AWB130 Dynamics 02 Modal - ANSYS

Customer Training Material

L t 2Lecture 2

Modal Analysisy

ANSYS MechanicalANSYS MechanicalLinear and Nonlinear Dynamics

L02-1ANSYS, Inc. Proprietary© 2011 ANSYS, Inc. All rights reserved.

Release 13.0March 2011

Dynamics

ANSYS Mechanical Linear and Nonlinear Dynamics

Customer Training MaterialModal AnalysisTopics covered:

A. Define modal analysis and its purpose.A. Define modal analysis and its purpose.

B. Discuss associated concepts, terminology, and mode extraction methodsmethods.

C. Learn how to do a modal analysis in Workbench.

D. Work on one or two modal analysis exercises.




Customer Training MaterialDescription & Purpose• A modal analysis is a technique used to determine the vibration

characteristics of structures:– natural frequencies

• at what frequencies the structure would tend to naturally vibrate– mode shapes

• in what shape the structure would tend to vibrate at each frequency– mode participation factors

• the amount of mass that participates in a given direction for each mode

M t f d t l f ll th d i l i t• Most fundamental of all the dynamic analysis types.




Customer Training MaterialDescription & PurposeBenefits of modal analysis• Allows the design to avoid resonant vibrations or to vibrate at a

specified frequency (speaker box, for example).• Gives engineers an idea of how the design will respond to different

types of dynamic loads.• Helps in calculating solution controls (time steps, etc.) for other p g ( p , )

dynamic analyses.

Recommendation: Because a structure’s vibration characteristics determine how it responds to any type of dynamic load, it is generally recommended to perform a modal analysis first before trying any other dynamic analysis.




Customer Training MaterialTerminology• A “mode” refers to the pair of one

natural frequency and corresponding mode shape.

A structure can have any number of

mode 1← {φ}1f = 109 Hz– A structure can have any number of

modes, up to the number of DOF in the model.

f1 = 109 Hz

mode 2← {φ}2f2 = 202 Hzf2 = 202 Hz

mode 3← {φ}3f3 = 249 Hz



f3 249 Hz


Customer Training MaterialAssumptions & Restrictions• The structure is linear (i.e. constant stiffness and mass).

• There is no damping.There is no damping.– Damped eigensolvers (MODOPT,DAMP or MODOPT,QRDAMP) may be

accessed using Commands Objects, but will not be covered here.

• The structure has no time varying forces, displacements, pressures, or temperatures applied (free vibration).




Customer Training MaterialDevelopment• Start with the linear general equation of motion:

[ ]{ } [ ]{ } [ ]{ } { }FuKuCuM =++ &&&

• Assume free vibrations, and ignore damping:

[ ]{ } [ ]{ } [ ]{ } { }uuCu

[ ]{ } [ ]}

{ } [ ]{ } { }}

[ ]{ } [ ]{ } { }0

00

=+=++

uKuMFuKuCuM

&&

&&&

• Assume harmonic motion:

[ ]{ } [ ]{ } { }0=+ uKuM

{ } { } ( ){ } { } ( ){ } { } ( )

iiii

iii

tutu

θωφωθωφ

+=+=

cossin

2

&



{ } { } ( )iiii tu θωφω +−= sin2&&


Customer Training MaterialDevelopment• Substitute and simplify

[ ]{ } [ ]{ } { }0=+ uKuM &&[ ]{ } [ ]{ } { }[ ]{ } ( ) [ ]{ } ( ) { }

[ ] [ ]( ){ } { }0

0sinsin0

2

2

=+

=+++−

=+

iiiiiii

KM

tKtMuKuM

φω

θωφθωφω

[ ] [ ]( ){ } { }0=+− ii KM φω

• This equality is satisfied if {φ}i = 0 (trivial, implies no vibration) or if

[ ] [ ]( ) { }0det 2 =− MK ω

• This is an eigenvalue problem which may be solved for up to neigenvalues ωi

2 and n eigenvectors {φ}i where n is the number of

[ ] [ ]( ) { }0det =MK iω



eigenvalues, ωi , and n eigenvectors, {φ}i, where n is the number of DOF.


