Page 148 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 5.2 Initial Conditions 135

the natural frequencies and mode shapes are

0)2 = y + 2---- ^
( 1) ( 2)
1.0 = -1.0

Thus, in the first mode, the two pendulums move in phase and the spring remains
unstretched. In the second mode, the two pendulums move in opposition and the
coupling spring is actively involved with a node at its midpoint. Consequently, the
natural frequency is higher.

5.2 INITIAL CONDITIONS
When the normal mode frequencies and mode shapes are known, it is possible to
determine the free vibration of the system for any initial conditions by the proper
summation of the normal modes. For example, we have found the normal modes
of the system of Fig. (5.1-1) to be
0.732
o), = y^0.654/c/m - 1.000

- 2.732 \
CÜ = ^2366k/m (¡>2 1.000/
2
For free vibration to take place in one of the normal modes for any initial
conditions, the equation of motion for mode i must be of the form
(/)
= sin + (/^,) i = 1,2 (5.2-1)

The constants c, and are necessary to satisfy the initial conditions, and (/> ,
ensures that the amplitude ratio for the free vibration is proportional to that of
mode i.
For initial conditions in general, the free vibration contains both modes
simultaneously and the equations of motion are of the form
0.732 -2.732;
1.000 sin (iOjf + <Ai) + C2 1.000 } ( ^ 2^ + ^ 2) (5.2-2)
=
where Cj, C2, iAi? the four necessary constants for the two differential
equations of second order. Constants Cj and C2 establish the amount of each
mode, and phases ij/^ and 1I/2 allow the freedom of time origin for each mode. To
solve for the four arbitrary constants, we need two more equations, which are