Introduction to Dynamics Module 1
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Introduction to Dynamics
Module 1
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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: www.ansys.com under “Training Services”.
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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.
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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.
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Module 1
Introduction to Dynamics
A. Define dynamic analysis and its purpose.
B. Discuss different types of dynamic analysis.
C. Cover some basic concepts and terminology.
D. Introduce the Variable Viewer in the Time-History Postprocessor.
E. Do a sample dynamic analysis exercise.
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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.
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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.
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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.
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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.
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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.
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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.
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– 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
Dynamics
… Types of Dynamic Analysis
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– 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
Dynamics
… Types of Dynamic Analysis
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Dynamics
C. Basic Concepts and Terminology
Topics discussed:
• General equation of motion
• Solution methods
• Modeling considerations
• Mass matrix
• Damping
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Dynamics - Basic Concepts & Terminology
Equation of Motion
• The general equation of motion is as follows.
tFuKuCuM
• 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(t), where X is the amplitude and is the frequency in radians/sec.
– Transient dynamic analysis: The above form is maintained.
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Dynamics - Basic Concepts & Terminology
Solution Methods
How do we solve the general equation of motion?
• Two main techniques:– Mode superposition– Direct integration
Mode superposition
• 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.
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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.
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Dynamics - Basic Concepts & Terminology
… Solution Methods
Explicit Method
• No matrix inversion
• Can handle nonlinearities easily (no convergence issues)
• Integration time step t 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.
Implicit Method
• Matrix inversion is required
• Nonlinearities require equilibrium iterations (convergence problems)
• Integration time step t can be large but may be restricted by convergence issues
• Efficient for most problems except where t needs to be very small.
• This is the topic covered in this seminar
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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.
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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
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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.
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[M] Consistent
xx0xx0
xx0xx0
00x00x
xx0xx0
xx0xx0
00x00x
ROTZ
UY
UX
ROTZ
UY
UX
2
2
2
1
1
1
[M] Lumped
00000
00000
00000
00000
00000
00000
x
x
x
x
x
x
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
BEAM3BEAM3
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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).
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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.
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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
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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 (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 , cc = 2m.
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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 is the coefficient of friction.
• Generally not considered in a dynamic analysis.
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Dynamics - Basic Concepts & Terminology
… Damping
ANSYS allows all three forms of damping.
• Viscous damping can be included by specifying the damping ratio , Rayleigh damping constant (discussed later), or by defining elements with damping matrices.
• Hysteresis or solid damping can be included by specifying another Rayleigh damping constant, (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).
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• In ANSYS damping is defined as
Dynamics - Basic Concepts & Terminology
… Damping
]C[C]K[]K)[(]M[]C[NEL
1kk
NMAT
1jjjc
[C]Mc
j [Ck] C
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)
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• Damping is specified in various forms:– Viscous damping factor or damping ratio – Quality factor or simply Q– Loss factor or Structural damping factor – Log decrement – Spectral damping factor D
• Most of these are related to DAMPING RATIO used in ANSYS
• Conversion factors are shown next
Dynamics - Basic Concepts & Terminology
… Damping
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• Conversion between various damping specifications:
Dynamics - Basic Concepts & Terminology
… Damping
MeasureDamping
ratioLoss Factor
Log Decrement
Quality Factor
Spectral Damping
Amplification Factor
Damping Ratio 1/(2Q) D/(4U) 1/2A
Loss Factor Q D/(2U) 1/A
Log Decrement Q D/(2U)
Quality Factor Q U/D
Spectral Damping U U 2U U/Q D U
Amplification Factor Q U/D
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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, can be calculated from a known value of (damping ratio) and a known frequency :
= 2
• Only one value of alpha is allowed, so pick the most dominant response frequency to calculate .
• Input using the ALPHAD command.
Frequency
Dam
pin
g R
atio
3
1
2
0.5
Effect of Alpha Damping on Damping Ratio (Beta Damping Ignored)
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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, can be calculated from a known value of (damping ratio) and a known frequency :
= 2/
• Pick the most dominant response frequency to calculate .
• Input using MP,DAMP or BETAD command.
Frequency
Dam
pin
g R
atio
0.004
0.003
0.001
0.002
Effect of Beta Damping on Damping Ratio (Alpha Damping Ignored)
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Dynamics - Basic Concepts & Terminology
… Damping
Rayleigh damping constants and
• Used as multipliers of [M] and [K] to calculate [C]:
[C] = [M] + [K]
/2 + /2 =
where is the frequency, and is the damping ratio.
• Needed in situations where damping ratio cannot be specified.
• Alpha is the viscous damping component, and Beta is the hysteresis (a.k.a. solid or stiffness) damping component.
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Dynamics - Basic Concepts & Terminology
… Damping
To specify both and damping:
• Use the relation /2 + /2 =
• Since there are two unknowns, assume that the sum of alpha and beta damping gives a constant damping ratio over the frequency range 1 to 2. This gives two simultaneous equations from which you can solve for and . = /21 + 1/2
= /22 + 2/2Frequency
Dam
pin
g R
atio
1 2
How to Approximate Rayleigh Damping Constants
Rayleigh Equation: the sum of the and terms is nearly constant over the range of frequencies
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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
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11 22 33 44 55 66 77 88
1313 1414
1717
Add variable buttonAdd variable button11
Delete variable buttonDelete variable button22
Graph variable buttonGraph variable button33
List variable buttonList variable button44
Properties buttonProperties button55
Import data buttonImport data button66
Export data buttonExport data button77
Export data typeExport data type88
Real/Imaginary ComponentsReal/Imaginary Components
Variable listVariable list
Variable name input areaVariable name input area
1111
Expression input areaExpression input area
1212
Defined APDL variablesDefined APDL variables
1313
Defined Post26 variablesDefined Post26 variables
1414
1515
CalculatorCalculator
Dynamics - PostProcessing
…Variable Viewer
1515 1616
99
1111
1010
99 Clear Time-History DataClear Time-History Data
1010 Refresh Time-History DataRefresh Time-History Data
1212
1616
1717
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100kg100kg
25kg25kg
k = 36kN/mk = 36kN/m
FF
0,0
0,4000
t
tNF
k = 36kN/mk = 36kN/m
xx
yy
Dynamics - PostProcessing
…Variable Viewer
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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
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