Vibrations of a complex-shaped panel

The free vibrations of flexible shallow shells with complex planform are studied. To analyze the natural frequencies and modes of linear vibrations, the R-function and Rayleigh–Ritz methods are used. A discrete model is obtained using the Bubnov–Galerkin method. The nonlinear vibrations are studied by combining the nonlinear normal mode method and the multiple-scales method. Skeleton curves of natural vibrations are drawn     

Using nonlinear normal modes to analyze forced vibrations

The paper proposes a method to analyze forced vibrations in nonlinear systems. The procedure combines Rauscher’s method and Pierre–Shaw nonlinear modes. Results from an analysis of the forced vibrations of a shallow arch are presented as an example Translated from Prikladnaya Mekhanika, Vol. 44, No. 12, pp. 102–110, December 2008.     

Parametric vibrations of cylindrical shells subject to geometrically nonlinear deformation: multimode models

The nonlinear parametric vibrations of cylindrical shell are described by the Donnell–Mushtari–Vlasov equations. The motions are represented as a mode expansion. Discretization is performed using the Bubnov–Galerkin method. The describing-function method is used to study traveling waves and nonlinear normal modes in systems with and without dissipation     

Nonlinear dynamics of one disk asymmetrical rotor supported by two journal bearings

Asymmetrical one disk rotor interacting with fluid films of short journal bearings is considered. Gyroscopic moments acting on a disk are taken into account. The forces of the journal bearing fluid film are derived analytically. The system of four nonlinear ordinary differential equations is obtained to study the rotor vibrations. The origination of self-sustained vibrations of rotor is studied by means of Shaw–Pierre nonlinear modes. The harmonic balance method is applied to study the self-sustained vibrations with large amplitudes.     

水轮发电机转子偏心引起的非线性电磁振动

由于机械和电磁相互耦合,水轮发电机的电磁振动具有强非线性特征。根据不平衡磁拉力与转子偏心的非线性函数关系,通过简化的各向同性的单圆盘转子系统,建立了水轮发电机转子电磁振动的非线性系统。利用非线性振动理论的多尺度方法,从理论上分析了该系统强迫振动的稳态响应,进而研究了水轮发电机转子偏心引起的电磁振动。研究发现不平衡磁拉力使系统的涡动频率下降,使运动的中心发生变化;并且会引起两倍转频的振动。最后用模型转子仿真试验的结果验证了这些理论分析的结论。     

An investigation on lateral and torsional coupled vibrations of high power density PMSM rotor caused by electromagnetic excitation

Electromagnetic excitation in high power density permanent magnet synchronous motors (PMSMs) due to eccentricity is a significant concern in industry; however, the treatment of lateral and torsional coupled vibrations caused by electromagnetic excitation is rarely addressed, yet it is very important for evaluating the stability of dynamic rotor vibrations. This study focuses on an analytical method for analyzing the stability of coupled lateral/torsional vibrations in rotor systems caused by electromagnetic excitation in a PMSM. An electromechanically coupled lateral/torsional dynamic model of a PMSM Jeffcott rotor is derived using a Lagrange–Maxwell approach. Equilibrium stability was analyzed using a linearized matrix of the equation describing the system. The stability criteria of coupled torsional–lateral motions are provided, and the influences of the electromagnetic and mechanical parameters on mechanical vibration stability and nonlinear behavior were investigated. These results provide better understanding of the nonlinear response of an eccentric PMSM rotor system and are beneficial for controlling and diagnosing eccentricity.     

Theoretical development and closed-form solution of nonlinear vibrations of a directly excited nanotube-reinforced composite cantilevered beam

The nonlinear equations of motion of planar bending vibration of an inextensible viscoelastic carbon nanotube (CNT)-reinforced cantilevered beam are derived. The viscoelastic model in this analysis is taken to be the Kelvin–Voigt model. The Hamilton principle is employed to derive the nonlinear equations of motion of the cantilever beam vibrations. The nonlinear part of the equations of motion consists of cubic nonlinearity in inertia, damping, and stiffness terms. In order to study the response of the system, the method of multiple scales is applied to the nonlinear equations of motion. The solution of the equations of motion is derived for the case of primary resonance, considering that the beam is vibrating due to a direct excitation. Using the properties of a CNT-reinforced composite beam prototype, the results for the vibrations of the system are theoretically and experimentally obtained and compared.     

