ABSTRACT
Consider free oscillations of a conservative system with quadratic nonlinearity, governed by
U¨(t) = f [U(t), U˙(t), U¨(t)], (12.1)
where t denotes the time, the dot denotes derivative with respect to t, and f [U(t), U˙(t), U¨(t)] is a known function of U(t), U˙(t), and U¨(t). Physically, free oscillation of conservative systems is a periodic motion. Let ω and a denote the frequency and amplitude of oscillation, respectively. Define the mean of motion
δ = 1 T
U(t)dt, (12.2)
where T = 2π/ω is the period of oscillation. For conservative systems with quadratic nonlinearity, the mean of motion δ is generally nonzero. This is the main difference between free oscillations of conservative system with odd nonlinearity and those with quadratic one. Obviously, both δ and ω have clear physical meanings. For conservative systems, the oscillation amplitude a is determined by initial conditions and is related to the total kinetic energy. Both ω and δ are dependent of a. Without the loss of generality, we consider oscillations with amplitude a under the initial conditions
U˙(0) = 0 U(0) = a + δ. (12.3)
Unlike perturbation techniques, we need not assume that Equation (12.1) contains any small/large parameters.
