ansys for structural engineers

7/18/2019 ANSYS for Structural Engineers

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FOR STRUCTURALENGINEERS



Course Contents:• I. Static Analysis.

•1. Stress Analysis.•2. Heat Transfer Analysis.

• II. Eigen Buckling Analysis. • II. Dynamic Analysis.

•1. Modal Analysis.

•2. Transient Analysis.•3. Harmonic Analysis.

•4. Spectral Analysis.

•5. Random Vibration Analysis.

prep by: Yasser ElJajeh



Analyses Types Offered by ANSYS

• ANSYS offers the following types of analysis:

•1. Static analysis.

•2. Mode analysis.•3. Harmonic analysis.

•4. Transient analysis.

•5. Spectral analysis.

•6. Eigen Buckling analysis.




•In order to set the units system we perform the following command:





Chapter1:Static Analysis



•This problem will be solved using area element, and a result check could be performed using beam element.

•Usually the algorithm graph of analyzing any model is shown below:




Home Work




Class Work




Due to symmetry we will model only a quarter of the beam.





Home Work



Geometry of the block: each edge is 10 in length.




The plate is 50x20 cm with three circular holes as follows:




Chapter2:

Eigen Buckling Analysis




Two ways are possible:

Either by issuing the command “ PSTRES,ON”.

Or by following this path:




Find the normal forces causing the column buckling for the first and

the second modes. Check the result manually. Euler force is given by

the following relation:


Class Work

22

)(kL

EI n P En



prep by: Yasser ElJajehprep by: Yasser ElJajeh

HomeworkFind the normal forces causing the column buckling for the first andthe second modes. Check the result manually.




Chapter3:Dynamic Analysis




Module 1

Introduction to Dynamics

Module 1




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.

Module 1

Introduction to Dynamics

Dynamics




What is dynamic analysis?• A technique used to determine the dynamic behavior of astructure or component, where the structure’s inertia (masseffects) 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’sdisplacements and stresses, for example).

▫ Effect of periodic (a.k.a. oscillating or random)loads.

Dynamics

… Definition & Purpose

Dynamics




y


• A static analysis might ensurethat the design will withstandsteady-state loadingconditions, but it may not besufficient, especially if theload varies with time.

• The famous Tacoma Narrows bridge (Galloping Gertie)collapsed under steady windloads during a 42-mph wind

storm on November 7, 1940, just four months afterconstruction.

Dynamics





• 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.

b Y ElJ j hDynamics



prep by: Yasser ElJajeh Dynamics

B. Types of Dynamic Analysis

Consider the following examples:▫ An automobile tailpipe assembly could shake apart if itsnatural frequency matched that of the engine. How can youavoid 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.

b Y ElJ j hDynamics



prep by: Yasser ElJajehy

… Types of Dynamic Analysis

▫ An automobile fender should be able to withstand low-speedimpact, but deform under higher-speed impact.

▫ A tennis racket frame should be designed to resist the impactof a tennis ball and yet flex somewhat.

to calculate atransient dynamic analysisdo a-Solution

structure’s response to time varying loads.

b Y ElJ j hDynamics



prep by: Yasser ElJajehy


▫ Rotating machines exertsteady, alternating forces on bearings and supportstructures. These forces causedifferent deflections and

stresses depending on thespeed of rotation.

Solution - do a harmonicanalysis to determine astructure’s response to steady,

harmonic loads.

prep by: Yasser ElJajehDynamics




▫ Building frames and bridge structures inan earthquake prone region should bedesigned to withstand earthquakes.

Solution - do a spectrum analysis todetermine a structure’s response to

seismic loading.

Courtesy: U.S. Geological Survey

y






▫Spacecraft and aircraft components must withstand

random loading of varying frequencies for a sustained timeperiod.

to determinerandom vibration analysisdo a-Solutionhow a component responds to random vibrations.

Courtesy: NASA

y






C. Basic Concepts and Terminology

Topics discussed:

• General equation of motion

• Solution methods

• Modeling considerations• Mass matrix

• Damping





Dynamics - Basic Concepts & Terminology

Equation of Motion

• The general equation of motion is as follows.

tFuK uCuM

• 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 beharmonic 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.

prep by: Yasser ElJajehDynamics Basic Concepts & Terminology



prep by: Yasser ElJajehDynamics - 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 thedisplacement solution.

• Can be used for transient and harmonic analyses.


D i B i C & T i l





… Solution Methods

Direct integration

• Equation of motion is solved directly, without the use of generalized coordinates.

• For harmonic analyses, since both loads and response areassumed 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 timeand can be solved using either an explicit or implicit method.




p p y jDynamics Basic Concepts & Terminology


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 durationtransients such as wavepropagation, shock loading, andhighly nonlinear problems suchas metal forming.

• ANSYS-LS/DYNA uses thismethod. Not covered in thisseminar.

Implicit Method• Matrix inversion is required

• Nonlinearities require equilibriumiterations (convergence problems)

• Integration time step Dt can belarge 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 thisseminar




p p y jy p gy

Modeling Considerations

Geometry and Mesh:

• Generally same considerations as a static analysis.

• Include as many details as necessary to sufficiently representthe 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, acoarse mesh may be sufficient.





… 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




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. Thatis, the initial state of the nonlinearity will be maintainedthroughout the solution.




March 14, 2003Inventory #001809

1-95

[M]Consistent

xx0xx0xx0xx0

00x00x

xx0xx0

xx0xx0

00x00x

ROTZUY

UX

ROTZ

UY

UX

2

2

2

1

1

1

[M]Lumped

00000

00000

00000

00000

00000

00000

x

x

x

x

x

x

y p gy

Mass Matrix

• Mass matrix [M] is required for a dynamic analysis and is calculatedfor each element from its density.

• Two types of [M]: consistent and lumped . Shown below for BEAM3,the 2-D beam element.

1 2

BEAM3BEAM3





… Mass Matrix

Consistent mass matrix• Calculated from element shape functions.

• Default for most elements.

• Some elements have a special form called the reduced massmatrix, 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).






… 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 structuresthat 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.






Damping

What is damping?

• The energy dissipation mechanism thatcauses vibrations to diminish over time andeventually stop.

• Amount of damping mainly depends on thematerial, velocity of motion, and frequency of vibration.

• Can be classified as:

▫ Viscous damping

▫ Hysteresis or solid damping

▫ Coulomb or dry-friction damping

Dampening of aResponse





… Damping

Viscous damping

• Occurs when a body moves through a fluid.

• Should be considered in a dynamic analysis since the dampingforce is proportional to velocity.

▫ The proportionality constant c is called thedamping constant .

• Usually quantified as damping ratio x (ratio of dampingconstant 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.




y a cs as c Co cepts & e ology

… 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 thesurface.

▫ Proportionality constant m is the coefficient of friction.

• Generally not considered in a dynamic analysis.


D i g



… Damping

ANSYS allows all three forms of damping.

• Viscous damping can be included by specifying the dampingratio 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 (notdiscussed in this seminar; see the ANSYS Structural AnalysisGuide for information).


Damping



• In ANSYS damping is defined as

… Damping

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

1k

k

NMAT

1 j

j jc x

bbba

[C]

aM

b

bc

K

b j

[Ck ]

Cx

structure damping matrix

constant mass matrix multiplier (ALPHAD)structure mass matrixconstant 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 andMP,DAMP)


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

… Damping


Damping



• 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


Damping



… Damping

Alpha Damping

• Also known as mass damping.

• Specified only if viscous damping isdominant, such as in underwaterapplications, shock absorbers, or objectsfacing wind resistance.

• If beta damping is ignored, a can becalculated from a known value of x (damping ratio) and a known frequency w:

a = 2xw

• Only one value of alpha is allowed, sopick the most dominant responsefrequency to calculate a.

• Input using the ALPHAD command.

Frequency

D a m p i n g R a t i o

a3

1

2

0.5

Effect of Alpha Damping on DampingRatio (Beta Damping Ignored)


Damping



… 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 becalculated from a known value of x (damping ratio) and a known frequency w:

b = 2x/w

• Pick the most dominant responsefrequency to calculate b.

• Input using MP,DAMP or BETADcommand.

Frequency


b0.004

0.003

0.001

0.002

Effect of Beta Damping on Damping Ratio(Alpha Damping Ignored)


Damping



… 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 thehysteresis (a.k.a. solid or stiffness) damping component.


Damping



… 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 constantdamping ratio x over the frequency range w1 to w2. This gives twosimultaneous equations from which you can solve for a and b.

x = a/2w1 + bw1/2x = a/2w2 + bw2/2

Frequency


ab

b

a

w1 w2

How to Approximate RayleighDamping Constants

Rayleigh Equation: the sum of

the a and b terms is nearly constantover the range of frequencies

prep by: Yasser ElJajehDynamics - PostProcessing

D Variable Viewer



D. Variable Viewer

• The Variable Viewer is a

specialized tool allowing one topostprocess results with respectto time or frequency.

