1. Introduction to Dynamics
Module 1
Introduction to Dynamics

2. Welcome to the Dynamics Training Course!
Introduction Welcome!
Welcome to the Dynamics Training Course!
This training course covers the ANSYS procedures required to perform dynamic analyses.
It is intended for novice and experienced users interested in solving dynamic problems using ANSYS.
Several other advanced training courses are available on specific topics. See the training course schedule on the ANSYS homepage: under “Training Services”.
March 14, 2003
Inventory #001809
1-2

3. Introduction Course Objectives
By the end of this course, you will be able to use ANSYS to:
Preprocess, solve, and postprocess a modal, harmonic, transient, and spectrum analysis.
Use a Restart Analysis to either add time points to an existing load history or recover from an unconverged solution.
Use the Mode Superposition method to reduce the solution time of either a transient or harmonic analysis.
Use ANSYS’s advanced modal analysis capabilities. These include prestressed modal, cyclic symmetry, and large deflection analyses.
March 14, 2003
Inventory #001809
1-3

4. Introduction Course Material
The Training Manual you have is an exact copy of the slides.
Workshop descriptions and instructions are included in the Workshop Supplement.
Copies of the workshop files are available (upon request) from the instructor.
March 14, 2003
Inventory #001809
1-4

5. Module 1 Introduction to Dynamics
Define dynamic analysis and its purpose.
Discuss different types of dynamic analysis.
Cover some basic concepts and terminology.
Introduce the Variable Viewer in the Time-History Postprocessor.
Do a sample dynamic analysis exercise.
March 14, 2003
Inventory #001809
1-5

6. Dynamics A. Definition & Purpose
What is dynamic analysis?
A technique used to determine the dynamic behavior of a structure or component, where the structure’s inertia (mass effects) and damping play an important role.
“Dynamic behavior” may be one or more of the following:
Vibration characteristics - how the structure vibrates and at what frequencies.
Effect of time varying loads (on the structure’s displacements and stresses, for example).
Effect of periodic (a.k.a. oscillating or random) loads.
March 14, 2003
Inventory #001809
1-6

7. Dynamics … Definition & Purpose
A static analysis might ensure that the design will withstand steady-state loading conditions, but it may not be sufficient, especially if the load varies with time.
The famous Tacoma Narrows bridge (Galloping Gertie) collapsed under steady wind loads during a 42-mph wind storm on November 7, 1940, just four months after construction.
March 14, 2003
Inventory #001809
1-7

8. Dynamics … Definition & Purpose
A dynamic analysis usually takes into account one or more of the following:
Vibrations - due to rotating machinery, for example.
Impact - car crash, hammer blow.
Alternating forces - crank shafts, other rotating machinery.
Seismic loads - earthquake, blast.
Random vibrations - rocket launch, road transport.
Each situation is handled by a specific type of dynamic analysis.
March 14, 2003
Inventory #001809
1-8

9. Dynamics B. Types of Dynamic Analysis
Consider the following examples:
An automobile tailpipe assembly could shake apart if its natural frequency matched that of the engine. How can you avoid this?
A turbine blade under stress (centrifugal forces) shows different dynamic behavior. How can you account for it?
Answer - do a modal analysis to determine a structure’s vibration characteristics.
March 14, 2003
Inventory #001809
1-9

10. Dynamics … Types of Dynamic Analysis
An automobile fender should be able to withstand low-speed impact, but deform under higher-speed impact.
A tennis racket frame should be designed to resist the impact of a tennis ball and yet flex somewhat.
Solution - do a transient dynamic analysis to calculate a structure’s response to time varying loads.
March 14, 2003
Inventory #001809
1-10

11. Dynamics … Types of Dynamic Analysis
Rotating machines exert steady, alternating forces on bearings and support structures. These forces cause different deflections and stresses depending on the speed of rotation.
Solution - do a harmonic analysis to determine a structure’s response to steady, harmonic loads.
March 14, 2003
Inventory #001809
1-11

