Vibration analysis of corrugated beams: The effects of temperature and corrugation shape

This paper deals with the vibration analysis of corrugated beams under large amplitude displacements and temperature variations. The dynamic response depends on the shape, the period and the amplitude of the corrugations as well as on the temperature variations. These different parameters were taken into account using a homogenization process. We coupled the harmonic balance method and the Galerkin technique to derive a frequency amplitude equation, which includes the corrugation shape and temperature. Finally, the influence of different shapes of corrugation (sinusoidal, triangular and square) on the nonlinear vibration response of a corrugated beam was compared to results obtained for a planar beam.     

Natural vibrations of corrugated orthotropic shells of revolution

The torsional and longitudinal–flexural vibrations of corrugated orthotropic shells are investigated. Relations, including the equations of motion in forces and moments and Hooke’s relations, are obtained using the Kirchhoff–Love hypotheses. The influence of the geometric parameters of the shell (corrugation amplitude and length) on the eigenfrequencies and natural vibrations modes is studied for fixed-end shells. It is found that during torsional vibrations, increasing the corrugation amplitude and increasing the number of corrugations leads to a decrease in the resonant frequencies. In the case of torsional and longitudinal–flexural vibrations, the influence of the corrugation amplitude on the natural vibration modes is investigated.     

摩擦自激振动引起摩擦面波状磨耗的试验研究

在销-盘摩擦磨损试验机上进行金属干摩擦试验,研究摩擦自激振动对盘试件磨痕表面轮廓的影响,采用激光位移传感器测量盘试件摩擦表面轮廓尺寸,用加速度传感器测量销试件的摩擦自激振动.试验结果表明：在干摩擦状态下,销-盘系统容易发生持续的摩擦自激振动,当摩擦时间达到一定数值后,盘试件的磨痕表面轮廓会出现明显的波浪形磨耗,波浪形磨损机理主要是疲劳磨损.分析表明波浪形磨耗的波长近似等于摩擦自激振动的周期与滑动速度的乘积,由此推断摩擦自激振动引起了摩擦表面的波浪形磨耗.     

“Non-linear normal modes” and the generalized Ritz method in the problems of vibrations of non-linear elastic continuous systems

In the study of natural vibrations of non-linear elastic systems it is shown that the mode shape of the vibration can vary with the amplitude as well as the frequency, and that the amplitude frequency relation is strongly affected by constraints imposed on the mode shape in an approximate solution. A method is developed which assumes the approximate solution in the form of a truncated series in which, instead of the set of coefficients, the set of functions of spatial variables is unknown and then determined by a procedure that can be regarded as a generalization of the Ritz method. The problem of variations of the normal mode shapes and of the associated natural frequencies with the amplitude is illustrated by two examples of beams with non-linear boundary conditions, and the amplitude-frequency relation is compared to that corresponding to the a priori assumed linear normal mode solution. Further possible consequences of the mode shape amplitude variations in forced, resonant motion of nonconservative systems are also indicated.     

Equivalent models of corrugated panels

The design of corrugated panels has wide application in engineering. For example corrugated panels are often used in roof structures in civil engineering. More recently corrugated laminates have been suggested as a good solution for morphing aircraft skins due to their extremely anisotropic behaviour. The optimal design of these structures requires simple models of the panels or skins that may be incorporated into multi-disciplinary system models. Thus equivalent material models are required that retain the dependence on the geometric parameters of the corrugated skins or panels. An homogenisation-based analytical model, which could be used for any corrugation shape, is suggested in this paper. This method is based on a simplified geometry for a unit-cell and the stiffness properties of original sheet. This paper outlines such a modelling strategy, gives explicit expressions to calculate the equivalent material properties, and demonstrates the performance of the approach using two popular corrugation shapes.     

Design of corrugated plates with extreme stiffnesses

The maximum (minimum) effective stiffnesses of a corrugated plate with varying corrugation shape are determined.     

Symmetric and anti-symmetric Rayleigh–Lamb modes in sinusoidally corrugated waveguides: An analytical approach

The wave propagation analysis in corrugated waveguides is considered in this paper. Elastic wave propagation in a two-dimensional periodically corrugated plate is studied here analytically. The dispersion equation is obtained by applying the traction free boundary conditions. Solution of the dispersion equation gives both symmetric and anti-symmetric modes. In a periodically corrugated waveguide all possible spectral order of wave numbers are considered for the analytical solution. It has been observed that the truncation of the spectral order influences the results. Truncation number depends on the degree of corrugation and the frequency of the wave. Usually increasing frequency requires increasing number of terms in the series solution, or in other words, a higher truncation number. For different degrees of corrugation the Rayleigh–Lamb symmetric and anti-symmetric modes are investigated for their non-propagating ‘stop bands’ and propagating ‘pass bands’. To generate the dispersion equation for corrugated plates with a wide range of the degree of corrugation, appropriate truncation of the spectral orders has to be considered. Analytical results are given for three different degrees of corrugation in three plates. Resonance of symmetric and anti-symmetric modes in these plates, their ‘cut-off’, ‘cut-on’, ‘branch-point’, ‘change-place’, ‘mode conversion’ and ‘pinch points’ at various frequencies are also studied.     

