Dynamics 70 M1 Intro

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Introduction to Dynamics

Module 1


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Int roduct ion

Welcome!

Welcome to the DynamicsTraining 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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Int roduct ionCourse Object ives

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 ANSYSs advanced modal analysis capabilities. These include

prestressed modal, cyclic symmetry, and large deflection analyses.


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Int roduct ionCou rse Material

The Training Manualyou have is an exact copy of the slides.

Workshop descriptions and instructions are included in the

Workshop Supp lement.

Copies of the workshop files are available (upon request) from the

instructor.


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Module 1

In t roduct ion to Dynam ics

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. Def in i t ion & Purpose

What is dynamic analysis?

A technique used to determine the dynamicbehavior of a

structure or component, where the structures inert ia(mass

effects) and dampingplay 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 structures 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 Gert ie) collapsedunder 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 Dynam ic Analys is

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 analysisto determine a structures 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 analysisto calculate a structures

response to time varying loads.


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Dynamics


Rotating machines exert steady,

alternating forces on bearings andsupport structures. These forces

cause different deflections and

stresses depending on the speed of

rotation.

Solution - do a harmonic analysis todetermine a structures 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

structures response to seismic loading.

Courtesy: U.S. Geological Survey

Dynamics



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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



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Dynamics

C. Basic Concepts and Term ino logy

Topics discussed:

General equation of motion

Solution methods

Modeling considerations

Mass matrix

Damping


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Dynamics - Basic Concepts & Termino logy

Equat ion of Mot ion

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(wt), where X is the amplitude and w is the frequencyin radians/sec.

Transient dynamic analysis: The above form is maintained.


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Solut ion Method s

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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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 afunction 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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Solution Methods

Explicit Method

No matrix inversion

Can handle nonlinearities easily

(no convergence issues)

Integration time step Dt must besmall (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 Dt can be largebut may be restricted by

convergence issues

Efficient for most problems except

where Dt needs to be very small.

This is the topic covered in thisseminar


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Model ing Considerat ions

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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Modeling Considerations

Material properties:

Both Youngs modulus and density are required.

Remember to use consistent units.

For density, specify mass densi tyinstead ofweight d ensi tywhen usingBritish 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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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

xx0xx0xx0xx0

00x00x

xx0xx0

xx0xx000x00x

ROTZUY

UX

ROTZ

UYUX

2

2

2

1

1

1

[M]Lumped

0000000000

00000

00000

0000000000

x

x

x

x

x

x


Mass Matrix

Mass matrix [M] is required for a dynamic analysis and is

calculated for each element from its density.

Two types of [M]: consis tentand lumped. Shown below for

BEAM3, the 2-D beam element.

1 2

BEAM3BEAM3


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Mass Matrix

Consistent mass matrix

Calculated from element shape functions.

Default for most elements.

Some elements have a special form called the reducedmass

matrix, which has rotational terms zeroed out.

Lumped mass matrix

Mass is divided among the elements nodes. Off-diagonal terms

are zero.

Activated as an analysis option (LUMPM command).


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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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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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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 rat iox (ratio of damping constant cto 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.


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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.


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Damping

ANSYS allows all three forms of damping.

Viscous damping can be included by specifying the damping ratiox, Rayleigh damping constant a (discussed later), or by definingelements 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 Guid efor

information).


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In ANSYS damping is defined as


Damping

]C[C]K[]K)[(]M[]C[NEL

1k

k

NMAT

1j

jjc x

bbba [C]

aM

b

bcK

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)


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Damping is specified in various forms:

Viscous damping factor or damping ratio x Quality factor or simply Q

Loss factor or Structural damping factorh Log decrement D Spectral damping factor D

Most of these are related to DAMPING RATIO x used in ANSYS Conversion factors are shown next


Damping


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Conversion between various damping specifications:


Damping

MeasureDamping

ratioLoss Factor

Log

Decrement

Quality

Factor

Spectral

Damping

Amplification

Factor

Damping

Ratio x h D 1/(2Q) D/(4U) 1/2A

Loss Factor x h D Q D/(2U) 1/A

Log

Decrement x h D Q D/(2U)

Quality

Factor x h D Q U/D

Spectral

Damping Ux Uh 2UD U/Q D U

Amplification

Factor x h D Q U/D


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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 becalculated from a known value ofx(damping ratio) and a known

frequency w:a = 2xw

Only one value of alpha is allowed, sopick the most dominant response

frequency to calculate a. Input using the ALPHAD command.

Frequency

DampingRatio

a3

1

2

0.5

Effect of Alpha Damping on Damping

Ratio (Beta Damping Ignored)


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Damping

Beta Damping

Also known as structuralorst i f fness

damping.

Inherent property of most materials.

Specified per material or as a single,

global value.

If alpha damping is ignored, b can becalculated from a known value ofx(damping ratio) and a known frequencyw:

b = 2x/w Pick the most dominant response

frequency to calculate b. Input using MP,DAMP or BETAD

command.

Frequency

DampingRatio

b0.0040.003

0.001

0.002

Effect of Beta Damping on Damping

Ratio (Alpha Damping Ignored)


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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 orst i f fness)damping component.


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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 frequencyrange w1 to w2. This gives twosimultaneous equations from which

you can solve fora and b.x = a/2w1 + bw1/2x = a/2w2 + bw2/2 Frequency

DampingRatio ab

ba

w1 w2

How to Approximate Rayleigh

Damping Constants

Rayleigh Equation: the sumof the a and b terms is nearlyconstant over the range of

frequencies


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Dynamics - PostProcessin g

D. Variable Viewer

The Variable Viewer is a

specialized tool allowing one topostprocess results with respect

to time or frequency.

The Variable Viewer can be

started by:

Main Menu > TimeHist Postpro >Variable Viewer


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1 2 3 4 5 6 7 8

13 14

17

Add variable button1Delete variable button2

Graph variable button3

List variable button4

Properties button5

Import data button6

Export data button7

Export data type8

Real/Imaginary Components

Variable list

Variable name input area

11

Expression input area

12

Defined APDL variables

13

Defined Post26 variables

14

15

Calculator


Variable Viewer

15 16

9

11

10

9 Clear Time-History Data

10 Refresh Time-History Data

12

16

17


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100kg

25kg

k = 36kN/m

F

0,0

0,4000

t

tNF

k = 36kN/m

x

y


Variable Viewer


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Dynamics

E. Introductory Workshop

In this workshop, you will run a

sample dynamic analysis of theGalloping Gertie (Tacoma

Narrows bridge).

Follow the instructions in your

Dynamics Workshopsupplement

(Int rodu ctory Dynamics -Gallopin g 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 thisseminar.

Failure of Tacoma Narrows Bridge
http://localhost/var/www/apps/conversion/tmp/scratch_6/Dynamics_70_workshops.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_6/Dynamics_70_workshops.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_6/Dynamics_70_workshops.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_6/Dynamics_70_workshops.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_6/Dynamics_70_workshops.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_6/Dynamics_70_workshops.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_6/Dynamics_70_workshops.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_6/Dynamics_70_workshops.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_6/Dynamics_70_workshops.ppt


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