12. Dynamics … Types of Dynamic Analysis
Building frames and bridge structures in an earthquake prone region should be designed to withstand earthquakes.
Solution - do a spectrum analysis to determine a structure’s response to seismic loading.
Courtesy: U.S. Geological Survey
March 14, 2003
Inventory #001809
1-12

13. Dynamics … Types of Dynamic Analysis
Spacecraft and aircraft components must withstand random loading of varying frequencies for a sustained time period.
Solution - do a random vibration analysis to determine how a component responds to random vibrations.
Courtesy: NASA
March 14, 2003
Inventory #001809
1-13

14. Dynamics C. Basic Concepts and Terminology
Topics discussed:
General equation of motion
Solution methods
Modeling considerations
Mass matrix
Damping
March 14, 2003
Inventory #001809
1-14

15. Dynamics - Basic Concepts & Terminology Equation of Motion
The general equation of motion is as follows.
Different analysis types solve different forms of this equation.
Modal analysis: F(t) is set to zero, and [C] is usually ignored.
Harmonic analysis: F(t) and u(t) are both assumed to be harmonic in nature, i.e, Xsin(wt), where X is the amplitude and w is the frequency in radians/sec.
Transient dynamic analysis: The above form is maintained.
March 14, 2003
Inventory #001809
1-15

16. Dynamics - Basic Concepts & Terminology Solution Methods
How do we solve the general equation of motion?
Two main techniques:
Mode superposition
Direct integration
The frequency modes of the structure are predicted, multiplied by generalized coordinates, and then summed to calculate the displacement solution.
Can be used for transient and harmonic analyses.
Covered in Module 6.
March 14, 2003
Inventory #001809
1-16

17. Dynamics - Basic Concepts & Terminology … Solution Methods
Direct integration
Equation of motion is solved directly, without the use of generalized coordinates.
For harmonic analyses, since both loads and response are assumed to be harmonic, the equation is written and solved as a function of forcing frequency instead of time.
For transient analyses, the equation remains a function of time and can be solved using either an explicit or implicit method.
March 14, 2003
Inventory #001809
1-17

18. Dynamics - Basic Concepts & Terminology … Solution Methods
Implicit Method
Matrix inversion is required
Nonlinearities require equilibrium iterations (convergence problems)
Integration time step Dt can be large but may be restricted by convergence issues
Efficient for most problems except where Dt needs to be very small.
This is the topic covered in this seminar
Explicit Method
No matrix inversion
Can handle nonlinearities easily (no convergence issues)
Integration time step Dt must be small (1e-6 second is typical)
Useful for short duration transients such as wave propagation, shock loading, and highly nonlinear problems such as metal forming.
ANSYS-LS/DYNA uses this method. Not covered in this seminar.
March 14, 2003
Inventory #001809
1-18

19. Dynamics - Basic Concepts & Terminology Modeling Considerations
Geometry and Mesh:
Generally same considerations as a static analysis.
Include as many details as necessary to sufficiently represent the model mass distribution.
A fine mesh will be needed in areas where stress results are of interest. If you are only interested in displacement results, a coarse mesh may be sufficient.
March 14, 2003
Inventory #001809
1-19

20. Dynamics - Basic Concepts & Terminology … Modeling Considerations
Material properties:
Both Young’s modulus and density are required.
Remember to use consistent units.
For density, specify mass density instead of weight density when using British units:
[Mass density] = [weight density]/[g] = [lbf/in3] / [in/sec2] = [lbf-sec2/in4]
Density of steel = 0.283/386 = 7.3 x 10-4 lbf-sec2/in4
March 14, 2003
Inventory #001809
1-20

21. Dynamics - Basic Concepts & Terminology … Modeling Considerations
Nonlinearities (large deflections, contact, plasticity, etc.):
Allowed only in a full transient dynamic analysis.
Ignored in all other dynamic analysis types - modal, harmonic, spectrum, and reduced or mode superposition transient. That is, the initial state of the nonlinearity will be maintained throughout the solution.
March 14, 2003
Inventory #001809
1-21

