Vibration analysis of a beam with PCLD Patch

An analytical approach for vibration response analysis of a beam with single passive constrained layer damping (PCLD) patch is presented. The governing equation of motion of the beam is firstly derived on the basis of an energy approach and the Lagrange equation. The noval contribution is that a third admissible function is introduced to represent the longitudinal displacements of the constraining layer in the PCLD patch when the assumed-modes method is applied for discretizing the governing equation. In conventional analytical approaches, only two admissible functions are used together with a longitudinal static equilibrium equation of a section of base beam or constraining layer. Comparison of the computational results from the proposed analytical approach and the conventional analytical approach as well as a commercial FEM code reveals that the proposed analytical approach can describe the vibration responses of the damped beam more accurately for commonly used viscoelastic material (VEM) layer in the PCLD patch while the conventional analytical approach, in general, overestimates the damping effects of the PCLD patch. The advantages and disadvantages of the proposed analytical approach and conventional analytical approach are discussed through some case studies.     

Enhanced ACLD treatment using stand-off-layer: FEM based design and experimental vibration analysis

The main disadvantage of active constrained layer damping treatment is the reduced transmissibility of active forces. This problem can be solved up to certain extent by using edge anchors. These edge anchors or stiffeners increase the transmissivity of forces only at very high feedback gains but decrease the effectiveness of passive constrained layer damping (PCLD) treatment. The efficiency of the passive constrained layer damping treatment can be improved drastically by adding the stand-off-layer (SOL) between the viscoelastic layer and the base beam. This technique has additional advantages as well. Firstly, it increases the viscoelastic strain so that more energy is dissipated via viscoelastic layer. Secondly, it enhances the effect of active forces and moments even without using edge anchors because the shear modulus of the SOL is in the range 10⁸–10⁹ N/m². Hamilton’s principle in conjunction of finite element method is used to derive the equations of motion. The complex eigenvalue is developed and solved numerically by using simple proportional feedback control strategy. Results are compared with ordinary active constrained layer damping (ACLD) treatment in order to highlight the effectiveness of the proposed technique. Validity of the proposed treatment has also been verified experimentally.     

FINITE ELEMENT FORMULATION AND ACTIVE VIBRATION CONTROL STUDY ON BEAMS USING SMART CONSTRAINED LAYER DAMPING (SCLD) TREATMENT

This work deals with the active vibration control of beams with smart constrained layer damping (SCLD) treatment. SCLD design consists of viscoelastic shear layer sandwiched between two layers of piezoelectric sensors and actuator. This composite SCLD when bonded to a vibrating structure acts as a smart treatment. The sensor piezoelectric layer measures the vibration response of the structure and a feedback controller is provided which regulates the axial deformation of the piezoelectric actuator (constraining layer), thereby providing adjustable and significant damping in the structure. The damping offered by SCLD treatment has two components, active action and passive action. The active action is transmitted from the piezoelectric actuator to the host structure through the viscoelastic layer. The passive action is through the shear deformation in the viscoelastic layer. The active action apart from providing direct active control also adjusts the passive action by regulating the shear deformation in the structure. The passive damping component of this design eliminates spillover, reduces power consumption, improves robustness and reliability of the system, and reduces vibration response at high-frequency ranges where active damping is difficult to implement. A beam finite element model has been developed based on Timoshenko's beam theory with partially covered SCLD. The Golla-Hughes-McTavish (GHM) method has been used to model the viscoelastic layer. The dissipation co-ordinates, defined using GHM approach, describe the frequency-dependent viscoelastic material properties. Models of PCLD and purely active systems could be obtained as a special case of SCLD. Using linear quadratic regulator (LQR) optimal control, the effects of the SCLD on vibration suppression performance and control effort requirements are investigated. The effects of the viscoelastic layer thickness and material properties on the vibration control performance are investigated.     

Complex power distribution analysis in plates covered with passive constrained layer damping patches

The vibration of a plate partially covered with a passive constrained layer damping (PCLD) patch is studied from an energetic point of view. The damped plate is excited by an acoustic plane wave. The study is done with a numerical two-dimensional multilayer plate model. Results of the present model are compared to those obtained with three-dimensional finite element models. It is shown that the present model gives accurate results, even for the layer's inner behavior. It is less expansive in terms of computational cost; hence, it can simulate efficiently the structure for higher frequencies. Mathematical formulas for complex mechanical power are presented, and the link with strain and kinetic energies and dissipated power is detailed. Both local and global complex power balance are established, and corresponding expressions for the discretized problem are formulated. Conservative and dissipative powers are studied for the PCLD damped plate. After a global balance analysis versus frequency, a local study has been carried out in order to quantify the relative contribution of the components of strain and stress tensors to the total strain energy and dissipated power; the individual layer's contributions is also investigated. The in-plane distributions of powers are mapped, showing the location where dissipative phenomenon occurs and where strain energy is stored. Finally, three criteria based on the previous power quantities are proposed in order to quantify the mechanical damping efficiency of the patch.     

