a guide to linear dynamic damping - fea for all |and all...
TRANSCRIPT
A Guide to linear dynamic
analysis with Damping
This guide starts from applications of linear dynamic responseand its role in FEA simulation. Fundamental concepts andprinciples will be introduced such as equation of motion, types ofvibrations, role of damping in engineering and its types, lineardynamic analyses, etc. Finally we will discuss how to chooseappropriate solution type for damping and introduce strength ofmidas NFX for solving dynamic problems.
Courtesy of Faculty of Mechanical Engineering and Mechatronics; West Pomeranian University of Technology, Poland
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1. Dynamic Analysis Application
Dynamic analysis is strongly related to vibrations.
Vibrations are generally defined as fluctuations of a mechanical or structural system about an
equilibrium position. Vibrations are initiated when an inertia element is displaced from its
equilibrium position due to an energy imported to the system through an external source.
Vibrations as the science is one of the first courses where most engineers to apply the knowledge
obtained from mathematics and basic engineering science courses to solve practical problems.
Solution of practical problems in vibrations requires modeling of physical systems. A system is
abstracted from its surroundings. Usually assumptions appropriate to the system are made.
Basic engineering science, mathematics and numerical methods are applied to derive a computer
based model.
Vibrations induced by an unbalanced helicopter blade while rotating at high speeds can lead to
the blade's failure and catastrophe for the helicopter. Excessive vibrations of pumps,
compressors, turbomachinery, and other industrial machines can induce vibrations of the
surrounding structure, leading to inefficient operation of the machines while the noise produced
can cause human discomfort.
Vibrations occur in many mechanical and structural
systems. Without being controlled, vibrations can
lead to catastrophic situations.
Vibrations of machine tools or machine tool chatter
can lead to improper machining of parts. Structural
failure can occur because of large dynamic
stresses developed during earthquakes or even
wind induced vibrations.
Figure 1: 1940 Tacoma Narrows Bridge failure
Figure 2: Failed compressor components
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2. Dynamic Analysis Equation
The mathematical modeling of a physical system results in the formulation of a mathematical
problem. The modelling is not complete until the appropriate mathematics is applied and a
solution obtained.
The type of mathematics required is different for different types of problems. Modeling of any
statics, dynamics, and mechanics of solids problems leads only to algebraic equations.
Mathematical modeling of vibrations problems leads to differential equations.
In mathematical physics, equations of motion are equations that describe the behavior of
a physical system in terms of its motion as a function of time.
Exact analytical solutions, when they exist, are preferable to numerical or approximate solutions.
Exact solutions are available for many linear problems, but for only a few nonlinear problems.
)(tum )(tuc )(tku )(tp
APPLIED FORCEINERTIA FORCE DAMPING FORCE RESTORING FORCE
Equations of motion are consisted of inertial force, damping force (energy dissipation) and
elastic (restoring) force.
The overall behavior of a structure can be grasped through these three forces.
Inertia Force is generated by accelerated mass.
Damping Force describes energy dissipation mechanism which induces a force that is a
function of a dissipation constant and the velocity. This force is known as the general viscous
damping force.
The final induced force in the dynamic system is due to the elastic resistance in the system and
is a function of the displacement and stiffness of the system. This force is called the elastic force,
restoring force or occasionally the spring force.
The applied load has been presented on the right-hand side of equation and is defined as afunction of time. This load is independent of the structure to which it is applied.
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2.1 Single Degree of Freedom System
)()()()( tkutuctumtp
2.2 Equation of Motion for Single Degree of Freedom System
System parameters are represented in the model, and their values should be known in order to
determine the response of the system to a particular excitation.
State variables are a minimum set of variables, which completely represent the dynamic state of a
system at any given time t. For a simple SDOF oscillator an appropriate set of state variables would
be the displacement u and the velocity du/dt.
The equation of motion for SDOF mechanical system may be derived using the free-body diagram
approach.
Simple mechanical system is schematically shown in Figure 3. The inputs (or excitation) applied to
the system are represented by the force p(t). The outputs (or response) of the system are
represented by the displacement u(t). The system boundary (real or imaginary) demarcates the
region of interest in the analysis. What is outside the system boundary is the environment in which
the system operates.