Customer Training MaterialExtraction & Normalization• Note that the equation

[ ] [ ]( ) { }0det 2 =− MK iω

has one more unknown than equations; therefore, an additional equation is needed to find a solution

[ ] [ ]( ) { }0det MK iω

equation is needed to find a solution.– The addition equation is provided by mode shape normalization.

• Mode shapes can be normalized either to the mass matrix

{ } [ ]{ } 1 =iTi M φφ

or to unity, where the largest component of the vector {φ}i is set to 1.• Workbench displays results normalized to the mass matrix.• Because of this normalization, only the shape of the DOF solution



Because of this normalization, only the shape of the DOF solution has real meaning.


Customer Training MaterialEigenvalues & Eigenvectors• The square roots of the eigenvalues

are ωi, the structure’s natural circular frequencies (rad/s).

mode 1← {φ}1f = 109 Hz

• Natural frequencies fi can then calculated as f = ω /2π (cycles/s)

f1 = 109 Hz

calculated as fi = ωi/2π (cycles/s).– It is the natural frequencies, fi in Hz,

that are input by the user and output by Workbench.

mode 2← {φ}2f2 = 202 Hz

• The eigenvectors {φ}i represent the d h i th h

f2 = 202 Hz

mode shapes, i.e. the shape assumed by the structure when vibrating at frequency fi.

mode 3← {φ}3f3 = 249 Hz



f3 249 Hz


Customer Training MaterialEquation Solvers

( )• The equation

can be solved using one of two solvers available in Workbench

[ ] [ ]( ) { }0det 2 =− MK iω

Mechanical:– Direct (Block Lanczos)

• To find many modes (about 40+) of large models.• Performs well when the model consists of shells or a combination of shells• Performs well when the model consists of shells or a combination of shells

and solids.• Uses the Lanczos algorithm where the Lanczos recursion is performed with a

block of vectors. Uses the sparse matrix solver.

– Iterative (PCG Lanczos)• To find few modes (up to about 100) of very large models (500,000+ DOFs).• Performs well when the lowest modes are sought for models that are

d i t d b ll h d 3 D lid l tdominated by well-shaped 3-D solid elements.• Uses the Lanczos algorithm, combined with the PCG iterative solver.

• In most cases the Program Controlled option selects the optimal



• In most cases, the Program Controlled option selects the optimal solver automatically.


Customer Training MaterialParticipation Factors (Solution Information)• The participation factors are calculated by

{ } [ ]{ }DMTii φγ =

where {D} is an assumed unit displacement spectrum in each of the global Cartesian directions and rotation about each of these axes.

– This measures the amount of mass moving in each direction for each mode.Th “R ti ” i i l th li t f ti i ti f t li d t th– The “Ratio” is simply another list of participation factors, normalized to the largest.

• The concept of participation factors will be important in later chapters.




Customer Training MaterialParticipation Factors (Solution Information)• A high value in a direction indicates that the mode will be excited by forces in

that direction.mode 1 mode 3 mode 5




Customer Training MaterialEffective Mass (Solution Information)• Also printed out is the effective mass.

{ } [ ]{ }{ } [ ]{ } 1 if,2

2

=== iTiiT

iieff MM φφγγ

• Ideally, the sum of the effective masses in each direction should equal total mass of structure but will depend on the number of modes extracted

{ } [ ]{ }{ } [ ]{ },, iii

iTi

ieff Mφφγ

φφ

mass of structure, but will depend on the number of modes extracted.• The ratio of effective mass to total mass can be useful for determining

whether or not a sufficient number of modes have been extracted.




