12.6.2 Stationary Response of Oscillations with Nonlinear 
Damping to Random Excitation 
As was mentioned at the beginning of this section, the Fokker—Planck equation has 
been solved only for restricted cases, such as nonlinear systems with nonlinear restoring 
forces or systems under white noise excitation. Roberts*? removed these restrictions and 
studied the behavior of the Fokker—Planck equation for a system with nonlinear damping. 
For a nonlinear system with nonlinear damping, the equation of motion is 
X + BF(x, x) + G(x) = n(t), (12.84) 
where f is small, F is an odd function of % (odd as was used by Dalzell®! is more 
convenient to manipulate), and n(t) is white noise, G(x) being the restoring term. 
The two—dimensional Fokker—Planck equation for 
P(x, x) = p(x, yl%o, Yo; 2) 
is the same as Eq. 12.30, for Eq. 12.20 in the form, 
op op 0 Jf ap 
a SS SJ SS F ———- 12. 
an ae + ay [6 (x,y) + G(x)|p] + BER (12.85) 
and is difficult to solve for general F(x, y), so Roberts inverted this two—dimensional 
Fokker—Planck equation into a one-dimensional Fokker—Planck equation, introducing a 
physical variable called the energy envelope V(t) 
V(t) = +2 + U(x), (12.86) 
x 
therefore x = /2V(t)— U(d) where U(x) = | G(&)dé. 
0 
The one-dimensional Fokker—Planck equation for p(v) = P(v\vo; t) is in the form 
ap a ji InP 
neh Syne Biv) -— ee 5 
a ey (Oa iP aan (12.87) 
W, 
where / is the strength of excitation as ] = oa in Eq. 12.22 or Eq. 12.29, and 
Biv), C(v) areas in Eqs. 12.75—12.77 
347 
