Natural vibration of a sandwich beam on an elastic foundation

Kubenko, V. D.; Pleskachevskii, Yu. M.; Старовойтов, Э. И.; Леоненко, Д. В.

doi:10.1007/s10778-006-0118-8

Cited by 22 publications

(13 citation statements)

References 9 publications

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“…Omitting the description of these functions, we just point out that Y r 0 ( ) l and K r 0 ( ) l have a logarithmic singularity at the origin of coordinates [8], i.e., at the center of the plate. Therefore, it is necessary to set Ñ Ñ…”

Section: Introductionmentioning

confidence: 99%

Resonant effects of local loads on circular sandwich plates on an elastic foundation

Старовойтов

Леоненко

2010

Int Appl Mech

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The axisymmetric forced vibrations of a circular sandwich plate on an elastic foundation are studied. The plate is subjected to axisymmetric surface and mechanical loads with frequency equal to one of the natural frequencies of the plate. The foundation reaction is described by the Winkler model. To describe the kinematics of an asymmetric sandwich, the hypothesis of broken normal is used. The core is assumed to be light. The analytical solution of the problem is obtained and numerical results are analyzed Keywords: circular sandwich plate, axisymmetric forced vibrations, Winkler elastic foundation, axisymmetric surface, mechanical loadIntroduction. Analytic and numerical results on the vibrations of circular sandwich plates not bonded to an elastic foundation were obtained in [1][2][3]. The nonlinear vibrations of layered plates were studied in [4]. The quasistatic deformation of sandwich structures on an elastic foundation was studied in [5]. The natural vibrations of a sandwich rod bonded to a Winkler foundation were studied in [6].We will consider an asymmetric (throughout the thickness) circular elastic sandwich plate bonded to an elastic foundation and undergoing small axisymmetric transverse vibrations excited by surface and mechanical resonant loads, i.e., loads with frequency equal to one of the natural frequencies of the plate.1. Problem Formulation. We will use a cylindrical coordinate system r, j, z (Fig. 1) fixed to the midsurface of the core. For isotropic face layers of thickness h 1 and h 2 , the Kirchhoff hypotheses are accepted. The incompressible core (h 3 = 2ñ) is light, i.e., the work of the tangential stresses s rz in the tangential direction can be neglected. The deformed normal of the core remains straight, but turns through some additional angle y. The displacements at the boundaries of the layers are continuous. There is a rigid diaphragm at the edge of the plate to prevent the relative movement of the layers. The external vertical load does not depend on the coordinate j: q = q(r, t). The outside surface of the second face layer sustains the reaction q R of the elastic foundation.For symmetry reasons, the tangential displacements of the layers are zero, and the deflection w of the plate, the relative shear y in the core, and the radial displacement u of the coordinate surface do not depend on the coordinate j, i.e., u(r, t), y(r, t), w (r, t). Hereafter these functions are unknown. The thickness and density of the kth layer are denoted by h k and r k , respectively.The relationship between the reaction and the deflection is described by the Winkler model (q w R = k 0 , k 0 is the foundation modulus (modulus of subgrade reaction)).The system of partial differential equations describing the forced transverse vibrations of a circular sandwich plate not bonded to an elastic inertialess foundation and disregarding the reduction and rotary inertia of the normal in the layers is obtained from Lagrange's variational principle taking into account the variation of the work of inertial forces [1,3...

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Section: Introductionmentioning

confidence: 99%

Resonant effects of local loads on circular sandwich plates on an elastic foundation

Старовойтов

Леоненко

2010

Int Appl Mech

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“…The zig-zag hypothesis is valid for the three-layer beam, i.e. Kirchhoff hypothesis is valid for bearing layers, as well as the normal in the beam filler remains straight and turning by the angle [13][14][15][16][17]. Let us connect Cartesian coordinate with the center of beam filler medium surface in an undisturbed state.…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…Let us connect Cartesian coordinate with the center of beam filler medium surface in an undisturbed state. The rigid diaphragms, hindering the relative layers shift, but not impeding the deformation from its plane, are supposed to be situated at the beam edges [15][16][17]. Hence, the three-layer beam oscillations are caused by foundation vibration, while deformations of the plate are considered to be small.…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…References [15][16][17][18] consider the statics and dynamics of three-layer beams and plates under the local and distributed loads of various natures. The investigation of elastic three-layer plate dynamic interaction with a viscous liquid layer is of theoretical importance, while its results are of practical interest for computing and analyzing the new technology objects.…”

Section: Introductionmentioning

confidence: 99%

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Mathematical modeling of three-layer beam hydroelastic oscillations

Mogilevich¹,

Popov²,

Popova³

et al. 2017

Vib. proced.

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Abstract. The problem of hydroelastic oscillations of three-layer beam interacting with viscous liquid layer is set up and analytically solved. The problem presents the equation system of a three-layer beam and Navier-Stokes equations. The following boundary conditions are chosen: the no-slip conditions, the conditions for pressure at the edges, the simply supported edges conditions. The problem is solved for the steady-state harmonic regime. The frequency dependent distribution functions of the beam deflection are constructed. The given function allows investigating the resonance hydroelastic oscillations of a three-layer beam, as well as its deflected mode.

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“…For example, references [1][2][3][4] study the oscillations and stability of multi-layer beams and plates, rested on an elastic foundation under the influence of local and distributed loads of various origins. Winkler and Pasternak models are used for foundation reactions modeling.…”

Section: Introductionmentioning

confidence: 99%

Hydroelastic oscillation of a plate resting on Pasternak foundation

Kondratov¹,

Mogilevich²,

Popov³

et al. 2017

Vib. proced.

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Abstract. The bending oscillations of the plate, resting in Pasternak foundation and interacting with a vibrating stamp through a thin layer of viscous incompressible liquid, are investigated. On the basis of hydroelasticity problem solution, the laws of the plate deflections and pressure in the liquid are found. The functions of the deflections amplitude distribution and liquid pressure along the channel are constructed. The obtained results allow to define oscillations resonance frequencies and to study viscous liquid interaction with elastic plates, resting on Pasternak foundation.

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Natural vibration of a sandwich beam on an elastic foundation

Cited by 22 publications

References 9 publications

Resonant effects of local loads on circular sandwich plates on an elastic foundation

Resonant effects of local loads on circular sandwich plates on an elastic foundation

Mathematical modeling of three-layer beam hydroelastic oscillations

Hydroelastic oscillation of a plate resting on Pasternak foundation

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