Numerical study of free oscillations of a beam with oscillators

Free oscillations of a free-ended beam of variable cross section and mass with spring-suspended point masses (oscillators) are considered. It is found that parametric resonances are possible in this oscillating system. The effectiveness of the proposed calculation procedure is confirmed numerical calculations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 150–156, May–June, 2006.     

Forced flexural-and-torsional vibrations of a cantilever beam of constant cross section

The paper addresses the forced flexural-and-torsional vibrations of a cantilever beam of constant cross section. The relevant boundary-value problem is solved. The system of two partial differential equations of the fourth order that describes these vibrations is analyzed in a vector-function space and is subjected to an equivalent transformation to obtain one vector equation of the fourth order with two matrices as coefficients. One is an idempotent matrix; the other is a diagonal matrix. This makes it much easier to construct a Cauchy vector function as an analytic function of these matrices __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 8, pp. 102–114, August 2007.     

Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass

In this study, a slightly curved Euler Bernoulli beam carrying a concentrated mass was handled. The beam was resting on an elastic foundation and simply supported at both ends. Effects of the concentrated mass on nonlin- ear vibrations were investigated. Sinusoidal and parabolic type functions were used as curvature functions. Equations of motion have cubic nonlinearities because of elongations during vibrations. Damping and harmonic excitation terms were added to the equations of motion. Method of mul- tiple scales, a perturbation technique, was used for solving integro-differential equation analytically. Natural frequen- cies were calculated exactly for different mass ratios, mass locations, curvature functions, and linear elastic foundation coefficients. Amplitude-phase modulation equations were found by considering primary resonance case. Effects of nonlinear terms on natural frequencies were calculated. Frequency-amplitude and frequency-response graphs were plotted. Finally effects of concentrated mass and chosen curvature function on nonlinear vibrations were investigated.     

Large free vibrations of a beam carrying a moving mass

The focus of this work is to develop a technique to obtain numerical solution over a long range of time for non-linear multi-body dynamic systems undergoing large amplitude motion. The system considered is an idealization of an important class of problems characterized by non-linear interaction between continuously distributed mass and stiffness and lumped mass and stiffness. This characteristic results in some distinctive features in the system response and also poses significant challenges in obtaining a solution.

In this paper, equations of motion are developed for large amplitude motion of a beam carrying a moving spring–mass. The equations of motion are solved using a new approach that uses average acceleration method to reduce non-linear ordinary differential equations to non-linear algebraic equations. The resulting non-linear algebraic equations are solved using an iterative method developed in this paper. Dynamics of the system is investigated using a time-frequency analysis technique.     

变截面多稳态梁结构减振吸能分析与优化设计

基于多稳态梁结构具有吸能且可重复使用的特点,本文研究包含变截面多稳态梁的单胞结构及其周期性排布的减振吸能效应及其优化设计方法。对多稳态结构进行考虑几何非线性的位移加载/卸载有限元仿真,根据其载荷-位移曲线分析多稳态结构的减振吸能原理,并研究串联与并联周期性排布形式对结构整体吸能特性的影响规律。研究基于多参数调控的变截面梁结构形状表征方法,根据多稳态结构储能特点建立变截面多稳态单胞结构的结构优化模型,通过求解优化问题获得总质量不变条件下最优变截面梁结构形状。进一步地通过对优化结果的有限元分析验证优化的有效性,并对结构进行瞬态冲击荷载下动响应分析,证明多稳态结构的冲击保护作用。     

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.     

Studying the free vibrations of multilayer plates with a complex planform

An efficient method is developed to solve the free-vibration problems for arbitrarily shaped orthotropic multilayer plates in a refined formulation. The method is based on the R-function and Ritz methods. Sequences of coordinate functions satisfying kinematic boundary conditions are constructed in an analytic form. The method is used to solve the vibration problem for multilayer square and arbitrarily shaped plates. The results obtained for square plates are analyzed. A comparison of these results with those available in the literature demonstrates the efficiency of the method __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 119–125, January 2006.     

THE GENERAL SOLUTION FOR DYNAMIC RESPONSE OF NONHOMOGENEOUS BEAM WITH VARIABLECROSSSECTION

ＴＨＥＧＥＮＥＲＡＬＳＯＬＵＴＩＯＮＦＯＲＤＹＮＡＭＩＣＲＥＳＰＯＮＳＥＯＦＮＯＮＨＯＭＯＧＥＮＥＯＵＳＢＥＡＭＷＩＴＨＶＡＲＩＡＢＬＥＣＲＯＳＳＳＥＣＴＩＯＮＪｉＺｈｅｎ－ｙｉ（纪振义）（ＡｎｈｕｉＡｒｃｈｉｔｅｃｔｕｒａｌＩｎｄｕｓｔｒｙＣｏｌｌ...     

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.     

On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle

Existence and uniqueness results are established for weak formulations of initial-boundary value problems which model the dynamic behavior of an Euler-Bernoulli beam that may come into frictional contact with a stationary obstacle. The beam is assumed to be situated horizontally and may move both horizontally and vertically, as a result of applied loads. One end of the beam is clamped, while the other end is free. However, the horizontal motion of the free end is restricted by the presence of a stationary obstacle and when this end contacts the obstacle, the vertical motion of the end is assumed to be affected by friction. The contact and friction at this end is modelled in two different ways. The first involves the classic Signorini unilateral or nonpenetration conditions and Coulomb's law of dry friction; the second uses a normal compliance contact condition and a corresponding generalization of Coulomb's law. In both cases existence and uniqueness are established when the beam is subject to Kelvin-Voigt damping. In the absence of damping, existence of a solution is established for a problem in which the normal contact stress is regularized.The work of the last two authors was supported in part by Oakland University Research Fellowships.     

