International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 127238 | 19 pages | https://doi.org/10.5402/2012/127238

On Vibration and Noise Dissipation in Ship and FPSO Structures with Smart Systems

Academic Editor: C. F. Gao
Received10 Jan 2012
Accepted12 Mar 2012
Published07 Aug 2012

Abstract

Ships and floating structure production systems are widely deployed for deep and ultradeep waters operation. Active vibration reduction and noise control in such structures can significantly improve their hydrodynamic performance and stability during navigation, exploration, and exploitation activities. One way to minimise or reduce the transmission of vibration in these moving offshore structures is to exploit the mechanism of interfacial slip in press fit joints or layered structural laminates in their internal hull configurations to dissipate vibration energy. In this paper, slip damping with heterogeneous sandwich composite viscoelastic beam-plate smart systems as a model for dissipation of vibration and active noise control mechanism in ship and floating structures is investigated. For this problem, a boundary value partial differential equation is formulated for the case of linear and nonlinear hydrodynamic wave loadings. In particular, the effect of pressure distribution variation at the interface of the layered smart system on the energy dissipation, logarithmic damping decrement, and spatial transfer function is analyzed and presented for design application and selection of appropriate stabilizers.

1. Introduction

Ships and floating structures are widely employed in the course of deep and ultradeep waters navigation as well as oil and gas engineering operations, because of their high adaptability, relatively low construction cost, and good stability. With the renewed interest in off-shore oil and gas exploration, huge resources are being invested in the development of the structures.

These structures (mostly ship shaped) are either purposely built or merely converted from existing tankers with good hull structural scantling design. One of the factors that influence the design of these structures is the environment in which they have to operate. Nevertheless, FPSOs have unique features and characteristics. One new innovation in the design of ships and floating structures employs the application of sandwich plate system (SPS) technology with built-in vibration and insulation mechanism, as illustrated in Figure 1. The structures are usually sited at specific locations and are subjected to dynamic loading that is quite different from those arising from unrestricted service conditions, Moan et al. [1].

In particular, such structures and systems are constantly under the influence of hydrodynamic forces that prevail in such locations. Within the context of the design of these structures and systems, most of the analyses currently employ probabilistic, statistical, and empirical models along with validating experiments and field observations. This makes the analysis and design of ships, floating production systems (FPS), and FPSOs a challenging process. One problem such designs have to address is adequate damping of vibration and active noise generation or transmission.

Vibration damping and active noise control in engineering structures and systems have attracted the attention of mathematical physicist and engineers for many years, (Lazan [2]; Nakra [3]; Sun and Lu [4]; Garibaldi and Onah [5]; Meador and Mead [6]; Vydra and Shogren [7]). With the application of surface damping and laminates in ships, the FPS and the FPSOs are part of recent outcome of advancement in manufacturing processes that are cost-effective and suitable for high volume production. Contrary to the conventional laminating press procedure, it is now possible to manufacture multilayered damped laminates consisting of two metal skins with a viscoelastic core via continuous process in coil form using existing equipment and technology.

Several methods, classified into active, semiactive, and passive control methods, have, over the years, been effectively adapted to control noise and reduce vibration in various dynamic systems. Certain active elements have been employed in the past, such as speakers, actuators, and microprocessors to produce an β€œout-of-phase” signal that can electronically cancel the disturbance. For fluid-borne noise, passive control methods that have been deployed include the use of elements such as absorbers, barriers, mufflers, and silencers. Whilst for structurally borne vibration and noise, changing of the systems modal parameters can significantly alter the resonance frequencies, provided the excitation frequencies remain constant. Nonetheless, the most effective means for handling this problem is the vibration damping or isolation through the use of damping materials. In fact, semi-active methods use active control methods such as electrorheological (ER), magnetorheological (MR) fluids and active constrained layer damping (ACLD) where the traditional constraining layer is replaced with a smart structure to enhance the damping properties of passive elements as discussed by Rao [8].

