Static Analysis:
Natural Frequency Analysis
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Section II – Static Analysis
Objectives
Module 5 - Natural Frequency Analysis
Page 2
The objective of this module is to develop the equations and numerical
methods used to compute the natural frequencies and mode shapes of
a linear system.
This will be accomplished by specializing the incremental equations
developed for general static analysis to linear systems.
 The equations for linear static systems will then be extended to include
inertia and viscous damping terms.
 The free vibration problem will then be used to reveal the eigenvalue
problem used to compute the natural frequencies and mode shapes.
 Numerical methods for efficiently computing the natural frequencies and
mode shapes will then be presented.

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Section II – Static Analysis
Incremental Equations for Static Analysis
Module 5 - Natural Frequency Analysis
Page 3

The governing equations used to compute the displacement history
of a static system were developed in Modules 1 through 4. These
equations are
KT u  Fext  Rint   Runb.

There is no need to break the solution into small increments if the
system is linear. The above equation reduces to the following when
there is only one increment
K u  Fext .
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Section II – Static Analysis
Equations of Motion
Module 5 - Natural Frequency Analysis
Page 4

Inertia and viscous damping effects can be added to yield the
equation
M u Cu K u  F t .

This equation of equilibrium governs the motion of any linear system
undergoing time dependent motion.

Because these equations control the system response they are called
the Equations of Motion.
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Section II – Static Analysis
Undamped Natural Frequencies
Module 5 - Natural Frequency Analysis
Page 5
The undamped natural
M u Cu K u  F t 
frequencies and mode shapes
are obtained from the equations
of motion by setting the
M u K u  0
damping and the forcing
function to zero.
it

u  e
 We seek solutions of the form
u  eit where   describes
2
it
it










M

e

K

e
 0
it
the deformed shape and e
defines the magnitude as a
function of time.
K    2 M    0


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
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Section II – Static Analysis
Eigenvalue Problem
Module 5 - Natural Frequency Analysis
Page 6




This is a system of homogeneous
equations because the right hand
side of the equations are zero.
  can only be determined when
the determinant is equal to zero.
There are specific values of ,
called the natural frequencies,
that make the determinant zero.
Once the natural frequencies are
determined, the array   , called
the mode shape, can be found for
each natural frequency.
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K   M   0
2


det K    M   0


2
The determinant will yield a
polynomial with  as the
unknown parameter.
The polynomial will be the
same order as the number of
equations.
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Section II – Static Analysis
Important Parameters
Module 5 - Natural Frequency Analysis
Page 7




The natural frequencies and
mode shapes of a system are
important parameters.
The natural frequencies describe
the frequencies at which the
system wants to vibrate.
The mode shape describes the
shape the system takes at these
preferred frequencies.
The mode shapes form a basis
for the system and all solutions
to the equations of motion can
be written as a linear
combination of them.
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u  11  2 2   3 3   n n 
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n
u    i i 
i 1
or
u   
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Section II – Static Analysis
Constant Stiffness
Module 5 - Natural Frequency Analysis
Page 8



Natural frequencies and mode
shapes are computed using the
stiffness matrix at a particular
instance.
If the stiffness matrix is linear or
constant at all instances then the
natural frequencies and mode
shapes are constants.
In a stressed structure, the
stiffness matrix is typically based
on the stress at the end of a
static solution that is used to
compute the stresses.
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

Natural frequencies and mode
shapes are not computed for
problems that have intermittent
contact, gaps, etc.
When these non-linear effects
are present, the natural
frequencies and mode shapes
are not constants but change as
each gap opens or closes.
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Section II – Static Analysis
Mass Orthonormal
Module 5 - Natural Frequency Analysis
Page 9




The mode shapes can only be
determined to within a constant.
They describe the shape that the
structure takes while vibrating at
a particular natural frequency,
but don’t provide the magnitude
of the deformation.
It is convenient to scale the
mode shapes such that the third
equation is satisfied.
This leads to the last equation.
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K   M   0
2
K i    M i 
2
i
i  M i   1
T
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i  K i   
T
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i
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Section II – Static Analysis
Example
Module 5 - Natural Frequency Analysis
Page 10

Simulation was used to compute the first five natural frequencies and
corresponding mode shapes of the thin cantilevered beam.
Fixed end
boundary
condition
Brick elements with mid-side nodes
are used to improve bending accuracy
through the thin section. 0.0625 inch
element size.
1 inch wide x 12 inch long x 1/8 inch thick.
Material - 6061-T6 aluminum.
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Section II – Static Analysis
Example – Computed Results
Module 5 - Natural Frequency Analysis
Page 11
Mode 1, 28 Hz, 1st bending mode about weak axis
Mode 2, 175 Hz, 2nd bending mode about weak axis
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Section II – Static Analysis
Example – Computed Results
Module 5 - Natural Frequency Analysis
Page 12
Mode 3, 222 Hz, 1st bending mode about the stiff axis
Mode 4, 592 Hz, 3rd bending mode about the weak axis
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Section II – Static Analysis
Example – Computed Results
Module 5 - Natural Frequency Analysis
Page 13
Mode 5, 618 Hz, 1st torsional mode

The shape of each mode is clearly seen in the preceding plots.