Effect of a scratch on curved rail on initiation and evolution of plastic deformation induced rail corrugation

A calculation model is put forward to analyze the effect of a scratch on the running surface of a curved rail on initiation and growth of plastic deformation induced rail corrugation when a wheelset is steadily and repeatedly curving. The numerical method considers a combination of Carter’s two-dimensional contact theory, a two-dimensional elastic–plastic finite element model and a vertical dynamics model of railway vehicle coupled with a curved track. A concept of feedback between the corrugation development and the vertical coupling dynamics of the wheelset and track is involved. The cyclic ratchetting effect of the rail material under repeated contact loadings is taken into account. The numerical results indicate that when a vehicle runs on rails with a scratch the contact vibration between the wheel and rail occurs at large amplitude, and rail corrugation due to plastic deformation initiates and develops. The corrugation has a tendency to move along the running direction and its evolution rate decays as wheelset passages increase. The passing frequencies of the plastic deformation induced corrugation depend on the natural frequencies of the track. The residual stresses stabilize after a limited number of wheelset passages. The residual strains increase at a reduced rate with increasing wheelset passages.     

DYNAMIC CHARACTERISTICS OF AXIALLY-SYMMETRICAL ANNULAR CORRUGATED SHELL PIEZOELECTRIC TRANSDUCERS

The coupled extensional and flexural vibrations of an annular corrugated shell piezoelectric transducer consisting of multiple circularly-annular surfaces smoothly connected along the interfaces were investigated in the paper. Only a time-harmonic voltage is applied across two electrodes of the piezoelectric shell as the external loading. A theoretical solution was obtained using the classical shell theory. Based on the solution, basic vibration characteristics of resonant frequencies, mode shapes were calculated and examined.     

Nonlinear vibration of corrugated shallow shells under uniform load

Based on the large deflection dynamic equations of axisymmetric shallow shells of revolution, the nonlinear forced vibration of a corrugated shallow shell under uniform load is investigated. The nonlinear partial differential equations of shallow shell are reduced to the nonlinear integral-differential equations by the method of Green's function. To solve the integral-differential equations, expansion method is used to obtain Green's function. Then the integral-differential equations are reduced to the form with degenerate core by expanding Green's function as series of characteristic function. Therefore, the integral-differential equations become nonlinear ordinary differential equations with regard to time. The amplitude-frequency response under harmonic force is obtained by considering single mode vibration. As a numerical example, forced vibration phenomena of shallow spherical shells with sinusoidal corrugation are studied. The obtained solutions are available for reference to design of corrugated shells     

Bending of corrugated thick rectangular plates

An approximate analytical solution describing the bending of hinged corrugated thick plates is obtained using the second variant of the boundary shape perturbation method and taking into account the first three approximations. The effect of the shape of the boundary surfaces in the zone of maximum external load on the magnitude and nonlinear variation of the displacements and stresses throughout the thickness of a corrugated thick plate depending on the corrugation amplitude and spatial frequency is analyzed. The results are compared to the exact solution for a flat plate     

含45°斜裂纹圆柱薄壳动态特性试验研究

测量了含45°斜裂纹圆柱薄壳的固有频率并拍摄了相应的激光全息振型图．实验表明斜裂纹比轴向和环向裂纹对壳体动态特性的影响更大,致使振型图发生了严重畸变而显得相当复杂,利用传统思路难以找到裂纹长度对壳体动态特性的影响规律．为此,把裂纹周围的振动看作为一种独立的局部振动,从而把含斜裂纹壳体的各种复杂振型划分为3类:纯局部振动振型、纯原振动振型、局部振动和原振动耦合振型．其中前两种振型的固有频率皆随裂纹的加长而降低,但对于耦合振型有时会出现“随裂纹加长频率反而升高的现象”, 这是由于把壳体原振动的频率和局部振动的频率相混淆而产生的错觉．     

含45度斜裂纹圆柱薄壳动态特性试验研究

测量了含45度斜裂纹圆柱薄壳的固有频率并拍摄了相应的激光全息振型图. 实验 表明斜裂纹比轴向和环向裂纹对壳体动态特性的影响更大，致使振型图发生了严重 畸变而显得相当复杂，利用传统思路难以找到裂纹长度对壳体动态特性的影响规律. 为此， 把裂纹周围的振动看作为一种独立的局部振动，从而把含斜裂纹壳体的各种复杂振型划 分为3类：纯局部振动振型、纯原振动振型、局部振动和原振动耦合振型. 其中前两种振型 的固有频率皆随裂纹的加长而降低，但对于耦合振型有时会出现``随裂纹加长频率反而升高 的现象'，这是由于把壳体原振动的频率和局部振动的频率相混淆而产生的错觉.     

