Nonlinear saturation controller for vibration supersession of a nonlinear composite beam

In this paper, a study for nonlinear saturation controller (NSC) is presented that used to suppress the vibration amplitude of a structural dynamic model simulating nonlinear composite beam at simultaneous sub-harmonic and internal resonance excitation. The absorber exploits the saturation phenomenon that is known to occur in dynamical systems with quadratic non-linearities of the feedback gain and a two-to-one internal resonance. The analytical solution for the system and the nonlinear saturation controller are obtained using method of multiple time scales perturbation up to the second order approximation. All possible resonance cases were extracted at this approximation order and studied numerically. The stability of the system at the worst resonance case (Ω = 2ω _s and ω _s = 2ω _c ) is investigated using both frequency response equations and phase-plane trajectories. The effects of different parameters on the system and the controller are studied numerically. The effect of some types of controller on the system is investigated numerically. The simulation results are achieved using Matlab and Maple programs.     

Dynamic analysis of bending–torsion coupled composite beams using the Flexibility Influence Function Method

In this work the dynamic behaviour of symmetrical laminated beams was studied, taking into account the effect of bending–torsion coupling by a one-dimensional model. This model includes the influence of the shear force and rotatory inertia. To solve the equations of motion, the Flexibility Influence Function Method (FIFM) was used. The dynamic displacements (deflection, bending rotation, and torsional rotation) were calculated for a beam in which the deflection and torsional rotation were restricted at its both ends, allowing the bending rotation. The accuracy of this method was determined by using a Three-Dimensional Finite Element Method (FEM3D) model to compare the dynamic displacements. The need was shown to incorporate coupling in the one-dimensional model in order to calculate the dynamic deflection and bending rotation of a composite beam.     

Dynamic analysis of composite beam subjected to harmonic moving load based on the third-order shear deformation theory

The response of an infinite Timoshenko beam subjected to a harmonic moving load based on the third-order shear deformation theory (TSDT) is studied. The beam is made of laminated composite, and located on a Pasternak viscoelastic foundation. By using the principle of total minimum potential energy, the governing partial differential equations of motion are obtained. The solution is directed to compute the deflection and bending moment distribution along the length of the beam. Also, the effects of two types of composite materials, stiffness and shear layer viscosity coefficients of foundation, velocity and frequency of the moving load over the beam response are studied. In order to demonstrate the accuracy of the present method, the results TSDT are compared with the previously obtained results based on first-order shear deformation theory, with which good agreements are observed.     

A simplified analysis method for composite beams with interlayer slip

A simplified static procedure is proposed for analysing and designing composite beams with interlayer slip. The method is parallel to the Eurocode 5 method, but it is general in nature and can be applied to arbitrary boundary and loading conditions. In contrast with Eurocode 5, a general and correct way of choosing the effective beam length of the problem is given by the present procedure, which is that the effective beam length equals the buckling length that is found in the corresponding column buckling problem. The procedure predicts the deflections and internal actions and stresses, in principle by replacing the fully composite bending stiffness (EI_∞) with the effective (partially) composite bending stiffness (EI_eff) in the expressions for these quantities in the corresponding fully composite beam. This effective bending stiffness depends on two non-dimensional parameters: the composite action parameter (shear connection stiffness) and the relative bending stiffness parameter. The method is applied to a number of simple practical cases and the results obtained have been compared with the exact values. The applicability of the simplified analysis procedure was found to be very good, except for interlayer shear stresses. The error in the Eurocode 5 procedure, as compared with the method proposed in this paper, can in some cases be up to almost 30% depending on the boundary conditions.     

