Active Constrained Layer Damping of Smart Skew Laminated Composite Plates Using 1–3 Piezoelectric Composites

Abstract

This paper deals with the analysis of active constrained layer damping (ACLD) of smart skew laminated composite plates. The constraining layer of the ACLD treatment is composed of the vertically/obliquely reinforced 1–3 piezoelectric composites (PZCs). A finite element model has been developed for accomplishing the task of the active constrained layer damping of skew laminated symmetric and antisymmetric cross-ply and antisymmetric angle-ply composite plates integrated with the patches of such ACLD treatment. Both in-plane and out-of-plane actuations by the constraining layer of the ACLD treatment have been utilized for deriving the finite element model. The analysis revealed that the vertical actuation dominates over the in-plane actuation. Particular emphasis has been placed on investigating the performance of the patches when the orientation angle of the piezoelectric fibers of the constraining layer is varied in the two mutually orthogonal vertical planes. Also, the effects of varying the skew angle of the substrate laminated composite plates and different boundary conditions on the performance of the patches have been studied. The analysis reveals that the vertically and the obliquely reinforced 1–3 PZC materials should be used for achieving the best control authority of ACLD treatment, as the boundary conditions of the smart skew laminated composite plates are simply supported and clamped-clamped, respectively.

1. Introduction

Extensive research on the use of piezoelectric materials for making distributed actuators and sensors of light weight flexible smart structures has been carried out during the past several years [1–16]. The distributed piezoelectric actuators and sensors are either mounted on or embedded into the host flexible light weight structures. When they are activated with proper control voltage, the resulting structures attain self-controlling and self-sensing capabilities. Such flexible structures having built-in mechanism for self-controlling and self-sensing capabilities are customarily called “smart structures.” In most of the work on smart structures, the distributed actuators were considered to be made of the existing monolithic piezoelectric materials. The magnitudes of the piezoelectric coefficients of the existing monolithic piezoelectric materials are very small. Hence, the distributed actuators made of these materials need large control voltage for satisfactory control of smart structures. The further research on the efficient use of these low-control authority monolithic piezoelectric materials led to the development of the active constrained layer damping (ACLD) treatment [17]. The ACLD treatment consists of a constraining layer made of the piezoelectric materials and a constrained viscoelastic layer. The flexural vibration control by the constrained layer damping treatment is attributed to the dissipation of energy in the constrained viscoelastic layer due to its transverse shear deformations. The constraining layer of the activated ACLD treatment increases the transverse shear deformations of the viscoelastic constrained layer over its passive counterpart resulting in improved damping of the host structures. The control voltage required to cause transverse shear deformations in the low-stiff constrained viscoelastic layer of the ACLD treatment is compatible with the low-control authority of the monolithic piezoelectric materials. Hence, the piezoelectric materials perform much better to attenuate the vibrations of smart structures when they are used for the constraining layer of the ACLD treatment than when they are directly bonded to the flexible host structures. If the constraining piezoelectric layer of the ACLD treatment is not activated with the control voltage, the treatment causes the standard passive constrained layer damping of the smart structure. Thus, the ACLD treatment being under operation provides the attributes of both passive and active dampings and acts as an inbuilt fail-safe mechanism. Since its inception, extensive research has been carried out to investigate the performance of the ACLD treatment for active damping of smart structures [18–25].

Piezoelectric composite (PZC) materials have been emerged as the new class of smart materials. Such PZC materials are composed of piezoelectric fiber reinforcements and epoxy matrix. These PZC materials provide wide range of effective material properties, good conformability, and strength integrity [26]. Among the various PZC materials studied by the researchers, the vertically and the obliquely reinforced 1–3 PZC materials are commercially available [27] and are being effectively used for underwater transducers, medical imaging applications, and high frequency ultrasonic transducers [26]. The constructional feature of a lamina made of the vertically reinforced 1–3 PZC material is that the piezoelectric fibers are vertically aligned across the thickness of the lamina. In case of the obliquely reinforced 1–3 PZC, the piezoelectric fibers are obliquely aligned in the vertical plane across the thickness of the lamina. These PZCs are characterized by improved mechanical performance, electro-mechanical coupling characteristics, and acoustic impedance matching over the existing monolithic piezoelectric materials [26]. Research on PZC materials is mainly concerned with the micromechanical analysis of these materials [28–34]. Recently, Ray and his coworkers [35–37] investigated the performance of these 1–3 PZC materials for active damping of linear and nonlinear vibrations of composite beams, plates and shells.

Skew laminated composite plates are widely used in engineering applications such as aircraft wings and marine industries and are highly prone to undergoing vibrations. Researchers extensively investigated the free vibrational characteristics of such skew plates. For example, Krishnan and Deshpande [38] carried out the free vibration analyses of cantilevered skew isotropic platesand three-layered symmetric cross-ply laminates by deriving two finite element models. Krishna Reddy and Palaninathan [39] developed a general high precision triangular plate bending finite element for the free vibration analysis of laminated skew plates. Babu and Kant [40] developed shear deformable finite element models for the buckling analysis of skew composite plates and panels. Garg et al. [41] developed a simple C° isoparametric finite element model, based on a higher order shear deformation theory for the free vibration analysis of isotropic, orthotropic, and layered anisotropic composite and sandwich skew laminates.

The review of the existing literature on the skew plates reveals that the attention has not yet been focused on investigating the active control of vibrations of skew laminated composite plates using piezoelectric composites. In this paper, authors intend to investigate the active constrained layer damping (ACLD) of skew laminated composite plates. For such investigation, three-dimensional analysis of ACLD of skew laminated composite plates integrated with the patches of ACLD treatment has been carried out by the finite element method. The constraining layer of the ACLD treatment is considered to be made of the vertically or the obliquely reinforced 1–3 PZC materials. Particular emphasis has been placed on investigating the effect of variation of piezoelectric fiber orientation angle on the performance of the ACLD patches for controlling the vibrations of the skew laminated composite plates.