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.     

波纹板结构动力学建模及其自由振动特性分析

波纹板结构是高速列车中较为常见的型材结构,因其具有轻质、高强度和高稳定性等一系列优点,所以广泛应用于车体结构轻量化设计。首先,基于一阶剪切变形理论和中厚壳(圆弧壳)理论,采用微分求积有限元法(DQFEM)建立一般边界条件下五自由度波纹板结构的通用动力学分析模型;紧接着对五自由度波纹板结构动力学模型进行收敛性验证,确定最佳微分节点数和惩罚因子的取值;然后,采用有限软件(ABAQUS)验证了建立模型的准确性;最后根据已建立模型,研究波纹板结构参数对其振动特性的影响。结果表明,波纹板结构厚度和固支边界条件会增加其固有频率,提高波纹板结构的稳定性,使得共振频率向高频移动;增加波纹板结构长度、参与耦合圆弧壳半径和参与耦合圆弧壳弧度会降低其固有频率,使得共振频率向低频移动。     

Calculation of the effective stiffnesses of corrugated plates by solving the problem on the plate cross-section

It is shown that for corrugated, in particular, multilayer plates, the tree-dimensional cell problem of averaging can be reduced to the two-dimensional problem on the cross section of the periodicity cell of the plate. This significantly increases the accuracy of numerical calculation of the effective stiffnesses of corrugated plates. Numerical calculations of the stiffnesses of a plate with a sinusoidal corrugation are performed, and the results are compared with available data.     

Analysis of curved beams using a new differential transformation based curved beam element

Differential transformation method is used to obtain the shape functions for nodal variables of an arbitrarily non-uniform curved beam element including the effects of shear deformation considering axially functionally graded material. The proposed shape functions depend on the variations in cross-sectional area, moment of inertia, curvature and material properties along the axis of the curved beam element. The static and free vibration of axially functionally graded tapered curved beams including shear deformation and rotary inertia are studied through solving several examples. Numerical results are presented for circular, parabolic, catenary, elliptic and sinusoidal beams (both forms—prime and quadratic) with hinged-hinged, hinged-clamped and clamped-clamped and clamped-free end restraints. Three general taper types (depth taper, breadth taper and square taper) for rectangular cross section are studied. Out of plane vibration is studied and the lowest natural frequencies are calculated and compared with the published results. Out of plane buckling is investigated for circular beams due to radial load.     

Experimental full field investigations of resonant vibrations for piezoceramic plates by an optical interferometry method

The experimental measurement of resonant frequencies for piezoelectric material is generally performed by impedance analysis. In this paper we employ an optical interferometry method, called amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI), to investigate the vibration characteristics of piezoceramic plates. This method demonstrates its advantages of combining noise reduction, like the subtraction method, and high fringe sensitivity, like the time-averaged method. As compared with the film recording and optical reconstruction procedures used for holographic interferometry, the interferometric fringes of AF-ESPI are produced instantly by a video recording system. Based on the fact that clear fringe patterns measured by the AF-ESPI method will be shown only at resonant frequencies, both the resonant frequencies and corresponding mode shapes are obtained experimentally at the same time. Excellent quality for the interferometric fringe patterns of the mode shapes is demonstrated. We find from experimental results that the out-of-plane vibration modes (type A) with lower resonant frequencies cannot be measured by impedance analysis and only the in-plane vibration modes (type B) will be shown. However, both the out-of-plane (bending) and in-plane (extension) vibration modes of piezoceramic plates are obtained by the AF-ESPI method. Finally, numberical finite element calculations are also performed, and the results are compared with the experimental measurements. Excellent agreement for the resonant frequencies and mode shapes are obtained from both results.     

Dynamics of co-rotating vortices in a flow around a bio-inspired corrugated airfoil

Bio-inspired corrugated airfoils show favourable aerodynamic characteristics such as high coefficient of lift and delayed stall at low Reynolds numbers. Two-dimensional (2D) direct numerical simulation has been performed here on a corrugated airfoil at various angles of attack (0°, +5°, -5°) and Reynolds number of 280 to 6700. The objective is to analyse the pressure variation inside the corrugations and correlate it to the vortex movement across the corrugations and the overall aerodynamic characteristics of the corrugated airfoil. The flow characteristics have been examined based on the local Strouhal numbers in the corrugations of the airfoil. It is observed that the pressure variation in each corrugation is the result of vortex merging and separation in the corrugation which plays a major role in changing the flow characteristics. The Strouhal number of the flow is dictated by the most dominant local Strouhal number. The numerical results are further compared with experimental results obtained using particle image velocimetry, and the two set of results are found to match well. These results are significant because they elucidate the effect of corrugation, angle of attack, and Reynolds number on flow over a corrugated airfoil.     

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.