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Loading factors within the strength calculation procedure for involute marine gears with parallel axes

Davor Mrsić University College for Practical Studies (student), Split, Croatia

davor_mrsic@live.com

Marko Vulić Adria Winch Group, Split, Croatia

vulic@adriawinch.com

Nenad Vulić Faculty of Maritime Studies, Split, Croatia

nenad.vulic@pfst.hr

Abstract. Unified Requirement UR M56, prescribing the calculation of geometry, surface durability (pitting) and tooth root bending strength of gear pairs implemented in marine propulsion and auxiliary machinery was revised by the International Association of Classification Societies (IACS) in 2013. In the previous paper, published at the CIET 2014 Conference, the author who had taken place in the development of the UR M56 within the task of the IACS Machinery Panel, presented the background of that task, the results, its final outcome and the benefits gained. The most important benefit was the development of the verified and validated MS Excel/VBA calculation program based upon UR M56, developed by that author, which was handed over to the IACS Machinery Panel members to verify the UR M56 proposed calculation procedure itself. One of the drawbacks of the calculation procedure prescribed in UR M56 is that it does not present the way to calculate the loading factors (so called K-factors) for a gear pair, other than referring to the ISO 6336-1 standard. This standard has also inherited the problem of certain ambiguities in the calculations of loading factors. The calculation procedure itself is rather extensive and complicated, so the K-factors were also omitted in the then developed version of the MS Excel/VBA calculation program. For this reason, additional efforts have been made later to improve the program introducing the entire calculation of loading factors, with all the necessary details. The aim of the present paper is to present the methodology used in the development of this part of the program, methodology of the verification (based upon the recently published technical report ISO/TR 6336-30:2017) and the future validation of the program results.

Key words: gearing, IACS Unified Requirement UR M56, gear strength, ISO 6336 standards series

1. Introduction

International Association of Classification Societies (IACS), gathering all the internationally recognised ships and maritime objects classification and technical supervision organisations, published the Unified Requirement UR M56 Marine gears–load capacity of involute parallel axis spur and helical gears, Rev. 3 (2015) [1]. The document covers the strength calculation procedure for gear pairs operating in marine propulsion and auxiliary machinery. The procedure may be applied to gear pairs of cylindrical shape with straight or helical, external or internal teeth. It prescribes calculation of geometry, surface durability (pitting) actual and allowable stresses, tooth root bending actual and allowable stress, as well as the strength related to surface durability and tooth root bending, obtained by comparison of calculated and minimal safety factors. The calculation procedure is mainly based upon the actual edition of international standard series ISO 6336, published in 2006, as corrected in 2008.

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The previous paper entitled Amended calculation procedure for involute marine gears with parallel axis, published at CIET 2014 conference [2], described the why-and-how of the development of the mentioned IACS UR M56 revision of from 2011 to 2013. That paper also describes the development, testing, verification and validation of the Microsoft Excel/Visual Basic for Applications (abbr. Excel/VBA) computer program, needed to check the proposed UR M56 on practical examples. However, at that time, the developed program did not contain calculation of the service loading factors, so-called the K-factors. The reason was that the procedure prescribed by the ISO 6336 standards to calculate K-factors is extensive and complicated. Consequently, programming of the calculation of these factors would have consume too much time, which would not fit within the time frame scheduled IACS Machinery Panel to complete the task of revising the IACS UR M56. In addition to this, regarding calculation of the K-factors, the revised edition of IACS UR M56, which was finally adopted by IACS Council, provides only a reference to ISO 6336 standards, not describing the entire procedure (with a single exception, to be mentioned later on).

The present paper describes the on-going development of the Excel/VBA computer program, presently entitled as S10CylGears_IACS. The program has been jointly developed within a final assignment task of the first author at the Mechanical Engineering B. Eng. study at the College of Practical Studies of the University of Split. The completed finalised program is expected to be implemented in teaching courses at the Marine Engineering M. Eng. study programme at the Faculty of Maritime Studies of the University of Split, as well as at Adria Winch Group Design and Development Division (in their engineers’ everyday tasks of design and strength calculations of cylindrical gear pairs) and by the Croatian Register of Shipping technical specialists in the Machinery and Automation Department, working on their plan approval tasks of marine gearboxes designs.

