Effect of Torsion and Warping on the Free Vibration of Sandwich Beams

The problem on free vibrations of wide sandwich beams is tackled in this paper. Torsional and warping effects in addition to flexure are included in the formulation of the dynamic problem. In order to show the effects of bending-torsion coupling and warping on the natural frequencies and the corresponding vibration modes, three cases are considered. First, the warping and torsion effects are neglected, second, the effect of warping on the coupling terms is neglected, and third, the effect of warping on the coupling terms is included. The viscoelastic core is modeled by elastic translational and rotational springs. The finite-difference method is used to solve the partial differential equations of motion with different boundary conditions for the top and bottom layers. Results for different materials, fiber orientations, depth-to-width ratios, and boundary conditions are found. The natural frequencies and the corresponding vibration modes obtained are in a good agreement with those cited in the literature. If the bending-torsion coupling is pronounced, the inclusion of warping affects the natural frequency considerably.__________Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 2, pp. 163–176, April–May, 2005.     

Vibration of a Reissner–Mindlin–Timoshenko plate–beam system

In this paper, we consider a plate–beam system in which the Reissner–Mindlin plate model is combined with the Timoshenko beam model. Natural frequencies and vibration modes for the system are calculated using the finite element method. The interface conditions at the contact between the plate and beams are discussed in some detail. The impact of regularity on the enforcement of certain interface conditions is an important feature of the paper.     

An approach to studying nonsteady-state vibrations of a finite length cylinder

A method is given for analyzing the nonsteady-state vibrations of a finite length cylinder based on using Laplace transforms and approaches developed previously for determining the vibration natural frequencies and forms. A numerical method is demonstrated for determining residues with reference to expressions for displacements and stresses. Quantitative estimates are given for stresses in a cylinder for a specific principle of the change in load with time and features of the time-dependent behavior of displacements in relation to the magnitude of the area of loading at the end are studied.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 129–133, 1987.     

On the determination of natural frequencies and mode shapes of cable-stayed bridges

An accurate analysis of the natural frequencies and mode shapes of a cable-stayed bridge is fundamental to the solution of its dynamic responses due to seismic, wind and traffic loads. In most previous studies, the stay cables have been modelled as single truss elements in conventional finite element analysis. This method is simple but it is inadequate for the accurate dynamic analysis of a cable-stayed bridge because it essentially precludes the transverse cable vibrations. This paper presents a comprehensive study of various modelling schemes for the dynamic analysis of cable-stayed bridges. The modelling schemes studied include the finite element method and the dynamic stiffness method. Both the mesh options of modelling each stay cable as a single truss element with an equivalent modulus and modelling each stay cable by a number of cable elements with the original modulus are studied. Their capability to account for transverse cable vibrations in the overall dynamic analysis as well as their accuracy and efficiency are investigated.     

Various methods of determining the natural frequencies and damping of composite cantilever plates. 1. Exact solution for the binomial model of deformation

On the basis of the classical theory of thin anisotropic laminated plates the article analyzes the free vibrations of rectangular cantilever plates made of fibrous composites. The application of Kantorovich's method for the binomial representation of the shape of the elastic surface of a plate yielded for two unknown functions a system of two connected differential equations and the corresponding boundary conditions at the place of constraint and at the free edge. The exact solution for the frequencies and forms of the free vibrations was found with the use of Laplace transformation with respect to the space variable. The magnitudes of several first dimensionless frequencies of the bending and torsional vibrations of the plate were calculated for a wide range of change of two dimensionless complexes, with the dimensions of the plate and the anisotropy of the elastic properties of the material taken into account. The article shows that with torsional vibrations the warping constraint at the fixed end explains the apparent dependence of the shear modulus of the composite on the length of the specimen that had been discovered earlier on in experiments with a torsional pendulum. It examines the interaction and transformation of the second bending mode and of the first torsional mode of the vibrations. It analyzes the asymptotics of the dimensionless frequencies when the length of the plate is increased, and it shows that taking into account the bending-torsion interaction in strongly anisotropic materials type unidirectional carbon reinforced plastic can reduce substantially the frequencies of the bending vibrations but has no effect (within the framework of the binomial model) on the frequencies of the torsional vibrations.Institute of Engineering Science Russian Academy of Sciences, St. Petersburg, Russia. St. Petersburg State University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 6, pp. 759–769, November–December, 1996.     

