Active-passive hybrid constrained layer for structural damping augmentation

[0002] This application continuation of co-pending provisional application Ser. No. 60/210,876 filed Jun. 9, 2000, the disclosure of which is hereby specifically incorporated by reference.

[0004] 1. Field of the Invention

[0005] This invention relates generally to an apparatus and methods for reducing vibrations. More particularly, this invention relates to an active-passive hybrid constrained layer.

[0006] 2. Background of the Invention

[0007] Controlling structural vibration is important in numerous technical fields. Examples of such fields include, without limitation, aerospace, automotive, machinery, and numerous other specific applications where vibration control is desirable. A background discussion of active controls and passive controls is found in U.S. Pat. No. 5,838,092, hereby incorporated by reference in its entirety.

[0008] Active constrained layer (ACL) damping treatments have been studied by various researchers. See Agnes, G. and Napolitano, K., 1993, “Active Constrained Layer Viscoelastic Damping,” Proceedings34^thSDM Conference, pp. 3499-3506; Baz, A., 1993, “Active Constrained Layer Damping,” Proceedings of Damping93, San Francisco, Calif., Vol.3, pp. IBB 1-23; Shen, I. Y., 1993, “Intelligent Constrained Layer: An Innovative Approach,” Intelligent Structures, Materials, and Vibrations, ASME DE-Vol. 58, pp. 75-82; Huang, S. C., Inman, D. J. and Austin, E. M., 1996, “Some Design Considerations for Active and Passive Constrained Layer Damping Treatments”, Smart Materials and Structures, Vol. 5, pp. 301-313; Liao, W. H. and Wang, K. W., 1997, “On the Analysis of Viscoelastic Materials for Active Constrained Layer Damping Treatments,” Journal of Sound and Vibration, Vol. 207, pp. 319-334; and Liao, W. H. and Wang, K. W., 1997, “On the Active-Passive Hybrid Control Actions of Active Constrained Layers,” ASME Journal of Vibration and Acoustics, Vol. 119, pp. 563-572.

[0009] Such systems generally consist of a piece of viscoelastic material (VEM) sandwiched between an active piezoelectric layer and a host structure. The main purpose of using a piezoelectric coversheet is that its active action enhances the viscoelastic layer damping ability by increasing the VEM shear angle during operation (the so-called enhanced passive damping action). When the active action fails, significant passive damping could still exist in ACL (it becomes a passive constrained layer (PCL) configuration), which would be important for fail-safe reasons. On the other hand, because the VEM will reduce the active authority of the piezoelectric layer, it would be more effective to use the Enhanced Active Constrained Layer (EACL) concept disclosed in Liao, W. H. and Wang, K. W., 1996, “A New Active Constrained Layer Configuration with Enhanced Boundary Actions,” Smart Materials and Structures, Vol. 5, pp. 638-648; Liao, W. H. and Wang, K. W., 1998, “Characteristics of Enhanced Active Constrained Layer Damping Treatments with Edge Elements, Part 2: System Analysis,” ASME Journal of Vibration and Acoustics, Vol. 120, pp. 894-900; and Liu, Y., and Wang, K. W., 1999a, “A Non-dimensional Parametric Study of Enhanced Active Constrained Layer Damping Treatments,” Journal of Sound and Vibration, Vol. 223, No.4, pp. 611-644; or the separate active and passive designs of (Lam, M. J., Inman, D. J. and Saunders, W. R., 1998, “Variations of Hybrid Damping,” Proceedings of SPIE on Smart Structures and MaterialsVol. 3327, pp. 32-43, if high active action is needed. Therefore, it has been recognized that the applications best suitable for ACL treatments are those that can utilize significant damping from the VEM, rather than from direct piezoelectric-structure interactions.

[0010] Given the above observations, it would be important and desirable to optimize the open-loop characteristics (the so-called baseline structure without active action) of the ACL treatment, as well as the system's closed-loop behavior from the enhanced passive damping action. In both cases, the constraining layer material property plays an important role. An ideal constraining layer for ACL would be a material with high stiffniess, lightweight, and high active authority. So far, the constraining layer in ACLs has been limited to piezoelectric materials (e.g., PZT ceramics or PVDF polymer) because of their active features. PZT materials are in general much better than PVDF polymers for this purpose. Nevertheless, having a density similar to steel (relatively heavy) and a modulus close to aluminum (moderate stiffness), PZTs are not ideal as constraining materials (Lam, M. J., Inman, D. J. and Saunders, W. R., 1998, “Variations of Hybrid Damping,” Proceedings of SPIE on Smart Structures and MaterialsVol. 3327, pp. 32-43). Due to this limitation in the original baseline structure, the open-loop damping ability of an ACL system, in general, is less than that of an optimally designed PCL system.

[0011] As can be seen from the foregoing discussion, problems remain with active constraining layers. Thus a need in the art exists for a constraining layer that improves upon the prior art.

[0012] It is therefore an object of the present invention to provide a method and apparatus for a constrained layer damping treatment that improves over the state of the art.

[0013] A further object of the present invention to provide a method and apparatus for an improved constrained layer that optimizes the open-loop characteristics of an ACL treatment.

[0014] A further object of the present invention to provide a constrained layer that improves the systems closed-loop behavior from an active-passive damping action.

[0015] A further object of the present invention to provide a constrained layer that is capable of using piezoelectric materials.

[0016] A still further object of the invention is to provide a constrained layer capable of meeting damping requirements while requiring less added weight.

[0017] A still further object of the invention is to provide a constrained layer capable of use with different host structure strain distributions.

[0018] Other objects of the invention will become apparent from the description of the invention in that which follows.

[0019] The invention is a novel method and apparatus for an active-passive hybrid constraining layer (HCL) surface damping treatment. The invention improves the damping performance of traditional active constrained layer systems. Instead of using a pure piezoelectric constraining layer or other active layer, passive and active materials are used together to constrain the viscoelastic material layer.

[0020] One aspect of the invention provides for selecting a passive constraining material and assigning appropriate lengths for the active and passive constraining parts in order to optimize the damping effect of the constraining layer.

[0021] Another aspect of the invention provides for separating the passive constraining material and/or the active constraining material into one or more sections in creating the constrained layer in order to improve the damping effect.

[0022] FIG. 1 is a diagram of a generic one-dimensional structure treated by HCL.

[0023] FIG. 2 is a free body diagram of an active constraining layer.

[0024] FIG. 3A is a graph showing open-loops loss factor versus characteristic length for treatment with a PZT constraining layer.

[0025] FIG. 3B is a graph of open-loop loss factor versus characteristic length for a steel constraining layer.

[0026] FIG. 4 is a graph of open-loop loss factor versus active material coverage ratio.

[0027] FIG. 5A is diagram of loss factors versus control gain G for treatment without the anchor.

[0028] FIG. 5B is a graph of loss factors versus control gain G for treatment with an anchor.

[0029] FIG. 6A is a graph of loss factors versus active material coverage ratio for treatment without the anchor (K=0).

[0030] FIG. 6B is a graph of loss factors versus active material coverage ratio for treatment with the anchor (K=100).

[0031] FIG. 7 is a graph of shear distribution of the closed- loops systems in the VEM layer.

[0032] FIG. 8 is a graph of dimension with optimal control gain G versus the active material coverage ratio α.

[0033] FIG. 9A is a graph of HCL hybrid loss factor η_sand open-loop loss factor η_opversus α for different S_pfor treatment without the anchor.

[0034] FIG. 9B is a graph of HCL hybrid loss factor η_sand open-loop loss factor η_opversus α for different S_pfor Treatment with the anchor.

[0035] FIG. 10 is a diagram of one experimental set-up.

[0036] FIG. 11 is a graph of the affect of self-sensing control gain on the first model peek response of the systems.

[0037] FIG. 12 is a graph of impulse responses of the open-loop systems.

[0038] FIG. 13 is a graph of impulse responses of the closed-loop systems.

[0039] FIG. 14 is diagram of a simply supported beam with fully covered HCL with three sections.

[0040] FIG. 15A is a graph of the resonant responses versus magnitude of VEM shear modulus |G_v|.

[0041] FIG. 15B is a graph of resonant frequencies versus magnitude of VEM shear modulus |G_v|.

[0042] FIG. 16A is a graph of loss factor of 3M ISD112.

[0043] FIG. 16B is a graph of storage modulus of 3M ISD112.

[0044] FIG. 17 is a a graph of frequency responses of open-loop ACL and PCL.

[0045] FIG. 18A is a graph of the effect of constraining layer configuration on the first mode of the open-loop system.

[0046] FIG. 18B is also a graph of the effect of constraining layer configuration on the first mode of the open-loop system.

[0047] FIG. 19A graph of the effect of constraining layer configuration on the second mode of the open-loop system.

[0048] FIG. 19B is also a graph of the effect of constraining layer configuration on the second mode of the open-loop system.

[0049] FIG. 20A is a graph of the effect of constraining layer configuration on the third mode of the open-loop system.