结构密集模态非线性耦合作用引起的线性模型失效问题研究

通过对两自由度密频近线性系统主共振响应的分析,研究了结构密集模态弹性非线性耦合作用引起的一些典型的非线性动力学现象,如幅频曲线弯曲Ｈｏｐｆ分叉导致的概周期响应。研究结果表明,存在密频内共振时,线性模型失效,必须采用非线性模型来描述结构的动力学响应。     

黏弹性梁弯曲振动的复模态分析

发展复模态分析研究黏弹性梁的弯曲振动。将梁的控制方程写作状态变量的形式，然后利用复模态的正交性可解耦为无穷多个彼此独立的常微分方程组。基于固有频率和模态函数，可以得到黏弹性梁对于任意初始条件和外激励的响应。在固支梁的边界条件下确定黏弹性梁的固有频率、衰减系数和模态函数，并计算梁受两种典型的外激励时的响应。     

复杂载荷变刚度静不定梁通用力学模型及变形方程

对复杂载荷下变刚度静不定梁建立了一种通用力学模型，得出梁任一截面的变矩方程。通过对变矩方程的积分和积分后方程的递推，推导出梁任意截面的转角和挠度变形的一般方程，并给出工程计算实例。该方程易于采用计算机处理，为复杂载荷下变刚度静不定梁程序化求解提供基础。     

Exact static analysis of partially composite beams and beam-columns

The ordinary differential equations and general solutions for the deflection and internal actions and, especially, the pertaining consistent boundary conditions for partially composite Euler–Bernoulli beams and beam-columns are presented. Static loading conditions, including transverse and axial loading and first- and second-order analyses are considered. The theoretical procedure is applicable to general loading and boundary conditions for uniform composite beams and beam-columns with interlayer slip. Further, the exact closed form characteristic equations and their associated exact buckling length coefficients for composite columns with interlayer slip are derived for the four Euler boundary conditions. It is shown that these coefficients are the same as those for ordinary fully composite (solid) columns, except for the Euler clamped-pinned case. For the clamped-pinned case, the difference between the exact buckling length coefficient and the corresponding value for solid columns is less than 1.8% depending on the so-called composite action parameter and relative bending stiffness parameter. Correspondingly, the maximum deviation between the exact and approximate buckling load is at most 2.5%. These small differences can in most practical cases be neglected. Also, the maximum theoretical range for the relative bending stiffness for partially composite beams and beam-columns is derived. An effective bending stiffness, valuable in the determination of the critical buckling load for partially composite members, is derived. This effective bending stiffness is also suitable for analysing approximate deflections and internal actions or stresses in composite beams with flexible shear connection. The beam-column analysis is applied to a specific case. The difference in the approaches to the first- and second-order analysis is illustrated and the results clearly show the magnification in the actions and displacements due to the second-order effect. The magnification of the internal axial forces is different from magnifications obtained for the other internal actions, since only that portion of an internal axial force that is induced by bending is magnified by the second-order effect.     

TUNING OF NORMAL MODES BY MULTI-AXIAL BASE EXCITATION

Normal mode testing is commonly performed on large aerospace structures in order to identify their eigenfrequencies, mode shapes, modal damping values, and generalized masses. Normally, multiple-point sinusoidal excitation is applied. For each normal mode, the exciter configuration is adapted in location and magnitude in order to compensate for the internal damping forces of the test structure. An alternative to multiple-point excitation is a driven-base test using a multi-axial shaker table. Multi-axis shaker facilities have been developed for structural dynamic qualification. However, they can be used for normal mode testing as well, whereby the overall structure is accelerated in space in order to compensate for the internal damping forces. This requires a suitable combination of all six spatial dof of the base excitation. In the case of resonance, the structure responds in quadrature related to the base excitation. If this criterion is fulfilled, the structure vibrates in an eigenmode of vibration in the sense of a sdof system. This article gives the theoretical background of the test method with particular emphasis on the damping estimation, on the determination of the generalized mass, and on finding a suitable base axis combination. A suitable test rig for the normal mode testing by base excitation, DLR's Multi-Axis Vibration Simulator (MAVIS) in Göttingen (FRG), is presented. The test procedure and test data evaluation as applied to a test structure characterized by closely spaced eigenfrequencies is presented. The advantages and disadvantages of normal mode testing by means of base excitation are pointed out.     

