopis Konstruisanje mašina / Journal of Mechanical Engineering Design, Vol. 9, No.1-2006, pp. 6–15.
Cycloidal reducers with two eccentric
shafts A
rc m
in
Case Rollers Cams
Beads
Output shaft
Planetary gears
Part of the output shaft
Toothed input shaft
Camshaft
7
6
5
4
3
2
1
0 Cycloidal reducers
with three eccentric shafts
38
NABTESCO, www.nabtescomotioncontrol.com (accessed on 20.05.2012)
NEUGART, www.neugartusa.com (accessed on 20.05.2012)
OGNJANOVIĆ, M. (2000). Machine Design. University of Belgrade, Faculty of Mechanical Engineering, Belgrade.
ONDRIVE PRECISION MANUFACTURING, www.ondrives.com (accessed on 20.05.2012)
SUMIMOTO DRIVE, www.sumitomodriveeurope.com (accessed on 20.05.2012)
CORRESPONDENCE
Sinisa KUZMANOVIC, Prof. Dr. University of Novi Sad Faculty of Technical Sciences Trg Dositeja Obradovica 6 21000 Novi Sad, Serbia [email protected]
Milan RACKOV, M. Sc. Assistant University of Novi Sad Faculty of Technical Sciences Trg Dositeja Obradovica 6 21000 Novi Sad, Serbia [email protected]
Klara RAFA, M. Sc. Assistant University of Novi Sad Faculty of Technical Sciences Trg Dositeja Obradovica 6 21000 Novi Sad, Serbia [email protected]
39
Balkan Journal of Mechanical Transmissions
(BJMT)
Volume 1 (2011), Issue 1, pp. 39-44
ISSN 2069–5497
GEARBOX MODIFICATION TO OBTAIN CONTINUOUS CONTROLLABILITY
Heikki MARTIKKA
ABSTRACT. This paper presents results of modifying a commercial gearbox to obtain continuous controllability of the total transmission ratio. The goal is to explore the innovative advantages what these modifications may offer. The methods are analytical gearing theory, simulation program and building of the prototype based on the ideas obtained from theory. A conventional gearbox with three planetary gears is modified to be controlled by one annulus activated by a motor with inverter speed control. Main power input is from an electric motor at constant speed and power. Output power is fed to a generator. The transmission ratio can be continuously varied by the control motor. Analytical machine models are formulated for simulating the kinematics behaviour and transmission ratio. Dynamic simulation is made with the Dymola program. Both methods gave the same results at check points. A combination of all these methods is synergically useful for developing new concepts to prototypes and then to commercial products.
KEYWORDS. Transmission design, gear, continuous control
NOMENCLATURE
Symbol Description Ak Gear ratios Lk Tooth ratio parameters x Transmission ratio T Torque zk Number of teeth of gear with code k
1. INTRODUCTION
There is a constant need for better mechanical and electrical power transmissions and their optimal control. The background need is to save energy by smooth and continuous conversion of various energies to various storages and back. This is a difficult task using conventional mechanical power transmission. In these gear transmissions, clutches and brakes are assembled sequentially as separate groups. Modern trend is to strive for integration already at the concept design stage.
Some innovations have been made along this trend.
One innovation is the stepless drive system for new Mini as discussed by Hall, Pour, Mathiek and Gueter (2002).
The design of wind turbine drive train based on a mechanically continuously variable transmission (CVT) is studied by Miltenovic et.al (2011). This opens a new solution approach using the advantages offered by using mechanical control elements.
Next stage after concept generation is to do detailed design. Machine design principles are found in texts by Pahl, Beitz, Feldhusen and Grote (2007). Machine elements can be in detail dimensioned using texts of Decker (2009) and Shigley (1989). Basic theory of gears is discussed by Muller (1971). Consideration of
the effect of contact stress on gears important and is discussed by Johnson (1987) and tribology by Satchowiak and Batchelor (2000).
In this project the goal was to get a continuously controlled gearbox. The realisation idea was obtained by studying analytically the basic kinematics of planetary gears. The idea emerged that a commercial gearbox with three planetary gears can be modified by making one annulus controllable by an electric motor and inverter. Input is from an electric motor at constant speed and power. Output is fed to a generator. Then this concept was chosen to be realised. The third goal was to verify the analytical models for simulating the kinematic behaviour. To study the dynamic behaviour the commercial Dymola simulation program was chosen to be used.
2. MODELLING OF THE MACHINERY
2.1. Megatrends for guiding concept innovation
The motive for developing new innovations has consistently been based on a new rising global megatrend of a new energy source. These interrelations are illustrated in Fig. 1. Fossil based energies have been found and used up. Now the trend of combining many kinds of eco-energies is rising. The need to utilise it efficiently gives an impetus for developing of cost-effective energy conversions. One of them is the continuous gear shifting possibility.
2.2. Model description and test set up
The basic principles of mechanics are used to design mechanical and electrical power transmissions and connections between them. The sign conventions for energy and torque balances and rotations are shown in Fig. 2.
ROmanian
Association of
MEchanical
Transmissions
(ROAMET)
Balkan Association of
Power Transmissions
(BAPT)
40
Main modules of the transmission are shown in Figs. 3 and 4.
The experimental set up is shown in Fig.5.
2.3. Functioning principles of the machine
Torque balance is expressed using the sign convention in Fig.2. The input torque is positive and others opposite to it. 0
2E1=−Σ− TTT (1)
Power balance is expressed using the sign convention that all powers are positive or flowing into the machine. P P P1 E 2+ + =Σ 0 (2)
Rotational sign convention is that they are all in positive direction. Using these conventions the power balance can be written as function of torque’s and speeds as ( )( ) ( )( ) ( )( )+ + + − + + − + =T T T1 2 0ω ω ω1 E E 2Σ (3)
This may be simplified to: T T T1 2 0ω ω ω1 E E 2− − =Σ (4)
2.4. One controlled variable
Now only one controlled variable is considered. It can be chosen as some one of the annuli. Power balance can be written: 0TTT 22 EE11 =ω−ω−ω . (5)
By dividing this with input speed ω1 one obtains the relationship for the transmission ratio: 0xTsTT 2 EE1 =−− . (6)
Now the annulus a1 can be fixed or controlled. From equation (6) one can conclude that the relative speed sE might be a linear function of the transmission ratio x or:
xBAss a1
1
a1
1
EE +==
ωω
=ωω
= .
(7)
Here A and B are assumed to be constants which depend on the fixed gearing parameters.
Using Eq.(7) the power balance becomes: ( ) 02 E1 =−+− xTBxATT . (8)
From this the transmission ratio x is solved:
BTT
ATTx
E2
E1
+−
= . (9)
Fig. 1. Global megatrends of energy sources
Eco= Ecoenergies, ecogas,biomass. solar,windpower.
InnoEco = innovations related to econenergies.
Eco
Market share
Time
Coal Oil Gas
1
T1 = T2 = 55
0
Now 2012
InnoEco
Fig.2. Principle of function and main components
A B C
1EP
1ET
1Eω
2EP
2ET
2Eω
1EP
1ET
1Eω
2EP
2ET
2Eω
Fig. 3. Sketch of the machine
T1
T2
Ta1 = TE
Tr
Fig. 4. Gearbox in the study
a) Opened box. b) The modified box after assembly.
a) b)
Fig. 5. The experimental set up
41
If the annulus a1 is fixed then the relationship between A and B and the constant transmission ratio x0 is obtained. ( ) 000a1 ⇒+= xBAxs
BxA 0−=⇒ .
(10)
Substitution of this relationship B vs. A into equation (9) gives:
BTT
xBTTx
E2
0E1
++
= . (11)
From the torque balance the input torque is expressed as function of others 2E12E1 ,0 TTTTTT +=⇒=−− . (12)
The total transmission is now obtained. This result is used to check the simulation results at selected times when the torques are known. ( )
.,
1
11
0
2
E
2
E
BxA
BT
T
AT
T
x −=+
−+
= (13)
2.5. Controlling of the transmission by a
modification
The gearbox sketch is shown in Fig.6.
The constant parameters are
.36
144111
108
36
;54
162111
108
54
;12
120111
108
12
33a3
s33
22a2
s22
11a1
s11
==−→−=−=
==−→−=−=
==−→−=−=
LAz
zA
LAz
zA
LAz
zA
(14)
The Dymola simulation model is shown in Fig.7.
Kinematic equations for the planetaries 3,2,1 are obtained from the Fig.6.
Planetary 3 equation is:
T2
ω1
T1 input
zs1=12 zs2 =54
zs3 =36
za1 =108
za2 =108
za3 =108
ω2, out
ωc1 = ωs2
ωc2 =ωs3
ωc3 = ω2
ω1 = ωs1
Ta1 Ta2 Ta3
M
L
Fig.6. Cross section of the gearbox.
Fig.7. The Dymola simulation flow chart
42
331a3
1s3 11
1
LAx
x−==
−
−
ωωωω
. (15)
Planetary 2 equation is
22s3a2
s3s2 11
1
LA−==
−−ωωωω
(16)
Planetary 1 equation is
11s2a1
s21 11
1
LA−==
−−ωωωω
(17)
2.6. All annuli are locked
Initial constant transmission ratio is obtained when all annuli are locked or immobile: ;0a1 =ω
;0a2 =ω
0a3 =ω .
(18)
The input rotation is applied to sun s1: ω ω1 s1= . (19)
These are substituted to the transmission ratio model x.
For planetary 3 one obtains:
⇒=−
−
31
1s3 1
0 Ax
x
ωωω
31
s3 1
Lx=
ωω
.
(20)
For planetary 2 one obtains:
⇒=−
−
2s3
s3s2 1
0 Aωωω
2s3
s2 1
L=
ωω
.
(21)
For planetary 1 one obtains:
⇒=−
−
1s2
s21 1
0 Aωωω
1s2
1 1
L=
ωω
.
(22)
Multiplying all these gives the inverse transmission:
0123s2
1
s3
s2
1
s3
0
11111
xLLLx⇒=
ωω
ωω
ωω
. (23)
Thus the transmission ratio with all annuli locked is
120
1
120
12
162
54
144
36LLLx 1230 === . (24)
2.7. One annulus is controlled and others are
locked The speed of the first annulus a1 is controlled using
the torque applied to it, TE giving ωE. Other annuli are held fixed at zero speed: ;Ea1 ωω =
;0a2 =ω
.0a3 =ω
(25)
Substitution gives for equation (15)
→=
−
−
3
1s3 1
0 Ax
x ωω
331
s3 111
LAx=−=
ωω
. (26)
and for equation (16)
→=−−
2s3
s3s2 1
0 Aωωω
22s3
s2 111
LA=−=
ωω
. (27)
Multiplication of these equations gives
⇒=ωω
321
s2
LL
1
x
32
s2
1 LLx
1=
ωω
.
(28)
The equation (17) for the planetary 1 can be written as:
.1
1
11
11
1
1
1
1
132
1a1
321
s2
1
1
a1
s2
1
s2
a1
s2
1
LLL
xs
LLx
a
a −=−
−
=−
−
=−
−
ωω
ωωωω
ωωωω
(29)
Where annulus a1 speed ratio is:
1
a11 ω
ω=as . (30)
Thus: ( ) ( ) 3211321a1a1a LLLL1LLssx +−= ;
( ) 3211a LLL0x = . (31)
From this equation (31) one may solve sa1:
1a1a1a1a xBAs += . (32)
The coefficients are obtained as follows. The first term is constant:
( ) 9
1
120
121120
12
L1
LA
1
11a −=
−
−=
−−
= . (33)
The second term depends on x with a constant factor:
( ) .9
120
120
121
144
36
162
54
1
1
1
1321 =
−
=−
=LLL
Ba
(34)
3. TWO CONTROLLERS
More degrees of freedom for controlling are obtained if two simultaneous controllers E1 and E2 are used. This increases also the complexity and costs. This option was not now realised. Now the input rotation from motor 1 is applied to sun s1 and the output is from sun s3: s32s11 ωωωω ⇒⇒ . (35)
The power balance for two controllers E1 and E2 is:
02E2E11 =+++ PPPP . (36)
43
The signs are determined by the rules in Fig.2. 022E2E2E1E111 =−−− ωωωω TTTT (37)
Division by input speed gives speed ratios:
01
22
1
E2E2
1
E1E11 =−−−
ωω
ωω
ωω
TTTT . (38)
If inputs are expressed in powers then one obtains:
01
2
1
E2
1
E11 =+++
ωωωPPP
T . (39)
Here the total transmission ratio x and the two controlled transmission ratio variables are:
1
2
ωω
=x ;
1
E1E1 ω
ω=s ;
1
E2E2 ω
ω=s .
(40)
Thus the power balance becomes: 0xTsTsTT 2E2E2E1E11 =−−− . (41)
The T2 torque may be solved from the torque balance equation:
E2E1122E2E11 0 TTTTTTTT −−=⇒=−−− . (42)
Substitution gives:
1T1TT
sTsTTx
E2E11
E2E2E1E11
⋅−⋅−⋅−⋅−
= . (43)
This formula suggests possibilities for innovative solutions.
One option is to keep all torque constant. Then the powers can be changed to change speed ratios to get the desired x:
021
E2
1
E11 =−++ xT
PPT
ωω. (44)
From this one may solve the ratio x as:
( ) .,,,2
2
1
E2
1
E11
1
E11
2E2E11T
T
PPT
PT
PPPPx
ωωω
++++
= (45)
One option is to set all control powers to zero and leave the input output power acting. Then the transmission ratio reduces to the fixed annuli x0 ratio given by equation (24): ( ) ⇒==== 0
2
2
1
1
1
2
2
121 ,0,0, x
P
P
T
TPPx
ωωω
ω
;021 =− PP
( ) .LLLx0,0,0x 32103a2a1a0 ===ω=ω=ω
(46)
4. DYMOLA SIMULATION RESULTS AND
COMPARISON WITH ANALYTIC RESULTS
The simulation model flowchart is shown in Fig. 7. Using the model many simulation results are obtained. One selection is shown in Fig. 8. Some data points are
selected for comparison with analytical model predictions as shown in Fig. 9.
Comparison of simulation results is shown in Table 1
Table 1. Torques (Nm) TE and T2. Torque T1 is increased in 10 sec to T1=10Nm. Transmission ratio xa1 when annulus a1 is controlled by torque TE.
Time (s) 5 12 80
xa1, analytical 1/28.56 1/14.1 1/12.182
xa1, Dymola 1/28.56 1/14.1 1/12.182
TE, torque at E 3 10 10
T2, torque at 2 -2 -1.576 -1.576
The equation (13) was used to calculate the analytical prediction for the transmission ratio at two time steps.
At time t = 5 sec the total transmission is:
( )( )
;56.28
1
9
120
2
31
9
11
2
31
1
11
12
E
12
E
1 =
−+
−−
−+
⇒+
−+
=
a
a
a
BT
T
AT
T
analx (47)
Fig.9. Datapoints for comparison check of analytical predictions and Dymola simulation results
1T
ET
2T
Time, s
Fig. 8. Dymola output in the model of Fig. 7
2T = -1.576, t=12
T1
1/x=12.182, t=80
1/x =28.56, t=5, x = 1/28.56
1/x=14.1, t=12
0
ET =3,
t=5
ET =10,
t =12
2T = -2
t=5
ET
5 10 12 80 Time t, s
44
( )56.28
11 ≈Dymolaxa .