[0050] FIG. 20B is also a graph of the effect of constraining layer configuration on the third mode of the open-loop system.

[0051] FIG. 21A is a graph showing the first resonant peak response versus control gain.

[0052] FIG. 21B is a graph showing first modal frequency versus control gain.

[0053] FIG. 22A is a graph showing second modal peak response versus control gain.

[0054] FIG. 22B is a graph showing second modal frequency versus control gain.

[0055] FIG. 23A is a graph showing third modal peak response versus control gain.

[0056] FIG. 23B is a graph showing third modal frequency versus control gain.

[0057] FIG. 24A is a graph showing first modal peak response versus active coverage ratio in the closed loop system.

[0058] FIG. 24B is a graph showing first modal frequency versus active coverage ratio in the closed loop system.

[0059] FIG. 25A is a graph showing second modal peak response versus active coverage ratio in the closed loop system.

[0060] FIG. 25B is a graph showing second modal frequency versus active coverage ratio in the closed loop system.

[0061] FIG. 26A is a graph showing third modal peak response versus active coverage ratio in the closed loop system.

[0062] FIG. 26B is a graph showing third modal frequency versus active coverage ratio in the closed loop

[0063] The present invention is a method and apparatus for an active-passive hybrid control layer for structural damping augmentation. To aid in describing the invention, the following nomenclature is used throughout:

[0064] b: Width of the host structure

[0065] D_b: Bending stiffniess ratio between host structure and active constraining layer

[0066] D_p: Bending stiffniess ratio between passive and active constraining layers

[0067] E_b: Young's modulus of host structure

[0068] E_p: Young's modulus of passive constraining material

[0069] E_a: Young's modulus of active constraining material

[0070] G: Non-dimensional control gain

[0071] G_v*: Complex shear modulus of VEM

[0072] G_vl: Storage shear modulus of VEM

[0073] h_b: Thickness of host structure

[0074] h_p: Thickness of passive constraining material

[0075] h_a: Thickness of active constraining material

[0076] h_v: Thickness of VEM layer

[0077] h_L: Distance between the neutral axis of host structure to the neutral axis of active constraining layer

[0078] h_R: Distance between the neutral axis of host structure to the neutral axis of the passive constraining layer

[0079] K: Non-dimensional anchor stiffness

[0080] K_eq: Equivalent anchor stiffness

[0081] K_g: Control gain

[0082] L: Length of damping treatment

[0083] Re[ ]: Real part of the complex number inside bracket

[0084] S_b: Extensional stiffness ratio between host structure and active constraining layer

[0085] S_p: Extensional stiffness ratio between passive and active constraining layers

[0086] u_b: Longitudinal deformation at the neutral axis of the host structure

[0087] u_p: Longitudinal deformation at the neutral axis of the passive constraining layer

[0088] u_a: Longitudinal deformation at the neutral axis of the active constraining layer

[0089] {overscore (U)}_b: Dimensionless displacement at the neutral axis of the host structure defined as U_b/L

[0090] {overscore (U)}_p: Dimensionless displacement at the neutral axis of the passive constraining material defined as U_p/L

[0091] {overscore (U)}_a: Dimensionless displacement at the neutral axis of the active constraining material defined as U_a/L

[0092] w: Transverse displacement of the system

[0093] {overscore (W)}: Dimensionless transverse displacement of the system defined as W/L

[0094] W_p: Energy dissipated per cycle through VEM shear deformation

[0095] W_a: Energy dissipated per cycle through direct action control action

[0096] W_s: Maximum strain energy stored in system

[0097] {overscore (x)}: Non-dimensional position coordinate defined as x/L

[0098] α: Coverage ratio of the active constraining layer

[0099] σ_a, σ_p: Longitudinal stress at the neutral axis of the active and passive constraining layer, respectively

[0100] τ: Shear stress in VEM ρ_a, ρ_p: Densities of active and passive constraining materials

[0101] Λ: Strain of the active constraining material directly induced by the electric field

[0102] γ: Shear strain in VEM layer

[0103] η_v: VEM loss factor

[0104] η_p, η_a, η_s: Passive, active and hybrid loss factors of the damping treatment η_op: Open-loop loss factor of damping treatment

[0105] Γ_a: Dimensionless characteristic length of the treatment when α=1

[0106] Γ_p: Dimensionless characteristic length of the treatment when α=0

[0107] θ_a: Dimensionless thickness of active constraining layer defined as h_a/L

[0108] θ_v: Dimensionless VEM thickness defined as h_v/L

[0109] θ_L: Dimensionless parameter defined as h_L/L

[0110] θ_R: Dimensionless parameter defined as h_R/L

[0111] ω: Vibration frequency of the system in radians

[0112] Ω_p: Non-dimensional characteristic frequency of the passive constraining part

[0113] Ω_a: Non-dimensional characteristic frequency of the active constraining part

[0114] A hybrid constrained layer (HCL) damping treatment is described. In this novel configuration, the constraining layer consists of two different kinds of materials. One is an active material such as PZT, and the other is a passive material. By selecting a proper passive constraining material and adjusting the dimensions of the active and passive constraining layers, the baseline structure and the system's closed-loop performance are both improved. In other words, this new configuration not only uses the active characteristics of an active layer, but also improves the coversheet's ability to function as a constraining layer.

[0115] A dimensionless mathematical model of the HCL-based structure is described. Using a self-sensing control algorithm for the active component, the closed-form solutions to the equations of motion are obtained. The loss factors of the treatments are defined and used as indices to discuss the open-loop and closed-loop damping properties of the treatments. Experimental results are then shown to verify and validate the mathematical model of the invention. It should be appreciated by one skilled in the art, that the present invention is in no way limited to the particular example used to verify the created mathematical model, but this experimental setup and particular variables selected is merely one example or embodiment of the invention. The present invention includes other applications of the model discovered. As shown in FIG. 1, a generic one-dimensional base structure 12 is treated by HCL 11. That is, the VEM 14 is constrained by an active-passive hybrid constraining layer 11. The active part 16 is made of PZT ceramics. The active constraining part 16 and passive constraining part 18 are mechanically connected such that the displacement and force are continuous at the connecting point 19. However, they are isolated electrically so that the passive constraining layer 18 will not affect the function of the active constraining layer 16. This is not a concern if the passive material 18 is not conductive. The present invention contemplates that if the passive material is conductive then an insulative material may be placed between the active material and the passive material, the active material and the passive material are still mechanically connected, even though they are not in direct physical contact. At one end of the active layer 16, an anchor 20 is optionally used to F connect the active material 16 and the base structure 12. Results are verified and validated for both an HCL treatment with an anchor and an HCL treatment without the anchor. The following assumptions are made in deriving the model:

[0116] a) There is only shear deformation in VEM.

[0117] b) Passive damping is only generated by the shear deformation of VEM.

[0118] c) Interfaces are perfect; no slip occurs between the layers.

[0119] d) There is no relative transverse motion between different layers.

[0120] e) The displacement (δ) between the anchor location on the host structure and the corresponding end of the PZT layer is zero.

[0121] f) Applied voltage is assumed uniform along the PZT.

[0122] g) Only harmonic, steady state vibration is considered.

[0123] h) Linear theories of elasticity, viscoelasticity, and piezoelectricity are used.

[0124] As can be appreciated by one skilled in the art, the model may be adjusted to include additional or different assumptions while still in accordance with the invention. The selection of particular assumptions would depend upon the particular application or use of the hybrid constrained layer of the present invention.

[0125] Using the above given assumptions and considering the equilibrium of the PZT section as shown in FIG. 2, the following equation of motion can be derived:

[0126] p_a∂²u_a/∂t²=∂σ_a/∂x−τ/h_a0<x<αL  (1)

[0127] where σ_ais the longitudinal stress at the neutral axis of the PZT, which, according to the constitutive equation of the PZT, can be expressed as: $\begin{array}{cc}{\sigma }_a=E_a\left(\frac{\partial u_a}{\partial x}-\Lambda \right)& \left(2\right)\end{array}$

[0128] Here, σ_ais the strain of the PZT directly induced by the electric field, which is dependent on the control law used. τ is the shear stress in the VEM. In steady state, it can be written as

[0129] τ=G_v*γ=G_vl(1+iη_v)γ  (3)

[0130] where G_v* is the complex shear modulus of the VEM, G_vlis the VEM storage modulus, and η_vis the material loss factor.