2. Finite Element Model of Smart Skew Laminated Composite Plate

Figure 1(a) illustrates the schematic diagram of a smart skew laminated composite plate composed of 𝑁 number of orthotropic layers. The length and the skew width of the plate are denoted by 𝑎 and 𝑏, respectively. The top surface of the plate is integrated with the skewed patches of the ACLD treatment. The constraining layer of the ACLD treatment is made of the vertically/obliquely reinforced 1–3 piezoelectric composite (PZC) material. The notations ℎ, ℎ_𝑝, and ℎ_𝑣 represent the thicknesses of the substrate skew laminated plate, the piezoelectric composite layer, and the viscoelastic layer, respectively. The skew angle of the plate is denoted by 𝛼. The middle plane of the substrate composite plate is considered as the reference plane. The origin of the laminate coordinate system (𝑥𝑦𝑧) is located at one corner of the reference plane such that the lines 𝑦 = 0 and 𝑦=𝑏cos𝛼 represent the two opposite boundaries of the substrate skew composite plates, while the lines 𝑥=𝑦tan𝛼 and 𝑥=𝑎+𝑦tan𝛼 describe the two opposite skewed edges of the plate. Denoting by 𝑘(=1,2,3,…,𝑁+2) the layer number of any layer of the overall plate, the thickness coordinates 𝑧 of the top and the bottom surfaces of any (𝑘th) layer are represented by ℎ_𝑘+1 and ℎ_𝑘, respectively. The fiber orientation angle in any layer of the substrate plate in the plane (𝑥𝑦) of the lamina with respect to the laminate coordinate system is denoted by 𝜃. The piezoelectric fibers in the constraining layer made of the obliquely reinforced 1–3 PZC material are coplanar with the vertical 𝑥𝑧- or 𝑦𝑧-plane making an angle 𝜓 with respect to the 𝑧-axis as shown in Figure 1(b). If the value of 𝜓 is 0°, the layer becomes vertically reinforced 1–3 PZC layer. The overall skew composite plate being studied here is thin, and consequently, the first order shear deformation theory (FSDT) can be used to model the axial displacements in all the layers of the overall plate. In Figure 2, the kinematics of axial deformations of the overall plate based on the FSDT has been illustrated. Displayed in this figure, the variables 𝑢₀ and 𝑣₀ represent the generalized translational displacement of a point (𝑥,𝑦) on the reference plane (𝑧=0) along 𝑥- and 𝑦-directions, respectively; 𝜃_𝑥, 𝜙_𝑥, and 𝛾_𝑥 denote the generalized rotations of the normal to the middle planes of the substrate plate, the viscoelastic layer and the 1–3 PZC layer, respectively, about the 𝑦-axis, while 𝜃_𝑦,  𝜙_𝑦 and 𝛾_𝑦 represent the generalized rotations of the same about the 𝑥-axis, respectively. According to the kinematics of deformations illustrated in Figure 2, the axial displacements 𝑢 and 𝑣 at any point in any layer of the overall plate along 𝑥- and 𝑦-directions, respectively, can be expressed as 𝑢𝑥,𝑦,𝑧,𝑡=𝑢₀+ℎ𝑥,𝑦,𝑡𝑧−𝑧−2𝜃_𝑥+ℎ𝑥,𝑦,𝑡𝑧−2−𝑧−ℎ_𝑁+2𝜙_𝑥+𝑥,𝑦,𝑡𝑧−ℎ_𝑁+2𝛾_𝑥,𝑣𝑥,𝑦,𝑡𝑥,𝑦,𝑧,𝑡=𝑣₀+ℎ𝑥,𝑦,𝑡𝑧−𝑧−2𝜃_𝑦+ℎ𝑥,𝑦,𝑡𝑧−2−𝑧−ℎ_𝑁+2𝜙_𝑦+𝑥,𝑦,𝑡𝑧−ℎ_𝑁+2𝛾_𝑦𝑥,𝑦,𝑡, (1)(2) in which a function within the bracket ⟨⋅⟩ represents an appropriate singularity function, which satisfies the continuity of displacements between two adjacent continua. Since the transverse actuation of the constraining layer of the ACLD treatment will be used for the flexural vibration control of the plate, the transverse normal strain in the overall plate must be considered in the model. Hence, as the plate considered here is thin, the transverse displacements (𝑤) at any point in the substrate plate, the viscoelastic layer, and the 1–3 PZC layer are assumed to be linearly varying across their thicknesses. Thus, similar to the axial displacement, the transverse displacements at any point in the overall plate can be expressed as 𝑤𝑥,𝑦,𝑧,𝑡=𝑤₀+ℎ𝑥,𝑦,𝑡𝑧−𝑧−2𝜃_𝑧+ℎ𝑥,𝑦,𝑡𝑧−2−𝑧−ℎ_𝑁+2𝜙_𝑧+𝑥,𝑦,𝑡𝑧−ℎ_𝑁+2𝛾_𝑧𝑥,𝑦,𝑡, (3) in which 𝑤₀ refers to the transverse displacement at any point on the reference plane, and 𝜃_𝑧, 𝜙_𝑧, and 𝛾_𝑧 are the generalized displacements representing the gradients of the transverse displacement in the substrate plate, the viscoelastic layer, and the 1–3 PZC layer, respectively, with respect to the thickness coordinate (𝑧).

(a)

(b)

Figure 1 

(a) Schematic representation of a skew laminated composite plate integrated with the patches of ACLD treatment composed of vertically/obliquely reinforced 1–3 piezoelectric composite constraining layer. (b) Schematic diagram of layers of obliquely reinforced 1–3 piezoelectric composite.

Figure 2 

Kinematics of deformations.

For the ease of analysis, the generalized displacement variables are grouped into the following two vectors: 𝑑_𝑡=𝑢₀𝑣₀𝑤₀^𝑇,𝑑_𝑟=𝜃_𝑥𝜃_𝑦𝜃_𝑧𝜙_𝑥𝜙_𝑦𝜙_𝑧𝛾_𝑥𝛾_𝑦𝛾_𝑧^𝑇. (4) In order to implement the selective integration rule for computing the element stiffness matrices corresponding to the transverse shear deformations, the state of strain at any point in the overall plate is divided into the following two strain vectors {𝜀_𝑏} and {𝜀_𝑠}: 𝜀_𝑏=𝜀_𝑥𝜀_𝑦𝜀_𝑥𝑦𝜀_𝑧^𝑇,𝜀_𝑠=𝜀_𝑥𝑧𝜀_𝑦𝑧^𝑇 (5) in which 𝜀_𝑥, 𝜀_𝑦, and 𝜀_𝑧 are the normal strains along 𝑥-, 𝑦-, and 𝑧-directions, respectively; 𝜀_𝑥𝑦 is the in-plane shear strain; 𝜀_𝑥𝑧 and 𝜀_𝑦𝑧 are the transverse shear strains. Using the displacement fields given by (1)–(3) and the linear strain-displacement relations, the vectors {𝜀_𝑏}_𝑐, {𝜀_𝑏}_𝑣, and {𝜀_𝑏}_𝑝 defining the state of in-plane and transverse normal strains at any point in the substrate skew composite plate, the viscoelastic layer, and the active constraining layer, respectively, can be expressed as 𝜀_𝑏𝑐=𝜀_𝑏𝑡+𝑍₁𝜀_𝑏𝑟,𝜀_𝑏𝑣=𝜀_𝑏𝑡+𝑍₂𝜀_𝑏𝑟,𝜀_𝑏𝑝=𝜀_𝑏𝑡+𝑍₃𝜀_𝑏𝑟. (6) Similarly, the vectors {𝜀_𝑠}_𝑐, {𝜀_𝑠}_𝑣, and {𝜀_𝑠}_𝑝 defining the state of transverse shear strains at any point in the substrate composite plate, the viscoelastic layer, and the active constraining layer, respectively, can be expressed as 𝜀_𝑠𝑐=𝜀_𝑠𝑡+𝑍₄𝜀_𝑠𝑟,𝜀_𝑠𝑣=𝜀_𝑠𝑡+𝑍₅𝜀_𝑠𝑟,𝜀_𝑠𝑝=𝜀_𝑠𝑡+𝑍₆𝜀_𝑠𝑟. (7) The various matrices appearing in (6) and (7) have been defined in the Appendix while the generalized strain vectors are given by𝜀_𝑏𝑡=𝜕𝑢₀𝜕𝑥𝜕𝑣₀𝜕𝑦𝜕𝑢₀+𝜕𝑦𝜕𝑣₀0𝜕𝑥^𝑇,𝜀_𝑠𝑡=𝜕𝑤₀𝜕𝑥𝜕𝑤₀𝜕𝑦^𝑇,𝜀_𝑏𝑟=𝜕𝜃_𝑥𝜕𝑥𝜕𝜃_𝑦𝜕𝑦𝜕𝜃_𝑥+𝜕𝑦𝜕𝜃_𝑦𝜃𝜕𝑥^𝑧𝜕𝜙_𝑥𝜕𝑥𝜕𝜙_𝑦𝜕𝑦𝜕𝜙_𝑥+𝜕𝑦𝜕𝜙_𝑦𝜙𝜕𝑥^𝑧𝜕𝛾_𝑥𝜕𝑥𝜕𝛾_𝑦𝜕𝑦𝜕𝛾_𝑥+𝜕𝑦𝜕𝛾_𝑦𝛾𝜕𝑥^𝑧𝑇,𝜀_𝑠𝑟=𝜃_𝑥𝜃_𝑦𝜙_𝑥𝜙_𝑦𝛾_𝑥𝛾_𝑦𝜕𝜃_𝑧𝜕𝑥𝜕𝜃_𝑧𝜕𝑦𝜕𝜙_𝑧𝜕𝑥𝜕𝜙_𝑧𝜕𝑦𝜕𝛾_𝑧𝜕𝑥𝜕𝛾_𝑧𝜕𝑦^𝑇. (8)

