A mixed system of equations of elasticity

Shul’ga, M. O.

doi:10.1007/s10778-010-0306-4

Cited by 6 publications

(1 citation statement)

References 3 publications

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“…In [10,11], the canonical equations in one spatial coordinate were deduced from a study of problems of harmonic bending vibrations of plates with parameters periodic in one coordinate. Similar investigations were also performed in the theory of vibrations of beams with periodic parameters (see, e.g., [4,5]). In the present work, the Hamilton formalism is developed for the Timoshenkotype equations describing the bending of plates.…”

Section: Introductionmentioning

confidence: 77%

Application of the hamilton formalism in a timoshenko-type theory of vibrations of plates

Shul’ga

2012

J Math Sci

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We carried out a general analysis of the mixed systems of six equations of a Timoshenko-type theory of vibrations of plates in rectangular and polar coordinates. It is shown that these systems can be represented in the operator canonical (Hamiltonian) form with respect to the spatial coordinate under the appropriate choice of "canonical" variables and the operator Hamilton function. Some functionals for canonical systems are constructed. For a mixed Hamiltonian system, a reduced Hellinger-Reissner functional of the spatial coordinate in the rectangular coordinates is obtained. The variation of the functional yields six canonical equations.

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Section: Introductionmentioning

confidence: 77%

Application of the hamilton formalism in a timoshenko-type theory of vibrations of plates

Shul’ga

2012

J Math Sci

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Harmonic thickness vibrations of inhomogeneous elastic layers with curved boundaries

2011

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The thickness vibrations of elastic inhomogeneous bodies of different geometry under dynamic harmonic loading are studied. The dependence of the amplitude-frequency characteristics of homogeneous and inhomogeneous bodies on excitation frequency is analyzed in detail. The frequency spectra for plane, cylindrical, and spherical layers are determined Introduction. Lame's static and dynamic problems for elastic homogeneous (and piecewise-homogeneous) cylinders and spheres have exact analytic solutions [1, etc.]. However, engineering practice deals with continuously inhomogeneous layers (plane, cylindrical, spherical, etc.). Such inhomogeneities are caused by various structural, technological, operational, climatic, and other factors. This issue for cylindrical bodies was discussed in [4][5][6][8][9][10][11]. To study the effect of various radial inhomogeneities on the frequency and amplitude-frequency characteristics (AFCs) of elastic layers with curved boundaries undergoing harmonic vibrations, we need not only efficient mathematical and computational algorithms for solving boundary-value problems of elasticity, but also a certain strategy for physical analysis of the solution found.Here we present a unified approach to the analysis of the harmonic vibrations of a plate, a cylinder, and a sphere with transverse anisotropy and inhomogeneity. We will use the developed algorithm to compare the AFCs and natural frequencies of inhomogeneous bodies of different geometry.1. Problem Statement. To study the ADCs of inhomogeneous elastic plane, cylindrical, and spherical layers undergoing thickness vibrations, we start with the equations of motion

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Comparative analysis of the electroelastic thickness vibrations of layers with curved boundaries

Shul’ga

Grigor’eva

2011

Int Appl Mech

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The three-dimensional equations of electroelasticity in Cartesian, cylindrical, and spherical coordinates are represented in Hamiltonian form with respect to the thickness coordinate. The boundary-value problem with a harmonic potential difference and zero mechanical load given on the boundaries is solved numerically. The amplitude-frequency characteristics and natural frequencies are compared. The resonant and antiresonant frequencies of the current and the dynamic electromechanical coupling coefficient are determined Keywords: piezoelectric layer, sphere, plate with transverse polarization, electroelastic thickness vibrations, amplitude-frequency characteristics, resonant and antiresonant frequencies Introduction. Piezoelectric elements (plates, cylinders, spheres, etc.) are widely used in various devices [1,3,9]. The study of their dynamic characteristics is of scientific and applied importance. The exact analytic solution describing the thickness vibrations of a plate-like vibrator is well known [1, 6, etc.], whereas the analytic solutions for cylindrical and spherical vibrators with arbitrary boundary conditions are formal because they require evaluating integrals of special functions. When curvature is small, it is expedient to seek solutions in the form of ordinary power series of linear functions of the radial coordinate. This approach was followed to solve many problems of wave propagation and vibration in elastic and piezoelectric cylindrical and spherical bodies [6, 10 11]. The electroelastic vibrations of cylinders were studied using the variational difference and finite-element methods [6, etc.]. The method of lines and spline functions (for approximation in one coordinate) were used in [7,8] to study the vibrations of electroelastic cylinders. The problems addressed in these papers were partially solved earlier, but no comparison with their solutions was made. As for the accuracy of the solutions, no particulars were given in [7,8], just generalities.The present paper outlines a unified approach to the solution of the problem for a plate, a cylinder, and a sphere. We will reduce the three-dimensional equations of electroelasticity to an operator Hamiltonian system with respect to the thickness coordinate and analyze the properties of this system. In the case of a harmonic electric potential, we will numerically determine the amplitude-frequency characteristics, resonant and antiresonant frequencies, and dynamic electromechanical-coupling coefficient. The results will be subject to comparative analysis.1. Problem Formulation. To study the harmonic electroelastic vibrations of a piezoelectric plate, cylinder, and sphere, we will start with the equations of mechanical vibrations [6]

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A mixed system of equations of elasticity

Cited by 6 publications

References 3 publications

Application of the hamilton formalism in a timoshenko-type theory of vibrations of plates

Application of the hamilton formalism in a timoshenko-type theory of vibrations of plates

Harmonic thickness vibrations of inhomogeneous elastic layers with curved boundaries

Comparative analysis of the electroelastic thickness vibrations of layers with curved boundaries

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