Determination of Natural Frequencies of Multilayered Plates by Mixed Finite Elements

Natural frequencies for multilayer plates are calculated by mixed finite element method. The main object of this paper is to use the mixed model for multilayer plates, analyzing each layer as an isolated plate, where the continuity of displacements is achieved by Lagrange multipliers （representing static variables）. This procedure allows us to work with any model for single plate （so as to ensure the proper behavior of each layer）, and the complexity of the multilayer system is avoided by ensuring the condition of displacements by the Lagrange multipliers （static variables）. The plate is discretized by finite element modeling based on a primary hybrid model, where the domain is divided by quadrilateral, both for the displacement field and static variables. This mixed element for plates was implemented and several examples of vibrations have been verified successfully by the results obtained by other methods in the literature.     

Natural frequencies of fluid-filled transversely isotropic cylindrical shells

The free vibration of a simply supported, transversely isotropic cylindrical shell filled with compressible fluid is exactly analyzed based on three-dimensional elasticity. The displacement separation technique as well as the orthogonal series expansion method are employed. Bessel functions with complex arguments are directly used in the solution for the case of complex eigenvalues. Frequency equation of the coupled vibration problem is obtained by taking account of the effect of the inner fluid. Numerical results are finally presented and compared with those based on three typical shell theories.     

Free vibration of a class of solids with cavities

The Ritz method is used to obtain an eigenvalue equation for the free vibration of a class of solids. Each solid is modelled by means of a segment which is described in terms of Cartesian coordinates and is bounded by the yz, zx and xy orthogonal coordinate planes as well as by two curved surfaces which are defined by polynomial expressions in the coordinates x, y and z. Simple algebraic polynomials which satisfy the boundary conditions at the five surfaces of the segment are used as trial functions. By exploiting symmetry, the range of problems which can be treated is substantially broadened and includes a variety of problems of significant interest in structural analysis, such as thick or very thick shells with various shaped cavities. In order to demonstrate the accuracy of the approach, natural frequencies are given for a sphere with a spherical cavity (a thick spherical shell) as calculated by using the present analysis and by using an exact formulation. The versatility of the approach is then demonstrated by the treatment of several other hollow solids of differing geometry, including a thick cylinder with end plates, a cubic box, a cube with a spherical cavity and a cylinder with a conical inclusion.     

The influence of corner stress singularities on the vibration characteristics of rhombic plates with combinations of simply supported and free edges

This paper offers accurate flexural vibration solutions for rhombic plates with simply supported and free edge conditions. A cornerstone here is that the analysis explicitly considers the bending stress singularities that occur in the two opposite, hinged–hinged and/or hinged–free corners having obtuse angles of the rhombic plates. These singularities become significant to the vibration solution as the rhombic plate becomes highly skewed (i.e. the obtuse angles increase). The classical Ritz method is employed with the assumed normal displacement field constructed from a hybrid set of (1) admissible and mathematically complete algebraic polynomials, and (2) comparison functions (termed here as “corner functions”) which account for the bending stress singularities at the obtuse hinged–hinged and/or hinged–free corners. It is shown that the corner functions accelerate the convergence of solutions, and that these functions are required if accurate solutions are to be obtained for highly skewed plates. Accurate nondimensional frequencies and normalized contours of the vibratory transverse displacement are presented for rhombic plates having a large enough skew angle of 75° (i.e. obtuse angles of 165°), so that the influence of the stress singularities is large. Frequencies and mode shapes of isosceles triangular, hinged–free plates are also available from the data presented.     