Remember that a mode shape represents only the shape that the
beam takes as it vibrates.

The mode shape is not the deformation due to an external
disturbance.
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Section II – Static Analysis
Stressed Systems
Module 5 - Natural Frequency Analysis
Page 14



The natural frequencies and
associated mode shapes are
sometimes sensitive to the
stresses in the system.
The frequency at which a guitar
string vibrates can be changed
by increasing or decreasing the
tension in the string.
In this type of system, the linear
stiffness matrix must be
augmented with the stress
stiffness matrix derived in
Module 3.
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Linear System
K   M   0
2


det K    M   0
2
Stressed System
K  K    M   0
2


det K   K     2 M   0
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Section II – Static Analysis
Eigenvalue Solution Methods
Module 5 - Natural Frequency Analysis
Page 15



It is not practical to solve large
eigenvalue problems using
methods that find the
determinant.
A variety of numerical methods
have been developed to find the
natural frequencies and mode
shapes of large systems.
Eigenvalue Solution Methods
Vector Iteration Methods
Transformation Methods
Polynomial Iteration Methods
Characteristic Polynomial Methods
All methods for computing
natural frequencies and mode
shapes are iterative.
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Section II – Static Analysis
Inverse Iteration
Module 5 - Natural Frequency Analysis
Page 16




Inverse iteration is a vector iteration
method.
K i    M i 
2
i
It is often part of a more advanced
method such as Sub-space iteration.
It begins by making an initial estimate
for the mode shape and then
performing a series of iterations to
improve the estimate.
Once the mode shape has converged,
the natural frequency is found from
Assume
x1   i 
2
i
K x j 1  M x j 
Solve forx j 1and then make
orthogonal to the mass
matrix

x 
x 
x  M x 
j 1
j 1
T
j 1
j 1
x  K x   .
T
j 1
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j 1
2
i
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Section II – Static Analysis
Frequency Shifts
Module 5 - Natural Frequency Analysis
Page 17

Frequency shifts are often used
by numerical algorithms to:
extract frequencies in a specific
range,
 improve poorly conditioned
matrices encountered during the
solution process, or
 improve convergence rate.



The natural frequencies of the
shifted and original systems are
related by the equation
 2  2  2
When a shift is used, the
standard equation becomes
K   M    M 
2
2
where  is a shift frequency.
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Section II – Static Analysis
Simulation Analysis Parameters
Module 5 - Natural Frequency Analysis
Page 18
Number of frequencies and
modes to be computed.
Used as a search range by
the numerical algorithm
Check if structure is not
constrained. Rigid body
modes require special
handling in numerical
algorithm.
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Section II – Static Analysis
Simulation Analysis Parameters
Module 5 - Natural Frequency Analysis
Page 19
Lanczos based algorithm
that uses multiple
processors commonly found
on workstations.
Alternate solver Sub-Space
Iteration uses inverse
iteration and Sturmsequence properties of
characteristic equation.
Recommended only for small
problems.
Percent of RAM used to
read the element data and
to assemble the matrices.
Sparse solver requires less
than the Sub-space
Iteration solver.
Use all processors during solution phase  fastest approach.
Using a subset of the available processors will let you work on
other things while the problem is running.
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Section II – Static Analysis
Simulation Analysis Parameters
Module 5 - Natural Frequency Analysis
Page 20
If zero, the number of
Lanczos vectors used
by the Sparse solver is
two times the number
of frequencies being
computed.
This box needs to be
checked if you are not
going to use the results in
a subsequent analysis
(modal superposition).
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If checked, solver will correct
any negative diagonals
encountered. Negative
diagonals can occur when rigid
body modes are included.
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Used to improve the
solution if rigid body
modes are included or
modes in a particular
range are needed.
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Section II – Static Analysis
Module Summary
Module 5 - Natural Frequency Analysis
Page 21

All linear dynamic systems encountered in solid mechanics have
natural frequencies at which they prefer to vibrate.

Each natural frequency has a corresponding mode shape that the
system takes when vibrating at that natural frequency.

The natural frequencies and corresponding mode shapes are found
by solving an eigenvalue problem.

All possible solutions to a forced response problem can be written in
terms of the mode shapes.

This property will be used in Module 6: Modal Superposition.
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