System Response
(Outputs)
System Excitation
(Inputs)
Environment
Dynamic System State
of Variables
Parameters (m, c, k)
System Boundary
u(t)
Figure 3: A Mechanical dynamic system
m – mass
c – damping coefficient
k – stiffness coefficient
p(t) – applied force
u(t)– mass displacement
u(t)– mass velocity
u(t)– mass acceleration
Figure 4: SDOF free body diagram.
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2.3 Single Degree of Freedom System Responses
Free Undamped Response
Free Damped Response
0)()( tkutum
)()()( tptkutum )()()()( tptkutuctum
0)()()( tkutuctum
Forced Undamped Response
tutu
tu nn
n
cossin)( 00
][ s
rad
m
kn
entities
Condition Initial an are and 00 uu
Natural frequency:
mcteBtAtu 2/)()(
Critically Damped System Overdamped SystemUnderdamped System
ncr mkmcc 22 crcc
General solution:
)cossin()( / tBtAetu dd
mct 2
21 nd
crc
c
tBtAtu nn sincos)(
General solution:
crcc
Underdamped solution:
Damped natural
frequency
Damping ratio
Forced Damped Response
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3. Dynamic Analysis Types in midas NFX
Structure
Mass StiffnessNatural
Frequencies
Normal Mode Shapes
Analysis of the Normal Modes or Natural
Frequencies of a structure is a search for it’s
resonant frequencies. By understanding the
dynamic characteristics of a structure
experiencing oscillation or periodic loads, we can
prevents resonance and damage of the structure.
Natural Frequency – the actual measure of frequency, [Hz] or similar units
Normal Mode shape – the characteristic deflected shape of a structure as it resonates
Constraints
3.1 Eigenvalue Analysis/Normal Modes/Modal
Load in Time Domain
Structure
Mass Stiffness
Response in Time Domain
Constraints
DampingExecuted in the time domain, it obtains the
solution of a dynamic equation of equilibrium
when a dynamic load is being applied to a
structure.
Though the load and boundary conditions
required for a transient response analysis are
similar to those of a static analysis, a difference is
that load is defined as a function of time.
3.2 Transient Analysis
Load in Freq. Domain
Structure
Mass Stiffness
Response in Freq. Domain
Constraints
DampingA technique to determine the steady state
response of a structure according to sinusoidal (
harmonic) loads of known frequency.
Frequency Response is best visualized as a
response to a structure on a shaker table.
Adjusting the frequency input to the table gives a
range of responses.
)sin()()()( tFtkutuctum
3.3 Frequency Response Analysis
lead phase frequency driving or input
222
2
2-1 )/()(
)sin(/)(
n
n
tkFtu
Figure 5: SDOF frequency
response - Displacement
Figure 6: Shaking table
for frequency response
experiment
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4. Dynamic Loading Types
Dynamic loads can be applied directly on your model or as time or frequency dependent
static loadings.
- Time domain based loading types - Frequency domain based loading types
Figure 8. Dynamic Analysis Solution Types and Methods.
Dynamic Analysis
Direct Integration Method
Modal Superposition Method
Normal Modal Analysis
(Eigenvalue Analysis)
Transient Response AnalysisFrequency Response
AnalysisShock and Response Analysis*
Linear or Nonlinear
Analysis
Linear Analysis
*Not Covered in this guide
4.1 Solution Methods
Figure 7. Domain dependent Static Loading.
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Damping is the phenomenon by which mechanical energy is dissipated (usually by conversion
into internal thermal energy) in dynamic systems. Knowledge of the level of damping in a dynamic
system is important in the utilization, analysis, and testing of the system. In structural systems,
damping is more complex, appearing in several forms.
5. Damping
Mechanical Damping
- Internal
- External
- Distributed
- Localized
Damping Models
General Viscous Damping
Structural /
HystereticCoulomb Fluid
Internal damping refers to the structural material
itself. Internal (material) damping results from
mechanical energy dissipation within the material
due to various microscopic and macroscopic
processes.
All damping ultimately comes from frictional effects, which may however take place at different
scales. If the effects are distributed over volumes or surfaces at macro scales, we speak of
distributed damping. Damping devices designed to produce beneficial damping effects, such as
shock absorbers, represent localized damping.
