## Multiple Degree of Freedom Systems: Forced Vibrations of Undamped Two Degree of Freedom Systems

The general form of the equations of motion for an undamped two degree of freedom system

subjected to external loading are

The solution to this general form of the equations of motion is beyond the scope of this course (but only slightly). However, we will consider a special case in which the forcing functions and are harmonic, at the same frequency and in-phase with each other. In such a case the equations of motion become

(8.34)

or

To find the steady–state response of the system, we again look for solutions in which the masses are undergoing simple simultaneous harmonic motion at the forcing frequency

The equations of motion become

which to be valid for all times requires

(8.35)

or

where is defined as

Unlike the free vibration situation, the RHS of equation 8.35 is not zero, so that a unique solution for the amplitudes of each of the masses can be found as

(8.36)

which is valid provided . Since is a matrix, the inverse is

so we see from equation (8.36) that the steady–state amplitudes of the response are

(8.37)

with

Determine the steady state response for the system shown.