Mech Dynamics 14.5 L04 Modal

8/11/2019 Mech Dynamics 14.5 L04 Modal

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2012 ANSYS, Inc. March 28, 2013 1 Release 14.5

14.5 Release

Lecture 4

Modal Analysis

ANSYS MechanicalLinear and Nonlinear Dynamics


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Modal Analysis

Topics Covered

A. Definition and Purpose

B. Theory and Terminology

C. Eigen-frequencies, mode shapes

D. Participation factors, Effective Mass

E. Mode extraction methods- undamped

F. Contact

G. Analysis settings Workshop 4A


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Modal Analysis

Topics Covered

H. Damped Modal Analysis

I. Mode extraction methods- damped

Workshop 4B


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A. What is Modal Analysis

The modal analysis technique is used to determine the vibrationcharacteristics (i.e., natural frequencies and mode shapes) of linear

elastic structures.

The most fundamental of all dynamic analysis types.

Allows the design to avoid resonant vibrations or to vibrate at a specified

frequency.

Gives engineers an idea of how the design will respond to different types

of dynamic loads.

Helps in calculating solution controls for other dynamic analyses.


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... What is Modal Analysis

Assumptions and restrictions

The structure is linear (i.e. [M] and [K] matrices are constant).

No loads (forces, displacements, pressures, or temperatures) areallowed; i.e., free vibration.

Recommendation: Because a structures 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.


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B. Theory and Terminology

The linear equation of motion for free, un-damped vibration is

Assume harmonic motion:

Substituting{}

and{} In the governing equation gives an eigenvalue

equation:

0 uKuM

iiii

iii

tu

tu

sin

sin

2

02 i

MK


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... Theory and Terminology

This equality is satisfied if

i= 0

(trivial, implies no vibration)

2. or if =

This is an eigenvalue problem which may be solved for up to nroots

(12

,22

, ..,n2

).

These roots are the eigenvaluesof the equation

For each root (eigenvalue), there is a corresponding eigenvector

(1,2,.,n)

02

iMK



In Modal Analysis:

The eigenvalues the square of the natural circular frequency of the

structure i

The eigenvectors the corresponding mode shapes i

Mode shapes can be normalized either to the mass matrix

or to unity, where the largest component of the vector {}iis set to 1.

Workbench displays results normalized to the mass matrix.

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

1i

T

i M

C. Eigen-frequencies and mode shapes.



The square roots of the eigenvaluesare i, the structures naturalcircular frequencies (rad/s).

Natural frequenciesfican then

calculated asfi= i/2p(cycles/s).

It is the natural frequencies,fiin Hz,

that are input by the user and

output by Workbench.

The eigenvectors {

}irepresent themode shapes, i.e.the shape

assumed by the structure whenvibrating at frequencyfi.

... Eigen-frequencies and mode shapes.

mode 1

{

}1f1= 109 Hz

mode 2

{

}2f2= 202 Hz

mode 3

{

}3f3= 249 Hz



The participation factors are calculated by

where {D} is an assumed unit displacement spectrum in each of the global

Cartesian directions and rotation about each of these axes.

Measures the amount of mass moving in each direction for each mode.

A high value in a direction indicates that the mode will be excited by

forces in that direction.

The Ratio is simply another list of participation factors, normalized to

the largest.

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

D. Participation Factor, Effective Mass

DMTii


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... Participation Factor, Effective Mass

DMTii

The participation factors are calculated by


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The effective mass is calculated by

Ideally, the sum of the effective masses in each direction should equaltotal mass of structure, but will depend on the number of modesextracted.

The ratio of effective mass to total mass can be useful for determining

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


1if,2

2

, i

T

ii

i

T

i

iieff M

MM


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The effective mass is calculated by


1if,2

2

, i

T

ii

i

T

i

iieff M

MM


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E. Mode Extraction Methods - Undamped

Procedure Usages Applications

Block Lanczos

(Direct)

Symmetric Many modes (about 40+) of large models.

