Static Analysis: Modal Superposition

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Static Analysis:Modal Superposition


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Objectives

This module will present the equations and theory used to perform a Modal Superposition Analysis of a linear system.

The basis of Mode shapes will be discussed.

A system of single degree of freedom equations will be presented.

The individual modes and total response will be examined.

Section II – Static Analysis

Module 6 – Modal Superposition


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Equations of Motion

The equations of motion for a linear system developed in Module 5: Natural Frequency Analysis are

The mode shapes form a basis for the solution space of the undamped system and the displacements can be written as a linear combination of the mode shapes

is a matrix that has a mode shape in each column.



tFuKuCuM

xxxxu nn 2211


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Time Derivative Equations

The velocities and accelerations written in terms of the mode shapes and coefficients are

These equations can be used to transform the equations of motion from a coupled set of equations to an uncoupled set.

xu

xuxu

Note that the mode shapes are

not a function of time and only the coefficients, x, have time derivatives.




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Transformation Process

Substituting the time derivative equations into the equations of motion for an undamped system yields the equation

tFuKuM

tFxKxM

xu

xuxu

tFxKxM TTT




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Mass Orthonormal Mode Shapes

If the mode shapes are orthonormal to the mass matrix, then the following relationships hold (reference Module 5: Natural Frequency Analysis).

IMT I is the identity matrix

2 KT 2Square matrix with natural frequencies on the diagonal elements and zero for all non-diagonal elements




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Uncoupled Equations

Combining the equations on the previous two slides yields the equation

Each row of the this equation contains an equation of the form

.2 tFxxI T

tfxx iiii 2

where is equal to the i th row of times . tfi tF




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Modal Participation Factors

The i th row of the forcing function term can be written as

where each term is called a modal participation factor.

The subscripts are



tFtFtF ninii 2211

.ij

i th equation j th mode shape


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Physical Significance

If a force is applied at the end of a cantilevered beam, it has a large affect in exciting the first mode because the mode shape has a large value at this point.

If a force is applied at the point of zero displacement in the second mode, the force does not excite the second mode because the mode shape value at this point is zero.

The physical significance of the modal participation factors is best illustrated using an example.




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Integration

Each of the equations can be integrated separately to determine the variation of the coefficient as a function of time.

Once all of the equations have been integrated, the response of the total system can be obtained from the linear superposition equation

tfxx iiii 2

.txtu




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Practical Systems

The development up to this point has used all n-degrees of freedom of the system.

It is generally only necessary to include the response of a small subset of the total number of modes to accurately compute the response of the system.

The frequency content of the applied load and the points on the structure where loading is applied, play a large role in determining how many modes should be considered in determining the response.

It is not necessary to include mode shapes having natural frequencies that are significantly higher than the highest frequency in the excitation forces.




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Damping

Damping was omitted in the preceding development but can be included.

Damping can be attributed to a variety of material properties (viscoelastic materials have good damping properties) or displacement dependent mechanisms (slipping of bolted joints).

Damping is difficult to quantify analytically and must be determined experimentally.

Damping is generally different for each mode and can be nonlinear (i.e. the more a structure deforms, the higher the damping).




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Example ProblemSection II – Static Analysis


5 lb. force distributed over the 17 nodes on the upper edge of the free end

Fixed End

1 inch wide x 12 inch long x 1/8 inch thick.Material - 6061-T6 aluminum.

Brick elements with mid-side nodes are used to improve the bending accuracy through the thin section. 0.0625 inch element size.

Simulation is used to compute the step response of the cantilevered beam using the first five natural frequencies and mode shapes computed in Module 5.


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List of nodes on upper edge of free end

Forces become active at 0.05 seconds

Discussed on next slide

z is equal to 0.5% for all modes

Load curves are scaled by a factor of 1 in the y-direction

Example Analysis Parameters


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Example – Time Step

The time step is based on the natural frequencies in the system and the frequency content of the input.

The input force causes bending about the weak axis and will only excite modes that have bending about the weak axis.

The natural frequency of the third weak axis bending mode was computed in Module 5 to be 592 Hz.

The period of this frequency is 0.00169 seconds.

8-10 time steps per smallest period is generally used as a time step.



Mode 4, 592 Hz, 3rd bending mode about the weak axis

cyclescyclesfT sec

5921

sec

1

sec 000169.010

Tt


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Example – Restart Data

A modal superposition analysis uses natural frequency and mode shape data from a Natural Frequency Analysis.

The Design Scenario containing this information must be specified.

In this problem, the natural frequency analysis was performed in Design Scenario 2 that has been renamed to Nat Freq and Mode Shapes.




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Example - Load Curves

5 lb. acting in the negative y-direction is divided among 17 nodes

When activated after 0.05 seconds, the load will be constant




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Example – Fixed End Stress HistorySection II – Static Analysis


The higher frequency response can be seen superimposed on the dominant first mode response.

zz is plotted


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Example – Tip DisplacementSection II – Static Analysis


Start of Step Input

0.036 sec27.8 Hz

The response is dominated by the 1st bending mode.

This curve looks much smoother than the stress history curve. Why?

The stresses are based on strains that depend on the derivatives of the displacements.


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Module Summary

The equations and theory used to perform a Modal Superposition Analysis of a linear system are presented.

Mode shapes form a basis and all solutions can be written in terms of them.

This results in a system of single degree of freedom equations that can be integrated separately.

The results from the individual modes are then combined to determine the total response.

Often, only a small subset of the total number of modes need to be used to obtain accurate results.



Static Analysis: Modal Superposition

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