行波超声马达带齿定子固有振动频率解析

提出了一种用于计算行波超声马达带齿定子固有振动频率的解析算法。该法重点考虑了带齿梁弯曲变形的具体特点及高阶振动时定子齿的转动惯量对定子固有频率的影响 ,推导了相应的振动微分方程 ,解算出了带齿定子梁的固有振动频率 ,实验证明是一种行之有效的方法。该方法便于分析固有振动频率对各参数变化的敏感性 ,对马达最终驱动频率的确定、定子尺寸的设计及材质选取等优化问题具有指导意义     

基于CAE的YD-2型液压捣固机弹性结构的自振频率的研究

以YD-2型液压捣固机为模型.对其在作业中出现的有害振动形态,采用UG中的Nastran模块对整机结构进行分析,得到其自振频率.再和实验结果进行对比,排除了自振频率和其额定工作频率发生共振的情况.传统的结构自振频率的都是通过整机试验来得到,而利用CAE软件进行分析得到结论不但方便而且可以进行参数化设置、优化设计同步进行.软件计算结果验证了实验结果,对改进捣固振动性能提出了意见.针对试验结果和软件计算结果的差异进行讨论,模型的简化是主要原因.     

Stewart六自由度运动平台自振频率的分析与计算研究

提出了一种对重载的Stewart六自由度运动平台动力学模型局部线性化的新方法,将其六个作动杆简化成质量-弹簧系统.然后采用转移矩阵法将各局部坐标系中建立的质量矩阵、刚度矩阵、阻尼矩阵以及系统的位移和所受外力均转移到总的固定坐标系,运用D＇Alembert原理建立其振动的线性运动微分方程.最后通过结合Stewart平台的运动学反解计算方法,对该平台在运动过程中的自振频率进行了分析和计算,计算结果表明对于这种重载的Stewart运动平台,其各个自振频率在整个工作空间内的变化幅度不大,该结果为重载Stewart平台的设计和控制提供了理论依据,已被成功应用于 Study on the analysis and caculation of the natural frequencies of the stewart motion platform with six degree of freedom DAI Yu-liang　 ZHU Ying　 YU Ke-yun　 LIU Zu-yuan　 作者单位：戴余良（武汉理工大学,交通学院,湖北,武汉,430063;海军工程大学,船舶与动力学院,湖北,武汉,430033）     

Natural frequencies of composite plates with random material properties using higher-order shear deformation theory

Composites are known to display a considerable amount of scatter in their material properties due to large number of parameters associated with the manufacturing and fabrication processes. In the present work, the material properties have been taken as random variables for accurate prediction of the system behavior. Higher order shear theory including rotatory inertia effects has been accounted for in the system dynamic equations. A first order perturbation technique has been employed to obtain the solution of the governing equations. An approach has been outlined for obtaining closed form expressions for the variances of eigen solutions. The effects of side to thickness ratio and variation in standard deviation of the material properties have been investigated for cross-ply symmetric and anti-symmetric laminates. The mean and standard deviations of the first five natural frequencies have been worked out for laminated rectangular plates with all edges simply supported. The higher order shear deformation theory results have been validated with Monte Carlo simulation results and compared with the results based on classical laminate and first order shear deformation theories.     

基于瑞利-里兹法的低翅片管固有频率数值解

针对低翅片管几何形状复杂,质量和刚度分布不均匀,振动特征值不能有封闭解的问题,采用瑞利-里兹法对低翅片管的固有频率进行了数值研究,获得了低翅片管弯曲振动固有频率的数值计算模型。对于一定规格的低翅片管,应用该数值计算模型可求得其各阶固有频率的数值解。采用共振法测试了低翅片管的固有频率,前三阶固有频率的数值解与实测值相比,误差范围为-3.28%--3.51%。该数值计算模型可用于低翅片管固有频率的求解。     

