主动约束层阻尼梁有限元建模与动态特性研究

基于弹性、粘弹性和压电材料的本构关系,利用Hamilton原理,推导了主动约束层阻尼梁的有限元动力学模型．结合压电材料的机电耦合特性,采用自感电压的位移反馈,研究了主动约束层阻尼梁的闭环控制特性．求解了主动约束层阻尼简支梁的动态特性如固有频率、模态损耗因子及频率响应特性等．对被动控制、主动控制和主被动混合控制的控制效果进行了分析比较．研究了粘弹性层与约束层厚度等参数对减振控制效果的影响．     

Arbitrary active constrained layer damping treatments on beams: Finite element modelling and experimental validation

This paper concerns the analytical formulation and finite element modelling of arbitrary active constrained layer damping (ACLD) treatments applied to beams. A partial layerwise theory is utilized to define the displacement field of beams with an arbitrary number of elastic, viscoelastic and piezoelectric layers attached to both surfaces, and a fully coupled electro-mechanical theory is considered for modelling the behavior of the piezoelectric layers. The damping of the viscoelastic layers is modelled by the complex modulus approach. The weak forms of the analytical formulation, governing the motion and electric charge equilibrium, are presented. Based on the weak forms, a one-dimensional finite element (FE) model is developed, with the nodal mechanical degrees of freedom being the axial displacement, transverse displacement and the rotation of the mid-plane of the host beam and the rotations of the individual layers, and the electrical elemental degrees of freedom being the electrical potential difference of each piezoelectric layer. Frequency response functions were measured experimentally and evaluated numerically for a freely suspended aluminium beam with an ACLD patch. In order to validate the FE model the results are presented and discussed.     

The Galerkin element method applied to the vibration of rectangular damped sandwich plates

The Galerkin element method (GEM), which combines Galerkin orthogonal functions with the traditional finite element formulation, has previously been applied successfully to the vibration analysis of damped sandwich beams, and an improved iteration method was developed for its eigen solution. In the current paper, this promising method is extended to the vibration of damped sandwich plates. A quite different model is formulated which has both nodal coordinates and edge coordinates, while in the case of beams, there are only nodal coordinates. Displacement compatibility over the interfaces between the damping layer and the elastic layers is taken account of in order to ensure a conforming element and thereby guarantee good accuracy. The seed matrix method is proposed for simplifying the building of the element mass, stiffness and damping matrices. Numerical examples show that the application of the GEM to sandwich plate structures is computationally very efficient, while providing accurate estimates of natural frequencies and modal damping over a wide frequency range.     

Topology optimization of microstructures of viscoelastic damping materials for a prescribed shear modulus

Damping performance of a passive constrained layer damping (PCLD) structure mainly depends on the geometric layout and physical properties of the viscoelastic damping material. Properties such as the shear modulus of the damping material need to be tailored for improving the damping of the structures. This paper presents a topology optimization method for designing the microstructures in 2D, i.e., the structure of the periodic unit cell (PUC), of cellular viscoelastic materials with a prescribed shear modulus. The effective behavior of viscoelastic materials is derived through the use of a finite element based homogenization method. Only isotropic matrix material was considered and under such assumption it is found that the effective loss factor of viscoelastic material is independent of the geometrical configuration of the PUC. Based upon the idea of a Solid Isotropic Material with Penalization (SIMP) method of topology optimization, the relative material densities of the elements of the PUC are considered as the design variables. The topology optimization problem of viscoelastic cellular material with a prescribed property and with constraints on the isotropy and volume fraction is established. The optimization problem is solved using the sequential linear programming (SLP) method. Several examples of the design optimization of viscoelastic cellular materials are presented to demonstrate the validity of the method. The effectiveness of the design method is illustrated by comparing a solid and an optimized cellular viscoelastic material as applied to a cantilever beam with the passive constrained layer damping treatment.     

Damping of beams. Optimal distribution of cuts in the viscoelastic constrained layer

To damp the flexural vibrations of homogeneous beams or plates in a large frequency range, one of the most efficient methods is the use of constrained viscoelastic layers. Since most of the damping ability is due to shearing stresses in the viscoelastic layer, it is interesting to determine the most appropriate distribution of the shear in the layers. This paper presents the influence of a new parameter involved in this repartition, i.e. the distribution of cuts in the elastic constraining layer. It will be demonstrated that modal damping may be significantly modified in this way. The number and the locations of the cuts may vary and are determined to optimize the damping. The vibrating beam modal analysis is performed by a finite element analysis using special finite elements which have variable d.o.f. in order to take into account the lack of continuity of the viscoelastic constrained displacement field. Using a genetic algorithm, an optimal distribution of the cuts has been determined for a maximum damping of one or serval flexural modes.     

