MODAL ANALYSIS
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Tacoma Narrows Galloping Gertie
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Flutter of Boeing 747 wings
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B52 parked
B52 flying
Note deflection of wings
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1DOF.SLDASM
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Every structure has its preferred frequencies of vibration, called resonant frequencies. Each such frequency is
characterized by a specific shape of vibration. When excited with a resonant frequency, a structure will vibrate in this
shape, which is called a mode of vibration.
Recall that structural static analysis calculates nodal displacements as the primary unknowns:
[K]d = F
where [K] is known as the stiffness matrix, d is unknown vector of nodal displacements and F is the known vector of
nodal loads.
In dynamic analysis we additionally have to consider damping [C] and mass [M]
In a modal analysis, which is the simplest type of dynamic analysis we investigate the free vibrations in the absence
of damping and in the absence of excitations forces. Therefore, the above equation reduces to:
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Non-zero solutions of a free undamped vibration present the eigenvalue problem:
 K  - ωi 2  M    = 0
i


ωi
2
eigenvalue (square of circular frequency)
 i
eigenvector (shape)
ω
circular frequency [rad/s]
ω
2π
1
f=
T
f=
frequency [Hz]
Solutions provide with eigenvalues and associated modal shapes of vibration
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Note, that the equation of free undamped vibrations can be re-written to show explicitly that in resonance inertial
forces cancel out with elastic forces.
Inertial forces
Ki = ωi Mi
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Elastic forces
In resonance, inertial stiffness subtracts from elastic stiffness and, in effect, the structure loses its stiffness.
The only factor controlling the vibration amplitude in resonance is damping. If damping is most often low, therefore,
the amplitude of may reach dangerous levels.
Note that even though any real structure has infinite number of degrees of freedom it has distinct modes of
vibration. This is because the cancellation of elastic forces with inertial forces requires a unique combination of
vibration frequency and vibration mode (shape).
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Material density must be defined in units derived from the unit of force and the unit of length.
[mm] [N]
unit of mass
unit of mass density
for aluminum
tonne
tonne/mm3
2.794x10-9
[m] [N]
unit of mass
unit of mass density
for aluminum
kg
kg / m3
2794
[in] [lb]
unit of mass
unit of mass density
For aluminum
lbf = slug/12
slug/12/in3
lbf s2/in4
2.614x10-4
Notice that the erroneous mass density definition (kg / m 3 instead of tonne / mm3) will results in part mass
being one trillion (1e12) times higher.
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U SHAPE BRACKET
model file
model type
material
U BRACKET
shell
alloy steel
thickness
2mm
restraints
hinge
load
none
hinge support
(no translations)
hinge support
(no translations)
objective
• demonstrate modal analysis
• study convergence of natural frequencies
• defining supports for shell element model
• properties of lower and higher modes of vibration
U BRACKET
SAE models
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cantilever beam.SLDPRT
04 models modal
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TUNNING FORK
PLASTIC PART
Chapter 6
Chapter 6
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truck.SLDPRT
car.SLDPRT
04 models modal
04 models modal
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EXERCISE helicopter blade
Model file
ROTOR
Model type
solid
Material
1060 Alloy
Supports
fixed to the I.D.
sym B.C. to hub
Loads
centrifugal load due to 300RPM
Units
mm, N, s
Objectives
• Modal analysis without pre-stress
• Modal analysis with pre-stress
Analysis is conducted on one blade only.
ROTOR
CHAPTER 21
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pendulum 02.SLDPRT
04 models modal
15