Forced Vibration Analysis of Euler-Bernoulli Double-Beam Systems by Means of Green’s Functions北大核心CSCD

Double-curved-beam (DCB) systems are usually seen in many engineering fields. Compared to straight double-beam systems, DCB systems are more efficient in noise and vibration control problems. To obtain closed-form solutions of steady-state forced vibrations of DCB systems, the classical Euler-Bernoulli curved beam (ECB) model was employed to model vibration equations for the DCB systems. Green’s functions and the Laplace transform methods were used to get the closed-form solutions to the vibration equations for the DCB systems. These solutions apply to arbitrary boundary conditions. Numerical tests were conducted to verify the present solutions with related results from previous literatures. Effects of some important geometric and physical parameters on vibration responses and the interaction between the elastic layer stiffness and the DCB system, were discussed. The results show that, the DCB system will degenerate to a straight double-beam system when the 2 radii approach infinity, moreover, the DCB system can be simplified as one comprising a straight beam and a curved beam. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

A new Orthonormal Polynomial Series Expansion Method in vibration analysis of thin beams with non-uniform thickness

In this article, OPSEM (Orthonormal Polynomial Series Expansion Method) is developed as a new computational approach for the evaluation of thin beams of variable thickness transverse vibration. Capability of the OPSEM in assessing the free vibration frequencies and mode shapes of an Euler–Bernoulli beam with varying thickness is discussed. Multispan continuous beams with various classical boundary conditions are included. Contribution of BOPs (Basic Orthonormal Polynomials) in capturing the beam vibrations is also illustrated in numerical examples to give a quantitative measure of convergence rate. Furthermore, OPSEM is adopted for the forced vibration of a thin beam caused by a moving mass. Dynamics of beams supported by flexible elastic base like free to free beam on elastic foundation are also regarded. Verifications are made via eigenfunction expansion method and GMLSM (Generalized Moving Least Square Method). The very close observed agreement between the results of the two recently mentioned methods and that of OPSEM can be regarded as a guarantee of validity for the newly introduced technique. In comparison with eigenfunction expansion method, the simplicity and handiness of OPSEM in coping with different boundary conditions of the beam can be considered as its benefit for engineering practitioners.     

Optimal design of an elastic beam with a unilateral elastic foundation: Semicoercive state problem

A design optimization problem for an elastic beam with a unilateral elastic foundation is analyzed. Euler-Bernoulli’s model for the beam and Winkler’s model for the foundation are considered. The state problem is represented by a nonlinear semicoercive problem of 4th order with mixed boundary conditions. The thickness of the beam and the stiffness of the foundation are optimized with respect to a cost functional. We establish solvability conditions for the state problem and study the existence of a solution to the optimization problem.     

A dynamic analysis of a coupled beam/slider system

The paper deals with transverse vibration of a beam with moving boundary conditions. In order to examine the transfer of energy between a moving support and a vibrating beam under assumption of zero slope of the beam elastic line at the moving support, dynamic analysis of a coupled beam/slider system is carried out. The moving support is modelled as a slider attached to a spring which realizes definite boundary conditions. Equations of motion are derived using Hamilton’s principle. Because length of the beam varies appropriate transformations of time and position variables are made to convert the solution space into a rectangle and, subsequently, to solve the partial differential equation of transverse vibration of the beam using the FDM. The phenomenon of the energy flow between the slider and the beam is a subject of the detailed analysis. The beam vibration with a fixed formula of length is examined, too. The dynamic characteristics of the system is brought forward from spectral analysis of numerical solutions.     

Green's functions for forced vibration analysis of bending-torsion coupled Timoshenko beam

A method based on Green's functions is proposed for the analysis of the steady-state dynamic response of bending-torsion coupled Timoshenko beam subjected to distributed and/or concentrated loadings. Damping effects on the bending and torsional directions are taken into account in the vibration equations. The elastic boundary conditions with bending-torsion coupling and damping effects are derived and the classical boundary conditions can be obtained by setting the values of specific stiffness parameters of the artificial springs. The Laplace transform technology is employed to work out the Green's functions for the beam with arbitrary boundary conditions. The Green's functions are obtained for the beam subject to external lateral force and external torque, respectively. Coupling effects between bending and torsional vibrations of the beam can be studied conveniently through these analytical Green's functions. The direct expressions of the steady-state responses with various loadings are obtained by using the superposition principle. The present Green's functions for the Timoshenko beam can be reduced to those for Euler–Bernoulli beam by setting the values of shear rigidity and rotational inertia. In order to demonstrate the validity of the Green's functions proposed, results obtained for special cases are given for a comparison with those given in the literature and they agree with each other exactly. The influences of external loading frequency and eccentricity on Green's functions of bending-torsion coupled Timoshenko beam are investigated in terms of the numerical results for both simply supported and cantilever beams. Moreover, the symmetric property of the Green's functions and the damping effects on the amplitude of Green's functions of the beam are discussed particularly.     

