Warping effect in free vibration analysis of unidirectional composite non-cylindrical helical springs

The differential equations of motion for unidirectional composite non-cylindrical helical springs including warping, which consist of 14 first-order partial differential equations with variable coefficients, are first derived based on arbitrary spatially curved anisotropic beam theory. An analytical formula for the warping function of Saint-Venant’s torsion of unidirectional composite beams with rectangular cross-section is also obtained. The natural frequencies are determined using improved Riccati transfer matrix method. The element transfer matrix is calculated by the use of the Scaling and Squaring method and Pad’e approximations. Comparisons are made with the EF-results on the natural frequencies of the springs, made from rectangular wire, with inclusion of the warping effect. Information is given on the effect on the natural frequencies of the ratio of radii of the minimum cylinder to the maximum cylinder, the helix pitch angle and the number of active turns. Numerical results reveal that the warping deformation has a significant influence on the natural frequencies, which should be considered in the free vibration analysis of such springs.     

Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method

The literature regarding the free vibration analysis of Bernoulli–Euler and Timoshenko beams under various supporting conditions is plenty, but the free vibration analysis of Reddy–Bickford beams with variable cross-section on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded and semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton’s principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed.     

Optimum design of band-gap beam structures

The design of band-gap structures receives increasing attention for many applications in mitigation of undesirable vibration and noise emission levels. A band-gap structure usually consists of a periodic distribution of elastic materials or segments, where the propagation of waves is impeded or significantly suppressed for a range of external excitation frequencies. Maximization of the band-gap is therefore an obvious objective for optimum design. This problem is sometimes formulated by optimizing a parameterized design model which assumes multiple periodicity in the design. However, it is shown in the present paper that such an a priori assumption is not necessary since, in general, just the maximization of the gap between two consecutive natural frequencies leads to significant design periodicity.The aim of this paper is to maximize frequency gaps by shape optimization of transversely vibrating Bernoulli–Euler beams subjected to free, standing wave vibration or forced, time-harmonic wave propagation, and to study the associated creation of periodicity of the optimized beam designs. The beams are assumed to have variable cross-sectional area, given total volume and length, and to be made of a single, linearly elastic material without damping. Numerical results are presented for different combinations of classical boundary conditions, prescribed orders of the upper and lower natural frequencies of maximized natural frequency gaps, and a given minimum constraint value for the beam cross-sectional area.To study the band-gap for travelling waves, a repeated inner segment of the optimized beams is analyzed using Floquet theory and the waveguide finite element (WFE) method. Finally, the frequency response is computed for the optimized beams when these are subjected to an external time-harmonic loading with different excitation frequencies, in order to investigate the attenuation levels in prescribed frequency band-gaps. The results demonstrate that there is almost perfect correlation between the band-gap size/location of the emerging band structure and the size/location of the corresponding natural frequency gap in the finite structure.     

Natural frequencies of nonlinear vibration of axially moving beams

Axially moving beam-typed structures are of technical importance and present in a wide class of engineering problem. In the present paper, natural frequencies of nonlinear planar vibration of axially moving beams are numerically investigated via the fast Fourier transform (FFT). The FFT is a computational tool for efficiently calculating the discrete Fourier transform of a series of data samples by means of digital computers. The governing equations of coupled planar of an axially moving beam are reduced to two nonlinear models of transverse vibration. Numerical schemes are respectively presented for the governing equations via the finite difference method under the simple support boundary condition. In this paper, time series of the discrete Fourier transform is defined as numerically solutions of three nonlinear governing equations, respectively. The standard FFT scheme is used to investigate the natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results are compared with the first two natural frequencies of linear free transverse vibration of an axially moving beam. And results indicate that the effect of the nonlinear coefficient on the first natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results also illustrate the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters.     

Maximum frequency design of laminated plates with mixed boundary conditions

A layerwise optimization (LO) approach is extended to accommodate the finite element analysis for optimizing the free vibration behavior of laminated composite plates with discontinuities along the boundaries. The classical non-conforming finite element is modified to fit into the LO procedure and is used to calculate natural frequencies of symmetrically laminated plates. This combined LO-FEM approach is applied to laminated rectangular plates with combinations of free, simply supported and clamped edges along parts of the plate boundary. The fundamental frequency, an object function for the present study, is maximized by optimizing design variables that are a set of fiber orientation angles in the layers. For illustrative purpose nine examples of square and rectangular plates with various types of mixed boundary conditions are considered, and a comprehensive set of results are presented for the optimum fiber orientation angles and the maximum fundamental frequencies of the 8-layer and 24-layer plates.     

