Free vibrations of a mass grounded by linear and nonlinear springs in series

Tellı, Sevda; Kopmaz, Osman

doi:10.1016/j.jsv.2005.02.018

Cited by 45 publications

(44 citation statements)

References 1 publication

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“…2. After transformation, the motion is governed by a nonlinear differential equation of motion [22]:…”

Section: Governing Equation Of Motion and Formulationmentioning

confidence: 99%

“…In this research the basic idea of the VIM is introduced and then its application in nonlinear equations is studied. Telli and Kopmaz [22] have attempted to solve the motion of a mechanical system associated with linear and nonlinear properties using analytical and D.D. Ganji ( ) · H. Babazadeh · M.H.…”

Section: Introductionmentioning

confidence: 99%

“…The linkage of the linear and nonlinear springs in series has been derived with cubic nonlinear characteristics in the equations of motion [22]. As mentioned in [22], although there exists a vast literature on discrete systems including either linear or nonlinear springs [7,20], one does not encounter publications on mechanical systems with single-degree-of-freedom containing flexible component consisting of linear-nonlinear spring in series, which occurs in technical applications. One similar mechanical model is the conservative Duffing equation with linear and cubic characteristics governed by a second-order differential equation…”

Section: Introductionmentioning

confidence: 99%

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Application of He’s Variational Iteration Method for Solving Nonlinear BBMB Equations and Free Vibration of Systems

et al. 2008

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A new analytical method called He's variational iteration method (VIM) is introduced to be applied to solve nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equations and free vibration of a nonlinear system having combined linear and nonlinear springs in series in this article. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers can be identified optimally via the variational theory. The results are compared with the results of the homotopy analysis method and also with the exact solution. He's Variational iteration method in this problem functions so better than the homotopy analysis method and exact solutions one of them in per section.

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“…2. After transformation, the motion is governed by a nonlinear differential equation of motion [22]:…”

Section: Governing Equation Of Motion and Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Application of He’s Variational Iteration Method for Solving Nonlinear BBMB Equations and Free Vibration of Systems

et al. 2008

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“…Recently 2-DOF oscillation systems have been investigated [5][6][7]. In all studies the system reduces to one decoupled equation and one coupled equation by using a transformation.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian Approach to Multiple Coupled Nonlinear Oscillators

2012

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In this paper, Hamiltonian approach is extended to investigate coupled nonlinear multi-degree-of-freedom oscillatory systems. At the beginning of the study, basic principles of Hamiltonian approach are provided for multiple coupled oscillators. In the next section, the amplitude-frequency relation for the two-degree-of-freedom nonlinear mechanical systems is obtained via Hamiltonian approach. The natural frequency expression is compared to show the agreement between the present and published results. Additionally, the approximate and numerical periodic solutions are plotted.

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“…Introducing suitable transformation variables, two secondorder differential equations can be simplified as a single non-homogeneous second-order differential equation with quadratic nonlinearity and solved using the Jacobi elliptic functions. As also presented by Telli and Kopmaz [13], the equations of motion of a mechanical system associated with linear and nonlinear properties was attempted and transformed into a set of differential equations using intermediate variables. These differential equations may be further derived as a nonlinear ordinary differential equation.…”

mentioning

confidence: 99%

Application of a modified Lindstedt–Poincaré method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation

Lim

Lai

et al. 2008

Arch Appl Mech

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An analytical approach is developed for the nonlinear oscillation of a conservative, two-degree-offreedom (TDOF) mass-spring system with serial combined linear-nonlinear stiffness excited by a constant external force. The main idea of the proposed approach lies in two categories, the first one is the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation. Another is the treatment a quadratic nonlinear oscillator (QNO) by the modified Lindstedt-Poincaré (L-P) method presented recently by the authors. The first-order and second-order analytical approximations for the modified L-P method are established for the QNOs with satisfactory results. After solving the nonlinear differential equation, the displacements of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, the modified L-P method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and classical harmonic balance methods. Two examples of nonlinear TDOF mass-spring systems excited by a constant external force are selected and the approximate solutions are verified with the exact solutions derived from the Jacobi elliptic function and also the numerical fourth-order Runge-Kutta solutions.

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Free vibrations of a mass grounded by linear and nonlinear springs in series

Cited by 45 publications

References 1 publication

Application of He’s Variational Iteration Method for Solving Nonlinear BBMB Equations and Free Vibration of Systems

Application of He’s Variational Iteration Method for Solving Nonlinear BBMB Equations and Free Vibration of Systems

Hamiltonian Approach to Multiple Coupled Nonlinear Oscillators

Application of a modified Lindstedt–Poincaré method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation

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