Exact solutions for free vibrations of orthotropic rectangular Mindlin plates

Exact closed-form solutions are obtained for free vibrations of orthotropic rectangular Mindlin plates by using the separation of variables method although it is difficult to solve them. The plates have two opposite edges simply supported and all possible combinations of classical boundary conditions at the other two edges. The exact solutions of orthotropic rectangular Mindlin plates are compared with those of isotropic ones and their differences are discussed. The exact solutions are validated through both mathematical proof and numerical comparisons with available p-Ritz solutions and the differential quadrature finite element method solutions calculated by the authors.     

Solution of the problem on the vibrations of beams with a variable cross section using the finite-difference method

We present a numerical calculation of the frequencies of natural vibrations of beams with a cross section varying linearly in height under different conditions of fixing their ends. The differential equation is derived using the finite difference method. The eigenvalues of the equation and the frequency parameters are determined. The results are obtained in several various steps of nodes of the approximation, with their refinement being performed using the extrapolation method. The accuracy of the results is confirmed by comparison with the analytical solution for the case of a beam with a uniform cross section.     

On the existence of periodic solution for equation of motion of thick beams having arbitrary cross section with tip mass under harmonic support motion

A cantilever beam having arbitrary cross section with a lumped mass attached to its free end while being excited harmonically at the base is fully investigated. The derived equation of vibrating motion is found to be a non-linear parametric ordinary differential equation, having no closed form solution for it. We have, therefore, established the sufficient conditions for the existence of periodic oscillatory behavior of the beam using Green’s function and employing Schauder’s fixed point theorem. The derived equation of vibration motion is found to be a non-linear parametric ordinary differential equation, having no closed form solution for it. To formulate a simple, physically correct dynamic model for stability and periodicity analysis, the general governing equations are truncated to only the first mode of vibration. Using Green’s function and Schauder’s fixed point theorem, the necessary and sufficient conditions for periodic oscillatory behavior of the beam are established. Consequently, the phase domain of periodicity and stability for various values of physical characteristics of the beam-mass system and harmonic base excitation are presented.     

Longitudinal vibrations of a rod with a coating

A solution is obtained to the problem of the longitudinal vibration of a rod with a coating. The solution is obtained in a refined formulation. The same theoretical model is also used to find the solution for the case of static tension. An analysis is made of the way in which the properties of a deformable rod are affected by the stiffness of the shear constraints, which is regarded as a physicomechanical characteristic of the coating-substrate system. A method is proposed for experimental determination of stiffness for both static and cyclic loading conditions. All of the relations needed to analyze the stress-strain state are presented, including those required for fatigue tests in tension-compression.Translated from Problemy Prochnosti, No. 10, pp. 68–75, October, 1994.     

Exact solutions for the in-plane vibrations of rectangular Mindlin plates using Helmholtz decomposition

Deriving frequency equations for in-plane vibration of a rectangular plate with different boundary conditions and uniform thickness in the elastic range is the goal of this research. To derive frequency equations, the kinetic and potential energy for in-plane behavior initially are obtained by using the stress–strain–displacement expressions according to the theory of Mindlin plates in Cartesian coordinates by applying the Hamilton’s principle, which leads to five sets of highly coupled differential equations for the equations of motion. Replacement of Helmholtz decomposition for the coupled differential equations creates uncoupled equations of motion. The hypothesis of a harmonic solution for the uncoupled equations lead to wave equations. The general solutions for the wave equations are obtained by using the separation of variables. Finally, the application of boundary conditions yields the frequency equations for the rectangular plate. The natural frequencies are compared and validated by finite element analysis and previously reported results.     

Static and harmonic BEM solutions of gradient elasticity problems with axisymmetry

An advanced boundary element method/fast Fourier transform (BEM/FFT) methodology for treating static and time harmonic axisymmetric problems in linear elastic structures exhibiting microstructure effects, is presented. These microstructure effects are taken into account with the aid of a simple strain gradient elastic theory proposed by Aifantis and co-workers [Aifantis (1992), Altan and Aifantis (1992), Ru and Aifantis (1993)]. Boundary integral representations of both static and dynamic gradient elastic problems are employed. Boundary quantities, classical and non-classical (due to gradient terms) boundary conditions are expanded in complex Fourier series in the circumferential direction and the problem is decomposed into a series of problems, which are solved by the BEM by discretizing only the surface generator of the axisymmetric body. The BEM integrations are performed by FFT in the circumferential directions simultaneously for all Fourier coefficients and by Gauss quadrature in the generator direction. All the strongly singular integrals are computed directly by employing highly accurate three-dimensional integration techniques. The Fourier transform solution is numerically inverted by the FFT to provide the final solution. The accuracy of the proposed boundary element methodology is demonstrated by means of representative numerical examples.The authors acknowledge with thanks for the support provided by I.K.Y. through the program IKYDA 2002 (scientific cooperation between the University of Patras, Greece and the Ruhr-University Bochum, Germany).     

