Study of the Resonant Vibrations and Dissipative Heating of Thin-Walled Elements Made of a Physically Nonlinear Material

An approximate formulation is given to a dynamic coupled thermomechanical problem for physically nonlinear inelastic thin-walled structural elements within the framework of a geometrically linear theory and the Kirchhoff–Love hypotheses. A simplified model is used to describe the vibrations and dissipative heating of inhomogeneous physically nonlinear bodies under harmonic loading. Nonstationary vibroheating problem is solved. The dissipative function obtained from the solution for steady-state vibrations is used to simulate internal heat sources. For the partial case of forced vibrations of a beam, the amplitude–frequency characteristics of the field quantities are studied within a wide frequency range. The temperature characteristics for the first and second resonance modes are compared.     

The forced transversal vibrations and vibroheating of a cylindrical bimorph panel made from a dissipative piezomaterial

The forced vibrations and dissipative heating of a bimorph cylindrical shell are considered. The shell is made from a dissipative piezomaterial and subjected to a harmonic potential difference. The edges of the panel are assumed hinged and perfectly heat-insulated. The dissipative properties of the material are considered on the basis of the concept of complex characteristics. The analystical solution of this problem is found. A finite-element method is developed to study the dynamic behavior and vibroheating temperature of bimorph shells, which are made from viscoelastic material and subjected to harmonic loading. The results of the analyses of the electromechanical vibrations of the panel performed by the finite-element method and by analytically solving the problem are compared. Translated from Prikladnaya Mekhanika, Vol. 36, No. 5, pp. 89–97, Mary, 2000     

Transverse vibrations and vibrational heating of a rectangular bimorphous plate of dissipative piezoelectric material

The problem of induced resonance vibrations and dissipation heating in a rectangular bimorphous plate made of a dissipative piezoelectric material under a harmonic potential difference is tackled. The edges of the plate are considered to be hinged and ideally thermally insulated. The dissipation properties of the material are taken into account on the basis of the concept of complex characteristics, which are assumed to be temperature-independent. An exact solution is found for the problem. The critical value of the load parameter is determined when the maximum temperature reaches the Curie point. A finite-element method has been developed for investigating the dynamic behavior and temperature of vibrational heating that bimorphous plates made of a viscoelastic material undergo under a harmonic load. The results obtained for the electromechanical vibrations of plates by finite-element calculations and by an analytical solution are compared. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 85–93, September, 1999.     

Harmonic vibrations and waves in a cylindrical helically anisotropic shell

A Kirchhoff-Love type applied theory is used to study the specific characteristics of harmonic waves and vibrations of a helically anisotropic shell. Special attention is paid to axisymmetric and bending vibrations. In both cases, the dispersion equations are constructed and a qualitative and numerical analysis of their roots and the corresponding elementary solutions is performed. It is shown that the skew anisotropy in the axisymmetric case generates a relation between the longitudinal and torsional vibrations which is mathematically described by the amplitude coefficients of homogeneous waves. In the case of a shell with rigidly fixed end surfaces, the dependence of the first two natural frequencies on the shell length and the helical line slope α, i.e., the geometric parameter of helical anisotropy, is studied. A boundary value problem in which longitudinal vibrations are generated on one of the end surfaces and the other end is free of forces and moments is considered to analyze the degree of transformation of longitudinal vibrations into longitudinally torsional vibrations. In the case of bending vibrations, two problems for a half-infinite shell are studied as well. In the first problem, the waves are excited kinematically by generating harmonic vibrations of the shell end surface in the plane of the axial cross-section, and it is shown that the axis generally moves in some closed trajectories far from the end surface. In the second problem, the reflection of a homogeneous wave incident on the shell end is examined. It is shown that the “boundary resonance” phenomenon can arise in some cases.     

Vibrations of a Rigid Body with Cylindrical Surface on a Vibrating Foundation

The analytic solution of the problem of forced vibrations of a rigid body with cylindrical surface on a horizontal foundation is given. It is assumed that the dry friction force acts at the point of contact between the cylindrical surface of the body and the foundation and the foundation moves by a harmonic law in the horizontal direction perpendicularly to the cylindrical surface element. The averaging method is used to determine the forced vibration mode near the natural frequency of the body vibrations on the fixed foundation. The results are presented as amplitude-frequency and phase-frequency characteristics.     