At time t = 12 sec the total transmission is
( )( )
;1.14
1
9
120
576.1
101
9
11
576.1
101
1
11
.
12
E
12
E
1 =
−+
−−
−+
⇒+
−+
=
a
a
a
BT
T
AT
T
analx
( )1.14
11 ≈Dymolaxa .
(48)
The analytical and Dymola results agree satisfactorily.
5. CONCLUSIONS
Demand for efficient and intelligent power transmissions is growing for utilising many ecoenergy applications. Planetary gears are popularly chosen since they can be versatilely interconnected to give new innovative possibilities and they can be controlled to obtain many desired functions and transmission ratios.
The first goal was to modify a basic commercial gearing to utilise it to control the output speed continuously for new applications. This goal was well achieved by controlling one annulus by an electric motor with inverter. The calculations showed that even two of three annuli can be in principle controlled to get a wide control space of the transmissions ratio.
The second goal was to utilise and compare and two analysis methods and by applying them to the system. The comparison criteria are cost effectiveness, ease of use and utility and accuracy of results.
Analytical method is good at making clear the kinematics and balance functioning. Kinematic and static modelling is easy and results are accurate. It is a good tool for doing system optimisation by proceeding systematically.
The first stage is to obtain design variables by redefining some chosen system parameters and connecting them with kinematic and geometric relationships.
The second stage is formulated the decision variables and their desired ranges and fuzzy satisfaction functions. Important DV’s are cost, mass, transmission ratio ranges, power, safety factors for pitting and bending endurances of teeth.
The third stage is to form the total satisfaction by multiplying satisfaction on each DV and maximising it.
The fourth stage is to obtain optimal choices of design variable.
This optimisation will be used in further studies.
Analytical dynamics is more challenging that static modelling. Dynamic simulation requires activation of some non-linear dynamic simulation program based on equations as is possible with Dymola.
The dynamic kinematics simulator program proved most efficient in studying dynamics. The modelling method was to describe the system with input output elements interconnected in feasible ways to each other. The interconnections and interactions are complex and require a thorough exploration and verification. Interpretations are not so straightforward due to complex interactions of the dynamically changing states of the components.
Comparison of calculated showed that the results agree satisfactorily at the chosen checkpoint.
REFERENCES
DECKER, K-H et al. (2009).Maschinenelemente, Funktion,Gestaltungund Berechnung,Hanser,ISBN 978-3-446-41759-5.
Dymola User Manual 5.3.Dynasim AB,2004.
HALL, W, POUR, R., MATHIEK,,D. GUETER C. (2002). Das stufenlose Automatikgetriebe fur den neuen Mini, ATZ 5/2002, pp. 458-463.
JOHNSON ,K.L.(1987). Contact mechanics, Cambridge University Press, ISBN 0-521-34796-3.
MILTENOVIC ,V.,VELIMIROVIC,M.M BANIC,M., MILTENOVIC,A. (2011). Design of windturbines drive train based on CVT, Balkan Journal of Mechanical transmission, Volume 1, Issue 1, pp. 46-56, ISSN 2069-5497.
MULLER, H. (1971). Die Umlaufgetriebe, Springer Verlag.
OJA, J. (2007). Mechatronical continuously adjustable vehicle power transmission design and simulation, Dept. of Mechanical Engineering, Lappeenranta University of Technology, MSc Thesis.
PAHL, G., BEITZ, W., FELDHUSEN, J.,GROTE, K. H. (2007). Engineering Design-A systematic Approach.3
rd Edition,New York, Springer-Verlag,
ISBN-10:1846283183.
SHIGLEY, J.E., MISCHKE, C.R.(1989). Mechanical engineering design, McGraw Hill, ISBN 0-07-056899-5.
STACHOWIAL, G. W., BATCHELOR, A. W. (2000). Engineering Tribology, Butterworth Heinemann, Boston , ISBN 0-7506-3623-8.
CORRESPONDENCE
Heikki MARTIKKA, Prof.Emeritus, D.Sc.(Tech.) CEO, Chief Engineer Himtech Oy Engineering Ollintie 4, FIN-54100 Joutseno, Finland [email protected]
Simulation of gear contact on a two-disk test rig
45
Balkan Journal of Mechanical Transmissions
(BJMT)
Volume 2 (2012), Issue 1, pp. 45-50
ISSN 2069–5497
SIMULATION OF GEAR CONTACT ON A TWO-DISK TEST RIG
Athanassios MIHAILIDIS, Stelios GATSIOS
ABSTRACT. An improved method for the experimental simulation on a two-disk test rig of the contact conditions occurring along the path of contact of spur gears is presented. The effect of roughness and its orientation, as well as the Λ ratio are considered. The friction coefficient measured on a two-disk test rig is compared to the average gear friction coefficient obtained by power loss measurements carried out in the FZG test rig. Satisfactory correla-tion is found. It is shown that by properly selecting the surface finishing technique of the disks, gear contacts can be simulated in a two-disk test rig.
KEYWORDS. Gear friction, two disks test rig, friction coefficient, power loss, FZG gear test rig
NOMENCLATURE
Symbol Description a viscosity-pressure coefficient
E ′
12 21 2
1 2
1 11
2
v v
E E
− − − +
reduced modulus
of elasticity G E ′⋅α dimensionless material parameter g line of action hc central film thickness Hc central non-dimensional film thickness hmin minimum film thickness Hmin minimum non-dimensional film thickness
k ellipticity of Hertzian contact
kFZG FZG load stage PVZ Power loss due to gear meshing pH Maximum Hertzian Pressure
Rq ( )1 22 21 2
/
q qR R+ composite surface rough-
ness
Rx
1
1 2
1 1
R R
−
+
reduced radius of curvature
in rolling direction
S 2 gV
VΣ
⋅ slide-to-roll ratio
U x
V
E R
η⋅′ ⋅
dimensionless speed parameter
V VΣ /2 VG |V1-V2| sliding velocity
VΣ V1+V2 sum velocity
W 2
N
x
F
E R′ ⋅ dimensionless load parameter
for non-line contacts
W ′ N
x
F
E R L′ ⋅ ⋅ dimensionless load parameter
for line contacts z teeth number η oil dynamic viscosity θm oil temperature λoil oil thermal conductivity µ friction coefficient µm average friction coefficient
1. INTRODUCTION
Improving gear efficiency is a challenge for engineers. A lot of theoretical and experimental research has been devoted on this effort, in order to minimize pow-er losses and prevent surface induced failures such as wear, scuffing and pitting. From the early days of studying gears, gear test rigs were used to simulate real operating conditions that were met in gear reduc-ers. They can provide useful information concerning gear failures modes and efficiency, but only poor in-formation concerning friction coefficient between the mating surfaces. Only the average friction coefficient can be obtained from power loss measurements.
Two-disk test rigs, on the contrary, allow for the study of the friction coefficient in a much more comprehen-sive way. Pressure, sum velocity and slide-to-roll ratio can be set easily and therefore experiments with disks can cover a wide rage of operating conditions that found in gears and ball bearings.
In the present work it is studied how accurately a gear contact can be simulated in a two-disk test rig, and an experimental procedure is proposed, that enhances the simulation.
2. EXPERIMENTAL APPROACH
2.1. Previous research
A previous effort on this matter was made by Höhn et al. (2001). Experiments were conducted on FZG gear test rig with type C gears using ISO VG 150 mineral oil without additives at 90 oC. Power loss was meas-ured for various load stages and rotational speeds and the average friction coefficients were calculated,
ROmanian Associa-
tion of MEchanical
Transmissions
(ROAMET) Balkan Association of
Power Transmissions
(BAPT)
46
as shown in Table 1. They also measured the traction curves of the same lubricant in a two-disk test rig. Then, they calculated average friction coefficient for the gears by extrapolating the lubricant traction curves. The results are shown in Fig. 1.
They found that for load stage FZGk = 5, the calculated
friction coefficient is in agreement with the experimen-tally obtained. In this load stage the pressure is low, the contact operates almost in the full film lubrication regime and friction is dominated by lubricant shearing. As the load stage was increased, pressure was higher and the contact operated in the mixed lubrication re-gime. Now, friction coefficient depends on both lubri-cant shearing and surface roughness interaction.
Table 1. Average friction coefficient calculated from the experiments performed in a FZG test rig using the
type C gearing
FZG load stage
FZGk =5 FZGk =7 FZGk =9 FZGk =11 wheelη
[rpm] EXPµ
350 0.050 0.052 0.055 0.058 700 0.043 0.044 0.047 0.048 1400 0.037 0.038 0.039 0.040 2100 0.032 0.033 0.037 - 3500 0.025 0.027 0.031 -
Fig. 1. Average friction coefficient for FZG Type C gears. Experimental values measured in an FZG Gear
Test Rig are compared with calculated values ex-trapolated from lubricant traction curves. Data are
from Höhn et al. (2001)
It was further observed that the calculated friction co-efficient was up to 25% higher than the experimentally obtained in the FZG gear test rig. In their conclusions they attributed this disagreement mainly to the surface roughness orientation (transversal in gears and longi-tudinal in disks) and in surface roughness height ( mR gears µ4.00 = , mR gdisksearsq µ1.0= ).
For the current study, the experiments reported in the work of Höhn et al. (2001) with type C gears in the FZG test rig were used as a reference. Similar condi-tions concerning pressure, sum and sliding velocity were applied on the disk experiments. Additionally, both roughness orientation and roughness size were
taken into account. Also, the same lubricant was used. Average friction coefficient was calculated from disks friction force measurements.
2.2. Experimental set-up
The design and the capabilities of the two-disk test rig of the LME&MD are described in detail in the work of Mihailidis et al. (2003). Therefore, only the most im-portant features will mentioned here. The principle of operation is shown schematically in Fig. 2.
Fig. 2. The two-disk test rig of the LME&MD
1 – crowned disk; 2 – cylindrical disk; 3 – load cell.
A pair of disks consists of a cylindrical and a crowned disk with rolling diameter d = 80 mm. Each disk is mounted at the free end of the corresponding shaft. If both disks were cylindrical then edge loading would occur due to shaft bending. As a result, pressure dis-tribution would be significantly different from Hertzian. In order to avoid this situation, one of the disks is crowned. This means that line contact cannot be ex-actly simulated and it has to be approximated by an elongated elliptical contact. The ellipticity of the con-tact is set by properly selecting the crown radius of the crowned disk. For the experiments of the current study, disks with crown radius R = 990 mm were manufactured, resulting in ellipticity k = 12, shown in Fig. 3.
Two inverter controlled AC motors drive the shafts independently and rotate them with 60 to 6000 rpm (1 Hz – 100 Hz) resulting in circumferential velocity of 0.25 to 25 m/s. Thus, slide-to-roll ratio and rolling ve-locity can be freely set.
Normal load is applied by pushing the upper shaft-disk against the lower by means of a lever and weights. Friction force is measured by a load cell. That permits the direct and accurate calculation of friction coeffi-cient simply by dividing friction force by the applied normal force, in contrast to other two-disk test rigs in which friction coefficient is determined indirectly by friction torque measurement.
2.3. Current approach
In order to choose properly the conditions of the disk experiments it is necessary to study type C gears. Geometry parameters are given in Table 2. Radius of
3
2
1 1
2
Simulation of gear contact on a two-disk test rig
47
Fig. 3. Disk specimens layout
Fig. 4. Radius of curvature, normal load, maximum pressure distribution and velocity across the path of
contact for type C gears
curvature, normal load, maximum pressure distribu-tion along with rolling and sliding velocity across the path of contact, were calculated for every load stage and rotational speed, as shown in Fig. 4.
The frictional power loss a gear pair is calculated us-ing formula (1):
( ) ( ) ( )1 a
f
g
VZ N Ge g
P g F g V g dgp
µ−
= ∫ . (1)
Table 2. Geometry parameters of type C gearing
Normal module [mm] 4.5
Pinion teeth number z1 16
Wheel teeth number z2 24
Addendum modification factor for the pinion x1
0.182
Addendum modification factor for the gear x2
0.172
Face width [mm] 20
Helix angle 0o
Normal pressure angle 20o
Centre distance [mm] 91.5
If we assume that friction between gear teeth is repre-sented by an average friction coefficient mµ then the
previous relation can be written in the following form:
( ) ( )1 a
f
g
VZ m N Ge g
P F g V g dgp
µ−
= ∫ . (2)
From equations (1) and (2) average friction coefficient is calculated:
( ) ( ) ( )
( ) ( )
a
f
a
f
g
N G
gm g
N G
g
g F g V g dg
F g V g dg
µ
µ−
−
=
∫
∫
(3)
If the path of contact is divided by n uniformly distrib-uted points, then the above equation can be written in the following discretized form:
1
1
n
i Ni Gii
m n
Ni Gii
F V
F V
µ
µ =
=
=∑
∑ (4)
In order to simulate the contact conditions of a given contact point by the disk contact, the following has to be considered.
• The geometry of the disks (rolling and transverse curvature) was chosen so as to create an elon-gated elliptical contact with ellipticity k=12, which simulates the linear contact found in type C gears with acceptable accuracy.
Crowned disk Cylindrical disk
A-A A-A
48
• The oil and its temperature were kept the same in disks as in gears.
• Every point in the path of contact corresponds in a certain maximum Hertzian pressure and in certain sum and sliding velocities for a given load stage and rotational speed. Proper load, different for each point of the contact path, was applied on disks so as to create the same maximum Hertzi-anpressure. Also, the proper rotational speeds were applied so as to have the same sum and sliding velocities for every point.
• Furthermore, in order improve the simulation and take into account the roughness effects, both con-tacts must operate in the same lubrication regime. Therefore, the combined role of film thickness and roughness has to be estimated at first.
From the gear experiments of Höhn et al. (2001) the ones carried out at load stages 7 and 9 with 700, 1400 and 2100 rpm wheel speed were chosen to be simu-lated.
Minimum and central film thickness across the path of contact were calculated according to the formulas for line contacts found in the literature (Hamrock et al., 2004):
( ) 128.0'568.0694.0minmin 714.1
−== WGU
R
hH
x
; (5)
( ) 166.0'47.0692.0922.2−
== WGUR
hH
x
cc . (6)
Then, film thickness was calculated for the same con-ditions of pressure and speed for disks using formulas for elliptical contacts from Hamrock et al. (2004):
( )k
x
eWGUR
hH 68.0073.049.068.0min
min 163.3 −− −== ;(7)
( )k
x
cc eWGU
R
hH 73.0067.053.067.0 61.0169.2 −== − .(8)
The effect of heating due to sliding for both linear and elliptical contacts was taken into account by the ther-mal correction factor:
( ) 64.0*83.0
42.0*'
23.21213.01
2.131
LS
LE
p
C
H
t
++
−= , (9)
where:
2
2
m oil
V
Lηθ λ
Σ
∗
∂ = −
∂ . (10)
Finally, minimum and central film thickness including the effect of heating is calculated with the following formulas:
min min x th H R C= ⋅ ⋅ ; (11)
c c x th H R C= ⋅ ⋅ . (12)
Results are shown in Fig. 5. It is obvious that the film thickness in the disk contact is significantly larger than the film thickness of the gear contacts.