• The Variable Viewer can bestarted by:

▫ Main Menu > TimeHistPostpro > Variable Viewer


Dynamics - PostProcessing

Variable Viewer



100kg

25kg

k = 36kN/m

F

0,0

0,4000

t

t N F

k = 36kN/m

x

y

…Variable Viewer




Module 2Modal Analysis

prep by: Yasser ElJajehModal Analysis

A D fi iti & P



A. Definition & Purpose

• What is modal analysis?• A technique used to determine a structure’s vibration

characteristics:

▫ Natural frequencies

▫ Mode shapes

▫ Mode participation factors (how much a givenmode participates in a given direction)

• Most fundamental of all the dynamic analysis types.


Modal Analysis





Benefits of modal analysis

• Allows the design to avoid resonant vibrations or to vibrate at aspecified frequency (speakers, for example).

• Gives engineers an idea of how the design will respond todifferent types of dynamic loads.

• Helps in calculating solution controls (time steps, etc.) for otherdynamic analyses.

Recommendation: Because a structure’s vibrationcharacteristics determine how it responds to any type of dynamic load, always perform a modal analysis first before trying any other dynamic analysis.


B Terminology & Concepts



• General equation of motion:

• Assume free vibrations and ignore damping:

• Assume harmonic motion ( i.e. )

• The roots of this equation are wi2, the Eigen values, where iranges from 1 to number of DOF. Corresponding vectors are{u}i, the eigenvectors.

0uMK 2 w

0uK uM

tFuK uCuM

B. Terminology & Concepts

)tsin(Uu w


Terminology & Concepts



… Terminology & Concepts

•The square roots of the eigenvalues are wi , the structure’snatural circular frequencies (radians/sec). Naturalfrequencies f i are then calculated as f i = wi /2 cycles/sec.It is the natural frequencies f i that are input by the user andoutput by ANSYS.

•The eigenvectors {u}i represent the mode shapes - the shapeassumed by the structure when vibrating at frequency f i.




Which method you choose depends primarily on the model size

(relative to your computer resources) and the particularapplication.

The Block Lanczos method is recommended for most applications.Efficient extraction of large number of modes (40+) in most modelsTypically used in complex models with mixture of solids/shells/beams

etc.Efficient extraction of modes in a frequency rangeHandles rigid-body modes well

When extracting a small number of modes (<40) in similar size models, thesubspace method can be more suitable.

Requires relatively less memory but large diskspaceMay have convergence problems when rigid body modes are present.Not recommended when constraint equations are present.Generally superseded by Block Lanczos




• For large (100K+ DOF) models and a small number of modes

(< 20), use the PowerDynamics method. It can be significantly faster than Block Lanczos or Subspace, but:

▫ Requires large amount of memory.

▫ May not converge with poorly shaped elements or an ill-conditioned matrix.

▫ May miss modes (No Sturm sequence check)▫ Recommended only as a last resort for large models.

For models in which lumping mass does not create a localoscillation, typically beams and spars, use the Reduced method.

Damping is normally ignored in a modal analysis, but if itseffects are significant, the Damped method is used.




Mode expansion:

You need to expand mode shapes if you want to do any of thefollowing:

Review mode shapes during postprocessing.Have element stresses calculated.Do a subsequent spectrum analysis.

Recommendation: Always expand as many modes as thenumber extracted. The cost of this is minimal.




Sec A-A

• Find the first ten modes for the airplane wing shown in thefigure, suppose that the wing is fixed at one end.

Homework

Dimensions are in cm




Module 3

Modal Analysis Advanced

Topics




A. Learn how to do a prestressed modal analysis

B. Learn how to perform a modal analysis following a largedeflection static analysis


A. Prestressed Modal Analysis



What is prestressed modal analysis and why?

• Modal analysis of a prestressed structure.

• Some structures show different dynamic behavior depending ontheir stress state.

▫ A guitar string or a drum head, for example, will

vibrate at higher frequencies as its tension isincreased.

▫ When a turbine blade spins, its natural frequenciestend to be higher because of the prestress caused

by centrifugal forces.• To properly design such structures, both stress-free andprestressed modal analyses may be required.




The Tacoma Narrows bridge, also known as the Galloping Gertie is famousfor its spectacular collapse in 1940. In this workshop, we will examine a

model of the bridge and calculate its natural frequencies and mode shapes.


Three main steps:



*Issue the command [pstres, on]

Three main steps:1.Build the model

2.Prestress the model with a static analysis3.Do the modal analysis with prestress

1.Build the Model:Same considerations as a normal modal analysis.

Remember to specify density with proper units..Priestess the model with a static analysis 2

*Loading: Apply the loads causing the prestress.*Postprocessing: Review the results to make sure that the proper

loads have been applied.

3.Do the modal analysis with prestressSame procedure as a normal modal analysis except thatprestress effects must be included when specifying analysisoptions.


S t th f ll i l i ti



Set the following analysis options.Solution > Analysis Type > Analysis Options...

accept the default (Block Lanczos)10 modes to extract10 modes to expandCalculate element stressesInclude prestress effects… press OK Accept defaults on the next dialog (Options for Block Lanczos

Modal Analysis)


Problem:



Description:

• Determine the first ten natural frequencies and mode shapes of theperforated aluminum disc shown. The disc is constrained at the centralhole both in the radial and out-of-plane directions. A pre-stress existsdue to a radial pressure load of -20 lbs/inch at the perimeter.Properties of the disc are as follows:

▫ Young’s modulus = 1.0 x 107 psi

▫ Density = 2.3 x 10-4 lbf-sec2/in4

▫ Poisson’s ratio = 0.27




Instructions

1. Clear the database and read input from disc.inp to create the modelgeometry and mesh.

2. Apply displacement constraints: UZ=0 and symmetry b.c. (forradial constraints) at the central hole. Hint: You will need to usetwo menus:

Solution > Define Loads >Apply > Structural > Displacement > On Lines forthe UZ constraint

Solution > Define Loads > Apply > Structural > Displacement > Symmetry B.C. >

On Lines for symmetry b.c.

To pick the lines easily, switch to front view and use Circle picking.




3. Apply the radial load as

pressure on the lines at theperimeter : -20pounds/inch on the outeredges of the disc.

Hint: Stay with the front view,use Circle picking to pick the

entire disc, then use Circleunpicking to unpick all exceptthe outer edges.

4. Activate pre-stress effects(using the Analysis Options

dialog box), obtain a staticsolution, and review results.

5. Switch to modal analysis,

activate pre-stress effects(again), and extract the first10 modes of the pre-stresseddisc using the Block Lanczosmethod.

6. Review the mode shapes.7. If time permits, do a second,

stress-free modal analysis(with pre-stress effects off)and compare results.?


B. Large Deflection Modal Analysis



What is large deflection analysis?





B. Large Deflection Modal Analysis



What is large deflection modal analysis?• Modal analysis of a structure that has undergone significantgeometry change due to loads.

• Some applications include

▫ When pressure or spin load is applied to a

relatively thin turbine blade, it tends to untwist theairfoil affecting natural frequency.

▫ Undersea pipeline installations where contact withseabed causes change in frequencies.

▫ Membranes under pressure load.


What is the difference between large deflection modal



What is the difference between large deflection modalanalysis and prestressed modal analysis?

Prestressed modal assumes that stresses will affect the naturalfrequencies but deflections are small; that is the geometry has notchanged.Large deflection modal assumes that the geometry is changing

significantly due to deformation and that this updated geometry (inaddition to the stress) will change the natural frequencies and modeshapes.

Contact and large deflection


Modal Analysis Workshop

… Plate with a Hole



Description:

Determine the first 10 naturalfrequencies of the plate with a hole

shown. Assume the plate to be

radially constrained at the hole.

The plate is made of aluminum,

with the following properties:

▫ Young’s modulus = 1.4x 105

MPa

▫ Density = 2400kg/cm3


▫ The plate is square of

200mmx200mm

prep by: Yasser ElJajehLarge Deflection Modal Analysis

Procedure



Five main steps:

• Build the model• Run static analysis with large deflection effects turned on

• Update geometry to the deformed geometry

• Perform modal analysis using partial solve procedures

• Review results

Build the Model:

• Same considerations as a normal modal analysis.

• Remember to specify density with proper units.


Large Deflection Modal Analysis

Run Static Analysis



Run Static analysis

• Choose analysis type & options: Be sure to activate both prestress and largedeflection effects.

• Loading: Apply the static

loads.• Solve

Solution > Unabridged MenuSolution > Analysis Type > Analysis

Options


Update Geometry to deformed shape



Update geometry

• Add displacements from static analysis to theoriginal geometry.

