Application of a refined theory of torsional vibrations to the calculation of natural frequencies and damping coefficients of orthotropic composite cantilever rods

For calculating the natural complex frequencies of torsional vibrations of rectangular orthotropic composite cantilever rods, a theory taking into account the normal stresses and inertial forces acting in the axial direction is employed. The results obtained are compared with those found by using the classical theory of torsional vibrations of rods, the theory of vibrations of thin orthotropic plates, and the FEM. It is found that the difference between the natural frequencies given by the classical and refined theories depends on relations between geometrical sizes of a rod and between its axial elastic modulus and shear moduli, and on the number of the mode of torsional vibrations.     

Resonance methods for determining the complex shear moduli of orthotropic composites

A survey of various methods for determining the complex elasticity and shear moduli from the resonant frequencies of flexural and torsional vibrations of rectangular rods cut out from a plate of an orthotropic composite is presented. The errors in the computed values of dynamic shear moduli caused by inaccuracies in the experimental determination of resonance frequencies are examined. A new variant of the resonance method is developed, which permits one to find three complex shear moduli of a composite from the resonant frequencies and the damping of torsional vibrations of three rods oriented along three symmetry axes of the material. For computing the moduli in the case of an overdetermined system, an algorithm of nonlinear optimization based on the least-squares method is recommended. From the results obtained it follows that, for determining the interlaminar shear moduli with a necessary accuracy, the rods must be sufficiently thick. It is shown that a good agreement alone between calculated and experimental frequencies of flexural and torsional vibrations of rods does not ensure a reliable determination of the moduli of interlaminar shear if experiments are carried out on wide test specimens cut out from a thin plate. Recommendations are given for the choice of geometrical sizes of test specimens for resonance experiments. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 721–744, November–December, 2007.     

A Technique for Determining the Elastic and Dissipative Properties of Orthotropic Composites in Shear

A new method is presented for the characterization of three principal complex shear moduli of linear viscoelastic orthotropic materials, which is based on the measurement of complex torsional vibration frequencies of three rods of rectangular cross section. The rod-type test specimens are cut out from a composite plate along the principal material axes in the reinforcement plane. It is shown that the torsional stiffness of an elastic rod can be calculated not only by means of the Saint-Venant torsion theory, but also using a relationship obtained from the Reissner-Mindlin theory of plates. The transfer to a viscoelastic model of the material with complex moduli is realized with the help of the correspondence principle. By applying a numerical sensitivity analysis of natural frequencies to the shear moduli, the advisable width-to-thickness ratios of the specimens are found. As an illustration of data processing, the dynamic shear moduli and the loss factors for a GFRP fabric with an epoxy matrix are calculated. A comparison of the method offered with some known static and dynamic methods for determining the shear moduli of orthotropic materials is given.     

Application of the resonance method for determining the elastic and dissipative properties of an orthotropic polymer composite

Experimentally determined resonant frequencies and damping of flexural and torsional vibrations of rod-type rectangular test specimens made of an orthotropic GFRP fabric with different ratios of cross-sectional sizes are used for calculating six principal complex elastic and shear moduli. The application of the classical theories of flexural and torsional vibrations, the theory of flexural vibrations of a Timoshenko beam, and a refined theory of torsional vibrations of free-free orthotropic rods is analyzed.     

Designing of layered rod structures from viscoelastic materials at a given service life

The problem on the rational design of a layered rod structure subjected to force and temperature actions under the conditions of creep is considered. The linear hereditary theory is used for the viscoelastic materials of layers. The rods are packages of several homogeneous layers with a rectangular cross section. The variable geometrical parameters of the rods are determined from the energy equal-strength condition at a given in stant of time. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 5, pp. 581–594, September–October, 2007.     