Fig. 5. Minimum and central film thickness for 7=FZGk and min/1400 rotnwheel =
This is mainly attributed to the difference in the equiv-alent radius of curvature which is constant and greater in disks than in gears, as it is shown in Fig. 6.
Fig. 6. Comparison between type C gears equivalent radius and disks equivalent radius
Since it is required that both contacts operate in the same lubrication regime, the disks need to have larger composite roughness. Additionally, they should also have the same, transversal, roughness orientation.
A special finishing technique was applied on several disk pairs, which resulted in a transversal roughness orientation. A part of the rolling surface of the crowned and cylindrical disk is shown in Fig. 7. The pair of disks used in the experiments had composite surface roughness mR disksq µ7.0= . Surface roughness
measurements are given in Fig. 8.
The next step is to estimate the Λ ratio across the path of contact for the gears and disks in order to ver-ify that disks contacts will operate in the same lubrica-tion regime as the gear contacts:
35
ρ [m
m]
][mmga
-8 -6 -4 -2 0 2 4 6 8 10 -10
30
25
20
15
10
5
0
Type "C" gears equivallent radius
of curvature
disks equivallent radius of
curvature
Type C gears equivalent radius of curvature Disks equivalent radius of curvature
Simulation of gear contact on a two-disk test rig
49
qR
hmin=Λ . (13)
Fig. 7. Surface roughness orientation on disk surfaces
Λ for gears was calculated using mR gearsq µ4.0= ,
taken from the work of Höhn et al. (2001). Results are shown in Fig. 9. It is concluded that for most points along the path of contact, Λ ratio differs less than 20%.
3. RESULTS AND DISCUSSION
33 contact points were chosen along the path of con-tact of type C gearing. For each one of them, a meas-urement was conducted on the two-disk test rig, ac-cording to the method described above. The results are presented in Fig. 10 and Fig. 11. In Fig. 12 the average friction coefficient as obtained by the disk experiments is compared to the measurements ob-
tained directly by the power loss measurements re-ported in the work of Höhn et al. (2001).
Fig. 9. Λ ratio comparison: gear and disk contact
The comparison is satisfactory. The assumptions that were made on the effect of roughness orientation, Λ ratio and lubrication regime on friction coefficient were confirmed, despite the errors induced in film thickness calculation by Hamrock’s formula. Average friction coefficient calculated from disks measurements differs less than 10%. It must also be mentioned that is high-er in any case. This is explained by the fact that Λ-ratio is lower by 10-20% in disks than in gears, as shown in Fig. 9. Consequently, a greater part of the normal load is supported by asperities contacts in disks than in gears resulting in higher friction coeffi-cient.
4. CONCLUSIONS
An experimental method was presented that permits the tribological simulation of the Hertzian contact found in a gear pair by a disk contact of a two-disk test rig. Experiments were conducted on the two-disk test rig of the LME&MD simulating experiments in an FZG test rig with type C gears. The above method was verified by comparing the average friction coeffi-cient. Very good agreement was found. Therefore it makes possible to study gear efficiency without the need of manufacturing gears and testing them in a gear test rig.
15 m
Rolling direction
15 mm
Rolling direction
Crowned disk Cylindrical disk
Measurement length [µm]
Fig. 8. Surface roughness measurements of the experimental pair of disks
a) Cylindrical disk
Cylindrical diskR q =0.526 µm mRq µ526.0=
0 1 2 3 4
3
2
1
0
-1
-2
-3
-4
Roug
hness
[µ
m]
b) Crowned disk
Measurement length [µm]
Crowned diskR q =0.463 µm mRq µ463.0=
0 1 2 3 4
3
2
1
0
-1
-2
-3
-4
Roug
hness
[µ
m]
Λ Disks
Λ Gears
kFZG = 7,
nwheel = 1400 rpm
-150
1.0 0.9
0.0
Λ=
hm
in/R
q
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
-100 -50 0 50 100 [%]
KFZG=7 nwheel=1400 rpm
Slide-to-Roll ratio
Gears Disks
Λ=hmin/Rq Gears
0.0 0.1 0.2 0.3 0.4 0.3
+20%
-20%
kFZG = 7,
nwheel = 1400 rpm
0.0
Λ=
hm
in/R
q D
isks
0.6
0.5
0.4
0.2
0.1
0.3
KFZG=7 nwheel=1400 rpm %20
%20−
50
Fig. 10. Friction coefficient measurements for load stage 7=FZGk
Fig. 11. Friction coefficient measurements for load stage 9=FZGk
Fig. 12. Comparison of the average friction coefficient
measurements
REFERENCES
HAMROCK, B. G., SCHMID, S. R., JACOBSON, Bo O. (2004) Fundamentals of Fluid Film Lubrication, 2nd Edition, Marcel Dekker, ISBN 0-8247-5371-2.
HÖHN, B. R., MICHAELIS, K., DOLESCHEL, A. (2001) Frictional behaviour of synthetic gear lubri-
cants. Tribology Series, Elsevier, Volume 39, Tri-bology Research: From Model Experiment to In-dustrial Problem, pp. 759-768.
MIHAILIDIS A., SALPISTIS C., DRIVAKOS N., PANAGIOTIDIS K. (2003) Friction behaviour of FVA reference mineral oils obtained by a newly designed two-disk test rig, Proc. of the Int. Conf. “Power Transmissions 03”, Varna, Bulgaria, pp. 32-37.
CORRESPONDENCE
Athanassios MIHAILIDIS, Prof. Dr. Aristotle University of Thessaloniki Faculty of Engineering Lab. of Machine Elements & Machine Design 54124 Thessaloniki, Greece [email protected]
Stelios GATSIOS, Dipl. Eng, Aristotle University of Thessaloniki Faculty of Engineering Lab. of Machine Elements & Machine Design 54124 Thessaloniki, Greece [email protected]
Slide-to-roll ratio [%] 50 100 150 -150 -100 -50 0
0.00
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0F
rict
ion c
oeff
icie
nt µ nwheel=1400 rpm
nwheel= 700 rpm
nwheel=2100 rpm
50 100 150 -150 -100 -50 0
Slide-to-roll ratio [%]
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Frict
ion c
oeff
icie
nt µ
nwheel=1400 rpm
nwheel=700 rpm
nwheel = 2100 rpm
51
Balkan Journal of Mechanical Transmissions
(BJMT)
Volume 2 (2012), Number 1, pp. 51-60
ISSN 2069–5497
DESIGN AND OPTIMISATION OF CYLINDRICAL GEAR DRIVES BASED ON IS0 6336
Peter NENOV, Velislav VARBANOV, Emilia ANGELOVA, Bojhidar KALOYANOV
ABSTRACT. The standard ISO 6336 offered possibilities for strength evaluation of cylindrical gear drives with comparison of the calculated and permissible stresses. In the same time the engineering community needs tools, which provide high quality design and optimal solutions. For gear with known geometry the authors have developed evaluating task to determine the full load capacity. An other task - for finding the optimum centre distance have been also organised. The present work extends the optimising process with the search of suitable modulus, number of teeth, profile shift coefficients and the face width of the gearing. The multi-parametrical 3-D modelling provides the highest possible caring capacity of gear drives, unachievable in any other ways.
KEYWORDS. Gear drives, Design, Optimizing, Program systems, ISO 6336.
1. INTRODUCTION
The standard ISO 6336:2006 summarised the modern achievements and offered good possibility for strength evaluation of involute cylindrical gear drives, based on the two criteria, comparing the calculated and the permissible stresses in meshing. In the same time the real design feels the need from systems, which not only simplify the usage of the complex methods during evaluation calculations, but also to provide high quality results of various optimisation and design tasks. For gears with known parameters the authors have developed evaluation task to a degree, where full assessment of the maximum load capacity is possible (Varbanov et al., 2006). The present work is focused on the organisation of design task, which finds gear with centre distance and modulus, close to the optimal.
The present work extends the optimisation process with the search of suitable modulus, additional variations with the number of teeth, profile shift coefficients, centre distance, and the face width. The conclusion form our calculations shows that multi-purpose parametric surface and special optimisation provides gear load capacity, unachievable in other ways. Such design approach has been successfully implemented by us whit methods, preceding the new version of ISO 6336:2006 (NENOV et al, 1982). It starts with the choice of relative width of the gear set (representing the ratio of the width of the meshing and the centre distance aw), and the finding of a start value of aw from contact strength. The initial value is a subject of verification, because it is based on approximate values of all parameters and coefficients, which cannot be calculated properly in the beginning. The bound among some of the coefficients,
characterising the load capacity, forced the task to pass through a procedure of harmonisation of the values with the transmitted load, at the design rotation speed (Fig. 1).
The International Standard does not provide a way to determine the maximum power, which a gear set, can transmit. Because of the influence of the force on the
values of vK , β)(FHK and α)(FHK the problem could
only be solved by an iteration procedure. A signal for the end of the iteration could be the finding of a
ROmanian Association of
MEchanical Transmissions
(ROAMET)
Balkan Association of Power
Transmissions (BAPT)
Fig. 1. Procedure for determination of βHK and βFK
(from Varbanov et al., 2006)
( )1== βαVHiKFF
iiFcF 11 =+
calciFF >
calciFF <
11 −+ =iiFF
iiFcF 21 =+
calciFF >
cycles
F
Cycles
52
peripheral force with calculated stresses deviating from the permissible with less than the predefined deviation.
It states that all symbols used in paper are according to standard ISO 6336.
2. DETERMINATION OF THE MAXIMUM LOAD CAPACITY OF CYLINDRICAL GEAR DRIVE BY ISO
6336:2006
The International Standard does not provide a way to determine the maximum power which a gear set can transmit. Because of the influence of the force on the
values of vK , β)(FHK and α)(FHK the problem could
only be solved by an iteration procedure. A signal for the end of the iteration could be the finding of a peripheral force with calculated stresses deviating from the permissible with less than the predefined
deviation (e.g. FF 005.0=∆ ).The highest force
maxHtF , which the gear can transmit based on its
contacts strength, is the force at which the calculated and the permissible stresses are equal. Because of
the bound of some of the coefficients, maxHtF could
be determined with the help of iteration procedure, based on the equation:
αβ
βε
σ
HHA
EHDB
HP
HtKKKK
u
ubd
ZZZZZF
V
1
1
2
)( +
=
,
(1)
where: ( )
( )( ).,,
;,
;
3
2
1
βα
β
HvHH
VtHH
tHV
KKFtfK
KFfK
FfK
=
=
=
(2)
By analogy the maximum force from the criteria bending strength could be determined:
αβ
β
σ
FFVA
DTBSF
nFP
FtKKKK
YYYYY
mb
F = . (3)
Here: ( )
( )( )( ).,
;,,
;,
;
2
3
4
1
VFtH
HvtFF
VHF
tHV
KFfK
KKFfK
KKfK
FfK
=
=
=
=
β
βα
ββ (4)
The designations used in the formulas correspond to the ones, used in ISO 6336:2006.
Having in mind that we seek such a force FtH(F), which fulfils the requirement: |F| max)(max)( tHFFtHFtH FF ∆±= . (5)
For the equalisation, only the product of
αβ )()( FHFHv KKK has influence, which will further be
replaced with the coefficient KVHβα and KVFβα (when calculating contact resp. bending stresses). The
analysis shows, that when all other conditions are
equal the maximum value of max)(FHtF corresponds to
KVH(F)βα = 1, which is taken as a starting point of the coefficients, in two separate cycles. In each of them with the value of “one” a temporary value of the peripheral force FtH(F)max is obtained, and then the coefficient KVH(F)βα is recalculated. With the recalculated value of KVH(F)βα the force FtH(F) is determined, and later compared to FtH(F)max. Because always KVH(F)βα > 1, during the first step FtH(F) is less than FtH(F)max during the second step. The dependency between the two forces is inversely proportional, and while the FtH(F)max diminishes, KVH(F)βα increases. This is a premise for organisation of a cycle in which the temporary value of the force FtH(F) gradually decreasing and tends to close on its limit – the maximum peripheral force FtH(F)max. With that the necessary levelling of the temporary value of KVH(F)α with the value corresponding to the standard is achieved. Levelling of the calculated and the permissible stresses is also achieved. This means that FtH(F) is the maximum tangential force the gear could transmit under the given conditions. For acceleration of the calculation process, illustrated on Fig.1 the procedures are divided in stages. During the first stage every temporary value FtH(F) is calculated though staged decrease of the preceding one: iFtHiFtHiFtH FFCF )()(11)( 0.9==+ (6)
The process continues until: iFtHFtH FCF )(1max)( > . (7)
This means that the required maximal value is jumped over. A step back to the previous value follows, and similar procedure is executed with more graduate decrease (for instance C2 = 0.09) – until FtH(F)max is found. With the corresponding coefficients, the maximum permissible power of the transitions is calculated.
3. AREA OF THE OPTIMUM CENTRE DISTANCE
From the theory is known, that with fixed centre distance and gear ratio the contact strength of the gear, in comparison with that of bending is influenced less from the modulus of the meshing. Regardless of the usefulness the recommendations, including one derived form computer experiments, the limits of this influence are difficult to predict for the different combinations of materials and gear ratios. The tendencies are seen from the results, gathered from research for the purposes of the developed algorithm (Table 1). In this case careful evaluation calculations are done. A broad scope of centre distances, rations, and materials are covered. All gears are with modulus m = 0.015 aw. Besides the results in the last column recommendations are given for the ratio between the modules and the centre distances, providing maximum load capacity of the gears. With fixed values of the given powers this ratios guarantees minimum centre distances.
For stabilisation of the design process, as well as finding of design resolutions, allowing modification and improvement in additional areas, the algorithm of
53
the design task starts with search of suitable centre distance with initial value of the modulus m = 0.015 aw.
It is taken that the working conditions, manufacturing conditions and the operational conditions are known and studied to a degree, necessary for the design process.
Throughout extensive dialog, the exit conditions of the design are entered, including the parameters:
• rotation speed of the input shaft – n1, min-1
;
• load capacity (given) P (power of the input shaft), kW;
• life (working time) h, hours;
• gear ratio, u.
The following is given as well:
• parameters of the rack profile of the gear (usually standard JIS B 1701-1:1999);
• material and heat treatment of the pinion;
• material and heat treatment of the gear;
• helix angle, β;
• relative mesh thickness, ψba.