平方非线性对屈曲薄板振动性能影响的研究

分析研究参数激励简支矩形薄板非线性振动中的超谐振动。依据Karman方程的动态比拟，运用Galeerkin法将控制薄板振动的微微分方程转化参数激励Duffing型方程。针对该方程进行的变换表明，屈曲薄板振动系统为带有平方和立方非线性的参数激励非线性动力系统。应用摄动分析研究系统中平方非线性因素对系统的调节机理以及平方非线性导致的2倍超谐振动。分析结果表明，由于平方非线性对系统的调节作用，在一定的参数域响应中自由振动项的幅值不会因阻尼的存在而衰减，自由振动以激励频率2倍的频率参与系统的响应。基于理论分析的试验研究证明，所讨论的参数激励屈曲薄板振动系统在一定的参数条件下将出现2倍超谐振动。由2倍超谐振动参与的系统的振动状态是稳定的周期运动。     

Modélisation de nœuds mixtes acier–béton soumis à des sollicitations sismiques

The paper deals with the three-dimensional finite element modelling of typical beam-to-column intersection zone of composite frames under earthquake horizontal loading. The modelling takes into account the non-linear material properties of the different components: column, longitudinal beam, transverse beam, reinforced concrete slab and shear connectors. The studied element is the longitudinal composite beam, which is chosen to dissipate the earthquake energy. The paper presents the basic modelling, its calibration on a subassembly tested experimentally and a part of the parametrical study performed with the model. The important results obtained for the behaviour of the beam subjected to sagging bending in an exterior node configuration and for the behaviour of the beam subjected to hogging bending at one side of the node and sagging bending at the other side in an interior node configuration are presented.     

Active robust vibration control of flexible composite beams with parameter perturbations

In this paper, a time domain approach is presented to treat the problem of active controlling simultaneously the bending and torsional vibration of flexible composite beams under both mode truncations and parameter perturbations. In the proposed approach, the residual model, which is known as unstructured uncertainty, is viewed as an additive perturbation to the controlled model. Based on a state space model, which incorporates both bending and torsional deformation effects, of the flexible composite beam with piezoelectric sensors and actuators, a robust stability condition is derived to guarantee that both bending and torsional vibration of the flexible composite beam, which is subject to both mode truncation and linear time-varying parameter perturbations, can be actively controlled by an observer-based controller. The proposed robust stability condition gives an insight into the relationship between the stability margins of the controlled and residual mode subsystems, spillover effects and additive time-varying parameter perturbations. Finally, an active robust vibration control problem of a cantilevered flexible composite beam with piezoelectric sensors and actuators is provided for illustration.     

Exact dynamic analysis of composite beams with partial interaction

The partial differential equations and general solutions for the deflection and internal actions and the pertaining consistent boundary conditions are presented for composite Euler-Bernoulli members with interlayer slip subjected to general dynamic loading. Both free and forced vibrations are treated. The solutions are shown to be unique and complete under certain conditions, and valid for all so-called restricted admissible boundary conditions. Specifically, the exact eigenmode length coefficients are derived for the four Euler BC. They differ from those valid for ordinary, fully composite (solid) beams, except for the pinned-pinned case. The maximum deviation for beams with the other three Euler BC is shown to be less than 2-6% with respect to the eigenmode length coefficient and 3-10% with respect to the eigenfrequency, respectively, depending on the two non-dimensional parameters, composite action or shear connector stiffness and relative bending stiffness parameters. However, these deviations occur in a rather narrow range of the determining parameters, so for most practical cases the eigenmode length coefficients given for solid (fully composite) beams can approximately be used also for partially composite beams. The procedures of analysing beam vibrations are applied to a specific case. These solutions illustrate the effect of interlayer connection on the peak velocity of the beam vibrations. The proposed analytical theory is verified by tests and finite element calculations.     

复合材料夹层梁弯曲分析的状态空间法

给出了复合材料夹层梁弯曲刚度和剪切刚度的表达式。导出了其弯曲的状态空间方程，并求出了它的通解，最后，给出了三个例子。本法求解问题简单，明显地优于以往的方法     

A bending theory for beams with vertical edge crack

In this paper a linear continuous theory for bending analysis of beams with an edge crack perpendicular to the neutral plane subject to bending has been developed. The model assumes that the displacement field is a superposition of the classical Euler-Bernoulli beam's displacement and of a displacement due to the crack. It is assumed that in bending the additional displacement due to crack decreases exponentially with distance from the crack tip. The strain and stress fields have been calculated using this displacement field and the bending equation has been obtained using equilibrium equations. Using a fracture mechanics approach the exponential decay rate has been calculated. There is a good agreement between the analytical results from solving the differential equation of cracked beam and those obtained by finite element method.     

