Dynamics 70 M7 Modesup

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Training Manual

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Module 7

Mode Superposi t ion

A. Define mode superposition.

B. Learn how to use the mode superposition method.

C. Work on a mode superposition exercise.


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Mode Superpos i t ion

A. Def ini t ion & Purpose

A solution technique for transient or harmonic analyses. It sums

factored mode shapes from a modal analysis to calculate the

dynamic response.

A fast, efficient method that can be used for l ineardynamics

problems.

The alternative is to use the direct integrat ionmethod which can

be time consuming. The two methods are compared next.


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General equation of motion:


Definition & Purpose

)t(fKuuCuM

Mode superposition assumes thatU(t) can be represented as a linear

combination of mode shapes of the structure.

}]{[)( ytu

where [] is the matrix of mode shapesf1 f2 f3 ... fm


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

The general equation of motion can be premultiplied by[]Tand written

as:



Orthogonality of natural modes means:

2][K][

1][M][

JJ

T

J

J

T

J

If proportional damping is specified, then:

JJJ

T

J 2][C][


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Defining m as the number of modes, this reduces the problem to

a system of m single DOF uncoupled equations :

These equations are then solved for yi(t) using an undampedsolver (e.g. the Frontal solver).

If non-proportional damping is specified, then the system of m

single DOF equations are coupled by the damping matrix. This

system of equations must be solved using the QR Damped solver.

The final solution (regardless of the specified damping) is:

)t(f][yy2y 2 T

JJJJJJJ

}]{[)(...)()()( 2211 ytytytytu mm fff




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Mode Superposition+ Fast solution regardless of whether

equations of motion are uncoupled

(proportional 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 to use

may be difficult. Too few modes may

give good displacements but poor

stresses

Direct Integration Fully coupled equation of motion.

Solution can be time consuming.

+ Effective for most problems.

No eigenvectors required. However,

most dynamic analyses begin with amodal solution.

+ Nonlinearities allowed in transient

analysis.

+ Easier to determine Dt, the

integration time step, than number of

modes.


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B. Procedure

Five main steps:

Build the model

Obtain the modal solution

Switch to harmonic or transient analysis

Apply loads and solve

Review results


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Bu i ld the Model

Model

Same considerations as a modal analysis.

Linear elements and materials only. Nonlinearities are ignored.

Remember density! Also, if material-dependent damping is

present, it must be defined in this step.

See also Model ing Con siderat ionsin Module 1.


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Obtain the Modal Solut ion

Build the model


Same procedure as a normal modal analysis.

A few differences, discussed next.


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Obtain the Modal Solution

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 required for

the mode superposition solution.

If QR damped mode extraction method is used, the damping must bespecified during preprocessing or in the modal analysis. Damping

specified during the mode superposition transient or harmonic

analysis will be ignored.


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Obtain the Modal Solution

Loads and BCs:

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 must be specified

in this step.

The solver

ignores the

loads for the

modal solution,

but writes a

load vector to

the .mode file.


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Switch to Harmon ic or Transient Analysis

Build the model



Exit and re-enter Solution

New analysis: Harmonic or Transient

Analysis options: Discussed next

Damping: Discussed next


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Switch to Harmon ic or Trans ient

Analysis options - same as for a full harmonic or transient except:

Solut ion method: Mode superposition

Maximum mod e numb er: Highest mode number to be used for solution.

Defaults to highest mode extracted.

Min imum mode number: Lowest mode number. Defaults to 1.

Also for harmonic analysis:

Solution clustering option for a smooth response curve.

Option to print mode contributions at each frequency.


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Switch to Harmon ic or Trans ient

Damping

Specified here if QR Damped mode

extraction method not used.

Damping in some form should be

specified in most cases.

All four forms are available for mode

superposition:

Alpha (mass) damping

Beta (stiffness) damping

both global and material dependent

Constant damping ratio

Frequency dependent damping ratio

(mo dal damping)


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Apply Loads and Solve

Build the model




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).


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Load vector

Gives a way to apply element loads

(pressures, accelerations, and

temperatures) in a mode

superposition analysis.

Calculated during the modal

solution based on loads specifiedin the modal analysis.

Can be applied with a scale factor

(which defaults to 1.0).


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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 frontal solver).

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.


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Solve

Same procedure as for a full transient or harmonic analysis.

Only displacement results are calculated during solution (no

stresses or reaction forces). The displacement solution is written

to:

jobnam e.rdsp for a transient analysis

jobnam e.rfrq for a harmonic analysis

Next step is to review results.


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Review Results

Build the model




Review results. Three steps: Review the displacement solution

Expand the solution

Review the expanded solution


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Review Results

Review displacement solution

Enter POST26, the time-history postprocessor.

First identify the results file -jobname.rdsp orjobnam e.rfrq.

TimeHist Postpro > Settings > File or FILE command

Define displacement variables at specific points in the model and obtain

displacement-versus-time (or frequency) plots.

Using graphs and

listings, identify the

critical time-points(or frequencies and

phase angles).


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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.

Solution > ExpansionPass

or EXPASS,ON


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2. Specify the solution or range of solutions to be

expanded. For harmonic analysis, remember tospecify the phase angle(s) or request expansion of

both real and imaginary parts (which can then be

combined in POST1 using the HRCPLX command).

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 (jobnam e.rst) and can

then be reviewed using POST1, the general

postprocessor.


Review Results


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Review the expanded solution

Use POST1, the general postprocessor.

Procedure is the same as for a full transient or harmonic analysis.

Read the desired results set from the results file, then plot deformed

shape, stress contours, etc.

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

imaginary parts, use the HRCPLX command to combine them at the

desired phase angle. (No need to do this if you chose to expand the

displacement solution at a specified phase angle.)


Review Results


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Review Results

Build the model




Review results


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Dynamics 70 M7 Modesup

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