Table 1. Load capacity of gears with modules m=0.015 wa with different set u, wa and materials
mmaw ,
Mate-rials
u Power 80 100 125 160 200 250 315 400 Observations
11 17.7 35.7 71.6 133 272 528 987 2.5 12.6 25.5 48.3 99.6 191 378 772 1580
>> 0.015 wa
(2F2.5)%·aw
7.01 11.3 23.7 48.4 86.2 176 337 652 4.0 6.49 12.9 26.4 55.7 101 203 405 805
>> 0.015 wa
(1.5F2)%·aw
4.49 6.65 13.7 28.8 50.5 103 199 384 Carb
on
ized
ste
els
6.3 3.32 5.91 11.6 26.2 46.6 90.0 176 366 >> 0.015 wa
~1.5%·aw
8.29 13.2 26.4 52.4 97.4 200 397 758 2.5 6.96 14.2 26.8 55.4 106 206 408 834
~ 0.015 wa
~1.5%·aw
4.96 8.02 17.4 35.0 60.9 131 252 425 4.0 3.64 7.24 14.6 30.9 56.6 112 224 425
~< 0.015 wa
(1.25F1.5)%·aw
3.25 4.70 9.65 20.4 35.7 72.7 141 271 Hard
en
ed
ste
els
6.3 1.82 3.30 6.45 14.5 25.9 49.9 97.6 203 < 0.015 wa
(1F1.25)%·aw
7.63 12.1 24.3 48.2 89.7 184 365 697 2.5 2.94 5.98 11.3 23.3 45.5 92.0 198 416
< 0.015 wa
1.25%·aw
4.68 7.57 16.5 33.0 57.0 111 239 436 4.0 1.55 3.07 6.27 13.2 24.0 43.8 99.0 204
<< 0.015 wa
(1F1.25)%·aw
3.23 4.68 9.61 20.3 35.5 72.2 140 270 Cast ste
els
6.3
P, kW
0.82 1.46 2.85 6.44 11.4 22.0 43.1 91.0 <<< 0.015 wa
1%·aw
Input conditions. Helix angle: β = 15o. Relative face width: 3.0=baψ . Life: L = 20 000 hours. Rotation speed:
11 min1000 −=n . Parameters of the gear rack: standard. Parameters of the quality of the meshing: standard.
Technology and other. No hardening, no sharp edges, good thermal treatment; roughness 8-8-8; bearing span – symmetrical for both wheels.
Materials (All materials by BulStand 17108-89 & GOST). Carbonized steels: 18HGT. MPas 750=σ . HRC1 58.
HRC2 56. Hardened steels: 40 Х. MPas 780=σ . HRC1 45. HRC2 40. HRCcore 40. Cast steel: MPas 780=σ .
HB1 =300 MPa. HB2=280 MPa.
The main tasks at this stage are to find (NENOV, 2002):
• the minimal possible centre distance aw min, chosen form a row for aw;
• results for all standard modules mi, from the interval mmin = 0.01aw min to mmax = 0.025aw min;
• results for a single gear pair with every modulus mi from the selected interval, with the numbed of teeth;
• z1i, z2i, providing minimal deviation from the required gear ratio ugiven.
4. DESIGN TASK - STRUCTURE
The algorithm of the design task in its complete variant is shown on Fig. 2, and includes some basic procedures:
Procedure A
With tasks:
54
4.1. Determination of the parameters of the gear with chosen centre distance awi including:
• Face width bw = ψbaaw
• Modulus of the meshing m
We find mcalc = 0.015awi
From the first row of the modulus a standard value, closer to mcalc is chosen;
• Number of teeth z1 and z2
From set u = z2 / z1 and
a = mj (z1+ z2 ) / (2 cos β) = mj z1 (1 + u ) / (2 cos β),
the result is:
z1calc = 2 a cos β / [mj (1 + ucalc )]; •z2calc = ucalc z1 calc;
a = mj zΣ/ 2 cos β)
αt = arctg (tgα /cosβ)
Fig. 2. Basic procedures of design task and its optimisation variants
Proc. A. Generation of geometry of the gears, and determination of the load capacity, while doing iterations in parallel
Result 1 – the maximal load capacity of the gear, calculated from the two strength criteria
Proc. B. Development of procedure for determination of the load capacity based on the centre distances – from the beginning till reaching the first (aw opt), which load capacity suffices the design Pgiven.
Result 2 geometrical and strength parameters of the first decision where m=0.015aw P ≥ Pgiven
Proc. C. Development of the results for all standard modulus from the range (0.010E0.025) mm for aw opt and the previous one
Result 3 the parameters of the gear with highest load capacity from the groups for aw opt and the previous aw
Additional procedures. Proc. D, Proc. E, Proc. F, Proc. G, for further development of the results of the optimal gear and in direction of optimization of the profile shift coefficients, number of teeth in the range of the allowable deviations from the gear ratio, with the change of the centre distance in the range of GBC, and towards lowering the face width, until the depletion of
the excessive power above the required.
7
1
Pgiven (known n1, ugiven, PGR, PQM,
materials, work conditions)
2
3
4
5
6
Proc. A
P
Proc. B Proc. C Proc. D Proc. E Proc. F Proc. G
55
αtw = arccos (a cosαt / awi)
xΣ = zΣ (invαtw - invαtw) / (2 tgα).
• Recommendations for the profile shift coefficients x1 and x2:
When xΣ < 0, then x1 = 0 and x2 = xΣ.
When xΣ = 0, then x1 = 0.3 and x2 = -0.3.
When 0 < xΣ ≤ 0.5, then x1 = xΣ and x2 = 0.
When 0.5 < xΣ, then x1 = 0.5 and x2 = xΣ – 0.5.
2. Determination of the permissible load capacity of the gear Р=min(PF1, PF2 , PH1 , PH2)
In accordance with the two criteria in ISO 6336 (it is done with the solving of Evaluation task, with application of the iteration procedure for harmonization of the interconnected coefficients).
4.3. Comparison of the load capacity of the gear P with the design power Pgiven
• The calculated number of teeth is converted into integers: z1 – with rounding; z2 – with stripping.
• Meshing angle αt and sum coefficient xΣ:
zΣ = z1 + z2
a = mj zΣ/(2 cos β)
αt = arctg [tgα/cosβ]
First cycle :When the Pgiven ≤ 1.05P – message that the gear could be with aw < from the initial value for the given row, and end to the calculations with message “Gear with aw lower that the initial”.
When Pgiven > 1,05P – development of the Proc. B, i.e. move to the next centre distance and repeat of Proc. A.
Second and following cycles
When Pgiven > 1,05P – continuation with the procedure Proc. B, i.e. move to the next centre distance and repeat of Proc. A.
When Pgiven ≤ 1.05P – Development of procedure Proc. C, where the parameters of all gears with standard modules from the range (0.010-0.025)aw, and from the scope from the previous centre distance. For solving this procedure (Result 3) the gear is represented with the highest load capacity, chosen from comparison between the two groups. The idea behind procedure Proc. C comes from the impossibility to predict the most promising modulus (which is mentioned in the comments in Table 1).
The same conclusion comes from the analysis of the results, calculated from the author’s system GEAR.
As you can see from Fig. 3, with given centre distance, the bending load capacity of the gear is influenced heavily by the modulus, while the contact load capacity remains relatively unchanged. It is shown that from a certain point the increase of the modulus does not lead to higher load capacity. In some cases there could be even degradation in gear
bending load quality. Similar results are produced by software using ISO 6336 (Fig. 4). The graphics are build from evaluation calculations for a series of gears with given gear ratio u=4.0 and relation between the modulus and the centre distance, (1- 2.5)%.
The smooth of the curves, representing the load capacity of the gears, is a proof for the stability of repeats the reciprocal evaluating and design tasks.
On Fig. 4 the results from the first procedures from the algorithm are shown. The results serve as a proof for the algorithm Proc. C, which provides optimization of the results based on the modulus. In this example, it drops the result from aw=200 mm to the lower preceding value (Fig. 5).
The aim of the next additional procedures. Proc. D, Proc. E, Proc. F, and Proc. G are clarified only in principle, and it is a subject of further discussion. The provided expansion connects the power factors with the more general geometrical parameters of the gears.
This is one of the advantages of the authors’ software, which is developed in comparison to the other versions. The system GEAR/ISO provides possibilities for flexible variation with the basic design parameters like ugiven, relative face width etc.
Table 2. Relation m/aw providing maximum strength
Materials u Optimal relation m/aw
Carbonized steels
2,5
4,0
6,3
>>0.015 wa , (2F2.5)% wa
>0.015 wa , (1.5F2)% wa
~0.015 wa , ~1.5% wa
Hardened steels
2,5
4,0
6,3
~0.015 wa , ~1.5% wa
~<0.015 wa ,(1.25F1.5)% wa
<0.015 wa , (1F1.25)% wa
Steels 2,5
4,0
6,3
<0.015 wa , 1.25% wa
<<0.015 wa , (1F1.25)% wa
<<<0.015 wa , 1% wa
5. FORMATION OF SOLUTIONS WITH THE SEARCH OF SUITABLE MODULUS AND PROFILE
SHIFT COEFFICIENTS
The idea behind procedure Proc. C comes from the impossibility to predict the most promising modulus (which is mentioned in the comments in Table 1).
Fig. 3. Example for the influence of the modulus of the gear on its load capacity (aw=250 mm, u =4,
carbonized steels; the rest like in Table 1)
FP HP P
kW
200
100
2.5 3.0 4.0 5.0 6.0 0
m, mm
56
The analysis of the results shows that with a given centre distance the load capacity on the criteria “bending strength” is influenced heavily by the change of the modulus, while the “contact strength” remains relatively stable. Besides that, after a certain value the increase of the modulus does not necessary mean better load capacity.
These conclusions, which come from another studies as well, have been confirmed with the new software
system. The smooth characteristic of the curves, representing the permissible load capacity with different centre distances is an indirect proof for the stability of the calculation process. Important results coming from the calculations are that the base procedures of the algorithm provide repeating of the results from evaluation and design calculations.
In the chosen example for calculation the target is to find gear set with minimum centre distance (for a
Fig. 4. Load capacity with equal materials, u and rotation speed and different m
m = 0.010 aw
FP
HP
m = 0.015 aw m = 0.020 aw m = 0.025 aw P
kW
700
600
500
400
300
200
100
0
HP HP HP
FP FP FP
160
200
250
315 wa , mm
100
160
200
250
315 wa , mm
100
160
200
250
315 wa , mm
100
160
200
250
315 wa , mm
100
Fig. 5. Visualization of the calculation process while designing with the system ISO 6336-RU-09 and final part of the procedure for finding optimal modulus and center distance aw for Proc. C
2 4 5 3
2.5 m , mm
1.5 3 4 2.5
2 m , mm
Pgiven 65 kW
P
kW
60
0
50
30
40
20
10
wa , mm 40 50 63 71 100 125 160 200
FP HP
b) wam 015.0= ,
.
a) wa c) ( ) wam 025.0...015.0= ,
mmaw 160= .
57
given working conditions, chosen maximum face width, and power Pgiven=65kW).
The maximum load capacity for gear dive with modulus m=2.5 mm and aw=160 mm (with ψba=0.3 and recommended geometry) is P=46 kW (Fig. 6).
With the next centre distance (aw=200 mm) the design load capacity of the gear is higher than the required for the larger modulus (m=3, 4 and 5 mm). The question if it is possible to achieve the required load capacity with lower centre distance (aw = 160 mm) with higher modulus remains unanswered. For the clarification of the problem the design process is focused on the formation of gears perspective number
of teeth and profile shift coefficients with another modulus from a given interval. The right part of Fig. 6 shows, that with modulus m=3 there is such solution (B-lines). The gear with modulus m=4 mm (z1=21, z2=82) is also interesting, because the optimisation of the profile shift coefficients will increase its load capacity (Proc. D).
The effect of the procedure for “x optimisation” with respect to the two strength criteria is shown with three examples (Fig. 6). With black dots the situation of the maximum load capacity and the profile shift coefficients is shown.
6. GENERATION OF GEARS WITH SIMILAR GEAR RATIOS
The introduced Proc. E could improve the load capacity. This is shown from the calculations on Fig. 7. Here the modulus is small and all combinations of z1, z2 are generated in the permissible deviation of the rear ratio. The search is for the one with highest load capacity.
The load capacity of the most suitable (the second one) is with 8% higher than the one from the base optimisation procedure (the third), the modulus of which is in the area of 0.015 aw.
With gears with higher modulus the load capacity under the two criteria is relatively similar, but again the base versions are not the better, but the sown with black dots (Fig. 8).
This makes Proc. E is necessary in all important cases.
P , kw 80 70 60 50 40 30
- 0.4 0 0.1 0.1 0.1 0.3
Fig. 6. Results from „x optimisation” of three base design gears with different modulus (carbonized steels) for aw=160 mm
m=2.5 mm; z1=25; z2=98;
m=3 mm; z1=21; z2=82;
m=4 mm; z1=15; z2=61.
P kW
90
80
70
60
50
40
30
- 0.6 - 0.4 - 0.2 0 0.2 0.4 0.6 x1
m = 2.5 mm
m = 3 mm
m = 4 mm
58
Fig. 7. Change of the load capacity trough variation with the number of teeth
aw=160 mm; m=2.5 mm; u=4±3%.
Fig. 8. Parameters of competing gears with modulus m=3, m=4 mm, similar centre distance and close gear
ratios (u=4±3%)
7.
INCREASE OF THE LOAD CAPACITY TROUGH VARIATIONS IN THE AREA OF THE GBC
The achieved results show that the variation with the GBC from the kind aw-x1 could lead to increase of the load capacity of gears, designed in accordance with ISO 6336. This procedure (Proc. F) should be targeted at gears, developed for optimal centre distance, especially the one with most promising modulus and among them – the one with the gear with most promising number of teeth. The development of the described procedure is shown on Table 2.
The task – to design optimal gear with load capacity of P=65 kW – passes trough different stages. It starts with the checking against the standard row of centre distances. It gradually reaches to aw=160 mm, with modulus m=2.5 mm (the closest to the base value of 0.015 aw). The load capacity of this gear is P=42.6 kW and is not a solution to the problem. The next standard distance (a =200 mm) provides the necessary load capacity. The calculation process, however, goes back to the previous centre distance, to search for missed gears. For that reason, a gear pair is generated for each of the modules with given promising profile shift coefficients.
Table 2.
Development of the parameters of the solution
The result is a gear, providing the necessary load capacity – with aw=160 mm, z1=22, z2=82, m=3 mm and P=66.3 kW. Other gears for this promising modulus are formed. For a solution is taken gear with the highest load capacity (P=68,2 kW z1=21, z2=83, and x optimisation). When multiple solutions are available, the end result could be a gear with lower
load capacity, if other parameters are better than the one from its competitors. For instance, additional criterion could be the overlap ratio. The most promising gear for m=4 mm (P= 62.1 kW, z1=21, z2=83) after ”x optimisation” reaches P= 68 kW. In the absence of limits for the centre distance, the calculations could continue with optimisation of the
m, mm
z1 z2 aw, mm
xΣ x1 proj P, kW x1 opt Popt Kw
25 98 160 0.337 0.337 42.1 0 46 2.5
25 99 160 -.185 0 49.6 0 49.6
21 82 160 0.017 0.017 66.3 0.1 67 3.0 Opt 21 83 160 -.484 Big εαtw -0.1 68.2
Development aw 169 2.856 1.0 69.7
15 61 160 0.696 0.500 62.1 0.04 68 4.0 Opt 15 60 160 1.292 -0.6 70.2
Development aw 161 1.593 Good εαtw -0.6 71.4
26 αtw 17 1.7 εα 1.2
m=4 1 2 3 m=3 : 1 2 3 4 5 6
P, 90 80 70 60 5
basic basic dr
59
centre distance. Similar analysis of the gears with modulus m=4 mm shows the gear with z1=15, z2 =60 to be the most promising. With optimum x1-x2, the load capacity could reach P= 70.2 kW and the next procedure optimisation “aw” targets it.
