AMPLITUDE 
the special forms as follows, the system is called a Duffing type or a Van del Pol type 
oscillator, respectively, as described in Chapter 10. 
Duffing type: 
gi\X(p} =c 
g2{X(t)| = a + BX? (13.32) 
Van del Pol type: 
gilX(0)} = —c{1-x(er| 
g2(X()} = a. (13.33) 
For the Duffing type oscillator, under harmonic excitement F coswt, 
X(t) +cX+aXx + BX? = Fcosat. (13.34) 
We know that, with harmonic excitement, the response shows damping phenomena, 
as indicated in Fig. 13.1, and in this case the natural frequency is dependent on amplitude. 
So we can call this oscillation an amplitude—dependent period shifting oscillation. 
SOFT SPRING TYPE 
HARD SPRING TYPE B<0 
| B>0 ~~ 
Ww 
Q 
E y) 
a) 
< 
ayy — Wo —>- o 
Fig. 13.1. Duffing type oscillator. 
For the Van del Pol oscillator, 
¥-cl1-X*}X +x =0, forc >0,a>0 (13.35) 
the system possesses a limit cycle, because when the amplitude X is small, the damping 
becomes negative and X starts to diverge, and when X(t) becomes large, the damping be- 
comes positive and the amplitude starts to decay. This system remains in oscillation 
without any excitation. 
When we have random noise excitation n(r) 
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