The present version (3.0) of the program S10CylGears_IACS, calculates all the individual gear and gears pair dimensions and other geometry values, as well as all the contact strength and tooth root bending strength related service stresses, allowable stresses and safety factors with a minimum entry of only the necessary input data. This also means that the program itself calculates all the factors (including the K-factors), without any need for entering their estimated value. For the reasons explained, the procedure for the calculation of K-factors will be presented in the present paper in more details than in the earlier paper.

The developed S10CylGears_IACS program has first been tested on examples presented in the previous (CIET 2014) paper and then verified on the eight examples provided in the ISO/TR 6336-30:2017 technical report [3]. It has to be pointed out that the International Organisation for Standardisation Technical Committees introduced an excellent practice of developing and publishing technical specifications (TS) and technical reports (TR) presenting practical calculation examples for their computationally extensive standards, providing thus basis to clear possible ambiguities and disputes. Verification results show an excellent agreement of the results obtained by S10CylGears_IACS program with those in the ISO/TR 6336-30. Owing to the possibility of implementation of Excel built in what-if analysis capability, the program enables the user not only to check the strength of an already developed gear pair, but also to obtain such a design solution meeting a certain prescribed criterion, what can be very important to cylindrical gear pair designers.

2. Gear dimensions and strength calculations

Design and strength calculation of a cylindrical gear pair, similarly to the design and strength calculation of any engineering design component, completely and practically relies upon the definition and selection of the following items:

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- Structural design shape; - Dimensions (geometry and scantlings); - Material and its production process; and - Service loading.

Within the present development task the structural design shape is the spur or helical gear pair (pinion and gear wheel) with external or internal toothing, where the term internal means that the pinion has external teeth and the gear wheel has internal teeth. Typical design shapes, all covered by the developed program, are presented in Figure 1.

Figure 1 Spur and helical gears design shapes

Dimensions (geometry and scantlings) comprise another important topic to be defined. These have been based upon definitions in the actual standard ISO 21771:2007 [4]. Figure 2 from this standard ilustrates a typical set of gear dimensions important for other geometry calculation values.

Figure 2 Typical gear geometric values – dimensions (diameters, pitches and angles [4]

The next item is the material and its production process. Material production process means both mechanical and thermic processing (heat treatment). The gear materials have been subdivided into eight categories as presented in Table 1.

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Table 1 Gear material types (material; abbreviation; surface hardness)

1 - case hardened steel; Eh; 660-800 HV

5 - alloyed through hardened steel; V; 200-360 HV

2 - nitriding steel, nitrided; NT(nitr.); 650-900 HV

6 - carbon through hardened steel; V; 135-210 HV

3 - through hardening steel, nitrided; NV(nitr.); 450-650 HV

7 - normalized low carbon steel; St; 110-210 HB

4 - flame or induction hardened steel; IF; 500-615 HV

8 - cast steel; St (cast); 140-210 HB

Service loading of gear pair is normally defined upon maximal continuous power transmitted by the gear set P, (in kW) and the rotational speed of pinion (n1, in rpm). Instead of power, the service loading can also been specified by means of the torque in way of pinion (T1, in kNm) or gear wheel (T2, in kNm). In addition to this, service loading strongly depends upon the characteristics of driving and driven machines, as explained further on.

Gear strength calculations, in accordance with the IACS UR M56 [1], as well as the ISO 6336 standards series, comprise calculation of tooth contact stresses (based upon the so-called Z-factors) as described in ISO 6336-2 [5] and tooth root stresses (based upon the Y-factors) from ISO 6336-3 [6]. Both types of strength calculations require the service loading factors (K-factors) to be calculated in accordance with the standard ISO 6336-1 [7].

