Frequency solutions for circular plates with internal supports and discontinuous boundaries

The paper presents a study of the free-flexural vibration analysis of circular plates continuous over point supports, partial internal curved supports, and with mixed-edge boundary conditions. An approximate model which combines the advantages of the Rayleigh-Ritz and the Lagrangian multiplier methods is developed for analyzing this class of circular plate problems. The Rayleigh-Ritz method is used to formulate plates with classical boundary conditions, such as free, simply-supported or clamped, while the Lagrangian multiplier method is used to handle plates with point supports, partial internal curved supports and mixed-edge boundary conditions. The admissible pb-2 Ritz function consists of the product of a two-dimensional polynomial and a basic function. The basic function is defined by the product of the equations of the prescribed piecewise-continuous boundary shape each raised to the power of 0, 1 or 2, corresponding to free, simply-supported or clamped edge, respectively. The set of functions automatically satisfies all the kinematic boundary conditions of the plate at the outset. The geometric boundary conditions associated with the internal supports and discontinuous edges are simulated using a sufficient number of closely-spaced point constraints. Numerical results for several selected plate problems are presented to demonstrate the various features and accuracy of the present method.     

Free vibration analysis of rectangular plates with internal columns and uniform elastic edge supports by pb-2 Ritz method

Free vibration analysis of rectangular plates with internal columns and elastic edge supports is presented using the powerful pb-2 Ritz method. Reddy's third order shear deformation plate theory is employed. The versatile pb-2 Ritz functions defined by the product of a two-dimensional polynomial and a basic function are taken as the admissible functions. Substituting these displacement functions into the energy functional and minimizing the total energy by differentiation, leads to a typical eigenvalue problem, which is solved by a standard eigenvalue solver. Stiffness and mass matrices are numerically integrated over the plate using the Gaussian quadrature. The accuracy and efficiency of the proposed method are demonstrated through several numerical examples by comparison and convergency studies. Many numerical results for reasonable natural frequency parameters of rectangular plates with different combinations of elastic boundary conditions and column supports at any locations are presented, which can be used as a benchmark for future studies in this area.     

Vibrations of point-supported rectangular plates with variable thickness using a set of static tapered beam functions

This paper studies the free vibrations of point-supported rectangular plates with variable thickness using the Rayleigh–Ritz method. The domain of the plate is bounded by x=αa′, a′ (0α<1); y=βb′, b′ (0β<1) in the Cartesian coordinate system. The thickness of the plate varies continuously and is represented by a power function (x/a′)^s(y/b′)^t. Varieties of tapered rectangular plates can be described by giving s and t different values. A set of static tapered beam functions which are the solutions of a tapered beam (a unit width strip taken from the particular plate under consideration in one or the other direction parallel to its edges) under a Taylor series of static loads, are developed as the admissible functions for the vibration analysis of point-supported rectangular plates with variable thickness in one or two directions. The eigenfrequency equation is derived through the Rayleigh–Ritz approach, supplemented by the zero deflection conditions at the point-supports. A very simple program in common use has been compiled. The convergence study shows a small computational cost and the comparison with known solutions for point-supported rectangular plates with uniform thickness demonstrates the accuracy of the present method. Finally, some new numerical results are given, which may serve as the benchmarks for future research on the aforementioned problem.     

Vibration of composite plates with internal supports

The free vibration of rectangular laminated composite plates with arbitrary support conditions along the edges, internal line supports and discrete point supports are studied using the Rayleigh-Ritz method. Polynomial approximation functions are selected to satisfy all essential boundary conditions along the edges of the plate and to vanish along line supports parallel to the co-ordinate axes. Straight line supports at an angle from the co-ordinate axes and curved line supports are modeled by introducing several point supports along the line. Zero displacement constraints at the point support locations are enforced using the Lagrange multiplier technique. The plate constitutive equations are expressed in terms of stiffness invariants and the fundamental natural frequency is maximized by selecting the appropriate lay-up. Several examples are presented to illustrate the versatility of the approach and provide results not previously available. The influence of the number of plies in the laminate, lay-up, material properties and plate aspect ratios are investigated.     

