Probabilistic analysis of a nonlinear pendulum
The equation of the noise-free pendulum is \(\ddot x+\sin x=\varepsilon f(x, \dot x,t)\), where \(f(x, \dot x,t)= f(x, \dot x,t+T)\) and \(0<\varepsilon\ll 1\). In order to obtain closed-form solutions, the function \(f(x,\dot x,t)\) is taken to be \(-\beta\dot x+F \cos\Omega t\). Period-1 solutions corresponding to the primary resonance of the system are looked for. The analysis is restricted to oscillatory motions only, and the purpose is to avoid the rotational-type motion. By averaging the fast oscillation of the response, the resonant orbits are mapped into equilibrium points in the space of the energy and resonant phase variables. Random excitations may create a transition from one regime to another, and in this way some failure can occur. Two types of random perturbations are introduced: a purely additive, and phase fluctuation of the resonant force. The averaging method is extended to stochastic excitations. Using the Fokker-Planck-Kolmogorov equation and solving it in the form of a WKB approximation, the author obtains the probability of the failure of the oscillatory regime. For certain numerical cases very instructive plots visualize a part of the results.
scientific article; zbMATH DE number 940958
scientific article