Vibration and dynamic stability of a traveling sandwich beam

The vibration and dynamic stability of a traveling sandwich beam are studied using the finite element method. The damping layer is assumed to be linear viscoelastic and almost incompressible. The extensional and shear moduli of the viscoelastic material are characterized by complex quantities. Complex-eigenvalue problems are solved by the state-space method, and the natural frequencies and modal loss factors of the composite beam are extracted. The effects of stiffness and thickness ratio of the viscoelastic and constrained layers on natural frequencies and modal loss factors are reported. Tension fluctuations are the dominant source of excitation in a traveling sandwich material, and the regions of dynamic instability are determined by modified Bolotin's method. Numerical results show that the constrained damping layer stabilizes the traveling sandwich beam.     

Damping of stiffened plates by multiple layer treatments

It is shown in this paper that the modal damping and resonant frequencies of a stiffened plate structure, with a multiple layer constrained damping treatment attached to the surface, can be predicted from a knowledge of the equivalent complex modulus properties of the treatment. The equations used represent a simple extension of the classical equations of Oberst for a free layer treatment applied to an unstiffened beam or plate, with terms accounting for the effect of the stiffeners. The equivalent complex modulus properties of the treatment depend on a shear parameter, a geometrical parameter, the stiffness of the constraining layer and the loss factor of the adhesive. Experimental results are discussed.     

Effect of ACLD treatment configuration on damping performance of a flexible beam

A clamped–free beam with partial active constrained layer damping (ACLD) treatment is modelled by using the finite element method. The Golla–Hughes–McTavish (GEM) method is employed to account for the frequency-dependent characteristic of the viscoelastic material (VEM). As the resultant finite element model contains too many degrees of freedom due to the introduction of dissipative coordinates, a model reduction is performed to bring the system back to its original size. Finally, optimal output feedback gains are designed based on the reduced models. Numerical simulations are performed to study the effect of different ACLD treatment configurations, with various element numbers, spacing and locations, on the damping performance of a flexible beam. Results are presented for damping ratios of the first two vibration modes. It is found that to enhance the second mode damping, without deteriorating the first mode damping, splitting a single ACLD element into two and placing them at appropriate positions of the beam could be a possible solution.     

TRANSVERSE VIBRATION OF ELASTIC-VISCOELASTIC-ELASTIC SANDWICH BEAMS: COMPRESSION-EXPERIMENTAL AND ANALYTICAL STUDY

Experimental and analytical results are presented from an investigation into the compressional vibration of an elastic-viscoelastic-elastic three-layer sandwich beam. Most analytical models make the fundamental assumption that shear deformation in the viscoelastic core yields the largest damping and compressional deformation is negligible. Experimental results from a cantilever beam with a constrained layer viscoelastic damping treatment driven with a sinusoidal input are given which show compressional deformation over a relatively wide driving frequency range. A new analytical model for compressional damping is presented and compared with experimental results, with the Mead and Markus shear damping model, and with the Douglas and Yang compressional damping model. These results indicate that the proposed compressional model is a better predictor of resonance frequencies for the cantilever beams tested and that all models show deficiencies in predicting damping     

上海光源储存环主支撑减振研究

 上海光源是一台在建的第三代同步辐射光源,对束流轨道稳定性要求很高。由磁铁和支架组成的支撑组件的机械稳定性是影响束流轨道稳定性的重要因素。对主支撑组件样机的测试结果表明,其最低共振频率处放大倍数为50左右,超过要求5倍。因此,需要研究相应的减振措施。利用阻尼减振原理设计了一种约束阻尼结构。在样机上的测试结果表明,安装该装置后,支撑组件的共振放大倍数最大可以降低91.8%,对应的功率谱密度的峰值可以降低25 dB。因此,该装置可以用来增加支撑的稳定性。     

Minimization of sound radiation from baffled beams through optimization of partial constrained layer damping treatment