Free vibration analysis of isotropic and orthotropic triangular plates

A highly accurate and computationally efficient numerical method is developed for the flexural vibration of isotropic and orthotropic triangular plates. A set of two-dimensional orthogonal plate functions is used as an admissible displacement function in the Rayleigh-Ritz method to obtain the natural frequencies and mode shapes for the plates. From a prescribed starting function satisfying the boundary conditions, the higher terms in the orthogonal plate functions are constructed using the Gram-Schmidt orthogonalization process. Natural frequencies and mode shapes are obtained for three triangular plates with different support conditions. The obtained numerical results are presented, and the isotropic case is verified with other numerical methods in the literature.     

The dynamic analysis of nonuniformly pretwisted Timoshenko beams with elastic boundary conditions

The coupled governing differential equations and the general elastic boundary conditions for the coupled bending–bending forced vibration of a nonuniform pretwisted Timoshenko beam are derived by Hamilton's principle. The closed-form static solution for the general system is obtained. The relation between the static solution and the field transfer matrix is derived. Further, a simple and accurate modified transfer matrix method for studying the dynamic behavior of a Timoshenko beam with arbitrary pretwist is presented. The relation between the steady solution and the frequency equation is revealed. The systems of Rayleigh and Bernoulli–Euler beams can be easily examined by taking the corresponding limiting procedures. The results are compared with those in the literature. Finally, the effects of the shear deformation, the rotary inertia, the ratio of bending rigidities, and the pretwist angle on the natural frequencies are investigated.     

Free vibration analysis of circularly curved multi-span Timoshenko beams by the pseudospectral method

The pseudospectral method is applied to the free vibration analysis of circularly curved multi-span Timoshenko beams. Each section of the beam has its own basis functions, and the continuity conditions at the intermediate supports as well as the boundary condition are treated as the constraints of the basis functions so that the number of degrees of freedom matches the number of the pseudospectral expansion coefficients. The computed natural frequencies are compared with those of existing literature, where it is shown that they are in good agreement. Numerical examples are provided for pinned-pinned, clamped-clamped and free-pinned boundary conditions for different numbers of sections and for different thickness-to-length ratios.     

Parametric instability of spinning twisted Timoshenko beams under compressive axial pulsating loads

The parametric instability on lateral bending vibrations of a spinning pretwisted beam under compressive axial pulsating forces is investigated. Equations of motion of the twisted beam are derived in the spinning twist coordinate frame using the Timoshenko beam theory and applying the Hamilton’s principle. The finite element method is employed to discretize the equations of motion into time-dependent ordinary differential equations with gyroscopic terms. A set of second-order ordinary differential equations with periodic coefficients of Mathieu-Hill type is formed to obtain the boundary frequencies of instability regimes. The influence of twist angle, spinning speed, static component of axial force, aspect ratio and restraint condition on the instability regions of the spinning twisted Timoshenko beam is discussed.     

Instability and vibration of a rotating Timoshenko beam with precone

A rotating blade with a precone angle is usually designed, but little literature has investigated the effect of the precone angle on vibration. This paper investigates divergence instability and vibration of a rotating Timoshenko beam with precone and pitch angles. It uses Hamilton's principle to derive the coupled governing differential equations and boundary conditions for a rotating Timoshenko beam. Analytical solution of an inextensional Timoshenko beam without taking into account the Coriolis force effect can be derived. Some simple relations among the parameters of rotating Timoshenko beams are revealed. Based on these relations, one can predict the natural frequencies and parameters of other systems from those of known systems. Moreover, the mechanism of divergence instability (tension buckling) is investigated. Finally, the effects of the parameters on natural frequencies, and the phenomenon of divergence instability are investigated.     