遗传递增演化算法配筋优化设计

递增演化结构算法(AESO)是在初始结构上,逐步增加有效材料,进而得到优化拓扑形状,其运算速度快,但容易陷入局部最优解。为提高寻找全局最优解的能力,把递增演化结构算法(AESO)和遗传算法(GA)相结合,提出遗传递增演化算法(GAESO),在增加有效材料的选择性上引入生物进化遗传理论,并以ANSYS有限元分析软件的非线性分析为平台,采用钢筋混凝土分离式模型,探讨遗传递增演化算法(GAESO)在钢筋混凝土复杂应力构件配筋优化设计上的应用,直观地完成在荷载作用和应力约束条件下简支深梁、简支开洞深梁和开洞剪力墙等钢筋混凝土复杂应力构件的配筋优化设计,所得结果符合受力机理,演化方向正确,钢筋布置明确,与遗传演化优化算法(GESO)所得结果进行比较验证,证实了算法的可行性、速敛性和稳定性,能为钢筋混凝土配筋优化设计提供参考。     

VIBRATION ANALYSIS OF MICROSCALE PLATES BASED ON MODIFIED COUPLE STRESS THEORY

A non-classical Kirchhoff plate model is developed for the dynamic analysis of microscale plates based on the modified couple stress theory in which an internal material length scale parameter is included. Unlike the classical Kirchhoff plate model, the newly developed model can capture the size effect of microscale plates. Two boundary value problems of rectangular micro- plates are solved and the size effect on the lowest two natural frequencies is investigated. It is shown that the natural frequencies of the microscale plates predicted by the current model are size-dependent when the plate thickness is comparable to the material length scale parameter.     

Initial value method for free vibration of axially loaded functionally graded Timoshenko beams with nonuniform cross section

Free vibration of nonuniform axially functionally graded Timoshenko beams subjected to combined axially tensile or compressive loading is studied. An emphasis is placed on the effect of tip and distributed axial loads on the natural frequencies and mode shapes for an inhomogeneous cantilever beam including material inhomogeneity and geometric non-uniform cross section. The initial value method is developed to determine the natural frequencies. The method’s effectiveness is verified by comparing our results with previous ones for special cases. Natural frequencies of standing/hanging Timoshenko beams are calculated for four different cross sections. The influences of shear rigidity, taper ratio, gradient index, tip force, and axially distributed loading on the natural frequencies of clamped-free beams are discussed. Material inhomogeneity and geometric non-uniform cross-section strongly affect higher-order vibration frequencies and mode shapes.     

FREE VIBRATION ANALYSIS OF LATTICE SANDWICH BEAMS UNDER SEVERAL TYPICAL BOUNDARY CONDITIONS

Free vibration problems of lattice sandwich beams under several typical boundary conditions are investigated in the present paper. The lattice sandwich beam is transformed to an equivalent homogeneous three-layered sandwich beam. Unlike the traditional analytical model in which the rotation angles of the face sheets and the core are assumed the same, different rotation angles are considered in this paper to characterize the real response of sandwich beams. The analytical solutions of the natural frequencies for several typical boundary conditions are obtained. The effects of material properties and geometric parameters on the natural frequencies are also investigated.     

Optimization of the Flutter Load by Material Orientation

Abstract

The present work concerns optimization of the stability of composite plates for supersonic flutter. A Finite element model of the structure is applied. The individual material orientation within each finite element defines the degrees of freedom in the design space. Design iterations are based on analytical sensitivity analyses, derived by Pedersen and Seyranian. Plaut's flutter instability condition is discussed. The condition implies the possibility of an accurate flutter analysis without reducing the eigenvalue problem. For one particular choice of material, an optimal design, in the case of a rectangular, simply supported plate, is found. Design iterations on a delta-shaped plate supported as a cantilever are discussed. A condition for when static divergence is not a possible consequence of the aerodynamic load for any design is derived.     

