M. Ghomeshi Bozorg, M. Keshmiri ,
Volume 34, Issue 1 (7-2015)
Abstract
In this paper, considering all the linear and nonlinear inertia terms of moving masses on a flexible beam, the dynamic response and dynamic stability of the beam are studied. Homotopy perturbation method is used to perform the analysis and results are provided in a stability map for the different values of mass and velocity of the moving masses. It is concluded that there is a borderline in the diagram that separates the stable and unstable regions. For the first time, this borderline is determined semi-analytically. Results of the stability analysis are validated using the Floquet theory. In addition to this borderline, it is also concluded that the Homotopy perturbation method is capable of evaluating the new critical values for mass and velocity which cause vibration resonance in the beam. The locus of these resonant points, which is totally a new finding in dynamic analysis of beam-moving mass problem, is determined semi-analytically. Finally, the effect of the friction between the beam and the moving mass is studied on the stability of the system and resonant conditions. Accuracy of the results for this case is also evaluated with a numerical simulation.
M. Ghomeshi Bozorg,
Volume 41, Issue 1 (9-2022)
Abstract
In this paper, dynamic instability due to parametric and external resonances of moderately thick functionally graded rectangular plates, under successive moving masses, is examined. Plate mass per unit volume and Young’s modulus are assumed to vary continuously through the thickness of the plate and obey a power-law distribution of the volume fraction of the constituent. The considered rectangular plates have two opposite simply supported edges while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. The governing coupled partial differential equations of the plate are derived based on the first-order shear deformation theory with consideration of the rotational inertial effects and the transverse shear stresses. All inertial components of the moving masses are considered in the dynamic formulation. Using the Galerkin procedure, the partial differential equations are transformed into a set of ordinary differential equations with time-dependent coefficients. The Homotopy Analysis Method (HAM) is implemented as a semi-analytical method to obtain stable and unstable zones and external resonance curves in a parameters space. The effects of the index of volume fraction, thickness to length ratio, and different combinations of the boundary conditions on the dynamic stability of the system are also investigated. The results indicate that decreasing the index of volume fraction, increasing thickness to length ratio, and higher degree of edge constraints (respectively from free to simply-supported to clamped) applied to the other two edges of the plate shift up the instability region and resonance curves in the parameters plane and, from a physical point of view, the system becomes more stable. In addition to using numerical simulations of the plate midpoint displacement, Floquet theory is also employed to validate the HAM results. Finally, the results of this study, in a particular case, are compared and validated with the results of other works.