[0131] For the system with longitudinal and transverse vibrations, the shear strain γ in VEM can be calculated from the deformation of the base structure and the hybrid constraining layer as (Liu, Y., and Wang, K. W., 1999a, “A Non-dimensional Parametric Study of Enhanced Active Constrained Layer Damping Treatments,” Journal of Sound and Vibration, Vol. 223, No.4, pp. 611-644): $\begin{array}{cc}\gamma =\{\begin{array}{cc}\frac{u_a-u_b+h_l\frac{\partial w}{\partial x}}{h_v}& 0\le x\le \alpha L\\ \frac{u_p-u_b+h_R\frac{\partial w}{\partial x}}{h_v}& \alpha L\le x\le L\end{array}& \left(4\right)\end{array}$

[0132] Substituting Equations (2)-(4) into Equation (1) and considering assumption (f), the equation of motion of the active constraining part can be derived as $\begin{array}{cc}\frac{{\partial }^2{u}_a}{\partial x^2}-\frac{G_v^{*}\left(u_a-u_b+h_L\frac{\partial w}{\partial x}\right)}{E_a{h}_a{h}_v}=\frac{{\rho }_a}{E_a}\frac{{\partial }^2{u}_a}{\partial t^2}0<x<\alpha L& \left(5\right)\end{array}$

[0133] Similarly, for the passive constraining part, the equation of motion can be derived as $\begin{array}{cc}\frac{{\partial }^2{u}_p}{\partial x^2}-\frac{G_v^{*}\left(u_p-u_b+h_R\frac{\partial w}{\partial x}\right)}{E_p{h}_p{h}_v}=\frac{{\rho }_p}{E_p}\frac{{\partial }^2{u}_p}{\partial t^2}\alpha L<x<L& \left(6\right)\end{array}$

[0134] It can be seen from Equations (5) and (6) that the longitudinal displacements in the constraining layers are related to the host structure deformation. In this study, the host structure deformation is assumed given. Therefore, Equations (5) and (6) can be used to derive the deformation in the constraining layers. To do that, four conditions are needed since Equations (5) and (6) are two second-order equations. Modeling the anchor as an equivalent spring with stiffness K_eq(FIG. 1), the boundary conditions for u_aand u_pcan be written as: $\begin{array}{cc}{E}_a{h}_{a}b\left(\frac{\partial u_a}{\partial x}-\Lambda \right)=K_{\mathrm{eq}}\left(u_a-u_b+h_L\frac{\partial w}{\partial x}\right)x=0& \left(7\right)\\ \frac{\partial u_p}{\partial x}=0x=L& \left(8\right)\end{array}$

[0135] At the connecting point of the active and passive constraining layers, the displacement and force are continuous. Therefore,

[0136] u_a=u_px=αL  (9)

[0137] $\begin{array}{cc}{E}_a{h}_a\left(\frac{\partial u_a}{\partial x}-\Lambda \right)=E_p{h}_p\frac{\partial u_p}{\partial x}x=\alpha L& \left(10\right)\end{array}$

[0138] Equations (5)-(1 0) are all the equations we need to solve for u_aand u_p. In steady state, considering the separation of the variables in the forms of

[0139] w(x,t)=W(x)e^iωt  (11)

[0140] u_b(x,t)=U_b(x)e^iωt  (12)

[0141] u_a(x,t)=U_a(x)e^iωt  (13)

[0142] u_p(x,t)=U_p(x)e^iωt  (14)

[0143] where ω is the excitation frequency in radians. Introducing the following non-dimensional parameters $\begin{array}{cc}{\Gamma }_p^2=\frac{G_v^{*}{L}^2}{E_p{h}_p{h}_v}& \left(15\right)\\ {\Gamma }_a^2=\frac{G_v^{*}{L}^2}{E_a{h}_a{h}_v}& \left(16\right)\\ {\Omega }_p^2=\frac{{\rho }_p{L}^2{\omega }^2}{E_p}& \left(17\right)\\ {\Omega }_a^2=\frac{{\rho }_a{L}^2{\omega }^2}{E_a}& \left(18\right)\\ K=\frac{K_{\mathrm{eq}}L}{E_a{h}_{a}b}& \left(19\right)\\ S_p=\frac{E_p{h}_p}{E_a{h}_a}=\frac{{\Gamma }_a^2}{{\Gamma }_p^2}& \left(20\right)\end{array}$

[0144] Equations (5)-(10) can be written as: $\begin{array}{cc}\frac{d^2{\stackrel{\_}{U}}_a}{d{\stackrel{\_}{x}}^2}-\left({\Gamma }_a^2-{\Omega }_a^2\right){\stackrel{\_}{U}}_a=-{\Gamma }_a^2\left({\stackrel{\_}{U}}_b-{\theta }_L\frac{d\stackrel{\_}{W}}{d\stackrel{\_}{x}}\right)0<\stackrel{\_}{x}<\alpha & \left(21\right)\\ \frac{d^2{\stackrel{\_}{U}}_p}{d{\stackrel{\_}{x}}^2}-\left({\Gamma }_p^2-{\Omega }_p^2\right){\stackrel{\_}{U}}_p=-{\Gamma }_p^2\left({\stackrel{\_}{U}}_b-{\theta }_R\frac{d\stackrel{\_}{W}}{d\stackrel{\_}{x}}\right)\alpha <\stackrel{\_}{x}<1& \left(22\right)\\ \frac{d{\stackrel{\_}{U}}_a}{d\stackrel{\_}{x}}-\frac{\Lambda }{e^{i\omega t}}=K\left({\stackrel{\_}{U}}_a-{\stackrel{\_}{U}}_b+{\theta }_L\frac{d\stackrel{\_}{W}}{d\stackrel{\_}{x}}\right)\stackrel{\_}{x}=0& \left(23\right)\\ \frac{d{\stackrel{\_}{U}}_p}{d\stackrel{\_}{x}}=0\stackrel{\_}{x}=1& \left(24\right)\\ {\stackrel{\_}{U}}_a={\stackrel{\_}{U}}_p\stackrel{\_}{x}=\alpha & \left(25\right)\\ \frac{d{\stackrel{\_}{U}}_a}{d\stackrel{\_}{x}}-\frac{\Lambda }{e^{i\omega t}}=S_p\frac{d{\stackrel{\_}{U}}_p}{d\stackrel{\_}{x}}\stackrel{\_}{x}=\alpha & \left(26\right)\end{array}$

[0145] Here, Γ_ais the non-dimensional complex characteristic length of the treatment when α=1 (characteristic length for the pure active constraining layer), and Γ_pis the non-dimensional complex characteristic length of the treatment when α=0 (characteristic length for the pure passive constraining layer). Ω_ais the non-dimensional characteristic frequency of the active constraining layer, and Ω_pis the non-dimensional characteristic frequency of the passive constraining layer. K is the non-dimensional anchor stiffniess. S_pis the ratio of the extensional stiffniess of the passive and active constraining layers. Larger S_pimplies that the passive constraining part is stiffer, relative to the active part. We will see later that S_phas a significant effect on the performance of the HCL treatment. Other parameters are defined in the nomenclature.

[0146] In general, for thin damping treatment and low frequency, the following approximations can be used. $\begin{array}{cc}\frac{{\Omega }_p^2}{{\Gamma }_p^2}<<1& \left(27\right)\\ \frac{{\Omega }_a^2}{{\Gamma }_a^2}<<1& \left(28\right)\end{array}$

[0147] Therefore, Equations (21) and (22) can be simplified as $\begin{array}{cc}\frac{d^2{\stackrel{\_}{U}}_a}{d{\stackrel{\_}{x}}^2}-{\Gamma }_a^2{\stackrel{\_}{U}}_a=-{\Gamma }_a^2\left({\stackrel{\_}{U}}_b-{\theta }_L\frac{d\stackrel{\_}{W}}{d\stackrel{\_}{x}}\right)0<\stackrel{\_}{x}<\alpha & \left(29\right)\\ \frac{d^2{\stackrel{\_}{U}}_p}{d{\stackrel{\_}{x}}^2}-{\Gamma }_p^2{\stackrel{\_}{U}}_p=-{\Gamma }_p^2\left({\stackrel{\_}{U}}_b-{\theta }_R\frac{d\stackrel{\_}{W}}{d\stackrel{\_}{x}}\right)\alpha <\stackrel{\_}{x}<1& \left(30\right)\end{array}$

[0148] The strain field in the host structure may be assumed to be a given input, which is linearly distributed along the thickness and length directions as: $\begin{array}{cc}ϵ\left(x,z,t\right)=\frac{\partial u_b}{\partial x}-z\frac{{\partial }^{2}w}{\partial x^2}=\left(C_{\mathrm{z0}}+C_{\mathrm{z1}}\stackrel{\_}{z}\right)\left(C_{\mathrm{x0}}+C_{\mathrm{x1}}\stackrel{\_}{x}\right)e^{i\omega t}& \left(31\right)\end{array}$

