Free vibration analysis of composite beams with two non-overlapping delaminations

An analytical solution to the free vibration of composite beams with two non-overlapping delaminations is presented. The delaminated beam is modeled as seven interconnected Euler-Bernoulli beams using the delaminations as their boundaries. The continuity and the equilibrium conditions are satisfied between adjoining beams. The analysis includes the differential stretching between the delaminated layers and the bending-extension coupling. The results of the present model agree well with the analytical and experimental data reported in the literature. Parametric studies show that the sizes and locations of the delaminations have significant effect on the natural frequencies and mode shapes. These results provide useful information in the study of the free vibration of delaminated composite beams.     

Exact solutions for the free in-plane vibrations of rectangular plates

All classical boundary conditions including two distinct types of simple support boundary conditions are formulated by using the Rayleigh quotient variational principle for rectangular plates undergoing in-plane free vibrations. The direct separation of variables is employed to obtain the exact solutions for all possible cases. It is shown that the exact solutions of natural frequencies and mode shapes can be obtained when at least two opposite plate edges have either type of the simply-supported conditions, and some of the exact solutions were not available before. The present results agree well with FEM results, which show that the present solutions are correct and the direct separation of variables is practical. The exact solutions can be taken as the benchmarks for the validation of approximate methods.     

Exact vibration results for stepped circular plates with free edges

A pontoon-type, very large floating structure (VLFS) is often modeled as a huge plate with free edges when performing a hydroelastic analysis under the action of waves. The analysis consists of separating the hydrodynamic analysis from the dynamic response analysis of the VLFS. The deflection of the plate is decomposed into vibration modes where as many higher modes as possible should be used to capture the actual deflection shapes and the stresses. It is generally accepted that finite element method and the Ritz-type energy method fail to model zones with steep gradients which are encountered in, for instance, the stress resultants near the free edges of plates [Journal of Engineering Mechanics 1983;109(2):537–56]. Moreover, the natural boundary conditions are not satisfied completely because they are not enforced a priori [International Journal of Solids and Structures 2001;38:6525–58, Journal of Computational Structural Engineering 2001;1(1):49–57, Journal of Structural Engineering ASCE 2002;128(2):249–57, Computers and Structures 2002:80(2):145–54]. Exact solutions for frequencies, mode shapes and modal stress resultants are thus very important as they provide valuable benchmarks for assessing the convergence, accuracy and validity of numerical results obtained using the finite element method. To this end, we present the exact vibration results for stepped circular plates with free edges. When employed in a hydroelastic analysis, these exact vibration solutions yield accurate deflections and stress resultants (stresses) for circular VLFSs with stepped drafts.     

Free vibration analysis of arbitrary shaped thick plates by differential cubature method

In this paper, a new numerical solution technique, the differential cubature method, is applied to solve the free vibration problems of arbitrary shaped thick plates. The basic idea of the differential cubature method is to express a linear differential operation such as a continuous function or any order of partial derivative of a multivariable function, as a weighted linear sum of discrete function values chosen within the overall domain of a problem. By using the differential cubature procedure, the governing differential equations and boundary conditions are transformed into sets of linear homogeneous algebraic equations. This is an eigenvalue problem, of which the eigenvalues can be calculated numerically. The subspace iterative method is employed in search of the free vibration frequency parameters. Detailed formulations are presented, and the method is examined here for its suitability for solving the vibration problems of moderately thick plates governed by Mindlin shear deformation theory. The applicability, efficiency and simplicity of the method are demonstrated through solving some example plate vibration problems of different shapes. The numerical accuracy of the method is ascertained by comparing the vibration frequency solutions with those of existing literatures.     

