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.     

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.     

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     

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.     

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.     

Acoustic radiation from the submerged circular cylindrical shell treated with active constrained layer damping

Based on the transfer matrix method of exploring the circular cylindrical shell treated with active constrained layer damping(i.e., ACLD), combined with the analytical solution of the Helmholtz equation for a point source, a multi-point multipole virtual source simulation method is for the first time proposed for solving the acoustic radiation problem of a submerged ACLD shell. This approach, wherein some virtual point sources are assumed to be evenly distributed on the axial line of the cylindrical shell, and the sound pressure could be written in the form of the sum of the wave functions series with the undetermined coefficients, is demonstrated to be accurate to achieve the radiation acoustic pressure of the pulsating and oscillating spheres respectively. Meanwhile, this approach is proved to be accurate to obtain the radiation acoustic pressure for a stiffened cylindrical shell. Then, the chosen number of the virtual distributed point sources and truncated number of the wave functions series are discussed to achieve the approximate radiation acoustic pressure of an ACLD cylindrical shell. Applying this method, different radiation acoustic pressures of a submerged ACLD cylindrical shell with different boundary conditions, different thickness values of viscoelastic and piezoelectric layer, different feedback gains for the piezoelectric layer and coverage of ACLD are discussed in detail. Results show that a thicker thickness and larger velocity gain for the piezoelectric layer and larger coverage of the ACLD layer can obtain a better damping effect for the whole structure in general. Whereas, laying a thicker viscoelastic layer is not always a better treatment to achieve a better acoustic characteristic.     

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.     

Geometric and number effect on damping capacity of Helmholtz resonators in a model chamber

An acoustic cavity was selected as a stabilization device to control high-frequency combustion instabilities in gas turbines or liquid rocket engine combustors, and the acoustic damping capacity of the acoustic cavity was investigated for various geometric configurations under atmospheric non-reacting conditions. The tuning frequency of the acoustic cavity and the acoustic responses of a model chamber with a single acoustic cavity were studied first. Damping capacity was initially quantified through the frequency width of two split modes and the amplitude-damped ratio. The results showed that the cavity with the largest orifice area or the shortest orifice length was the most effective in acoustic damping of the harmful resonant mode. The effect of the number of cavities on acoustic damping capacity was also studied. Damping capacity was improved by increasing the number of cavities. For a better evaluation of acoustic damping capacity, two quantified parameters; the acoustic absorption, meaning the damping efficiency, and acoustic conductance, meaning the acoustic power loss, were introduced. The case was observed that has had insufficient loss of acoustic power in spite of having the highest absorption efficiency. As a result, fine geometric tuning for the acoustic cavity is required for the sufficient passive control. Also, the choice of the number of cavities is important to optimize the damping efficiency and absolute damping loss in consideration of the restriction of the cavity volume.     

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.     

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.     

The spatio-temporal structure of wideband acoustic pulses in shallow sea

We study the influence of a bottom-sediment layer on the spatio-temporal structure of wideband acoustic pulses in shallow sea. The shallow sea is modeled as a uniform liquid layer above a layer of liquid sediments located on an elastic half-space. The influence of various acoustic parameters of the problem, in particular the thickness of the sedimentary layer, damping parameters, etc. on the structure of wideband pulses is considered. We analyze the features of constructive mode interference due to which the spatial structure of the pulses takes the form of beams.     

Structural synthesis of sandwich beams with outer layers of box-section

It is proved by model measurements that, for sandwich beams constructed from two rectangular tubes and a damping layer glued between them, the following calculation methods can be applied. Static bending and shear stresses as well as deflections of simply supported beams may be calculated by Allen's formulae for sandwich beams with flexurally stiff faces. The first eigenfrequency and the loss factor can be determined by using the diagrams given in reference [1]. For the loss factors Ungar's formula gives a suitable approximation. A minimum cost design procedure is presented for a sandwich beam with constant cross-section. The unknown dimensions of the cross-section are determined which satisfy the design constraints and minimize the material costs. In a numerical example, constraints relating to the maximal dynamic stresses and deflection as well as local buckling of plate elements are considered. In the optimization the backtrack combinatorial discrete programming method is applied. A numerical comparison shows that the material costs of a sandwich beam are lower than those of a homogeneous box one.     