[0149] By selecting different number combination for C_x0, C_x1, C_z0and C_z1, we can define longitudinal or transverse motion of the host structure. For example, when C_z1=C_x1=0, the strain in the host structure is constant and the structure is under pure longitudinal vibration (Liu, Y., and Wang, K. W., 1999b, “Surface Damping Treatment with an Active-Passive Hybrid Constraining Layer,” Proceedings of the 1999 ASME Design Engineering Technical Conferences, DETC/VIB-8301). We will let C_z0=C_x1=0, C_z1=C_x0=1, which represents pure bending of the host structure. From Equation (31), the following relations can be derived $\begin{array}{cc}{\stackrel{\_}{U}}_b-{\theta }_L\frac{d\stackrel{\_}{W}}{d\stackrel{\_}{x}}=\left(C_{\mathrm{z0}}+C_{\mathrm{z1}}{\theta }_L\right)\left(\mathrm{Const}+C_{\mathrm{x0}}\stackrel{\_}{x}+\frac{1}{2}{C}_{\mathrm{x1}}{\stackrel{\_}{x}}^2\right)& \left(32\right)\\ {\stackrel{\_}{U}}_b-{\theta }_R\frac{d\stackrel{\_}{W}}{d\stackrel{\_}{x}}=\left(C_{\mathrm{z0}}+C_{\mathrm{z1}}{\theta }_R\right)\left(\mathrm{Const}+C_{\mathrm{x0}}\stackrel{\_}{x}+\frac{1}{2}{C}_{\mathrm{x1}}{\stackrel{\_}{x}}^2\right)& \left(33\right)\end{array}$

[0150] Substituting Equations (32) and (33) into Equations (29) and (30), the general solutions to the equations of motion are $\begin{array}{cc}\begin{array}{c}{\stackrel{\_}{U}}_a=A\cosh ({\Gamma }_a\stackrel{\_}{x})+B\sin h({\Gamma }_a\stackrel{\_}{x})+\\ \left(C_{\mathrm{z0}}+C_{\mathrm{z1}}{\theta }_L\right)\left(\mathrm{Const}+\frac{C_{\mathrm{x1}}}{{\Gamma }_a^2}+C_{\mathrm{x0}}\stackrel{\_}{x}+\frac{C_{\mathrm{x1}}}{2}{\stackrel{\_}{x}}^2\right)\end{array}& \left(34\right)\\ \begin{array}{c}{\stackrel{\_}{U}}_p=C\cosh \left({\Gamma }_p\stackrel{\_}{x}\right)+D\sinh \left({\Gamma }_p\stackrel{\_}{x}\right)+\\ \left(C_{\mathrm{z0}}+C_{\mathrm{z1}}{\theta }_R\right)\left(\mathrm{Const}+\frac{C_{\mathrm{xl}}}{{\Gamma }_p^2}+C_{\mathrm{x0}}\stackrel{\_}{x}+\frac{C_{\mathrm{x1}}}{2}{\stackrel{\_}{x}}^2\right)\end{array}& \left(35\right)\end{array}$

[0151] Substituting Equations (32)-(35) into (4), the shear in VEM can be written as $\begin{array}{cc}\frac{\gamma }{e^{i\mathrm{\omega t}}}=\{\begin{array}{c}\frac{A\cosh \left({\Gamma }_a\stackrel{\_}{x}\right)+B\sinh \left({\Gamma }_a\stackrel{\_}{x}\right)+\frac{C_{\mathrm{x1}}}{{\Gamma }_a^2}+\left(C_{\mathrm{z0}}+C_{\mathrm{z1}}{\theta }_R\right)}{{\theta }_v}0\le \stackrel{\_}{x}\le \alpha \\ \frac{C\cosh \left({\Gamma }_p\stackrel{\_}{x}\right)+D\sinh \left({\Gamma }_p\stackrel{\_}{x}\right)+\frac{C_{\mathrm{x1}}}{{\Gamma }_p^2}\left(C_{\mathrm{z0}}+C_{\mathrm{z1}}{\theta }_R\right)}{{\theta }_v}\alpha \le \stackrel{\_}{x}\le 1\end{array}& \left(36\right)\end{array}$

[0152] Here, θ_vis the non-dimensional thickness of the VEM layer. A, B, C and D are constants to be determined by the boundary and connecting conditions.

[0153] In order to solve for constants A, B, C and D, the control law has to be decided first. Here, the self-sensing control method proposed by Dosch, J. J., Inman, D. J. and Garcia, E., 1992, “A Self-Sensing Piezoelectric Actuator for Collocated Control,” Journal of Intelligent Systems and Structures, Vol.3, pp. 166-185; Shen, I. Y., 1997, “A Variational Formulation, a Work-Energy Relation and Damping Mechanisms of Active Constrained Layer Treatments,” ASME Journal of Vibration and Acoustics, Vol. 119, pp. 192-199; and Baz, A., 1997, “Optimization of Energy Dissipation Characteristics of Active Constrained Layer Damping,” Smart Materials and Structures, Vol.6, pp. 360-368; is used: $\begin{array}{cc}\Lambda =K_g\left[\frac{\partial u_a\left(0,t\right)}{\partial t}-\frac{\partial u_a\left(\alpha L,t\right)}{\partial t}\right]=iG\left[{\stackrel{\_}{U}}_a\left(0\right)-{\stackrel{\_}{U}}_a\left(\alpha \right)\right]e^{i\mathrm{\omega t}}& \left(37\right)\end{array}$

[0154] where K_gis the control gain, and G is the non-dimensional control gain defined as

[0155] G=K_gLω  (38)

[0156] While more advanced control laws could be employed, this simple scheme guarantees the stability of the system so that we can concentrate on the performance of the treatments. Substituting Equations (32), (34), (35) and (37) into Equations (23)-(26), constants A, B, C and D can be derived.

[0157] For the self-sensing control law used here, the energies dissipated per cycle by passive damping (W_p) and by active control (W_a) can be expressed, respectively, as (Baz, A., 1997, “Optimization of Energy Dissipation Characteristics of Active Constrained Layer Damping,” Smart Materials and Structures, Vol.6, pp. 360-368; Shen, I. Y., 1997, “A Variational Formulation, a Work-Energy Relation and Damping Mechanisms of Active Constrained Layer Treatments,” ASME Journal of Vibration and Acoustics, Vol.119, pp. 192-199): $\begin{array}{cc}{W}_p=\pi {\eta }_v{G}_{\mathrm{v1}}{h}_v\mathrm{bL}{\int }_0^1{\left|\gamma \right|}^{2}d\stackrel{\_}{x}& \left(39\right)\end{array}$  W_a=πGE_ah_abL|{overscore (U)}_a(0)−{overscore (U)}_a(α)²  (40)

[0158] Passive loss factor η_pand active loss factor η_aof the HCL treatment are defined as $\begin{array}{cc}{\eta }_p=\frac{W_p}{2\pi W_s}& \left(41\right)\\ {\eta }_a=\frac{W_a}{2\pi W_s}& \left(42\right)\end{array}$

[0159] where W_sis the maximum strain energy stored in the host structure, active and passive constraining layers, viscoelastic material, and the anchor, which can be expressed as $\begin{array}{cc}{W}_s=\frac{E_a{h}_a\mathrm{bL}}{2}\left\{\frac{{\theta }_a^2}{12}\left(1-D_p\right)\left(C_{\mathrm{x0}}^2\alpha +C_{\mathrm{x0}}{C}_{\mathrm{x1}}{\alpha }^2+\frac{C_{\mathrm{xl}}^2}{3}{\alpha }^3\right)C_{\mathrm{z1}}^2+\left[S_b{C}_{\mathrm{z0}}^2+\frac{{\theta }_a^2{C}_{\mathrm{z1}}^2}{12}\left(D_b+D_p\right)\right](C_{\mathrm{x0}}^2+C_{\mathrm{x0}}{C}_{\mathrm{x1}}+\frac{C_{\mathrm{x1}}^2}{3})+{\int }_0^a{\left|\frac{d{\stackrel{\_}{U}}_a}{d\stackrel{\_}{x}}\right|}^{2}d\stackrel{\_}{x}+S_p{\int }_a^1{\left|\frac{d{\stackrel{\_}{U}}_p}{d\stackrel{\_}{x}}\right|}^{2}d\stackrel{\_}{x}+\mathrm{Re}\left[{\Gamma }_a^2\right]{\theta }_v^2{\int }_0^1{\left|\gamma \right|}^{2}d\stackrel{\_}{x}+K{\left|{\stackrel{\_}{U}}_a\left(0\right){\stackrel{\_}{U}}_b\left(0\right)+{\theta }_L\frac{d\stackrel{\_}{W}\left(0\right)}{d\stackrel{\_}{x}}\right|}^2\right\}& \left(43\right)\end{array}$

[0160] Here, S_bis the extensional stiffness ratio between the host structure and the active constraining layer. $\begin{array}{cc}{S}_b=\frac{E_b{h}_b}{E_a{h}_a}& \left(44\right)\end{array}$

[0161] D_bis the bending stiffness ratio between the host structure and the active constraining layer; and D_pis the bending stiffness ratio between the passive constraining layer and active constraining layer. $\begin{array}{cc}{D}_b=\frac{E_b{I}_b}{E_a{I}_a}=\frac{E_b{h}_b^3}{E_a{h}_a^3}& \left(45\right)\\ D_p=\frac{E_p{I}_p}{E_a{I}_a}=\frac{E_p{h}_p^3}{E_a{h}_a^3}& \left(46\right)\end{array}$

[0162] Total system (hybrid) loss factor η_sis the summation of the passive and active loss factors.