• This creates the new geometry on which modalanalysis will be carried out.


Perform modal analysis



• Perform modal analysis using partial solve procedures

Step 1. Analysis type and options

Step 2. Triangularize the matrix ( PSOLVE,TRIANG)

Step 3. Compute eigenvalues ( PSOLVE,EIGLANB)

Step 4. Expand mode shapes ( PSOLVE,EIGEXP)


… Perform Modal Analysis



Step 1. Analysis type and options

▫ Set analysis type to Modal▫ Select mode extraction method ( Block

Lanczos recommended)

▫ Select the number of modes to extract




i l i h i ( )



Step 2. Triangularize the matrix (PSOLVE,TRIANG)

Solution > Solve > Partial Solu




i l ( )



Step 3. Compute eigenvalues (PSOLVE,EIGLANB)


St E d d h





Step 4. Expand mode shapes

• This is done as a separate pass (issue FINISHand get back into Solution)

• Turn expansion key ON



… Perform modal analysis

S E d d h ( i d)



Step 4. Expand mode shapes (…continued)

• Select how many modes to expand



… Perform modal analysis

St E d d h ( ti d)



At this point the user will have a standard modal analysis results file.

Step 4. Expand mode shapes (…continued)

• Perform partial solution to expand modes




Module 4

Harmonic Analysis


A. Define harmonic analysis and its purpose.



. e e a o c a a ys s a d ts pu pose.B. Learn basic terminology and concepts underlying harmonic

analysis.C. Learn how to do a harmonic analysis in ANSYS.D. Work on a harmonic analysis exercise.

prep by: Yasser ElJajehHarmonic Analysis




What is harmonic analysis?

• A technique to determine the steady state response of astructure to sinusoidal (harmonic) loads of known frequency.

• Input:

▫ Harmonic loads (forces, pressures, and imposeddisplacements) of known magnitude and frequency.

▫ May be multiple loads all at the same frequency. Forces anddisplacements can be in-phase or out-of phase. Surface and body loads can only be specified with a phase angle of zero.

• Output:

▫ Harmonic displacements at each DOF, usually out of phase with the applied loads.

▫ Other derived quantities, such as stresses and strains.


Harmonic analysis is used in the design of:Supports fixtures and components of rotating equipment such as



Supports, fixtures, and components of rotating equipment such ascompressors, engines, pumps, and turbo machinery.Structures subjected to vortex shedding (swirling motion of fluids) suchas turbine blades, airplane wings, bridges, and towers.

Why should you do a harmonic analysis?To make sure that a given design can withstand sinusoidal loads at

different frequencies (e.g, an engine running at different speeds).To detect resonant response and avoid it if necessary (by usingdampers, for example).

prep by: Yasser ElJajehHarmonic Analysis - Terminology & Concepts

Equation of Motion



• General equation of motion:

• [F] and {u} are harmonic, with frequency w:

• Equation of motion for harmonic analysis:

FuK uCuM

ti

21

tii

max

ti

21

tii

max

e)uiu(eeuu

e)FiF(eeFF

ww

wwy

)FiF()uiu)(K CiM( 21212 ww


Nature of Harmonic Loads

• Sinusoidally varying at known



• Sinusoidally varying, at knownfrequencies.

• Phase angle y allows multiple,out-of-phase loads to be applied.Defaults to zero.

• All applied loads are assumed to be harmonic, includingtemperatures and gravity.

Real

I m a g i n

a r y

y


Complex Displacements



• Calculated displacements will be complex if:

▫ Damping is specified.▫ Applied load is complex (i.e, imaginary part is non-

zero).• Complex displacements lag by phase angle (with respect to the

applied force).• Results can be viewed in the form of real and imaginary parts or

amplitude and phase angle.


Solution Methods

Three methods of solving the harmonic equation of motion:



Three methods of solving the harmonic equation of motion:

• Full method ▫ Default method, easiest of all.

▫ Uses full structure matrices. Unsymmetric matrices (e.g.acoustics) are allowed.

• Reduced method* ▫ Uses reduced matrices, faster than full method.

▫ Requires master DOF selection, which results inapproximate [M] and [C].

• Mode superposition**

▫ Sums factored mode shapes from a preceding modalanalysis.

▫ Fastest of all methods.

prep by: Yasser ElJajehHarmonic Analysis Workshop

… Fixed-Fixed Beam



Description:

• Determine the harmonic response of a steel beam carrying tworotating machines which exert a maximum force of 170 KN atoperating speeds of 300 to 1800 rpm. The beam, 3 m long, is fully fixed at both ends, and the machines are mounted at its “one-third”points. Assume a damping ratio of 2%.

• Section Properties:• A= 60 cm2

• I zz

= 500 cm4


Four main steps:1.Build the model



2.Choose analysis type and options

3.Apply harmonic loads and solve4.Review results

1. Build The Model:

Linear elements and materials only.

Nonlinearities are ignored.Remember density!

2.Choose analysis type and options:Enter Solution and choose harmonicanalysis.

Main analysis option is solution method -discussed next.Specify damping - discussed next.


Analysis options



Analysis options

• Solution method - full, reduced, or modesuperposition. Defaults to full method.

• DOF printout format - mainly used in batch mode.

• Lumped mass matrix

▫ Recommended if the structure is small in one

dimension compared to the other twodimensions, e.g, slender beams and thin shells.


Damping



Damping

• Choose from alpha damping, beta damping, and damping ratio.

• Damping ratio is most commonly used.


3.Apply harmonic loads and solve



3 pp y All applied loads vary harmonically at the specified frequency

“Loads” consist of: Displacement constraints - zero or non-zero.ForcesPressures

Caution: If gravity and thermal loads are applied, they are also taken

as harmonically varying loads!Specifying harmonic loads requires:

Amplitude and phase angleFrequency Stepped vs. ramped specification

Amplitude and phase angleThe load value (magnitude) represents the amplitude Fmax.Phase angle Y is the phase shift between two or more harmonic loads.Not required if only one load is present. Non-zero Y only valid forforce and displacement harmonic loads.


r y F2

ANSYS does not allow direct input of amplitude and phase angle. Instead,



Real

I m a g i n a r

y

F2max you specify the real and imaginary

components.

For example, given two harmonicforces F1 and F2 that are out of phase by angle Y:

F1real = F1max (amplitude of F1)

F1imag = 0

F2real = F2maxcosY

F2imag = F2maxsinY

You can use APDL for these calculations, but make sure angularunits are set to degrees (default is radians).


•Utility Menu > Parameters > Angular Units …




Frequency of harmonic load:

Specified in cycles per second (Hertz) by a frequency range



• Specified in cycles per second (Hertz) by a frequency range

and number of substeps within that range.• For example, a range of 0-50 Hz with 10 substeps gives

solutions at frequencies of 5, 10, 15, …, 45, and 50 Hz. Samerange with 1 substep gives one solution at 50 Hz.


Stepped versus ramped loads:



pp p

• With multiple substeps, loads can be applied gradually (ramped ) or all at once in the first substep (stepped ).

• Harmonic loads are usually stepped since the load valuerepresents maximum amplitude.


• After applying the harmonic loads, thenext step is to start the solution



next step is to start the solution.

• Typically one load step, but you may usemultiple substeps within a single load step.The frequency range is defined over theentire load step.

4.Review resultsThree steps:

a. Plot displacement vs. frequency at specific points in the structure. b. Identify critical frequencies and

corresponding phase angles.c. Review displacements andstresses over entire structure atthe critical frequencies and phaseangles.

Use POST26, the time-

history postprocessor

Use POST1, the general

postprocessor


Displacement vs. frequency plots



p q y p

• First define POST26 variables.

▫ Tables of nodal or element data.

▫ Identified by a number 2.

▫ Variable 1 contains frequencies and is predefined.


• Define variables (cont'd.)



( )

▫ Pick nodes that might deform the most, thenchoose the DOF direction.

▫ List of defined variables is updated.


•Define variables.

Th h th



•Then graph them.

A Graphed Response in the Frequency Domain


Identify critical frequencies and phase angles

• Graph shows frequency at which highest amplitude occurs



• Graph shows frequency at which highest amplitude occurs.

• Since the displacements are out-of-phase with the applied loads(if damping is present), the phase angle at which the peak amplitude occurs needs to be determined.

▫ To do this, first choose amplitude + phase format ...


▫ … then list the variable(s).

• Notice that peak amplitude = 3 7 occurs at 48 Hz 85 7°



• Notice that peak amplitude = 3.7 occurs at 48 Hz, - 85.7 .

• Next step is to review displacements and stresses over the entiremodel at that frequency and phase angle (using POST1).