On Fig. 9 the projections of the two PBC’s on the centre distance is shown. The best solutions represent the biggest ordinates. With modulus m=3 mm there are good solutions in both ends of the interval. Additional information for the overlap ratio could gives advantage to the left side, with significant overlap ratio.
8. USAGE OF THE OPTIMISATION FOR LOWERING OF THE FACE WIDTH
The load capacity of a gear is function of many variables and its connection with the face width is not liner. The substitutions with simplified guidelines for its interconnection contain hidden dangers. The results shown on Fig. 10 are for gears which in general differ only in modulus. It is obvious that for a large modulus the lowering of the face width have bigger impact on the load capacity. The calculations are carried out for face width bw=48 mm, bw=46 mm, down to bw=34mm. Formally speaking such approach could be criticised. In practise despite the seemingly non optimal in such scheme, the choice of suitable steps and the recalculation only of the changing parameter provides the necessary speed in calculations.
10. CONCLUSIONS
The presented development of the concept for improvement of the quality of the results for the gears
during design time is based on heavy usage of calculation models, consistent with the need of the design practise. One of its main advantages is the fact that calculations are based on the new ISO standard, which gives high quality results. The program works with established in the former soviet block materials, which are known to the local designers. What differentiates it from the other software is that it gives competing results based on variations with the modulus of the gear. They include also the variant providing high load capacity, calculated after full synchronisation of the coefficients, with the input data. The acquired results are prepared in a way, which helps additional increase of their quality after further optimisations in other areas. The main characteristic of the presented strategy for optimal design is the constant interconnection between thorough and reliable calculation method, with broad system for generation and control of the geometrical parameters of the gear drives. So, we hope our philosophy and algorithms for designing the geometry of the gear drives in parallel with synchronising calculi of their caring capacity, based on creating of 3-D multi parametric models, could provides hay level of the design process, hardly achievable by other ways.
REFERENCES
ISO 6336:2006. Calculation of load capacity of spur and helical gears. Part 1 Part 3
JIS B 1701-1:1999. Cylindrical gears for general and heavy engineering - Part 1: Standard basic rack tooth profile.
NENOV P. (2002) Parametrical optimization of cylindrical gear drives, 2002, Sofia
NENOV P., ANGELOVA E. (1982). Program version of method simplified of ISO-TC60 for gear drive’s calculations. Ruse, UR
VARBANOV V., KALOYANOV B., ANGELOVA E., NENOV P. (2006). Automated building of complex parametrical models of external gear drives, Proceedings of the Int. Conf. “Power transmissions 2006”, Novi Sad.
CORRESPONDENCE
Velislav VARBANOV, PhD, Eng, MrSc, University of Ruse Faculty of Transport Dept. Machine Science, Machine Elements and Engineering Graphics 7017 Ruse, Bulgaria [email protected]
P,kw
70
60
50
40
30
155 160 165 aw
Fig. 9. Change of the load capacity in the boundaries of GBC (projection of PBC on aw)
Peter NENOV Professor, PhD University of Ruse Faculty of Transport Dept. Machine Science, Machine Elements and Engineering Graphics. 7017 Rousse, Bulgaria [email protected]
m=4 mm m=3 mm
bw 48 46 44 42 40 38 36 34;48 46 44 42 40 38 , mm
P, kW
70
60
50
40
30 m=4
m=3 mm
Fig. 10. Lowering of the face width until the load capacity limits are reached
60
Emilia ANGELOVA, Assoc. Prof., PhD, University of Ruse, Faculty of Transport Dept. Machine Science, Machine Elements and Engineering Graphics. 7017 Ruse, Bulgaria [email protected]
Bojhidar KALOYANOV, Sc.Res, University of Ruse, Faculty of Transport Dept. Machine Science, Machine Elements and Engineering Graphics 7017 Ruse, Bulgaria [email protected]
61
Balkan Journal of Mechanical Transmissions
(BJMT)
Volume 2 (2012), Issue 1, pp. 61-72
ISSN 2069–5497
AN EXTENSION OF THE ELECTROMECHANICAL ANALOGY IN THE DOMAIN OF HYDROSTATIC TRANSMISSIONS
Part I. THE ELECTROMAGNETIC AND ELECTROMECHANICAL ANALOGIES
Mircea RADULESCU ABSTRACT. The paper aims to expand the electromechanical analogy in other domains of technology: hydraulic, pneumatic, acoustic, sonic, and even in thermodynamics. In addition to the similarity of the equations and mathematical models, in the domain of fluidic systems we have highlighted the analogy of the circuit elements and some basic structures, for which the equivalent schemes are given. Analogy tables are presented, including the important sizes, units, symbols and generalized mathematical models applicable in all domains above and the advantages of the analogy and its limits of application are highlighted.
KEYWORDS. Electromechanic / hydraulic analogy, generalized parameters, electric / mechanical / hydraulic resistance / inductance / capacity / impedance, analogy of sizes / equations, limits of the analogy.
NOMENCLATURE Symbol Description
Ar
; = ∇×r rB A electrodynamic vector potential
−Ω 1[ ]eB electric susceptance
2[ / ]LB N m bulk modulus of elasticity of liquid
−= 1 [ ]e eC S F electric capacity / elastance
f Hz circular frequency
= ∆ −/ [ ]HI H L hydraulic slope
DK diffusion coefficient in electrochemistry
3[ ]HK m flow module in hydrodynamics
[ ]eL H electric inductance
−2 5[ ]HM s m hydraulic module in hydrodynamics
− n nondimensional exponent of HR -
hydraulic resistance to motion
∆ 3; [ / ]Q Q m s volumetric flow/leakage flow
∆ = −; i ep p p p pressure; drop/jump pressure
−= Ω1 [ ]e eR G electric resistance / conductance
[ ]ν −= −1Re vd Reynolds number in hydraulic pipes
5; ' [ / ]LR R Ns mlinear/linearised hydraulic resistance to fluid motion
+
3 2
n
N n
NsR
m
non-linear hydraulic resistance to motion of fluid
s [m] curvilinear coordinate; string deviation T [K] thermodynamic temperature U [Nm] scalar potential of the force
3[ / ]gV m rot Basic geometrical volume of rotary hydraulic volumic machines
V, ∆V [m3] volume; finite fluid volume variation
ω ω 0; /rad s angular/natural frequency
= +z x jy complex number ( 1)j = −
θ λ( ; ; )R spatial polar coordinates
τ apparent power mass density
ABBREVIATIONS P.C. particular case B.C.; I.C. boundary conditions/initial conditions S.C.P.; S.D.P. lumped/distributed parameters system E.S.F.; M.S.F. electric/magnetic quasi-stationary field L.F.; I.F. local (differential)form / integral form O.D.E.; P.D.E. ordinary/partial differential equations E.Q.S.;M.Q.S. electric/magnetic quasistationary
1. INTRODUCTION
The analogy is one of the basic methods used to research various areas of physics-specific phenomena, being based on the formal identity of mathematical models that describe the original phenomenon and the similar phenomenon. Therefore, the analogy method allows solving problems in a particular domain, using the results from another field, where the phenomena of the two domains have the same mathematical model (are analogue).
The scale of analogy is the constant ratio between the values of the two similar sizes, which belong to the two domains and must verify the (formally identical) mathematical models which express the conduct of the two phenomena. The analogy is partial, if only some of the quantities involved in the description of phenomena are analogue. During the dynamic modeling of complex systems, difficulties arise, regarding their direct analysis, so that sometimes the application of methods of study and indirect analysis
ROmanian
Association of
MEchanical
Transmissions
(ROAMET)
Balkan Association of
Power Transmissions
(BAPT)
62
are required, allowing the complete knowledge of the studied phenomenon, based on observations and experimental research performed on similar models.
The biunivocal correspondence between the original phenomenon and its model allows that, on the basis of rules and assumptions established a priori, the variables that can not be assessed on the primary (original) system can be determined on the analog model; based on information obtained from the model, some conclusions can be drawn on the original behavior. Frequently, the original system (a mechanical, acoustic, hydraulic, pneumatic or thermal system) is studied on an analog electric model, this providing the research of the phenomenon on an equivalent electric or electronic circuit, which allows the processing of results and the implementation of solving methods from the electric field in the area of interest. Similar models in the electric field are preferred, as their structure and operation have been improved based on results obtained in the theory of electrical circuits. In these circumstances, the analogy can be a process of synthesis of complex non-electric circuits, based on synthesis algorithms specific to electric or electronic circuits.
A detailed approach of the studies regarding the analogy is shown by Olson (1958), Kinsler (1962), Levi (1966), Hackenschmidt (1972), Stanomir (1989), and Fransua (1999).
2. REQUIREMENTS OF THE ELECTROMECHANICAL ANALOGY
The electromechanic analogy has the advantage that it can easily adapt to the following requirements specific to the study of physical phenomena by modeling (as shown by Stanomir, 1982).
1. The model should ensure a broad representation of the original, that is to allow highlighting its unknown properties; they must be better known than the ones of the original, or must be easily experimentally modeled;
2. The bonds of the non-electric system must be holonome, scleronome and linear and the system must have a linear equivalent graph and only elementary one-port structures.
3. The specific dynamic process of the original system to be studied on the analog model will strictly collapse (limit) to the domain of interest;
4. Based on the scheme of the original model, we can establish the equivalent electrical scheme; for example, the series / parallel connection of elements that goes into the original, must be replaced with similar elements of circuit, connected properly, in series / parallel;
5. The physical character of a quantity of the original system must be maintained to the study on the analogue system; for example, the hydraulic potential must have as analogous the electric potential, etc.
6. Any analogy must respect the principle of conservation of energy, so that the condition of compatibility from an analogy to the other derived from it, is to be a biunique correspondence of powers for the two domains of the similar phenomena; for this reason, the equations from the mathematical model of the analog electrical circuit must be isomorphic with the mathematical model equations of the studied (original) system, as defined in the biunique correspondence.
The analogy of non-electrical elements and circuits (mechanic, acoustic, thermic, hydraulic, pneumatic, sonic etc.) with the electrical ones allows the modeling of the original system, based on some active circuit elements (sources) connected to passive elements (resistances, inductances, capacities, perditances) through specific junction elements. Since the analogy of sizes and mathematical models also aims an analogy of the physical laws specific to analogue circuits (systems), some theorems and laws of the electric disciplines can be properly translated into laws and theorems expressing the phenomena of the domain of interest (the original domain); example: Kirchhoff’s theorems, Ohm’s law, the law of electromagnetic induction, the transfiguration
theorem, superposition theorem, Y↔∆ transfiguration etc. (as shown by Mocanu, 1979, Şora, 1982, Stanomir, 1989).
3. GENERALIZED PARAMETERS
The need to extend the electromechanical analogy to other non-electric domains, which derive from mechanics, becomes obvious with the approach of interdisciplinary researches with applicability in science vanguard fields like space flight engineering, mecatronics etc. There is also a need to establish a common language based on the essential role of the extended analogy between the domains corresponding to physical systems that aim the interdisciplinary science objectives.
The degree of generality of the analogy can be significantly increased if the quantities involved in the description of the phenomena are brought to dimensionless forms and if a common language, with general validity is adopted. In order to highlight the domain of application for some extended analogies, generalized power variables are introduced: f (flow) and e (effort), and on this basis we can define the generalized energy variables (as shown by Buculei, 1993, Rădulescu, 2005).
We can generalize some non-electric sizes, similar to those associated with passive circuit elements (resistances, inductances, capacities, perditances etc.); these quantities are introduced taking into account the analogy of some basic laws from the disciplines with non-electric profile, with laws and equations of circuit and electrical machines theory. Table 1 gives a generalization of the basic parameters that characterize some systems/devices and the elements of machines and circuits.
63
Table 1. Generalized parameters
Parameters
Genera
l
Displacement: ( ) ( ) ( )τ τ= =∫0
, 0 0
t
q t f d q
Pulse: ( ) ( ) ( )τ τℑ = ℑ =∫ , 0 0
t
e
t e d
Generalized power: = = Π P ef W ; Generalized energy: ( ) ( ) ( )τ τ= = ∫0
, 0 0
t
E t P d E J
Action: ( ) ( ) ( )= ℑ A t q t t J ; Impedance: −= = + = Ω = − 1/ ; 1Z e f R jX Y j
Admittance: − = − = Ω
1Y G jB S
Mechan
ical
Displacement: ϕ
− =
−
, ( . .)
, ( . .)m
x m stroke length for linear hydraulic volumic machines L Mq
rad angle for rotary hydraulic volumic machines R M
Velocity: ω ϕ
= − =
= −
&
&
/ , . .
/ , . .
v q m s linear velocity of the mobile equipment of the machine L Mw
rad s angular velocity of the rotative parts of the machine R M
Effort: φ −
= −
, . .
, . .
F N force reduced to the stroke of the machine L M
M Nm torque reduced to the rotor axis of the machine R M
Mass:
− = −
2
, . .
, . .
m kg mass reduced to the stroke axis L M
J Nm s momentum of gyration reduced to the rotor axis R MM
Damping constant :ω
− − =
− −
/ . .
. .( )
vw
m
k Ns m viscous friction constant L Mk
k Nms viscous friction constant R MR
Mechanical stiffness: ϕ
φ − = =
−
/ , , . .
/ , , . .
xq
k N m elastic constant for linear deformations L Mdk
dq k Nm rad elastic constant for angular deformations R M
Elementary work: δ φ δ= ⋅ L q J ; Overall energy: ( )= + = + 2 21
2c p mW W W w K q JM
Hydra
ulic
Geometric capacity:
π − −
= −
3
31 2
/ 2 / , . .
/ , . .
g gV m rad V basic geometricvolume R MK
A or A m m the pistonareaof theactivechamber L M
Hydraulic resistance to motion (friction):
( ) ( )( ( )+
<∆ = =
∆ ∈ >
5
3 2
[ / ] , Re Re
[ / ]; 1;2 , Re Re
L crH n n
N cr
R Ns m linear resistanced pR
Q R Ns m n nonlinear resistance
Hydraulic leakage conductance: = ∆
5/ /HG Q p m N
Hydraulic resistance to acceleration (hydraulic inertance/inductance):
= ∆ =
2 5/ /HL p Q H Ns m ; Hydraulic mobility: − =
1 5 2/H HM L m Ns
Hydraulic capacity: = ∆
5/ /HC Q p m N ;
Hydraulic resistance to deformation / hydraulic capacity: − =
1 5/H HD C N m
Hydraulic stiffness: φ −
=∂ = ⋅
∂2
0h QR Nm radq
Hydraulic compliance: − − −Λ = =2 1 1 1/ [ ]h H hC K R N m rad
64
4. THE ELECTRIC / MAGNETIC ANALOGY
Table 2 shows the analogy between the equations of the electric steady field and the ones of the magnetic field. Most of the sizes and definition relationships have the same shape in both domains (volumic density of the energy and forces, stored energy, flux / circulation of the fields, generalized forces etc.). Most laws and theorems are also similar (Kirchhoff’s theorems, the constitutive relationships, the uniqueness and superposition theorems etc.), as shown by Levi (1966) and Şora (1982).
