A dynamic stiffness element for free vibration analysis of composite beams and its application to aircraft wings

The dynamic stiffness matrix of a composite beam that exhibits both geometric and material coupling between bending and torsional motions is developed and subsequently used to investigate its free vibration characteristics. The formulation is based on Hamilton’s principle leading to the governing differential equations of motion in free vibration, which are solved in closed analytical form for harmonic oscillation. By applying the boundary conditions the frequency dependent dynamic stiffness matrix that relates the amplitudes of loads to those of responses is then derived. Finally the Wittrick-Williams algorithm is applied to the resulting dynamic stiffness matrix to compute the natural frequencies and mode shapes of an illustrative example. The results are discussed and some conclusions are drawn. The theory can be applied for modal analysis of high aspect ratio composite wings and can be further extended to aeroelastic studies.     

Free vibrational analysis of axially loaded bending-torsion coupled beams: a dynamic finite element

In this article, a dynamic finite element formulation for the free vibration analysis of axially loaded bending-torsion coupled beams is presented. Based on the Euler–Bernoulli and St. Venant beam theories, the exact solutions of the differential equations governing the uncoupled vibrations of an axially loaded uniform beam are found. Then, employing these solutions as basis functions, the analytical expressions for uncoupled bending and torsional dynamic shape functions are derived. Exploiting the principle of virtual work, together with the variable approximations based on the resulting shape functions, leads to a single frequency dependent element matrix which has both mass and stiffness properties. The application of the theory is demonstrated by an illustrative example of a bending-torsion coupled beam with cantilever end conditions, for which the influence of axial force on the natural frequencies is studied. The correctness of the theory is confirmed by the published results and numerical checks.     

Exact dynamic stiffness matrix of non-symmetric thin-walled beams on elastic foundation using power series method

Exact dynamic element stiffness matrix for the flexural–torsional free vibration analysis of the shear deformable thin-walled beam with non-symmetric cross-section on two-types of elastic foundation is newly presented using power series method based on the technical computing program Mathematica. For this, the shear deformable beam on elastic foundation theory is developed by introducing Vlasov's assumption and applying Hellinger–Reissner principle. This beam includes the shear deformation effects due to the shear forces and the restrained warping torsion and due to the coupled effects between them, and rotary inertia effects and the flexural–torsional coupling effects due to the non-symmetric cross-sections. And then equations of motion and force–deformation relations are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components and the exact dynamic element stiffness matrix is determined using force–deformation relationships. In order to verify the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element solutions using the isoparametric beam elements. Particularly the influences of the coupled shear deformation on the vibrational behavior of non-symmetric beam on elastic foundation are investigated.     

Free and forced vibration analysis of viscoelastic damped FG-CNT reinforced micro composite beams

In this paper, free and forced vibration analysis of viscoelastic microcomposite beam reinforced by functionally graded single-walled carbon nanotubes (FG-SWCNTs) is studied using the modified couple stress theory (MCST). The material properties of micro composite beam by generalized rule of mixtures carbon nanotubes are estimated. In addition, these properties are stated as uniform, and functionally graded (FG) distributions in the thickness direction. Energy method and Hamilton’s principle are employed to establish the governing equations of motion for the vibration of viscoelastic damped micro composite beam reinforced by SWCNTs based on the Kelvin–Voigt model. The influences of material length scale parameter, structural damping coefficient and different distributions of SWCNTs on non-dimensional complex natural frequency and amplitude vibration of the viscoelastic micro composite beam are investigated. The results reveal that the lowest vibration amplitude of FG microcomposite beam by the FG-X and the highest occurs by FG-◊. Moreover, in the presence of external periodic load and the absence of structural damping coefficient, the vibration amplitude increases and FG microcomposite beam becomes unstable, even though the amplitude of vibration decreases with increasing structural damping coefficient. It is shown that the natural frequency of SWCNT reinforced composite is more than the frequency of multi-walled carbon nanotubes because SWCNT have more stiffness. In addition, the results illustrate that the experimental data by Lei et al. agree well with those predicted by the MCST in the present work.

     

Finite element modal analysis of an HDD considering the flexibility of spinning disk–spindle, head–suspension–actuator and supporting structure

