Page 133 - Finite Element Analysis with ANSYS Workbench

124 Chapter 7 Vibration Analysis

T
        
M
Nd
N
N
where   is the element interpolation function matrix of the
element type selected. As an example, the mass matrix correspond-
ing to the two-node truss element with the length of L and cross-
sectional area of A in chapter 2 is,
 2 AL  1 
    1 6  2  
M
Similarly, the mass matrix of the two-node beam element with the
length of L and cross-sectional area of A in chapter 3 is,
  156 22L 54  13L  

 M   AL  22L 4L 2 13L  3L 2 
420  54 13L 156  22L
  13L   3L 2  22L 4L  
2
The mass matrices of other element types can also be determined in
the same way. It is noted that many symbolic manipulation
software packages, such as MATLAB, Mathematica, Maple,
Maxima, etc., can be used to derive the mass matrices for many
element types in closed-form expressions. The element matrices in
closed-form expressions can help reducing the computational time.
Several finite element computer programs have been
developed to analyze structural vibration and dynamics problems in
the past. We will employ ANSYS through its Workbench to
analyze an academic type and realistic problems in the following
sections.

7.3 Academic Example

7.3.1 Vibration of Thin Plate
A square plate with the dimensions of 11 m and
thickness of 0.01 m is shown in the figure. The plate is made from
a material that has the Young’s modulus of 10.92 10 N m 6 2 and
Poisson’s ratio of 0.3. The plate is clamped along its four edges.
We will employ ANSYS through the Workbench to determine its