Dynamics of a freely rotating cutting brush subjected to variable speed

Cutting and flicking brushes of street sweeping vehicles sweep the debris that lies in the road gutter. As most of the debris is found in the gutter, it is of interest to understand and improve the behaviour of these brushes. This article investigates the dynamics of a novel oscillatory cutting brush in free rotation, by means of an analytical model derived by the authors. Two oscillatory functions are dealt with: a sinusoidal function and a function developed by the authors. The latter function, called Vanegas Useche–Abdel Wahab–Parker (VAP), depends on a smoothness parameter, b, which controls the smoothness of the angular speed function and the maximum brush accelerations. The parameters that affect brush behaviour are identified. The effects of the type of function, frequency of oscillation, smoothness parameter, ratio between the alternating and mean angular speed, bristle length, mount angle and mount radius are studied. The model indicates that bristle deflections and stresses are proportional to the square of the mean angular speed, provided that the ratio between the alternating and mean angular speed is constant, and depend linearly on the mount radius. The results indicate that, for the cases studied, the first mode dominates bristle vibrations, and that there tends to be resonance when the frequency is close to the first natural frequency of the bristles. Notably, bristle dynamics exhibits a peculiar behaviour, a condition similar to resonance at odd fractions of the natural frequencies for the VAP function. The intensity of this phenomenon could be controlled by adjusting the value of b or changing the frequency of oscillation. It is noted that the results of this work are also applicable to a small-deflection cantilever beam, when this is subjected to a function that is the square of a triangle, sinusoidal, or VAP wave. As the VAP function can achieve a resonant behaviour not only at the natural frequency of the bristle but also at odd fractions of at least the first four natural frequencies, small frequencies may be needed to produce a high-frequency mode resonant characteristic.     

Influence factors on natural frequencies of composite materials

Frontiers of Mechanical Engineering - Compared with traditional materials, composite materials have lower specific gravity, larger specific strength, larger specific modulus, and better...     

An investigation into the vibration of harmonic drive systems

Harmonic drive systems are precise and specific transmission gear systems which are beneficial in terms of the high transmission ratio and almost zero backlash. These inherent and spectacular properties result in using this mechanism in robotic and space sciences where the precision and lightwieght play an important role. This paper presents a vibration analysis of harmonic drive systems using the shell theory. Equations of vibration for the flexspline and the circular spline of the system are derived and used to find the natural frequencies for both parts and, moreover, vibration response of the system under the operating condition is calculated. Also, obtained vibration equations are utilized to study the effects of different involved parameters such as the geometry of the flexspline and its gear tooth, eccentricity, and unbalancing on the vibrational behavior of the system.     

密集模态响应饱和与线性模型失效研究

通过对一两自由度密频近线性系统主共振响应分析 ,揭示了由于密频内共振的存在 ,密集模态坐标的运动将产生复杂的非线性响应饱和现象 ,此时 ,线性模型失效 ,必须采用非线性模型 ;同时 ,讨论了线性模型失效与激励强度、系统初始条件的依赖关系。     

刹车盘对双定子风力发电机主轴振动的影响

作为双定子风力发电机的关键部件,主轴的振动特性直接关系到机组的工作性能。为获得发电机主轴的振动特性,从转子动力学理论出发,应用有限元软件ANSYS对两种结构主轴进行了模态分析,研究了两种结构主轴的固有频率和振型、及振动的特点。分析得到：刹车盘对主轴结构的第一阶固有频率影响不大,但是对主轴结构的振型有较大影响,为发电机主轴及整体结构的设计提供了理论依据。     

Natural frequencies of a beam–mass system in transverse vibration: Rayleigh estimation versus eigenanalysis solutions

A frequency analysis of a beam carrying multiple point masses at various locations are presented by using an eigenanalysis and the Rayleigh's estimation. In the eigenanalysis, the frequency equation is generated by satisfying all boundary and mass-loading conditions. As for Rayleigh's method, the frequency is obtained by solving an algebraic expression involving a specified shape function. As compared with the general expression derived by virtue of Rayleigh method, the eigenfrequency equation is lengthy with products of transcendental functions. The Rayleigh's results with each respective shape function are compared with the eigenfrequencies to verify the effectiveness of Rayleigh's method with assumed shape functions. It is found that the Rayleigh method with the simple shape functions can provide good approximation and can thus replace the solving of complicated eigenfrequency equations. The time saving by using Rayleigh's method becomes more significant as the number of masses carried by the beam increases.     