3. Methods for the calculation of service loading factors

Strength of a cylindrical gear pair strongly depends upon its service loading. In the calculation procedure, according to the standards ISO 6336 series, service loading is expressed by the above mentioned nominal values of mechanical loading values (power, rotational speed, torque), which are increased by the service loading factors. These factors are defined by the standard ISO 6336-1:2006 Corr. 2008 [7]. The IACS UR M56 contains only the verbal reference to the ISO 6336-1 standard [1]. Loading factors refer to the gear pair, i.e. they are not calculated separately for the pinion or the wheel in the gear pair, though the input data of the calculation procedure requires some values entered separately for each gear in the pair. The service loading factors can be combined into the two values by the following equations:

KH = KA ∙ Kγ ∙ Kv ∙ KHα ∙ KHβ (1)

KF = KA ∙ Kγ ∙ Kv ∙ KFα ∙ KFβ (2)

where:

KH – combined service loading factor for the contact strength;

KF – combined service loading factor for the tooth root strength.

The particular service loading factors in the equations (1) and (2) will be briefly described hereafter. The complete explanation and reference to these factors can be found in the ISO 6336-1 standard [7].

3.1 Application factor KA

The application factor KA adjusts the nominal load F in order to compensate for incremental

gear loads from external sources. These additional loads are largely dependent on the

characteristics of the driving and driven machines, as well as the masses and stiffness of the

system, including shafts and couplings used in service.

The quote from ISO 6336-1 [7] states: For applications such as marine gears and others

subjected to cyclic peak torque (torsional vibrations) and designed for infinite life, the

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application factor can be defined as the ratio between the peak cyclic torques and the

nominal rated torque. The nominal rated torque is defined by the rated power and speed. It is

the torque used in the load capacity calculations.

From the above definition it is essential to understand why …it is recommended that the purchaser and manufacturer/designer agree on the value of the application factor… (as also stated in the ISO 6336-1 [7]). The gearbox designer designs gear pairs specifying nominal torque in way of pinion, T1 (which may be expressed by the nominal transmitted power and pinion corresponding rotational speed) and the application factor KA. In the gearbox design phase the application factor normally cannot be determined in accordance with its exact definition, i.e. ratio of torque maximal value (equal to nominal torque plus the torque amplitude due to torsional vibrations) and the nominal torque. At this phase, the complete mechanical system does not yet exist at all, so it is not possible to perform torsional vibration calculations to obtain the actual torque amplitude. Consequently, the gearbox will be specified, designed and delivered for the specified maximal application factor KA, which can be exactly determined and checked later on, once the whole mechanical system (e.g. all the components of the ship propulsion system) has been specified, with all the details necessary for the calculation of torsional vibrations.

3.2 Load sharing factor Kγ

This factor accounts for the unequal distribution of load in multiple-path transmissions (e.g. epicyclical or dual tandem, etc.) [8]. Calculation of this factor is based upon a very simple formula according to [7] or [8]. E.g. for epicyclical gearboxes with 3; 4; 5 or 6 planetary gears, load sharing factor amounts to 1,00; 1,25; 1,35 or 1,43 respectively.

3.3 Dynamic factor Kv

Dynamic factor accounts for internally generated dynamic loads due to vibrations of pinion and wheel against each other [7], taking into account internally generated dynamic loads due to mesh errors and tooth deformations [8].

The succession of factors in course of calculation is essential to understand the sequence of calculating each of the remaining service loading factors. It is based upon the nominal tangential loading force Ft (equal for both pinion and wheel in the gear pair), and shall be performed as follows [7]:

a) dynamic factor Kv calculated with the load Ft ∙KA;

b) face load distribution factors (KHβ and KFβ) with the load Ft ∙KA∙Kv;

c) transverse load distribution factors (KHα and KFα) with the load Ft ∙KA∙Kv∙KHβ.