[0163] η_s=η_p+η_a  (47)

[0164] The hybrid loss factor, η_sis the overall ability of the treatment to dissipate energy. The active loss factor, η_a, reflects the direct active control authority of the piezoelectric layer on the host structure. The passive loss factor, η_p, reflects the passive damping ability of the treatment through VEM shear deformation. For the closed-loop system (G>0), this includes the open-loop damping of the baseline structure (G=0) plus the enhanced passive damping due to the additional VEM deformation induced by the active PZT action. When the control gain G is zero (open-loop), η_awill be zero, and η_pwill only contain passive damping of the baseline structure, which is defined to be the open-loop loss factor, η_op. The value of η_opreflects the fail-safe ability of the treatment.

[0165] From the definition of the loss factors, it can be found that HCL damping properties are determined by independent parameters Γ_a, K, G, α, D_p, S_p, D_b, S_bθ_L, θ_R, and θ_a. The nominal values of these parameters used in the analysis are listed in the following table unless stated otherwise.

[0166] We will mainly focus on the effects of α, D_pand S_pon the open-loop and closed-loop damping properties of the HCL treatment. In FIGS. 3aand 3b, the open-loop loss factor η_op(η_op=η_psince G=0) is shown for the treatment with two different constraining layers. FIG. 3ashows a PZT constraining layer. FIG. 3bshows a steel constraining layer. In FIG. 3a, both treatment with an anchor 30 (K=1000) and treatment without an anchor 32 are shown. Similarly, in FIG. 3b, both treatment with an anchor 34 (K=1000) and treatment without an anchor 36 are shown. With both PZT and steel constraining layers, there is an optimal characteristic length |Γ_a| (or |Γ_p) to maximize η_op(Plunkett, R., and Lee, C. T., 1970, “Length Optimization for Constrained Viscoelastic Layer Damping,” Journal of the Acoustical Society of AmericaVol.48, No.1, Part 2, 150-161; Baz, A., 1997, “Optimization of Energy Dissipation Characteristics of Active Constrained Layer Damping,” Smart Materials and Structures, Vol.6, pp. 360-368 Liu, Y., and Wang, K. W., 1999a, “A Non-dimensional Parametric Study of Enhanced Active Constrained Layer Damping Treatments,” Journal of Sound and Vibration, Vol. 223, No.4, pp. 611-644). However, when both treatments are designed to operate at their optimal characteristic length, the treatment with a steel constraining layer obtains higher damping than the treatment with a PZT constraining layer.

[0167] It can be seen from both FIG. 3A and FIG. 3B that the anchor merely has the effect of shifting the optimal characteristic length of the treatment in the open-loop system. It will not change the maximum open-loop loss factor. It is also worth noting that the optimal characteristic length of the treatment when the system is under pure bending is different from that when the system is under pure axial motion. In general, the strain field of the host structure should be considered in the design of the optimal characteristic length for the open-loop system.

[0168] Since the treatment with a steel constraining layer can obtain higher open-loop damping than the treatment with a PZT constraining layer, the open-loop damping performance or fail-safe property of the damping treatment with a pure PZT coversheet can be improved by replacing part of the PZT coversheet by steel. In FIG. 4, the effect of the active material coverage ratio α on the open-loop loss factor is shown. It is clear that the smaller the a, the larger the loss factor η_op. In other words, mixing the stiffer passive material into the coversheet will enhance the fail-safe ability of the treatment. This is relationship is shown both with an anchor 38 and without an anchor 40.

[0169] Although smaller a would achieve higher open-loop damping, it is not desirable to design a treatment with a very small α since this could reduce the active and hybrid (closed-loop) damping actions. In other words, not only the open-loop damping but also the closed-loop damping of the system should be considered before a suitable a can be selected.

[0170] In a closed-loop system, the system hybrid loss factor is the combined effect of active and passive dampings as defined in Equation (47). For the self-sensing algorithm used in this embodiment, there is an optimal control gain G to maximize the hybrid loss factor. When this optimal gain is used, a range of a can be found (0<α<1) such that the HCL hybrid loss factor is larger than the hybrid loss factor of the system with a pure PZT coversheet (α=1). The HCL damping ability is further improved by selecting appropriate S_pand D_p. Details of the analysis are discussed in the following paragraphs.

[0171] FIG. 5A illustrates the effect of the control gain G on the active 46, passive 44 and hybrid 42 loss factors of treatments with a pure active coversheet (α=1) without anchor. FIG. 5B illustrates the effect of the control gain G on the active 52, passive 50 and hybrid loss 48 factors of treatments with a pure active coversheet (α=1) with anchor. Obviously, there exists an optimal G for maximizing η_s. When the control gain becomes sufficiently large, the active loss factor approaches zero and the passive loss factor saturates (Liu, Y., and Wang, K. W., 1999a, “A Non-dimensional Parametric Study of Enhanced Active Constrained Layer Damping Treatments,” Journal of Sound and Vibration, Vol. 223, No.4, pp. 611-644). In the following studies, optimal control gains for maximizing the hybrid loss factor are used, unless specified otherwise.

[0172] With the application of the optimal control gain, a range of α exists such that the HCL system can obtain higher damping than the system with a pure PZT coversheet. FIG. 6A shows the loss factors versus active material coverage ratio a for loss factors where the treatment is applied without an anchor (K=0). The hybrid loss factor 54, active loss factor 60, passive loss factor 56, and open-loop loss factor 58 are shown. Similarly, FIG. 6B shows the loss factors versus active material coverage ratio a for loss factors where treatment is applied with an anchor (K=1000). The hybrid loss factor 62, active loss factor 66, passive loss factor 64, and open-loop loss factor 68 are shown.

[0173] As shown in FIGS. 6A and 6B, when a is large enough but still less than 1, HCL has a higher hybrid loss factor than that of a system with α=1. The reason behind this is that we are gaining significantly from the increase in passive damping (open-loop damping plus the enhanced passive damping from additional VEM deformation), and are still doing well in terms of active damping (active action from PZT coversheet to host structure). Also from FIGS. 6A and 6B, when a is close to 1, HCL obtains higher passive damping than the system with a pure PZT coversheet (α=1). To better understand this phenomenon, FIG. 7 plots the shear deformation in VEM for the closed-loop system when the optimal control gain is applied for each selection of α. Shear distribution of the closed-loop systems in the VEM layer are plotted for α=1 with anchor 72, α=0.7 with anchor 70, α=1 without anchor 76, and α=0.9 without anchor 74. The inclusion of the stiffer passive material in the constraining layer has the effect of pushing the VEM shear deformation curve higher, which will increase the system's passive damping ability. On the other hand, the active damping for HCL is close to or higher than that of the case with α=1 as shown in FIGS. 6A and 6B. From Equation (40), the energy dissipated per cycle through the active control action is proportional to the non-dimensional control gain G and the square of the displacement difference at the ends of the active constraining layer. Although the implementation of the active-passive hybrid constraining layer could slightly reduce the displacement difference, the increase of the optimal control gain shown in FIG. 8 (both with anchor 80 and without anchor 78) in the case of hybrid constraining layer can actually compensate for that reduction, as long as the active material coverage ratio a is not too small, as shown in FIG. 6 for the active loss factor. Because of the combined effects of the passive and active loss factors, HCL can provide higher overall damping (higher than cases with purely active or passive constraining layers), when an optimal α(0<α<1) is used.

[0174] FIGS. 9A and 9B show the hybrid loss factor as a function of a when different S_pare selected. In this embodiment, it is assumed that the active and passive constraining layers have the same thickness. Of course, the present invention contemplates varying shapes and dimensions of the active and passive constraining layers. Therefore, D_pis also changed accordingly. For S_p=1, 3 and 6, it is comparable to using aluminum, steel, and aluminum oxide (Al₂O₃), respectively, as the passive constraining material with the same thickness as the active one. In FIG. 9A, plot line 82 shows S_p=6 for the HCL hybrid loss, plot line 88 shows S_p=6 for the open-loop loss, plot line 90 shows S_p=3 for the open-loop loss, plot line 84 shows S_p=3 for the HCL hybrid loss, plot line 92 shows S_p=1 for the open-loop loss, and plot line 86 shows S_p=1 for the HCL hybrid loss. In FIG. 9B, plot line 94 shows S_p=6 for the HCL hybrid loss, plot line 100 shows S_p=6 for the open-loop loss, plot line 102 shows S_p=3 for the open-loop loss, plot line 96 shows S_p=3 for the HCL hybrid loss, plot line 104 shows S_p=1 for the open-loop loss, and plot line 98 shows S_p=1 for the HCL hybrid loss.

