A Note on Free Vibration of a Double-beam System with Nonlinear Elastic Inner Layer

In this note, small amplitude free vibration of a double-beam system in presence of inner layer nonlinearity is investigated. The nonlinearity is due to inner layer material and is not related to large amplitude vibration. At first, frequencies of a double-beam system with linear inner layer are studied and categorized as synchronous and asynchronous frequencies. It is revealed that the inner layer does not affect higher modes significantly and mainly affects the first frequency. Then, equation of motion in the presence of cubic nonlinearity in the inner layer is derived and transformed to the form of Duffing equation. Using an analytical solution, the effect of nonlinearity on the frequency for simply-supported and clamped boundary conditions is analyzed. Results show that the nonlinearity effect is not significant and, in small amplitude free vibration analysis of a double-beam system, the material nonlinearity of the inner layer could be neglected

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In the present study, large amplitude free vibration of beams resting on variable elastic foundation is investigated. The Euler–Bernoulli hypothesis and the Winkler model have been applied for beam and elastic foundation, respectively. The beam is axially loaded and is restrained by immovable boundary conditions, which yields stretching during vibrations. The energy method and Hamilton’s principle are used to derive equation of motion, where after decomposition an ordinary differential equation with cubic nonlinear term is obtained. The second order homotopy perturbation method is applied to solve nonlinear equation of motion. An explicit amplitude-frequency relation is achieved from solution with relative error less than 0.07% for all amplitudes. This solution is applied to study effects of variable elastic foundation, amplitude of vibration and axial load on nonlinear frequency of beams with simply supported and fully clamped boundary conditions. Proposed formulation is capable to dealing with any arbitrary distribution of elastic foundation

Large amplitude free vibration of axially loaded beams resting on variable elastic foundation

In the present study, large amplitude free vibration of beams resting on variable elastic foundation is investigated. The Euler–Bernoulli hypothesis and the Winkler model have been applied for beam and elastic foundation, respectively. The beam is axially loaded and is restrained by immovable boundary conditions, which yields stretching during vibrations. The energy method and Hamilton’s principle are used to derive equation of motion, where after decomposition an ordinary differential equation with cubic nonlinear term is obtained. The second order homotopy perturbation method is applied to solve nonlinear equation of motion. An explicit amplitude-frequency relation is achieved from solution with relative error less than 0.07% for all amplitudes. This solution is applied to study effects of variable elastic foundation, amplitude of vibration and axial load on nonlinear frequency of beams with simply supported and fully clamped boundary conditions. Proposed formulation is capable to dealing with any arbitrary distribution of elastic foundation

Analytical approximations for a conservative nonlinear singular oscillator in plasma physics

AbstractA modified variational approach and the coupled homotopy perturbation method with variational formulation are exerted to obtain periodic solutions of a conservative nonlinear singular oscillator in plasma physics. The frequency–amplitude relations for the oscillator which the restoring force is inversely proportional to the dependent variable are achieved analytically. The approximate frequency obtained using the coupled method is more accurate than the modified variational approach and ones obtained using other approximate methods and the discrepancy between the approximate frequency using this coupled method and the exact one is lower than 0.31% for the whole range of values of oscillation amplitude. The coupled method provides a very good accuracy and is a promising technique to a lot of practical engineering and physical problems

Explicit formula to estimate natural frequencies of a double-beam system with crack

n the present study, an analytical formula to estimate natural frequencies of a simply supported double-beam system in the presence of open crack is derived. Euler-Bernoulli hypothesis was applied to beams, and Winkler model was used for inner layer. Material properties and cross section geometry of beams could be arbitrary and different from each other. To obtain natural frequencies, Eigenvalue problem solving finally yields an algebraic equation which must be solved numerically and does not show effects of different damage parameters in the explicit form. In this regard, Rayleigh method was applied to derive explicit formulation for natural frequencies. In the case of crack occurrence, the mode shapes of intact beam were modified by adding cubic polynomial functions to represent crack effect. The unknown coefficients of polynomial functions were calculated by using boundary conditions of the system and compatibility conditions at the crack section. Using the obtained admissible functions and Rayleigh method, an explicit formulation was achieved for natural frequencies. The problem one more time was solved using the differential transform method to approve the accuracy of the analytical formulation for the cracked double-beam system. Comparison of analytical and numerical results indicates good accuracy of derived formulation for natural frequencies of the cracked double-beam system