Effective widths of rectangular slabs stiffened along two opposite edges by prestressed edge beams

The present analysis is an attempt to determine the portion of a rectangular slab that is acting with its two parallel stiffening edge beams, through which prestressing loads are applied to the entire section, in resisting load. Employing the well known theories of bending of plates and beams, the constitutive equations governing the behaviour of this type of composite system are presented. In particular, the equation of compatibility of strains between the slab edges and the stiffening edge beams at their junctions is formulated. In doing this, the biaxial nature of the bending of the edge beams, ignored in earlier formulations [1], has been incorporated. The results of the present analysis show that, under transverse loading, the portion of the slab, called the effectiv width, that can be considered effective as a part of each of the stiffening edge beams in determining stresses and deflexions is not significantly different from that obtained for an unprestressed section or a simply reinforced section. The effective width of the slab when such a section is subjected to only prestressing loads however shows a significant difference. We conclude from this that a single table of effective widths could be adopted for design purposes when considering transverse bending of this type of composite system whether the section is prestressed or not. Typical stress distributions due to (i) prestress alone, (ii) transverse loading alone and (ii) combined prestress and transverse loading are presented to demonstrate that the present formulation is versatile enough to solve problems involving prestressed edge beams in this type of composite assembly.     

Vibration analysis of piezoelectric laminated slightly curved beams using distributed transfer function method

Piezoelectric laminated slightly curved beams (PLSCB) is currently one of the most popular actuators used in smart structure applications due to the fact that these actuators are small, lightweight, quick response and relatively high force output. This paper presents an analytical model of PLSCB, which includes the computation of natural frequencies, mode shapes and transfer function formulation using the distributed transfer function method (DTFM). By setting the radius of curvature of the proposed model to infinity, a piezoelectric laminated straight beams (PLSB) model can be obtained. The DTFM is applied and extended to carry out the transfer function formulation of the PLSCB and PLSB models. This method will be used to solve for the natural frequencies, mode shapes and transfer functions of the PLSCB and PLSB models in exact and closed form solution without using truncated series of particular comparison or admissible functions. The natural frequencies of the cantilevered PLSCB and PLSB are calculated by the DTFM and the Rayleigh–Ritz method. The analysis indicates that the stretching–bending coupling due to curvature has a considerable effect on the frequency parameters. Increasing the radius of curvature of the PLSCB has its largest effect on the natural frequencies. But the inhomogeneity of the boundary conditions does not have any effects on the natural frequencies or system spectrum due to the both receptance and boundary transfer functions have the same characteristic equations. The method can also be generalized to the vibration analysis of non-piezoelectric composite beams with arbitrary boundary conditions.     

Effects of End Conditions of Cross-Ply Laminated Composite Beams on Their Dimensionless Natural Frequencies

The effects of the end conditions of cross-ply laminated composite beams on their dimensionless natural frequencies of free vibration is investigated. The problem is analyzed and solved by using the energy approach, which is formulated by a finite element model. Various end conditions of beams are used. Each beam has either movable ends or immovable ends. Numerical results are verified by comparisons with other relevant works. It is found that more constrained beams have higher values of natural frequencies of transverse vibration. The values of the natural frequencies of longitudinal modes are found to be the same for all beams with movable ends because they are generated by longitudinal movements only.     

压电层合梁静动力学分析的高精度梁单元

给出了一个对复合材料压电层合梁进行数值分析的高精度压电层合梁单元。基于Shi三阶剪切变形板理论的位移场和Layer-wise理论的电势场,用力-电耦合的变分原理及Hamilton原理推导了压电层合梁单元列式。采用拟协调元方法推导了一个可显式给出单元刚度矩阵的两节点压电层合梁单元,并应用于压电层合梁的力-电耦合弯曲和自由振动分析。计算结果表明,该梁单元给出的梁挠度和固有频率与解析解吻合良好,并优于其它梁单元的计算结果,说明了本文所给压电层合梁单元的可靠性和准确性。研究结果可为力-电耦合作用下压电层合梁的力学分析提供一个简单、精确且高效的压电层合梁单元。     

Free vibration of non-uniform axially functionally graded beams using the asymptotic development method

