A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

Abstract

a b s t r a c tIn the present paper, a complicated strongly nonlinear oscillator with cubic and harmonic restoring force, has been analysed and solved completely by harmonic balance method (HBM). Investigating analytically such kinds of oscillator is very difficult task and cumbersome. In this study, the offered technique gives desired results and to avoid numerical complexity. An excellent agreement was found between approximate and numerical solutions, which prove that HBM is very efficient and produces high accuracy results. It is remarkably important that, second-order approximate results are almost same with exact solutions. The advantage of this method is its simple procedure and applicable for many other oscillatory problems arising in science and engineering. IntroductionNonlinear oscillations are important fact in physical science, mechanical structures and other engineering problems. Naturally, all differential equations involving engineering and physical phenomena are nonlinear. The methods of solutions of linear differential equations are comparatively easy and well established. On the contrary, the techniques of solutions of nonlinear differential equations (NDEs) are less available and have no exact solution and, in general, linear approximations are frequently used. Nowadays, NDEs have been the subject of all-embracing studies in various branches of nonlinear science and engineering. A special class of analytical solutions named strongly nonlinear oscillator with cubic and harmonic restoring force has a lot of importance, because, most of the phenomena that arise in mathematical physics and engineering fields can be described by NDEs. Therefore, investigating strongly nonlinear oscillator with cubic and harmonic restoring force solutions is becoming increasingly attractive in nonlinear sciences. Moreover, obtaining exact solutions for nonlinear oscillatory problems has many difficulties. It is very difficult to solve nonlinear problems and in general it is often more difficult to get an analytic approximation than a numerical one for a given nonlinear problem. To overcoming the shortcomings, many new analytical techniques have been successfully developed by diverse groups of mathematicians and physicists, of Modified Differential Transforms Method [13][14][15][16], and so on. Several other authors used many powerful analytical methods in the field of approximate solutions especially for strongly nonlinear oscillators like Max-Min Approach Method [17,18], Algebraic Method [30-33], Newton-harmonic Balancing Approach [34], and so on for solving NDEs. The HBM is another technique for solving strongly nonlinear systems. Borges et al. [35] and Bobylev et al. [36] first provided overviews of HBM. Mickens [37-39] was first applied HBM in truly nonlinear oscillators. Due to his contribution he is known as father of HBM. Afterwards, Belendez et al. [40] and others researchers [41-43] has significantly improved the HBM.The HBM provides a general technique for calculating approximations to the periodic solutio...

Generic numerical and analytical methods for solving nonlinear oscillators

“…Nonlinear oscillations have wide applications across various fields such as physics [1][2][3][4][5], mathematics [6][7][8][9], mechanics, and engineering [10][11][12][13][14][15]. In order to describe these oscillators accurately, nonlinear differential equations (NDEs) are often employed.…”

“…These methods include the harmonic balance method [16,17], variational iteration method [18,19], homotopy analysis method [20,21], homotopy perturbation method [22,23], Li-He's modified homotopy perturbation method [24], enhanced homotopy perturbation method [25], asymptotic method [26,27], energy balance method [28], differential transformation method [29], parameter expansion method [30,31], variational principle [32], and frequency-amplitude formulation [33], among others. In this paper, the nonlinear oscillation with cubic and harmonic restoring force [1,34,35] will be explored:…”

A simple modified harmonic balance method for strongly nonlinear oscillator with cubic non-linearity and harmonic restoring force

“…Homotopy perturbation method was applied by others 13,14 appropriately. Also, many researchers were applied the harmonic balance method, 15 the energy balance method, 16 He's frequency formulation method, 17,18 the variational iteration method, [19][20][21][22] the homotopy analysis method, 23,24 VIM-Pade technique 25,26 etc. successfully to solve strongly nonlinear problems.…”

The VIM-Padé technique for strongly nonlinear oscillators with cubic and harmonic restoring force

“…In the past decades, several methods were proposed for solving these nonlinear problems. Such methods include homotopy analysis method, 1 variational iteration method (VIM), 2,3 homotopy perturbation method, 4,5 modified homotopy perturbation method, 6 asymptotic method, 7,8 differential transformation method, 9 He’s energy balance method, 10 harmonic balance method, 11,12 modified harmonic balance method, 13 parameter expansion method 14,15 and other methods.…”

“…In this paper, we consider a strongly nonlinear oscillator with cubic and harmonic restoring force 13 where a and b are given constants, and the initial conditions are given by …”