Customer Training MaterialContact Regions• Contact regions are available in modal analysis; however, since this

is a purely linear analysis, contact behavior will differ for the nonlinear contact types, as shown below:

Contact Type Static AnalysisLinear Dynamic Analysis

Initially Touching Inside Pinball Region Outside Pinball Regiong

Bonded Bonded Bonded Bonded Free

No Separation No Separation No Separation No Separation Free

Rough Rough Bonded Free Free

Frictionless Frictionless No Separation Free Free

Frictional Frictionalμ = 0, No Separation

μ > 0, BondedFree Free

• Contact behavior will reduce to its linear counterparts.– It is generally recommended, however, not to use a nonlinear contact



It is generally recommended, however, not to use a nonlinear contact type in a linear-dynamic analysis


Customer Training MaterialUnconstrained Systems• An unconstrained system is one that has no constraints or supports

and can move as a rigid body in at least one direction.– Rigid-body motion can be considered to be a mode of oscillation with

zero frequency.– In practice, these modes may not have a frequency of exactly zero.

“rigid-body”or“zero” modeszero modes

• Note that a well-connected system can have at most six rigid-body modes.

Obt i i th i i id b d d i di t th t bli



– Obtaining more than six rigid-body modes may indicate that assemblies are not well connected.


M d lModal

Prestress Effects




Dynamics


Customer Training MaterialPrestress Effects• A prestressed modal analysis can be used to calculate the

frequencies and mode shapes of a prestressed structure, such as a spinning turbine blade.

– The prestress influences the stiffness of the structure through the stress-stiffening matrix contribution.




Customer Training MaterialPrestress Effects• In free vibration with prestress analyses, two solutions are required.

– A linear static analysis is initially performed:

[ ]{ } { } [ ]FK

– Based on the stress state [σ] from the static analysis, a stress stiffness matrix [S] is calculated (see Theory Reference for details):

[ ]{ } { } [ ]σ→= FuK

matrix [S] is calculated (see Theory Reference for details):

[ ] [ ]S→σ– The free vibration with pre-stress analysis is then solved, including the

[S] term:

[ ] [ ]( ){ } { }02+ MSK φω

• Note that the prestress only affects the stiffness of the system.i e the static prestress will not be added to the modal stress

[ ] [ ]( ){ } { }0=−+ ii MSK φω



– i.e. the static prestress will not be added to the modal stress


Customer Training MaterialPrestress Effects• Mechanical employs the new MAPDL linear perturbation technique

for all prestressed eigenvalue analysis

• Linear perturbation supports– Large deflection, nonlinear pre-stress– Prestressed modal from any restart pointy p– Handling true contact status from the static analysis– Cyclic symmetry

[K T][KiT]

• Any time point from a static or transient structural analysis can be perturbed.p

• Uses global tangent stiffness matrix [Ki

T] from the time point.• Restart files must be available.



ti


Customer Training MaterialPrestress Effects• Tangent Stiffness matrix

essLoadStiffni

Contacti

Mati

Ti KKKK ++=

[KiT]

fnessStressStifi

ingSpinSofteni

iiii

KK ++

• Material Stiffness [KiMat]

– For nonlinear materials, only the linear portion is used– For hyperelastic materials, the tangent material properties at the point of restart

are used

ti

are used

• Stress Stiffening [KiStressStiffness] / Spin softening [Ki

SpinSoftening]Effects included automatically in large deflection analyses– Effects included automatically in large deflection analyses

• Contact stiffness [KiContact]

– Contact behavior can be changed prior to the modal analysis



Contact behavior can be changed prior to the modal analysis– (more on this later)


M d lModal

Procedure




Dynamics


Customer Training MaterialProcedure• Drop a Modal (ANSYS) system into the project schematic.




Customer Training MaterialProcedure• Create new geometry, or link to

existing geometry.• Edit the Model cell to bring up the

Mechanical application.




Customer Training MaterialPreprocessing• Verify materials, connections, and mesh settings.

– This was covered in Workbench Mechanical Intro.




Customer Training MaterialPreprocessing• Optional: Specify “Cyclic Region”

– Matched mesh will be created at the low-high regions

Select Harmonic Indices of interest in

Analysis SettingsAnalysis Settings




Customer Training MaterialPreprocessing• Add supports to the model.

– Displacement constrains must have a magnitude of zero.




Customer Training MaterialSolution Settings• Choose the number of modes to

extract.• If needed, upper and lower bounds

on frequency may be specified to extract the modes within a specified range.