There is another important point to be stated: calculation method by the range of its complexity and accuracy. Methods are the following: Method A, Method B or Method C Method A factors are derived from the results of full scale load tests, precise measurements or comprehensive mathematical analysis of the transmission system on the basis of proven operating experience, or any combination of these [7]. In everyday engineering tasks, this method is obviously not of practical importance. Method B calculates factors with sufficient accuracy for most of the gear pair applications. However, this approach may also be very complex, especially with respect to the service loading factors. Method C provides simplified approximations for the calculations of some factors, but it can be subject to some presumptions and conditions, which are not always complied with.

In accordance with [1], the dynamic factor Kv can be calculated by either Method B or Method C, where the Method C is completely specified and described in [1], with all of its assumptions. However, Method B has only been referred to in [1], so just a short description

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will be given hereafter. The complete reference to the Method B has been provided in the standard ISO 6336-1 [7], with all the details.

Determination of the dynamic factor using Method B requires calculation of running speed ranges, because the loading excitation frequencies (e.g. tooth meshing frequency and its harmonics) may come very near to the natural frequency of vibration of the gearing system, thus causing resonance with an extremely high tooth loading.

For this reason, at the beginning of the calculation, the speed ranges shall be subdivided into the following running speed ranges: subcritical range, main resonance range and supercritical range [7]. These ranges are dependent upon the resonance speed of the pinion nE1, further on dependent upon the mean value of mesh stiffness per unit face width (cγα) and reduced gear pair mass per unit face width referenced to the line of action (mred) as follows [7]:

1

1

30000

E

red

cn

z m (3)

Dynamic factor in the e.g. subcritical range is calculated upon the expression [7]:

1 2 31 v V p V f V kK N C B C B C B (4)

In equation (4), factor N stands for the ratio of the pinion speed n1 to its resonance speed nE1. CV factors (CV1, CV2, and CV3) account for the pitch deviation effects, tooth profile deviation effects and cyclic variation effect in mesh stiffness respectively. These are dependent mostly upon the total overlap ratio εγ, obtained from the gear pair geometry calculations.

However, the non-dimensional B-factors in equation (4), i.e. Bp, Bf, and Bk, as defined in [7] require gear pair single pitch, fpb, and profile deviations, ffa, to be determined, prior to their evaluation. These deviations are determined from their maximal values specified in the ISO 1328-1:1995 standard [5] (obsolete and withdrawn, but used as the reference in the technical report ISO/TR 6336-30 [3], or their tolerances specified in the ISO 1328-1:2013 standard [6]. The approach based upon ISO 1328-1:1995 requires ranges of reference diameter, normal module and gear width to be taken into account, instead of the actual values. However, the technical report ISO/TR 6336-30, though published in 2017, has been based upon the withdrawn standard ISO 1328-1:1995 and actual values of diameters, modules and widths, instead of the ones based upon ranges, what is not in-line with the ISO 1328-1:1995.

For the explained reasons, finally developed calculation program shall contain calculations of deviations and tolerances, in accordance with both editions of ISO 1328-1, [9] and [10].

3.4 Face load distribution factors KHβ and KFβ

These factors account for the effects of non-uniform distribution of load across the face width. Calculation of the KHβ factor for the contact stress is very complex and is performed in accordance with the procedure specified as Method C as described in the ISO 6336-1 standard [7]. Method A and Method B, as specified in the standard [7] would be too complex for the normal engineering calculations), so the Method C essentials are briefly presented hereafter.

Factor KHβ is evaluated from the following formula [7]:

22; when 1

/ 2 /

1 2; when 12 / 2 /

y y

m mH

y y

m m

F c F c

F b F bK

F c F c

F b F b

(5)

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Evaluation of the values in equation (5), such as effective equivalent misalignment after running-in (Fβy) or mean value of mesh stiffness per unit face width, cγα, has to be performed in accordance with a complex procedure as specified in ISO 6336-1 [7]. This procedure also involves evaluation of gear teeth deviations and tolerances as described in 3.3. Additionally, the position of the pinion on its shaft, defined by bearings arrangement on the shaft, bearing span and bearing offset, shall also be taken into account. Further details about the calculation of KHβ factor are presented in [7] and are too complicated to be presented here.