Similar to the strain vectors given by (5), the state of stresses at any point in the overall plat is described by the following stress vectors: 𝜎_𝑏=𝜎_𝑥𝜎_𝑦𝜎_𝑥𝑦𝜎_𝑧^𝑇,𝜎_𝑠=𝜎_𝑥𝑧𝜎_𝑦𝑧^𝑇, (9) where 𝜎_𝑥, 𝜎_𝑦, and 𝜎_𝑧 are the normal stresses along 𝑥-, 𝑦- and 𝑧-directions, respectively; 𝜎_𝑥𝑦 is the in-plane shear stress; 𝜎_𝑥𝑧 and 𝜎_𝑦𝑧 are the transverse shear stresses. The constitutive relations for the material of any orthotropic layer of the substrate plate are given by 𝜎^𝑘_𝑏=𝐶^𝑘_𝑏𝜀^𝑘_𝑏,𝜎^𝑘_𝑠=𝐶^𝑘_𝑠𝜀^𝑘_𝑠(𝑘=1,2,3,…,𝑁), (10) where the elastic coefficient matrices are 𝐶^𝑘_𝑏=𝐶^𝑘₁₁𝐶^𝑘₁₂𝐶^𝑘₁₆𝐶^𝑘₁₃𝐶^𝑘₁₂𝐶^𝑘₂₂𝐶^𝑘₂₆𝐶^𝑘₂₃𝐶^𝑘₁₆𝐶^𝑘₂₆𝐶^𝑘₆₆𝐶^𝑘₃₆𝐶^𝑘₁₃𝐶^𝑘₂₃𝐶^𝑘₃₆𝐶^𝑘₃₃,𝐶^𝑘_𝑠=𝐶^𝑘₅₅𝐶^𝑘₄₅𝐶^𝑘₄₅𝐶^𝑘₄₄, (11) and 𝐶^𝑘_𝑖𝑗(𝑖,𝑗=1,2,3,…,6) are the transformed elastic coefficients with respect to the reference coordinate system. The material of the viscoelastic layer is assumed to be linearly viscoelastic and homogenous isotropic and is modeled by using the complex modulus approach. Thus, the shear modulus 𝐺 and Young’s modulus 𝐸 of the viscoelastic material are given by 𝐺=𝐺^′1+𝑖𝜂,𝐸=2𝐺(1+𝜈), (12) in which 𝐺^′ is the storage modulus, 𝜈 is Poisson’s ratio, and 𝜂 is the loss factor at a particular operating temperature and frequency. Employing the complex modulus approach, the constitutive relations for the material of the viscoelastic layer (𝑘=𝑁+1) can also be represented by (10) with 𝐶^𝑘_𝑖𝑗(𝑖,𝑗=1,2,3,…,6) being the complex elastic constants [29, 30]. The constraining PZC layer will be subjected to the applied electric field (𝐸_𝑧) acting across its thickness (i.e., along the 𝑧-direction) only. Accordingly, the constitutive relations for the 1–3 PZC material with respect to the laminate coordinate system (𝑥𝑦𝑧) can be expressed as 𝜎^𝑘_𝑏=𝐶^𝑘_𝑏𝜀^𝑘_𝑏+𝐶^𝑘_𝑏𝑠𝜀^𝑘_𝑠−𝑒_𝑏𝐸_𝑧,𝜎^𝑘_𝑠=𝐶^𝑘_𝑏𝑠𝜀^𝑘_𝑏+𝐶^𝑘_𝑠𝜀^𝑘_𝑠−𝑒_𝑠𝐸_𝑧𝐷_𝑧=𝑒_𝑏^𝑇𝜀^𝑘_𝑏+𝑒_𝑠^𝑇𝜀^𝑘_𝑠+𝜀₃₃𝐸_𝑧𝑘=𝑁+2. (13) Here, 𝐷_𝑧 represents the electric displacement along the 𝑧-direction and 𝜀₃₃ is the dielectric constant. It may be noted from the previous form of the constitutive relations that the transverse shear strains are coupled with the in-plane normal strains due to the orientation of piezoelectric fibers in the vertical 𝑥𝑧- or 𝑦𝑧-plane, and the corresponding coupling elastic constant matrices[𝐶^𝑁+2_𝑏𝑠] are given by 𝐶^𝑁+2_𝑏𝑠=𝐶^𝑁+2₁₅0𝐶^𝑁+2₂₅00𝐶^𝑁+2₄₆𝐶^𝑁+2₃₅0or𝐶^𝑁+2_𝑏𝑠=0𝐶^𝑁+2₁₄0𝐶^𝑁+2₂₄𝐶^𝑁+2₅₆00𝐶^𝑁+2₃₄ (14) as the piezoelectric fibers are coplanar with the vertical 𝑥𝑧- or 𝑦𝑧-plane, respectively. Note that if the fibers are coplanar with both 𝑥𝑧- and 𝑦𝑧-planes (i.e., 𝜓=0^∘), this coupling matrix becomes a null matrix. Also, the piezoelectric constant matrices {𝑒_𝑏} and {𝑒_𝑠} appearing in (13) contain the following transformed effective piezoelectric coefficients of the 1–3 PZC: 𝑒_𝑏=𝑒₃₁𝑒₃₂𝑒₃₆𝑒₃₃^𝑇,𝑒_𝑠=𝑒₃₅𝑒₃₄^𝑇. (15) The total potential energy 𝑇_𝑝 and the kinetic energy 𝑇_𝑘 of the overall plate/ACLD system are given by [35] 𝑇_𝑝=12^𝑁+2_𝑘=1Ω𝜀^𝑘_𝑏^𝑇𝜎^𝑘_𝑏+𝜀^𝑟_𝑠^𝑇𝜎^𝑘_𝑠−𝑑Ω_Ω𝐷_𝑧𝐸_𝑧𝑑Ω−_𝐴{𝑑}^𝑇𝑓𝑇𝑑𝐴,_𝑘=12^𝑁+2_𝑘=1Ω𝜌^𝑘𝑢̇²+𝑣̇²+𝑤̇²𝑑Ω, (16) in which 𝜌^𝑘 is the mass density of the 𝑘th layer, {𝑓} is the externally applied surface traction acting over a surface area 𝐴, and Ω represents the volume of the concerned layer. Since the plates under study are considered to be thin, the rotary inertia of the overall plate has been neglected in estimating the kinetic energy. The overall plate is discretized by eight-noded isoparametric quadrilateral elements. Following (4), the generalized displacement vectors, associated with the 𝑖th (𝑖=1,2,3,…,8) node of the element, can be written as 𝑑_𝑡𝑖=𝑢_0𝑖𝑣_0𝑖𝑤_0𝑖^𝑇,𝑑_𝑟𝑖=𝜃_𝑥𝑖𝜃_𝑦𝑖𝜃_𝑧𝑖𝜙_𝑥𝑖𝜙_𝑦𝑖𝜙_𝑧𝑖𝛾_𝑥𝑖𝛾_𝑦𝑖𝛾_𝑧𝑖^𝑇. (17) Thus, the generalized displacement vectors at any point within the element can be expressed in terms of the nodal generalized displacement vectors {𝑑^𝑒_𝑡} and {𝑑^𝑒_𝑟} as follows: 𝑑^𝑒_𝑡=𝑁_𝑡𝑑^𝑒_𝑡,𝑑^𝑒_𝑟=𝑁_𝑟𝑑^𝑒_𝑟, (18) in which 𝑑^𝑒_𝑡=𝑑^𝑒_𝑡1^𝑇𝑑^𝑒_𝑡2^𝑇⋯𝑑^𝑒_𝑡8^𝑇𝑇,𝑑^𝑒_𝑟=𝑑^𝑒_𝑟1^𝑇𝑑^𝑒_𝑟2^𝑇⋯𝑑^𝑒_𝑟8^𝑇𝑇,𝑁_𝑡=𝑁_𝑡1𝑁_𝑡2⋯𝑁_𝑡8^𝑇,𝑁_𝑟=𝑁_𝑟1𝑁_𝑟2⋯𝑁_𝑟8^𝑇,𝑁_𝑡𝑖=𝑛_𝑖𝐼_𝑡,𝑁_𝑟𝑖=𝑛_𝑖𝐼_𝑟, (19) while 𝐼_𝑡 and 𝐼_𝑟 are (3×3) and (9 × 9) identity matrices, respectively, and 𝑛_𝑖 is the shape function of natural coordinates associated with the 𝑖th node. Making use of the relations given by (6)–(8) and (18), the strain vectors at any point within the element can be expressed in terms of the nodal generalized displacement vectors as follows: 𝜀_𝑏𝑐=𝐵_𝑡𝑏𝑑^𝑒_𝑡+𝑍₁𝐵_𝑟𝑏𝑑^𝑒_𝑟,𝜀_𝑏𝑣=𝐵_𝑡𝑏𝑑^𝑒_𝑡+𝑍₂𝐵_𝑟𝑏𝑑^𝑒_𝑟,𝜀_𝑏𝑝=𝐵_𝑡𝑏𝑑^𝑒_𝑡+𝑍₃𝐵_𝑟𝑏𝑑^𝑒_𝑟,𝜀_𝑠𝑐=𝐵_𝑡𝑠𝑑^𝑒_𝑡+𝑍₄𝐵_𝑟𝑠𝑑^𝑒_𝑟,𝜀_𝑠𝑣=𝐵_𝑡𝑠𝑑^𝑒_𝑡+𝑍₅𝐵_𝑟𝑠𝑑^𝑒_𝑟,𝜀_𝑠𝑝=𝐵_𝑡𝑠𝑑^𝑒_𝑡+𝑍₆𝐵_𝑟𝑠𝑑^𝑒_𝑟, (20) in which the nodal strain-displacement matrices [𝐵_𝑡𝑏], [𝐵_𝑟𝑏], [𝐵_𝑡𝑠], and [𝐵_𝑟𝑠] are given by 𝐵_𝑡𝑏=𝐵_𝑡𝑏1𝐵_𝑡𝑏2⋯𝐵_𝑡𝑏8,𝐵_𝑟𝑏=𝐵_𝑟𝑏1𝐵_𝑟𝑏2⋯𝐵_𝑟𝑏8,𝐵_𝑡𝑠=𝐵_𝑡𝑠1𝐵_𝑡𝑠2⋯𝐵_𝑡𝑠8,𝐵_𝑟𝑠=𝐵_𝑟𝑠1𝐵_𝑟𝑠2⋯𝐵_𝑟𝑠8. (21) The submatrices of [𝐵_𝑡𝑏], [𝐵_𝑟𝑏], [𝐵_𝑡𝑠], and [𝐵_𝑟𝑠] have been explicitly presented in the Appendix. On substitution of (9)–(13) and (20) into (16), the total potential energy 𝑇^𝑒_𝑝 and the kinetic energy 𝑇^𝑒_𝑘 of a typical element augmented with the ACLD treatment can be expressed as 𝑇^𝑒_𝑝=12𝑑^𝑒_𝑡^𝑇𝐾^𝑒_𝑡𝑡𝑑^𝑒_𝑡+𝑑^𝑒_𝑡^𝑇𝐾^𝑒_𝑡𝑟𝑑^𝑒_𝑟+𝑑^𝑒_𝑟^𝑇𝐾^𝑒_𝑡𝑟^𝑇×𝑑^𝑒_𝑡+𝑑^𝑒_𝑟^𝑇𝐾^𝑒_𝑟𝑟𝑑^𝑒_𝑟−2𝑑^𝑒_𝑡^𝑇𝐹^𝑒_𝑡𝑝𝑉−2𝑑^𝑒_𝑟^𝑇𝐹^𝑒_𝑟𝑝𝑉−2𝑑^𝑒_𝑡^𝑇𝐹^𝑒−𝜀₃₃𝑉²ℎ_𝑝,𝑇^𝑒_𝑘=12𝑑̇^𝑒_𝑡^𝑇𝑀^𝑒𝑑̇^𝑒_𝑡. (22)(23) In (22), 𝑉 represents the voltage difference applied across the thickness of the PZC layer. The elemental stiffness matrices [𝐾^𝑒_𝑡𝑡], [𝐾^𝑒_𝑡𝑟] and [𝐾^𝑒_𝑟𝑟], the elemental electroelastic coupling vectors {𝐹^𝑒_𝑡𝑝} and {𝐹^𝑒_𝑟𝑝}, the