Page 129 - Finite Element Analysis with ANSYS Workbench

120 Chapter 7 Vibration Analysis

The idea above could be applied to problem with
complicated geometry. The frequency  of the external forces is
f
often difficult to control while the natural frequency  of the
system is controllable. Since we know that the natural frequency of
the system depends on the overall stiffness and mass, we can alter
either the system stiffness or mass. The system mass is not easy to
change in general but its stiffness can be altered by modifying,
(a) the model geometry
(b) the material
(c) the boundary conditions
The finite element method can provide natural frequency solutions
conveniently for different model configuration, materials and
boundary conditions. The method is thus suitable for vibration
analysis of complicated structures.
Since the structures often consist of the truss, beam,
plate and solid components, we will look at the differential
equations that govern their vibration behaviors as follows.

1D Truss

 2 u  2 u
 A  EA
t  2 x  2
The displacement u  u (, ) varies with the axial coordinate x of
x
t
the truss and time ,t  is the material density, A is the cross-
sectional area, and E is the material Young’s modulus.

1D Beam

 2 w  4 w
 A  EI
t  2 x  4
The deflection w  w (, ) varies with the axial coordinate x of the
t
x
beam and time ,t  is the material density, A is the cross-sectional
area, E is the material Young’s modulus and I is the moment of
inertia of area.