The method of fundamental solutions for problems of free vibrations of plates

In this paper a new boundary method for problems of free vibrations of plates is presented. The method is based on mathematically modelling of the physical response of a system to external excitation over a range of frequencies. The response amplitudes are then used to determine the resonant frequencies. So, contrary to the traditional scheme, the method described does not involve evaluation of determinants of linear systems. The method shows a high precision in simply and doubly connected domains. The results of the numerical experiments justifying the method are presented.     

Methods of fundamental solutions for harmonic and biharmonic boundary value problems

In this work, the use of the Method of Fundamental Solutions (MFS) for solving elliptic partial differential equations is investigated, and the performance of various least squares routines used for the solution of the resulting minimization problem is studied. Two modified versions of the MFS for harmonic and biharmonic problems with boundary singularities, which are based on the direct subtraction of the leading terms of the singular local solution from the original mathematical problem, are also examined. Both modified methods give more accurate results than the standard MFS and also yield the values of the leading singular coefficients. Moreover, one of them predicts the form of the leading singular term.     

Acoustic intensity for a long vessel with noncircular cross section

The acoustic intensity distribution around and within long vessels of noncircular cross section was investigated for parameters typical of biomedical ultrasound and blood vessels. We have developed a collocation method for finding the acoustic field when a uniform plane wave is obliquely incident on a long, not necessarily cylindrical, impedance interface. Results are presented for vessels of noncircular cross section and for vessels with thick walls of nonuniform thickness. The intensity in the vicinity of the vessel, throughout the lumen, and in the wall, is calculated for intermediate length scales, i.e., vessel radius and wall thickness in the range 1 to 10 wavelengths. The intensity distribution is an interference pattern, with complicated regions of increased and decreased intensity. These results are compared with approximate intensity obtained using ray theory. Effects not predicted by ray theory and intensity variations that will be significant in any close ultrasonic investigation of these vessels are revealed.     

The torsional stiffness of prismatic beams of isosceles triangular cross section: a survey of methods for small included angles

Six approximate methods of calculating the torsional stiffness of prismatic beams of isosceles triangular cross section are compared with the exact solution for the cross section in the form of a sector of a circle.     

New analytic-numerical solutions for the mutual inductance of two coaxial circular coils with rectangular cross section in air

In this paper, we present analytic-numerical expressions for the calculation of the mutual inductance of two axisymmetric circular coils with rectangular cross section in air. This original and new method may seem complicated but it is explicit, accurate, and fast, even though all expressions are obtained by the complete elliptic integrals of the first and second kind, Heuman's lambda function, and three terms that must be solved numerically. We confirm the validity of this approach by comparing it with other approaches (filament method and previously published data). We also compare the accuracy and the computational cost of this approach and that of the filament method. All results obtained by the various approaches are in excellent agreement.     

On the Laplace transform of a matrix of fundamental solutions for thermoelastic plates

The Laplace transform of a matrix D(x,t) of fundamental solutions for the partial differential operator describing the time-dependent bending of thermoelastic plates with transverse shear deformation is constructed, and its asymptotic behavior near the origin is investigated. The differential system is reduced to an algebraic one through the application of the Laplace and then Fourier transformations, and all possible cases of roots of the determinant of the latter system are considered. It is shown that in every case, the asymptotic expansion of near the origin has the same dominant term. This is an important step in the construction of boundary-element methods for the above time-dependent model because it determines the nature of the singularity of the kernel of the boundary-integral-equations associated with various initial-boundary-value problems for the governing system.     

Solution to certain contact problems without axial symmetry in the case of a punch with an elliptical cross section

A solution is obtained to two new contact problems in the class of problems considered earlier [3]. The analytically established relations are suitable for use in several mechanical and engineering applications as, for instance, in the problem of heat and electric current conduction through a contact between solid deformable bodies. The most essential parameters have been computed and the results are shown here in the form of graphs.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 24, No. 2, pp. 334–342, February, 1973.