The homotopy perturbation method for a nonlinear oscillator with a damping

Abstract

The homotopy perturbation method is extended to solve nonlinear oscillators with damping terms, and an explicit relationship between the frequency and amplitude is obtained, the main factor affecting the periodic property is revealed.

The harmonic balance method for Yao–Cheng oscillator

“…This paper shows that the harmonic balance method 1 is as effective as the homotopy perturbation method [10][11][12][13] for Yao-Cheng oscillator. The advantages of the harmonic balance method are: (1) no need for construction of the homotopy equation, which is a must in the homotopy perturbation method; (2) no expansion for the solution and the parameter involved in the linear term.…”

“…where k, b, and c are constants. Equation (1) can be effectively solved by the homotopy perturbation method, 12 which is generally effective for oscillators without a damping term. [13][14][15][16][17][18] The harmonic balance method…”

“…Equation 7is exactly same as that by the homotopy perturbation method. 12 A fractal modification of Yao-Cheng oscillator…”

He’s frequency–amplitude formulation for nonlinear oscillator with damping

“…which is the same as the algebraic equation obtained through homotopy perturbation method. 1 The approximate frequency of equation 1can be computed from equation 17x ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi…”

Frequency and solution of an oscillator with a damping

“…with a velocity-dependent damping. This oscillator can be solved by various methods, [10][11][12][13] among which the variational iteration method accompanied by Laplace transform 14 and the modified He's amplitude-frequency formulation 15 have been caught much attention due to their simple solution process and high accurate solutions.…”