Averaged description of a fluid containing gas bubbles

An averaged equation is derived that describes the irrotational flow of an ideal incompressible fluid due to the expansion and translational motion of bubbles at the sites of a periodic lattice. The calculation of the coefficients of the averaged equation reduces to the solution of a cell problem. An exact solution to the problem is constructed in the form of a series in periodic harmonic functions. An infinite system of equations is written down for the coefficients of the series, and the system is analyzed asymptotically at low volume concentrations c of the bubbles.     

Analytical solution for the cylindrical bending vibration of piezoelectric composite plates

An analytical solution for the cylindrical bending vibrations of linear piezoelectric laminated plates is obtained by extending the Stroh formalism to the generalized plane strain vibrations of piezoelectric materials. The laminated plate consists of homogeneous elastic or piezoelectric laminae of arbitrary thickness and width. Fourier basis functions for the mechanical displacements and electric potential that identically satisfy the equations of motion and the charge equation of electrostatics are used to solve boundary value problems via the superposition principle. The coefficients in the infinite series solution are determined from the boundary conditions at the edges and continuity conditions at the interfaces between laminae, which are satisfied in the sense of Fourier series. The formulation admits different boundary conditions at the edges of the laminated piezoelectric composite plate. Results for laminated elastic plates with either distributed or segmented piezoelectric actuators are presented for different sets of boundary conditions at the edges.     

Nonstationary vibrations of a discretely accreted thermoelastic parallelepiped

A procedure for determining nonstationary vibrations of a discretely accreted thermoelastic body in the approximation of small deformations and thermal flows is developed. A closed-form solution is constructed for a growing parallelepiped under “smoothly rigid” heat-insulated fixation conditions for the stationary faces and the growing load-free face. The temperature field on the growing face is analyzed numerically for various accretion scenarios.     

Infinite systems in problems for a stiffened rectangular plate

A method is proposed for obtaining analytic solutions of a set of infinite systems of linear algebraic equations arising in problems of elasticity for stiffened rectangular plates with stiffening ribs. The method is based on a transformation of a set of infinite systems to a single system and on determining a majorant of the function generating the system series with regard to the order of the unknowns. It is proved that the constructed solution satisfies the infinite system for large indices of the unknowns. The amount of computations is decreased, and the reliability of the results increases. Some realization examples are given.     

Nonlinear dynamics of a system of rigid bodies with controlled electromagnetic dampers

A nonlinear mathematical model of a system of n rigid bodies undergoing translational vibrations under inertial loading is constructed. The system includes ball supports as a seismic-isolation mechanism and electromagnetic dampers controlled via an inertial feedback channel. A system of differential dynamic equations in normal form describing accelerative damping is derived. The frequencies of small undamped vibrations are calculated. A method for analyzing the dynamic coefficients of rigid bodies subject to accelerative damping is developed. The double phase–frequency resonance of a two-mass system is studied     

On paradoxes of the formal approach to determining the eigenvalues of one-dimensional boundary value problems

The conventional approach to determining the eigenvalues of a one-dimensional boundary value problem consists in writing out the solution of the differential equation in general form containing indeterminate coefficients and constructing a system of homogeneous linear algebraic equations for these coefficients on the basis of the expressions for the boundary conditions. The eigenvalue is determined from the condition that the determinant of the system thus constructed is zero. In the classical problems (of string, rod, etc. vibrations), this method, as a rule, does not cause any difficulties, although several examples in which the zero value of the frequency satisfying the characteristic equation thus constructed is not an eigenfrequency were constructed and investigated, for example, in [1, p. 220]. We show that in some cases more complicated than the classical ones a similar situation can lead to paradoxical conclusions and erroneous results.     

On the general theory of thermo-elastic friction

Summary A theory of the thermo-elastic dissipation in vibrating bodies is developed, starting from the three-dimensional thermo-elastic equations. After a discussion of the basic thermodynamical foundations, some general considerations on the problem of the conversion of mechanical energy into heat are given. The solution of the coupled thermo-elastic equations is found in the form of series expansions in terms of normalised orthogonal eigenfunctions. For the coefficients an infinite system of algebraic equations with constants, which are complicated field integrals, is derived. An approximate solution of the infinite system is given. In some cases the coupling-constants can be calculated exactly, in other cases they have to be determined on the base of approximate theories.     

