512
13 Finite Difference Method for the Vibration Analysis …
Applying virtual work principles, we get
{X }
T
e {F I } =
v
(dF)
T
{F I }
(13.59)
where {F I } is the nodal inertia force parameters.
Combining Eqs. (13.56), (13.58) and (13.59) yields
{X }
T
e {F I } =
v
ρdV [N mw ]{ ¨
X } e
(13.60)
Equation (13.60) can be written as
{F I } = ρ
v
[N mw ]
T
[N mw ] dV { ¨
X } e = [M] e { ¨
X } e
(13.61)
where
[M] e = ρ
[N mw ]
T
[N mw ]dV
(13.62)
[M] e for a plate strip of uniform thickness t is given below when rotational
displacements are included
[M] e =
ρt
3
12
¨
[N mw ]
T
,x [N mw ] ,x + [N mw ]
T
,y [N mw ] ,y
dx dy
+ ρt
¨
[N mw ]
T
[N mw ] dx dy
(13.63)
Exercise 13
13.1 Using finite difference method, determine the fundamental frequency of the
uniform beam shown in the figure, by dividing the beam into four equal parts.
13.2 A simply supported square plate with mesh division is shown in the figure.
Determine the fundamental frequency. Take advantage of the symmetry about
both axes.