The response of simply supported rectangular plates carrying moving masses and resting on variable Winkler elastic foundations is investigated in this work. The governing equation of the problem is a fourth order partial differential equation. In order to solve this problem, a technique based on separation of variables is used to reduce the governing fourth order partial differential equations with variable and singular coefficients to a sequence of second order ordinary differential equations. For the solutions of these equations, a modification of the Struble's technique and method of integral transformations are employed. Numerical results in plotted curves are then presented. The results show that response amplitudes of the plate decrease as the value of the rotatory inertia correction factor R0 increases. Furthermore, for fixed value of R0, the displacements of the simply supported rectangular plates resting on variable elastic foundations decrease as the foundation modulus F0 increases. The results further show that, for fixed R0 and F0, the transverse deflections of the rectangular plates under the actions of moving masses are higher than those when only the force effects of the moving load are considered. Therefore, the moving force solution is not a safe approximation to the moving mass problem. Hence, safety is not guaranteed for a design based on the moving force solution. Also, the analyses show that the response amplitudes of both moving force and moving mass problems decrease both with increasing Foundation modulus and with increasing rotatory inertia correction factor. The results again show that the critical speed for the moving mass problem is reached prior to that of the moving force for the simply supported rectangular plates on variable Winkler elastic foundation.
The flexural motions of elastically supported rectangular plates carrying moving masses and resting on variable Winkler elastic foundations is investigated in this work In order to solve the fourth order partial differential equation governing the problem, a technique based on separation of variables is used to reduce the governing fourth order partial differential equations with variable and singular coefficients to a sequence of second order ordinary differential equations. These equations are then solved using a modification of the Struble's technique and method of integral transformations. Numerical results are then presented in plotted curves. The results show that response amplitudes of the plate decrease as the value of the rotatory inertia correction factor R o increases and for fixed value of R o , the displacements of the elastically supported rectangular plates resting on variable elastic foundations decrease as the foundation modulus F o increases. Also, for fixed R o and F o , the transverse deflections of the rectangular plates under the actions of moving masses are higher than those when only the force effects of the moving load are considered. Therefore, the moving force solution is not a safe approximation to the moving mass problem. Hence, safety is not guaranteed for a design based on the moving force solution. Furthermore, the results show that the critical speed for the moving mass problem is reached prior to that of the moving force for the elastically supported rectangular plates on Winkler elastic foundation with stiffness variation.
The dynamic response to moving concentrated masses of elastically supported rectangular plates resting on Winkler elastic foundation is investigated in this work. This problem, involving non-classical boundary conditions, is solved and illustrated with two common examples often encountered in engineering practice. Analysis of the closed form solutions shows that, for the same natural frequency (i) the response amplitude for the moving mass problem is greater than that one of the moving force problem for fixed Rotatory inertia correction factor R0 and foundation modulus F0, (ii) The critical speed for the moving mass problem is smaller than that for the moving force problem and so resonance is reached earlier in the former. The numerical results in plotted curves show that, for the elastically supported plate, as the value of R0 increases, the response amplitudes of the plate decrease and that, for fixed value of R0, the displacements of the plate decrease as F0 increases. The results also show that for fixed R0 and F0, the transverse deflections of the plates under the actions of moving masses are higher than those when only the force effects of the moving load are considered. Hence, the moving force solution is not a save approximation to the moving mass problem. Also, as the mass ratio Γ approaches zero, the response amplitude of the moving mass problem approaches that one of the moving force problem of the elastically supported rectangular plate resting on constant Winkler elastic foundation.
In this present study, the response characteristics of a flexible member carrying harmonic moving load are investigated. The beam is assumed to be of uniform cross section and has simple support at both ends. The moving concentrated force is assumed to move with constant velocity type of motion. A versatile mathematical approximation technique often used in structural mechanics called assumed mode method is in first instance used to treat the fourth order partial differential equation governing the motion of the slender member to obtain a sequence of second order ordinary differential equations. Integral transform method is further used to treat this sequence of differential equations describing the motion of the beam-load system. Various results in plotted curves show that, the presence of the vital structural parameters such as the axial force N, rotatory inertia correction factor r 0 , the foundation modulus F 0 , and the shear modulus G 0 , significantly enhances the stability of the beam when under the action of moving load. Dynamic effects of these parameters on the critical speed of the dynamical system are carefully studied. It is found that as the values of these parameters increase, the critical speed also increases. Thereby reducing the risk of resonance and thus the safety of the occupant of this structural member is guaranteed.
The dynamic response to variable magnitude moving distributed masses of simply supported nonuniform Bernoulli-Euler beam resting on Pasternak elastic foundation is investigated in this paper. The problem is governed by fourth order partial differential equations with variable and singular coefficients. The main objective of this work is to obtain closed form solution to this class of dynamical problem. In order to obtain the solution, a technique based on the method of Galerkin with the series representation of Heaviside function is rst used to reduce the equation to second order ordinary differential equations with variable coefficients. Thereafter the transformed equations are simplied using (i) The Laplace transformation technique in conjunction with convolution theory to obtain the solution for moving force problem and (ii) nite element analysis in conjunction with Newmark method to solve the analytically unsolvable moving mass problem because of the harmonic nature of the moving load. The nite element method is rst used to solve the moving force problem and the solution is compared with the analytical solution of the moving force problem in order to validate the accuracy of the nite element method in solving the analytically unsolvable moving mass problem. The numerical solution using the nite element method is shown to compare favorably with the analytical solution of the moving force problem. The displacement response for moving distributed force and moving distributed mass models for the dynamical problem are calculated for various time t and presented in plotted curves. Foremost, it is found that, the moving distributed force is not an upper bound for the accurate solution of the moving distributed mass problem, showing that the inertia term must be considered for accurate assessment of the response to moving distributed load of elastic structural members. Analyses further show that increase in the values of the structural parameters such as axial force N, shear modulus Gandfoundation stiffness K reduces the response amplitudes of the non-uniform Bernoulli-Euler beam under moving distributed loads. Finally, for the same natural frequency, the critical speed for the non-uniform Bernoulli-Euler beam traversed by moving distributed force is greater than that under the inuence of a moving distributed mass. Hence resonance is reached earlier in the moving distributed force problem.
Abstract:The variable velocity influence on the vibration of a simply supported Bernoulli-Euler beam, resting on a uniform foundation, under the action of an exponentially varying magnitude load moving with variable velocity is investigated in this work. The technique is based on the Finite Fourier Sine transformation and the Finite Difference method. Numerical results in plotted curves are presented. It is observed from the plotted curves that, when the foundation modulli 'K' and the axial force 'N' are fixed, the transverse deflections of the beam under load moving with variable velocity are higher than the deflections when the load moves with constant velocity. It is also observed that the response amplitudes in both cases of variable velocity and constant velocity decrease both with increasing 'K' and with increasing 'N'. Also, the analysis shows that the effects of the foundation modulli 'K' and the axial force 'N' are more pronounced in the vibration of the beam under load moving with variable velocity than in the case when the load moves with constant velocity, hence more reinforcement is needed when the load moves with variable velocity.
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