TABLE OF CONTENTS Now that we've seen how to put our system into normal form, we can go ahead and tidy up all the remaining loose ends. To start, let's consider what to do when our problem has an external forcing. In this case, the equations of motion are given by
If we go ahead and use X = [ U] H to decouple this set of equations, we'll obtain
where
This was derived a few pages back ((4.7.13) and (4.7.14)) and is simply rewritten here for convenience. If the modal matrix [ U] has been mass normalized then [ M ?] is equal to the identity matrix and [ K'] is equal to [ ? 2]. Thus our equations are given by
This is a completely decoupled problem. In this form, our problem is exactly equivalent to n individual, second-order equations. Taking the ith one at random, we have
where f ? i is simply the ith component of the modified forcing vector. We know from earlier work that ? i represents the degree of participation of the ith eigenvector in the response. Thus we can clearly see how the external force affects the various modes of the system.
1.
Consider the 2 DOF example we recently looked at (in which m 1 = m 2 = 1 and k 1 = k 2 = k 3
