Example 2.4: Obtain the unbalance response of a rotor system with following equations of motion.

Solution: Since equations of motion are uncoupled hence both equations can be solved independently. The first equation can be written as

in which the real part of right hand side term has only meaning. The solution can be assumed as

where in general X is a complex quantity. The above equation gives

On substituting assumed solutions in the equation of motion, we get

which gives

Hence the response in the x-direction becomes

Similarly, the second equation of motion can be written as

in which the real part of the right hand side term only has a meaning. The solution can be assumed as

On substituting in the equation of motion, we get

Hence the response in the y-direction becomes

Now the same problem is solved in the matrix form. We have

so that

The response in the x- and y-directions can be written as

It should be noted that for undamped forced vibration, equation takes the form [Z]{X} = 0. To get the natural frequency of the system we need to put  and it gives