An optimization study is presented with aim to minimize the sound power radiated by a simply supported, baffled beam with constrained layer damping (CLD) treatment. The governing equation of motion for the calculation of time-harmonic response of a partially CLD covered beam is derived first on the basis of energy approach. Assumed-modes method is used to solve the equation with obtained frequency response functions at different beam locations, which are further used for the calculation of its radiated sound power into half free-space by using Rayleigh’s integral. The optimization problem is then formulated to minimize the sound power radiated by the beam over a frequency range of interest covering multiple resonant modes. A genetic algorithm-based penalty function method is employed to search for the optimum of location/length of the CLD patch and the shear modulus of viscoelastic layer. Optimal results show that for a simply supported beam with a transverse force applied at its central location, it is not necessary to fully cover the structure using CLD patch in order to achieve the largest reduction in the sound power radiated by the beam over a frequency range. With inclusion of the amount of damping material to be minimized, the optimal CLD coverage length is only one-fourth of the base beam’s. Moreover, the optima of three design variables, the CLD coverage length, location on the beam and the shear modulus of viscoelastic layer, are highly relevant to each other.     

Dynamic characterization of high damping viscoelastic materials from vibration test data

The numerical analysis and design of structural systems involving viscoelastic damping materials require knowledge of material properties and proper mathematical models. A new inverse method for the dynamic characterization of high damping and strong frequency-dependent viscoelastic materials from vibration test data measured by forced vibration tests with resonance is presented. Classical material parameter extraction methods are reviewed; their accuracy for characterizing high damping materials is discussed; and the bases of the new analysis method are detailed. The proposed inverse method minimizes the residue between the experimental and theoretical dynamic response at certain discrete frequencies selected by the user in order to identify the parameters of the material constitutive model. Thus, the material properties are identified in the whole bandwidth under study and not just at resonances. Moreover, the use of control frequencies makes the method insensitive to experimental noise and the efficiency is notably enhanced. Therefore, the number of tests required is drastically reduced and the overall process is carried out faster and more accurately. The effectiveness of the proposed method is demonstrated with the characterization of a CLD (constrained layer damping) cantilever beam. First, the elastic properties of the constraining layers are identified from the dynamic response of a metallic cantilever beam. Then, the viscoelastic properties of the core, represented by a four-parameter fractional derivative model, are identified from the dynamic response of a CLD cantilever beam.     

Vibration and damping analysis of annular plates with constrained damping layer treatments

The natural frequencies and modal loss factors of annular plates with fully and partially constrained damping treatments are considered. The equations of free vibration of the plate including the transverse shear effects are derived by a discrete layer annular finite element method. The extensional and shear moduli of the viscoelastic material layer are described by the complex quantities. Complex eigenvalues are then found numerically, and from these, both frequencies and loss factors are extracted. The effects of viscoelastic layer stiffness and thickness, constraining layer stiffness and thickness, and treatment size on natural frequencies and modal loss factors are presented. Numerical results also show that the longer constrained damping treatment in radial length does not always provide better damping than the shorter ones.     

MODELLING AND VIBRATION CONTROL OF BEAMS WITH PARTIALLY DEBONDED ACTIVE CONSTRAINED LAYER DAMPING PATCH

A detailed model for the beams with partially debonded active constraining damping (ACLD) treatment is presented. In this model, the transverse displacement of the constraining layer is considered to be non-identical to that of the host structure. In the perfect bonding region, the viscoelastic core is modelled to carry both peel and shear stresses, while in the debonding area, it is assumed that no peel and shear stresses be transferred between the host beam and the constraining layer. The adhesive layer between the piezoelectric sensor and the host beam is also considered in this model. In active control, the positive position feedback control is employed to control the first mode of the beam. Based on this model, the incompatibility of the transverse displacements of the active constraining layer and the host beam is investigated. The passive and active damping behaviors of the ACLD patch with different thicknesses, locations and lengths are examined. Moreover, the effects of debonding of the damping layer on both passive and active control are examined via a simulation example. The results show that the incompatibility of the transverse displacements is remarkable in the regions near the ends of the ACLD patch especially for the high order vibration modes. It is found that a thinner damping layer may lead to larger shear strain and consequently results in a larger passive and active damping. In addition to the thickness of the damping layer, its length and location are also key factors to the hybrid control. The numerical results unveil that edge debonding can lead to a reduction of both passive and active damping, and the hybrid damping may be more sensitive to the debonding of the damping layer than the passive damping.     