Review results over entire structure



• Enter POST1 and list results summary to identify load step and

substep number of critical frequency.


• Use the HRCPLX command to read in results at desired



frequency and phase angle:

▫ HRCPLX, LOADSTEP, SUBSTEP, PHASE, ...

▫ Example: HRCPLX,2,4,-88.2754

• Plot deformed shape, stress contours, and other desired

results.




Module 5

Transient Dynamic Analysis

prep by: Yasser ElJajehTransient Dynamic Analysis



A. Define transient dynamic analysis and its purpose.

B. Learn basic terminology and concepts underlying transientanalysis.

C. Learn how to do a transient analysis in ANSYS.

D. Work on a transient analysis exercise.



prep by: Yasser ElJajehTransient Analysis




Transient dynamic analysis is used in the design of:

• Structures subjected to shock loads, such as automobile doorsand bumpers, building frames, and suspension systems.

• Structures subjected to time-varying loads, such as bridges,earth moving equipment, and other machine components.

• Household and office equipment subjected to “bumps and bruises,” such as cellular phones, laptop computers, and vacuum cleaners.

Topics covered:

• Equation of motion

• Solution methods• Integration time step

prep by: Yasser ElJajehTransient Analysis - Terminology & Concepts

Equation of Motion



• Equation of motion for a transient dynamic analysis is the sameas the general equation of motion.

• This is the most general form of dynamic analysis. Loading may be any arbitrary function of time.

• Depending on the method of solution, ANSYS allows all types of nonlinearities to be included in a transient dynamic analysis -large deformation, contact, plasticity, etc.

t F u K uC u M

prep by: Yasser ElJajehTransient Analysis - Terminology and Concepts

Solution Methods



Solving the equation of motion

Direct Integration Mode Superposition

Implicit Explicit

Full Reduced Full Reduced



Two methods of solving the equation of motion:



Two methods of solving the equation of motion:

• Mode superposition (discussed later)• Direct integration

▫ Equation of motion is directly integrated step by step over time. At each time point ( time = 0, Dt ,

2Dt, 3Dt,….) a set of simultaneous, staticequilibrium equations (F=ma) is solved.• An assumption (integration scheme) is made regarding how

displacement, velocity and acceleration will vary over Dt

• Various integration schemes are available in literature such as

Central difference, Average acceleration, Houbolt, WilsonQ,Newmark etc.


• ANSYS uses Newmark integration scheme

Transient Analysis - Terminology and Concepts




ANSYS uses Newmark integration scheme.

• Varying values of a and d causes integration scheme to change(implicit / explicit / average acceleration ).

• Newmark is an implicit scheme.

• ANSYS/LS-DYNA uses explicit scheme. See module 1 for adiscussion of implicit and explicit.

t t t

ut u

t u

t t u

t t t

ut ut

t u

t u

t t u

t F t t

u K t t

uC t t

u M

DD

D

DD

DD

D

D

D

])1[(

2])2/1[(

d d

a a




Integration Time Step

• An important concept in time integration techniques is the



An important concept in time integration techniques is the

integration time step (also ITS or Dt).▫ ITS = time increment Dt from one time point to the

next.

▫ Determines solution accuracy, so its value should

be chosen carefully.▫ ANSYS allows only a constant value of ITS forreduced and mode superposition transientanalyses.

▫ In a FULL transient analysis, ANSYS canautomatically vary the time step size within limitsset by user (discussed later).


• The integration time step ( ITS) size should be small enough to


… Integration Time Step



The integration time step ( ITS) size should be small enough to

capture the following:▫ the response frequency

▫ the contact frequency (if applicable)

▫ wave propagation effects (if applicable)

▫ Nonlinear response (plasticity, creep, contactstatus)





Response frequency• Different types of loads excitedifferent natural frequencies of the structure.

• Response frequency is the weighted average of all

frequencies excited by a givenload.

• The ITS should be small enoughto capture the responsefrequency .

• Twenty time points per cycleshould be sufficient, i.e,

Dt = 1/20f where f is the responsefrequency.

Response period


• Response frequency (continued)





▫ During solution, the full transient method(discussed in this seminar) prints the responsefrequency and the number of points per cycle atevery time point.

▫ The goal is to maintain about 20 points per cycle.

▫ By default, ANSYS automatically increases ordecreases ITS to maintain about 20 points per

cycle at the response frequency.



Contact frequency

• When two objects come in contact the



• When two objects come in contact, the

gap or contact surface is usually represented by a stiffness (gapstiffness).

• The ITS should be small enough tocapture the frequency of the gap

“spring.” • Thirty points per cycle are

recommended. This is sufficient tocapture the momentum transfer between the two objects. A larger ITS

might result in energy loss, and theimpact may not be perfectly elastic.

• The response frequency printed duringsolution includes contact frequency.

masseffectivem

stiffnessgapk

frequencycontactf

m

k

2

1f

f 30

1ITS

c

c

c



Wave propagation



p p g

• Caused by impact. Moreprominent in slenderstructures (such as a thinrod dropped on one end).

• Requires a very small ITS

and a fine mesh along thedirection of the wave.

• Explicit method (available in ANSYS-LS/DYNA) may be better suited for this.

densitymass

modulussYoung'

speedwaveelastic

directionwavealonglength

20/sizeelement

3

D

D

E

E c

L

L x

c

x ITS


Nonlinear response

• A full transient analysis can include any type of nonlinearity.





y y yp y

• Nonlinearities can be classified into 3 types:▫ Material nonlinearity (plasticity , creep, hyper-elasticity …)

▫ Geometric nonlinearity (large strain , large rotation, buckling)

▫ Element nonlinearity (contact , cable)

• Nonlinearities require an iterative solution at each time point.• These iterations are called equilibrium iterations.

Smaller ITS sizes generally help equilibrium iterations to convergequickly.Nonlinearities such as plasticity, creep and friction are non-conservative

in nature and require the load history to be followed accurately. A smallITS size helps in following the load history accurately.

A small ITS size is also required to capture changes in contact status.


Nonlinear response (continued)





At time = t At time = t+Dt

Example of ball hitting a plate• With large Dt the ball goes through the plate.

• If ball goes through too far then contact will not be detected(beyond pinball radius).


• So how do you choose an ITS?





y

• Recommended way is to activate automatic time stepping(AUTOTS), then provide Dtinitial , Dtmin , and Dtmax. ANSYS usesan automatic time stepping algorithm (AUTOTS) to determinethe optimum Dt value for a given problem.

• Example: If AUTOTS is on with Dtinitial= 1 sec, Dtmin= 0.01 sec,

and Dtmax= 10 sec; then ANSYS starts with an ITS= 1 sec andallow it to vary between 0.01 and 10 depending on thestructure’s response.


• AUTOTS is on by default for full transient analyses and is not





available for reduced and mode superposition methods.• AUTOTS will reduce the ITS (up to Dtmin) if:

▫ less than 20 points are being used at the responsefrequency

▫ solution is diverging

▫ solution takes a large number of equilibrium equations(slow convergence)

▫ plastic strain is accumulated in one time step exceeds15%

▫ Creep ratio exceeds 0.1▫ if contact status is about to change ( controlled by

KEYOPT(7) of most contact elements)


Description of Example1:

6 6 i h bl k i d d

Transient Analysis Workshop

… Bouncing Block



• A 6x6x1-inch block is dropped on a 100-

inch long beam from a height of 100

inches. Obtain a graph of the motion of

the point of beam under the block as it

bounces on the beam. Assume a gap

stiffness of 2000 lb/in. The beam is

fully fixed at both ends, and the only load is gravity, 386 in/sec2. The beam

and the block are made of the same

material:

▫ Young’s modulus = 1,000,000 psi

▫ Density = 0.001 lbf-sec2/in4


prep by: Yasser ElJajehTransient Analysis

C. Procedure

• We will discuss the Full method only in this section.

• Five main steps:



• Five main steps:

▫ Build the model

▫ Choose analysis type and options

▫ Specify BC’s and initial conditions

▫ Solve for each load step

▫ Review results

1. Build the Model All nonlinearities are allowed.

Remember density!


2.Choose analysis type andoptions:

•Enter Solution and choose



•Enter Solution and choose

transient analysis.•Choose Full transient

•Solution options - discussed next.

•Damping - discussed next.


Solution options

• Choose large displacementtransient or small



displacement transient .▫ When in doubt, choose

large displacementtransient

Specify time at end of load step.

Automatic time stepping

(discussed next)

Specify initial, min and max

values of Dt for this load

step.


Automatic time stepping

• An algorithm that automatically calculates appropriate ITS sizes



• An algorithm that automatically calculates appropriate ITS sizes

during the transient.

• Recommendation is to activate it and also specify minimum andmaximum values of ITS.

• If nonlinearities are present, use the “Program Chosen” option.