Nonlinear bending and buckling for strain gradient elastic beams

Nonlinear bending of strain gradient elastic thin beams is studied adopting Bernoulli–Euler principle. Simple nonlinear strain gradient elastic theory with surface energy is employed. In fact linear constitutive relations for strain gradient elastic theory with nonlinear strains are adopted. The governing beam equations with its boundary conditions are derived through a variational method. New terms are considered, already introduced for linear cases, indicating the importance of the cross-section area, in addition to moment of inertia in bending of thin beams. Those terms strongly increase the stiffness of the thin beam. The non-linear theory is applied to buckling problems of thin beams, especially in the study of the postbuckling behaviour.     

Utilization of characteristic polynomials in vibration analysis of non-uniform beams under a moving mass excitation

Vibration of non-uniform beams with different boundary conditions subjected to a moving mass is investigated. The beam is modeled using Euler–Bernoulli beam theory. Applying the method of eigenfunction expansion, equation of motion has been transformed into a number of coupled linear time-varying ordinary differential equations. In non-uniform beams, the exact vibration functions do not exist and in order to solve these equations using eigenfunction expansion method, an adequate set of functions must be selected as the assumed vibration modes. A set of polynomial functions called as beam characteristic polynomials, which is constructed by considering beam boundary conditions, have been used along with the vibration functions of the equivalent uniform beam with similar boundary conditions, as the assumed vibration functions. Orthogonal polynomials which are generated by utilizing a Gram–Schmidt process are also used, and results of their application show no advantage over the set of simple non-orthogonal polynomials. In the numerical examples, both natural frequencies and forced vibration of three different non-uniform beams with different shapes and boundary conditions are scrutinized.     

The variability of dynamic responses of beams resting on elastic foundation subjected to vehicle with random system parameters

This paper investigates the variability of dynamic responses of a beam resting on an elastic foundation, which is subjected to a vehicle with uncertain parameters, such as random mass, stiffness, damping of the vehicle and random fields of mass density, and the elastic modulus of the beam and stiffness of elastic foundation. The vehicle is modeled as a two-degree-of-freedom spring-damper-mass system. The equations of motion of the beam was constructed using a finite element method. The mass and elastic properties of the beam, and the stiffness of foundation are assumed to be Gaussian random fields and were simulated by the spectral represent method. Masses, stiffness of the spring, and the damping coefficient of the vehicle are assumed as Gaussian random variables. The numerical analyses were performed using the finite element method (FEM) in conjunction with the Monte Carlo simulation (MCS). The variability of dynamic responses of the beam were investigated with various cases of random parameters. For each sample, the equations of motions were solved with the Wilson-q integral method to find dynamic responses. The influence of random system parameters and their correlation on the response variability is discussed in detail.     

面内功能梯度三角形板等几何面内振动分析

基于平面应变理论,利用等几何有限元方法分析了弹性边界条件下面内功能梯度三角形板的面内振动特性.板的材料属性沿厚度方向呈均匀分布,而在面内方向呈任意指数梯度变化.采用非均匀有理B样条(NURBS)基函数对三角形结构进行等几何建模和位移描述,实现了三角形板几何设计和振动分析的无缝衔接.在三角形板边界上引入虚拟弹簧约束并通过调节虚拟弹簧刚度,实现任意边界条件的施加.通过不同的单元细化方案和对比算例,验证了等几何方法的灵活性、准确性和快速收敛性.系统研究了边界条件、材料属性和几何参数对三角形板振动特性的影响.同时给出了弹性边界条件下面内功能梯度三角形板的振动特性解,具有重要参考价值.     