Natural vibration of the beam-strip fabricated from a composite material with small-scale curvings in the structure

The frequency and forms of natural vibrations of a hinged beam-strip fabricated from a composite material with small-scale curved structures have been investigated using the plate theory in the framework of the Timoshenko hypothesis. The mechanical relationships of the composites are described by the Akbarov and Guz continuum theory. Using the corresponding variational principle, a method of solution of the considered problem is presented. Specific numerical investigations have been carried out for the case when the strip material consists of alternating equidistantly located and curved isotropic layers. The effect of existence of the curving in the beam-strip structure on its frequencies and forms of natural vibrations is studied. Cases of periodic and local curvings in the beam-strip material structure are considered separately.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October, 1995.Yilziz Technical University, Civil Engineering Faculty, 80750 Yildiz-Istanbul, Turkey. Published in Mekhanika Kompozitnykh Materialov, Vol. 32, No. 4, pp. 502–512, July–August, 1996.     

Investigation of the vibrations of relaxing systems by the averaging method

The method of averaging is used to investigate the damping of the vibrations of a one-dimensional, physically and geometrically linear viscoelastic system with one degree of freedom. The motion of this system under an external load is studied at resonance. The averaging principle is justified on both finite and infinite time intervals for nonlinear multidimensional systems.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 5, pp. 806–813, September–October, 1969.     

Features of the construction of correction functions in the analysis of the free vibrations of mechanical systems by Bubnov''s method

Free vibrations of mechanical systems are investigated by Bubnov's method, using coordinate functions that do not satisfy all the boundary conditions of the problem. To eliminate discrepancies in the boundary conditions, correction functions are introduced, orthogonal to all coordinate functions involved in the analysis of the natural vibrations of mechanical systems. This method of constructing correction functions reveals the existence of differences between nearly equal numbers in the frequency equation and in the expressions for the vibrational modes. Moreover, for a given number N of coordinate functions, the method guarantees increased precision in computing all the natural frequencies of the mechanical system up to and including the (N + 1)th partial frequency as the number of terms of the power series in the frequency parameter representing the correction functions is increased. In the limit, as the number of terms of the power series tends to infinity, one is assured of obtaining exact values of the frequencies and vibrational modes in the indicated frequency range.     

Free vibrations of a conical shell of slightly varying thickness with annular strengthening

Free vibrations are considered for truncated conical shells of rotation of slightly varying thicknesses with annular strengthening. A solution is built up on the basis of linear shell and bar theory using the Ritz method. Qualitative estimates are given for the effect of rigidly joined solid bodies, and the number, stiffness, and placing of strengthening elements on the change in frequency spectrum and the natural forms of vibration. Theoretical results are confirmed by experimental data obtained by the holographic interferometry method.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 88–93, 1989.     

Finite element construction on the basis of analytic solutions of elasticity theory problems

A finite element in the form of a part of a circular ring loaded by a uniformly distributed radial load is constructed on the basis of the exact solution of an elasticity theory problem. This permits a system of resolving equations to be obtained by the finite element method with a three-diagonal matrix. An example is presented that illustrates the accuracy and efficiency of utilization of the elements mentioned.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 85–90, 1988.     

Various methods of determining the natural frequencies and damping of composite cantilever plates. 2. Approximate solution by Galerkin's method for the trinomial model of damping

The application of Kantorovich's method to a trinomial model of deformation taking into account transverse bending of a plate leads to a connected system of three ordinary differential equations of fourth order with respect to three unknown functions of the longitudinal coordinate and to the coresponding boundary conditions for them at the fixed end and on the free edge. For the approximate calculation of the frequencies and forms of natural vibrations Galerkin's method is used, and as coordinate functions we chose orthogonal Jacobi polynomials with weight function. The dimensionless frequencies depend on the magnitude of the four dimensionless complexes, three of which characterize the anisotropy of the elastic properties of the composite. For the fibrous composites used at present we determined the possible range of change of the dimensionless complexes d₁₆ and d₂₆ attained by oblique placement. The article examines the influence of the angle of reinforcement on some first dimensionless frequencies of a plate made of unidirectional carbon reinforced plastic. It also analyzes the asymptotics of the frequencies when the length of the plate is increased, and it shows that for strongly anisotropic material with the structure []_T the frequencies of the flexural as well as of the torsional vibrations may be substantially lower when flexural-torsional interaction is taken into account.For Communication 1 see [4].Institute of Engineering Science of the Russian Academy of Sciences, St. Petersburg, Russia. St. Petersburg State University, Russia. Translated from Mekhanika Kompozitnykh Materialov, No. 1. pp. 23–33, January–February, 1997.     