Note: The global solution controls switch [SOLCONTROL] isON by default. We recommend leaving it as is. Moreimportantly, do not turn this switch on and off between load

steps.


Output controls

• Used to determine what is written to the results file.



• Use the OUTRES command or choose Solution > Sol’n Control.. > Basic in the menu

• Typical choice is to write all items at every substep to theresults file.

▫ Allows smooth plots of results vs. time.

▫ Might cause results file to be large.


Turn transient effects on/off

▫ useful for setting up



▫ useful for setting upinitial conditions(discussed later)

• Ramp or Step apply load

• Specify damping (discussed

next)• Use default values for time

integration parameters


DampingBoth alpha damping and beta damping are available.



In many cases, alpha damping (viscous damping) is ignored andonly beta damping (damping due to hysteresis) is specified:

b = 2x/w

where x is the damping ratio and w is the dominantresponse frequency (rad/sec).

Material damping (e.g. rubber) and element damping (e.g. shock absorber) are also available.


• Choose solver

▫ By default ANSYS chooses Sparse solver



▫ By default ANSYS chooses Sparse solver

▫ For large problems (>100000 dofs) use PCG solver


3.Specify BC’s and initial conditions (4. solve for load steps)BC’s in this case are loads or conditions that remain constant



throughout the transient, e.g:Fixed points (constraints)Symmetry conditionsGravity

Initial conditions

Transient analyses require initial displacement (u0) and initial velocity(v 0) to be specified.By default, u0= v 0 = a0 = 0.Examples where non-zero initial conditions may be required:

Aircraft landing gear (v 00).

A golf club striking a ball (v 00).Drop test of an object (u0= v 0 =0 , a00).


Two ways to apply initial conditions:Start with a static load step

Useful when initial conditions need to be applied on only a portion



of the model, such as “plucking” the end of a cantilever beam withan imposed displacement (u0 is known , v 0 =0)Required for applying a non-zero initial acceleration.

Use the IC commandSolution > Define Loads > Apply > Initial Condit’n > Define

Useful when a non-zero initial displacement or velocity needs to beapplied on the entire body.




5.Review Results

• Consists of three steps:



p

▫ Plot results vs. time at specificpoints in the structure.

▫ Identify critical time points.

▫ Review results over entirestructure at those time points.



Use POST1, the

general postprocessor


To plot results vs. time:

• First define POST26 variables in the Variable Viewer.



▫ Tables of nodal or element data.▫ Identified by a number 2.

▫ Variable 1 contains time-points and is predefined.


• Define variables (cont'd)

▫ Pick nodes that might deform the most, then choose the



DOF direction.▫ List of defined variables is updated.

Once the variables are defined, you cangraph them or list them.




Review results over entire structure at critical time points

• Enter POST1, read results “By Time/Freq...”, and enter



, y / q ,

appropriate time value.• Plot deformed shape and stress contours.

prep by: Yasser ElJajehTransient Analysis Workshop

… Functional Load


• A steel frame is hit by an impulsive load



A steel frame is hit by an impulsive load

at the top of it, the load is given by the

following function:

• F= 70000xt2 (sin62.83t-30)

• Study the response of the frame in the

range of time between 0 and 3 sec.

• Find out the extremes and display the

deformed shape.

▫ Young’s modulus = 2.1 x 1011 GPa

▫ Density = 7800 kg/m3



1. Build the Model All nonlinearities are allowed.Remember density!

Procedure



2.Choose analysis type and options:• Enter Solution and choose transient analysis.

• Choose Full transient

• Choose large displacement transient or small displacement transient. When in doubt, choose large displacement transient


Specify initial, min and max values of Dt for this load step.

3.Specify BC’s BC’s in this case are loads or conditions that remain constant

throughout the transient, e.g:Fixed points (constraints)


4.Apply time-history loads and solveTime-history loads are loads that vary with time.



In this example loads will be applied using theFunction tool

Load

t

Load

t

Load

t


Function Tool

• Allows you to apply complicated boundary conditions. Toaccess the function editor, choose Solution > Define Loads >Apply > Functions > Define/Edit



Apply Functions Define/Edit

• Recommendation: do not use the Function Tool if the boundary conditions can be expressed directly with tabularinput

For more informationrefer to “ApplyingLoads Using FunctionBoundary Conditions”in the Basic AnalysisGuide.


5.Review Results




▫ Plot results vs. time at specificpoints in the structure.


▫ Review results over entirestructure at those time points.



Use POST1, the



… Tabular Load




• A steel frame is hit by an impulsive loadat the top of it, the load is given by the

following table:



• Find out the extremes and display thedeformed shape.




Load (ton)Time (sec)

00

400.25200.4

00.6



Procedure











4.Apply time-history loads and solveTime-history loads are loads that vary with time.In this example loads will be applied using the



tabular input

Tabular input Allows you to define a table of load vs. time(using array parameters) and apply the table as

a load. Very convenient, especially if there are severaldifferent loads, each with its own time history.For example, to apply the force-vs-time curveshown:

1. Choose Solution > Define Loads >

Apply > Structural > Force/Moment >On Nodes, then pick desired nodes.


2. Choose the force directionand “New table”, then OK.

3. Enter table name and no. of rows (no. of time points),



then OK.

4. Fill in time and load values,then File > Apply/Quit.


5. Specify ending time and integration time step.Solution > Load Step Opts > Time/Frequenc > Time - Time Step

There is no need to specify the stepped or ramped condition. It



is implied by the load curve.6. Activate automatic time stepping, specify output controls, andsolve (discussed later.)

5.Review Results


▫ Plot results vs. time at specific points inthe structure.


▫ Review results over entire structure at

those time points.



Use POST1, the



… Multi-Load step method


• A steel frame is hit by an impulsive load




at the top of it, the load is given by the

following function:



• Find out the extremes and display the

deformed shape.



▫ Poisson’s ratio = 0.3 Force

t

22.5

10

0.5 1.0 1.5



Procedure




2. Define load step 1: Apply force = 22.5 units at the desired nodes.Specify the ending time (0.5), integration time step, and



ramped loads. Activate automatic time stepping, specify output controls,and either solve or write the load step to a load step file.

3. Define load step 2:Change force values to 10.0.

Specify the ending time (1.0). No need to re specify theintegration time step or ramped condition.Solve or write the load step to a load step file.

4. Define load step 3:Delete the forces or set their values to zero.Specify the ending time (1.5) and stepped loads.Solve or write the load step to a load step file.


5.Solution

• Use SOLVE command (or LSSOLVE if load step files were written).



• At each time step, ANSYS calculates load values based on the load-vs-time curve.

5.Review Results


▫ Plot results vs. time at specific points inthe structure.


▫ Review results over entire structure at

those time points.



Use POST1, the





Spectrum Analysis

Module 6


A. Define a spectrum analysis and itspurpose.



B. Understand the underlying concepts andterminology.C. Learn how to do a response spectrum

analysis.D. Guidelines for spectrum analysis.

E. Random Vibration Analysis

prep by: Yasser ElJajehSpectrum Analysis


What is spectrum analysis?

• A technique to compute a structure’s



response to transient excitationsthat contain many frequencies.

• Excitations could be from sourcessuch as earthquakes, aircraft noise/

flight history, missile launches.• A spectrum is a representation of a

load’s time history in the frequency domain.

• This is also referred to as responsespectrum.

prep by: Yasser ElJajehSpectrum Analysis


)1940El Centro Earthquake (



Acceleration vs. time Acceleration spectrum (G vs. Hz)

A structure subject to the El Centro earthquake can be analyzed

using either a Transient analysis or spectrum analysis.


• Spectrum analysis follows a modal analysis.

C h i f h

Spectrum Analysis




• Computes the maximum response of the structureto a given spectrum at each natural frequency. Thismaximum response is computed as scalefactor*mode shape.

• These maximum responses are then combined togive a total response of the structure.


• An alternative is to perform a transient analysis.

• Transient analysis is generally more time consuming,

i ll h b f d l d

Spectrum Analysis




especially when a number of components and loadconditions have to be considered.

• However, transient analysis is more accurate.

• In spectrum analysis the focus is to get the maximum

response quickly, and some information is lost (phase).


• Used in the design of:

N l l (b ildi d )

Spectrum Analysis




▫ Nuclear power plants (buildings and components)▫ Airborne Electronic equipment (aircraft / missile)

▫ Spacecraft components

▫ Aircraft components

▫ Any structure or component that is subjected toseismic or other erratic loads

▫ Building frames and bridges

prep by: Yasser ElJajehSpectrum Analysis… Definition & Purpose

• ANSYS allows four types of spectrum analysis:

• Single-point response spectrum**

A i l t it ll ifi d i t i th



** Covered in this seminar

▫ A single response spectrum excites all specified points in themodel.