elemental load vector {𝐹^𝑒}, and the elemental mass matrix [𝑀^𝑒] appearing in (22) and (23) are given by 𝐾^𝑒_𝑡𝑡=𝐾^𝑒_𝑡𝑏+𝐾^𝑒_𝑡𝑠+𝐾^𝑒_{𝑡𝑏𝑠𝑝𝑏}+𝐾^𝑒_{𝑡𝑏𝑠𝑝𝑠},𝐾^𝑒_𝑡𝑟=𝐾^𝑒_𝑡𝑟𝑏+𝐾^𝑒_𝑡𝑟𝑠+12𝐾^𝑒_{𝑡𝑟𝑏𝑠𝑝𝑏}+𝐾^𝑒_{𝑟𝑡𝑏𝑠}^𝑇_𝑝𝑏+𝐾^𝑒_{𝑡𝑟𝑏𝑠𝑝𝑠}+𝐾^𝑒_{𝑟𝑡𝑏𝑠}^𝑇_𝑝𝑠,𝐾^𝑒_𝑟𝑡=𝐾^𝑒_𝑡𝑟^𝑇,𝐾^𝑒_𝑟𝑟=𝐾^𝑒_𝑟𝑟𝑏+𝐾^𝑒_𝑟𝑟𝑠+𝐾^𝑒_{𝑟𝑟𝑏𝑠𝑝𝑏}+𝐾^𝑒_{𝑟𝑟𝑏𝑠𝑝𝑠},𝐹^𝑒_𝑡𝑝=𝐹^𝑒_𝑡𝑏𝑝+𝐹^𝑒_𝑡𝑠𝑝,𝐹^𝑒_𝑟𝑝=𝐹^𝑒_𝑟𝑏𝑝+𝐹^𝑒_𝑟𝑠𝑝,𝐹^𝑒=^𝑎𝑒₀^𝑏𝑒₀𝑁_𝑡^𝑇𝑓𝑀𝑑𝑥𝑑𝑦,_𝑒=^𝑎𝑒₀^𝑏𝑒₀𝑚[𝑁]^𝑇[𝑁]𝑑𝑥𝑑𝑦,𝑚=^𝑁+2_𝑘=1𝜌^𝑘ℎ_𝑘+1−ℎ_𝑘, (24) where 𝐾^𝑒_𝑡𝑏=_𝐴𝐵_𝑡𝑏^𝑇𝐷_𝑡𝑏+𝐷_𝑡𝑏𝑣+𝐷_𝑡𝑏𝑝𝐵_𝑡𝑏𝐾𝑑𝑥𝑑𝑦,_𝑒𝑡𝑠=_𝐴𝐵_𝑡𝑠^𝑇𝐷_𝑡𝑠+𝐷_𝑡𝑠𝑣+𝐷_𝑡𝑠𝑝𝐵_𝑡𝑠𝐾𝑑𝑥𝑑𝑦,_{𝑒𝑡𝑏𝑠𝑝𝑏}=_𝐴𝐵_𝑡𝑏^𝑇𝐷_{𝑡𝑏𝑠𝑝}𝐵_𝑡𝑠𝐾𝑑𝑥𝑑𝑦,_{𝑒𝑡𝑏𝑠𝑝𝑠}=_𝐴𝐵_𝑡𝑠^𝑇𝐷_{𝑡𝑏𝑠𝑝}𝐵_𝑡𝑏𝐾𝑑𝑥𝑑𝑦,_{𝑒𝑡𝑟𝑏}=_𝐴𝐵_𝑡𝑟𝑏^𝑇𝐷_𝑡𝑟𝑏+𝐷_{𝑡𝑟𝑏𝑣}+𝐷_{𝑡𝑟𝑏𝑝}×𝐵_𝑡𝑟𝑏𝐾𝑑𝑥𝑑𝑦,_{𝑒𝑡𝑟𝑏𝑠𝑝𝑏}=_𝐴𝐵_𝑡𝑏^𝑇𝐷_{𝑡𝑟𝑏𝑠𝑝}𝐵_𝑟𝑠𝐾𝑑𝑥𝑑𝑦,_{𝑒𝑟𝑡𝑏𝑠𝑝𝑏}=_𝐴𝐵_𝑟𝑏^𝑇𝐷_{𝑟𝑡𝑏𝑠𝑝}𝐵_𝑡𝑠𝐾𝑑𝑥𝑑𝑦,_{𝑒𝑡𝑟𝑏𝑠𝑝𝑠}=_𝐴𝐵_𝑡𝑠^𝑇𝐷_{𝑟𝑡𝑏𝑠𝑝}𝐵_𝑟𝑏𝐾𝑑𝑥𝑑𝑦,_{𝑒𝑟𝑡𝑏𝑠𝑝𝑠}=_𝐴𝐵_𝑟𝑠^𝑇𝐷_{𝑡𝑟𝑏𝑠𝑝}𝐵_𝑡𝑏𝐾𝑑𝑥𝑑𝑦,_{𝑒𝑡𝑟𝑠}=_𝐴𝐵_𝑡𝑟𝑠^𝑇𝐷_𝑡𝑟𝑠+𝐷_{𝑡𝑟𝑠𝑣}+𝐷_{𝑡𝑟𝑠𝑝}×𝐵_𝑡𝑟𝑠𝐾𝑑𝑥𝑑𝑦,_{𝑒𝑟𝑟𝑏}=_𝐴𝐵_𝑟𝑟𝑏^𝑇𝐷_𝑟𝑟𝑏+𝐷_{𝑟𝑟𝑏𝑣}+𝐷_{𝑟𝑟𝑏𝑝}×𝐵_𝑟𝑟𝑏𝐾𝑑𝑥𝑑𝑦,_{𝑒𝑟𝑟𝑠}=_𝐴𝐵_𝑟𝑟𝑠^𝑇𝐷_𝑟𝑟𝑠+𝐷_{𝑟𝑟𝑠𝑣}+𝐷_{𝑟𝑟𝑠𝑝}×𝐵_𝑟𝑟𝑠𝐾𝑑𝑥𝑑𝑦,_{𝑒𝑟𝑟𝑏𝑠𝑝𝑏}=_𝐴𝐵_𝑟𝑏^𝑇𝐷_{𝑟𝑟𝑏𝑠𝑝}𝐵_𝑟𝑠𝐾𝑑𝑥𝑑𝑦,_{𝑒𝑟𝑟𝑏𝑠𝑝𝑠}=_𝐴𝐵_𝑟𝑠^𝑇𝐷_{𝑟𝑟𝑏𝑠𝑝}𝐵_𝑟𝑏𝐹𝑑𝑥𝑑𝑦,_{𝑒𝑡𝑏𝑝}=_𝐴𝐵_𝑡𝑏^𝑇𝐷_𝑡𝑏𝑝𝐹𝑑𝑥𝑑𝑦,_{𝑒𝑡𝑠𝑝}=_𝐴𝐵_𝑡𝑠^𝑇𝐷_𝑡𝑠𝑝𝐹𝑑𝑥𝑑𝑦,_{𝑒𝑟𝑠𝑝}=_𝐴𝐵_𝑟𝑠^𝑇𝐷_𝑟𝑠𝑝𝑑𝑥𝑑𝑦. (25) The various rigidity matrices and the rigidity vectors for electro-elastic coupling appearing in the previous elemental matrices are given by 𝐷_𝑡𝑏=^𝑁_𝑘=1^ℎ𝑘+1_ℎ𝑘𝐶^𝑘_𝑏𝐷𝑑𝑧,_𝑡𝑟𝑏=^𝑁_𝑘=1^ℎ𝑘+1_ℎ𝑘𝐶^𝑘_𝑏𝑍₁𝐷𝑑𝑧,_𝑟𝑟𝑏=^𝑁_𝑘=1^ℎ𝑘+1_ℎ𝑘𝑍₁^𝑇𝐶^𝑘_𝑏𝑍₁𝐷𝑑𝑧,_𝑡𝑠=^𝑁_𝑘=1^ℎ𝑘+1_ℎ𝑘𝐶^𝑘_𝑠𝐷𝑑𝑧,_𝑡𝑟𝑠=^𝑁_𝑘=1^ℎ𝑘+1_ℎ𝑘𝐶^𝑘_𝑏𝑍₄𝐷𝑑𝑧,_𝑟𝑟𝑠=^𝑁_𝑘=1^ℎ𝑘+1_ℎ𝑘𝑍₄^𝑇𝐶^𝑘_𝑏𝑍₄𝐷𝑑𝑧,_𝑡𝑏𝑣=ℎ_𝑣𝐶_𝑏^𝑁+1,𝐷_{𝑡𝑟𝑏𝑣}=^ℎ𝑁+2_ℎ𝑁+1𝐶_𝑏^𝑁+1𝑍₂𝐷𝑑𝑧,_{𝑟𝑟𝑏𝑣}=^ℎ𝑁+2_ℎ𝑁+1𝑍₂^𝑇𝐶_𝑏^𝑁+1𝑍₂𝐷𝑑𝑧,_𝑡𝑠𝑣=ℎ_𝑣𝐶_𝑠^𝑁+1,𝐷_{𝑡𝑟𝑠𝑣}=^ℎ𝑁+2_ℎ𝑁+1𝐶_𝑠^𝑁+1𝑍₅𝐷𝑑𝑧,_{𝑟𝑟𝑠𝑣}=^ℎ𝑁+2_ℎ𝑁+1𝑍₅^𝑇𝐶_𝑏^𝑁+1𝑍₅𝐷𝑑𝑧,_𝑡𝑏𝑝=ℎ_𝑝𝐶_𝑏^𝑁+2,𝐷_{𝑡𝑟𝑏𝑝}=^ℎ𝑁+3_ℎ𝑁+2𝐶_𝑏^𝑁+2𝑍₃𝐷𝑑𝑧,_{𝑟𝑟𝑏𝑝}=^ℎ𝑁+3_ℎ𝑁+2𝑍₃^𝑇𝐶_𝑏^𝑁+2𝑍₃𝐷𝑑𝑧,_𝑡𝑠𝑝=ℎ_𝑝𝐶_𝑠^𝑁+2,𝐷_{𝑡𝑟𝑠𝑝}=^ℎ𝑁+3_ℎ𝑁+2𝐶_𝑠^𝑁+2𝑍₆𝐷𝑑𝑧,_{𝑟𝑟𝑠𝑝}=^ℎ𝑁+3_ℎ𝑁+2𝑍₆^𝑇𝐶_𝑏^𝑁+2𝑍₆𝐷𝑑𝑧,_{𝑡𝑏𝑠𝑝}=^ℎ𝑁+3_ℎ𝑁+2𝐶^𝑁+2_𝑏𝑠𝐷𝑑𝑧,_{𝑡𝑟𝑏𝑠𝑝}=^ℎ𝑁+3_ℎ𝑁+2𝐶^𝑁+2_𝑏𝑠𝑍₆𝐷𝑑𝑧,_{𝑟𝑡𝑏𝑠𝑝}=^ℎ𝑁+3_ℎ𝑁+2𝑍₃^𝑇𝐶^𝑁+2_𝑏𝑠𝐷𝑑𝑧,_{𝑟𝑟𝑏𝑠𝑝}=^ℎ𝑁+3_ℎ𝑁+2𝑍₃^𝑇𝐶^𝑁+3_𝑏𝑠𝑍₆𝐷𝑑𝑧,_𝑡𝑏𝑝=^ℎ𝑁+3_ℎ𝑁+2−𝑒_𝑏ℎ_𝑝𝐷𝑑𝑧,_𝑟𝑏𝑝=^ℎ𝑁+3_ℎ𝑁+2−𝑍₃^𝑇𝑒_𝑏ℎ_𝑝𝐷𝑑𝑧,_𝑡𝑠𝑝=^ℎ𝑁+3_ℎ𝑁+2−𝑒_𝑠ℎ_𝑝𝐷𝑑𝑧,_𝑟𝑠𝑝=^ℎ𝑁+3_ℎ𝑁+2−𝑍₆^𝑇𝑒_𝑠ℎ_𝑝𝑑𝑧. (26) Applying the dynamic version of the virtual work principle [32], the following open-loop equations of motion for an element of the overall plate integrated with the ACLD treatment are obtained: 𝑀^𝑒𝑑̈^𝑒_𝑡+𝐾^𝑒_𝑡𝑡𝑑^𝑒_𝑡+𝐾^𝑒_𝑡𝑟𝑑^𝑒_𝑟=𝐹^𝑒_𝑡𝑝𝑉+𝐹^𝑒,𝐾^𝑒_𝑡𝑟^𝑇𝑑^𝑒_𝑡+𝐾^𝑒_𝑟𝑟𝑑^𝑒_𝑟=𝐹^𝑒_𝑟𝑝𝑉. (27) Since the elastic constant matrix of the viscoelastic layer is complex, the stiffness matrices of an element integrated with the ACLD treatment are complex. In case of an element not integrated with the ACLD treatment, the electro-elastic coupling matrices become null vectors and the elemental stiffness matrices are to be computed without considering the piezoelectric and viscoelastic layers. It should also be noted that as the stiffness matrices associated with the transverse shear strains are derived separately, one can employ the selective integration rule in a straight forward manner to avoid shear locking problem in case of thin plates. The restrained boundary conditions at the skew edges of the plate are such that the displacements of the points located at the skew edges are to be restrained along 𝑥^′-, 𝑦^′-, and 𝑧^′-directions (Figure 1(a)). Hence, in order to impose the boundary conditions in a straight forward manner, the generalized displacement vectors of a point lying on the skew edge are to be transformed as follows: 𝑑_𝑡=𝐿_𝑡𝑑^′_𝑡,𝑑_𝑟=𝐿_𝑟𝑑^′_𝑟, (28) where {𝑑^′_𝑡} and {𝑑^′_𝑟} are the generalized displacement vectors of the point with respect to 𝑥^′𝑦^′𝑧^′coordinate system and are given by 𝑑^′_𝑡=𝑢^′_𝑜𝑣^′_𝑜𝑤^′_𝑜^𝑇,𝑑^′_𝑟=𝜃^′_𝑥𝜃^′_𝑦𝜃^′_𝑧𝜙^′_𝑥𝜙^′_𝑦𝜙^′_𝑧𝛾^′_𝑧𝛾^′_𝑦𝛾^′_𝑧^𝑇. (29) Also, the transformation matrices [𝐿_𝑡] and [𝐿_𝑟] are given by 𝐿_𝑡=,𝐿𝑐𝑠0−𝑠𝑐0001_𝑟=,𝑐𝑠0000000−𝑠𝑐0000000001000000000𝑐𝑠0000000−𝑠𝑐0000000001000000000𝑐𝑠0000000−𝑠𝑐0000000001 (30) in which 𝑐=cos𝛼 and 𝑠=sin𝛼. Thus, the elemental matrices of the element containing the nodes lying on the skew edge are to be augmented as follows: 𝐾^𝑒_𝑡𝑡=𝑇₁^𝑇𝐾^𝑒_𝑡𝑡𝑇₁,𝐾^𝑒_𝑡𝑟=𝑇₁^𝑇𝐾^𝑒_𝑡𝑟𝑇₂,𝐾^𝑒_𝑟𝑟=𝑇₂^𝑇𝐾^𝑒_𝑟𝑟𝑇₂,𝑀^𝑒=𝑇₁^𝑇𝑀^𝑒𝑇₁. (31) The forms of the transformation matrices [𝑇₁] and [𝑇₂] are given by 𝑇₁=𝐿_𝑡𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝐿_𝑡𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝐿_𝑡𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝐿_𝑡𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝐿_𝑡𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝐿_𝑡𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝐿_𝑡𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝑜̂𝐿_𝑡,𝑇₂=𝐿_𝑟𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝐿_𝑟𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝐿_𝑟𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝐿_𝑟𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝐿_𝑟𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝐿_𝑟𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝐿_𝑟𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝐿_𝑟, (32) in which 𝑜̂ and 𝑜 are (3×3) and (9×9) null matrices, respectively. It may be noted that in case of a node of such element which does not lie on the skew edge, the generalized displacement vectors of this node with respect to 𝑥𝑦𝑧 coordinate system are not required to be transformed. Hence, the matrices [𝐿_𝑡] and [𝐿_𝑟] corresponding to such node turn out to be (3×3) and (9×9) identity matrices, respectively.