Timoshenko梁的变形场重构及传感器位置优化

针对KO位移理论仅适用于重构单方向位移场问题,提出一种适用于六自由度位移场重构的新方法,称之为“多维积分法”。依据Timoshenko梁的静力学平衡方程,建立了位移、转角与外载荷之间的数学模型。并针对不同的外载荷环境,推导出相应的应变场函数和位移场函数,建立了表面应变与截面应变之间的转换关系。为了提升该方法的容差性,以重构位移场的精确性和稳定性为优化目标,建立了关于应变传感器位置的多目标粒子群优化模型。以机翼框架为试验平台,对其进行有限元分析,建立优化目标模型,给出优化后的应变传感器的布置方案。并以此方案为依据,分别利用有限元分析结果和实测梁表面应变值来重构位移场。试验结果表明,提出的“多维积分法”在两种不同形式的外载荷作用下均呈现出较高的重构精度。     

Identification of boundary conditions of tapered beam-like structures using static flexibility measurements

The integrity and reliability of the beam-like structures are dependent in part on their boundary conditions, which can vary with time due to damage or aging, thus the identification of boundary conditions might be one of the most significant aspects for damage detection of such structures. This paper investigates a direct method for identifying the boundary conditions of tapered beam-like structures using static flexibility measurements. The beam is modeled by a flexible tapered beam, which is constrained at one end by translational and rotational springs. The translational and rotational springs are utilized to simulate the boundary conditions of the tapered beam, and the purpose of this paper is to identify the stiffnesses of the translational and rotational springs, i.e. translational root stiffness and rotational root stiffness. It is theoretical proved that the static flexibility measured on the beam can be expressed as a function of the flexural rigidity of the beam at its constrained end, translational root stiffness and rotational root stiffness. Then, a set of linear equations for identifying the translational and rotational root stiffnesses are formed by three or more different static flexibility measurements. Finally, the proposed method is validated using both simulative and experimental examples.     

Natural frequencies,modes and critical speeds of axially moving Timoshenko beams with different boundary conditions

In this paper, natural frequencies, modes and critical speeds of axially moving beams on different supports are analyzed based on Timoshenko model. The governing differential equation of motion is derived from Newton's second law. The expressions for various boundary conditions are established based on the balance of forces. The complex mode approach is performed. The transverse vibration modes and the natural frequencies are investigated for the beams on different supports. The effects of some parameters, such as axially moving speed, the moment of inertia, and the shear deformation, are examined, respectively, as other parameters are fixed. Some numerical examples are presented to demonstrate the comparisons of natural frequencies for four beam models, namely, Timoshenko model, Rayleigh model, Shear model and Euler–Bernoulli model. Finally, the critical speeds for different boundary conditions are determined and numerically investigated.     

Finite expansion of a thick compressible spherical elastic shell

A method is proposed for the analysis of the finite symmetrical expansion, due to internal pressure, of a thick-walled sphere of compressible hyperelastic material. Solutions can be obtained by this method for any admissible strain energy function and numerical results are presented for three strain energy functions.     

基于Timoshenko梁单元的径向波箔轴承箔片变形分析

为实现对径向波箔轴承箔片变形的更准确分析,基于Timoshenko梁单元建立箔片变形模型,在考虑库仑摩擦效应影响的基础上,计算不同载荷下箔片模型的变形,并与文献中的Heshmat公式、Iordanoff公式以及NDOF模型的计算结果进行对比,验证Timoshenko梁模型计算的准确性;改变摩擦系数的大小计算不同情况下的...     

A comparative study on the dynamics of high speed spindles with respect to different preload mechanisms

This paper presents the effects of bearing preload mechanisms on the dynamic performance of high speed spindles. The comparisons of two main types of bearing preload????constant?? and ??rigid????mechanisms are provided using a mathematical model as well as experiments. Based on the Timoshenko beam element theory coupled with a nonlinear model of angular contact ball bearings, the dynamics of the spindle shaft, housing, and bearings system is modeled as a nonlinear function of preload mechanism and amplitude, spindle speed, and external cutting loads. The mathematical model of the spindle is experimentally validated by comparing the predicted and measured static displacements, mode shapes, frequency response functions, and natural frequencies under different conditions. The performance of spindles under rigid and constant force preload is investigated systematically using a mathematical model under various conditions. It is shown, among other things, that at high speeds and under cutting loads the rigid preload mechanism is more efficient in maintaining the dynamic stiffness of spindles than constant preload.     