Minimum material cost design of five-layer sandwich beams

Welded aluminium box beams have very low vibration damping capacity. Regarding damping it is better to use a three-layer beam constructed from two rectangular hollow sections and a rubber layer glued between them. These three-layer beams have good damping capacity, but the dynamic deflection is large due to shear deformation of the rubber layer. To decrease this deflection two fiber-reinforced plastic layers are used. The minimum material cost design is worked out for such five-layer sandwich beams using the Rosenbrock Hillclimb mathematical programming method. Constraints on stress and local buckling are considered. A comparison is made between the optimized versions of a simple welded aluminium beam, three-layer and five-layer beams. It is shown that the deflection of the five-layer beams is smaller than that of the three-layer ones, but the cost of the five-layer beams is greater.     

Design of laminated composite plates for optimal dynamic characteristics using a constrained global optimization technique

The lamination arrangements of moderately thick laminated composite plates for optimal dynamic characteristics are studied via a constrained multi-start global optimization technique. In the optimization process, the dynamical analysis of laminated composite plates is accomplished by utilizing a shear deformable laminated composite finite element, in which the exact expressions for determining shear correction factors were adopted and the modal damping model constructed based on an energy concept. The optimal layups of laminated composite plates with maximum fundamental frequency or modal damping are then designed by maximizing the frequency or modal damping capacity of the plate via the multi-start global optimization technique. The effects of length-to-thickness ratio, aspect ratio and number of layer groups upon the optimum fiber orientations or layer group thicknesses are investigated by means of a number of examples of the design of symmetrically laminated composite plates.     

A finite element formulation for composite laminates with smart constrained layer damping

A composite laminate with active constrained-layer damping treatment is studied. The interface element for viscoelastic damping layers has been developed based on the relative displacements between composite plates and piezoelectric constraining layers. As an example, the problem of forced oscillations of the laminated composite structure with a smart constrained damping treatment is solved and the vibration response of the composite plate with smart damping layers is calculated using the presently developed procedure.     

Finite element free vibration analysis of damped stiffened panels

Free vibration characleristics of a damped stiffened panel with applied viscoelastic damping on the flanges of the stiffeners are studied using finite element method. The complex nature of the rotational and transverse stiffnesses of the stringers is taken into consideration while deriving the stiffness and mass matrices of the damped stiffener element. The finite element method consists of representing the panel by rectangular plate elements of 12 d.o.f. and the stiffeners by beam elements of 8 d.o.f. which allow for bending, torsional and warping effects. Numerical results showing the effect of the geometric and material properties of the damping layer treatment on the resonant frequencies and loss factors of the composite panel are presented.     

Generation and properties of banded viscous damping matrices

Numerical analysis of discrete vibratory systems subjected to broad band excitation usually requires specification of a viscous damping matrix. A method of determining the entries in such matrices directly is presented in this paper. The modal damping ratios, which can normally be estimated, are used to produce a fully populated damping matrix. To facilitate numerical integration of the equations of motion, a procedure termed stripping has been established for reducing the matrix to banded form. Additionally, using a procedure termed replacement, small element matrices were identified, extracted from the fully populated damping matrix and utilized (much as element mass and stiffness matrices are) to construct the global damping matrix. The effects of stripping or replacement were evaluated for a number of boundary conditions on plates, beams, and shells. Errors caused by variation of the number of system degrees of freedom, the number of diagonal rows remaining in the stripped matrix, and the magnitude and mo dal distribution of damping ratios were studied. Results were, in general, good. Damping ratios, damped natural frequencies, and eigenvectors were closely approximated by both type of systems.     

A new method for the vibration of thin-walled beams

This paper presents a new method for thin-walled beams with constrained torsional vibration. Based on the differential equation for torsional vibration, which includes the effect of cross-sectional warping, the shape functions are determined and, in turn, the frequency-dependent mass and stiffness matrices are derived. As an application of the new method, the dynamic finite element method, a numerical example is presented. The results show that this method gives more accurate results in the high-frequency range than those obtained by the static finite-element method.     

An experimental analysis of electrostatically vibrated array of polysilicon cantilevers

Micromechanical cantilevers are one of the most fundamental and widely studied structures in micro-electromechanical systems. Dynamic response of such cantilevers has long been an interesting subject to researchers and different analytical and experimental approaches have been reported to determine it. Theoretical estimation of different damping mechanisms have been reported over years which are relevant particularly in studying the dynamics of the micro-mechanical structures. Most properties and functionalities of the MEMS devices are invariably dependant on the dynamic response of the devices, which in turn depends on the quality factor of the devices or in other words the overall damping present in the system. This paper presents a thorough experimental analysis of vibration characteristics of micro-mechanical cantilevers of different dimensions. Arrays of polysilicon micro-cantilevers of different dimensions have been designed and fabricated using surface micromachining process. The beams are resonated by electrostatic actuation and their vibration characteristics have been observed using Laser Doppler Vibrometer. Also a thorough analysis of modal behaviour of the beams is presented using analytical approach and finite element method based simulation. Different damping mechanisms have been critically reviewed and a semi-analytical estimation of the overall damping is presented. The results are compared with experimental values.     