Recommended when poorly shaped solid and shell elements exist.

Shells or a combination of shells and solids.

PCG Lanczos

(Iterative)

Symmetric

(but not applicable

for buckling)

Few modes (up to about 100) of very large models (500,000+

degrees of freedom).

Well-shaped 3-D solid elements.

Unsymmetric Unsymmetric

matrices

Uses the full [K] and [M] matrices.

When K and M are unsymmetric (i.e., acoustic fluid-structure

interaction problems).

Supernode Symmetric

(but not applicable

to buckling)

Many modes (up to 10,000).

Used for 2-D plane or shell/beam structures (100 modes or more)

and for 3-D solid structures (250 modes or more).

In most cases, the Program Controlledoption selects the optimal solverautomatically.


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Contact regions are available in modal analysis; however, since this is apurely linear analysis, contact behavior will differ for the nonlinear

contact types, as shown below:

Contact behavior will reduce to its linear counterparts.

Contact Type Static Analysis

Linear Dynamic Analysis

Initially TouchingInside Pinball

RegionOutside Pinball Region

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 Frictional

= 0, No Separation

> 0, Bonded

Free Free

F. Contact in Modal Analysis


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Number of modes You need to specify the number of frequencies of interest. The default is to

extract thefirst 6 natural frequencies.

The number of frequencies can be specified in two ways:

1. The first N frequencies (N > 0), or

2. The first N frequencies in a selected range of frequencies.

G. Analysis Settings

1 2


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Solver Control Two settings are available in this controlDamped and Solver Type.

Damped >> No (Default; i.e., undamped system)

Yes.

Depending on the selection made for Damped, different solver options are

provided accordingly.

... Analysis Settings


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Output Control By default only mode shapes are calculated.

Stress and Strain results can be requested to be calculated.

stress results only show the relative distribution of stress in the structure

and are not real stress values. You can also choose whether or not to have

these results stored for faster result calculations in linked systems.

... Analysis Settings


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14.5 Release


Workshop 4A

Modal Analysis(Plate with a Hole)


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The linear equation of motion for free, damped vibration is

The eigenvalues are complex

The imaginary part of the eigenvalue is the natural frequency The real part of eigenvalue is a measure of stability

* Positive = unstable * Negative = stable

H. Damped Modal Analysis

0 uKuCuM

Undamped Damped =


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The complete expression for the structural damping matrix, [C], is

dampingStructural

1

dampingMass

1

mb

ma

N

j

j

m

j

N

ii

m

i

KK

MMC

... Damped Modal Analysis


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The value of and can be input using the following:

[1] Mater ial-dependent damping v alue

(Mass-Matrix Damping Multiplier, and k-Matrix Damping

Multiplier)

. Structural Damping Matrix [C]

mbma N

j j

m

j

N

i i

m

i

KMC11

22

i

i

i

Equivalent damping


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[2] Direct ly as glo bal damp ing v alue

(Details section of Analysis Settings)

. Structural Damping Matrix [C]

KMC 22

i

i

i

Equivalent damping


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I. Mode Extraction Methods - Damped

Procedure Usages Applications

Damped Symmetric or

unsymmetric

damped systems

To find all modes of small to medium models (less than 10,000

degrees of freedom).

Uses full matrices ([K], [M], and the damping matrix [C])Reduced

Damping

Symmetric or

unsymmetric

damped systems

Approximately represent the first few complex damped eigenvalues by

modal transformation using a small number of eigenvectors of the

undamped system. After the undamped mode shapes are evaluated by

using the real eigensolution (Block Lanczos method), the equations of

motion are transformed to these modal coordinates.

In most cases, the Program Controlledoption selects the optimal solverautomatically.


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2012 ANSYS Inc March 28 2013 25 Release 14 5

14.5 Release


Workshop 4B

Modal Analysis

(Model Airplane Wing)

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Mech Dynamics 14.5 L04 Modal