扭-纵耦合对曲轴系统固有特性的影响

曲轴系统具有特殊的结构和力学特点,因此从动力学耦合的角度研究曲轴系统的振动将更能准确地把握曲轴系统的动力学特性。利用集总参数法建立了曲轴系统扭转-纵向耦合振动力学模型,基于该模型分析了曲轴系统扭转-纵向耦合作用对曲轴系统固有特性的影响。数值计算结果表明:扭转-纵向耦合振动将改变曲轴系统的固有频率,使其系统临界转速降低,并产生一些新的耦合振型。由于系统模态将更加密集,设计中应合理选择曲轴系统的额定工作区以避免发生共振。     

An exact approach for free vibration analysis of rectangular plates with line-concentrated mass and elastic line-support

An exact approach for free vibration of an isotropic rectangular plate carrying a line-concentrated mass and with a line-translational spring support or carrying a line-spring-mass system is presented in this paper. The mode shape function of vibration of such a plate is expressed in terms of the four fundamental solutions derived in this paper. The main advantage of the proposed method is that the resulted frequency equation for such a rectangular plate can be conveniently obtained from a second-order determinant. The proposed method is thus computationally efficient due to the significant decrease in the determinant order as compared with previously developed procedures which usually led to an eighth-order determinant for solving the title problem. Two numerical examples are given to illustrate the efficiency of the proposed method and to investigate the effects of the location and the magnitude of a line-concentrated mass and elastic line-support as well as the influence of the aspect ratio on the natural frequencies of a rectangular plate.     

轴盘系统扭振固有频率的精确计算

把带有圆盘的轴分为若干段,对圆盘厚度范围段,可视为一特殊的段,在这一段内,不考虑圆盘的变形,截面极惯性矩就等于轴的截面极惯性矩,转动惯量为轴和盘转动惯量之和。每一段的扭振方程可精确求出,然后,根据边界条件和连续条件可将扭振固有频率计算出来。     

数控弧齿锥齿轮磨齿机主轴系统的有限元分析

利用有限元分析软件ANSYS ,对YK2 0 4 5型数控弧齿锥齿轮磨齿机的主轴系统进行静力分析和模态分析 ,得出其变形和应力的大小及分布 ,分析了跨距对静刚度的影响及最低三阶模态的振型和固有频率 ,为进一步的谐响应分析提供了条件。     

确定阶梯轴扭振固有频率的混合状态方法

提出的新方法将阶梯轴分为若干段,每段可视为等截面轴,每段内所满足的扭振混合状态方程可精确地求出,然后利用两段之间的连续条件和边界条件将固有频率求出。     

Exact solutions for vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses

Exact solutions are presented for the free vibration and buckling of rectangular plates having two opposite edges (x=0 and a) simply supported and the other two (y=0 and b) clamped, with the simply supported edges subjected to a linearly varying normal stress σ_x=−N₀[1−α(y/b)]/h, where h is the plate thickness. By assuming the transverse displacement (w) to vary as sin(mπx/a), the governing partial differential equation of motion is reduced to an ordinary differential equation in y with variable coefficients, for which an exact solution is obtained as a power series (the method of Frobenius). Applying the clamped boundary conditions at y=0 and b yields the frequency determinant. Buckling loads arise as the frequencies approach zero. A careful study of the convergence of the power series is made. Buckling loads are determined for loading parameters α=0,0.5,1,1.5,2, for which α=2 is a pure in-plane bending moment. Comparisons are made with published buckling loads for α=0,1,2 obtained by the method of integration of the differential equation (α=0) or the method of energy (α=1,2). Novel results are presented for the free vibration frequencies of rectangular plates with aspect ratios a/b=0.5,1,2 subjected to three types of loadings (α=0,1,2), with load intensities N₀/N_cr=0,0.5,0.8,0.95,1, where N_cr is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes are also shown.     

