System with non resonant hard excitations

In the previous lecture a single degree of freedom nonlinear system is considered when the amplit
ude of the external excitation ( f ) is one order less than the linear term (i.e. ). In the present lecture the forcing term is assumed to be of the same order as that of the linear term. So the equation of motion considered in this case is

.................................................................................(6.6.1)

Following similar procedure of method of multiple scales, one may write

..............................................................................(6.6.2)

Now separating the terms with different order of ε one obtains the following equations.

.............................................................................................( 6.6.3 )

.................................................................... ( 6.6.4 )

The solution of Eq. (6.6.3) can be written as

.................................................................... (6.6.5)

Where

It may be noted that unlike the previous lecture, where only the complementary part of the solution was present, in this case both complimentary and particular integral parts are present in the solution of . Now substituting Eq. (6.6.5) in Eq. (6.6.4) one obtains the following equation.

( 6.6.6 )