Spatial braiding of cylindrical articles

The effective deformative characteristics of spatially reinforced composites made by spatial braiding along the generatrices of a one-sheet hyperboloid are analyzed. The geometrical relationships determining the structure of a unit cell of a braided composite are derived. The effective thermoelastic characteristics are calculated by the method of orientational averaging. The dependences of the bending and torsional stiffnesses of thick-walled cylindrical rods — made by the method suggested and by winding — on the braiding/winding angle are compared. The numerical estimations are given for rods made of carbon (CFRP) and aramid (AFRP) epoxy plastics. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompzitnykh Materialov, Vol. 36, No. 3, pp. 341–354, May–June, 2000.     

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.     

The vibrations of a thin elastic orthotropic circular cylindrical shell with free and hinged edges

The problem of the existence of natural oscillations of a thin elastic orthotropic circular closed cylindrical shell with free and hinge-mounted ends and of an open cylindrical shell with free and hinge-mounted edges, when the two boundary generatrices are hinge-mounted is investigated. Dispersion equations and asymptotic formulae for finding the natural frequencies of possible vibration modes are obtained using the system of equations corresponding to the classical theory of orthotropic cylindrical shells. A mechanism is proposed by means of which the vibrations can be separated into possible types. Approximate values of the dimensionless characteristic of the natural frequency and the attenuation characteristic of the corresponding vibration modes are obtained using the examples of closed and open orthotropic cylindrical shells of different lengths.     

Forced vibration of an initially stressed thick rectangular plate made of an orthotropic material with a cylindrical hole

The forced vibration of an initially statically stressed rectangular plate made of an orthotropic material is studied. The plate is simply supported along all its edges and contains an internal across-the-width cylindrical hole of rectangular cross section with rounded corners. The initial stresses are created by uniformly distributed normal forces applied to opposite end faces of the plate. Because of the hole, these stresses are not uniform in the plate and significantly affect the stress field caused by additional time-harmonic dynamical forces acting on the upper face of the plate. Hence, for solving the boundary-value problem considered, the superposition principle is unsuitable. Therefore, our investigations are carried out within the framework of the three-dimensional linearized theory of elastic waves in initially stressed bodies. The corresponding boundary- value problems on determining the initial and additional, dynamical stress states are solved by using the three-dimensional finite-element method. Numerical results on stress concentrations around the cylindrical hole and the fundamental frequencies, and on the influence of the initial stresses on the frequencies are presented.     

Asymptotic solutions of non-classical boundary-value problems of the natural vibrations of orthotropic shells

The natural vibrations of orthotropic shells are considered in a three-dimensional formulation for different versions of the boundary conditions on the faces: rigid clamping rigid clamping, rigid clamping free surface, and mixed conditions. Asymptotic solutions of the corresponding dynamic equations of the three-dimensional problem of the theory of elasticity are obtained. The principal values of the frequencies of natural vibrations are determined. It is shown that three types of natural vibrations occur in the shell: two shear vibrations and a longitudinal vibration, which are due solely to the boundary conditions on the faces. It is proved that each boundary layer has its own natural frequency. The boundary-layer functions are determined and the rates at which they decrease with distance from the faces inside the shell are established.     

Investigation of the natural vibrations of plates made of composite materials with the help of the finite-element method

The modes and frequencies of small natural vibrations in a cavity of thin plates are investigated on the basis of the method of finite elements in displacements. The effect of the rigidity characteristics of a material on the natural frequencies and modes of vibrations of flat cantilever vanes whose tapered edge is restrained is studied. An example is given of the use of the finite-element method to determine the natural vibrations of an orthotropic plate in the shape of an airplane wing. Quadrangular and modified triangular bending finite elements are used to simulate a continuous system. A mass-element matrix is constructed on the basis of a four-term ("contracted") polynomial determining the deflection that permits a significant reduction of the order of the solving system.Translated from Mekhanika Polimerov, No. 2, pp. 284–288, March–April, 1976.     

Torsional vibrations of anisotropic rods

By the method of eigenfunction expansions we solve the dynamic problem of the torsion of an anisotropic rod. As an example we study a prismatic rod with a square cross section. The natural vibration frequencies are computed. Two figures. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 104–107.     