This paper presents a finite element method to analyze the free vibration of a flexible HDD (hard disk drive) composed of the spinning disk–spindle system with fluid dynamic bearings (FDBs), the head–suspension–actuator with pivot bearings, and the base plate with complicated geometry. Finite element equations of each component of an HDD are consistently derived with the satisfaction of the geometric compatibility in the internal boundary between each component. The spinning disk, hub and FDBs are modeled by annular sector elements, beam elements and stiffness and damping elements, respectively. It develops a 2-D quadrilateral 4-node shell element with rotational degrees of freedom to model the thin suspension efficiently as well as to satisfy the geometric compatibility between the 3-D tetrahedral element and the 2-D shell element. Base plate, arm, E-block and fantail are modeled by tetrahedral elements. Pivot bearing of an actuator and air bearing between spinning disk and head are modeled by stiffness elements. The restarted Arnoldi iteration method is applied to solve the large asymmetric eigenvalue problem to determine the natural frequencies and mode shapes of the finite element model. Experimental modal testing shows that the proposed method well predicts the vibration characteristics of an HDD. This research also shows that even the vibration motion of the spinning disk corresponding to half-speed whirl and the pure disk mode are transferred to a head–suspension–actuator and base plate through the air bearing and the pivot bearing consecutively. The proposed method can be effectively extended to investigate the forced vibration of an HDD and to design a robust HDD against shock.     

多跨下承式拱桥面内全局动力学理论与自由振动研究

为弥补高维工程结构有限元分析对结构参数优化设计等的局限,本文基于欧拉伯努利梁理论与传递矩阵法,研究大跨度拱桥面内自由振动问题.首先,基于多跨拱桥的刚度分析,建立系统的全局动力学模型.其次,基于传递矩阵法建立系统的全局动力学理论,最后,以一座四跨下承式拱桥的平面力学模型,求解其面内自由振动时的固有频率与振型,并将所得结果与用同样参数建模的有限元分析结果对比,证明本文所建理论对求解该类问题的有效性与精确性.此外,通过整桥各跨矢跨比、吊杆截面面积、拱肋惯性矩等的多组参数组合分析了该系统面内自振频率的分布规律.结果表明：随着拱桥矢高增大,系统固有频率减小,因拱桥质量的快速增加,对整桥面内刚度影响显著;增大吊杆截面面积可在一定程度内增大拱桥的面内刚度,导致系统频率在一定范围内增大,且观察到Veering现象.     

Mathematical observations in structural dynamics

In dynamic structural analysis, the basic relations between forces and displacements for a beam element subjected to axial, torsional or flexural vibration are obtained either by solving the appropriate equation of motion or by using an approximate method. The exact equation leads to the dynamic stiffness matrix while the approximate method results in the superposition of elastic and inertial forces represented respectively by the stiffness and mass matrices.The common procedure in finding the natural frequencies is to set the determinant of the dynamic stiffness matrix for the system equal to zero. The approximate method leads to an eigenvalue type problem while the exact method results in a transcendental equation of trigonometric and hyperbolic functions. The natural frequencies in a region of interest are found by a systematic search in the determinntal function.The purpose of this paper is to show that the search technique cannot be applied for certain values of the argument at which the determinantal function is not defined. It is proved that the natural frequencies of any isolated member in the system are critical values for the determinantal function. A practical method is given to obviate the difficulty in order to find the natural frequencies from the determinant, including the critical values at which the dynamic stiffness matrix is not defined. Also, as part of this investigation, the mathematical relation is established between the dynamic stiffness matrix derived by the approximate finite element method and the results obtained from the exact Bernoulli-Euler equation for flexural vibration or the wave equation for axial or torsional vibration.     

变截面铁木辛柯梁振动特性快速计算方法

提出了一种快速计算变截面铁木辛柯梁横向振动特性的方法.基于铁木辛柯梁理论建立的变截面梁的横向振动方程,其梁的截面参数如有效剪切面积、密度、弯曲刚度、转动惯量等沿梁轴线连续或非连续变化；首先将变截面梁等效为多段均匀阶梯梁；然后基于相邻两段连接处的位移(位移、转角)和力(弯矩、剪力)连续条件,建立相邻两段模态函数间相互关系,并递推出首段段与末段模态函数相互关系,利用边界条件得到相应特征方程,使用Newton-Raphson方法计算其固有频率；最后针对梁常见边界条件,得到计算变截面铁木辛柯梁横向振动固有频率特征     

In-plane vibration of plates by continuous mass matrix method

The continuous mass matrix method derived for frameworks is extended to the analysis of in-plane vibration of plates. A continuous mass distribution which is the same as the actual mass distribution of the plate is considered over each rectangular finite element. Taking into account that the rigid body movement produces inertial forces in dynamic analysis for a rectangular plate element eight independent conditions are provided to satisfy eight independent freedoms. Each condition is obtained from an independent displacement distribution satisfying the equations of motion at any point of the element and not only at the nodes of the rectangle. The dynamic element stiffness matrix thus obtained is a function of the natural circular frequency. The limit of the dynamic element stiffness matrix when the value of the natural circular frequency tends to zero is the static, stress compatible element stiffness matrix. The analysis of plates under forcing forces is performed by modal analysis after the natural circular frequencies and the corresponding modal shapes have been obtained from the free vibrations, for all the forcing forces are assumed to be function of the same time variation. Otherwise one must recur to a numerical analysis. The effect of the sizes, number of the meshes, the additional static load on the plate and the rigidity of the boundaries on the vibration of the plate is discussed. Few example problems are solved in order to illustrate the above mentioned effects. The numerical results obtained by continuous mass matrix method are compared with those of consistent mass matrix method. The convergence in terms of the sizes of meshes and the limit of convergence are examined.     