Estimation of modal damping ratio of impact-damped flexible beams using ANFIS

This paper presents the development of adaptive neuro-fuzzy inference systems (ANFIS) model for estimating the modal damping ratio of impact-damped flexible beams. The experimental results obtained from the literature are used as training and testing data for developing the model. Five of six parameters namely the gap, mass, modal amplitude, frequency, and peak value of the imaginary part of the frequency response functions were used as input, whereas the modal damping ratio of impact-damped flexible concrete beams was used as output parameter for the model. The Sugeno-type fuzzy rules were constituted with the Gaussian-type membership functions in the model. In the Sugeno-type fuzzy inference model, the modal damping ratio of impact-damped flexible concrete beams is estimated according to these input parameters. ANFIS model results for modal damping ratio are in good agreement with the experimental results than those obtained from formula presented in the literature.     

嵌入多层黏弹性胶膜复合材料阻尼工字梁的多目标设计优化

针对嵌入多层黏弹性胶膜的复合材料阻尼工字梁,传统的混合单元法在进行结构动态分析与设计优化时存在很大困难的问题,采用基于离散层理论的多层梁单元建模分析复合材料阻尼工字梁.通过对正交各向异性铺层和腹板进行等效处理的方法,对工字梁凸缘嵌入单层阻尼层模型进行参数化分析;分别对嵌入多层或单层阻尼层的模型建立多目标优化模型,优化目标为模态损耗因子和固有频率最大化,设计变量为阻尼层层数（厚度）及其嵌入位置;应用多目标遗传算法进行优化求解.结果表明：基于离散层理论的阻尼梁单元计算精度好且易于优化,对于嵌入单层较厚阻尼和嵌入多层较薄阻尼的复合材料工字梁,获得的阻尼效果与动刚度损失基本相当,但对于高阻尼的方案,前者比后者的动刚度损失更大.     

Forced harmonic response of viscoelastic structures by an asymptotic numerical method

This work presents an asymptotic numerical method for forced harmonic vibration analyses of viscoelastic structures. A mathematical formulation that may account for various viscoelastic models is presented. Power series expansions and Padé approximants of the displacement and frequency are developed and the finite element method is used for numerical solution. Only some matrix inversions and a few iterations are needed for large frequency ranges. Iterations of the process lead to a powerful continuation method for harmonic responses of viscoelastic structures with constant and frequency dependent coefficients. For numerical tests, undamped, viscoelastic and sandwich viscoelastic beams and plates are considered. Passive control, response curves and equivalent damping characteristics are obtained for various frequency ranges, excitation amplitudes and viscoelastic models.     

Optimal structural damping of skis using a genetic algorithm

Two problems of the optimal damping of skis with the use of an improved genetic algorithm (GA) are presented. The first problem is the optimal location of viscoelastic constrained layers; the second problem is the optimization of the stacking sequence with or without variable layer thicknesses. It is demonstrated that the GA is a very attractive and powerful method for these problems in which the objective function has no derivatives and several optimal solutions simultaneously exist.     

Damping optimization of viscoelastic laminated sandwich composite structures

Recent developments on the optimization of passive damping for vibration reduction in sandwich structures are presented in this paper, showing the importance of appropriate finite element models associated with gradient based optimizers for computationally efficient damping maximization programs. A new finite element model for anisotropic laminated plate structures with viscoelastic core and laminated anisotropic face layers has been formulated, using a mixed layerwise approach. The complex modulus approach is used for the viscoelastic material behavior, and the dynamic problem is solved in the frequency domain. Constrained optimization is conducted for the maximization of modal loss factors, using gradient based optimization associated with the developed model, and single and multiobjective optimization based on genetic algorithms using an alternative ABAQUS finite element model. The model has been applied successfully and comparative optimal design applications in sandwich structures are presented and discussed.     

A constrained assumed modes method for dynamics of a flexible planar serial robot with prismatic joints

In the present study, for the first time, flexible multibody dynamics for a three-link serial robot with two flexible links having active prismatic joints is presented using an approximate analytical method. Transverse vibrations of flexible links/beams with prismatic joints have complicated differential equations. This complexity is mostly due to axial motion of the links. In this study, first, vibration analysis of a flexible link sliding through an active prismatic joint having translational motion is considered. A rigid-body coordinate system is used, which aids in obtaining a new and rather simple form of the kinematic differential equation without the loss of generality. Next, the analysis is extended to include dynamic forces for a three-link planar serial robot called PPP (Prismatic, Prismatic, Prismatic), in which all joints are prismatic and active. The robot has a rigid first link but flexible second and third links. To model the prismatic joint, time-variant constraints are written, and a motion equation in a form of virtual displacement and virtual work of forces/moments is obtained. Finally, an approximate analytical method called the “constrained assumed modes method” is presented for solving the motion equations. For a numerical case study, approximate analytical results are compared with finite element results, which show that the two solutions closely follow each other.     

An improved dynamic relaxation method for the analysis of plate bending problems

An improved dynamic relaxation method is used to solve the set of simultaneous equations resulting from the application of finite element method for plate bending problems. Different weights are used as multiplying factors for each mass (m) and damping factor (C) in each equation representing one of the three degrees of freedom at each node. The optimum values of these weights are obtained for different cases of the bending of cantilever plates stiffened with edge beams with different sizes of stiffening edge beams.