Damping of longitudinal vibrations of an elastic rod by a piezoelectric element

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Possible damping of longitudinal vibrations of a thin homogeneous elastic rod under the influence of a normal force in the cross section is studied. This time-varying force, which can be excited, for example, by using piezoelectric elements, is uniformly distributed along the length on a given segment of the cantilevered rod and is equal to zero outside it. Those placements of the ends of the segment are presented in which the excited force does not affect the amplitude of certain modes. The minimum time in which the oscillations of all other modes can be damped is found, and based on the Fourier method, the corresponding law of the damping force is obtained in the form of a series. A generalized formulation of the boundary value problem on moving the rod during this time to the zero terminal state is given, for which an algorithm for exact solution is proposed in the case of rational relations on the geometric parameters. Unknown functions of the rod state are sought in the form of a linear combination of the traveling wave and normal force functions, which are determined from a linear system of algebraic equations following from boundary relations and continuity conditions. The solutions obtained in series by the Fourier method and in the form of d’Alembert traveling waves are compared.

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Sobre autores

G. Kostin

Ishlinsky Institute for Problems in Mechanics RAS

Autor responsável pela correspondência
Email: kostin@ipmnet.ru
Rússia, Moscow

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2. Fig. 1. Diagram of a rod with a control element

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3. Fig. 2. Grid in the space-time domain D for nx = 4

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4. Fig. 3. Control u*(t) for λ = 1/4: x- = 0 (solid curve), x- =1/4 (dashed), x- =1/2 (dash-dotted), x- = 3/4 (dotted).

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5. Fig. 4. Force f(t,188) for x- = 0 and λ =1/4: exact solution (solid curve) and 8-mode approximation (dashed curve).

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6. Fig. 5. Distribution of the dynamic potential r(t, x) under critical control for x- = 0 and λ =1/4

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7. Fig. 6. Distribution of elastic displacements v(t, x) under critical control for x- = 0 and λ =1/4

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