Natural vibration analysis of rail track as a system of elastically coupled beam structures on Winkler foundation

An analytical model for analyzing the vertical free vibration of a rail track is presented. The track structure is represented as a system of elastically coupled beam structures resting on a Winkler foundation. The rail and the tie beams are described by any combination of the two existing beam theories, the Bernoulli-Euler type, and the Timoshenko type, while the rail is assumed to be periodically supported at discrete points on cross-track tie beams. A generalized track element, which consists of a rail span (beam segment), two adjacent ties, and the coupling spring stiffnesses, is established to discretize the track system into identical units. A concept of an equivalent frequency-dependent spring coefficient for the rail support system is introduced to formulate the dynamic stiffness matrix of the track element. Solutions are provided for the natural frequencies of the track and the associated mode shapes of the rail and the ties under transversely (cross-track) symmetric vibration. The free vibration results are used to obtain the dynamic receptance response of a typical field track and to compare them with an existing model and field experimental data.     

低速滚转弹道导弹的气动弹性特性与稳定技术

滚转是弹道导弹的一种新的突防手段,在主动段进行滚动飞行,可以减少强激光照射的驻留时间,有效对抗激光反导.滚转导致弹体气动弹性特性与以正常姿态飞行时相比发生重大的改变,突出表现在弹体固有横振频率以滚转角速度值降低,另一方面,刚性弹体姿态的测量装置与控制装置使弹性信号产生调制现象.由于导弹在飞行中滚速不能保持完全恒定,因此在稳定弹性振动时,必须引入变参数滤波网络以适应振动频率的变化.     

Dynamic stiffness matrix of non-symmetric thin-walled curved beam on Winkler and Pasternak type foundations

Using the technical computing program Mathematica, the dynamic stiffness matrix for the spatially coupled free vibration analysis of thin-walled curved beam with non-symmetric cross-section on two-types of elastic foundation is newly presented based on the power series method. For this, the elastic strain energy considering the axial/flexural/torsional coupled terms, the kinetic energy including the rotary inertia effect, and the energy due to the elastic foundation are introduced. Then, equations of motion are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components. Finally, the exact dynamic stiffness matrix is determined using force–displacement relations. In order to demonstrate the validity and the accuracy of this study, the natural frequencies of thin-walled curved beams with mono-symmetric and non-symmetric cross-sections are evaluated and compared with the analytical solutions and finite element solutions using Hermitian curved beam elements and ABAQUS’s shell elements. In addition, some results by a parametric study are reported.     

Transverse shear effect on vibration of laminated plate using higher-order plate element

An approach using a higher-order plate element to include the effect of transverse shear deformation on free vibration of laminated plate is presented. The total displacement of the element is expressed as the sum of the displacement due to bending and that due to shear deformation. The double-sized stiffness and mass matrices due to the separation of bending and shear displacements are then reduced to the size as if only the total deflection was considered. Numerical results for natural frequencies for a range of different isotropic and anisotropic plates with various thickness-to-length ratios are obtained and compared with solutions available in the literature. The effect of transverse shear deformation on natural frequencies of higher modes of laminated plates is also discussed.     

Effect of boundary conditions on natural frequencies for rotating composite cylindrical shells with orthogonal stiffeners

The effect of the boundary conditions on the natural frequencies for rotating composite cylindrical shells with the orthogonal stiffeners is investigated using Love’s shell theory and the discrete stiffener theory. The frequency equation is derived using the Rayleigh–Ritz procedure based on the energy method. The considered boundary conditions are four sets, namely: (1) clamped–clamped; (2) clamped–simply supported; (3) clamped–sliding; and (4) clamped–free. The beam modal function is used for the axial vibration mode and the trigonometric functions are used for the circumferential vibration mode. The composite shells are stiffened with uniform intervals and the stiffeners have the same material. By comparison with the previously published analytical results for the rotating composite shell without stiffeners and the orthogonally stiffened isotropic cylindrical shells, it is shown that natural frequencies can be determined with adequate precision.     

Axisymmetric vibration of bimodulus thick circular and annular plates

The fundamental natural frequencies of axisymmetric circular and annular plates subjected to a combination of a pure bending stress and extensional stress in the plane of the plate are investigated. The thick plate ring element model which includes the effect of transverse shear deformation is created for axisymmetric free vibration problems. The obtained results of non-dimensional natural frequency coefficient compared with the closed form solutions for ordinary plates are shown to be very good. The effects of various parameters on the natural frequency and neutral surface locations are studied. The bimodulus properties are shown to have significant influence on the natural frequency.     