变系数曲线支承矩形厚板自由振动的傅里叶分析

本文给出了变系数曲线支承的Ａｍｂａｒｓｕｍｉａｎ矩形厚板自由振动问题的级数解，将位移和剪力在板域内展成重傅里叶级数，将其导数在边界上展成单傅里叶级数，通过傅里叶变换将控制微分方程和边界条件转化成关于位移级数的系数的一组无穷线性代数方程，最终将板的自由振动问题转化为矩阵特征值问题。     

Axisymmetric Collision Problem for Two Identical Elastic Solids of Revolution

A direct central collision of two identical bodies of revolution is studied. A nonstationary mixed boundary-value problem with an unknown moving boundary is formulated. Its solution is represented by a series in term of Bessel functions. An infinite system of Volterra equations of the second kind for the unknown expansion coefficients is derived by satisfying the boundary conditions. The basic characteristics of the collision process are determined depending on the curvature of the frontal surface of the bodies     

A Pressure Field in an Acoustic Medium Containing a Cylindrical Solid and a Sphere Oscillating in a Predetermined Manner

A problem on the interaction of a spherical body oscillating in a predetermined fashion and a rigid cylinder is formulated. The bodies do not intersect, are immersed into an ideal compressible liquid, and their centers are in one plane. The solution is based on the possibility of representing the partial solution of the Helmholtz equation, written in cylindrical coordinates, in terms of partial solutions in spherical coordinates, and vice versa. An infinite system of linear algebraic equations is obtained by satisfying the boundary conditions on the sphere and cylinder surfaces. The system is intended for determining the coefficients of the expansion of the velocity potential into a series in terms of spherical and trigonometric functions. The system obtained is solved by the reduction method. The appropriateness of this method is substantiated. The hydrodynamic characteristics of the liquid surrounding the spherical and cylindrical bodies are determined. A comparison is made with the problem on a sphere oscillating in an infinite incompressible liquid that contains also a cylinder and in a compressible liquid that contains nothing more. Two types of motion of the sphere — pulsation and oscillation — are considered     

Nonlinear bending of rectangular orthotropic plate with two free edges and the other edges having nonuniform rotational flexibility

this paper presents a series solution to von Kármán nonlinear equations of a rectilinearly orthotropic rectangular plate under the combined action of lateral load and in-plane tension with the titled edge restraints. In the formulation the edge moments are replaced by an equivalent pressure near the edges. Using generalized double Fourier series for the deflection and stress function, governing equations are reduced to an infinite set of algebraic equations for coefficients in these series. Numerical results for deflections, bending moments and in-plane forces are graphically presented for various values of aspect ratio and material properties and for different edge conditions.     

Plane Collision Problem for Two Identical Elastic Parabolic Bodies—Direct Central Impact

A direct central collision of two identical infinite cylindrical bodies is studied. A nonstationary plane elastic problem is solved. The variable boundary of the contact area is determined. A mixed boundary problem is formulated. Its solution is represented by Fourier series. An infinite system of Volterra equations of the second kind for the unknown expansion coefficients is derived by satisfying boundary conditions. The basic characteristics of the collision process are determined numerically depending on the curvature of the frontal surface of the bodies     

An exact axisymmetric solution for a simply supported piezoelectric cylindrical shell

Summary Three-dimensional axisymmetric solution is presented for a simply supported piezoelectric cylindrical shell. The variables are expanded in Fourier series to satisfy the boundary conditions at the ends. The solution of the governing differential equations with variable coefficients is constructed as a product of an exponential function and a power series. The coefficients of terms of all degrees in the governing equations are set to zero, yielding a characteristic equation for the exponent and recursive relations for the coefficients of the power series. Results are presented illustrating the effect of thickness parameter of the shell. An inverse problem of inferring the applied temperature from the measured potential difference has been solved. Accepted for publication 26 July 1996     

SCATTERING OF CIRCULAR CAVITY IN RIGHT-ANGLE PLANAR SPACE TO STEADY SH-WAVE

Complex function method and multi-polar coordinate transformation technology are used here to study scattering of circular cavity in right-angle planar space to SH-wave with out-of-plane loading on the horizontal straight boundary. At first, Green function of right-angle planar space which has no circular cavity is constructed; then the scattering solution which satisfies the free stress conditions of the two right-angle boundaries with the circular cavity existing in the space is formulated. Therefore, the total displacement field can be constructed using overlapping principle. An infinite algebraic equations of unknown coefficients existing in the scattering solution field can be gained using multi-polar coordinate and the free stress condition at the boundary of the circular cavity. It can be solved by using limit items in the infinite series which can give a high computation precision. An example is given to illustrate the variations of the tangential stress at the boundary of the circular cavity due to different dimensionless wave numbers, the location of the circular cavity, the loading center and the distributing range of the out-of-plane loading. The results show the efficiency and effectiveness of the method introduced here.     