Free vibration of thin walled open section beams with constrained damping treatment

Free vibration characteristics of a thin walled, open cross-section beam, with constrained damping layers at the flanges, are investigated. Both uncoupled transverse vibration and the coupled bending-torsion oscillations, of a beam of a top-hat section, are considered. Numerical results are presented for natural frequencies and modal loss factors in the first two modes of simply supported and clamped-clamped beams. For the uncoupled mode the constrained damping treatment is more effective than an unconstrained one, but for the coupled mode the effect is just the opposite.     

Vibration characteristics of a rotating flexible arm with ACLD treatment

In this paper, the vibration behavior and control of a clamped–free rotating flexible cantilever arm with fully covered active constrained layer damping (ACLD) treatment are investigated. The arm is rotating in a horizontal plane in which the gravitational effect and rotary inertia are neglected. The stress–strain relationship for the viscoelastic material (VEM) is described by a complex shear modulus while the shear deformations in the two piezoelectric layers are neglected. Hamilton's principle in conjunction with finite element method (FEM) is used to derive the non-linear coupled differential equations of motion and the associated boundary conditions that describe the rigid hub angle rotation, the arm transverse displacement and the axial deformations of the three-layer composite. This refined model takes into account the effects of centrifugal stiffening due to the rotation of the beam and the potential energies of the VEM due to extension and bending. Active controllers are designed with PD for the piezosensor and actuator. The vibration frequencies and damping factors of the closed-loop beam/ACLD system are obtained after solving the characteristic complex eigenvalue problem numerically. The effects of different rotating speed, thickness ratio and loss factor of the VEM as well as different controller gain on the damped frequency and damping ratio are presented. The results of this study will be useful in the design of adaptive and smart structures for vibration suppression and control in rotating structures such as rotorcraft blades or robotic arms.     

Dynamic stability of a rotating beam with a constrained damping layer

The dynamic behavior and dynamic instability of the rotating sandwich beam with a constrained damping layer subjected to axial periodic loads are studied by the finite element method. The influences of rotating speed, thickness ratio, setting angle and hub radius ratio on the resonant frequencies and modal system loss factors are presented. The regions of instability for simple and combination resonant frequencies are determined from the Mathieu equation that is obtained from the parametric excitation of the rotating sandwich beam. The regions of dynamic instability for various parameters are presented.     

Active structural acoustics control of beams using active constrained layer damping through loss factor maximization

Vibration and acoustic control of beams with classical boundary conditions using active constrained layer damping is presented. The control input that maximizes the loss factor of the active constrained layer damping is determined through taking the first variation of the loss factor with respect to the control input. Although the loss factor is a positive definite quantity, the first variation yields control input that maximizes the factor. The resulting control input significantly reduces the vibration and acoustic response of the beams at their resonant frequencies.     

Constrained layer damping with vitreous enamel

The concept of constrained layer damping with vitreous enamel has been experimentally evaluated. The constraining layer markedly broadens the free layer damping peak. The broadening has been explained on the basis of two simultaneous energy dissipation mechanisms and is related to the vitreous enamel's loss factor and viscosity.     

Damping characteristics of a spray-deposited shape memory alloy beam

The damping characteristics of an Ni–Ti shape memory alloy (SMA) beam are theoretically and experimentally studied with interest in identifying an appropriate damping model for the material. The SMA beam is manufactured by a spray deposition method followed by heat treatment and found to have nanocrystalline structure in which damping capacity is high. The beam is then tested to obtain an impulse response and the frequency response function (FRF). By using the Hilbert transform technique it is shown that damping of the beam is almost amplitude independent in the tested range of displacement. It is also shown from the FRF that the damping of the spray-deposited shape memory alloy beam is well represented by a model including both linear viscous and hysteretic dampings.     

Physical origin of large positive and negative lateral optical beam shifts in prism-waveguide coupling system

Large lateral beam shift in prism-waveguide coupling system is theoretically analyzed from the viewpoint of interference between multiple reflected beam constituents. It is shown that the reflected beam is a result of interference between two beams: the beam directly reflected from the prism and the total leaky beam coming from guided mode. The thickness of coupling layer determines the amplitude of the total leaky beam, and further determines the sign (positive or negative) of the reflected beam shift. Because of interference between two beams, intrinsic damping itself plays an important role in deciding the distortion of the reflected beam.