Reliability domain of the Timoshenko model in dynamics of rods and plates

Demochkin, N. I.; Morgachev, K. S.; Fridman, L. I.

doi:10.3103/s0025654408060137

Cited by 2 publications

(1 citation statement)

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“…Timoshenko [4,5] derived the equation of influence of effects of a higher order in comparison with the classical theory of beams, which is currently of great interest. Refined theories of the Timoshenko type have been extensively developed and are discussed in many studies, for example [6][7][8][9][10][11][12][13][14]. In these and other articles, equations via forces and moments are traditionally used, the boundary conditions on the long sides of the beam or on the front sides of the beam (shell) are not satisfied, and conditions of the beam type are set at the ends.…”

Section: Introductionmentioning

confidence: 99%

Low Frequency Vibrations of a Long Elastic Strip

Zveryaev

2021

Mech. Solids

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The application of the iterative Saint-Venant-Picard-Banach method is described on the example of constructing the solution of the system for differential equations of motion of the theory of elasticity with a small parameter for a long strip at perturbation frequencies commensurate with the frequencies of transverse vibrations of the beam. The system is transformed in such a way that the equations are integrated sequentially and without increasing in order. The unknowns are calculated using the first-order Picard operators so that the previously calculated unknowns are input for the next equation, etc. The integrals of all unknowns for problem that make it possible to fulfill all the boundary conditions on the long and short sides are found. The convergence of the solution is ensured using a small thin-walled parameter in accordance with the Banach's contraction principle. Satisfaction of the boundary conditions at the long edges leads to two equations for slowly and two singular ones for rapidly varying components of the solution, which depend only on the longitudinal coordinate. It is shown that when these equations are reduced to one with the loss of a rapidly changing component, the Timoshenko equation is obtained. The presentation is illustrated by two examples of loading a strip with a transverse distributed load and a concentrated force. A technique for establishing orders of magnitude by a small parameter relative to the load is described.

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