On the other hand, calculation of the KFβ factor for the tooth root bending stress requires only a simple formula, taking into account the tooth height (h) and the gear width (b), once the KHβ factor has been evaluated, specified as follows [7]:

2

1; where

1 / /

FN

F H FK K Nh b h b

(6)

3.5 Transverse load distribution factors KHα and KFα

Transverse load distribution factors, KHα for contact stress and KFα for tooth root stress, account for the effects of pitch and profile errors on the transversal load distribution between two or more pairs of teeth in mesh [1]. Both factors mainly depend upon the total mesh stiffness, total tangential load, base pitch error, tip relief and running-in allowances. They are determined by Method B in the ISO 6336-1 standard [7] from the following equation:

0,9 0, 4 ; when 2 2 /

2 10,9 0,4 ; when 2

/

pb

t A v H

Ha Fa

pb

t A v H

c f y

F K K K bK K

c f y

F K K K b

(7)

At the phase of calculation of both transverse load distribution factors, the values of mesh stiffness, cγα, base pitch deviation, fpb, and running-in allowance, yα have been already calculated within the scope of the KHβ factor calculation. Moreover, the formula (7) explicitly requires the KHβ factor in the applied total tangential load.

4. Methods for the calculation of actual stresses, allowable stresses and safety factors

Calculation of actual contact stresses, allowable contact stresses and the relevant safety factor has been specified in the ISO 6336-2 standard [5]. Calculation of actual tooth root stresses, allowable tooth root stresses and the relevant safety factor follows the ISO 6336-3 standard [6]. These calculations have been presented in the previous conference CIET 2014 paper [2] and have been completely covered by the-then developed program S10CilZ_IACS, so they will not be repeated again.

5. Results and discussion

The first important outcome of this research was the program S10CylGears_IACS version 3.0 itself. This is actually the previously developed program S10CilZ_IACS, from [2] upgraded with the calculation of tolerances and deviations according to both editions of ISO 1328-1 standards, [9] and [10], as well as with the additional input data and function subprograms necessary for the calculation of all of the service loading factors.

Based upon the input data, all the required factors, including the service loading factors, are now calculated automatically in the program.

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The program has been developed in Excel/VBA, where all of the factors have been calculated by means of the Function type VBA subprograms. The use of Sub type subprograms has been deliberately avoided, in order to keep up with the possibility of using What-If, previously Goal-Seek analysis in Excel. Thus all the Function VBA subprograms may be treated only as an Excel extensions, allowing the use of any of Excel advanced analysis and optimisation features, important to gear designers in this case.

Typical results are presented in Tables 2 to 8 presented hereafter.

Table 2 Geometry input data

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Table 3 Geometry results

Table 4 Tolerances and deviations according to ISO 1328-1:2013 standard

Table 5 Input data for strength calculations

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Table 6 Gear material and loading data

Table 7 Service loading factors for contact stresses

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Table 8 Service loading factors for tooth root stresses

The developed program S10CylGears_IACS, version 3.0, has been verified on the eight examples showed in the technical report ISO/TR 6336-30 [7]. The first example in [7] is presented there with all the calculation and formulae details. All the results show excellent agreement with the ones in [7], so there is no need to comment them separately hereafter.

Program validation is still pending. It is expected to be performed in Adria Winch Group and the Croatian Register of Shipping on actual practical examples in everyday work of their designers and technical specialists. The authors will use feedback from this validation to improve the program in future.

6. Conclusion

The aim of this paper is to present important issues and solutions to problems that were faced with during the development, testing and verification of the Excel/VBA computer program S10CylGears_IACS, for the calculation of spur (straight teeth) or helical (oblique teeth) cylindrical gear pairs with external or internal teeth. The task was to develop the program capable of calculation of gear pair dimensions, overlap ratios, tolerances and deviations required for the strength calculations, as well as the calculation of actual service stresses, allowable stresses and safety factors, both for the contact strength and tooth root strength.