Table 3 presents the summarized analogy by comparing the definitions and properties of the electrostatic, electrokinetic and magnetic fields. For particular cases (P.C.) we have assumed the three mediums are linear, isotropic and homogenous. In this case there is also a similarity between some definitions and laws of the three domains (as mentioned by Stan, 2005).
The analogy between the stationary electric and magnetic fields can help solving some theoretical and experimental problems regarding the study of electric circuits using already known results in electrostatics / magnetostatics or vice versa. A good example would be determining some electrostatic characteristics using experimental models in electro-kinetics using electrolytic tanks or electroconductive paper. When choosing the physical model one must take into consideration the condition that the two analogue models have similar configurations.
The previous observations mentioned prove their utility mainly in the didactic field and mostly in interdisciplinary practical courses (electromechanics, magnetohydrodynamics, mecatronics, robotics etc.
5. THE ELECTROMECHANICAL ANALOGY
The electromechanical analogy (E.M.A.) is used in the study of simple oscillatory electric systems, on the basis of some models of elementary mechanic systems and in some cases the more complex discrete mechanic systems are analyzed using analogue electrical networks, considering the formal equivalence between Lagrange’s equations and Kirchhoff’s laws, as well as the practical possibilities of measurement of the state parameters of electrical circuits. In table 4 is presented the E.M.A. of the basic sizes for the I and II - type analogies.
In the E.M.A., the ideal passive elements that form a mechanical lumped-parameter (discrete) system are represented using analogue symbols of the elements R, L, C specific to electrical lumped-parameter circuits as mentioned by Nedelcu (1978), Iacob (1980) and Stanomir (1989).
The ideal active elements (the mechanical sources) are adopted in analogy to the sources specific to electric circuits and are associated a positive sense and polarities which correspond to the positive sign of the mechanical power in the circuit (as shown by
Fransua, 1999).
On the basis of this rationing, the ideal passive elements of the mechanical lumped-parameter system, noted Rm, Lm, Cm, as well as the sources of generalized force (Φm) or velocity (wm) are represen-ted using symbols presented in the second part of this paper (as mentioned by Fransua, 1999 and Radulescu, 2005).
When adopting the mechanical scheme one must have in mind the compliance between d’Alembert’s principle and Kirchhoff’s first law, as well as the correspondence between the passive elements of the two analogue circuits. When drafting the analogue scheme for the circuit corresponding to a mechanic system one can distinguish the following stages:
1. The compounding bodies are reduces to material points which correspond to nodes of mechanic network with the velocity in ratio to a standard branch point;
2. The elements of circuit are inserted between the nodes;
3. The equations of equilibrium of forces (φkm ) are
written for every node;
4. The mathematical model is solved (the integral - differential equations) taking into consideration the initial conditions and the velocity of the node wj is obtained.
The electromechanical analogy allows solving issues related to vibrations of complex mechanical systems by replacing them with equivalent electrical network, which can be easily studied on the basis of “standard” results obtained in the theory of electrical circuits.
Using this analogy is only possible if the mechanical system studied is linear, if its vibration amplitudes are small enough. It is essential to obtain the correct configuration of the electrical scheme of the vibrating mechanical system, given the following recommendations (as shown by Fransua, 1999; Stanomir, 1989):
• for the analogy of impedances, to a mechanical series assembly corresponds a parallel electrical circuit and vice versa.
• for the analogy in admittances, to a mechanical parallel assembly corresponds a parallel electric circuit and to a mechanic series assembly corresponds a series electric circuit.
Usually, the equivalent electromechanical scheme is determined, which corresponds to the mathematical model, based on which one can determine the mechanical impedance or mechanical mobility.
For example, Fig. 1 presents an elementary mechanical system (autovehicle and trailer) in translation movement (a) and its analogue mechanical scheme (b).
65
Table 2. The electric - magnetic analogy of parameters
Electrics Denomination Magnetism
( )r r
; [ / ]E r t V m Strength of the vectorial field
( )r r
; [ / ]H r t A m
r r 2( ; ) [ / ]D r t C m Flux density r r( ; ) [ ]B r t T
[ ]⋅r
ep C m Moment [ ]r
/mp Nm T
=r r
/p eP dp d V Polarization / magnetization
=r r
/mM dp d V
= ×r rr
e eC p E Torque = ×rr
m mC p B
= ∇ ⋅r rr
( )e eF p E Force = ∇ ⋅r rr
( )m mF p B
( )ρ ε= − ∇r r
21/ 2e vf E E ( ( )ε τ′ = 0 ) Volumetric force ( ) µ= × − ∆r r r
21/ 2mf J B H ( ( )µ τ′ = 0 )
( )∂= −
∂0;e
kk
W q xX
x
Theorem of generalized forces
( )ϕ∂= −
∂
;mk
k
W xX
x
( ) ( )= ⋅ − ⋅r r r r rr r1
2nT E D n D E n
⇒= ⋅
r renT n T
Maxwellian tensions ( ) ( )= ⋅ − ⋅r r r r rr r1
2nT H B n H B n
⇒= ⋅
r rmnT n T
[ ]ε0
/F m Vacuum permittivity/ permeability
[ ]µ0 /H m
χe Susceptivity χm
ε χ= +1r e Relative permittivity/ permeability
µ χ= +1r m
( )ε ε − ′= 10diff D E Differential permittivity/
permeability ( )µ µ− ′= 1
0diff B H
ε⇒
= ⋅r r
0 etP X E Law of polarization/ magnetization
⇒= ⋅
r r
mtM X H r
pP Permanent polarization/ magnetization
r
pM
( )= ⋅∫
r r
em
C
u E dr Electro / magnetomotive force ( )
= ⋅∫r r
mm
C
u H dr
( )ψ σ
Σ
= ⋅∫∫r rD d Flux
( )φ σ
Σ
= ⋅∫∫r rB d
( )ψ σ
Σ
= ⋅∫∫r r
* E d Flux of the field ( ε = .ct )
( )φ σ
Σ
= ⋅∫∫r r
* H d
( )=r;e eV V r t Scalar potential ( )=
r;m mV V r t
∇× =r r
0E ; = −∇r
eE V Irotational field ∇× =r r
0H ; = −∇r
mH V
= −1 2e e eu V V Voltage / magnetic
tension = −
1 2m m mu V V
∇ ⋅ = =r
0 ( 0)eD q Solenoid field ∇ ⋅ =r
0B
= ⋅r r1
2ew E D
Volumetric density of the energy = ⋅
r r1
2mw H B
=1
2e e eW q V
Energy in the capacitor/ inductor φ=
1
2mW i
ρ ε −∆ + =10 0e vV ;
scalar: = −∇r
eE V
Poisson’s equation Potential
µ∆ + =r r
0A J ;
vector: = ∇×r rB A
ρ = ⇒ ∆ =0 0v eV P.C.: Laplace’s equation = ⇒ ∆ =r r r r
0 0J A
Fig. 2 shows an example of elementary mechanical system in rotary movement (gear box): the cinematic scheme (a) and the analogue calculus scheme (b).
The two schemes include specific notations for the two types of motion, but they can also be written in a generalized form, as shown in Table 1.
66
Table 3. The electrostatic-electrokinetic-magnetic analogy
Electrostatic field Electrokinetic field Magnetic field
Differential equation of the field lines
× =r rr
0dr E × =r rr
0dr J × =r rr
0dr H
Characteristics of the material
Electric permittivity
ε ε ε= 0 [ / ]r F m
Electric conductivity/resistivity
σ ρ ρ ρ ρ ρ−= = = Ω( )
10; [ ]
not
e e r m
Magnetic permeability/reluctivity
µ µ µ ν −= = 10 [ / ]r H m ;
ν - magnetic reluctivity
Global energetic relations
Electrostatic energy
( ) ( )ρ ρ σ
Σ
= + ∫∫∫ ∫∫
1
2e v e s e
D
W V dV V d
Law of degradation of electric energy
( )= ∫∫∫J J
D
W w dV ; τ= ⋅∫r r
0
t
Jw E Jd
(Joule-Lenz)
Magnetic energy
( )( )
( )σ
Σ
= ⋅ + ∇ ⋅ × ∫∫∫ ∫∫
r r r r1
2m
D
W A JdV A H d
Particular forms of the Maxwell’s equations
Constitutive relations
ε⇒
= +r r r
pD E P
PC: ε ε ε⇒
→ = = ⇒ =r r r r
; 0pct P D E
The electric conduction law
( )σ= +r r r
iJ E E (Ohm’s law)
PC: σ ρ= ⇒ = ⇔ =r r r r r
0iE J E E J
Constitutive relations
µ µ⇒
= ⋅ +r r r
0 pB H M
PC: µ µ µ⇒
→ = = ⇒ =r r r r
; 0pct M B H
Electric flux law (Gauss’s law)
(L.F.) ρ∇ ⋅ = ⇔r
vD
( )σ
Σ
⇔ ⋅ =∫∫r r
eD d q (I.F.)
P.C.: ( )ρ = ⇒ ∇ ⋅ = ⇔r
0 0 L.F.v D
( )σ
Σ
⇔ ⋅ =∫∫r r
0D d (I.F.)
Law of electricity conservation
(L.F.) ρ∇ ⋅ = − ⇔r
&vJ
( )( )σ
Σ
⋅ = −∫∫r r
& eJ d q t (I.F.)
P.C.: ( )ρ = ⇒ ∇ ⋅ = ⇔r
& 0 0 L.F.v J
( )σ
Σ
⇔ ⋅ =∫∫r r
0J d (I.F.)
Magnetic flux law (Gauss’s law)
(L.F.) ∇ ⋅ = ⇔r
0B
( )σ
Σ
⇔ ⋅ =∫∫r r
0B d (I.F.)
P.C.: the flux of the field vector through one field tube (stationary field)
( )σ
Σ
⋅ = Ψ∫∫r rD d
( )σ
Σ
⋅ =∫∫r rJ d i
( )σ
Σ
⋅ = Φ∫∫r rB d
Law of electrostatic potential
( eV ); irrotational field:
∇× = ⇒r r
0E ( )= −∇ ⇔r
L.F.eE V
⇔ ⋅ = −∫r r
1 2e eE dr V V (I.F.)
E.Q.S. field:
Faraday’s law ( )=r
0v
( )∂∇× = − ⇔
∂
rr
L.F.B
Et
∂Φ⇔ ⋅ = −
∂∫r r
E drt
(I.F.)
PC: = ⇒ = −∇r r r&
0 eB E V ;
eV - scalar electric potential
M.Q.S. field: law of magnetic circuits (Ampere’s law)
( )∇× = ⇔r r
D.F.H J
( ) ( )σ
Σ
⇔ ⋅ = ⋅∫ ∫∫r rr r
C
H dr J d (I.F.)
PC: = ⇒ = −∇r r
0 ;mJ H V
mV - scalar magnetic potential
Usual circuit elements (stationary field)
Electric capacitor:
= = −1 2
;ee C e e
C
qC u V V
u
Electric conductance:
( )
−= = = ⋅∫r r1;e e R
R C
iG R u E dr
u
Magnetic permeance:
( )
−ΦΛ = = = ⋅∫
r r1;mm m m
m C
R u H dru
Electric elastance:
ε− − = =
∫ 1 1
0
l
e eds
S C FS
Electric resistance:
[ ]σ
−= = Ω∫ 1
0
l
e eds
R GS
Magnetic reluctance:
µ−
= = Λ ∫ 1
0
l
m mds A
RA Wb
67
When drafting these schemes, we have used the II -
type analogy (φ ↔m i ; ↔m ew w ) which assures the
compatibility of d’Alembert’s principle and Kirchhoff’s first law. If the I - type analogy would have been used
(φ ↔m eu ; ↔mw i ), the two analogue circuits would
have had different configurations, because d’Alembert’s principle would have been incorrectly associated to Kirchhoff’s second law.
Table 4. The electromechanic analogy of parameters
Generalised parameters in mechanics Electric analogy
I type (Z) II type (Y)
Displacement: ( ) ( )τ τ= ∫0
t
m mq t w d Charge: ( ) ( )τ τ= −∫0
t
eq t i d Flux: ( )τ τΦ = −∫0
t
e eu d
Velocity: ( )=o
m mw q t Current: Voltage: ue (t)
Acceleration: =o oo
m mw q Velocity of i: i (t) Velocity of ue: ( )o
eu t
Mass: M Inductance: Le Capacitance: Ce
Stiffness: kq Inverse of capacitance: −1eC Inverse of inductance: −1
eL
Damping coefficient: kw Resistance: Re Conductance: Ge
Inertial: ( )φ =o
i t wM Inductive: ( )=o
L eu L i t Capacitive: ( )=o
c e ei C u t
Elastic: φ =q q mk q Capacitive: ( )−= 1C e eu C q t Inductive: −= Φ1L e ei L E
ffort
Damping: φ =w w mk w
Vo
lta
ge
Resistive: ( )=R eu R i t
curr
ent
Conductive: =G e ei G u
Irrotational field
φ = −∇r
m mU = −∇r
eE V = −∇r
mH V
Generalized force
∂= ⋅
∂
rr
k
ik i
m
rX F
q
∂= +
∂ e
ek
k q
WX
x
Φ
∂= −
∂ e
mk
k
WX
x
Stationary characteristic
φ φ= ( )m m mw = ( )e eu u i = Φ( )m m eu u
Impulse of the body:
( )ℑ =r r
t wM Flux of the inductor: Φ =e Li
Charge of the capacitor:
=e e eq C u
Mechanic power: φ=m m mP w Electric power: =e eP u i
1M
2M
2vk AM
Aω
2ω
2z
2J
TM
Tω
1z
1ω 11; vJ k
2ek
-
+
a
b
AM
Aω 1J 1vk
1N′
1N 2N
2N′
2vk
1ek
1ω
1z 2z
2ω
2J
Fig. 2. Example of equivalent scheme for a mechanical system in rotation movement
1m 2m
( )1v t ( )2v t
( )F t ek
vk
1vk 2vk
1m 2m
ek
vk
1N 2N
1vk 2vk
1N′ 2N′
F
-
+
a
b
Fig. 1. Example of equivalent scheme for a mechanical system in translation movement
68
Table 4. Continuation
Elementary electric work Elementary mechanical work:
δ φ δ=m m mL q δ δ=e e eL u q δ δφ=e e eL i
Mechanical reactance:
ω ω−= 1m qX kM -
Electric reactance:
ω ω− −= 1 1e e eX L C-
Susceptance: =2e
e
e
XB
Z
Mechanical impedance: φ
= mm
m
Zw
= +m w mZ k jX
Electric impedance: = ee
UZ
I
= +e e eZ R jX
Admittance: =ee
IY
U
= −e e eY G jB
Kinetic: = 21
2k mW wM Solenoid: = 21
2L eW L i Capacitor: =
21
2
cC
e
qW
C
Potential: = 21
2p q mW k q Capacitor: =
21
2
eC
e
qW
C = 21
2L eW L i
Mechan
ic e
nerg
y
Dissipative: τ= ∫ 2
0
t
d wW k w d
Resistance:
( )τ τ= ∫ 2
0
t
J eW R i d
Ele
ctr
ic e
nerg
y:
Conductance:
( )τ τ= ∫ 2
0
t
J e RW G u d
Total energy
( )= + +2 21
2t d m q mW W w k qM
= + +
221
2
et J e
e
qW W L i
C
Action
( ) ( )= ℑr r
m mA t q t ( ) ( )= Φe eA t q t
Table 5 and Table 6 give two commonly used mathematical models with generalized (unified) notations corresponding to the previously defined generalized parameters. Table 5 shows the basic features of the linear oscillator with damping and harmonic excitation, seen as a model for the study of systems with concentrated parameters.
