Approximate Methods For A Class Of Nonlinear Jerk Equations

The phenomenon of nonlinear vibration is very common in engineering and technology. It is important for the application of the project technology. Most of nonlinear vibration problems can be described by the second-order nonlinear differential equations, but some nonlinear vibration problems or nonlinear dynamic systems, such as the Rossler system and the Lorenz system, can be described by the third-order nonlinear differential equations. In 2004, the Australian scholar Gottlieb [H.P.W.Gottlieb. Harmonic balance approach to periodic solutions of nonlinear jerk equations [J]. Journal of Sound and Vibration, 2004, 271: 671-683] used the lowest-order harmonic balance method to calculate the analytical approximate periodic solutions of the Jerk equation with cubic nonlinearities. Since the accuracy of Gottlieb's solutions is not high, many scholars have been interested in the high-order approximate solutions of the Jerk equation.In this paper, the classical Multiple Scale Method, the modified Multiple Scale Method, the modified two-variable expansion method, the new iteration procedure and the Homotopy Analysis Method are applied to determine the approximate solutions of the Jerk equation with cubic nonlinearities. The approximate solutions obtained have high accuracy. The main content and innovative points of the dissertation as follows:To begin with, the classical Multiple Scale Method and the modified Multiple Scale Method are, respectively, used to solve the nonlinear Jerk equation that does not have the linear part of the velocity. By comparison, we find out that the classical method is no longer suitable to solve this nonlinear Jerk equation. However, the modified method combining the Multiple Scale Method and Lindstedt-Poincarétechniques is still valid for the nonlinear Jerk equation that does not have the linear part of the velocity.Then, the modified two-variable expansion method is applied to calculate the analytical approximate solution of a nonlinear Jerk equation. Comparing the modified two-variable expansion method with the Multiple Scale Method, we can find that the procedure of calculating the approximations of the nonlinear Jerk equation by the two-variable expansion method is simple.Furthermore, a new iteration procedure is applied for the nonlinear Jerk equation. The example shows that we only need solve simple algebraic equations to determine the approximate angular frequency when introducing new variablesτ=ωt, y =ωx and applying the new iteration procedure. The second-order approximate solution obtained is more accurate than the result obtained by the Homotopy Perturbation Method.Finally, the Homotopy Analysis Method is used to yield the analytical approximate solutions of the nonlinear Jerk equation. The example illustrates that inducing new variablesτ=ωt, y =ωx can simplify the initial condition and diminish computation. The important advantage of the Homotopy Analysis Method is that it can control solution convergence region and convergence speed through the parameters. The second-order approximation obtained has a high accuracy.