Nonlinear problems of the dynamics of elastic shells partialiy filled with a liquid

Theoretical and experimental investigations of the nonlinear vibrations and dynamic stability of thin shells partially filled with a liquid are reviewed. The paper deals with the basic laws governing the dynamic high-deflection deformation of carrying shell structures and the considerable vibrations of the free liquid surface due to the natural, forced, and parametrically excited vibrations of the combined system and also due to impulse loads acting on the carrying object. The nonlinear dynamic interaction of shells with a liquid filler is analyzed with allowance for the wave motions of the free liquid surface. S.P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 3–34, April, 2000.     

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     

Resonant vibrations of a hinged viscoelastic rectangular plate

The paper examines the effect of dissipative heating on the performance of a sensor in a hinged thermoviscoelastic rectangular plate undergoing resonant flexural vibrations. The thermoviscoelastic behavior of materials is described using the concept of complex characteristics. The coupling of the electromechanical and thermal fields is taken into account. The nonlinear problem is solved by the variational and Bubnov–Galerkin methods. The effect of the dissipative-heating temperature and the dimensions of the sensor on its performance during resonant vibrations of the plate is analyzed.     

Basic equations of the theory of thermoviscoelastic plates with distributed sensors and actuators

The basic equations of the theory of thermoviscoelastic thin-walled plates with piezoelectric sensors and actuators under monoharmonic mechanical and electric loading are derived using the Kirchhoff–Love hypotheses. The thermomechanical behavior of passive and piezoactive materials is described using the concept of complex characteristics. Methods of solving nonlinear problems of active damping of thermomechanical vibrations of plates with sensors and actuators are considered. The effect of dissipative heating on the damping of axisymmetric vibrations of a thermoviscoelastic solid circular plate is analyzed as an example     

Stability of a plane conducting fluid layer under convection in a vertical magnetic field

The nonlinear convective instability of a plane horizontal conducting fluid layer placed in a uniform vertical magnetic field is studied [1]. A similar problem was previously considered in [2] but with allowance only for so-called weakly nonlinear third-order effects. In the present paper attention is concentrated on the study of the finite-amplitude instability mechanisms associated with the "hard" excitation of vibrations. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 23–28, January–February, 1998.     

Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system

Thermo-mechanical vibrations of a simply supported spring-mass-beam system are investigated analytically in this paper. Taking into account the thermal effects, the nonlinear equations of motion and internal/external boundary conditions are derived through Hamilton’s principle and constitutive relations. Under quasi-static assumptions, the equations governing the longitudinal motion are transformed into functions of transverse displacements, which results in three integro-partial differential equations with coupling terms. These are solved using the direct multiple-scale method, leading to closed-form solutions for the mode functions, nonlinear natural frequencies and frequency–response curves of the system. The influence of system parameters on the linear and nonlinear natural frequencies, mode functions, and frequency–response curves is studied through numerical parametric analysis. It is shown that the vibration characteristics depend on the mid-plane stretching, intra-span spring, point mass, and temperature change.     

Vibrational Convection in the Hele-Shaw Cell. Theory and Experiment

The influence of high-frequency horizontal vibrations on convection in the Hele-Shaw cell located in a uniform gravity field is considered experimentally and theoretically. Nonlinear regimes of vibrational convection in the supercritical region are examined. It is shown that horizontal vibrations (directed toward the wide sides of the cell) decrease the threshold of quasi-equilibrium stability. Regions of existence of one- and two-vortex steady flows are found, and unsteady regular and random regimes of thermal vibrational convection are considered. New random regimes in the Hele-Shaw cell are found, which result from nonlinear interaction of the “lower” modes responsible for the formation of regular supercritical convective regimes. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 40–48, March–April, 2006.     