The mechanical system with concentrated parameters shows a basic structure which materializes the basic effects (inertial, elastic, dissipative) and is characterized by the constitutive equation that indicates the dependence between a kinematic size (displacement, velocity or acceleration) and an effort size (force or torque); based on this dependency we can identify the mechanical impedance.
The mechanical system with distributed parameters can be one-dimensional or two-dimensional, where waves arise (in particular small oscillations) for which the equations of the mathematical model are linear. The modeling of non-electric systems can be based on electric systems with distributed parameters whose mathematical model contains the telegraph, the second order differential equations with partial deri-vates which relate the parameters u (t, x) and i (t, x).
Table 6 shows the basic features of systems with distributed parameters, taking into account the two generalized basic sizes involved in relationships (1) and (2), intensive and complementary sizes: e (effort) and f (flow); these sizes can be identified in each of the areas subject to observation in this work (electric, namely, non-electric), so based on the mathematical
model in which they occur, we can obtain the dynamic characteristics in the frequency domain, as mentioned by Olson (1958), Iacob (1980) and Stanomir (1989).
The mechanic phenomena specific to continuous material spectrums can be modeled on the basis of the equations of the electromagnetic field, using the calculus of variations and having in view the Lagrange density function and Hamilton’s equations. Therefore, Table 6 summarizes the model with distributed parameters, which has a general character, allowing the study of analog phenomena in several fields (electrical, mechanical, hydraulic, acoustic, sonic, etc.).
Table 7 presents the study of a mechanical circuit (Rm, Lm, Cm) compared to the analogue electrical circuit; in both assemblies (series / parallel) are shown two types of analogies (in impedance - analogy type I) and (in admittance - analogy type II), based on the mathematical model in Table 5.
From the above it is confirmed that the base criterion for choosing the right type of analogy is the correspondence between Kirchhoff’s theorems and the two base equations which interfere in the description of the analogue (non-mechanic) system. One can observe that the I-type analogy regards Kirchhoff’s first theorem and the mechanics equation resulting from Newton’s first law (the equilibrium of sizes e). The II-type analogy regards Kirchhoff’s second theorem and the mechanics equation which results from Newton’s second law (the balance of the sizes f). A particular case is the electroacoustic analogy (E.A.A.), which allows modeling the acoustic
69
systems using elementary electric systems. For example, Fig. 3 shows the acoustic (a) and the analogue electric scheme (b) for a classic microphone (as shown by Iacob, 1980). The schemes have been adopted considering the II - type analogy (ue ↔ v; i ↔pa) and the notations correspond to those in Table 8. The mathematical model of the scheme corresponds to an oscillatory system without losses, whose scheme is reduced to a series combination
( +eqe eC L ) connected in parallel to
3eC .
The E.A.A. is frequently used in electroacoustics (as shown by Stanomir, 1989) where various inhomogeneous converting subsystems interfere, specific to unconventional conversion microdrives.
Table 8 presents the study of a section of acoustic line (pipe) with distributed parameters based on the mechanical-acoustic analogy (Type I - direct, respectively, type II - reverse), obtained from the reference mathematical model from Table 6.
For acoustic systems with distributed parameters the geometric dimensions of the system are comparable
to the wavelength λu , for which there is a phase shift,
due to propagation. When designing technical systems a mechanic-mechanic analogy is sometimes needed, being regarded in some cases also as a similitude. Next, we present an example of mechanic-mechanic analogy.
Table 5. The basic features of the linear oscillator (for S.C.P.)
Parameter Definition and calculus equations Other formulae, notations Resonance values
Pulsation Natural: ( )ω −= =1/21/2
0 ( ) /m m m mM C K M ; pseudo-pulsation: ω ω δ ω ζ= − = −2 2 20 0 1p
Mechanical resistance
(damping factor) Rm
Damping factor: δ
ζω
= =0cr
m
m
R
R ω= 02
crm mR M
Damping coefficient
δ ζω= = 0/ 2m mR M δ ω ς= ≅ 0/ 2crrez m mR M
Driving frequency
ω γ ω ω= 0/ γ ζ= − 21 2rez
Quality factor ω
ωδ
= = =00
1
2
mm m
m m
MQ M K
R R
ζ=0
1
2Q
Static displacement
φ ω φ= = =20 0 0/ / ; /st m mq K p p M φ0 - amplitude of perturbation ( )ζ= 1 max
2stq q
Generalized mathematical model: δ ω ω+ + =&& & 202 sinq q q p t ;
Transitory component: ( )δ ω ϕ−= +0 0sintt pq q e ; Permanent component: ( )ω ϕ= −1 sinpq q t
Characteristic equation
δ ω+ + =&& & 202 0x x x ; solution: δ δ ω= − ± −2 2
1,2 0x
Amplitude ( ) ( )ω ω δω
=
− +1
2 22 20 2
pq
( )γ ζ γ
=
− +1
22 2 21 4
stqq ( )
ζ ζ
ζ δω
= ≅−
≅ =
1 max 2
0
2 1
/ 2 / 2
st
st
q p
Phase δω
ϕω ω
=−2 2
0
2arctg
γϕ
γ=
− 2(1 )arctg
Q ϕ ζ − = −
2 2rez arctg
Dynamic amplification
factor ξ γ
−= = −
12
1 / 1stq q ξγ γ ζ
=− +2 2 2 2
1
(1 ) 4
ξξ ζ
= ≅−
02
1
2 1rez Q
Logarithmic decrement ( )δ +=1 1ln /k kq q ; ( )δ πζ ζ πζ ζ= − ≅
21 2 / 1 2 1 ; π δ ζ=1/ 1/ 2 - over amplitude coefficient
Dynamic elastic constant
φ ω ω ω= = − + =2/ m m mD mK Z K M j K j Z ; = = + = +1 ; ; m mmq z z x jy Z R jX
Energy π ω
ω γ γ −=
− +
2 2 2
4 2 2 2 20
2
(1 )
pW
Q
γ
γ γ= − +
2
2 2 2(1 )rez
W
W Q π
ω=
20
20
2rez
QW
2ek
a b
±
m
1N eL 1ek
3ek F
0ek
F±
m
0eC
1eC 2eC
3eC
2N
N
Fig. 3. The mechanical scheme and the analogue electric scheme of the microphone
70
Table 6. The basic features of mechanical systems with distributed parameters (for S.D.P.)
Definitions, formulas, mathematical models and their solutions Quantities; characteristic
constants General case (R’; L’; C’; G’ ≠ 0) Without perditance (G’ = 0) Without losses (R’ = G’ = 0)
Without distortion
Wave period
ω ω πδ
− ′= = = =
′1
0 0 0 01
1; 2
2
LT f
R
Consta
nts
Damping coeff.
δ′
=′;
2
R
Hδ
′=
′1 ;2
G
Cδ δ
′= =
′2 2R
H δ =1 0 δ δ= =1 0
δδ δ= = 2
12
Time: δ =3 [ ]x
sa
; = ⇒ =pL
x L Ta
; wave propagation speed: =′ ′
1a
H C; =
′ ′00
1a
H C; wavelength:λ = =0
0u
aaT
f
Operatio-nal
γ ′ ′ ′ ′= + +( ) ( )( )s R H s G C s γ ′ ′ ′= +( ) ( )s R H s C s γ ′ ′= =( )s
s s HCa
δ
γ = + 2( )s sa
Pro
pagation
facto
r
Complex γ ω ω ω γ δ ω′ ′ ′ ′= + + = +( ) ( )( );j R H j G C j j
γ ω
ω ω
=
′ ′ ′ ′= − +2
( )j
H C j R C ( ) ω
γ ω =j
ja
Operatio-nal
δδ
+′ ′ ′ ′= + + =
+01
2( ) ( )( )
2c c
sZ s R H s G C s Z
s
δ= +
0
21c cZ Z
s →
0c cZ Z
Impedance
Complex ω
ωω ω
′ ′+=
′ ′ ′ ′ ′ ′ ′ ′− + +2( )
( )c
R j HZ j
R G C H j R C H G
; ρ′
= =′0
0c
aHZ
C s - wave impedance
Load (terminal) impedance ω
′ ′ ′= + = ⇒ = + + ′
221
; ( )2
s s sc c s
RZ R jx Z Z R H H
C;
ω′ ′ ′= − + + ′
221
( )2
sR
X H HC
Coupled eq.
∂ ∆ ∂ ∆ ′ ′= − ∆ + ∂ ∂
( ) ( )e fR f H
x t; ∂ ∆ ∂ ∆ ′ ′= − ∆ + ∂ ∂
( ) ( )f eR e H
x t
Math
em
atical m
odel
Decoupled eq.
∂ ∆ ∂ ∆ ∂ ∆′ ′ ′ ′ ′ ′ ′ ′= + + + ∆
∂∂ ∂
2 2
2 2
( ) ( ) ( )( )
e e eH C R C G H R G e
tx t
∂ ∆ ∂ ∆ ∂ ∆′ ′ ′ ′ ′ ′ ′ ′= + + + ∆
∂∂ ∂
2 2
2 2
( ) ( ) ( )( )
f f fH C R C G H R G f
tx t
Systemic model
∆′ ′ ′ ′− + + ∆ =
2
2
( )( )( ) 0
d eR H s G C s e
dx;
∆′ ′ ′ ′− + + ∆ =
2
2
( )( )( ) 0
d fR H s G C s f
dx
General ( ) γ γ−∆ = +1 2, x xe s x C e C e
( ) γ γ−∆ = +1 2, x xf s x C e C e
Solu
tion
Final
γ γγ
∆ = ∆ + ∆ − 2 11
( , ) ( ) ( )( )
e s x e sh x e sh L xsh L
γ γγ
∆ = ∆ + ∆ − 2 11
( , ) ( ) ( )( )
f s x f sh x f sh L xsh L
Opera
tional
( )( ) ( ) ( )
( )( )
γ γ ωω
γ γω ω
∆ ∆ = ∆ ∆
1
1
( ) ( )
1
c
c
ch x Z sh xe je j
sh x ch xf j f jZ
;
Matrix input - output complex
relationship
Com
ple
x
γ ω ω ω γ ωω ω
ω ω γ ω γ ωω ω
∗ ∗
∗ ∗
∆ ∆ = ∆ ∆
0
0
2 1
2 1
( ( ) ) [ ( ) / ( )] ( ( ) )( ) ( )
[ ( ) / ( )] ( ( ) ) ( ( ) )( ) ( )
c c
c c
ch j L Z j Z j sh j Le j e j
Z j Z j sh j L ch j Lf j f j
71
Table 7. Comparison between a mechanical system, an electric series circuit and an electric parallel circuit
Mechanica l s ys tem / dev ice Type o f ana logy Analogue e lec t r i c c i rcu i t
Ph
ys
ica
l
Type I (direct) - in impedance
↔m eZ Z
f ↔ u; v ↔ i; Rm ↔Re;
Mm ↔Le; − = ↔1m m eK C C
Ma
the
ma
tic
al
( ) ( ) ( )τ τ+ + =∫&
0
1t
m m km
M v t R v t v d fC
( ) ( )ω ϕ= +cosm
Fv t t
Z;
ωω
= =00
1mm
m m m
MQ
R R C
ω ωϕ
ω
−=
20( / ) 1
m m
arctgR C
( ) ( ) ( )τ τ+ + =∫&
0
1t
e e ke
L i t R i t i d uC
( ) ( )ω ϕ= +cose
Ui t t
Z;
ωω
= =00
1em
e e e
LQ
R R C
ω ωϕ
ω
−=
20( / ) 1
e e
arctgR C
Ph
ys
ica
l
Type I (direct) - in impedance
↔m eZ Z
f ↔ u; v ↔i; Rm ↔ Re;
Mm ↔ Le; − = ↔1m m eK C C
Mo
de
l o
f th
e m
ec
ha
nic
sy
ste
m /
de
vic
e a
nd
of
the
an
alo
gu
e e
lec
tric
cir
cu
it
Ma
the
ma
tic
al ( ) ( ) ( )τ τ
•+ + =∫
0
1t
m m km
C f t G f t f d vM
ω ϕ= +( ) cos( )mf t Z V t
ω ωϕ
ω
−=
20( / ) 1
mm
arctg RM
ωω
= + −
1Z R j L
C (impedance)
ωω
= + −
1Y G j C
L (admittance)
ω −= 1/20 ( )LC - natural frequency
( ) ( ) ( )τ τ•
+ + =∫0
1t
e e ie
C u t G u t u d vL
ω ϕ= +( ) cos( )eu t Z I t
ω ωϕ
ω
−=
20( / ) 1
ee
arctg RL
Table 8. The mathematical model of an acoustic line with D.P., established using the electroacoustic analogy
Type of osc i l lat ing - harmonic system Domain Type
Mechanical system with longitudinal motion
Acoustic system with constant section (S=ct.)