• Multi-point response spectrum **

▫ Different response spectra excite different points in themodel.

• Dynamic design analysis method (DDAM)

▫ A specific type of spectrum defined by the U.S. NavalResearch Laboratory to evaluate shock resistance of shipboard equipment.

• Power Spectral Density (PSD)**

▫ A probabilistic approach used in random vibration analysis.


Topics covered:

*Definition of a spectrum*How a response spectrum is used to *calculate a

* t t ’ t th it ti



*structure’s response to the excitation 1.Participation factor2.Mode coefficient3.Mode combination

prep by: Yasser ElJajehSpectrum Analysis - Terminology & Concepts

Definition of spectrum

What is a spectrum?

• A curve representing the maximum response of an idealized

t t it ti Th b l ti



system to an excitation. The response may be acceleration, velocity, displacement, or force.

• Consider, for example, four single-DOF spring-mass systemsmounted on a shaker table. Their frequencies are f1, f2, f3, andf4, with f1 < f2 < f3 < f4.

1 2 3 4


• If the shaker table is excited atfrequency f1 and the displacement

f th f t i

u

Spectrum Analysis - Terminology & Concepts

… Definition of spectrum



response of the four systems isrecorded, it will look as shown on theright.

• Now add a second excitation of frequency f3 and record the

displacement response. Systems 1and 3 will each reach their peak response.

• If now a general excitationcontaining several frequencies is

applied and only the peak responsesare recorded, we might get the curveshown. This curve is the spectrum,specifically a response spectrum.

f

f

u

f

u




• For each mode of the structure, a participation factor gi iscalculated in the excitation direction.

The participation factor is a function of the mode shape and the


… Approach - Participation factor



• The participation factor is a function of the mode shape and thedirection of excitation.

• This is a measure of how much a mode will contribute to thedeflections (and hence stresses) in the direction of excitation.


• For example, consider the cantilever beam shown.

• If an excitation is applied in Y direction, mode 1 will have the

highest PF and mode 2 a lower PF Mode 3 will have zero PF


… Approach - Participation factor



highest PF and mode 2 a lower PF. Mode 3 will have zero PF.• If the excitation is in the X direction, then modes 1 and 2 will

have zero PF, whereas mode 3 will have a high PF.

mode 3

mode 2

mode 1

Y

X


• The mode coefficient is the “scale factor” used to multiply themode shapes to get the maximum response.

• The mode coefficient Ai for each mode is Ai Sigi *


… Approach - Mode coefficient



• The mode coefficient A i for each mode is A i = Sigi *Si is the response spectrum value at frequency wi

gi is the participation factor for mode i• The maximum modal response is then computed as

{U}i max = A i {i

*A different formula is used for acceleration, velocity and forcespectra; see the ANSYS Theory Manual .


• Once the maximum response at each mode is known for a givenresponse spectrum, these need to be combined in some way to

get the total response


… Approach - Mode combination



get the total response.• The simplest combination is to add all the maximum modal

responses. However, it is highly unlikely that all the maximummodal responses will occur at the same time.

• Several standard combination methods are published in the

literature. Usually each industry’s regulating authority recommends or enforces a technique most suitable for thatindustry.


• Six different combination methods are available in the ANSYSprogram:

▫ Complete Quadratic Combination (CQC) method


… Approach - Mode combination



▫ Complete Quadratic Combination (CQC) method▫ Grouping Method (GRP)

▫ Double Sum method (DSUM)

▫ Square Root of the Sum of the Squares (SRSS) method

▫ Naval Research Laboratory (NRL) sum method(DDAM)

▫ Power Spectral Density method


• We will discuss the procedure for a single-point responsespectrum analysis.

• In the following discussion we will use the term “response

Spectrum Analysis

… Terminology & Concepts



• In the following discussion, we will use the term responsespectrum” to mean single-point response spectrum.

• To learn about multi-point response spectrum and DDAM,please refer to the ANSYS Structural Analysis Guide.


Description:

Determine the displacements

and stresses in a workbench

Response Spectrum Workshop

… Workbench Table



and stresses in a workbenchtable due to the accelerationspectrum shown below.

A c c e l e r a t i o n

Frequency

20 80 200 300

217 217

79.5

150.2

prep by: Yasser ElJajehC. Procedure

Five main steps:

• 1.Build the model 2.Obtain the modal solution

• 3.Switch to spectrum analysis type

• 4 Define the response spectrum 5 Solve and review results



• 4.Define the response spectrum 5.Solve and review results

… Obtain the Modal Solution• Mode extraction:

▫ Only valid methods are Block Lanczos, subspace, or reduced.

▫ Block Lanczos strongly recommended

▫ Extract enough modes to cover the spectrum’s frequency content.

▫ Expand all modes. Only expanded modes can be used for thespectrum solution.

• Loads and BC’s: For a base excitation, be sure to constrain theappropriate DOFs.

• Files: The .mode file contains the eigenvectors and is neededfor the spectrum solution.



prep by: Yasser ElJajehResponse Spectrum Procedure

… Switch to Spectrum Analysis TypeDamping

• Available forms of damping are:

▫ Beta (stiffness) damping

Constant damping ratio Can



▫ Constant damping ratio. Can be material dependent butonly if specified as a materialproperty* in the modal step.

▫ Frequency dependentdamping ratio (modal damping)

• Some form of damping must bespecified for the CQC modecombination method.

*Material property DAMP in thiscase is damping ratio, not betadamping.


Define the Response Spectrum

Build the model

Obtain the modal solution

Switch to spectrum analysis type




Define the response spectrum

• Settings: type of spectrum and excitation direction

• Table of spectral value versus frequency • Mode combination method


… Define the Response SpectrumSettings:

• Type of spectrum

▫ Seismic or force (not PSD)S i i



Seismic or force (not PSD)▫ Seismic spectra -

automatically applied at the base

▫ Force spectrum - manually

applied at desired nodes as aforce

• Excitation direction (globalCartesian)

▫ Specified by a unit vector for

seismic spectra: 1,0,0 meansX; 0,1,0 means Y; 0,0,1means Z.

▫ Implied by FX, FY, or FZlabels for force spectrum.


… Define the Response Spectrum

Spectral value vs frequency table

• First define frequency table. Up to

20 points are allowed.



20 points are allowed.• Then define corresponding spectral

values.

▫ Specify damping ratio only formultiple spectral curves.

▫ For a force spectrum, thespectral values can be scaled by the applied force value.


Mode combination method

• Determines how the individual modal

responses are combined.

Response Spectrum Procedure




responses are combined.• Five methods are available:

▫ CQC (Complete QuadraticCombination)

▫ GRP (Grouping)

▫ DSUM (Double Sum)

▫ SRSS (Square Root of Sum of Squares)

▫ NRLSUM (Naval Research Laboratory Sum)

Which method you choose typically depends on company or governmentstandards being followed.


Mode combinations (continued)

• The significance threshold allows you to include only significant

modes in the mode combination. It is the ratio of the mode





modes in the mode combination. It is the ratio of the modecoefficient of a mode to the maximum mode coefficient. Use azero value to include all modes.

• Type of output allows calculation of different responsequantities: displacement, velocity, or acceleration.


Solve and Review Results

Build the model





Switch to spectrum analysis typeDefine the response spectrum

Solve and review results

• Solve the current load step.

• Mode combination calculations are written asPOST1 commands to the .mcom file.

• Review results: discussed next.



… Solve and Review Results

Review results:

• Enter POST1 (general postprocessor).

• Perform mode combinations



Perform mode combinations▫ Commands to do this are written to .mcom file during

solution.

▫ Read the file jobname.mcom using Utility Menu > File >

Read Input from...

• Review deformed shape.

• Plot and list stresses and strains.

prep by: Yasser ElJajehD. Spectrum Analysis Guidelines• Modal analysis

▫ Make sure you extract and expand enough modes in themodal analysis to cover the frequency range of interest.

▫ For example, if the spectrum extends from 1 to 1000 Hz, a



For example, if the spectrum extends from 1 to 1000 Hz, arule of thumb is to extract and expand modes up to 1500 Hz.

▫ Block Lanczos extraction technique recommended

▫ If you have material dependent damping ratio, this should bespecified in the modal analysis.

Spectrum analysisRemember that no results file is written in a spectrum analysis.Instead the instructions for mode combination are written to jobname.mcom.Most combination methods involve squaring operations causing

the component stresses to lose their signs. Hence derivingequivalent or principal stresses from these unsigned components will be non-conservative and incorrect.


If equivalent or principal stresses and strains are of interest thenyou need to issue the command SUMTYPE,PRIN ( General

Spectrum analysis

D. Spectrum Analysis Guidelines



you need to issue the command SUMTYPE,PRIN ( General

Postprocessor > Load Case > Calc Options > Stress Options) beforereading in jobname.mcom. This causes direct operation onderived quantities leading to more conservative results.

