CHAPTER 10 
APPROXIMATION METHODS FOR THE ANALYSIS OF NONLINEAR 
SYSTEM IN RANDOM EXCITEMENT 
10.1 INTRODUCTION 
Nonlinear vibration systems have been studied for a long time. The equivalent linea- 
rization is the most popular method for handling the weakly nonlinear system, and the 
perturbation method has been used to obtain the equivalent linear expressions. However, 
most of the studies have been concerned with systems under deterministic excitation. In 
this chapter, systems under random or stochastic excitation are treated. The equivalent 
linearization method and the perturbation method are treated independently for 
convenience of explanation. 
10.2 EQUIVALENT LINEARIZATION METHOD 
Suppose there is a weakly damped, slightly nonlinear oscillation expressed as 
X + ax + Ble + wox + koe = FO). (10.1) 
When, for example in case of rolling with moderate amplitude, the damping term 
includes the quadratic term Blxlx for viscous damping in addition to the linear damping 
ax, the restoring term includes the cubic term 4° that comes from the shape of the right- 
ing arm curve (stability curve). Generally when k20, this system is called a hard or soft 
spring oscillation system. 
In the deterministic case, or when this system is exposed to a harmonic excitation 
F(t), this nonlinear damping can be linearized with the equivalent linear damping 
8a 
Qe=at+a,=at eed) Bxo (10.2) 
LA 
using the amplitude of oscillation Xp. 
This relation was derived by Jacobson (1930),’* equating the energy dissipation by 
the damping term ax + Blilt and by @_* during one period of oscillation at amplitude Xo. 
In the same way, equating the work done by the restoring term at amplitude xo, the equiv- 
alent linear restoring coefficient we, is as shown in Fig. 10.1, 
1 
We = 0 +5 ko. (10.3) 
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