Environmental effects on the identified natural frequencies of the Dowling Hall Footbridge

Continuous monitoring of structural vibrations is becoming increasingly common as sensors and data acquisition systems become more affordable, and as system and damage identification methods develop. In vibration-based structural health monitoring, the dynamic modal parameters of a structure are usually used as damage-sensitive features. The modal parameters are often sensitive to changing environmental conditions such as temperature, humidity, or excitation amplitude. Environmental conditions can have as large an effect on the modal parameters as significant structural damage, so these effects should be accounted for before applying damage identification methods. This paper presents results from a continuous monitoring system installed on the Dowling Hall Footbridge on the campus of Tufts University. Significant variability in the identified natural frequencies is observed; these changes in natural frequency are strongly correlated with temperature. Several nonlinear models are proposed to represent the relationship between the identified natural frequencies and measured temperatures. The final model is then validated using independent sets of measured data. Finally, confidence intervals are estimated for the identified natural frequencies as a function of temperature. The ratio of observed outliers to the expected rate of outliers based on the confidence level can be used as a damage detection index.     

Free vibration of multi-layered spherically isotropic hollow spheres

A three-dimensional analysis is carried out for the free vibrations of a multi-layered spherically isotropic hollow sphere using a state-space method. By the introduction of three displacement functions and two stress functions, two independent state equations with varying coefficients are derived. Taylor's expansion theorem is then employed to obtain solutions to the two state equations and relationships between the state variables at the upper and lower surfaces of each lamina are established. A variable substitution technique is particularly used to make the derivation more natural and simpler. By virtue of the continuity conditions between two adjacent layers, two sets of linear algebraic equations about the boundary variables at the inner and outer surfaces of a multi-layered hollow sphere are obtained. Frequency equations for the free vibrations are then presented.     

密频内共振对1／2亚谐共振的抑制作用

本文通过两自由度密频近线性系统１／２亚谐共振的分析，揭示出由于密频内共振的作用，发生１／２亚谐共振时，其自由振动项幅度受到有效的抑制。通过定常响应（非平凡解）的稳定性分析，可以看出定常响应的参数稳定区域受密频内共振的影响而呈现较复杂的情况。     

Design of a nonlinear vibration absorber using three-to-one internal resonances

A weakly nonlinear vibration absorber is designed to attenuate the primary resonance vibrations of a single-degree-of-freedom weakly nonlinear oscillator having cubic nonlinearity. The linearised natural frequency of the nonlinear absorber is tuned to be approximately one-third the linearised natural frequency of the primary nonlinear oscillator. The low frequency mode for the absorber is favourably considered based on the fact that the nonlinear absorber can be easily realised in practice by using a light-weight mass attachment with small values of linear and nonlinear stiffness of coupling. For a given primary nonlinear oscillator and absorber mass, implementation of three-to-one internal resonances requires the smallest value of the absorber linear stiffness among three options for utilising internal resonances to design nonlinear absorber. The method of multiple scales is used to obtain the averaged equations that determine the amplitudes and phases of the first-order approximate solutions to the vibrations of the primary nonlinear oscillator and nonlinear absorber. It is found that the absorber response may admit either forced vibration having the forcing frequency or a combination of forced vibration and free-oscillation term having one third the forcing frequency. The nonlinear absorber can effectively suppress the amplitude of primary resonance response and eliminate saddle-node bifurcations occurring in the frequency-response curves of the primary nonlinear oscillator. Numerical results are given to show the effectiveness of the nonlinear absorber for suppressing nonlinear vibrations of the primary nonlinear oscillator under primary resonance conditions.     

含横向裂纹管道固有频率的变化规律研究

基于应变能释放原理和线性断裂力学原理,推导了裂纹尖端方向与管道轴线方向成任意角度的横向裂纹的等效刚度的计算方程,得到了裂纹与管道固有频率的定量关系。进而研究了不同裂纹位置、深度和角度工况下结构的前3阶固有频率变化规律;通过对含裂纹管道进行实验,验证了含横向裂纹管道固有频率计算方法的有效性。