The elemental equations of motion are now assembled to obtain the open-loop global equations of motion of the overall skew plate as follows: [𝑀]𝑋̈+𝐾_𝑡𝑡{𝑋}+𝐾_𝑡𝑟𝑋_𝑟=^𝑞_𝑗=1𝐹^𝑗_𝑡𝑝𝑉^𝑗+{𝐹},𝐾_𝑡𝑟^𝑇{𝑋}+𝐾_𝑟𝑟𝑋_𝑟=^𝑞_𝑗=1𝐹^𝑗_𝑟𝑝𝑉^𝑗, (33) where [𝐾_𝑡𝑡], [𝐾_𝑡𝑟], and [𝐾_𝑟𝑟] are the global stiffness matrices of the overall skew laminated composite plate, [𝑀] is the global mass matrix, {𝐹_𝑡𝑝} and {𝐹_𝑟𝑝} are the global electro-elastic coupling vectors, {𝑋} and {𝑋_𝑟} are the global nodal generalized displacement vectors, {𝐹} is the global nodal force vector, 𝑞 is the number of patches, and 𝑉^𝑗 is the voltage applied to the 𝑗th patch. Since the elemental stiffness matrices of an element augmented with the ACLD treatment are complex, the global stiffness matrices become complex and the energy dissipation characteristics of the overall plate are attributed to the imaginary part of these matrices. Hence, the global equations of motion as derived previously also represent the modeling of the passive (uncontrolled) constrained layer damping of the substrate plate when the constraining layer is not subjected to any control voltage following a derivative control law.