5.1 Damping Models
It is not practical to incorporate detailed microscopic representations of damping in the dynamic
analysis of systems. Instead, simplified models of damping that are representative of various types
of energy dissipation are typically used.
External damping comes from boundary
effects. An important form is structural
damping, which is produced by rubbing
friction: stick-and-slip contact or impact.
Another form of external damping is fluid
damping.
Modal damping
Rayleigh damping
Figure 10. Localized damping examples
Figure 11. General Damping models review
Figure 9. Damping forms
The following types of damping are available inside the midas NFX:
- Proportional Damping (Rayleigh; classic)
- Hysteretic/Structural Damping
- Direct Damping values
- Frequency dependent damping
- Modal Damping
- Coulomb damping, requires special modelling techniques
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- BUSH 1D element Property
- Damper element Property
- Rayleigh Damping (Proportional)
5.2 Damping Models in midas NFX
5.3 Damping Model input for:
- Structural Damping
- Discrete Viscous Damping
- Defined via Material Card
- Defined via Material Card and Analysis
Case Manager
- Overall and Elemental Damping
- Modal Damping
- Defined via Analysis Case Manager and
Functions
- Coulomb Damping
- Defined by combination of elements and
features
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midas NFX are used in various projects to investigate dynamic characteristics of a large range of
industrial products. Some of the examples are shown below.
6. Project Applications of Dynamic Analysis in midas NFX
Mode Frequency
1ND 1,046 Hz
2ND 1,778 Hz
3ND 1,801 Hz
4ND 9,457 Hz
5ND 9,679 Hz
6ND 10,217 Hz
In this project, the
performance of a cellphone
speaker is evaluated under
different sound pressure
levels.
Displacement / Frequency
Firstly modal analysis was performed to
determine natural frequencies of the speaker
components. Then frequency response
analysis was performed to calculate stresses
and deformation shapes of the speaker
components according to different frequency
spectrums.
From the right image, we can see different
mode shapes of the structure. We can also
observe that maximum displacement was 0.4
mm, it occurred at around 1000Hz. At this
frequency the suspension structure reached
its maximum stress of 250MPa.
Stress
Case 2
Resonance avoidance of ultra large AC servo robot(midas NFX application from YUDO STAR Automation)
In this case a dynamic characteristics of a
servo robot arm was investigated both to
avoid resonance during machine operation
and to ensure structural safety during
earthquakes.
From the image we can confirm the necessity
for resonance avoidance design in order to
avoid the resonance which happens at 15Hz
under repetitive load.
Stresses were equally calculated under
seismic load through response spectrum
analysis.
Resonance frequency range
ModeNatural
Frequency
1st Mode 6.0Hz
2nd Mode 8.2Hz
3rd Mode 14.6Hz
Stress distribution under seismic load
Case 1
Performance evaluation of a mobile speaker through sound
pressure level (SPL) analysis(midas NFX application from KEYRIN TELECOM)
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Case 3
Brake Disc Squeal Analysis(midas NFX application)
In this project the dynamic characteristics of brake disc were reviewed to avoid squeal noise
caused by vibration. Through frequency response analysis, we can observe that at around 7000
Hz, frequencies of Nodal Diameter Mode and In-Plane Compression Mode are very close,
where squeal noise is most likely to occur. Therefore, a design modification is needed to
separate 2 frequencies to avoid squealing problems.
Case 4
Safety analysis of marine refrigeration machine under vibration(midas NFX application)
In this project, natural frequency analysis and frequency response analysis were performed to
predict the happening of cracks on the body and piping of a marine refrigeration machine under
vibration loads.