Uniform one-dimensional acoustic system
Ph
ys
ica
l
mo
de
l
Equ
ations
∂ ∂∂ ∂
0' 'm m
f v+ R v + M =
x t
∂ ∂∂ ∂
0' 'aa a
p q+ R q + M =
x t
∂ ∂∂ ∂
0' 's s
p v+ R v + M =
x t
Ty
pe
-I-a
na
log
y (
in i
mp
ed
an
ce
)
Ma
the
ma
tic
al
mo
de
l
Syste
m p
ara
me
ters
−1' 'm mM = ρS; C = (ES)
' 'm m
u ' 'm m
R + jωMZ =
G + jωC
0u
EZ = ρcS; c =
ρ
' 'a a
ρ SM = ; C = ;
S E
' 'a a
u ' 'a a
R + jωMZ =
G + jωC
0u
ρc EZ = ; c =
S ρ
−1' 's sM = ρ; C = E ;
' 's s
u ' 's s
R + jωMZ =
G + jωC
0u
EZ = ρc; c =
ρ
+
p v x
sG′
sC′
;s sR M′ ′ +
p q x
aG′
aC′
;a aR M′ ′ +
f v x
mG′
mC′
;m mR M′ ′
Type II (inverse) - in admittance
f ↔ i; v ↔ u; Rm ↔ Ge=1
eR− ;
Mm ↔ Ce; 1
m m eK C L− = ↔ ;
m eZ Y↔
72
Table 8. Continuation
Ph
ys
ica
l
mo
de
l
Equ
ations
∂ ∂∂ ∂
0' 'm m
v f+ G f + C =
x t
∂ ∂∂ ∂
0' 'a a
q p+ G p + C =
x t
∂ ∂∂ ∂
0' 's s
v p+ G p + C =
x t
Ty
pe
- II
-an
alo
gy
(in
ad
mit
tan
ce
)
Ma
the
ma
tic
al
mo
de
l
Syste
m p
ara
me
ters
−1( )' 'm mM = ρS; C = ES ;
' 'm m
u ' 'm m
G + jωCZ =
R + jωM
0 1u
EZ = ; c =
ρcS ρ
' 'a a
ρ SM = ; C = ;
S E
' 'a a
u ' 'a a
G + jωCZ =
R + jωM
0u
S EZ = ; c =
ρc ρ
−1' 's sM = ρ; C = E ;
' 's s
u ' 's s
G + jωCZ =
R + jωM
0 1u
EZ = ; c =
ρc ρ
ρ- density; E - Young’s modulus; p - acoustic pressure; q - acoustic flow; v - velocity of the medium
REFERENCES
BUCULEI, M., RADULESCU, M. (1993) Hydraulic Drives and Automations. University of Craiova.
FRANSUA, Al. (1999). Conversia electromecanica a energiei. Technics Publ. House, Bucharest.
HACKENSCHMIDT, M. (1972). Strömungsmechanik, VEB Deutscher Verlag fur Grünindustrie, Leipzig.
IACOB, C. (1980). Dicţionar de mecanică, EDP Bucharest.
KINSLER, L.E. (1962). Fundamentals of acoustics. Wiley & Sons, New York.
LEVI, E., PANZER, M. (1966). Electromechanical Power Conversion. McGrow-Hill Book Company, New York.
MOCANU, C.I. (1979). Teoria circuitelor electrice, EDP, Bucharest.
NEDELCU, V.N. (1978). Teoria conversiei electromecanice. Technics Publishing House, Bucharest.
OLSON, H.F. (1958). Dynamical analogies, Princeton, D. van Nostrand Comp.
RADULESCU, M. (2005). The Electrohydromechanic Analogy, ISIRR8 Szeged.
STAN, M.F. (2005). Tratat de inginerie electrica, „Bibliotheca” Publishing House, Bucharest.
STANOMIR, D. (1982). The Physical Theory of Electromechanical Systems. The Academy Publishing House, Bucharest.
STANOMIR, D. (1989). Teoria fizică a sistemelor electromecanice, The Acad. Publishing House, Bucharest.
ŞORA, C. (1982). Bazele electrotehnicii. EDP, Bucharest.
CORRESPONDENCE
Mircea RADULESCU Assist. Prof. University of Craiova Faculty of Mechanics 13, Alexandru Ioan Cuza Street 060042 Craiova, Romania [email protected]
+
v p x
sR′
sM ′
;s sC G′ ′ +
q p x
aR′
aM ′
;a aC G′ ′ +
v f x
mR′
mM ′
;m mC G′ ′
73
Balkan Journal of Mechanical Transmissions
(BJMT)
Volume 2 (2012), Issue 1, pp. 73-77
ISSN 2069–5497
CONTRIBUTIONS CONCERNING THE ROLLER GEARING TOOTH PROFILE GENERATION
Adrian Dănuţ VEJA
ABSTRACT. The objective of the present scientific paper concerns the theoretical and practical issues of the roller gears tooth profile on the driven wheel. In the specialized literature, we can’t find defined data regarding the step teeth calculation on the driven wheel. The transmission for the precise displacement with roller gears, conceived by the author of this scientific paper, and which makes the object of an patent proposal, consists of an pinion roller by wrapping with tightness on a hub of an a precision chain drive with short roller chain links, as it is presented in SR ISO 606-2000. The next step targets the mounting on the drive shaft of the drive motor an cycloidal gear wheel profile, which will involve the pinion roller when it is in rotational motion of the driven element.
In order to achieve the practical realization of this gear, it is necessary to determine theoretically with sufficient accuracy the calculation of the step teeth wheel of cycloidal profile, the flanks being conjugates profiles on the cylindrical surface of the roller.
KEYWORDS. Mechanical transmission. Precise movements. Roller gearing tooth.
NOMENCLATURE
Symbol Description
i Transmission ratio '1p
Linear pitch of chain
'2p Linear pitch of gear
1r The rolling circles radius of the pinion
2r The rolling circles radius of the cycloid wheel
s Arc tooth circle division
MM yx , Coordinates of tooth head
BB yx , Coordinates of tooth foot
1χ Angular pitch of pinion
2χ Angular pitch of gear
1φ Angle between O1O2 axial distance and center roll O1C1
2φ Angle between O1O2 axial distance and OX axis
ρ Radius of roll
1.1. INTRODUCTION
Nearly all practical gears now have involute tooth profiles. Originally, however, cycloidal profiles were used. Cycloidal profiles (generated by a circle rolling around the outside and inside of another circle, respectively) are as technically suitable as involute profiles, perhaps even slightly superior taken into account some aspects (e.g. cycloidal teeth does not undercut or interfere with its mating teeth, lesser number of teeth can be possible which facilitates large reduction ratios, cycloidal teeth are stronger as opposite to involute ones etc.). Also according to Zhou and Chen (2001)–precision cycloidal gearings are used in different applications such as industry robots
and machine tools. Nevertheless, many modern engineering texts and handbooks which are available on this moment, at most only mention cycloidal gears in passing. In order to remove these shortcomings in the last decades a series of researchers have reported solutions on this problem.
Chen et al. (2008) described a general method (called enveloping method) to generate hypocycloid and epicycloid. The correct meshing condition for cycloid pin-wheel gearing is provided, and the contact line and the contact ratio are also discussed. Feng et al. (2011) present a study upon a pin-rack gearing (i.e. a particular type of cycloidal gearing which transforms a rotation motion to a linear one) in order to determine the relationship among design parameters which define this mechanical transmission. Yangfang et al. (2010) describe a calculation method for profile curves in a cycloidal pump (i.e. an internal gear pump) based on a meshing principle. Zhu et al. (2010) propose a new kind of axis-fixed cycloid transmission in order to solve the shortcomings (such as low efficiency, low bearing capacity and poor rigidity) of the cycloidal-pin wheel transmission. Davoli et al. (2007) investigate the structural characteristics and the kinematical principles of a cycloidal reducer. A theoretical approach based on the envelope theory (following Litvin's approach) it is proposed and compared with a development of Blanche and Yang's approach. Also a simplified procedure to calculate force distribution on cycloid drive elements, as well as its power losses and theoretical mechanical efficiency is presented. A new cycloidal gearing tooth form (where is used an elliptic grinding wheel) to generate the cycloidal wheel is presented by Zhou and Chen (2001). The kinematics relationship is analytically defined, and subsequently analyzed in detail via a computer-aided procedure and
ROmanian
Association of
MEchanical
Transmissions
(ROAMET)
Balkan Association of
Power Transmissions
(BAPT)
74
FEA program.
This paper presents a method for generate a cycloidal gearing tooth form. In Section 2 a detailed discussion about this method is presented. Also in this section an experiment has carried out in order to compare the measured and the calculated backlash. Eventually, some suggestions regarding the possible extensions of the results of this study are presented.
2. PROFILING THE CYCLOID AND STEP DRIVEN WHEEL TEETH
The mechanism for achieving precise rotary motion consists of the pinion 1 and the gear wheel 2 (Fig. 1 and Fig. 2) of cycloidal profile. This mechanism was developed and tested.
Achieving this transmission requires solving theoretical problems related to teeth profile of the cogwheel, in such a way that it obtained simultaneously contact of two or more rollers, the contact being on opposite sides of the teeth to eliminate backlash (Zhou et al., 2001).
To establish the equations of a point of this curve, one must start from the equations of the epicycloid described by the center C1 of the roll (Fig. 3), adopting the axis coordinate system OXY, the point O being in the center circle of radius r2 (Chen et al., 2009, Feng et al., 2011, Chen el al. 2008).
The axis Ox crosses through the turning point of the epicycloid:
( )
+−+= 2
2
2112211 coscos φφ
r
rrrrrxc ; (1)
( )
+−+= 2
2
2112211 sinsin φφ
r
rrrrryc . (2)
Point coordinates of the curve M will be (Fig. 3):
βρ cos1 −= cM xx ; (3) βρ cos1 −= cM xx . (4)
where:
2/)2(90 210 φφβ +−= . (5)
It results from Fig. 3 and the definition of the epicycloid:
1
2
12 φφ
r
r= . (6)
Replacing (3), (4) and (6) in (1) and (2), it results:
( )][+
+
+−+=
1
)1(coscos)1( 2
21i
iirxM
φφ (7)
Fig. 1. Pinion gear roller – wheel 1 – Pinion; 2 – cycloid wheel.
Fig. 2. Roller pinion 3 – Chain; 4 – flanges; 5 – screw.
Fig. 3. Construction of wheel tooth head
75
+−
++ 2
2 )1(902
cos φφ
ρ ii
( )+
+
+−+=
1
1sinsin)1( 2
21i
iiryM
φφ
+−
++ 2
2 )1(902
sin φφ
ρ ii
.
(8)
where: 090>θ ; ( )01 902 −= θφ .
Taking θ as basic parameter it determines
• 1φ , 2φ as mentioned in literature (Csibi et al.,
2006; Litvin, 2009; Veja and Sucala, 2011);
• Mx and My .
The equations (7) and (8), are the coordinates of a point of the left flank of the gear profile. In the same way, we can determine the point equations of the right flank of the teeth of gear wheel. If in the equations (7) and (8), results the equation of an epicycloid.
The coordinates of the tooth foot profile could be determined from ∆ABC (presented in Fig. 4) as follows:
θρ cos2 −= rxB ; (9) θρ sin=By . (10)
These equations (9) and (10) correspond to the angle θ of the foot area of the tooth profile in the arc with radius equal to the radius roller. Angle θ values vary between limits of 00 .... 900 (for foot tooth profile).
The problem to be solved is to avoid fj lateral
backlash during engagement. This can be achieved by building the gear in such a way that two rollers
have to be in simultaneously contact with opposite sides of the same tooth. Accordingly, for the calculation of the length of string appropriate to the
linear step '2p , considered involvement of two rolls
with opposite tooth profiles, the center line being perpendicular to the step chain (Fig. 5) as mentioned in literature (Litvin, 2009; Veja and Sucala, 2011)
.
The relation for calculating the pitch pinion is:
18011
1χπ ⋅⋅
=r
p , (11)
in which:
n
0
1360
=χ , (12)
n being the number of rolls of the pinion.
Knowing that 21 pp = and 12 rir ⋅= , one can write:
Fig. 4. Construction of wheel tooth foot
Fig. 5. Determination of tooth thickness
76
18022
2χπ ⋅⋅
=r
p , (13)
where:
2
22
180
r
p
⋅
⋅=
πχ . (14)
The linear step of the gear is:
2sin2 2
2'2
χrp ⋅= . (15)
The linear thickness of the tooth in the mentioned position has the formula:
αcos221 ⋅⋅=⋅= MCMAs , (16)
where:
ρα −= sin2 1rMC . (17)
The arc tooth circle division is (Fig. 6):
1802λ
πrs = . (18)
The angle λ is calculated with the relation:
22
2φ
χλ −= , (19)
where i/12 φφ = .
The angle 1φ is determined from ∆O1CO3, with the
relation:
−=
21
2
12
1arccosr
ρφ . (20)
The equations (7), (8), (9), (10) and (18) are used for the determination of coordinates points of cycloid tooth profile, side view of the eliminate backlash, coordinates are used in processing these wheels.
3. CONCLUSIONS
This article gives us the possibility to draw the next conclusions: - determining relatively calculation necessary to the tooth profile generation of the gear at the transmition of the roller gear pinion. - determining the thickness of the tooth in such a way that the backlash must be eliminated.
References:
CHEN, B., FANG, T., LI, C., WANG, S. (2008). Gear geometry of cycloid drives. Science in China, Series E: Technological Sciences, Volum 51, Issue 5, pp. 598-610.
CHEN, B. K., H.U., J. Z., LI, C. Y. (2009). Computerized numerical control programming system of cycloidal-pin gear based on double-arc method. Journal of Chongqing University, Volume 32, Issue 11, pp. 1246-1251.
CSIBI, V., SÂRBU, M., CRIŞAN, N., HERCIU, D., TOADER, U., SUDRIJAN, M., CIUREA, C. (2006). Angrenaje cicloidale şi scule pentru danturare (Cycloidal gears and tools for toothing ). Editura SemnE, Bucureşti.
DAVOLI, P., GORLA, C., ROSA, F., LONGONI, C., CHIOZZI, F., SAMARANI, A. (2007). Theoretical and experimental analysis of a cycloidal speed reducer. Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2007, pp. 1043-1051
FENG, Z.H., GOU, M.K., WU, S. (2011). Research on geometry relationship of pin-rack gearing. Applied Mechanics and Materials, Volume 86, pp. 547-551.
LITVIN, F.L. (2009). Geometria angrenării şi teorie aplicată (Engagement geometry and applicable theory). Editura Dacia, Cluj-Napoca.
VEJA, A., SUCALĂ, F. (2011). Propunere de brevet de invenţie cu denumirea “Mecanism pentru realizarea precisă a mişcării de rotaţie”. înregistrat la OSIM în data de 04 martie 2011, cu număr de înregistrare A/10011/2011 (Proposal of patent named “Mechanism for achieving precise rotational movement”, registered to OSIM on march four 2011, registration number A/10011/2011).
YANGFANG W., CHUNLIN X., YONGGANG S. (2010). Calculation method for the profile curves of cycloidal pump. Advanced Materials Research, Volume 136 , pp. 126-130.
Fig. 6. Determination of tooth thickness on the circle division (Veja and Sucala, 2011)
77
ZHOU, J., CHEN, Z. (2001). Computerized design of cycloidal gear drive with improved gear tooth modificationProceedings of the International Conference on Mechanical Transmissions (ICMT 2001), pp. 167-170
ZHU, C., LIU, M., DU, X., XIAO, N., ZHANG, B. (2010). Analysis on transmission characteristics of new axis-fixed cycloid gear. Advanced Materials Research, Volume 97, Issue 101, pp. 60-63.
CORRESPONDENCE
Adrian Danut VEJA PhD, Engineer Electrolux Romania SA Str. Traian nr 23-29, 440078 Satu Mare, Romania [email protected]