Measurement of nonlinear damping in a Gr/Al metal-matrix composite

This paper is concerned with the measurement of nonlinear (i.e., strain amplitude dependent) intrinsic material damping in continuous-fiber-reinforced metal-matrix composites (MMC). The particular MMC studied is a four-ply [±θ] _s P55Gr/6061 Al composite with θ=0, 15, 30, 45, 60, 75 deg. A popular method for measuring damping is the free-decay of flexural vibrations of a cantilevered beam. However, the strain field in a cantilevered beam is inhomogeneous. Therefore, for materials whose damping is nonlinear, the measured specimen damping is not equal to the intrinsic material damping. Using an elementary algorithm develeped by Lazan, the authors extract nonlinear intrinsic material damping from the nonlinear specimen damping.     

Nonlinear vibration analysis of viscoelastic plates based on a refined Timoshenko theory

Nonlinear vibrations of viscoelastic orthotropic and isotropic shells are mathematically modeled using a geometrically nonlinear Timoshenko theory. Nonlinear problems are solved by using the Bubnov-Galerkin method and a numerical method based on quadrature formulas. Results obtained from different theories are compared and analyzed. For each problem, the Bubnov-Galerkin method is tested for convergence. The influence of the viscoelasticity and inhomogeneity of materials on the vibrations of plates is demonstrated __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 120–131, May 2006.     

Studying the harmonic vibrations of a cylindrical shell made of a nonlinear elastic piezoelectric material

The paper examines the harmonic vibrations of an infinitely long thin cylindrical shell made of a nonlinear elastic piezoceramic material and subjected to periodic electric loading. Amplitude-frequency characteristics are plotted for different amplitudes of the load. Points of these characteristics are analyzed for stability. The transients occurring while harmonic vibrations attain the steady state are studied __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 101–106, April 2008.     

Stability analysis for nonplanar free vibrations of a cantilever beam by using nonlinear normal modes

Stability analysis of nonplanar free vibrations of a cantilever beam is made by using the nonlinear normal mode concept. Assuming nonplanar motion of the beam, we introduce a nonlinear two-degree-of-freedom model by using Galerkin’s method based on the first mode in each direction. The system turns out to have two normal modes. Using Synge’s stability concept, we examine the stability of each mode. In order to check the validity of the stability criterion obtained analytically, we plot a Poincaré map of the motions neighboring on each mode obtained numerically. It is found that the maps agree with the stability criterion obtained analytically.     

Damper modules with adapted stiffness ratio

A mechanism for the excitation of piston rod vibrations in automotive damper modules is discussed by a simple model. An improved nonlinear model based on elasticity effects leads to good simulation results. It is shown theoretically and experimentally that the adaptation of the stiffness of the piston rod bushing to the “stiffness” of the damper force characteristic can eliminate the piston rod oscillations completely.     

Energy analysis in a nonlinear hybrid system containing linear and nonlinear subsystems coupled by hereditary element

Energy transfer between subsystems coupled by standard light hereditary element in hybrid system is very important for different engineering applications, especially for dynamical absorption. An analytical study of the energy transfer between coupled linear and nonlinear oscillators in the free vibrations of a viscoelastically connected double-oscillator system as a new hybrid nonlinear system with two and half degrees of freedom is pointed out. The analytical study shows that the viscoelastic–hereditary connection between oscillators causes the appearance of like two-frequency regimes of subsystem's vibrations and that the energy transfer between subsystems appears. The Lyapunov exponents corresponding to each of two eigenmodes of the hybrid system, as well as to the subsystems are obtained and expressed by using energy of the corresponding eigentime components. The Lyapunov exponents are measures of the vibration processes stability in the hybrid system and in component subsystem vibrations. In Honor of Giuseppe Rega and Fabrizio Vestroni on the Occasion of their 60th Birthday.