Harmonic thickness vibrations of inhomogeneous elastic layers with curved boundaries

The thickness vibrations of elastic inhomogeneous bodies of different geometry under dynamic harmonic loading are studied. The dependence of the amplitude–frequency characteristics of homogeneous and inhomogeneous bodies on excitation frequency is analyzed in detail. The frequency spectra for plane, cylindrical, and spherical layers are determined     

Harmonic Vibrations of a Viscoelastoplastic Sandwich Cylindrical Shell

The free and forced harmonic vibrations of viscoelastoplastic sandwich shells are analyzed. An equation for approximate determination of the amplitudes of near-resonance vibrations is derived. As an example, the problem is solved for a sandwich circular cylindrical shell     

Resonance Vibrations and Dissipative Heating of Thin-Walled Laminated Elements Made of Physically Nonlinear Materials

The dynamic thermomechanical problem for thin-walled laminated elements is formulated based on the geometrically linear theory and Kirchhoff–Love hypotheses. A simplified model of vibrations and dissipative heating of structurally inhomogeneous inelastic bodies under harmonic loading is used. The mechanical properties of materials are described using strain-dependent complex moduli. A nonstationary vibration-heating problem is solved. The dissipative function, derived from the stationary solution, is used to specify internal heat sources. The amplitude–frequency characteristics and spatial distributions of the main field variables are studied for a sandwich beam subjected to forced vibrations     

Numerical simulation of forced nonlinear vibrations of a thread

We consider 3D nonlinear vibrations of an elastic thread in the case of a plane harmonic motion of one of its ends. The other end of the thread is fixed. To simulate the vibrations, a monotone finite-difference ENO-scheme of second-order accuracy is used. We study amplitude-frequency characteristics, vibration beats in the horizontal and vertical planes, and the thread shape and trajectories. On the basis of analysis of the thread motion, the possibility of using the one-mode approximation for describing the thread dynamics is discussed.     

Radial Vibrations and Heating of a Ring Piezoplate under Electric Excitation Applied to Nonuniformly Electroded Surfaces

Radial vibrations and dissipative heating of a polarized piezoceramic ring plate are studied. The plate is excited by a harmonic electric field applied to nonuniformly electroded surfaces of the plate. The viscoelastic behavior of piezoceramics is described in terms of complex quantities. An analytical solution is found in the case of quasistatic harmonic loading. The dynamic nonlinear problem of coupled thermoviscoelasticity is solved with regard for the temperature dependence of the properties of piezoceramics by step-by-step integration in time, using the numerical methods of discrete orthogonalization and finite differences. A numerical analysis is conducted for TsTStBS-2 piezoceramics to study the influence of partial electroding on the stress–strain distribution, natural frequency, and amplitude–frequency and temperature–frequency characteristics     

Basic vibrations and anharmonic analysis of a vibration

We prove that, besides the simple harmonic vibrations, some anharmonic vibrations are basic as well, because a general vibration can be considered as a superposition of such vibrations with different frequencies. The results in this paper are a generalization of Fourier analysis and a new theory of vibration analysis.     

Flexural Vibrations and Heating of a Circular Bimorph Piezoplate under Electric Excitation Applied to Nonuniformly Electroded Surfaces

The flexural vibrations and dissipative heating of a circular bimorph piezoceramic plate are studied. The plate is excited by a harmonic electric field applied to nonuniformly electroded surfaces. The viscoelastic behavior of piezoceramics is described in terms of temperature-dependent complex moduli. The nonlinear coupled problem of thermoviscoelasticity is solved by step-by-step integration in time, using the discrete-orthogonalization method to solve the mechanics equations and the finite-differences method to solve the heat-conduction equations. A numerical analysis is conducted for TsTStBS-2 piezoceramics to study the influence of the nonuniform electroding on the resonant frequency, amplitude, and modes of flexural vibrations and the amplitude- and temperature-frequency characteristics of the plate __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 9, pp. 94–100, September 2005.     

Stability of high-frequency periodic motions of a heavy rigid body with a horizontally vibrating suspension point

The motion of a heavy rigid body one of whose points (the suspension point) executes horizontal harmonic high-frequency vibrations with small amplitude is considered. The problem of existence of high-frequency periodic motions with period equal to the period of the suspension point vibrations is considered. The stability conditions for the revealed motions are obtained in the linear approximation. The following three special cases of mass distribution in the body are considered; a body whose center of mass lies on the principal axis of inertia, a body whose center of mass lies in the principal plane of inertia, and a dynamically symmetric body.     