微型压电单晶俘能器研究

建立了一曲梁压电单晶MEMS压电俘能器的分析模型.利用曲梁理论建立该矩形截面梁的动力学方程.采用ANSYS软件对该曲梁的结构参数进行了初步的分析.分析结果表明压电曲梁的第一阶谐振频率与其结构尺寸之间存在较复杂的关系.该仿真结果可为压电俘能器的设计提供参考.     

车体垂弯振型节点位置对其弹性振动的影响

根据车体的结构属性和质量分布将其考虑为多段变截面欧拉梁,建立包含车体一阶垂弯模态的车辆垂向动力学模型,研究车体一阶垂弯振型的节点位置对高速列车振动舒适度的影响,提出改善车体弹性振动的措施。基于变截面欧拉梁模型分析车体各截面的质量和抗弯刚度分布对模态振型的影响,发现不同截面之间的抗弯刚度和质量分布对整体模态振型影响显著,提高车体中部结构的抗弯刚度并减小其质量,可以增大节点间距和提高模态频率,而传统均直等截面梁模型则不能准确描述振型的幅值和节点位置。采用频域分析方法计算车辆在轨道随机激励下的振动响应,将车体垂弯振型节点调整到转向架二系上方附近时,车体的弹性振动水平显著降低,在车速为300km/h时车辆舒适度指标可降低50%。     

Dynamic finite element method for generally laminated composite beams

A dynamic finite element method for free vibration analysis of generally laminated composite beams is introduced on the basis of first-order shear deformation theory. The influences of Poisson effect, couplings among extensional, bending and torsional deformations, shear deformation and rotary inertia are incorporated in the formulation. The dynamic stiffness matrix is formulated based on the exact solutions of the differential equations of motion governing the free vibration of generally laminated composite beam. The effects of Poisson effect, material anisotropy, slender ratio, shear deformation and boundary condition on the natural frequencies of the composite beams are studied in detail by particular carefully selected examples. The numerical results of natural frequencies and mode shapes are presented and, whenever possible, compared to those previously published solutions in order to demonstrate the correctness and accuracy of the present method.     

Large deformations of thin curved plane beam of constant initial curvature

A theory for the large deformations of a thin curved plane beam of constant initial curvature is presented, based on the hypothesis that only the longitudinal component of the strain tensor exists in the beam. In this case, five of the six compatibility equations are identically satisfied, while the remaining one requires the axial strain to vary parabolically over the cross-section of the curved beam. The non-linear strain-displacement relations are then solved for two in-plane displacement components in terms of two arbitrary functions of the longitudinal co-ordinate, angle . It is shown that, as a consequence of our hypothesis, the displacements vary linearly over the cross-sections of the beam. The obtained functional form of the displacement components leads to equilibrium equations of a beam on a deformed configuration expressed in terms of deformation functions. For the special case of a linearly elastic material and circular cantilever beam subjected to conservative and non-conservative point loads these equations have been numerically integrated. A number of numerical examples, including the bending of a C-shaped spring, are presented.     

Saturation-based active absorber for a non-linear plant to a principal external excitation

The use of active non-linear absorber to control the high-amplitude vibration of the non-linear plant subjected to primary external excitation is investigated. The absorber exploits the saturation phenomenon that is known to occur in dynamical systems with quadratic non-linearities and a two-to-one internal resonance. The dynamic behaviour of the plant is described by an oscillator, which includes velocity-dependent damping forces, polynomial and differential-polynomial non-linearities. First, the approximate solutions to the governing equations are obtained by using the method of multiple scales. Then a bifurcation analysis is conducted to examine the stability of the system and the performance of the control strategy is studied. A parametric investigation is carried out to see the effects of changing the damping ratio of the absorber, the value of the feedback gain and the detuning frequency of the absorber on the responses of the plant and absorber. The instantaneous power of the control law is also calculated. Finally, the numerical simulations are performed to demonstrate and validate the saturation control method.