The application of piezoelectric materials in the control of vibration and noise in flexible structures has also been a subject of investigation for several decades. Piezoelectric ceramics provide a cheap, reliable, and nonintrusive means of actuation and sensing in flexible structures. Such a structure commonly described in the literature as a smart or intelligent/adaptive system is simply constructed by sandwiching piezoelectric actuators and/or sensors with flexible or elastic structures; Moheimani et al. [9] and Bailey and Hubbard [10] provide excellent reviews. The above referenced materials strain when exposed to a voltage or alternatively produce voltage when strained as pointed out by Alberts et al. [11]. For detailed descriptions of the unique electromechanical behaviour of these materials refer to Tzou [12].

Historically, the mechanism of damping as a means of controlling the undesirable effects of vibration has been treated over the years either in the context of aerodynamic/hydrodynamic structures or within the machine tool industry. In fact, there are several ways of effecting such damping, including the introduction of constrained, unconstrained, and even viscoelastic layers. One such technique is layered construction made possible by externally applied pressure that holds the members together at the interface. Under such circumstances, the profile of the interface pressure assumes a significant role, especially in the presence of slip, to dissipate the vibration energy. Another way of getting rid of unwanted vibration is through material damping as characterized by the strain energy of the structural members.

Within the context of the cantilever beam, one of the earliest works on slip damping is attributed to Goodman and Klumpp [13]. The nature of the interface pressure profile across the beam layer is a separate and important issue that has also received some attention over the years. There are several ways of simulating such interfacial pressure including mechanisms such as bonded (welded) connections, or the use of bolted connections, and even bonded-bolted connections placed at appropriate locations along the laminate interface.

With the introduction of composite materials and the possible beneficial effects these materials can have on slip damping, several authors have revisited the problem of layered or jointed structures subjected to uniform pressure distribution. In this regard, Nanda [14] studied the effect of structural members under controlled dynamic slip while Nanda and Behera [15] examined the problem of slip damping of jointed structures with connection bolts as found in machine structures. One of the difficulties encountered in earlier analysis of this problem is the assumption of the uniform pressure profile at the interface of the layers, as experiments and earlier analysis had clearly shown that this was rarely the case. The effect of nonuniform interface pressure distribution on the mechanism of slip damping for layered elastic beams was recently examined by Damisa et al. [16, 18], and Olunloyo et al. [17]. In particular, whereas the investigation by Damisa et al. [16] was limited to the case of the linear pressure profile, the analysis in Olunloyo et al. [17] included other forms of interfacial pressure distributions such as polynomial or hyperbolic representations. In addition, the investigation in Damisa et al. [18] has also been extended to account for the effects of frequency of excitation and viscoelastic coefficients on the damping properties of layered viscoelastic beam plate in Olunloyo et al. [19]. The results obtained in Olunloyo et al. [17] demonstrated that the effects of nonlinearities in interfacial pressure distributions as compared with the linear profile were largely incremental in nature, and no fundamental differences were found. In a related development, the effects of smart layers and material properties on the damping properties of sandwich heterogeneous elastic beams with varying thicknesses were reported in Olunloyo et al. [20]. The results from these earlier studies provide adequate justification for the linear pressure profile selected for the present investigation.

This paper is organized as follows. Section 1 introduces the problem under investigation with appropriate notations. In the next section, the essential fluid-structure interaction mechanics is discussed. In Section 3, these relationships are incorporated into an analytical model for the computation of the response of the smart system under harmonic excitation. In Section 4, the energy dissipation is computed while Section 5 gives the procedural method for the computation of logarithmic damping decrement. Section 6 is concerned with analysis of spatial transfer functions for active vibration and noise control design. Following in Section 7 are the simulated results showing the modulating role of the piezoelectric parameter with respect to design analysis and application. Finally, in Section 8, the paper ends with the summary and conclusion.

2. Essential Fluid-Structure Interaction Mechanics

For an isentropic (ocean state) inviscid fluid flow problem, flow in the domain of fluid-structure interface for the case of linear hydrodynamic wave loading is governed by the differential equationsβˆ‡2𝑃Φ=0,β„ŽπœŒπ‘€+πœ•Ξ¦1πœ•π‘‘+𝑔𝑧+2||||βˆ‡Ξ¦2=0.(1a) In view of the elliptical hull geometry as illustrated in Figures 2(a) and 2(b) under consideration, the well-known transformation to take us from Cartesian (π‘₯0,𝑦0,𝑧0) to elliptical cylindrical coordinates (πœ‰,πœ‚0,𝑧0) holds, namely, π‘₯0=𝑐coshπœ‰cosπœ‚0,𝑦0=𝑐sinhπœ‰sinπœ‚0,𝑧0=𝑧0,(1b)where πœ‰β‰₯0,0β‰€πœ‚0<2πœ‹,βˆ’βˆž<𝑧0<∞.