悬臂压电叠层复合梁的弯曲振动固有频率

压电叠层复合梁是一种三叠片对称结构。这里以压电陶瓷-金属-压电陶瓷复合结构为研究对象,给出了悬臂式对称压电复合结构的压电振子弯曲振动共振频率表达式,并建立了相应的有限元模型;计算分析了压电陶瓷-金属复合梁弯曲振动的固有频率值与几何尺寸之间的关系,并与相应的有限元模态分析结果进行了比较。结果表明,两种分析方法得到的压电振子的固有频率值基本符合,计算公式和模型是精确有效的;压电叠层复合梁愈细长则表达式的计算结果愈准确。     

Rotary ultrasonic machining: effects of tool natural frequency on ultrasonic vibration amplitude

Rotary ultrasonic machining (RUM) is a hybrid machining process that combines the material removal mechanisms of grinding and ultrasonic machining. RUM has been applied to hole-making for a wide range of materials. It is known that ultrasonic vibration amplitude has significant effects on cutting force, torque, and surface finish in RUM. One experimental observation that has been reported in the literature multiple times states that different tools show different vibration amplitudes on the same ultrasonic power level. However, no analyses can be found in the literature to explain this observation. The existence of this knowledge gap makes it difficult to explain some experimentally obtained trends or to conduct more realistic physics-based modeling work. The objectives of this research are to understand the effects of tool natural frequency on ultrasonic vibration amplitude in RUM, to provide an explanation to the observation and verification of measurement methods, and also to guide tool design and selection in RUM. Ultrasonic vibration amplitudes of tools are measured by three methods and compared. It is found that tool natural frequency significantly affects ultrasonic vibration amplitude. The tool with its natural frequency closest to that of the ultrasonic power supply (20?kHz) generates the highest ultrasonic vibration amplitude on every ultrasonic power level tested.     

气动式振动台理想激励信号的特性

气动式振动台是一类重要的可靠性振动强化试验设备,目前国内尚未对其关键技术及自主研发展开全面研究。根据薄板振动理论建立气动式振动台的力学模型,采用能量法(Rayleigh-Ritz法)分析和计算振动台面的固有频率和正则振型,将用于机械系统动力学分析的传递函数引入到振动台系统,推导气锤安装位置与振动台面任意响应位置之间的力―加速度传递函数,在此基础上,构建气动式振动台的动力学模型,揭示振动台面受到的激励力信号与其加速度响应信号之间的关系,针对振动台的“理想”加速度响应信号,进一步计算与之对应的“理想”激励力信号,并分析该激励信号的特性,为该类设备的性能改善乃至自主研发提供理论指导。     

Analysis of fluid induced vibration of cryogenic pipes in consideration of the cooling effect

The purpose of system analysis using fluid induced vibration is to identify the problems of the system in advance by analyzing the vibration behavior of the system excited by fluid flow. Fluid-induced vibration analysis methods, developed so far, generally use the numerical analysis method to analyze the fluid flowing inside the pipe and the infinitesimal elements at normal temperature on the basis of the governing equation obtained by applying Newton’s Second Law and the momentum equation. However, as the fluid temperature changes greatly at low temperature, fluid-induced vibration analysis methods for normal temperature cannot be applied. This study investigated methods of analyzing fluid-induced vibration in consideration of the cooling effect. In consideration of the changes in the properties of the fluid and system relative to temperature, vibration behavior was analyzed numerically by means of the equation of motion. As a result, the natural frequency of the system tends to change because of the changes of the properties of materials even when the flux is constant inside the pipe, and the vibration behavior of the system was compared to that in case of normal temperature to analyze how much influence the cooling effect has on the vibration behavior of the system. This paper was recommended for publication in revised form by Associate Editor Seockhyun Kim     

Axisymmetric free vibration of thick annular plates

This paper presents a numerical analysis of the axisymmetric free vibration of moderately thick annular plates using the differential quadrature method (DQM). The plates are described by Mindlin’s first-order shear-deformation theory. The first five axisymmetric natural frequencies are presented for uniform annular plates, of various radii and thickness ratios, with nine possible combinations of free, clamped and simply supported boundary conditions at the inner and outer edges of the plates. The accuracy of the method is established by comparing the DQM results with some exact and finite element numerical solutions and, therefore, the present DQM results could serve as a benchmark for future reference. The convergence characteristics of the method for thick plate eigenvalue problems are investigated and the versatility and simplicity of the method is established.     