Flexural vibrations of a moving rod

The problem of the transverse natural vibrations of part of a rod between two coaxially fixed guides, moving with an arbitrary constant velocity, is investigated. The conditions of rigid clamping are taken as the boundary conditions. Additional shear stresses, due to longitudinal tension or compression are taken into account. Relations defining the natural frequencies and forms are constructed in an exact formulation by Fourier method. The dependence of the natural frequencies and forms of the lowest vibration modes on the rate of displacement, unknown in the literature, are constructed, and their features are established. A modelling and animation of unusual wave motions of the rod are presented. The main characteristics for the higher vibration modes are constructed.     

Electroelastic thickness vibrations of a quartz SC-cut plate

Analytic solutions are obtained of the boundary value problem of steady motions of a thick piezocrystalline SC-cut quartz plate describing thickness modes of harmonic elastic vibrations. The plate endfaces have no electrodes and adjoin a vacuum or a rarefied gas. The measure of the difference between the found values of the five lowest eigenfrequencies of the thickness vibrations of a plate without electrodes and the values of the corresponding frequencies of a plate whose endfaces are covered by massive short-circuited electrodes is determined.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 83–85, 1988.     

Problems in dynamics of viscoelastic multilayer anisotropic shells and plates. Part 3: Optimization of vibration immunity characteristics of composite cylindrical shell

Conclusions Equations have been proposed which describe steady-state vibrations of multilayer shells made of composite materials. On the basis of these equations some simplest modes of deformation have been analyzed. The problem of optimizing the vibration immunity has been solved for a cylindrical shell serving as a component of a simple mechanical system.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 258–262, March–April, 1982.     

Radiation and Scattering by Complex Conformal Antennas on a Circular Cylinder

A technique for characterizing and designing complex conformal antennas flush-mounted on a singly-curved surface is presented. This approach is based on the hybrid finite element–boundary integral (FE–BI) method. A related method was proposed in the past utilizing cylindrical-shell finite element and roof-top rectangular basis functions for the boundary integral. Although that method proved very powerful for analyzing cylindrical–rectangular patch arrays flush-mounted to a circular cylinder, the requirement for uniform meshing in the aperture ultimately limited its usefulness. In this present formulation, tetrahedral elements are used to expand the volumetric electric fields while similar basis functions are used for the boundary integral. The curvature of the aperture is explicitly included via the use of the circular cylinder dyadic Green's function. After presentation of the formulation and validation using several well-understood examples, an example is presented that illustrates the capabilities of this method for modeling complex conformal antennas heretofore not examined by rigorous methods due to inherent limitations of the various published methods.     

A method for determining the dynamic Young's modulus of curvilinear specimens

On the basis of a numerical solution of the problem of the natural frequencies of a rod shaped as the are of a circle, the authors suggest a method for determining the dynamic Young's modulus of a curvilinear specimen. They derive an approximate expression suitable for practical use. The results agree well with experiment.Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 335–339, March–April, 1974.     

Natural vibrations of composite viscoelastic cylinders

Free damping spatial and axisymmetrical vibrations of short cylinders consisting of elastic and viscoelastic layers are considered. A numerical study is made of the dependence of natural frequencies and damping coefficients on the ratio of geometric cylinders.Translated from Dinamicheskie Sistemy, No. 5, pp. 55–58, 1986.     

Calculation of layered structures by the semianalytic method of finite elements

A semianalytic method of finite elements is developed for calculating layered composite structures. The variables are separated on the basis of the Reissner variation principle. A number of plate deformation problems are solved, and a comparison with the known solutions is made. The method can be efficient when dealing with research, as well as engineering, problems.Ukrainian Transport University, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 6, pp. 781–785, November–December, 1997.     

Application of fuzzy sets to transient analysis of space structures

An application of fuzzy sets, in conjunction with finite elements, to the transient analysis of a precision-deployable space structure is presented. The structural members are modeled by using beam finite elements, and the structure's latch joint is modeled by using a spring–damper–Coulomb friction element. Two types of transient response simulations are performed: slow transient load–deflection response and transient impulse response. The first simulation is used to evaluate the stiffness and buckling loads at the structure's tip. The second simulation is used to evaluate the structure's natural frequencies, mode shapes and the precision of the final shape. For each simulation the possibility distributions of various response quantities are obtained. Fuzzy sets are used to represent three beam properties, namely: damping coefficient, bending stiffness, and axial stiffness; as well as two joint parameters: Coulomb friction force and damping coefficient. Fuzzy set techniques provide an insight into the range of possible responses associated with the combined selected variations in the system parameters.