Thereafter, the first part of (1a) takes the form 1𝑐2ξ€·sinh2πœ‰+sin2πœ‚0ξ€Έξ‚΅πœ•2Ξ¦πœ•πœ‰2+πœ•2Ξ¦πœ•πœ‚02ξ‚Ά+πœ•2Ξ¦πœ•π‘§02=0.(2) The decomposition of Ξ¦ into Ξ¦=πœ™(πœ‰,πœ‚0)𝑓(𝑧0,𝑑) via variable separation leads to a set of two equations namely, 1𝑐2ξ€·cosh2πœ‰βˆ’cos2πœ‚0ξ€Έξƒ©πœ•2πœ™ξ€·πœ‰,πœ‚0ξ€Έπœ•πœ‰2+πœ•2πœ™ξ€·πœ‰,πœ‚0ξ€Έπœ•πœ‚02ξƒͺ+π‘˜2πœ™ξ€·πœ‰,πœ‚0ξ€Έπœ•=0,(3)2𝑓𝑧0ξ€Έ,π‘‘πœ•π‘§2βˆ’π‘˜2𝑓𝑧0ξ€Έ,𝑑=0,(4) where π‘˜2 is a constant (wave number) to be determined from the fluid kinematics boundary conditions. The solution of (4) within the context of fluid structure interaction boundary conditions is reported in Olunloyo et al. [21], while, πœ™(πœ‰,πœ‚0), which can be further decomposed as πœ™(πœ‰,πœ‚0)=𝐹(πœ‰)𝐺, leads to the well-known Mathieu and the modified Mathieu differential equations, namely, 𝑑2πΊξ€·πœ‚0ξ€Έπ‘‘πœ‚02+ξ€·π‘Žβˆ’2π‘žcos2πœ‚0ξ€ΈπΊξ€·πœ‚0𝑑=0,(5)2𝐹(πœ‰)π‘‘πœ‰2βˆ’ξ€·π‘Žβˆ’2ξ€Έπ‘žcosh2πœ‰πΉ(πœ‰)=0,(6) here the separation constant π‘Ž is any characteristic eigenvalue while π‘ž=π‘˜2𝑐2/4.

The solution to (6) and (7) is strongly influenced by the numerical value of π‘ž and for deep and ultradeep water problems π‘˜ is significantly small. Under such circumstances, the regime of linear and weakly nonlinear wave propagation can be satisfactorily assumed as the first-order approximation without loss of generality. In the meantime, (6) and (7) become amenable to parameter perturbation method. Consequently, 𝐹(πœ‰),𝐺(πœ‚), and π‘Ž can all be expanded in perturbation series in terms of the parameter π‘ž.

While noting the above, it is sufficient to state that the hydrodynamic wave loading force around a vibrating and translating ship or floating structure during exploration and exploitation or naval surveillance must satisfy the expression ξ‹πΉβ„Ž=ξ€π‘†π‘ƒβ„ŽΜ‚π‘’π‘§π‘‘π‘†,(7) where π‘ƒβ„Ž=βˆ’πœŒπ‘€ξƒ©πœ™ξ€·πœ‰,πœ‚0ξ€Έξ€·π‘§πœ•π‘“0ξ€Έ,𝑑1πœ•π‘‘+𝑔𝑧+2π‘ˆ2ξƒͺ(8) and ̂𝑒𝑧 is the outward normal unit vector for the element 𝑑𝑆 of the surface of the ship or floating structure.