22. Dynamics - Basic Concepts & Terminology Mass Matrix
Mass matrix [M] is required for a dynamic analysis and is calculated for each element from its density.
Two types of [M]: consistent and lumped. Shown below for BEAM3, the 2-D beam element. 1 2
BEAM3
March 14, 2003
Inventory #001809
1-22

23. Dynamics - Basic Concepts & Terminology … Mass Matrix
Consistent mass matrix
Calculated from element shape functions.
Default for most elements.
Some elements have a special form called the reduced mass matrix, which has rotational terms zeroed out.
Lumped mass matrix
Mass is divided among the element’s nodes. Off-diagonal terms are zero.
Activated as an analysis option (LUMPM command).
March 14, 2003
Inventory #001809
1-23

24. Dynamics - Basic Concepts & Terminology … Mass Matrix
Which mass matrix should you use?
Consistent mass matrix (default setting) for most applications.
Reduced mass matrix (if available) or lumped [M] for structures that are small in one dimension compared to the other two dimensions, e.g, slender beams or very thin shells.
Lumped mass matrix for wave propagation problems.
March 14, 2003
Inventory #001809
1-24

25. Dynamics - Basic Concepts & Terminology Damping
What is damping?
The energy dissipation mechanism that causes vibrations to diminish over time and eventually stop.
Amount of damping mainly depends on the material, velocity of motion, and frequency of vibration.
Can be classified as:
Viscous damping
Hysteresis or solid damping
Coulomb or dry-friction damping
Dampening of a Response
March 14, 2003
Inventory #001809
1-25

26. Dynamics - Basic Concepts & Terminology … Damping
Viscous damping
Occurs when a body moves through a fluid.
Should be considered in a dynamic analysis since the damping force is proportional to velocity.
The proportionality constant c is called the damping constant.
Usually quantified as damping ratio x (ratio of damping constant c to critical damping constant cc*).
Critical damping is defined as the threshold between oscillatory and non-oscillatory behavior, where damping ratio = 1.0.
*For a single-DOF spring mass system of mass m and frequency w, cc = 2mw.
March 14, 2003
Inventory #001809
1-26

27. Dynamics - Basic Concepts & Terminology … Damping
Hysteresis or solid damping
Inherently present in a material.
Should be considered in a dynamic analysis.
Not well understood and therefore difficult to quantify.
Coulomb or dry-friction damping
Occurs when a body slides on a dry surface.
Damping force is proportional to the force normal to the surface.
Proportionality constant m is the coefficient of friction.
Generally not considered in a dynamic analysis.
March 14, 2003
Inventory #001809
1-27

28. Dynamics - Basic Concepts & Terminology … Damping
ANSYS allows all three forms of damping.
Viscous damping can be included by specifying the damping ratio x, Rayleigh damping constant a (discussed later), or by defining elements with damping matrices.
Hysteresis or solid damping can be included by specifying another Rayleigh damping constant, b (discussed later).
Coulomb damping can be included by defining contact surface elements and gap elements with friction capability (not discussed in this seminar; see the ANSYS Structural Analysis Guide for information).
March 14, 2003
Inventory #001809
1-28

29. Dynamics - Basic Concepts & Terminology … Damping
In ANSYS damping is defined as
[C] a
[M] b bc
[K] bj
[Ck]
[Cx]
structure damping matrix
constant mass matrix multiplier (ALPHAD)
structure mass matrix
constant stiffness matrix multiplier (BETAD)
variable stiffness matrix multiplier (DMPRAT)
structure stiffness matrix
constant stiffness matrix multiplier for material j (MP,DAMP)
element damping matrix (element real constants)
frequency-dependent damping matrix (DMPRAT and MP,DAMP)
March 14, 2003
Inventory #001809
1-29

30. Dynamics - Basic Concepts & Terminology … Damping
Damping is specified in various forms:
Viscous damping factor or damping ratio x
Quality factor or simply Q
Loss factor or Structural damping factor h
Log decrement D
Spectral damping factor D
Most of these are related to DAMPING RATIO x used in ANSYS
Conversion factors are shown next
March 14, 2003
Inventory #001809
1-30

