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Approximations to the free response of a damped non-linear system Chan, Paul Tsang-Leung


In the study of many engineering systems involving nonlinear elements such as a saturating inductor in an electrical circuit or a hard spring in a mechanical system, we face the problem of solving the equation ẍ + 2εẋ + x + μx³ = 0 which does not have an exact analytical solution,. Because a consistent framework is desirable in the course of the study, we can assume that the initial conditions are x(0) = 1 and ẋ(0) = 0 without loss of generality. This equation is studied in detail by using numerical solutions obtained from a digital computer. When ε and μ are small, classical methods such as the method of variation of parameters and averaging methods based on residuals provide analytical approximations to the equation and enable the engineer to gain useful insight into the system. However, when ε and μ are not small, these classical methods fail to yield acceptable results because they are all based on the assumption that the equation is quasi-linear. Therefore, two new analytical methods, namely: the parabolic phase approximation and the correction term approximation, are developed according to whether ε < 1 or ε ≥1, and are proven to be applicable for values of ε and μ far beyond the limit of classical methods.

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