Accurate approximations of the nonlinear vibration of couple-mass-spring systems with linear and nonlinear stiffnesses

An analytical technique has been developed based on the harmonic balance method to obtain approximate angular frequencies. This technique also offers the periodic solutions to the nonlinear free vibration of a conservative, couple-mass-spring system having linear and nonlinear stiffnesses with cubic nonlinearity. Two real-world cases of these systems are analysed and introduced. After applying the harmonic balance method, a set of complicated higher-order nonlinear algebraic equations are obtained. Analytical investigation of the complicated higher-order nonlinear algebraic equations is cumbersome, especially in the case when the vibration amplitude of the oscillation is large. The proposed technique overcomes this limitation to utilize the iterative method based on the homotopy perturbation method. This produces desired results for small as well as large values of vibration amplitude of the oscillation. In addition, a new suitable truncation principle has been used in which the solution achieves better results than existing solutions. Comparing with published results and the exact ones, the approximated angular frequencies and corresponding periodic solutions show excellent agreement. This proposed technique provides results of high accuracy and a simple solution procedure. It could be widely applicable to other nonlinear oscillatory problems arising in science and engineering

Solving linear and non-linear stiff system of ordinary differential equations by multistage adomian decomposition method

In this paper, linear and non-linear stiff systems of ordinary differential equations are solved by the classical Adomian decomposition method (ADM) and the multistage Adomian decomposition method (MADM). The MADM is a technique adapted from the standard Adomian decomposition method (ADM) where standard ADM is converted into a hybrid numeric-analytic method called the multistage ADM (MADM). The MADM is tested for several examples. Comparisons with an explicit Runge-Kutta-type method (RK) and the classical ADM demonstrate the limitations of ADM and promising capability of the MADM for solving stiff initial value problems (IVPs)

High-order approximate solutions of strongly nonlinear cubic-quintic Duffing oscillator based on the harmonic balance method

In this paper, a new reliable analytical technique has been introduced based on the Harmonic Balance Method (HBM) to determine higher-order approximate solutions of the strongly nonlinear cubicquintic Duffing oscillator. The application of the HBM leads to very complicated sets of nonlinear algebraic equations. In this technique, the high-order nonlinear algebraic equations are approximated in the form of a power series solution, and this solution produces desired results even for small as well as large amplitudes of oscillation. Moreover, a suitable truncation formula is found in which the solution measures better results than existing results and it saves a lot of calculation. It is highly noteworthy that using the proposed technique, the third-order approximate solutions gives an excellent agreement as compared with the numerical solutions (considered to be exact). The proposed technique is applied to the strongly nonlinear cubic-quintic Duffing oscillator to reveals its novelty, reliability and wider applicability

An analytical technique to obtain higher-order approximate periods for nonlinear oscillator

ABSTRACT: This paper presents simulation results of the influence of wide range modulation index values ( ) in carrier-based PWM strategy for application in generating the stepped waveform. The waveform is tested for application in single-phase half-bridge modular multilevel converters (MMCs) topology. The results presented in this paper include a variation of the fundamental component (50 Hz) in the voltage output.  It also studies total harmonic distortion of the output voltage (THDv) and the output current (THDi) when the modulation index is changed over the linear-modulation region, 0 < < 1. It also explores the effect of a modulation index greater than 1. Moreover, different output voltage shapes, as a consequence of varied on MMCs, are also illustrated for showing the effect of varying the value of on sub-module of MMCs. ABSTRAK: Penulisan ini berkenan simulasi pengaruh pelbagai nilai indeks modulasi     ( ) dalam strategi PWM berasaskan aplikasi dalam menghasilkan bentuk gelombang yang bertingkat. Bentuk gelombang ini diuji untuk aplikasi dalam topologi MMCs. Penilaian dan hasil dari artikle ini termasuk variasi komponen asas (50 Hz) dalam voltan keluar. Ia juga meneliti jumlah penyelarasan harmonik voltan keluar (THDv) dan arus keluaran (THDi) apabila indeks modulasi ditukar dalam rantau modulasi linear, 0 < <1. Ia juga meneroka kesan indeks modulasi lebih daripada 1. Selain itu, bentuk voltan keluar yang berbeza sebagai akibat dari pelbagai  pada MMCs juga digambarkan untuk menunjukkan kesan berbeza-beza nilai  pada sub-modul MMCs

An analytical approximation technique for the duffing oscillator based on the energy balance method

In this paper, an analytical approximation technique has been presented of obtaining higher-order approximate solutions for highly nonlinear Duffing oscillator based on the energy balance method (EBM). Higher-order approximate natural frequencies have been obtained in a novel analytical way. The accuracy of the solution method is evaluated within as error analysis. It is highly remarkable that using the presented technique, the approximation solutions produces desired results even for large oscillation as compared with the exact ones. Moreover, the solution method yields much better results than existing solutions after using a suitable truncation formula. The presented technique is applied to well-known Duffing oscillator to illustrate its novelty, reliability and wider applicabilit

Approximate solutions of the equation of motion’s of the rigid rod which rocks on the circular surface without slipping

In this paper, a modified harmonic balance method based an analytical technique has been developed to determine approximate solutions for a strongly nonlinear oscillator with a discontinuous term which is arising from the motion of rigid rod on the surface without slipping. Usually, a set of nonlinear algebraic equations is solved in this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of a large oscillation. We have been compared the solution results of this method with the numerical solution in order to validate the approach and assess the accuracy of the solutions has been demonstrated and discussed. We found that, a second order modified harmonic balance method works very well for the whole range of initial amplitudes. The advantage of the using method is its simple procedure and gives almost similar results in comparison with the exact solution

A new analytical technique based on harmonic balance method to determine approximate periods for Duffing-harmonic oscillator

The Duffing-harmonic oscillator is a common model for nonlinear phenomena in science and engineering. In this paper, a new analytical technique has been presented to determine approximate periods of a strongly nonlinear Duffing-harmonic oscillator. Generally, a set of difficult nonlinear algebraic equations appear when harmonic balance method is imposed. The power series solutions of these equations are invalid. The proposed idea avoids this limitation and the necessity of numerically solving such nonlinear algebraic equations with very complex nonlinearities. In this technique, different parameters for the same nonlinear problems are found, for which the power series solution yields desired results. Besides a suitable truncation formula is found in which the solution measures better results than existing solutions. It is remarkable that this procedure is simple and takes less computational effort for determining second and higher order periods of oscillation for such nonlinear problems and shows a good agreement compared with the exact ones

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

In 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

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

In 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