Free transverse vibration of an axially loaded non-uniform spinning twisted Timoshenko beam using differential transform

This study investigates the vibration problems of an axially loaded non-uniform spinning twisted Timoshenko beam. First, using the Timoshenko beam theory and Hamilton's principle, we derive the governing equations and boundary conditions of the beam. Secondly, the differential transform method is used to solve these equations with appropriate boundary conditions. Finally, the effects of the twist angle, spinning speed, and axial force on the natural frequencies of a non-uniform Timoshenko beam are investigated and discussed.     

Thermal post-buckling analysis of slender columns using the concept of coupled displacement field

Thermal post-buckling analysis of columns with an axially immovable ends is studied using the Rayleigh-Ritz (R-R) method, where the admissible displacement functions are chosen based on the concept of coupled displacement field (CDF) criteria. Geometric non-linearity is considered using the von-Karman strain displacement relations of the beam. Furthermore, the displacement fields derived from CDF criteria are used in an intuitive formulation, where the thermal post-buckling behavior can be predicted by using two parameters namely tension developed in the column and linear buckling load. An exhaustive set of column boundary conditions are considered namely classical such as hinged-hinged, clamped-clamped, clamped-hinged and non-classical such as clamped-guided and hinged-guided. Post-buckling analysis results are presented in the form of closed form expressions, where the ratio of post-buckling load to linear buckling load parameter is expressed as a function of central amplitude of the column for all the boundary conditions considered. The amount of non-linearity predicted using the present formulations (R-R method and intuitive method) based on the concept of coupled displacement field (CDF) criteria shows an excellent agreement with the available literature results for both classical and non-classical boundary conditions.     

More general expression for the torsional warping of a thin-walled open-section beam

In this study, an analytical formulation for the torsional warping function of a thin-walled open-section beam is built based on the combination of the Vlasov assumption and the Kirchhoff assumption of plate/shell theory, essentially due to Goodier and Gjelsvik. Vlasov, Timoshenko and many authors (follow Vlasov) only consider the contour warping function as the real warping function. Goodier and Gjelsvik (follow Goodier) not only consider the contour warping but also the thickness warping. For some cross-sectional constants related to the warping function such as the torsional constant, the adoption of the warping function based on Vlasov's theory or Timoshenko's theory may cause them incorrectness. On the other hand, the torsional constant based on the Goodier's theory is consistent with the one based on the membrane analogy. Thus, Goodier's theory is a good approximation for the torsional warping of a thin-walled open-section beam. To the authors’ limited knowledge, the analytical expression for the complete torsional warping of a thin-walled open-section beam has not been found in the literature. In this study, a more general expression for the torsional warping of a thin-walled open-section beam is presented. The position formulas to determine the twist center are also given.     

连续热镀锌生产线的简化与数学建模

为研究气刀处的带钢大幅抖动原因,试对带钢连续热镀锌系统进行动力学建模。整个镀锌生产线是一个复杂的大系统,动力学建模时必须对其进行适当简化,以得到一个仅包含热张紧辊、退火段带钢、沉没辊、出锌锅上行段带钢、塔顶辊、下行段带钢、转向辊和八个弹簧支承的比较简单的混杂系统力学模型。先将简化后的混杂系统分成多个子模型,用带钢张力和附加惯性载荷代替边界条件,然后基于牛顿第二定律对这些子模型进行建模,以期得到整个系统的控制方程。此项研究可为混杂系统建模提供一种新参考。     

Vibration analysis of a beam with partially distributed internal viscous damping

In this paper, governing equations of vibration for a beam with distributed internal viscous damping are established by using Timoshenko beam theory and Hamilton's principle. Then, the transfer matrix method is applied to obtain the frequency equations for the beam. The results reveal, when the internal viscous damping fully distributes along the beam, that the natural frequency decreases with the increasing damping and drops to a zero value at a certain critical damping. While the damping is locally distributed, damped frequency, mode shape and transient response time are affected most significantly by locating the damped segment at the position with maximum bending moment. The flexural amplitudes and phase angles of a beam excited by the resonant harmonic load can be effectively predominated by tuning the damping value.