Effect of fiber orientation and cross section of composite tubes on their energy absorption ability in axial dynamic loading

The crushing behavior of composite tubes in axial impact loading is investigated. Tubes of circular and rectangular cross section are simulated using an LS-DYNA software. The effect of fiber orientation on the energy absorbed in laminated composite tubes is also studied. The results obtained show that rectangular tubes absorb less energy than circular ones, and their maximum crushing load is also lower. The composite tubes with a [+θ/ -θ] lay-up configuration absorb a minimum amount of energy at θ = 15°. The simulation results for a rectangular composite tube with a [+30/–30] lay-up configuration are compared with available experimental data. Cylindrical composite tubes fabricated from woven glass/polyester composites with different lay-ups were also tested using a drop-weight impact tester, and very good agreement between experimental and numerical results is achieved.     

Vibrations of Piezoceramic Rods

Presently, most of the research on vibrations of monolithic piezoelectric rods at weak electric fields is restricted to longitudinal oscillations of such structural members where free and forced vibrations have been dealt with and in the case of resonance conditions not only linear but also nonlinear effects within the constitutive relations have been incorporated. On the other hand, bending and torsional vibrations of piezoceramic one‐parametric rods have not been examined yet. The present contribution develops a linear vibration theory of rods with a focus to bending vibrations taking into account rotatory inertia and shear deformation. The governing boundary value problem for beams with both longitudinal and as well transversal polarization is derived, in particular free vibrations are analyzed. Also nonlinear extensions not only of physical nature but also geometrical ones are addressed. A possible technical application is given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Scattering of plane harmonic waves on a cylindrical cavity with an elliptical cross section in an orthotropic medium

The diffraction of a plane longitudinal harmonic wave on a cavity with a smooth curvilinear cross section in a rectilinearly orthotropic medium is solved by using small perturbations in the elastic moduli and introducing generalized wave potentials. Results are presented from a numerical analysis of the dynamic stresses in the near diffraction field and at the boundary of an elliptical cavity including variations in the relative incident wavelength, eccentricity of the cavity, and degree of anisotropy of the medium. Donetsk State University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 102–110, 1999.     

Collective vibrations in superconductors with low carrier density and overlapping energy bands on the fermi surface

Collective vibrations in two-band superconductors with low carrier densities are studied for the case described by the BCS superconductivity theory and for the case where localized pairs form a Bose condensate (the Schaffroth theory of superconductivity). It is shown that in both cases, there exists an acoustic mode of collective vibrations and an exciton-type collective mode caused by interband electron-electron interactions. For systems with low carrier densities, the frequencies of the collective vibrations are significantly influenced by the mixing of the phase and amplitude fluctuations of the order parameters for different bands. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 162–175, January, 1997.     

Buckling and free vibration analyses of laminated composite plates by using two new hyperbolic shear-deformation theories

Two new hyperbolic displacement models, HPSDT1 and HPSDT2, are used for the buckling and free vibration analyses of simply supported orthotropic laminated composite plates. The models contain hyperbolic expressions to account for the parabolic distributions of transverse shear stresses and to satisfy the zero shear-stress conditions at the top and bottom surfaces of the plates. The equation of motion for thick laminated rectangular plates subjected to in-plane loads is deduced through the use of Hamilton’s principle. Closed-form solutions are obtained by using the Navier technique, and then the buckling loads and the fundamental frequencies are found by solving eigenvalue problems. The accuracy of the models presented is demonstrated by comparing the results obtained with solutions of other higher-order models given in the literature. It is found that the theories proposed can predict the fundamental frequencies and buckling loads of cross-ply laminated composite plates rather accurately. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 217–230, March–April, 2008.     

Nonlinear vibrations of circular plates with notches. Method of <Emphasis Type="Italic">R</Emphasis>-functions

We investigate vibrations of geometrically nonlinear circular plates with two notches. For the determination of natural frequencies of vibrations, the method of R-functions is used. Nonlinear vibrations of a plate are expanded in eigenmodes of linear vibrations containing R-functions. As a result of using the Bubnov–Galerkin method, we obtain a dynamic system with three degrees of freedom, which is investigated by the method of multiple scales.