Page 147 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

134 Systems with Two or More Degrees of Freedom Chap. 5

At this point, we introduce the matrix notation, writing Eqs. (5.1-8) in the
concise form:

0 \ k , + k ^) -^2
(5.1-9)
0 J2 “ ^2 (^2 + ^ 3)
By following the rules for matrix operations in Appendix C, the equivalence of the
two equations can be easily shown.
A few points of interest should be noted. The stiffness matrix is symmetric
about the diagonal and the mass matrix is diagonal. Thus, the square matrices
are equal to their transpose, i.e., [kY = [k\ and [mY = [m]. In addition, for
the discrete mass system with coordinates chosen at each mass, the mass
matrix is diagonal and its inverse is simply the inverse of each diagonal element,
i.e., [m]~^ = [1 /m].
Example 5.1-3 Coupled Pendulum
In Fig. 5.1-4 the two pendulums are coupled by means of a weak spring k, which is
unstrained when the two pendulum rods are in the vertical position. Determine the
normal mode vibrations.
Solution: Assuming the counterclocl:wise angular displacements to be positive and taking
moments about the points of suspension, we obtain the following equations of motion
for small oscillations

= —mgW^ —ka^{6^ —O2)

ml^d2 = -mgl02 + ka^(6^ - 62)
which in matrix notation becomes
1 0 (M. ( -t- mgl) -ka^
w/2 (5.1-10)

0 1 —ka^ ( ka^ + mgl)
Assuming the normal mode solutions as
6, = cos (ot
Aj cos cot

Figure 5.1-4. Coupled pendulum.