Calculations of the service loading factors (K-factors) follow the extensive and complicated procedure described in ISO 6336-1 [7]. Calculations also require evaluation of deviations and tolerances based upon the ISO 1328-1 standards [9], [10], as they are used to calculate several K-factors. Due to this, K-factors calculations were not implemented in the previous program versions, as described in CIET 2014 paper [10], but the K-factors had to be supposed and entered directly into the program. For this reason, the calculation procedures for the K-factors is presented in some more details necessary to understand its basic concepts.

Developed program S10CylGears_IACS (version 3.0) enables the user (a mechanical or a marine engineering student, a design engineer who designs gears and/or gearboxes, or a class society engineering technical specialists) to quickly obtain all the necessary geometry definition and strength check calculation values. The program is developed in Microsoft Excel/Visual Basic for Applications (Excel/VBA) implementing only Function type subprograms in VBA. Regardless of the complexness of these subprograms, they can in each case be understood as the extension of the Excel itself, thus leaving enabled the possibility of using some extended Excel features of its what-if-analysis. This is of a special practical importance during the conceptual design phase of the gears and gear pairs.

The program has been tested on all the examples presented in the technical report ISO/TR 6336-30:2017 [7]. The results obtained by the program have been compared with the results in the technical report and show excellent agreement for each and every of the eight calculation results presented in the report.

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This proves that the idea of assigning such a development task to the mechanical engineering student as the final assignment, proposing the Excel/VBA as the development tool, finally brought out a useful tool for mechanical engineers in their practical everyday life tasks of designing and checking cylindrical gears and gear pairs strength and safety.

Further development of the program will rely upon the requirements of the engineers that will be using it, as well as the problems they will encounter in their work using the program. There are no obstacles to implement a similar development concept to a similar program for bevel and hypoid gear pairs, based upon other ISO standards. This has already been commenced with a group of Marine Engineering students attending their final year of the master’s degree study at the Faculty of Maritime Studies, Split.

REFERENCES

[1] ... (2015) IACS Unified Requirement UR M56 Marine gears–load capacity of involute parallel axis spur and helical gears, (Rev. 3, October 2015). IACS Blue Book, pp. 1-23, Retrieved April 15, 2018, from http://www.iacs.org.uk/blue-book/

[2] Vulić, N. (2014) Amended calculation procedure for involute marine gears with parallel axis (paper TR03_P01). Contemporary Issues in Economy & Technology (CIET 2014), June 19th -20th 2014, Conference Proceedings (pp. P303-P312), Split, Croatia

[3] ... ISO/TR 6336-30 (2017) Calculation of load capacity of spur and helical gears-Part 30: Calculation examples for the application of ISO 6336 parts 1,2,3,5. Geneva, Switzerland: International Organization for Standardization.

[4] ... ISO 21771 (2007), Gears-Cylindrical involute gears and gear pairs-Concepts and geometry. Geneva, Switzerland: International Organization for Standardization.

[5] ... ISO 6336-2:2006 & Corr. 1 (2008) Calculation of load capacity of spur and helical gears-Part 2: Calculation of surface durability (pitting). Geneva, Switzerland: International Organization for Standardization.

[6] ... ISO 6336-3:2006 & Corr. 1 (2008) Calculation of load capacity of spur and helical gears-Part 3: Calculation of tooth bending strength. Geneva, Switzerland: International Organization for Standardization

[7] ... ISO 6336-1:2006 & Corr. (2008) Calculation of load capacity of spur and helical gears-Part 2: Calculation of surface durability (pitting). Geneva, Switzerland: International Organization for Standardization.

[8] ... (2013) Rules for the Classification of Ships, Part 9-Machinery. Split, Croatia: Croatian Register of Shipping.

[9] ... ISO 1328-1 (1995) Cylindrical gears - ISO system of accuracy - Part 1: Definitions and allowable values of deviations relevant to corresponding flanks of gear teeth. Geneva, Switzerland: International Organization for Standardization.

[10] ... ISO 1328-1 (2013) Cylindrical gears - ISO system of flank tolerance classification - Part 1: Definitions and allowable values of deviations relevant to flanks of gear teeth. Geneva, Switzerland: International Organization for Standardization.