Analytical Approximations for Dry Friction-induced Stick-Slip and Pure-Slip Vibration Amplitudes of a Self-Excited SD Oscillator

2021 ◽  
Author(s):  
Junfeng Yan ◽  
Zehao Huang
Keyword(s):  
Perturbation Method ◽  
Friction Force ◽  
Dry Friction ◽  
Homotopy Perturbation Method ◽  
Perturbation Methods ◽  
Stick Slip ◽  
Homotopy Perturbation ◽  
Analytical Approximations ◽  
Sd Oscillator ◽  
Analytical Expressions

Abstract An analytical and numerical investigation into pure-slip and stick-slip oscillations induced by dry friction between a rigid mass linked by an inclined spring, modeled by the archetypal self-excited smooth and discontinuous (SD) oscillator, and the classical moving rigid belt, is presented. The friction force between surface contacts is modeled in the sense of Stribeck effect to formulate the friction model that the friction force firstly decreases and then increases with increasing relative sliding speed. Some perturbation methods are considered into this system for establishing the approximate analytical expressions of the occurring conditions, vibration amplitudes, and base frequencies of dry friction-induced stick-slip and pure-slip oscillations. For pure-slip oscillations, two different approaches are applied to analyze this self-excited SD oscillator. One of them is the homotopy perturbation method by constructing the nonlinear amplitude and frequency. Based on the multiple-scales homotopy perturbation method, a nonlinear equation for amplitude of the analytical approximate solution is constructed, which containing all parameters of problem. For stick-slip oscillations, the analytical approximations for amplitude and frequency are obtained by perturbation methods for finite time intervals of the stick phase, which is linked to the subsequent slip phase under the conditions of continuity and periodicity. The accuracy of analytical approximations is verified by the comparison between analytical approximations and numerical simulations. These analytical expressions are needed for gaining a deeper understanding of dry friction-induced pure-slip and stick-slip oscillations for the friction system with geometric nonlinearity.

Application of Homotopy Perturbation Method for Harmonically Forced Duffing Systems

2011 ◽  
Vol 110-116 ◽  
pp. 2277-2283 ◽  
Author(s):  
Xiang Meng Zhang ◽  
Ben Li Wang ◽  
Xian Ren Kong ◽  
A Yang Xiao
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Primary Resonance ◽  
Effective Technique ◽  
Duffing System ◽  
First Order ◽  
Homotopy Perturbation ◽  
Analytical Approximations ◽  
Case Based

In this paper, He’s homotopy perturbation method (HPM) is applied to solve harmonically forced Duffing systems. Non-resonance of an undamped Duffing system and the primary resonance of a damped Duffing system are studied. In the former case, the first-order analytical approximations to the system’s natural frequency and periodic solution are derived by HPM, which agree well with the numerical solutions. In the latter case, based on HPM, the first-order approximate solution and the frequency-amplitude curves of the system are acquired. The results reveal that HPM is an effective technique to the forced Duffing systems.

scholarly journals The Approximate Solution of Fractional Fredholm Integrodifferential Equations by Variational Iteration and Homotopy Perturbation Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Abdelouahab Kadem ◽  
Adem Kilicman
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Iteration Method ◽  
Homotopy Perturbation Method ◽  
Perturbation Methods ◽  
Variational Iteration Method ◽  
Integrodifferential Equations ◽  
Variational Iteration ◽  
Homotopy Perturbation ◽  
Constant Coefficients

Variational iteration method and homotopy perturbation method are used to solve the fractional Fredholm integrodifferential equations with constant coefficients. The obtained results indicate that the method is efficient and also accurate.

scholarly journals Application of Perturbation Theory in Heat Flow Analysis

2021 ◽  
Author(s):  
Neelam Gupta ◽  
Neel Kanth
Keyword(s):  
Partial Differential Equations ◽  
Heat Flow ◽  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Perturbation Methods ◽  
Separation Of Variables ◽  
Flow Analysis ◽  
Partial Differential ◽  
Homotopy Perturbation