[0175] FIGS. 9A and 9B illustrate that the inclusion of a passive material in the constraining layer can provide more flexibility in the surface damping treatment design to satisfy different requirements. FIG. 9A shows treatment without an anchor (K=0). FIG. 9B shows treatment with an anchor (K=100). For example, for systems with severe weight constraints, one can reduce the added weight of a treatment with a pure PZT coversheet by replacing part of the PZT (density is 7500 kg/m³) with aluminum (density is 2700 kg/m³) as constraining material. Since the aluminum has a Young's modulus (E=0.9×10¹⁰) that is almost the same as that of PZT (E=6.3×10¹⁰for PZT 850 from APC) but is much lighter, one can achieve better damping properties with less add-on weight (see FIG. 9 for S_p=1). On the other hand, for systems with higher damping requirements but less weight constraints, steel (E=2.0×10¹¹) or aluminum oxide (E=3.9×10¹¹) can be used as the passive constraining materials. Steel (density is 7600 kg/m³) is almost the same weight as PZT, and aluminum oxide (density is 4000 kg/m³) is lighter than PZT. The damping performance of such HCL treatments can be much better than the treatments with purely active PZT coversheets, as shown in FIG. 9 for S_p=3 and S_p=6. Another advantage of selecting a stiffer passive constraining material is that the optimal value of a where HCL obtains the maximum η_sbecomes smaller. This indicates that such HCLs will also have higher fail-safe ability since larger η_opcan be obtained at where optimal a is selected (compare the S_p=3 and 6 cases with S_p=1 in FIG. 9). It is worth noting that a larger S_pcan also be achieved by increasing the passive coversheet thickness.

[0176] The mathematical models of the hybrid damping of the present invention were verified through experimental efforts and test results.

[0177] The experiments are conducted on cantilever beams. As shown in FIG. 10, one aluminum beam 108 is treated by the hybrid constrained layer (coversheet with 60% PZT and 40% aluminum oxide). The second beam, with the same material and dimensions as the first one, is treated by an active constrained layer with a pure PZT coversheet. The treatments are of the same overall size and at the same location of the beams. Detailed parameters are listed in the following table.

[0178] The self-sensing control law is implemented using the bridge circuit proposed by Dosch, J. J., Inman, D. J. and Garcia, E., 1992, “A Self-Sensing Piezoelectric Actuator for Collocated Control,” Journal of Intelligent Systems and Structures, Vol.3, pp. 166-185. Capacitors and dummy PZTs (Anderson and Hagood, 1994) are used to balance the bridge. The capacitance of the pure PZT constraining layer is about 44.2 nF. The capacitance of the PZT part of the hybrid constraining layer is about 28 nF. The resistors used in the bridge are 1053 Ω(1%). The differential amplifier 116 is built from the OP470 operational amplifiers. The 790 series power amplifiers 110, 128 from PCB Piezotronics, Inc. are used in the test. The HP35670A spectrum analyzer 130 is used for signal processing. A fiber optical displacement sensor 126 (Philtec Incorporation) is located at the end of the treatment 10 to measure the beam transverse vibration. The PZT patch attached to the bottom of the test beam is used as an exciter 122 to excite the structure. A four-pole Butterworth dual electronic filter 112 from DL Instruments (Model 4302) is used as the low pass filter with a cutoff frequency of 80 Hz (the first resonant frequency of the system is about 30 Hz).

[0179] In FIG. 11, the first resonant responses of the two systems are plotted with respect to the self-sensing control gain. Data points 152 and plot line 150 correspond to treatment with a pure PZT cover sheet and data points 156 and plot line 154 correspond to treatment with a PZT-Al₂O₃cover sheet. The frequency response functions of the systems with different control gains are measured, and the response amplitudes at the first resonant frequencies are used to generate the plot. The input to the system is the voltage applied to the PZT exciter and the output is the beam displacement measured by the optical sensor. It can be seen from FIG. 11 that each system has an optimal control gain. The system with the pure PZT coversheet has an optimal control gain at about 2500 and the system with the hybrid coversheet has an optimal control gain at about 4000. The results shown in FIG. 11 verify that the treatment with an active-passive hybrid constraining layer can outperform the treatment with a purely active constraining layer, both in terms of their open-loop and closed-loop performances. Each with their optimal gains, the closed-loop response of the system with the hybrid coversheet is lower than that of the system with the pure PZT coversheet, as illustrated in FIG. 11. On the other hand, comparing the open-loop responses (control gain is zero), the system with the hybrid coversheet has a peak value of 1.0×10⁻⁶m/V, and the system with the pure PZT coversheet has a peak value of 1.4×10⁻⁶m/V. That is, the treatment with the active-passive hybrid constraining layer also has better fail-safe (open-loop) property than the treatment with the purely active coversheet. These observations are consistent with the analytical results.

[0180] The same conclusion can also be drawn from the impulse responses of the two systems. In this case, the same electric impulse is generated from the signal analyzer as inputs to the power amplifiers connected to the PZT exciters. The time responses of the beams are measured at the optical sensor location. FIG. 12 shows the impulse responses of the open-loop systems. The damping ratios (derived using the logarithmic decrement method [Inman, D. J., 1994, Engineering Vibration, Prentice Hall]) for the treatment with the hybrid coversheet 160 and that with the PZT coversheet 162 are 1.5% and 3.4%, respectively. FIG. 13 shows the impulse responses of the closed-loop systems. The damping ratios are 5.6% for the pure active constraining layer 166 and 8.3% for the HCL 164.

[0181] Note that the experimental condition here is not exactly the same as that in the analytical model. The input to the system is force (voltage to the PZT exciters) instead of the strain field of the host structure. Nevertheless, the test results still match the trend observed in the analysis.

[0182] The present invention has been further tested in still other embodiments. The present invention contemplates numerous other designs, as may be apparent for a particular environment, application or use. For example, the present invention contemplates that multiple sections of active material or passive material may be used in the HCL. Further, it should be apparent that the host structure need not have a uniform strain distribution within the section treated by HCL and may be of various mode shapes.

[0183] If The effects of multiple active/passive sections in the constraining layer have been further examined. Although the present invention contemplates any number of sections, a mathematical model for a three-section constraining layer is used. One skilled in the art will appreciate that this model may be naturally extended to other numbers of sections, other shapes of sections, other configurations of sections, and other placements of sections. Each section can be made of either active or passive materials. In this embodiment, we will focus on four cases where there are three separate sections and where the active material is located at (1) the left section, (2) the right section, (3) the center section, and (4) the two-end sections. To consider the effect of various strain fields in the host structure, the first three modes of a simply supported beam (which consist of both symmetric and anti-symmetric mode shapes) treated by the multi-section HCL are examined. The effects of the in-phase and out-of-phase strain fields on the HCL damping performance can thus be clearly demonstrated using this model and these modes.

[0184] The closed form transfer function matrix of the system to external inputs is derived in this section. To simplify the mathematics, the hybrid constraining layer is assumed to consist of only three sections as shown in FIG. 14. Although only three sections are shown, the present invention contemplates numerous numbers of sections and shapes of sections. FIG. 14 shows a left section 160, a center section 164, and a right section 162. Each section is modeled as an active material with its unique properties and control gain. When zero control gain is applied and passive material properties are set for a specific section, it essentially becomes a passive segment.

[0185] Going through the same procedure, the equations of motion in Laplace domain for the system shown in FIG. 14 can be written as $\begin{array}{cc}-{\rho }_i{h}_i{s}^2{u}_i+E_i{h}_i\frac{{\partial }^2{u}_i}{\partial x^2}-\frac{G_v\left(s\right)}{h_v}\left(u_i-u_{\mathrm{bi}}+h\frac{\partial w_i}{\partial x}\right)=0,{\alpha }_{i-1}L<x<{\alpha }_{i}L& \left(48\right)\\ -{\rho }_b{h}_b{s}^2{u}_{\mathrm{bi}}+E_b{h}_b\frac{{\partial }^2{u}_{\mathrm{bi}}}{\partial x^2}+\frac{G_v\left(s\right)}{h_v}\left(u_i-u_{\mathrm{bi}}+h\frac{\partial w_i}{\partial x}\right)=0,{\alpha }_{i-1}L<x<{\alpha }_{i}L& \left(49\right)\\ -{\left(\rho h\right)}_i{s}^2{w}_i-\frac{{\left(\mathrm{EI}\right)}_i}{b}\frac{{\partial }^4{w}_i}{\partial x^4}+h\frac{G_v\left(s\right)}{h_v}\left(\frac{\partial u_i}{\partial x}-\frac{\partial u_{\mathrm{bi}}}{\partial x}+h\frac{{\partial }^2{w}_i}{\partial x^2}\right)=0,{\alpha }_{i-1}L<x<{\alpha }_{i}L& \left(50\right)\end{array}$

[0186] with

[0187] (ph)_i=p_ih_i+p_bh_b+p_vh_v  (51)

[0188] (EI)_i=E_i+E_bI_b  (52)

[0189] where i=1, 2 and 3 indicate the three sections of the beam, and α₀=0 and α₃=1, u_i, u_biand w_iare the longitudinal displacement of the cover sheet, the longitudinal displacement of the host beam and the transverse displacement of the composite beam in the i^thsection, respectively. E, G_v, I, and p are the Young's modulus, VEM shear modulus (in s domain), bending stiffniess and density of the corresponding materials, respectively, as shown in FIG. 1. b is the width of the beam and L is the length of the beam. The thickness of the different constraining sections is assumed equal (h₁=h₂=h₃). Therefore, h is the same for different sections. Other parameters are shown in FIG. 1.