边界条件对梁类结构系统刚度的影响

车身骨架是由大量梁类结构单元构成的.除了这些梁的截面形状和尺寸外,边界条件也对系统刚度影响很大.讨论了各种边界条件和载荷模式下的梁系统的合成刚度.基于两端固支的均匀截面梁的弯曲和扭转刚度,研究了各联结刚度的大小对系统刚度的贡献,并绘制了相应的影响曲线.最后,通过上述解析公式和有限元法计算了某汽车仪表板横梁系统的实际弯曲和扭转刚度.文中获得的静态刚度公式对其它梁类结构也适用。     

Modelling of the forced motions of an elastic beam using the method of integrodifferential relations

A variational approach to the numerical modelling of forced lateral motions of an Euler–Bernoulli elastic beam is developed for a number of linear boundary conditions using the method of integrodifferential relations. A class of linear boundary actions is considered. A family of quadratic functionals, connecting the displacement field of points of the beam with the bending-moment functions in the cross section and the momentum density is proposed. Variational formulations of the original initial-boundary value problem on the motion of the beam are given and the necessary conditions for the functionals introduced to be stationary are analysed. The integral and local quality characteristics of the admissible approximate solutions are determined. The relation between the variational problems, formulated for the beam model, with the classical Hamilton–Ostrogradskii variational principles is demonstrated. An algorithm for constructing approximate systems of ordinary differential equations is developed, the solution of which yields stationary (minimum) values of the functionals introduced on a specified set of displacement fields, moments and momenta. Examples of calculations of the displacements for an elastic beam and an analysis of the quality of the numerical solutions obtained are presented.     

The dependence of the natural frequencies of a one-dimensional elastic system on its length

The dependence of the natural frequencies and modes of the oscillations of distributed elastic system with characteristics of the stiffness and density that are variable along a coordinate of the cross section for arbitrary boundary conditions is investigated. It is proved that the presence of an external elastic medium, described by the Winkler model, may lead to an increase in the natural frequencies of the lower oscillation modes when the length of a one-dimensional elastic system is increased. The fine properties of the change in the natural frequencies as a function of the length of the system and the number of the oscillation mode are also established. A numerical-analytical investigation of examples which illustrate the characteristic anomalous behaviour of the lowest natural frequencies is presented.     

Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models

The free vibration response of single-walled carbon nanotubes (SWCNTs) is investigated in this work using various nonlocal beam theories. To this end, the nonlocal elasticity equations of Eringen are incorporated into the various classical beam theories namely as Euler-Bernoulli beam theory (EBT), Timoshenko beam theory (TBT), and Reddy beam theory (RBT) to consider the size-effects on the vibration analysis of SWCNTs. The generalized differential quadrature (GDQ) method is employed to discretize the governing differential equations of each nonlocal beam theory corresponding to four commonly used boundary conditions. Then molecular dynamics (MD) simulation is implemented to obtain fundamental frequencies of nanotubes with different chiralities and values of aspect ratio to compare them with the results obtained by the nonlocal beam models. Through the fitting of the two series of numerical results, appropriate values of nonlocal parameter are derived relevant to each type of chirality, nonlocal beam model, and boundary conditions. It is found that in contrast to the chirality, the type of nonlocal beam model and boundary conditions make difference between the calibrated values of nonlocal parameter corresponding to each one.     

Free vibration analysis of a rigid bar supported by arbitrary elastic beams

For convenience, a two-node conventional elastic beam element (C beam element) with the displacements of its 2nd node replaced by those of center of gravity (c.g.) of the joined rigid bar is called the modified beam element (M beam element). The objective of this paper is to present a modified finite element method (modified FEM) such that the free vibration characteristics of a rigid bar supported by a number of elastic beams can be easily determined. First of all, the displacements for the 2nd node of a C beam element joined with the rigid bar are determined in terms of those for the c.g. of the joined rigid bar to establish the M beam element. Next, the mass and stiffness matrices for the M beam element are derived based on the displacements for the 1st node of the C beam element and those for the c.g. of the joined rigid bar. Then, the overall property matrices of the entire unconstrained vibrating system (i.e. a rigid bar supported by a number of elastic beams) can be determined by using the assembly technique of the conventional FEM and considering the effects of lumped mass and rotary inertia of the rigid bar. Finally, the boundary (supporting) conditions are imposed to produce the effective property matrices of the constrained vibrating system and then the free vibration characteristics are determined with the standard approach. In order to confirm the presented theory and the developed computer program, the rigid bar is modeled by a number of C beam elements with bigger Young’s modulus (E_R) and the conventional FEM is used to determine the natural frequencies and associated mode shapes of the vibrating system. It is found that the latter will converge to the corresponding ones obtained from the presented modified FEM when the magnitude of E_R increases to certain values.     