3. Closed-Loop Model

In the active control approach, the active constraining layer of each patch is activated with a control voltage negatively proportional to the transverse velocity of a point. Thus, the control voltage for each patch can be expressed in terms of the derivatives of the global nodal degrees of freedom as follows: 𝑉^𝑗=−𝑘^𝑗_𝑑𝑤̇=−𝑘^𝑗_𝑑𝑈^𝑗_𝑡𝑋̇−𝑘^𝑗_𝑑ℎ2𝑈^𝑗_𝑟𝑋̇_𝑟, (34) where [𝑈^𝑗_𝑡] and [𝑈^𝑗_𝑟] are the unit vectors describing the location of the velocity sensor for the 𝑗th patch and 𝑘^𝑗_𝑑 is the control gain for the 𝑗th patch. Substituting (34) into (33), the final equations of motion governing the closed-loop dynamics of the overall skew laminated composite plate/ACLD system can be obtained as follows: [𝑀]𝑋̈+𝐾_𝑡𝑡{𝑋}+𝐾_𝑡𝑟𝑋_𝑟+^𝑚_𝑗=1𝑘^𝑗_𝑑𝐹^𝑗_𝑡𝑝𝑈^𝑗_𝑡𝑋̇+^𝑚_𝑗=1𝑘^𝑗_𝑑ℎ2𝐹^𝑗_𝑡𝑝𝑈^𝑗_𝑟𝑋̇_𝑟={𝐹},𝐾_𝑡𝑟^𝑇{𝑋}+𝐾_𝑟𝑟𝑋_𝑟+^𝑚_𝑗=1𝑘^𝑗_𝑑𝐹^𝑗_𝑡𝑝𝑈^𝑗_𝑡𝑋̇+^𝑚_𝑗=1𝑘^𝑗_𝑑ℎ2𝐹^𝑗_𝑡𝑝𝑈^𝑗_𝑟𝑋̇_𝑟=0. (35)

4. Results and Discussion

In this section, the numerical results obtained by the finite element model derived in the previous section have been presented. Symmetric as well as antisymmetric cross-ply and antisymmetric angle-ply laminated skew substrate plates integrated with the two patches of ACLD treatment are considered for presenting the numerical results. As shown in Figure 1, the locations of the patches have been selected in such a way that the energy dissipation corresponding to the first few modes becomes maximum. PZT-5H/spur composite with 60% fiber volume fraction has been considered for the material of the constraining layer of the ACLD treatment. Using the micromechanics model and the material properties of the constituent phases [35], the effective elastic and piezoelectric material properties of the 1–3 PZC with respect to the principle material coordinate system are computed as follows: 𝐶₁₁=9.29GPa,𝐶₁₂=6.18GPa,𝐶₁₃𝐶=6.05GPa,₃₃=35.44GPa,𝐶₂₃=𝐶₁₃,𝐶₄₄𝐶=1.58GPa,₆₆=1.54GPa,𝐶₅₅=𝐶₄₄,𝑒₃₁=−0.1902C/m²,𝑒₃₂=𝑒₃₁,𝑒₃₃=18.4107C/m²,𝑒₂₄=0.004C/m²,𝑒₁₅=𝑒₂₄. (36) The material of the orthotropic layers of the substrate plates is a graphite/epoxy composite and its material properties are [42] 𝐸_𝐿𝐸=172GPa,^𝐿𝐸_𝑇=25,𝐺_𝐿𝑇=0.5𝐸_𝑇,𝐺_𝑇𝑇=0.2𝐸_𝑇,𝜈_𝐿𝑇=𝜈_𝑇𝑇=0.25, (37) in which the symbols have the usual meaning. Unless otherwise mentioned, the aspect ratio (𝑎/ℎ) and the thickness of the skew substrate plates are considered as 100 and 0.003 m, respectively, while the orthotropic layers of the skew substrate plates are of equal thickness. Also, the length and the skew sides of the substrate are equal (i.e., 𝑎=𝑏). The loss factor of the viscoelastic layer considered for numerical results remains invariant [20] within a broad band (0–400 Hz) of frequency range, and the values of the complex shear modulus, Poisson ratio, and the density of the viscoelastic layer are considered as 20(1+𝑖) MNm⁻², 0.49, and 1140 kg/m⁻³, respectively [20]. The thicknesses of the viscoelastic layer and the 1–3 PZC layer are considered as 50.8 μm and 250 μm, respectively. The value of the shear correction factor is used as 5/6. Three points of Gaussian integration rule are considered for computing the element matrices corresponding to the bending deformations while two points of that are used for computing the element matrices corresponding to transverse shear deformations. The simply supported boundary conditions at the edges of the overall plate are invoked as follows: 𝑣at𝑥=𝑦tan𝛼,𝑥=𝑎+𝑦tan𝛼:_′𝑜=𝑤^′_𝑜=𝜃^′_𝑦=𝜙^′_𝑦=𝛾^′_𝑦=𝜃^′_𝑧=𝛾^′_𝑧𝑢=0at𝑦=0,𝑦=𝑏cos𝛼:_𝑜=𝑤_𝑜=𝜃_𝑥=𝜙_𝑥=𝛾_𝑥=𝜃_𝑧=𝛾_𝑧=0. (38)

To verify the validity of the present finite element model (FEM), the natural frequencies of the skew laminated composite plates integrated with the inactivated patches of negligible thickness are first computed and subsequently compared with the existing analytical results [41] of the identical plates without being integrated with the patches. Such comparisons for the simply supported and clamped-clamped skew laminated plates are presented in Tables 1 and 2, respectively. It may be observed from these tables that the results are in excellent agreement, validating the model of the plate derived here.

