In this paper a solution to the free vibration problem of composite circular and annular membranes is presented. The vibrations of membranes whose material densities and/or thicknesses varied step-wise with the radial co-ordinate are considered. This approach is applied to approximate the solution to the vibration problem of a membrane with continuously varying density and/or thickness with the radial co-ordinate. The obtained analytical solutions are used in numerical investigations into the effect of parameters characterizing the composite membranes on their eigenfrequencies.
In the paper, the process of identification of crack parameters occurring in the cantilever beam with the variable cross-sectional area has been presented. For identification, the non-destructive vibration method has been applied. The analytical solution of the free vibration problem of the beam described according to the Bernoulli-Euler theory has been obtained with the help of Green’s functions.
The paper presents the problem of free vibration of the cantilever Bernoulli-Euler beam with a crack. The beam along the length has a variable cross-sectional area, and the crack parameters (thickness, depth and location) undergo changes. The work includes an analytical solution of the vibration problem and the results obtained with the use of FEM and CATIA program.
The paper presents the solution of a fourth order differential equation with various coefficients occurring in the vibration problem of the Euler-Bernoulli beam. The concerning equation is written as a first order matrix differential equation. To solve the equation, the power series method is proposed.
In the paper, the solution of second order differential equations with various coefficients is presented. The concerning equations are written as first order matrix differential equations and solved with the use of the power series method. Examples of application of the proposed method to the equations occurring in the technical problems are presented.
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In the paper, the derivation of Green's functions for Helmholtz equation in circular, annular and exterior circular domains is presented. The Green's functions are assumed of the form a cosine series. An example of application of the Green's functions in frequency analysis of a composite membrane is presented.
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In this paper, the free vibration of an annular membrane consisting of three concentric segments is considered. The frequency equation and mode shapes are obtained by the use of the Green's function method. A numerical example to vibration problem of nonhomogeneous annular membrane is presented.
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In this paper, the object of consideration is a membrane consisting of one circular and two annular segments. The Green's functions method for the free vibration of this membrane is applied. A numerical example illustrating this method is given.
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In the present paper, a Green's matrix used to solve vibration problems of Timoshenko beams is determined. The problem formulation includes beam vibrations which are described by differential equations with variable parameters. To determine the Green's matrix, the power series method was used.
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In this paper the Green's function method for the free vibration problem of nonuniform Bernoulli-Euler beams is presented. To find the Green's function of the fourth order differential operator, occurring at the beam's equation of motion, the power series method is proposed.
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In this paper is presented a solution of the free vibration problem of multistepped Timoshenko beam with attachments. The Green's functions method is used to obtain the solution in an form. The presented approach can be used for the vibration analysis of beams consisting of an arbitrary number of segments and an arbitrary number of attached discrete elements.
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The purpose of this paper is free longitudinal vibration problem of a non-uniform rods coupled by spring-mass systems. The solution of the boundary problem is obtained by the use of the Green's function method. In the paper examples of the Green's functions corresponding to the second order differential operator are presented.
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