Page 128 - Finite Element Analysis with ANSYS Workbench

7.1 Basic Equations 119


ut  A sin t  B cos t
()
where A and B are constants that can be determined from the initial
conditions. As an example, if the initial displacement and velocity
are u and zero, respectively, the mass movement behavior is as
0
shown in the figure.
For an oscillating cycle of T, the frequency f that
represents the cycles per second, or Hertz, is,
1
f 
T
Thus, the circular frequency  and the frequency f are related by,



 2
f
  

(rad sec) (2 rad cycle) (cycles sec)
The value of  above is also known as the natural circular
frequency. The oscillation in this classical example is called free
vibration.

If the mass is subjected to an
external force ()Ft in the form,
t
k F ()  F 0 sin f t
then, the governing differential equation of
the mass-spring system becomes,
m
2
ut m du  k u  F 0 sin f t
()
Ft dt 2
()

The general solution of this differential equation is,
( sF i n  ) t k
() 
ut A sin t   B cos t   0 f
1( f ) 2
The last term in the solution above suggests that the oscillating
magnitude ()ut becomes very large if the applied forcing frequency
 is closed to the natural frequency  of the system. Knowing
f
the natural frequency  of the system is thus important to avoid an
uncontrollable vibration caused by the external force.