[0190] The boundary conditions of the system are $\begin{array}{cc}\frac{\partial u_1}{\partial x}-{\Lambda }_1=0,x=0& \left(53\right)\end{array}$

[0191] U₁=0, x=0  (54)

[0192] W₁=0, x=0  (55)

[0193] $\begin{array}{cc}{\left(\mathrm{EI}\right)}_1\frac{{\partial }^2{w}_1}{\partial x^2}=m_0,x=0& \left(56\right)\\ \frac{\partial u_3}{\partial x}-{\Lambda }_3=0,x=L& \left(57\right)\\ \frac{\partial u_{\mathrm{b3}}}{\partial x}=0,x=L& \left(58\right)\end{array}$

[0194] W₃=0, x=  (59)

[0195] $\begin{array}{cc}{\left(\mathrm{EI}\right)}_3\frac{{\partial }^2{w}_3}{\partial x^2}=m_1,x=L& \left(60\right)\end{array}$

[0196] Where m₀and m₁are the moments applied at the left and right ends of the beam and Λ_iis the free strain of the i-th constraining section (i=1,2,3).

[0197] The connecting conditions at x=α_iL (i=1, 2) are

[0198] u_i=u_i−1, x=α_iL  (61)

[0199] u_bi=u_bi+1, x=α_iL  (62)

[0200] w_i=w_i+1, x=α_iL  (63)

[0201] $\begin{array}{cc}\frac{\partial w_i}{\partial x}=\frac{\partial w_{i-1}}{\partial x},x={\alpha }_{i}L& \left(64\right)\\ E_i{h}_i\left(\frac{\partial u_i}{\partial x}-{\Lambda }_i\right)=E_{i+1}{h}_{i+1}\left(\frac{\partial u_{i+1}}{\partial x}-{\Lambda }_{i+1}\right),x={\alpha }_{i}L& \left(65\right)\\ \frac{\partial u_{\mathrm{bi}}}{\partial x}=\frac{\partial u_{\mathrm{bi}+1}}{\partial x},x={\alpha }_{i}L& \left(66\right)\\ {\left(\underset{.}{\mathrm{EI}}\right)}_i\frac{{\partial }^2{w}_i}{\partial x^2}={\left(\mathrm{EI}\right)}_{i+1}\frac{{\partial }^2{w}_{i+1}}{\partial x^2},x={\alpha }_{i}L& \left(67\right)\\ {\left(\underset{.}{\mathrm{EI}}\right)}_i\frac{{\partial }^3{w}_i}{\partial x^3}={\left(\mathrm{EI}\right)}_{i+1}\frac{{\partial }^3{w}_{i+1}}{\partial x^3},x={\alpha }_{i}L& \left(68\right)\end{array}$

[0202] Note that the Equations 61-64 are essentially displacement continuity and Equation 65-68 are force continuity at the connecting points.

[0203] Define $\begin{array}{cc}D\left(x,s\right)=\left\{\begin{array}{c}{w}_i\\ \frac{\partial w_i}{\partial x}\\ u_i\\ u_{\mathrm{bi}}\end{array}\right\},Q_i\left(x,s\right)=\left\{\begin{array}{c}\frac{{\partial }^2{w}_i}{\partial x^2}\\ \frac{{\partial }^3{w}_i}{\partial x^3}\\ \frac{\partial u_i}{\partial x}\\ \frac{\partial u_{\mathrm{bi}}}{\partial x}\end{array}\right\},y_i\left(x,s\right)=\left\{\begin{array}{c}D\left(x,s\right)\\ Q_i\left(x,s\right)\end{array}\right\},i=1,2,3& \left(69\right)\end{array}$

[0204] The homogenous equations of motion 48-51 can be written as $\begin{array}{cc}\frac{\partial y_i}{\partial x}=F_i\left(s\right)y_i\left(x,s\right),i=1,2,3& \left(70\right)\end{array}$

[0205] where $\begin{array}{cc}{F}_i\left(s\right)=\left[\begin{array}{cccccccc}0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 1& 0& 0\\ \frac{-{{\mathrm{bs}}^2\left(\rho h\right)}_i}{{\left(\mathrm{EI}\right)}_i}& 0& 0& 0& \frac{{\mathrm{bh}}^2{G}_v}{{\left(\mathrm{EI}\right)}_i{h}_v}& 0& \frac{\mathrm{bh}{G}_v}{{\left(\mathrm{EI}\right)}_i{h}_v}& \frac{-\mathrm{bh}{G}_v}{{\left(\mathrm{EI}\right)}_i{h}_v}\\ 0& \frac{h{G}_v}{E_i{h}_i{h}_v}& \frac{G_v+{\rho }_i{h}_i{h}_v{s}^2}{E_i{h}_i{h}_v}& \frac{-G_v}{E_i{h}_i{h}_v}& 0& 0& 0& 0\\ 0& \frac{-h{G}_v}{E_b{h}_b{h}_v}& \frac{-G_v}{E_b{h}_b{h}_v}& \frac{G_v+{\rho }_b{h}_b{h}_v{s}^2}{E_b{h}_b{h}_v}& 0& 0& 0& 0\end{array}\right]& \left(71\right)\end{array}$

[0206] Follow the procedure in Yang, B., 1994, “Distributed Transfer Function Analysis of Complex Distributed Parameter Systems,” Journal of Applied Mechanics, Vol. 61, pp. 84-92, the boundary conditions 53-60 can be written as

[0207] B₁(s)y₁(0,s)=κ₁(s)Λ₁+κ_m0(s)m₀  (72)

[0208] B₃(s)y₃(L,s)=κ₃(s)Λ₃+κ_m1(s)m₁  (73)

[0209] where $\begin{array}{cc}{B}_1\left(s\right)=\left[\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\end{array}\right]& \left(74\right)\\ B_3\left(s\right)=\left[\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\\ 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\end{array}\right]& \left(75\right)\\ {\kappa }_1={\kappa }_3={\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]}^T,{\kappa }_{\mathrm{m0}}={\left[\begin{array}{cccc}0& 0& 0& \frac{1}{{\left(\mathrm{EI}\right)}_1}\end{array}\right]}^T,{\kappa }_{\mathrm{m1}}={\left[\begin{array}{cccc}0& 0& 0& \frac{1}{{\left(\mathrm{EI}\right)}_3}\end{array}\right]}^T& \left(76\right)\end{array}$

[0210] Define the following matrices $\begin{array}{cc}{M}_1=\left[\begin{array}{c}{B}_1\\ 0_{4\times 8}\end{array}\right],N_1=\left[\begin{array}{cc}0& 0\\ I& 0\end{array}\right],{\gamma }_1=\left\{\begin{array}{c}{\kappa }_1{\Lambda }_1+{\kappa }_{\mathrm{m0}}{m}_0\\ D\left({\alpha }_{1}L\right)\end{array}\right\}& \left(77\right)\\ M_2=\left[\begin{array}{cc}I& 0\\ 0& 0\end{array}\right],N_2=\left[\begin{array}{cc}0& 0\\ I& 0\end{array}\right],{\gamma }_2=\left\{\begin{array}{c}D\left({\alpha }_{1}L\right)\\ D\left({\alpha }_{2}L\right)\end{array}\right\}& \left(78\right)\\ M_3=\left[\begin{array}{cc}I& 0\\ 0& 0\end{array}\right],N_3=\left[\begin{array}{c}0\\ B_3\left(s\right)\end{array}\right],{\gamma }_3=\left\{\begin{array}{c}D\left({\alpha }_{2}L\right)\\ {\kappa }_3{\Lambda }_3+{\kappa }_{\mathrm{m1}}{m}_1\end{array}\right\}& \left(79\right)\end{array}$

[0211] The pseudo boundary conditions for each section can be written as

[0212] M_i(s)y_i(α_i−1L,s)+N_i(s)y_i(α_iL,s)=γ_i, i=1,2,3  (80)

[0213] According to Yang, B. and Tan, C. A., 1992, “Transfer Functions of One-Dimensional Distributed Parameter Systems,” Journal of Applied Mechanics, Vol. 59, pp. 1009-1014, the solutions to the homogenous equations of motion 70 and boundary conditions 80 are $\begin{array}{cc}{y}_i\left(x,s\right)\equiv \left\{\begin{array}{c}D\left(x,s\right)\\ Q_i\left(x,s\right)\end{array}\right\}=H_i\left(x,s\right){\gamma }_i\left(s\right),i=1,2,3& \left(81\right)\end{array}$