Modeling and optimization of local constraint elastomer treatments for vibration and noise reduction

This paper presents the vibroacoustic study of a constrained elastomer treatment used in the industry for reducing noise. It can be trimmed and bonded conveniently to vibrating structures for reducing radiated noise. First, an identification of the elastomer viscoelastic characteristics is carried out with a program that models damped vibrations, a conjugate gradient search technique and experimental data extracted rom two contact-free modal analyses. The first modal analysis, adapted to dissipation characterization, is made on a partially covered suspended plate. The second modal analysis, adapted to identifying the elastomer stiffness behavior, concerns a cantilever beam that has almost been covered by a large treatment. The complete dynamic characterization is finally deduced from an iterative procedure that combines information from both experiments. The procedure highlights the influence of the treatment bonding quality on the achieved elastomer damping. Second, practical rules are deduced from a number of parametric studies on beams with baffled radiation conditions. In particular, a design criterion is introduced to help positioning patches where the elastomer damping can be maximized. A threshold, for which an optimal acoustic performance with a minimum of elastomer can be fulfilled, is also identified.     

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.     

An analysis of interlaminar stresses in active constrained layer damping treatments

This paper presents an analysis of the interlaminar stresses in active constrained layer (ACL) damping treatments. The primary objective of this study is to provide in-depth understanding of the delamination of ACL damping treatment and, to establish guidelines to lower the risk of delamination without sacrificing performance. Two major issues are addressed in this investigation. First, the effects of feedback control schemes on interlaminar stresses are analyzed. The proportional (P) and the derivative (D) control laws are selected for comparison. It is found that for the system under consideration, for similar vibration reduction, the derivative control scheme introduces lower interlaminar stresses than proportional control. Also, the derivative control scheme has lower voltage requirements. Second, the ACL treatment is compared with the purely active configuration (without the viscoelastic layer). In addition to the damping performance and control effort requirement (which have been analyzed and compared by researchers in the past), the interlaminar stresses are now included in the comparison. It is shown that the ACL configuration could have significantly lower interlaminar stresses than the purely active configuration, for similar levels of vibration reduction. Hence, in applications where system durability is a concern, the ACL treatment should be preferred over purely active configuration because it has lower interlaminar stress as-well-as lower axial stresses in the piezoelectric cover sheet.     

Shear and compressional damping effects of constrained layered beams

A theory is formulated for combined shear and compressional damping effects of contrained layered beam structures with complicated cross section areas. The theory is applied to some selected theoretical examples. The calculation results indicate that the loss factor values of these beams are larger over a wider frequency range than could be expected from corresponding shear damping effect only or compressional damping treatment only.     

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.     

Optimum thickness distribution of unconstrained viscoelastic damping layer treatments for plates

This paper concerns the optimum thickness distribution of unconstrained viscoelastic damping layer treatments for plates. The system loss factor is expressed in terms of the mechanical properties of the plate and damping layer and the layer/plate thickness ratio. Optimum distributions of the thickness ratio that maximize the system loss factor are obtained through sequential unconstrained minimization techniques. Results are presented for both simply-supported and edge-fixed rectangular plates with aspect ratios of 1·0 to 4·0. These results indicate that the system loss factor can be increased by as much as 100%, or more, by optimizing the thickness distribution of the damping treatment. Also revealed are the regions of the plate where added damping treatments are most effective.     

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.     

An electro-acoustics impedance error criterion and its application to active noise control

Characteristics of radiation impedance and its inducing variation of electrical impedance for a controllable source have been investigated. An impedance-based error criterion has been proposed and its application to active noise control is demonstrated through a coil driven loudspeaker. A general formula of radiation impedance is derived for two control strategies, according to the criterion of total acoustic power output. The radiation impedances of some commonly used sound sources are calculated. We discuss in detail the relation between variation of the input electrical impedance and radiation impedance for the two control strategies. An AC-bridge circuit is designed to measure the weak variation of electrical impedance resulted from radiation impedance. The input electrical impedance of a loudspeaker was measured and the experimental result is consistent with that of theoretical analysis. An impedance-based error criterion is proposed since the AC-bridge relative output is unique for a certain control strategy. The implementation of this criterion applied to an active control system is analyzed by simulations. An analogue control system is set up and experiments are carried out in a semi-anechoic chamber to verify the new control approach.