Here, π‘ˆ is the translational velocity of the approaching ocean water for the case of a vibrating localised ship or floating structure or relative translational velocity for a moving ship or floating structure. For such a problem, a closed-form expression for 𝑓(𝑧0,𝑑) in the Laplace domain has been reported by Olunloyo et al. [21–23] and can be written as 𝑓𝑧0ξ€Έξ‚‹π‘Šξ€·π‘₯,𝑠=βˆ’π‘ 0,𝑦0ξ€Έ,π‘ ξ€·π‘˜sinhπ‘˜β„Žπ·ξ€Έξ€·coshπ‘˜π‘§0ξ€Έ,(9) where ξ‚‹π‘Š(π‘₯0,𝑦0,𝑠) and β„Žπ· represent the dynamic interaction response and the depth of immersion of the ship or floating structure, respectively.

For the case of nonlinear hydrodynamic wave loading force, the appropriate modifications in (9) to the π‘œ(πœ€π‘ 2) yield π‘ƒβ„Ž=βˆ’πœŒπ‘€ξƒ©ξ€·πœ™0ξ€·πœ‰,πœ‚0ξ€Έ+πœ€π‘ πœ™1ξ€·πœ‰,πœ‚0Γ—πœ•ξ€·π‘“ξ€Έξ€Έ0𝑧0ξ€Έ,𝑑+πœ€π‘ π‘“1𝑧0,𝑑1πœ•π‘‘+𝑔𝑧+2ξ€·π‘ˆ20+πœ€π‘ π‘ˆ20ξ€Έξƒͺ(10) while 𝑓0𝑧0ξ€Έξ‚‹π‘Š,𝑠=βˆ’π‘ 0ξ€·π‘₯0,𝑦0ξ€Έ,π‘ π‘˜0ξ€·π‘˜sinh0β„Žπ·ξ€Έξ€·π‘˜cosh0𝑧0ξ€Έ,𝑓1𝑧0ξ€Έξ‚‹π‘Š,𝑠=βˆ’π‘ 1ξ€·π‘₯0,𝑦0ξ€Έ,π‘ π‘˜1ξ€·π‘˜sinh1β„Žπ·ξ€Έξ€·π‘˜cosh1𝑧0ξ€Έ,(11) as reported by Olunloyo et al. [23].

Thus, the amplitude of the loading force 𝐹0 as illustrated in Figure 3(a) must satisfy the expression 𝐹0‖‖𝐹β‰₯maxβ‹…β„Žβ€–β€–.(12)

3. Problem Definition

As illustrated in Figure 3(a), the proposed sandwich composite smart structure consists of four layers of unequal thickness of the same length and width, respectively. In this case, the upper piezoelectric layer (actuator) and the lower piezoelectric layer (sensor) are assumed to be perfectly bonded to the surfaces of two dissimilar layers of elastic beam-plate structures. Furthermore, piezoelectric layers are also assumed to be of much smaller thickness than those of the respective laminates to which they are bonded. That is, β„Žπ‘Žβ‰ͺβ„Ž1;β„Žπ‘ β‰ͺβ„Ž2. On the other hand, the proposed internal hull configuration is contrived as an assembly of cascade of composite smart structure layers as illustrated in Figure 3(b).

The problem here is to examine analytically the effect of the nature of load, ocean water frequency variation, piezoelectric variables, and the pressure gradient on(i)the dynamic response of the clamped sandwich composite cantilever elastic beamplate, (ii)the profile of interfacial slip,(iii)the slip energy dissipation under dynamic conditions,(iv)logarithmic damping decrement associated with mechanism of slip damping in such layered structures, (v)spatial transfer function for vibration and active noise control design.