During the spectrum analysis the effective mass for each modeas well as the sum of all the effective mass is printed out.For a lumped mass system the sum of the effective massesshould approach the total mass of the structure as the number of modes used in the spectrum analysis is increased.The total effective mass is an indicator of whether enough modes

are included in the spectrum analysis.


***** RESPONSE SPECTRUM CALCULATION SUMMARY

CUMULATIVE

MODE FREQUENCY SV PARTIC.FACTOR MODE COEF. M.C. RATIO EFFECTIVE MASS MASS FRACTION

D. Spectrum Analysis Guidelines



1 2.37E-04 10 -1.18E-20 -5.34E-14 0 1.40E-40 3.07E-38

2 474 21.099 6.22E-02 1.48E-07 1 3.87E-03 0.85132

3 1182 10 1.14E-15 2.07E-22 0 1.30E-30 0.85132

4 1182 10 3.42E-16 6.20E-23 0 1.17E-31 0.85132

5 1881 10 -5.08E-16 -3.64E-23 0 2.58E-31 0.851326 2361 10 3.52E-11 1.60E-18 0 1.24E-21 0.85132

7 2361 10 -2.60E-02 -1.18E-09 0.007981 6.76E-04 1

8 3044 10 -4.39E-13 -1.20E-20 0 1.93E-25 1

9 3044 10 1.27E-12 3.48E-20 0 1.62E-24 1

10 4011 10 5.08E-12 8.00E-20 0 2.58E-23 1

SUM OF EFFECTIVE MASSES 4.55E-03




Random Vibration Analysis

Module 7


F. Random Vibration Analysis

Topics covered:

• Definition and purpose

• Overview of ANSYS capabilities



p• ANSYS procedure

prep by: Yasser ElJajehRandom Vibration Analysis

Definition and Purpose

What is random vibration analysis?

▫ A spectrum analysis technique based on probability and

statistics.f l d h l i l d i k l h



▫ Meant for loads such as acceleration loads in a rocket launchthat produce different time histories during every launch .


• Transient analysis is not an option since the time history is notdeterministic.

• Instead, using statistics the sample time histories are convertedto Power Spectral Density function (PSD) a statistical


… Definition and Purpose



, g pto Power Spectral Density function (PSD), a statisticalrepresentation of the load time history.

Image from “Random Vibrations Theory and Practice” by Wirsching, Paez and Ortiz.




Input:

▫ The structure’s natural frequencies and mode

shapes


… Definition and Purpose



p▫ The PSD curve (explained next)

Output:

▫ 1s displacements and stresses that can be used for

fatigue life prediction.▫ Response PSD curves that show the frequency

content of any output quantity ( RPSD ).

▫ Undocumented (FPAS and RISK ) life prediction

capability.


• Loading:

▫ Base or nodal excitation

▫ Single-point excitation


Overview of ANSYS Capabilities



Single point excitation

e.g. Single PSD excitation applied to all ground nodes

▫ Multi-point (i.e., multi-spectra) excitation

Uncorrelated Partially correlated

Fully correlated

▫ Partial correlation in terms of spatial coordinates

▫ Partial correlation in terms of a traveling wave


• Solution:

▫ Relative or absolute 1s output

▫ Option for calculating 1s forces/stresses etc.


… Overview of ANSYS Capabilities



p g /▫ Solution for complete structure i.e., results can be

contoured.

▫ Output in form of 1s displacements, velocities or

accelerations


• Post processing:

▫ 1s results can be contoured like any other analysis.

▫ Response PSD can be computed for any result quantity ( e.g.stress or nodal force at a node of an element) or cross


… Overview of ANSYS Capabilities



stress or nodal force at a node of an element) or crossresponse spectra can be computed between any twoquantities ( RPSD).

This enables the user to look at the frequency content of

output.▫ Covariance between any two quantities can be computed

(CVAR).

▫ Undocumented commands RISK and FPAS allow user tocompute equivalent stress / predict life.


Random Vibrations Workshop

… Model Airplane Wing

Description:

Determine the displacements and stresses of the model airplane

wing due to an acceleration PSD applied to the base of the wingin Y direction Assume the wing to be fully fixed at Z=0



in Y direction. Assume the wing to be fully fixed at Z=0.

A c c e l e r a t i o n ( G 2 / H

z )

Frequency (Hz)

20 100 400 600

0.1 0.1

0.025

0.075

prep by: Yasser ElJajehRandom Vibrations Procedure

Six main steps:

• Build the model

• Obtain the modal solution



• Switch to spectrum analysis type

• Define and apply the PSD excitation

• Solve

• Review results

prep by: Yasser ElJajehRandom Vibrations

Build the Model

Model

• Same considerations as a modal analysis.

• Linear elements and materials only. Nonlinearities arei d



ignored.

• Remember density! Also, if material-dependent dampingis present, it must be defined in this step.


Obtain the Modal Solution


• Same procedure as a normal modal

analysis.• A few differences discussed next



• A few differences, discussed next.

Mode extraction:Only valid methods are Block Lanczos, subspace, or reduced.

Block Lanczos strongly recommendedExtract enough modes to cover thespectrum’s frequency content.Expand all modes. Only expanded

modes can be used for thespectrum solution.


• Loads and BC’s:

▫ For a base excitation, be sure to constrain the

appropriate DOFs.F PSD l th d i d

Random Vibrations

… Obtain the Modal Solution



▫ For a pressure PSD, apply the pressures on desiredsurfaces in this step.

• Files: The .mode file contains the eigenvectors and is needed

for the spectrum solution.

prep by: Yasser ElJajehRandom Vibrations Switch to Spectrum Analysis Type


• Exit and re-enter Solution

• New analysis: SpectrumA l i ti Di d t



• Analysis options: Discussed next

• Damping: Discussed next




Damping

• All four forms are available.

▫ Alpha (mass) damping▫ Beta (stiffness) damping

Random Vibrations

… Switch to Spectrum Analysis Type



▫ Beta (stiffness) damping

▫ Constant damping ratio

▫ Frequency dependent dampingratio (modal damping)

• If no damping is specified, ANSYSuses a 1% constant damping ratio asdefault.


Define and Apply the PSD Excitation

Define and apply the PSD excitation

• Specify PSD settings

• Define PSD versus frequency table• Apply excitation at desired nodes



• Apply excitation at desired nodes

PSD settingsSpectrum type (units)

Acceleration (normal units org2/Hz) Velocity DisplacementForce

Pressure

Table number defaults to 1. Used formultiple PSD curves


PSD versus frequency table

• Specify table number (usually 1).

• Then enter frequency and PSD value pairs.

Random Vibrations

… Define and Apply the PSD Excitation




PSD versus frequency table (continued)

• Graph the PSD table to verify the input.

Random Vibrations





Apply the PSD

• Procedure depends on the type

of PSD.• Acceleration velocity or

Random Vibrations




Acceleration, velocity, ordisplacement PSD:

▫ These are base excitations andcan be applied only at

previously constrained nodes.▫ Apply as a constraint in UX,

UY, or UZ (excitationdirection) with a value of 1.0.

Pick nodes...




Solve

• Solve

• Activate PSD mode combinationmethod



• Specify items to be calculated*

• Calculate participation factors*

• Initiate PSD solution*

*Discussed next




… Solve

Calculate participation factors:

• Must be done for each PSD table defined.

• Specify base or nodal excitation.Initiate PSD solution:



Initiate PSD solution:

• Results are written to the .rst file.


Review Results

Review results

• Plot and list 1s quantities (POST1)

• Generate a response PSD (POST26)Life prediction



• Life prediction


Random Vibrations- Review Results

Review 1-Sigma Stresses

• Random vibration results are 1s quantities: 1s displacements, 1s

stresses, etc.• All quantities assume a Gaussian

Gaussian

(normal)Distribution



• All quantities assume a Gaussian(normal) distribution with zeromean. For example, a maximumdisplacement of Umax = 0.15

indicates a 68% probability (1s)that Umax will be 0.15 or less. Italso indicates:

▫ a 95% probability (2s) that Umax will be 0.15x2 = 0.3 or less.

▫ a 98% probability (3s) that Umax will be 0.15x3 = 0.45 or less.

1s

2s

3s


To review 1s displacements & stresses:

• Enter POST1 (General Postproc).

• Read results from load step 3, which is where 1s results arestored on the results file.

Random Vibrations- Review Results… Review 1-Sigma Stresses



▫ Note: 1s velocities and 1s accelerations, if requested, arestored in load steps 4 and 5, respectively.