车-简支梁桥耦合振动系统中全局最大动力响应研究

本文采用四分之一车辆模型和等截面简支梁模型,建立了车-桥耦合振动计算模型,分析了车辆匀速驶过桥梁时,桥梁动挠度、动弯矩、动剪力的全局最大值及发生的位置.进一步分析计算了车速、车距和桥梁模态截断阶数对桥梁动挠度、动弯矩、动剪力全局最大值的影响.结果表明,车速对桥梁动挠度、动弯矩、动剪力全局最大值影响较大,全局最大动挠度和全局最大动弯矩均出现在跨中附近,车速不同位置也不同,而全局最大动剪力均出现在车辆下桥的梁端;两车同时上桥时,前后车车距越大,桥梁动挠度、动弯矩、动剪力全局最大值越小,当达到一定车距时,三者不再减小且与单车情况相同;为提高桥梁动挠度、动弯矩、动剪力全局最大值计算精度,桥梁模态截断阶数宜分别大于3阶、6阶、7阶.     

Vibrations of grids by the finite element method

The finite element method is applied to the free vibration analysis of grids with arbitrary configuration. Grid bars are of solid or thin-walled doubly symmetric cross-section. Stiffness and consistent mass matrices for flexural behavior include the effects of shear deformation and rotary inertia in bending. The torsional behavior of solid sections is approximated by a linear displacement field, and of thin-walled sections, by a cubic. Rotary inertia in torsion is included in both cases and warping inertia, in the latter.

The computer program performs the free vibration analysis starting from the element stiffness and consistent mass matrices. A numerical solution of a thin-walled beam and a parametric solution of an orthogonal and a skew grid with solid and thin-walled bars are presented.     

尾段主梁刚度变化对T型尾翼颤振特性的影响

由于T型尾翼结构和气动布局的特殊性,其颤振特性的分析比较复杂.T型尾翼安装在机身尾部,这体现在有限元结构模型中即为T型尾翼与尾段主梁相连接.为了研究尾段主梁的刚度变化对T型尾翼颤振特性的影响,以某T型尾翼飞机的尾段为研究对象,根据其原始刚度,分别改变其垂直弯曲刚度和扭转刚度,并计算分析相应刚度下的固有振动特性与颤振特性.最后分别以这两个刚度为变量,另一个刚度的原始值为常量进行分析.结果表明尾段主梁的刚度变化对垂尾弯扭耦合颤振的影响较为显著.     

Critical rotational speed,critical velocity of fluid flow and free vibration analysis of a spinning SWCNT conveying viscous fluid

In this article, the influences of rotational speed and velocity of viscous fluid flow on free vibration behavior of spinning single-walled carbon nanotubes (SWCNTs) are investigated using the modified couple stress theory (MCST). Taking attention to the first-order shear deformation theory, the modeled rotating SWCNT and its equations of motion are derived using Hamilton’s principle. The formulations include Coriolis, centrifugal and initial hoop tension effects due to rotation of the SWCNT. This system is conveying viscous fluid, and the related force is calculated by modified Navier–Stokes relation considering slip boundary condition and Knudsen number. The accuracy of the presented model is validated with some cases in the literatures. Novelty of this study is considering the effects of spinning, conveying viscous flow and MCST in addition to considering the various boundary conditions of the SWCNT. Generalized differential quadrature method is used to approximately discretize the model and to approximate the equations of motion. Then, influence of material length scale parameter, velocity of viscous fluid flow, angular velocity, length, length-to-radius ratio, radius-to-thickness ratio and boundary conditions on critical speed, critical velocity and natural frequency of the rotating SWCNT conveying viscous fluid flow are investigated.     

旋转Timoshenko输流管道的固有频率和稳定性分析

基于Timoshenko梁模型,本文研究了旋转输流管道在自由振动状态下的流固耦合振动特性.考虑流体压力、重力、初始轴应力作用,基于Hamilton原理和欧拉角转换,推导得到了旋转Timoshenko输流管道的偏微分方程.根据Galerkin截断法将运动方程进行离散,通过求解系统的特征方程即可得到输流管一阶复频率的实部和虚部,实部代表固有频率,虚部代表能量变化.在流速较高时,研究发现必须考虑4阶及以上Galerkin截断,才能得到稳定的结果.通过与EulerBernoulli梁模型对比,验证了本文的结果正确性.研究发现针对短粗型管道,Timoshenko梁模型更加精确.此外研究了多种参数对旋转Timoshenko输流管道固有频率和振动稳定性的影响.研究结果表明质量比、流速、剪切系数对Timoshenko输流管道流固耦合振动的稳定性影响显著,而转动惯量、重力、流体压力和初始轴应力在一定程度上也会影响管道振动的频率和稳定性.转速的出现将管道频率分为两个量值,但转速并不影响系统能量变化.