[0214] where $\begin{array}{cc}{H}_i\left(x,s\right)\equiv \left[\begin{array}{cc}{H}_{\mathrm{iaa}}& H_{\mathrm{iab}}\\ H_{\mathrm{iba}}& H_{\mathrm{ibb}}\end{array}\right]={e^{F_i\left(s\right)x}\left[M_i\left(s\right)e^{F_i{\alpha }_{i-1}L}+N_i\left(s\right)e^{F_i{\alpha }_{i}L}\right]}^{-1},i=1,2,3& \left(82\right)\end{array}$

[0215] Note that because of Equation 78, the displacement continuity conditions 61-64 have already been satisfied. All we need to do now is to enforce the force continuity conditions. Equations 65-68 can be written into the following matrix form

[0216] S_iQ_i(α_iL,s)−S_i+1Q_i+1(α_iL,s)=r_iΛ_i−r_i+1Λ_i+1, i=1,2  (83)

[0217] where $\begin{array}{cc}{S}_i=\left[\begin{array}{cccc}0& 0& \frac{E_i{h}_i}{E_2{h}_2}& 0\\ 0& 0& 0& 1\\ \frac{{\left(\mathrm{EI}\right)}_i}{{\left(\mathrm{EI}\right)}_2}& 0& 0& 0\\ 0& \frac{{\left(\mathrm{EI}\right)}_i}{{\left(\mathrm{EI}\right)}_2}& 0& 0\end{array}\right],r_i=\left\{\begin{array}{c}\frac{E_i{h}_i}{E_2{h}_2}\\ 0\\ 0\\ 0\end{array}\right\},i=1,2,3& \left(84\right)\end{array}$

[0218] From Equations 81, we have

[0219] Q₁(α₁L)=H_1ba(α₁L)(κ₁Λ₁+κ_m0M₀)+H_1bb(α₁L)D(α₁L)  (85)

[0220] Q₂(α₁L)=H_2ba(α₁L)D(α₁L)+H_2bb(α₁L)D(α₂L)  (86)

[0221] Q₂(α₂L)=H_2ba(α₂L)D(α₁L)+H_2bb(α₂L)D(α₂L)  (87)

[0222] Q₃(Ε₂L)=H_3ba(α₂L)D(α₂L)+H_3bb(α₂L)(κ₃Λ₃+κ_m1M₁)  (88)

[0223] Substitute Equations 85-88 into 83, the displacements at the connecting points can be written as $\begin{array}{cc}{K}_{\mathrm{op}}\left\{\begin{array}{c}D\left({\alpha }_{1}L\right)\\ D\left({\alpha }_{2}L\right)\end{array}\right]=R_1{\Lambda }_1+R_2{\Lambda }_2+R_3{\Lambda }_3+R_{\mathrm{m0}}{m}_0+R_{\mathrm{m1}}{m}_1& \left(89\right)\end{array}$

[0224] where $\begin{array}{cc}{K}_{\mathrm{op}}=\left[\begin{array}{cc}{S}_1{H}_{1\mathrm{bb}}\left({\alpha }_{1}L\right)-S_2{H}_{2\mathrm{ba}}\left({\alpha }_{1}L\right)& -S_2{H}_{2\mathrm{bb}}\left({\alpha }_{1}L\right)\\ S_2{H}_{2\mathrm{ba}}\left({\alpha }_{2}L\right)& S_2{H}_{2\mathrm{bb}}\left({\alpha }_{2}L\right)-S_3{H}_{3\mathrm{ba}}\left({\alpha }_{2}L\right)\end{array}\right]& \left(90\right)\\ R_1=\left\{\begin{array}{c}{r}_1-S_1{H}_{1\mathrm{ba}}\left({\alpha }_{1}L\right){\kappa }_1\\ 0_{4\times 1}\end{array}\right\},R_2=\left\{\begin{array}{c}-r_2\\ r_2\end{array}\right\},R_3=\left\{\begin{array}{c}{0}_{4\times 1}\\ -r_3+S_3{H}_{3\mathrm{bb}}\left({\alpha }_{2}L\right){\kappa }_3\end{array}\right\}& \left(91\right)\\ R_{\mathrm{m0}}=\left\{\begin{array}{c}-S_1{H}_{1\mathrm{ba}}\left({\alpha }_{1}L\right){\kappa }_{\mathrm{m0}}\\ 0_{4\times 1}\end{array}\right\},R_{\mathrm{m1}}=\left\{\begin{array}{c}{0}_{4\times 1}\\ S_3{H}_{3\mathrm{bb}}\left({\alpha }_{2}L\right){\kappa }_{\mathrm{m1}}\end{array}\right\}& \left(92\right)\end{array}$

[0225] For passive constraining materials or the open-loop system, the free strains κ₁, κ₂and κ₃will be zero and Equation 89 can be used to solve for D((α₁L) and D(α₂L). The system response can then be found from Equation 81.

[0226] If any one of the constraining sections is an active material, the free strain Λ_iwill be related to the control algorithm used. Here, a self-sensing actuation is utilized [Dosch, J. J., Inman, D. J. and Garcia, E., 1992, “A Self-Sensing Piezoelectric Actuator for Collocated Control,” Journal of Intelligent Systems and Structures, Vol.3, pp. 166-185; Shen, I. Y., 1997, “A Variational Formulation, a Work-Energy Relation and Damping Mechanisms of Active Constrained Layer Treatments,” ASME Journal of Vibration and Acoustics, Vol.119, pp. 192-199; and Baz, A., 1997, “Optimization of Energy Dissipation Characteristics of Active Constrained Layer Damping,” Smart Materials and Structures, Vol.6, pp. 360-368] to ensure stability. Each section is controlled with a local control law and the free strain can be expressed as

[0227] Λ_i=g_is[u_i(α_i−1L)−u_i(α_iL)]=g_isT[D(α_i−1L)−D(α_iL)], i=1,2,3  (93)

[0228] with

[0229] T=[0 0 1 0]  (94)

[0230] Consider Equation 81, we have

[0231] D(0,s)=H_1aa(0)(κ₁Λ₁+κ_m0m₀)+H_1ab(0)D(α₁L)  (95)

[0232] D(L,s)=H_3aa(L)D(α₂L)+H_3ab(L)(κ₃Λ₃+κ_m1m₁)  (96)

[0233] Substitute Equations 95 and 96 into Equation 93, the free strains of the constraining sections can be written as $\begin{array}{cc}{\Lambda }_1=C_{\mathrm{m0}}{m}_0+C_{\mathrm{d1}}\left\{\begin{array}{c}D\left({\alpha }_{1}L\right)\\ D\left({\alpha }_{2}L\right)\end{array}\right\},{\Lambda }_2=C_{\mathrm{d2}}\left\{\begin{array}{c}D\left({\alpha }_{1}L\right)\\ D\left({\alpha }_{2}L\right)\end{array}\right\},{\Lambda }_3=C_{\mathrm{m1}}{m}_1+C_{\mathrm{d3}}\left\{\begin{array}{c}D\left({\alpha }_{1}L\right)\\ D\left({\alpha }_{2}L\right)\end{array}\right\}& \left(97\right)\end{array}$

[0234] where $\begin{array}{cc}{C}_{\mathrm{m0}}=\frac{g_1{\mathrm{sTH}}_{1\mathrm{aa}}\left(0\right){\kappa }_{\mathrm{m0}}}{1-g_1{\mathrm{sTH}}_{1\mathrm{aa}}\left(0\right){\kappa }_1},C_{\mathrm{m1}}=-\frac{g_3{\mathrm{sTH}}_{3\mathrm{ab}}\left(L\right){\kappa }_{\mathrm{m1}}}{1+g_3{\mathrm{sTH}}_{3\mathrm{ab}}\left(L\right){\kappa }_3}& \left(98\right)\\ C_{\mathrm{d1}}=-\frac{g_1\mathrm{sT}\left[H_{1\mathrm{ab}}\left(0\right)-I\right]\left[I0\right]}{1-g_1{\mathrm{sTH}}_{1\mathrm{aa}}\left(0\right){\kappa }_1},C_{\mathrm{d2}}=g_2\mathrm{sT}\left[I-I\right],C_{\mathrm{d3}}=\frac{g_3\mathrm{sT}\left[I-H_{3\mathrm{aa}}\left(L\right)\right]\left[0I\right]}{1+g_3{\mathrm{sTH}}_{3\mathrm{ab}}\left(L\right){\kappa }_3}& \left(99\right)\end{array}$

[0235] Substitute Equation 97 into Equation 89, the displacements at the connecting points in the closed-loop system can be derived as $\begin{array}{cc}{K}_{\mathrm{c1}}\left\{\begin{array}{c}D\left({\alpha }_{1}L\right)\\ D\left({\alpha }_{2}L\right)\end{array}\right\}=\left[R_{\mathrm{m0}}+R_1{C}_{\mathrm{m0}}\right]m_0+\left[R_{\mathrm{m1}}+R_3{C}_{\mathrm{m1}}\right]m_1& \left(100\right)\end{array}$