Table 1 

Comparison of fundamental natural frequencies (𝜔) of simply supported skew laminated composite plates integrated with the patches of negligible thickness.

Table 2 

Comparison of fundamental natural frequencies (𝜔) of clamped-clamped skew laminated composite plates integrated with the patches of negligible thickness.

In order to assess the performance of the vertically/obliquely reinforced 1–3 PZC as the material of the distributed actuators of smart skew laminated composite plates, frequency responses for ACLD of the substrate skew laminated composite plates are investigated. To compute the frequency response functions, the skew laminated composite plates are harmonically excited by a force of amplitude 1 Newton applied to a point (𝑎/2,(𝑎/4)cos𝛼,ℎ/2) on the skew substrate plates. The control voltage supplied to the constraining layer of the first patch is negatively proportional to the velocity of the point (𝑎/2,(𝑎/4)cos𝛼,ℎ/2) and that applied to the constraining layer of the second patch is negatively proportional to the velocity of the point (𝑎/2,(3𝑎/4)cos𝛼,ℎ/2).

Figures 3, 4, and 5 illustrate the frequency response functions for the transverse displacement of a point of a simply supported symmetric cross-ply (0°/90°/0°) plate with 15°, 30°, and 45° skew angles, respectively, while the fiber orientation angle (𝜓) in the constraining layer is 0°. These figures display both uncontrolled and controlled responses and clearly show that the active constraining layer made of the vertically reinforced 1–3 PZC material being studied here significantly attenuates the amplitude of vibrations, enhancing the damping characteristics of the overall laminated skew composite plates over the passive damping (uncontrolled). Similar responses are also obtained for antisymmetric cross-ply (0°/90°/0°/90°) and antisymmetric angle-ply (45°/−45°/45°/−45°) skew laminated composite plates. However, for brevity these results are not presented here. The maximum values of the control voltages required to compute the controlled responses presented in Figures 3–5 have been found to be quite low and are illustrated in Figure 6 in case of the symmetric cross-ply (0°/90°/0°) plate with skew angle 15° only. In order to examine the contribution of the vertical actuation in improving the damping characteristics of the skew laminated composite plates as demonstrated in Figures 3–5, active control responses of the symmetric cross-ply laminated composite plate with 15° skew angle are plotted in Figure 7 with and without considering the value of 𝑒₃₃ and 𝑒₃₁. It may be noted that when the values of 𝑒₃₁ is zero and that of 𝑒₃₃ is nonzero, the vertical actuation of the active constraining layer of the ACLD treatment is responsible for increasing the transverse shear deformations of the viscoelastic constrained layer over the passive counterpart resulting in improved damping of the smart plate over its passive damping. On the other hand, if 𝑒₃₃ is zero and 𝑒₃₁ is nonzero, the in-plane actuation of the active constraining layer causes the transverse shear deformation of the viscoelastic core of the ACLD treatment leading to the active damping of the smart plate. It is evident from Figure 7 that the contribution of the vertical actuation by the constraining vertically reinforced 1–3 PZC layer is significantly larger than that of the in-plane actuation by the same for controlling the modes displayed in Figure 7. Though not shown here, similar results are also obtained for antisymmetric cross-ply and angle-ply skew plates.

Figure 3 

Frequency response functions for the transverse displacement 𝑤 (𝑎/2,(𝑎/4)cos𝛼,ℎ/2) of a simply supported skew symmetric cross-ply (0°/90°/0°) plate (𝑎/ℎ=100, 𝜓=0∘, 𝛼=15∘).

Figure 4 

Frequency response functions for the transverse displacement 𝑤 (𝑎/2, (𝑎/4)cos𝛼, ℎ/2) of a simply supported skew symmetric cross-ply (0°/90°/0°) plate (𝑎/ℎ=100, 𝜓=0∘, 𝛼=30∘).

Figure 5 

Frequency response functions for the transverse displacement 𝑤 (𝑎/2, (𝑎/4)cos𝛼, ℎ/2) of a simply supported skew symmetric cross-ply (0°/90°/0°) plate (𝑎/ℎ=100, 𝜓=0∘, 𝛼=45∘).

Figure 6 

Frequency response functions for the control voltages for active damping of the simply supported skew symmetric cross-ply (0°/90°/0°) plate (𝑎/ℎ=100, 𝜓=0∘, 𝛼=15∘).

Figure 7 

Frequency response functions demonstrating the contribution of vertical actuation on the controlled responses for the transverse displacement 𝑤 (𝑎/2,(𝑎/4)cos𝛼,ℎ/2) of the skew symmetric cross-ply (0°/90°/0°) plate (𝐾𝑑=600, 𝑎/ℎ=100, 𝜓=0∘, 𝛼=15∘).

Figures 8, 9, and 10 demonstrate the effect of variation of piezoelectric fiber orientation angle on the performance of the patches for improving the damping characteristics of simply supported symmetric cross-ply (0°/90°/0°), antisymmetric cross-ply (0°/90°/0°/90°), and angle-ply (45°/−45°/45°/−45°) substrate skew composite plates with 30° skew angle, respectively, while the orientation angle (𝜓) of the piezoelectric fibers is varied in the vertical 𝑥𝑧-plane. These figures illustrate that the attenuating capability of the patches becomes maximum when the value of the orientation angle (𝜓) of the fibers in the 𝑥𝑧-plane is 0°. Similar effect is also obtained when the fiber orientation angle (𝜓) in the constraining layer of the ACLD treatment is varied in the vertical 𝑦𝑧-plane as illustrated in Figures 11, 12, and 13 for the symmetric cross-ply (0°/90°/0°), antisymmetric cross-ply (0°/90°/0°/90°), and angle-ply (45°/−45°/45°/−45°) substrate skew composite plates with 30° skew angle, respectively. In this regard, it should be noted that for investigating the effect of variation of piezoelectric fiber orientation angle (𝜓) on the performance of the ACLD patches, the value of 𝜓 has been smoothly varied to compute the controlled responses. However, for the sake of clarity in the plots, the responses corresponding to the four specific values of 𝜓 have been presented such that the optimum performance of the patches can be evaluated. The significant effect of the variation of the piezoelectric fiber orientation angle in the vertical 𝑥𝑧- and 𝑦𝑧-planes has been noticed when all the boundaries of the overall plates are clamped as shown in Figures 14–19. When the piezoelectric fibers are coplanar with the vertical 𝑥𝑧-plane, the frequency response functions plotted in Figures 14, 15, and 16 reveal that the control authority of the patches becomes maximum for improving the damping characteristics of the clamped-clamped skew cross-ply and angle-ply substrate plates if the value of the piezoelectric fiber orientation angle (𝜓) is 30°. The same is also found to be true if the piezoelectric fibers are coplanar with the vertical 𝑦𝑧-plane as shown in Figures 17, 18, and 19 for the previously mentioned skew plates. Similar responses are also found for the effect of piezoelectric fiber orientation angle on the active damping of other skew laminated plates with different skew angles. However, for the sake of brevity, they are not presented here.

Figure 8 

Effect of variation of piezoelectric fiber orientation angle (𝜓) on the controlled responses of a simply supported symmetric skew cross-ply (0°/90°/0°) plate when the piezoelectric fibers of the constraining layer are coplanar with the 𝑥𝑧-plane (𝐾𝑑=600, 𝑎/ℎ=100, 𝛼=30∘).