Analytic solutions for vibration characteristics of a multi-beam structure

The first purpose of this paper is to mathematically analyze the vibration characteristics of four parallel and uniform beams that are joined by a tip body at their free ends. This beam structure is often found in many mechanical structures such as robots, space constructions, and optical pickup actuators in optical disc drives (ODDs). To analyze the vibration characteristics of this beam structure, this paper considers the motions in all six directions that are coupled to each other. Mathematical expansion using the energy conservation law applied to this beam structure yields four characteristic equations, which are related to a pure-axial (PA) vibration, a coupled bending-torsional (CBT) vibration, and two coupled axial-bending (CAB) vibrations. In addition, the correctness of the equations is verified by showing that the natural frequencies from these equations match those from a finite element (FE) analysis.The second purpose of this paper is to contribute to the performance improvement of an optical pickup actuator for ODDs by analyzing key parameters of the characteristic equations. This is one of the few papers that analytically derives solutions for the natural frequencies and modes of optical pickup actuators in all six directions. From the perspective of practical usefulness for the design of optical pickup actuators, this paper describes how to reduce rotation in the two CAB beam vibrations by investigating the relationship between design parameters and rotating properties.     

Exact solutions for vibration of cylindrical shells with intermediate ring supports

This paper presents new exact solutions for vibration of thin circular cylindrical shells with intermediate ring supports, based on the Goldenveizer–Novozhilov shell theory (Theory of thin shells; The theory of thin elastic shells). An analytical method is proposed to study the vibration behaviour of the ring supported cylindrical shells. In the proposed method, the state-space technique is employed to derive the homogenous differential equation system for a shell segment and a domain decomposition approach is developed to cater for the continuity requirements between shell segments. Exact frequency parameters are presented in tables and design charts for circular cylindrical shells having multiple intermediate ring supports and various combinations of end support conditions. These exact vibration frequencies may serve as important benchmark values for researchers to validate their numerical methods for such circular cylindrical shell problems.     

楔形杆轴纵扭固有振动的摄动解

采用摄动法研究了楔形杆轴的固有纵扭振动 ,并且推导出了其固有纵扭振动的振型函数及频率特征方程。通过实例计算证明了该摄动解不仅计算简便 ,而且计算精度与 Bessel函数解的计算精度相当 ,对强、弱楔度楔形杆轴纵扭固有振动都适用。     

Transverse vibration of a rotating twisted Timoshenko beams under axial loading using differential transform

This study introduces the concept of a differential transform to solve the free vibration problems of a rotating twisted Timoshenko beam under axial loading. First, the concept of differential transform is briefly introduced. Second, taking a differential transform of a Timoshenko beam vibration problem, a set of difference equations is derived. Performing some simple algebraic operations on these equations, we can determine the jth natural-frequency, the closed form series solution of the jth mode shape. Finally, three cases—twist, axial force and rotation—are investigated to illustrate the accuracy and efficiency of the present method.     

Tension determination of stay cable or external tendon with complicated constraints using multiple vibration measurements

A novel concept of combining the mode shape ratios with the modal frequencies was recently introduced to develop an accurate method for the determination of stay cable force. A crucial restriction of symmetric boundary constraints at both ends was imposed in this method for the optimization of effective vibration length. However, there certainly exist a number of cases with apparent unsymmetrical boundary constraints such as the case of stay cables with supplementary dampers and that of external tendons affected by intermediate diaphragms. To deal with such difficulties, this method is further generalized in the current study by introducing a shifting parameter in the shape function to effectively consider the unsymmetrical boundary constraints. The numerical difficulty of the more complicated nonlinear optimization process associated with this new formulation is first discussed, followed by extensive numerical investigations, practical verifications, and real applications.