Mathematical analogy of a beam on elastic supports as a beam on elastic foundation

This paper presents the mathematical hypothesis that a beam on equidistant elastic supports (BOES) can be considered as a beam on an elastic foundation (BOEF) in static and free vibration problems. This modeling of BOES as BOEF is presumed to be applicable to a limited range of support stiffness, spacing and flexural rigidity of the beam. The authors investigate the applicability of the modeling of BOEF from the property of characteristic solutions obtained from governing equations of both BOES and BOEF. In this study, the formulation of BOES leads to governing difference equations, and the motions of BOEF are expressed by differential equations. This is because exact solutions must be employed in order to verify their analogy accurately. The characteristic solutions obtained from these two governing equations are compared to each other in order to investigate the relationship between them.     

A mixed differential quadrature and finite element free vibration and buckling analysis of thick beams on two-parameter elastic foundations

As a first endeavor, a mixed differential quadrature (DQ) and finite element (FE) method for boundary value structural problems in the context of free vibration and buckling analysis of thick beams supported on two-parameter elastic foundations is presented. The formulations are based on the two-dimensional theory of elasticity. The problem domain along axial direction is discretized using finite elements. The resulting system of equations and the related boundary conditions are discretized in the thickness direction and in strong-form using DQM. The method benefits from low computational efforts of the DQ in conjunction with the effectiveness of the FE method in general geometry and systematic boundary treatment resulting in highly accurate and fast convergence behavior solution. The boundary conditions at the top and bottom surface of the beams are implemented accurately. The presented formulations provide an effective analysis tool for beams free of shear locking. Comparisons are made with results from elasticity solutions as well as higher-order beam theory.     

Free vibration,buckling and dynamic stability of bi-directional FG microbeam with a variable length scale parameter embedded in elastic medium

In this paper, size dependent free vibration, buckling and dynamic stability of bi-directional functionally graded (BDFG) microbeam embedded in elastic medium are investigated. The material properties vary along both thickness and axial directions. In particular, the material length scale parameter of microbeam is considered as a function of spatial coordinates and varies with the material gradient parameters. The system of differential equations with variable coefficients governing the motion of BDFG microbeam is derived employing Hamilton’s principle, the modified couple stress theory and third-order shear deformation beam theory. The differential quadrature method (DQM) is utilized to solve the static and dynamic problem. Three different models evaluating the material length scale parameter of BDFG microbeam are presented for comparison. Parametric studies are carried out to show the influence of gradient parameters, size effect, stiffness of elastic medium on the free vibration, buckling and dynamic stability characteristic of BDFG microbeam. Results show that the variation of material length scale parameter should be considered in the analysis of BDFG microbeam.     

带控制器的弹性振动系统的能观性、能控性

<正> 在工作[1]中提出了带有控制迴路的分布参数反馈系统的模型.在工作[2]中讨论了以弹性樑的角速度、角度和线加速度作反馈信号输入到控制器,由控制器的输出端输出信号到舵的执行机构以实现反馈控制弹性振型的镇定问题.那里用的是线性算子的谱扰动方法.在[3]中讨论了不带控制器的弹性振动系统的能观测性和能控性问题,得到了能观测、能控的必要充分条件.在现代控制理论中,一个系统是否能控、能观测,无论在实际工     

Free vibration analysis of cracked composite beams subjected to coupled bending–torsion loads based on a first order shear deformation theory

In this paper, free vibration analysis of cracked composite beam subjected to coupled bending–torsion loading is presented. The composite beam is assumed to have an open edge crack of length a. A first order shear deformation theory is applied to count for the effect of shear deformations on natural frequencies as well as the effect of coupling in torsion and bending modes of vibration. Governing equations and boundary conditions are derived using Hamilton principle. Local flexibility matrix is used to obtain the additional boundary conditions of the beam in cracked area. After obtaining the governing equations and boundary conditions, generalized differential quadrature (GDQ) method is applied to solve the obtained eigenvalue problem. Finally, some numerical results of beams with various boundary conditions and different fiber orientations are given to show the efficiency of the method. In addition, to study the effect of shear deformations, numerical results of the current model are compared with previously given results in which shear deformations were neglected.