On the approximate analytical solution to non-linear oscillation systems
Mahmoud Bayat,Iman Pakar +1 more
TLDR
In this article, a new implementation of Variational Approach (VA) is presented to obtain highly accurate analytical solutions to free vibration of conservative oscillators with inertia and static type cubic nonlinearities.Abstract:
This study describes an analytical method to study two well-known systems of nonlinear oscillators. One of these systems deals with the strongly nonlinear vibrations of an elastically restrained beam with a lumped mass. The other is a Duffing equation with constant coefficients. A new implementation of the Variational Approach (VA) is presented to obtain highly accurate analytical solutions to free vibration of conservative oscillators with inertia and static type cubic nonlinearities. In the end, numerical comparisons are conducted between the results obtained by the Variational Approach and numerical solution using Runge-Kutta's [RK] algorithm to illustrate the effectiveness and convenience of the proposed methods.read more
Citations
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Mechanical properties soil stabilized with nano calcium carbonate and reinforced with carpet waste fibers
TL;DR: In this paper, the authors examined the effect of the addition of nano calcium carbonate (0, 0.6% by weight of the soil) as a reinforcement agent on the soil behavior.
Journal ArticleDOI
Effect of granulated rubber on shear strength of fine-grained sand
TL;DR: In this article, the shear behavior of mixtures of fine-grained sand and 1-5mm granulated rubber was investigated and 60 direct shear tests were conducted on sand-granulated rubber mixtures with various rubber contents (0, 5, 10, 20% and 30%) at different relative densities (50, 70% and 90%) and different normal stresses.
Journal ArticleDOI
Nonlinear vibration of an electrostatically actuated microbeam
TL;DR: In this paper, a new class of critical technique that called the He's Variational Approach (VA) was considered to solve the nonlinear vibration of an electrostatically actuated microbeam.
Journal ArticleDOI
Approximate solutions and their stability of a broadband piezoelectric energy harvester with a tunable potential function
TL;DR: Results demonstrate that the energy harvesting performance of the proposed PEH could be effectively tuned by the pre-deformation of the spring, a criterion for solution stability analysis of congeneric nonlinear systems with coupled higher-order terms.
Journal ArticleDOI
Nonlinear free vibration of systems with inertia and static type cubic nonlinearities: An analytical approach
TL;DR: In this article, the free vibration of a mass grounded by linear and nonlinear springs in series is studied and a nonlinear ordinary differential equation with inertia and static type cubic nonlinearities represents the governing equation of the system.
References
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Book
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
TL;DR: In this article, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.
A Reflection on Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
J. Guckenheimer,P. J. Holmes +1 more
TL;DR: In this paper, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.
Book
Mathematical Methods of Classical Mechanics
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
Book
Handbook of Exact Solutions for Ordinary Differential Equations
A. D. Poli︠a︡nin,V. F. Zaĭt︠s︡ev +1 more
Abstract: Annotation Foreward Some Remarks and Notation First Order Differential Equations Simplest Equations with Arbitrary Functions Integrable in a Closed Form Riccati Equations: g(y)y'x = f2(x)y2 + f1(x)y + f0(x) Abel Equations of the Second Kind Equations Containing Polynomial Functions of y Nonlinear Equations of the Form f(x,y)y'x = g(x,y) Containing Arbitrary Parameters Equations Not Solved for Derivative Equations of the Form F(x,y)y'x = G(x,y) Containing Arbitrary Functions Equations of the Form F(x,y,y'x) = 0 Not Solved for the Derivative and Containing Arbitrary Functions Second Order Differential Equations Linear Equations Autonomous Equations y"xx = F(y,y'x) Emden-Fowler Equation y"xx = Axnym Equations of the Form y"xx = A1xn1ym1 + A2xn2ym2 Generalized Emden-Fowler Equation y"xx = Axnym(y'x)l Equations of the Form y"xx = A1xn1ym1(y'x)l1 + A2xn2ym2(y'x)l2 Equations of the Form y"xx = f(x)g(y)h(y'x) Some Nonlinear Equations with Arbitrary Parameters Equations Containing Arbitrary Functions Third Order Differential Equations Linear Equations Equations of the Form y'"xxx = Axayss(y'x)g(y"xx)d Equations of the Form y'"xxx = f(y)g(y'x)h(y"xx) Some Nonlinear Equations with Arbitrary Parameters Nonlinear Equations Containing Arbitrary Functions Fourth Order Differential Equations Linear Equations Nonlinear Equations Higher Order Differential Equations Linear Equations Nonlinear Equations Supplement 1. Some Elementary Functions and Their Properties Trigonometric Functions Hyperbolic Functions Inverse Trigonometric Functions Inverse Hyperbolic Functions Some Conventional Symbols Supplement 2. Some Special Functions Gamma-Function Bessel Functions Jn and Yn Modified Bessel Functions In and Kn Degenerate Hypergeometric Functions Legendre Functions The Weierstrass Function References Index
Book Review: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
John Guckenheimer,Philip Holmes +1 more
TL;DR: Guckenheimer and Holmes as discussed by the authors survey the theory and techniques needed to understand chaotic behavior of ODEs and provide a user's guide to an extensive and rapidly growing field.