On inverse coefficient problems of poroelasticity

General approaches to the inverse coefficient problems of poroelasticity on the basis of a modified Biot model are considered. A generalized reciprocity relation is constructed, and an iteration process for determining the unknown coefficients is stated. By way of example, the problem of steady-state longitudinal vibrations of an inhomogeneous poroelastic layered system is considered, and integral equations for the direct and inverse problems are derived. The results of computational experiments where the elastic modulus and the Biot modulus were reconstructed for various laws of variation are given.     

Constructing invariant tori for two weakly coupled van der Pol oscillators

An algorithm is developed for the construction of an invariant torus of a weakly coupled autonomous oscillator. The system is put into angular standard form. The determining equations are found by averaging and are solved for the approximate amplitudes of the torus. A perturbation series is then constructed about the approximate amplitudes with unknown coefficients as periodic functions of the angular variables. A sequence of solvable partial differential equations is developed for determining the coefficients. The algorithm is applied to a system of nonlinearly coupled van der Pol equations and the first order coefficients are generated in a straightforward manner. The approximation shows both good numerical accuracy and reproducibility of the periodicities of the van der Pol system. A comparitive analysis of integrating the van der Pol system with integrating the phase equations from the angular standard form on the approximate torus shows numerical errors of the order of the perturbation parameter =0.05 for integrations of up to 10,000 steps. Applying FFT to the numerical periodicities generated by integrating the van der Pol system near the tours reveals the same predominant frequencies found in the perturbation coefficients. Finally an expected rotation number is found by integrating the phase equations on the approximate torus.Contribution of the National Institute of Standards and Technology, a Federal agency.     

Coupled frequencies of a rectangular hydroelastic system with two fluids

This paper deals with the study of behaviour of an idealized 2D hydroelastic system involving two inviscid liquids with an elastic rectangular container. The main objective is to investigate the influence of the physical parameters on eigenfrequencies and eigenmodes of the system. The study extends the previous results obtained for hydroelastic systems with one fluid. The governing equations describing the behaviour of the system are analyzed by using the concept of normal modes and their solutions presented in the form of infinite series. The expansion coefficients for the velocity potentials are calculated by employing a new inner product that allows the orthogonalisation of the normal modes. An eigenfrequency equation is then derived from the existence condition of a nontrivial solution. The numerical calculations are performed by varying only some relevant parameters.     

Interaction of a plane harmonic wave with a thin rigid inclusion of the shape of a cylindrical shell

The paper presents the solution of the problem of determining the stress state in an elastic matrix containing a rigid inclusion of the shape of a thin cylindrical shell. It is assumed that harmonic vibrations occur in the matrix under the conditions of axial symmetry (the symmetry axis is the inclusion axis) and the conditions of full adhesion between the inclusion and the matrix are satisfied. The vibrations are caused by the propagation of a plane wave whose front is perpendicular to the inclusion axis. The solution method is based on representing the displacements in the matrix as discontinuous solutions of the equations of axisymmetric oscillations of an elastic medium with unknown stress jumps on the inclusion surface. The realization of the boundary conditions for these jumps leads to a system of integral equations. Its solution is constructed numerically by the mechanical quadrature method with the use of special quadrature formulas for specific integrals. It is numerically investigated how the ratio of the inclusion geometric dimensions and the propagating wave frequency affect the stress concentration near the inclusion.     

Theory of dynamic processes in mechanical systems and materials of regular structure

The theory of vibrations and waves in natural and synthesized materials of regular structure is analyzed. Models based on different averaging and continualization methods are outlined. Emphasis is on periodically inhomogeneous structures. The exact solutions are obtained and analyzed using the closed-form solution of infinite algebraic systems, representing equations in Hamiltonian operator form and solving them based on the theory of differential equations with periodic coefficients, mode selection rule, and methods of drawing wave shapes at limit and arbitrary frequencies     

直角平面内弹性圆夹杂对入射平面SH波的散射

利用复变函数法、多极坐标移动技术及傅立叶级数展开求解二维直角平面内圆形弹性夹杂对稳态入射平面SH波的散射问题。首先写出直角平面内不含夹杂时的入射波场和反射波场;其次建立直角平面内含夹杂时夹杂外的散射波解和夹杂内的驻波解,并利用叠加原理写出问题的总波场,借助夹杂边界处应力和位移的连续条件建立求解散射波解和驻波解中未知系数的无穷代数方程组并求解,通过算例具体讨论了直角平面水平边界点的位移幅度比和夹杂边界处径向应力集中系数随不同无量纲波数、入射角及圆孔位置的变化情况,结果表明了算法的有效实用性。