Many physical and engineering problems can be modeled using partial differential equations such as heat transfer through conduction process in steady and unsteady state. Perturbation methods are analytical approximation method to understand physical phenomena which depends on perturbation quantity. Homotopy perturbation method (HPM) was proposed by Ji Huan He. HPM is considered as effective method in solving partial differential equations. The solution obtained by HPM converges to exact solution, which are in the form of an infinite function series. Biazar and Eslami proposed new homotopy perturbation method (NHPM) in which construction of an appropriate homotopy equation and selection of appropriate initial approximation guess are two important steps. In present work, heat flow analysis has been done on a rod of length L and diffusivity α using HPM and NHPM. The solution obtained using different perturbation methods are compared with the solution obtained from most common analytical method separation of variables.

scholarly journals Perturbation Methods to Analysis of Thermal, Fluid Flow and Dynamics Behaviors of Engineering Systems

2021 ◽  
Author(s):  
Gbeminiyi M. Sobamowo
Keyword(s):  
Fluid Flow ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Perturbation Methods ◽  
Thermal Contact ◽  
Engineering Systems ◽  
Regular Perturbation ◽  
Homotopy Perturbation ◽  
Thermal Fluid Flow ◽  
Nonlinear Elastic

This chapter presents the applications of perturbation methods such as regular and homotopy perturbation methods to thermal, fluid flow and dynamic behaviors of engineering systems. The first example shows the utilization of regular perturbation method to thermal analysis of convective-radiative fin with end cooling and thermal contact resistance. The second example is concerned with the application of homotopy perturbation method to squeezing flow and heat transfer of Casson nanofluid between two parallel plates embedded in a porous medium under the influences of slip, Lorentz force, viscous dissipation and thermal radiation. Additionally, the dynamic behavior of piezoelectric nanobeam embedded in linear and nonlinear elastic foundations operating in a thermal-magnetic environment is analyzed using homotopy perturbation method which is presented in the third example. It is believed that the presentation in this chapter will enhance the understanding of these methods for the real world applications.

scholarly journals A Novel Effective Approach for Solving Fractional Nonlinear PDEs

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Hossein Aminikhah ◽  
Nasrin Malekzadeh ◽  
Hadi Rezazadeh
Keyword(s):  
Partial Differential Equations ◽  
Perturbation Method ◽  
Population Model ◽  
Homotopy Perturbation Method ◽  
Analytical Procedure ◽  
Nonlinear Pdes ◽  
Fractional Partial Differential Equations ◽  
Homotopy Perturbation ◽  
Analytical Approximations ◽  
Biological Population

The present work introduces an effective modification of homotopy perturbation method for the solution of nonlinear time-fractional biological population model and a system of three nonlinear time-fractional partial differential equations. In this approach, the solution is considered a series expansion that converges to the nonlinear problem. The new approximate analytical procedure depends only on two iteratives. The analytical approximations to the solution are reliable and confirm the ability of the new homotopy perturbation method as an easy device for computing the solution of nonlinear equations.

Optimal Homotopy Perturbation Method for a Non-Conservative Dynamical System of a Rotating Electrical Machine

2012 ◽  
Vol 67 (8-9) ◽  
pp. 509-516 ◽  
Author(s):  
Nicolae Herişanu ◽  
Vasile Marinca
Keyword(s):  
Dynamical System ◽  
Perturbation Method ◽  
Excellent Agreement ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Numerical Results ◽  
Electrical Machine ◽  
Unknown Parameters ◽  
Homotopy Perturbation ◽  
Analytical Expressions

A version of the optimal homotopy perturbation method (OHPM) is applied in this study to derive highly accurate analytical expressions for the solutions to a non-conservative dynamical system of a rotating electrical machine. The main advantage of this procedure consists of providing us with a convenient and rigorous way to control the approximate solutions by means of some initially unknown parameters which are optimally determined later. Comparisons with numerical results reveal an excellent agreement, which demonstrates the effectiveness of the proposed method in analyzing non-conservative oscillators.

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