Customer Training MaterialSolution Settings

• If the Program-Controlled solver

• Stress and strain results may be turned on under Output Controls.

selection is not appropriate, the solver type can be changed to either Direct or Iterative.




Customer Training MaterialPostprocessing• Total-deformation results may be

quickly inserted by highlighting multiple rows in the tabular view or histogram viewhistogram view.




Customer Training MaterialPostprocessing• If stress/strain were requested, these results may also be access from the

Solution Toolbar.




Customer Training MaterialPostprocessing• If cyclic symmetry was activated, results will be automatically expanded.

Cyclic ExpansionCyclic Expansion

Modes listed

Harmonic Index

Control

by Harmonic Indices



Interactive Frequency-Spectrum Display


M d lModal

Prestressed Procedure




Dynamics


Customer Training MaterialProcedure• The procedure to do a prestressed

modal analysis is essentially the same as a regular modal analysis, except that you first need toexcept that you first need to prestress the structure by doing a static analysis.

• The static analysis results in a stressed structure, which is used as the initial condition for the modal

l ianalysis.




Customer Training MaterialProcedure• Drop a Static Structural (ANSYS) system into the project schematic.




Customer Training MaterialProcedure• Drop a Modal (ANSYS) system onto

the Solution cell of the Modal system.

• Note the circular-ended connector, indicating a data transfer from the Static to the Modal analysis.




Customer Training MaterialProcedure• Create new geometry, or link to

existing geometry.• Edit the Model cell to bring up the

Mechanical application.




Customer Training MaterialStatic: Preprocessing• In the Static Structural system, insert the loads and supports that will cause

the prestressed-state to occur.




Customer Training MaterialPreprocessing• Optional: Specify “Cyclic Region”

– Matched mesh will be created at the low-high regions

Select Harmonic Indices of interest in

Analysis SettingsAnalysis Settings




Customer Training MaterialStatic: Analysis Settings• Setup restart controls of parent static structural analysis according

to requirements.

– Default is to retain a restart point for the last substep of the last load step.– Recall: any substep can be perturbed, providing there is a restart point.




Customer Training MaterialStatic: Postprocessing• Review the static results before

proceeding.




Customer Training MaterialAnalysis Settings: Prestress Environment• In the modal analysis, user can specify

which restart point from the static analysis will be used as the prestressenvironment.

• This can be specified by either(1)End of load step(2)Time




Customer Training MaterialAnalysis Settings: Contact Status• The user also has some control over the

contact status.

• Available options are:(1)Use True Status

– Use the status at the restart point.Use the status at the restart point.(2)Force Sticking

– For frictional contact with mu>0, set sliding elements to sticking.sliding elements to sticking.

(3)Force Bonded– Use bonded contact for all contact

pairs that are closed.pairs that are closed.




Customer Training MaterialPostprocessing• The modal results may be reviewed as described in the previous section.




Customer Training MaterialPostprocessing• Note that the prestressed state increased the frequencies of this structure.

– e.g. the first mode in this example increased from 108.3 Hz to 274.6 Hz

Not Prestressed PrestressedNot Prestressed Prestressed

• A prestress may not always increase the natural frequencies; a compressive load will decrease the frequencies.

In fact a sufficiently-high compressive load will result in a natural frequency of



– In fact, a sufficiently-high compressive load will result in a natural frequency of zero, effectively replicating the results of a buckling analysis.


Customer Training MaterialD. Workshop - Modal Analysis This workshop consists of two problems:

1. Modal analysis of a plate with a hole1. Modal analysis of a plate with a hole– A step-by-step description of how to do the analysis.– You may choose to run this problem yourself, or your instructor may

show it as a demonstration.(WS2A: Modal Analysis - Plate with a Hole).

2 Pre-stressed Modal analysis of a model airplane wing2. Pre stressed Modal analysis of a model airplane wing– This is left as an exercise to you.

(WS2B: Modal Analysis - Model Airplane Wing).



AWB130 Dynamics 02 Modal - ANSYS

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