• Then plot and list the desired quantities.

prep by: Yasser ElJajehRandom Vibrations- Review Results





1s results are typically used for:

• Fatigue calculations

▫ In PSD analyses, the average frequency of excitation(number of cycles/second) is given by 1s velocity / 1s





( y / ) g y y /displacement.

▫ Using normal distribution the stress level is at 1s 68%of the time, at 2s 27% of the time (95-68), and at 3s

3% of the time (98-95).▫ Knowing the above two quantities, fatigue life can be

predicted using usual S-N diagram procedures.


Response PSD

• Gives engineers an idea of how a response quantity

(stress, for example) varies with frequency.• Results file contains 1s values which is the square root of


Response PSD



Results file contains 1s values, which is the square root of the area under the PSD curve.

• POST26, the time-history postprocessor, is used tocalculate response PSD.


To calculate response PSD

1. Enter POST26 and first store the frequency vector.

▫ You can use 1 to 10 additional data points on either side of anatural frequency for a smoother frequency curve. Default is5

Random Vibrations- Review Results… Response PSD



5.

▫ Variable 1 is automatically assigned to the frequency vector.


2. Identify results quantities for which response PSD is to becalculated.

▫ TimeHist Postpro > Variable Viewer ▫ Can be any nodal or element result item.


… Response PSD



Can be any nodal or element result item.

Choose category,

then pick node...


3. Calculate and plot the response PSD.

▫ TimeHist Postpro > Calc Resp PSD...

▫ TimeHist Postpro > Graph Variables…


… Response PSD




M d l 8



Mode Superposition

Module 8

prep by: Yasser ElJajehModule 8

Mode Superposition

A. Define mode superposition.

B. Learn how to use the mode superposition method.

C. Work on a mode superposition exercise.Mode Superposition



Mode Superposition


• A solution technique for transient or harmonic analyses. It

sums factored mode shapes from a modal analysis to calculatethe dynamic response.

• A fast, efficient method that can be used for linear dynamicsproblems.

• The alternative is to use the direct integration method which

can be time consuming. The two methods are compared next.


)t(f ][}y]{[K ][}y]{[C][}y]{[M][TTTT

The general equation of motion can be pre-multiplied by [T and written as:

Mode Superposition… Definition & Purpose



Orthogonality of natural modes means:

2][][

1][][

JJ

T

J

J

T

J

w

K

M

If proportional damping is specified, then:

JJJTJ wx 2][C][

prep by: Yasser ElJajehMode Superposition


Mode Superposition

+ Fast solution regardless of

whether equations of motionare uncoupled (proportionaldamping only ) or coupled

Direct Integration

– Fully coupled equation of

motion. Solution can be timeconsuming.

+ Effective for most problems



damping only ) or coupled(non-proportional damping).

+ Effective when only a few modes are needed to describe

response.± Requires eigenvectors from a

modal solution.

– Linear only, no nonlinearities.

– Deciding how many modes touse may be difficult. Too few

modes may give gooddisplacements but poor stresses

+ Effective for most problems.

± No eigenvectors required.However, most dynamic

analyses begin with a modalsolution.

+ Nonlinearities allowed intransient analysis.

+ Easier to determine Dt, theintegration time step, than

number of modes.

prep by: Yasser ElJajehMode SuperpositionB. Procedure

Five main steps:

• Build the model

• Obtain the modal solution• Switch to harmonic or transient analysis



• Apply loads and solve

• Review results

Build the Model

Model

• Same considerations as a modal analysis.

• Linear elements and materials only. Nonlinearities are ignored.

• Remember density! Also, if material-dependent damping ispresent, it must be defined in this step.

• See also Modeling Considerations in Module 1.


Obtain the Modal Solution


• Same procedure as a normal modal analysis.

• A few differences, discussed next.

Mode extraction:



Mode extraction:Only valid methods are Block Lanczos, subspace, reduced,powerdynamics, or QR damped.Extract all modes that may contribute to the dynamic response.

Mode expansion is needed to view mode shapes but not requiredfor the mode superposition solution.If QR damped mode extraction method is used, the damping must be specified during preprocessing or in the modal analysis.Damping specified during the mode superposition transient or

harmonic analysis will be ignored.


Mode Superposition

… Obtain the Modal Solution

• Loads and BC’s:

▫ All displacement constraints must be applied in this step.Zero valued only; non-zero displacements are not allowed.

▫ If element loads (pressures, temperatures, and accelerations)are to be applied in the harmonic or transient analysis they



are to be applied in the harmonic or transient analysis, they must be specified in this step.

The solver

ignores theloads for themodal solution, but writes aload vector to

the .mode file.


Switch to Harmonic or Transient Analysis

Switch to harmonic or transientanalysis

• Exit and re-enter Solution• New analysis: Harmonic orT i t



Transient

• Analysis options: Discussed next

• Damping: Discussed next


Mode Superposition

… Switch to Harmonic or Transient

Damping

• Specified here if QR Damped modeextraction method not used.

• Damping in some form should bespecified in most cases



specified in most cases.

• All four forms are available for modesuperposition:

▫ Alpha (mass) damping▫ Beta (stiffness) damping

both global and materialdependent

▫ Constant damping ratio

▫ Frequency dependent damping ratio(modal damping)

prep by: Yasser ElJajehMode SuperpositionApply Loads and Solve

Apply loads and solve

• Only forces, no non-zero displacements.

• Load vector from modal analysis (discussed next).• Conditions for initial static solution in a transient



analysis (discussed next).

• Integration time step is constant throughout transient.

• Start solution calculations (SOLVE).

prep by: Yasser ElJajehMode Superposition… Apply Loads and Solve

Load vector

• Gives a way to apply element

loads (pressures, accelerations,and temperatures) in a modesuperposition analysis



superposition analysis.

• Calculated during the modalsolution based on loads specifiedin the modal analysis.

• Can be applied with a scale factor(which defaults to 1.0).

prep by: Yasser ElJajehMode Superposition… Apply Loads and Solve

Initial static solution in a transient analysis

• The initial solution (at time = 0) in a mode superposition

transient analysis is always a static solution (using the frontalsolver).

• Can take a long time and much disk space for large models



• Can take a long time and much disk space for large models.

• To avoid it (and get {U}t=0 = {0}), do not apply any loads at time= 0.

SolveSame procedure as for a full transient or harmonic analysis.Only displacement results are calculated during solution (no stressesor reaction forces). The displacement solution is written to:

jobname.rdsp for a transient analysis

jobname.rfrq for a harmonic analysisNext step is to review results.


Review Results

Review results. Three steps:

• Review the displacement solution

• Expand the solution• Review the expanded solution



p

prep by: Yasser ElJajehMode Superposition… Review Results

Review displacement solution

• Enter POST26, the time-history postprocessor.

• First identify the results file - jobname.rdsp or jobname.rfrq.TimeHist Postpro > Settings > File or FILE command

D fi di l t i bl t ifi i t i th d l



• Define displacement variables at specific points in the modeland obtain displacement-versus-time (or frequency) plots.

Using graphs andlistings, identify thecritical time-points(or frequencies and

phase angles).

prep by: Yasser ElJajehMode Superposition… Review Results

Expand the solution

• A process in which derived data (stresses, reaction forces, etc.)

are calculated from the primary data (displacement solution).• Three steps:

1 Enter Solution and activate the expansion pass



1. Enter Solution and activate the expansion pass.

Solution > ExpansionPass or EXPASS,ON


2. Specify the solution or range of solutions to beexpanded. For harmonic analysis, remember tospecify the phase angle(s) or request expansion of both real and imaginary parts (which can then becombined in POST1 using the HRCPLX command).

Mode Superposition… Review Results



g

▫ Solution > Load Step Opts > ExpansionPass > Single

Expand >

3. Start expansion pass solution

▫ Solution > Solve > Current LS or SOLVE

▫ Results are written to the .rst file ( jobname.rst)and can then be reviewed using POST1, thegeneral postprocessor.


Review the expanded solution

• Use POST1, the general postprocessor.

• Procedure is the same as for a full transient or harmonicanalysis.

▫ Read the desired results set from the results file then plot

Mode Superposition

… Review Results



▫ Read the desired results set from the results file, then plotdeformed shape, stress contours, etc.

▫ For a harmonic analysis, if you chose to expand both real and

imaginary parts, use the HRCPLX command to combinethem at the desired phase angle. (No need to do this if youchose to expand the displacement solution at a specifiedphase angle.)

prep by: Yasser ElJajeh… References:

1. Engineering Analysis with ANSYS Software.(T.Stolarsky, Y.Nakasone & S.Yoshimoto).

2. ANSYS – Introduction to Dynamics- Training Manual.3. Three Dimensional Static and Dynamic Analysis.( EdwardWilson)



Wilson).

4. ANSYS Modeling and Meshing Guide.


ansys for structural engineers

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