Page 151 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

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

1 )

Figure 5.2-2. Demonstration model
for exchange of energy by beating.
(Courtesy of UCSB Mechanical Engi­
neering Undergraduate Laboratory.)

5.3 COORDINATE COUPLING
The differential equations of motion for the 2-DOF system are in general coupled,
in that both coordinates appear in each equation. In the most general case, the two
equations for the undamped system have the form

(5.3-1)
m2\X^ + 0^22^2 E k2\X^ + ^22-^2 ^ ^
These equations can be expressed in matrix form (see Appendix C) as

m , , m , 2 ■ ^ n kn 0
(5.3-2)
^ 2 1 "* 2 2 ki\ ki2 0
which immediately reveals the type of coupling present. Mass or dynamical
coupling exists if the mass matrix is nondiagonal, whereas stiffness or static
coupling exists if the stiffness matrix is nondiagonal.
It is also possible to establish the type of coupling from the expressions for
the kinetic and potential energies. Cross products of coordinates in either expres­
sion denote coupling, dynamic or static, depending on whether they are found in T