Modal Testing and Analysis

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Modal Testing and Analysis

• Undamped MDOF systems
• Saeed Ziaei-Rad

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Undamped MDOF systems
M and K are NN mass and stiffness
matrices. f(t) is N1 force vector
x2
x1
k3
k1
k2
m1
m2
f1
f2
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Undamped 2DOF system
or
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Undamped 2DOF system
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MDOF- Free Vibration
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MDOF- Free Vibration
Natural Frequencies
Mode shapes
Or in matrix form
Modal Model
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2DOF System- Free Vibration
Solving the equation
Numerically
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Orthogonality Properties of MDOF
The modal model possesses some very important
Properties, stated as
Modal mass matrix
Modal stiffness matrix
Exercise Prove the orthogonality property of MDOF
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Mass-normalisation
The mass normalized eigenvectores are written
as And have the following property
The relationship between mass normalised mode
shape and its more general form is
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Mass-normalisation of 2DOF
Clearly
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Multiple modes

• The situation where two (or more) modes have the
  same natural frequency.
• It occurs in structures with a degree of
  symmetry, such as discs, rings, cylinders.
• Free vibration at such frequency may occur not
  only in each of the two modes but also in a
  linear combination of them.

c
a
a Vertical mode b Horizontal mode c Oblique mode
b
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Forced Response of MDOF
Or by rearranging
Which may be written as
Response model
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Forced Response of MDOF

• The values of matrix H can be computed easily
  at each frequency point. However, this has
  several advantages
• It becomes costly for large N.
• It is inefficient if only a few FRF expression is
  required.
• It provides no insight into the form of various
  FRF properties.
• Therefore, we make use of modal properties for
  deriving the FRF parameters instead of spatial
  properties.

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Forced Response of MDOF
Premultiply both sides by and postmultiply
by
Inverse both sides
Equation 1
Note that
Diagonal matrix
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Forced Response of MDOF
As H is a symmetric matrix then
or
Principle of reciprocity
Using equation 1
or
Modal constant
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Forced Response of 2DOF
Which gives
Numerically
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Forced Response of 2DOF
Or numerically
Receptance FRF ( ) for 2dof system