It may be noted from Eq. (6.6.6) that when the exponent terms of the marked mixed secular terms are equal to a resonance condition will occur. Hence, resonance will be observed in the system when

For the non resonant case, i.e., when the external frequency is away from 0, , or 3 , from Eq. (6.6.6) eliminating the secular terms yield the following equation.

........................................................................( 6.6.7 )

Now using in Eq.(6.6.7) and separating the real and imaginary parts following reduced equations are obtained.

.............................................................(6.6.8)

and (6.6.9)

... ( 6.6.10 )

The free oscillation solution decays with time and hence the steady state response consists of forced solution only similar to the linear case.

Superharmonic Resonance ( )

To express the nearness of the external excitation frequency to one third of the natural frequency one may use the detuning parameter ( σ) as follows.