ModeNatural
frequency
2ND 1027 Hz
3ND 2394 Hz
4ND 3897 Hz
5ND 5468 Hz
1st IPC 7014 Hz
6ND 7045 Hz
7ND 8592 Hz
8ND 10068 Hz
2nd IPC 10285 Hz
1st IPC
6ND
0.0001
0.001
0.01
0.1
1
10
100
0 10 20 30 40 50 60 70 80 90 100Frequency
Frequency Response Function
X-direction
Y-direction
Z-direction
0.0001
0.001
0.01
0.1
1
10
100
0 10 20 30 40 50 60 70 80 90 100Frequency
Frequency Response Function
X-direction
Y-direction
Z-direction
0.0001
0.001
0.01
0.1
1
10
100
0 10 20 30 40 50 60 70 80 90 100Frequency
Frequency Response Function
X-direction
Y-direction
Z-direction
Pipe① Part frequency response Pipe② Part frequency response
Pipe③ Part frequency response
Resonant displacement distribution
1
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Resonant stress distribution
7.1 Easiness to be learned and adopted
Despite all the benefits that simulation can bring to product design, adopting “simulation driven
design” approach in a well established team can be sometimes challenging.
To fully leverage the capabilities of FEA simulation and create maximum values, design
engineers need to learn FEA software as well as some specialized engineering knowledge.
Sometime design engineers need to work together with analysis specialist in the team to
complete a more complex analysis.
midas NFX is well-known for its fast learning curve to both analysis specialists and product
designers. midas NFX divides its interface into Designer Mode and Analyst Mode to fit the
needs of simple and powerful analysis in one software.
Designer mode aims to help design engineers who aspire to take that intellectual leap from CAD
design to FEA simulation. It is equipped with automation tools such as automatic simplification,
auto-meshing, analysis wizards, etc., as well as powerful meshing algorithms and high-
performance solvers.
Analyst mode provides experienced analyst full flexibility to build and analyze finite element
models in the way you want. One can use different types of elements for modeling, and also
generate mesh manually. There are also tools to create and edit geometry that can help the user
tweak with the geometry without the inconvenience of returning to the CAD platform.
With the “auto-update” function, user can create “ analysis template” with which when model is
redesigned, analysis can be directly performed on the model without having to assign all the
analysis condition again and again. Besides time saving, you can also extend knowledge of the
analysis specialist by asking a specialist to create the template so that the design engineers can
use it later in the “design – simulation” iterations.
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7. Advantages of NFX for Dynamic Analysis
7.2 Easiness for modeling
New product are designed in CAD tools, many CAD tools are available on the market, and even
in one team different CAD tools are usually being used. Therefore, CAD compatibility is another
critical capacity to consider.
midas NFX is fully compatible with CAD, 13 most commonly used CAD formats are supported,
including Parasolid, CATIA V4/V5, UG, Pro/E, SolidWorks, Soild Edge, Inventor, STEP, IGES,
ACIS. Nastran format (.nas; .bdf) can also be import for analysis.
Models directly taken from CAD designs are usually with excessive detailing. Especially
electronic components, machine parts include many small holes, fillets, lines, faces which are
not only unnecessary but also will lead to inefficient analysis. Common practice is to clean up the
geometry model first before meshing and analyzing it.
midas NFX is equipped with automatic and manual simplification tools to handle effectively the
clean up work. With several clicks such detailing can be automatically detected, selected and
removed. This will save you considerably amount of time.
Furthermore, midas NFX is equipped with geometry creation and editing tools. Simple
modifications to the model can be directly done in midas NFX without going back to CAD tools.
7.3 Convenient graphical output and reporting
Dynamic analysis provides the output related to the nature of the problem:
Natural vibration analysis can extract natural vibration frequencies and the associated mode
shapes
The response analysis refers to the calculation of the response, which can be displacements,
strain, or stress, when the system is subjected to time or frequency varying excitation forces
A good software provides result visualization in the most appropriate manner, as well provides
tools to extract flexibly any result at any location of interest. Advanced requirements such as
drawing value tables and graphs should also be satisfied.
Result visualization is one of the biggest strengths of midas NFX. The software provides
different kinds of contour maps. With “ISO Surface”, overheated locations can be easily
discovered. Interactive graph for incremental solution provides an excellent possibility to
investigate and validate your model during solution phase.
midas NFX has the provision for automatic report generation. The reports include all the
information related to the model, right from the geometry to the result graphs and tables. The
automatic reports can be generated in MS-Word or 3D PDF format. The former can use default
or custom templates whereas the latter is an interactive report with animated illustrations of the
model in a 3D view.
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