The asymptotic development method is applied to analyze the free vibration of non-uniform axially functionally graded(AFG) beams, of which the governing equations are differential equations with variable coefficients. By decomposing the variable flexural stiffness and mass per unit length into reference invariant and variant parts, the perturbation theory is introduced to obtain an approximate analytical formula of the natural frequencies of the non-uniform AFG beams with different boundary conditions.Furthermore, assuming polynomial distributions of Young's modulus and the mass density, the numerical results of the AFG beams with various taper ratios are obtained and compared with the published literature results. The discussion results illustrate that the proposed method yields an effective estimate of the first three order natural frequencies for the AFG tapered beams. However, the errors increase with the increase in the mode orders especially for the cases with variable heights. In brief, the asymptotic development method is verified to be simple and efficient to analytically study the free vibration of non-uniform AFG beams, and it could be used to analyze any tapered beams with an arbitrary varying cross width.     

运动车辆横向振动半车模型分析

通过半车模型, 数值研究平滑路面上运动车辆车体的前两阶横向振动频率. 将车体模型化为两端自由的Euler-Bernoulli梁, 半车模型的车轮模型化为两个弹性不等的弹簧. 建立半车模型的数学模型描述车体的横向振动. 以两端自由的静态梁的模态为试函数和权函数, 通过高阶Galerkin截断计算车体横向振动的频率, 并研究车辆运行速度、车体刚度、弹簧刚度等参数对车体振动频率的影响.     

Vibration analysis of a circular disk tensioned by rolling using finite element method

Summary The paper proposes a method in finite element analysis for estimating natural frequencies of a disk tensioned by rolling, without the use of eigenvalue analysis. The natural frequencies of a disk vary when the localized plastic deformation caused by roll-tensioning induces residual stresses. Tensioning is used for improving the dynamic stability of circular saws; the optimal condition of rolling can be predicted from natural frequency characteristics. In the proposed method, the natural frequencies after rolling are easily estimated from the mode shapes of the disk before rolling and the stress distribution after rolling. The method is based on ideas similar to thermal stress and sensitivity analysis rather than on eigenvalue analysis. The effectiveness of the method is shown by comparing the natural frequency characteristics obtained by this method with those by eigenvalue analysis. Received 18 June 1998; accepted for publication 8 April 1999     

An evolutionary search technique to determine natural frequencies and mode shapes of composite Timoshenko beams

Natural frequencies and mode shapes of composite Timoshenko beams are determined by a diversity guided evolutionary algorithm (DGEA) with different boundary conditions. After applying boundary conditions, frequency equation is obtained in determinant form. Then, natural frequencies and consequently mode shapes are obtained using DGEA where the absolute value of determinant is the subject of optimization. Advantages of employing DGEA are: first, all natural frequencies are produced in a simple run, second, its simplicity for implementation and third, the procedure is not computationally prohibitive. Results clearly show the applicability of the proposed method for obtaining natural frequencies and mode shapes.     

Vibration of a variable cross-section beam

Vibration of an isotropic beam which has a variable cross-section is investigated. Governing equation is reduced to an ordinary differential equation in spatial coordinate for a family of cross-section geometries with exponentially varying width. Analytical solutions of the vibration of the beam are obtained for three different types of boundary conditions associated with simply supported, clamped and free ends. Natural frequencies and mode shapes are determined for each set of boundary conditions. Results show that the non-uniformity in the cross-section influences the natural frequencies and the mode shapes. Amplitude of vibrations is increased for widening beams while it is decreased for narrowing beams.     

地震栓波器假频本质的动力稳定性解释

地震栓波器是用来栓测竖向地震波的仪器.本文根据它的实际结构,提出了一个较精确的力学模型——双层三悬丝模型.把悬丝作为曲杆,用动力稳定性理论进行分析,揭示了地震仪栓波器由于竖向地震波以外的因素而产生输出的所谓假频现象产生的原因:位于悬丝所在水平面内的激励在一定条件下将导致悬丝在面外动力失稳而产生强烈振动,从而引起虚假输出.计算结果说明了影响假频的诸因素,为激励速度峰值、激励相对于栓波器的方向和曲杆面内振动的固有频率等的作用,并与实验值作了比较.