Dynamics of interaction between a squeezed layer of a viscous incompressible fluid and an elastic three-layer plate

We study the hydrodynamic response of a thin layer of a viscous incompressible fluid squeezed between impermeable walls. We consider the distribution of pressure and force dynamic characteristics of the fluid layer in the case of forced flows along the gap between a vibration generator (which is a rigid plane) exhibiting harmonic vibrations and a stator (which is an elastic freely supported three-layer plate). The inertial forces of the viscous fluid motion and the stator elastic properties are taken into account. The amplitude and phase frequency characteristics of the elastic three-layer plate are found from the solution of the plane problem.     

Forced vibrations of a thin closed spherical and an infinitely long cylindrical piezoelectric shells with allowance for thermal depolarization

The paper deals with a conjugate problem of harmonic electromechanical vibrations and dissipative heating of a thin closed spherical and an infinitely long cylindrical piezoelectric shells with allowance for the temperature dependence of the viscoelastic properties and the phenomenon of thermal depolarization. The presence of an acoustic medium inside and outside the shell is taken into account. An explicit expression is derived for the critical electrical load for the case of constant electromechanical characteristics. Using a spherical shell as an example, we study the effect that the frequency of the harmonic load, the geometric parameters, and the external acoustic medium have on the critical electrical load as well as the temperature dependence of the electromechanical properties, including the Curie point, on the temperature-frequency characteristics. Translated from Prikladnaya Mekhanika, Vol. 35, No. 11, pp. 62–67, November, 1999.     

The determination of elastic parameters of a half-space using a monochromatic vibration field at the surface

The paper presents a method to determine Lamé parameters λ, μ and density ϱ in a layered half-space, using monochromatic vibrations of its surface, excited by a harmonic source which is assumed to be known. The equations governing the vibrations are reduced to the Sturm-Liouville problem in scalar form for Love-type displacements, and in matrix form for the Rayleigh type. The scalar and matrix potentials of the Sturm-Liouville equations can be recovered from the corresponding impedance. Explicit formulas are given to construct the potentials by amplitudes and wavenumbers of normal (progressive) modes and attenuated standing waves. The potentials then are used to determine the elastic parameters and the density. The method can also be used for the acoustic equation.     

Nonlinear modes of cylindrical panels with complex boundaries. <Emphasis Type="Italic">R</Emphasis>-function method

Nonlinear vibrations of cylindrical panels with complex base are analyzed. The Donnell-Mushtari-Vlasov equations with respect to displacements are used to study vibrations of shallow shell with geometrical nonlinearity. R-function method is applied to satisfy the panel boundary conditions. The Rayleigh-Ritz method is used to obtain the linear vibrations eigenmodes, which contain R-function. The nonlinear vibrations of panel are expanded by using these eigenmodes. The harmonic balance method and nonlinear normal modes are used to study the free nonlinear vibrations.     

Harmonic vibrations of a half-space with a surface-breaking crack under conditions of out-of-plane deformation

The paper presents a solution of the problem of determining the stress state in an elastic isotropic half-space with a crack intersecting its boundary under harmonic longitudinal shear vibrations. The vibrations are excited by a regular action of a harmonic shear load on the crack shores. The solution method is based on the use of the discontinuous solution of the Helmholtz equation, which allows one to reduce the original problem to a singular integro-differential equation for the unknown jump of the displacement on the crack surface. The solution of this equation is complicated by the existence of a fixed singularity of its kernel. Therefore, one of the main results is the development of an efficient approximate method for solving such equations, which takes into account the true asymptotics of the unknown function. The latter allows one to obtain a high-precision approximate formula for calculating the stress intensity factor.     

Vibrations of rod-rigid element systems with a local non-linearity

The paper deals with vibrations of systems consisting of non-coaxial rods connected by rigid bodies and of a local non-linearity. The motion of the rods is described by classical wave equation and the solution of the d’Alembert type is applied in the study. This leads to solving ordinary differential equations with a retarded argument. The local non-linearity is described through irrational functions and in a special case it includes the polynomial of the third degree. Detailed considerations are given for a system consisting of three rods and two rigid bodies. In numerical analysis non-linear effects are discussed. The results concerning harmonic vibrations are presented for the local non-linearities having characteristics of a soft type as well as of a hard type.     

Two mode sub-harmonic and harmonic vibrations of a non-linear beam forced by a two mode harmonic load

The non-linear transverse vibrations of a uniform beam with ends restrained to remain a fixed distance apart and forced by a two mode function which is harmonic in time, are studied by a corresponding two mode approach. The existence of sub-harmonic response of order 1/3 and harmonic response in the sub-harmonic resonance region of the forcing frequency is proved. Approximate solutions are found by Urabe's numerical method applied to Galerkin's procedure and by an analytical harmonic balance-perturbation method. Error bounds of the Galerkin approximations are given.