31. Dynamics - Basic Concepts & Terminology … Damping
Conversion between various damping specifications:
March 14, 2003
Inventory #001809
1-31

32. Dynamics - Basic Concepts & Terminology … Damping
Alpha Damping
Also known as mass damping.
Specified only if viscous damping is dominant, such as in underwater applications, shock absorbers, or objects facing wind resistance.
If beta damping is ignored, a can be calculated from a known value of x (damping ratio) and a known frequency w:
a = 2xw
Only one value of alpha is allowed, so pick the most dominant response frequency to calculate a.
Input using the ALPHAD command.
Effect of Alpha Damping on Damping Ratio (Beta Damping Ignored)
a=3 2 1
0.5
March 14, 2003
Inventory #001809
1-32

33. Dynamics - Basic Concepts & Terminology … Damping
Beta Damping
Also known as structural or stiffness damping.
Inherent property of most materials.
Specified per material or as a single, global value.
If alpha damping is ignored, b can be calculated from a known value of x (damping ratio) and a known frequency w:
b = 2x/w
Pick the most dominant response frequency to calculate b.
Input using MP,DAMP or BETAD command.
Effect of Beta Damping on Damping Ratio (Alpha Damping Ignored)
b=0.004
0.003
0.002
0.001
March 14, 2003
Inventory #001809
1-33

34. Dynamics - Basic Concepts & Terminology … Damping
Rayleigh damping constants a and b
Used as multipliers of [M] and [K] to calculate [C]:
[C] = a[M] + b[K]
a/2w + bw/2 = x
where w is the frequency, and x is the damping ratio.
Needed in situations where damping ratio x cannot be specified.
Alpha is the viscous damping component, and Beta is the hysteresis (a.k.a. solid or stiffness) damping component.
March 14, 2003
Inventory #001809
1-34

35. Dynamics - Basic Concepts & Terminology … Damping
To specify both a and b damping:
Use the relation
a/2w + bw/2 = x
Since there are two unknowns, assume that the sum of alpha and beta damping gives a constant damping ratio x over the frequency range w1 to w2. This gives two simultaneous equations from which you can solve for a and b.
x = a/2w1 + bw1/2
x = a/2w2 + bw2/2
How to Approximate Rayleigh Damping Constants
Rayleigh Equation: the sum of the a and b terms is nearly constant over the range of frequencies w1 w2
a+b b a
March 14, 2003
Inventory #001809
1-35

36. Dynamics - PostProcessing D. Variable Viewer
The Variable Viewer is a specialized tool allowing one to postprocess results with respect to time or frequency.
The Variable Viewer can be started by:
Main Menu > TimeHist Postpro > Variable Viewer
March 14, 2003
Inventory #001809
1-36

37. Dynamics - PostProcessing …Variable Viewer
Add variable button 1 9 10
Delete variable button 1 2 3 4 5 6 7 8 2 11
Graph variable button 3
List variable button 4
Properties button 12 5
Import data button 6
Export data button 7
Export data type 8
Clear Time-History Data 9 13 14
Refresh Time-History Data 10 15 16
Real/Imaginary Components 11
Variable list 12
Variable name input area 13 17
Expression input area 14
Defined APDL variables 15
Defined Post26 variables 16
Calculator 17
March 14, 2003
Inventory #001809
1-37

38. Dynamics - PostProcessing …Variable Viewer
k = 36kN/m
100kg x
k = 36kN/m
25kg y F
March 14, 2003
Inventory #001809
1-38

39. Dynamics E. Introductory Workshop
In this workshop, you will run a sample dynamic analysis of the “Galloping Gertie” (Tacoma Narrows bridge).
Follow the instructions in your Dynamics Workshop supplement (Introductory Dynamics - Galloping Gertie, Page W-5 ).
The idea is to introduce you to the steps involved in a typical dynamic analysis. Details of what each step means will be covered in the rest of this seminar.
Failure of Tacoma Narrows Bridge
March 14, 2003
Inventory #001809
1-39