Nonlinear motion of a beam

The investigation reported herein analyzes the vibration of a uniform beam with hinged ends which are restrained. The beam is subjected to a linearly-varying distributed load which has a maximum intensity w ₀ at the center and is released from rest when the load is suddenly removed. The motion is found to be inherently nonlinear, even for small vibrations, and there is dynamic mode-coupling. The mode frequencies are functionally related to initial conditions, particularly the amplitudes of all modes.     

各向异性非均质梁截面特性计算的映射法

Ｇｉａｖｏｔｏ建立了确定各向异性梁截面特性和翘曲函数的二维有限元法。在此基础上，本文建立了一种映射法。在利用Ｇｉａｖｏｔｏ方法计算得某一具体截面的特性和翘曲函数后，与该截面具有相同形状不同尺寸的任意截面的特性和翘曲函数可用该截面的相应量通过显式确定，使计算大为简化。     

Nonlinear vibrations of a viscoelastic plate with concentrated masses

The problem of vibrations of a viscoelastic plate with concentrated masses is studied in a geometrically nonlinear formulation. In the equation of motion of the plate, the action of the concentrated masses is taken into account using Dirac δ-functions. The problem is reduced to solving a system of Volterra type ordinary nonlinear integrodifferential equations using the Bubnov-Galerkin method. The resulting system with a singular Koltunov-Rzhanitsyn kernel is solved using a numerical method based on quadrature formulas. The effect of the viscoelastic properties of the plate material and the location and amount of concentrated masses on the vibration amplitude and frequency characteristics is studied. A comparison is made of numerical calculation results obtained using various theories. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 158–169, November–December, 2007.     

Free vibrations of rectangular orthotropic shallow shells with varying thickness

The paper proposes a numerical-analytic approach to studying the free vibrations of orthotropic shallow shells with double curvature and rectangular planform. The approach is based on the spline-approximation of unknown functions. Calculations are carried out for different types of boundary conditions. The influence of the mid-surface curvature and variable thickness on the behavior of dynamic characteristics is studied __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 102–115, June 2007.     

作大范围空间运动柔性梁的刚-柔耦合动力学

研究带中心刚体的作大范围空间运动梁的刚-柔耦合动力学问题．从精确的应变-位移关系式出发,在动力学变分方程中,考虑了横截面转动的惯性力偶和与扭转变形有关的弹性力的虚功率,用速度变分原理建立了考虑几何非线性的空间梁的刚-柔耦合动力学方程,用有限元法进行离散．通过对空间梁系统的数值仿真研究扭转变形和截面转动惯量对系统动力学性态的影响．     

Transverse Vibrations of Laminated Beams in Three-Dimensional Formulation

A new asymptotic method is proposed to describe the free and forced transverse vibrations of elastic laminated beams of arbitrary cross section using the three-dimensional elastic equations without additional hypotheses and constraints. For beams with layers of equal Poisson's ratio, the zero-order natural frequencies are equal to the natural frequencies predicted by classical beam theory based on the Bernoulli hypothesis. The method makes it possible to calculate the frequencies of free vibrations and amplitudes of forced vibrations with prescribed accuracy for the first natural modes __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 56–71, June 2005.     

On the free vibrations of a piezoceramic hollow sphere

The aim of the paper is to analyze the free vibrations of a piezoceramic hollow sphere with radial polarization. Using the cnoidal method and a genetic algorithm solves the equations of a radially inhomogeneous spherically isotropic piezoelastic medium. The Reddy and the cosine laws represent the functionally graded property of material. It is seen that for a piezoceramic hollow sphere, the piezoelectric effect consists in increasing the values for the natural frequencies in the specified classes of vibrations.     

Exact solution for finite deformation problems of cantilever beam with variable section under the action of arbitrary transverse loads

This paper deals with finite deformation problems of cantilever beam with variable section under the action of arbitrary transverse loads. By the use of a method of variable replacement, the nonlinear differential equation with varied coefficient for the problem can be transformed into an equation with variable separable. The exact solution can be obtained by the integration method. Some examples are given in the paper, and the results of these examples show that this exact solution includes the existing solutions in references as special cases. Projects Supported by the Science Foundation of the Chinese Academy of Sciences.     

Consideration of the masses of helical springs in forced vibrations of damped combined systems

The common point about many systems modeled as Bernoulli-Euler beams with attachments is that the own masses of the helical springs are neglected. Some researchers accounted for the masses of the springs during free vibrations of those systems. Further to these studies, the present study deals with the investigation of the effect of not taking into account the masses of the helical springs in damped combined systems during their forced vibrations. It is shown that the spring mass effect may be important in regions near to the resonance frequencies. Further, this effect influences the phase angles more than the amplitudes.     

Numerical investigation of nonlinear vibrations of viscoelastic plates and cylindrical panels in a gas flow

Nonlinear vibrations of viscoelastic elements of aviation structures are studied. A method and an algorithm for the numerical solution of integrodifferential equations are proposed. The critical velocity of the flow past viscoelastic plates is determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 156–162, March–April, 2007.