Figure 9 

Effect of variation of piezoelectric fiber orientation angle (𝜓) on the controlled responses of a simply supported skew antisymmetric cross-ply (0°/90°/0°/90°) plate when the piezoelectric fibers are coplanar with the 𝑥𝑧-plane (𝐾𝑑=600, 𝑎/ℎ=100, 𝛼=30∘).

Figure 10 

Effect of variation of piezoelectric fiber orientation angle (𝜓) on the controlled responses of a simply supported skew antisymmetric angle-ply (45°/−45°/45°/−45°) plate when the piezoelectric fibers are coplanar with the 𝑥𝑧-plane (𝐾𝑑=600, 𝑎/ℎ=100, 𝛼=30∘).

Figure 11 

Effect of variation of piezoelectric fiber orientation angle (𝜓) on the controlled responses of a simply supported skew symmetric cross-ply (0°/90°/0°), plate when the piezoelectric fibers of the constraining layer are coplanar with the 𝑦𝑧-plane (𝐾𝑑=600, 𝑎/ℎ=100, 𝛼=30∘).

Figure 12 

Effect of variation of piezoelectric fiber orientation angle (𝜓) on the controlled responses of a simply supported skew antisymmetric cross-ply (0°/90°/0°/90°) plate when the piezoelectric fibers of the constraining layer are coplanar with the 𝑦𝑧-plane (𝐾𝑑 = 600, 𝑎/ℎ=100, 𝛼=30∘).

Figure 13 

Effect of variation of piezoelectric fiber orientation angle (𝜓) on the controlled responses of a simply supported skew antisymmetric angle-ply (45°/−45°/45°/−45°) plate when the piezoelectric fibers of the constraining layer are coplanar with the 𝑦𝑧-plane (𝐾𝑑=600, 𝑎/ℎ=100, 𝛼=30∘).

Figure 14 

Effect of variation of piezoelectric fiber orientation angle (𝜓) on the controlled responses of a clamped-clamped skew symmetric cross-ply (0°/90°/0°) plate when the piezoelectric fibers of the constraining layer are coplanar with the 𝑥𝑧-plane (𝐾𝑑=600, 𝑎/ℎ=100,𝛼=30∘).

Figure 15 

Effect of variation of piezoelectric fiber orientation angle (𝜓) on the controlled responses of a clamped-clamped skew antisymmetric cross-ply (0°/90°/0°/90°) plate when the piezoelectric fibers of the constraining layer are coplanar with the 𝑥𝑧-plane (𝐾𝑑=600, 𝑎/ℎ=100, 𝛼=30∘).

Figure 16 

Effect of variation of piezoelectric fiber orientation angle (𝜓) on the controlled responses of a clamped-clamped skew antisymmetric angle-ply (45°/−45°/45°/−45°) plate when the piezoelectric fibers of the constraining layer are coplanar with the 𝑥𝑧-plane (𝐾𝑑=600, 𝑎/ℎ=100, 𝛼=30∘).

Figure 17 

Effect of variation of piezoelectric fiber orientation angle (𝜓) on the controlled responses of a clamped-clamped skew symmetric cross-ply (0°/90°/0°) plate when the piezoelectric fibers of the constraining layer are coplanar with the 𝑦𝑧-plane (𝐾𝑑=600, 𝑎/ℎ=100, 𝛼=30∘).

Figure 18 

Effect of variation of piezoelectric fiber orientation angle (𝜓) on the controlled responses of a clamped-clamped skew antisymmetric cross-ply (0°/90°/0°/90°) plate when the piezoelectric fibers of the constraining layer are coplanar with the 𝑦𝑧-plane (𝐾𝑑=600, 𝑎/ℎ=100, 𝛼=30∘).

Figure 19 

Effect of variation of piezoelectric fiber orientation angle (𝜓) on the controlled responses of a clamped-clamped skew antisymmetric angle-ply (45°/−45°/45°/−45°) plate when the piezoelectric fibers of the constraining layer are coplanar with the 𝑦𝑧-plane (𝐾𝑑=600, 𝑎/ℎ=100, 𝛼=30∘).

5. Conclusions

In this paper, a study has been carried out to investigate the performance of the ACLD treatment for active damping of smart skew laminated composite plates when vertically/obliquely reinforced 1–3 PZC materials are used as the materials for the constraining layer of the ACLD treatment. A layerwise FSDT-based finite element model has been developed to describe the dynamics of the skew laminated composite plates integrated with the patches of ACLD treatment. Unlike the existing finite element models of smart structures integrated with ACLD treatment, the derivation of the present finite element model includes the transverse deformations of the substrate skew laminated composite plate, the constrained viscoelastic layer, and the constraining 1–3 PZC layer of the ACLD treatment along the thickness (i.e., 𝑧) direction such that both vertical and in-plane actuations by the constraining layer of the patches can be utilized for active damping of the plates. The frequency responses of the symmetric cross-ply, antisymmetric cross-ply, and angle-ply skew composite plates indicate that the active constraining layer of the ACLD treatment being made of the vertically reinforced 1–3 PZC material significantly enhances the damping characteristics of the plates over the passive damping. The analysis revealed that if the vertically reinforced 1–3 PZC material is used for the constraining layer of the ACLD treatment, then the contribution of the vertical actuation by the constraining layer alone for improving the active damping characteristics of the smart skew laminated composite plate is significantly larger than that due to the in-plane actuation by the constraining layer alone. It is important to note from the present investigation that the variation of the orientation angle of the piezoelectric fibers of the constraining layer in the vertical 𝑥𝑧- and 𝑦𝑧-planes as well as the skew angle and boundary conditions of the overall plates significantly affects the performance of the patches. For the simply supported laminated substrate plates, the performance of the patches becomes maximum when the piezoelectric fiber orientation angle (𝜓) is 0°, while in case of the clamped-clamped laminated skew substrate plates, the maximum control authority of the patches is achieved when the value of 𝜓 is 30° irrespective of the cases in which the fibers are oriented in 𝑥𝑧- and 𝑦𝑧-planes.

Appendix

In (6) and (7), the matrices [𝑍₁], [𝑍₂], [𝑍₃], [𝑍₄], [𝑍₅], and [𝑍₆] are given by 𝑍₁=𝑍₁00,𝑍₂=ℎ2𝐼𝑍₂0,𝑍₃=ℎ2𝐼ℎ_𝑣𝐼𝑍₃,𝑍₄=𝐼00𝑧𝐼00,𝑍₅=0𝐼0ℎ2𝐼ℎ𝑧−2𝐼0,𝑍₆=00𝐼ℎ2𝐼ℎ_𝑣𝐼𝑧−ℎ_𝑁+2𝐼, (A.1) in which 𝑍₁=,𝑧0000𝑧0000𝑧00001𝑍₂=ℎ𝑧−20ℎ000𝑧−2ℎ0000𝑧−20,0001𝑍₃=𝑧−ℎ_𝑁+20000𝑧−ℎ_𝑁+20000𝑧−ℎ_𝑁+20,[𝐼]=,00011000010000100000𝐼=,10010=,0=0000.0000 (A.2) The various submatrices 𝐵_𝑡𝑏𝑖, 𝐵_𝑟𝑏𝑖, and 𝐵_𝑟𝑠𝑖, appearing in (21), are given by 𝐵_𝑡𝑏𝑖=𝜕𝜂_𝑖0𝜕𝑥00𝜕𝜂_𝑖0𝜕𝑦𝜕𝜂_𝑖𝜕𝑦𝜕𝜂_𝑖0𝜕𝑥000,𝐵_𝑡𝑠𝑖=00𝜕𝜂_𝑖𝜕𝑥00𝜕𝜂_𝑖,𝜕𝑦𝐵_𝑟𝑏𝑖=𝜕𝜂_𝑖0𝜕𝑥00𝜕𝜂_𝑖0𝜕𝑦𝜕𝜂_𝑖𝜕𝑦𝜕𝜂_𝑖0𝜕𝑥001,𝐵_𝑟𝑏𝑖=𝐵^⌢_𝑟𝑏𝑖0^⌢0_⌢0𝐵^⌢_𝑟𝑏𝑖0_⌢0^⌢0𝐵_𝑟𝑏𝑖,𝐵_𝑟𝑠𝑖=^⌢𝐼0̆0̆0̆^⌢𝐼0̆0̆0̆^⌢𝐼𝐵_𝑟𝑠𝑖0̆0̆0̆𝐵_𝑟𝑠𝑖0̆0̆0̆𝐵_𝑟𝑠𝑖,^⌢𝐼=,100010 (A.3) wherein ^⌢0 and 0̆ are the (3×3) and (2×3) null matrices, respectively.