After excitation, the postslip geometries for both the upper and lower-layers are shown in Figures 3(c) and 3(d). Now for the contrived boundary value problem, the transverse displacement π‘Š, same for each layer, satisfies the following PDE:𝑐1+𝑐24ξ‚Άπœ•5π‘Šπœ•π‘‘πœ•π‘₯4+πœ•4π‘Šπœ•π‘₯4+𝛽1π’œ+𝛽2ℬ+𝛽3+𝛽4ξ€Έ4πœ•2π‘Šπœ•π‘‘2=𝛼1π’œ+𝛼2ℬ4πœ•π‘ƒ(π‘₯,0)+πΆπœ•π‘₯π‘Ž4πœ•2π‘‰π‘Žπœ•π‘₯2,(13) where π’œ denotes (1βˆ’πœ21), ℬ denotes (1βˆ’πœ22) and the following parameters have been defined viz: 𝛼1=6πœ‡πΈeq1β„Ž2eq1,𝛽1=𝜌1π‘β„Ž1𝐸1𝐼1,𝛽3=πœŒπ‘Žπ‘β„Žπ‘ŽπΈπ‘ŽπΌπ‘Ž1,β„Žeq1=ξ€·β„Ž1+β„Žπ‘Žξ€Έ,𝛼2=6πœ‡πΈeq2β„Ž2eq2,𝛽2=𝜌2π‘β„Ž2𝐸2𝐼2,𝛽4=πœŒπ‘ π‘β„Žπ‘ πΈπ‘ πΌπ‘ ,β„Žeq2=ξ€·β„Ž2+β„Žπ‘ ξ€Έ,πΆπ‘Ž=12πΈπ‘Žπ‘‘31π‘ξ€·β„Žπ‘Ž+β„Žπ‘ ξ€Έ,(14) following Alberts et al. [11], here, 𝑑31 is the electric charge constant of the film in (m/v).

We next introduce the Laplace transform, namely, ξ€œ(β€’)=∞0(β€’)π‘’βˆ’π‘ π‘‘1𝑑𝑑,(β€’)=ξ€œ2πœ‹π‘–πœ‚+π‘–βˆžπœ‚βˆ’π‘–βˆž(β€’)𝑒𝑠𝑑𝑑𝑠(15) on (13) to obtain 𝑐1+𝑠1+𝑐24𝑑4ξ‚‹π‘Š(π‘₯,𝑠)𝑑π‘₯4+𝛽1π’œ+𝛽2ℬ+𝛽3+𝛽4ξ€Έ4×𝑠2̇=ξ€·π›Όπ‘Š(π‘₯,𝑠)βˆ’π‘ π‘Š(0)βˆ’π‘Š(0)1π’œ+𝛼2ℬ4π‘ πœ•π‘ƒ(π‘₯,0)+πΆπœ•π‘₯π‘Ž4𝑑2𝑉(π‘₯,𝑠)π‘Žπ‘‘π‘₯2.(16) By limiting the analysis to the case of linear pressure variation along the laminate interface, namely, 𝑃(π‘₯,0)=𝑃0ξ‚€πœ€1+𝐿π‘₯,(17) Equation (16) simplifies to the form 𝑐1+𝑠1+𝑐24𝑑4ξ‚‹π‘Š(π‘₯,𝑠)𝑑π‘₯4+𝛽1π’œ+𝛽2ℬ+𝛽3+𝛽4ξ€Έ4×𝑠2̇=ξ€·π›Όπ‘Š(π‘₯,𝑠)βˆ’π‘ π‘Š(0)βˆ’π‘Š(0)1π’œ+𝛼2β„¬ξ€Έπœ€π‘ƒ0+𝐢4π‘ πΏπ‘Ž4𝑑2ξ‚π‘‰π‘Ž(π‘₯,𝑠)π‘Žπ‘‘π‘₯2.(18) In (18), the spatial distribution of the actuating layer is considered to be uniform with respect to π‘₯. Under this assumption, the term 𝑑2ξ‚π‘‰π‘Ž(π‘₯,𝑠)π‘Ž/𝑑π‘₯2 following Alberts et al. [11] becomes 𝑑2ξ‚π‘‰π‘Ž(π‘₯,𝑠)π‘Žπ‘‘π‘₯2=ξ€·π›Ώξ…ž(π‘₯βˆ’0)βˆ’π›Ώξ…žξ€Έπ‘‰(π‘₯βˆ’πΏ)π‘Žπ‘ ,(19) where 𝛿′(β‹…) is the first derivative of the Dirac delta function.

Introducing the Fourier finite sine transform [β€’]=ξ€œπΏ0[β€’]sinπ‘›πœ‹π‘₯𝐿[β€’]=2𝑑π‘₯,πΏβˆžξ“π‘›=1[β€’]sinπ‘›πœ‹π‘₯𝐿(20) in conjunction with the following boundary conditions in the Laplace transform plane namely, ξ‚‹π‘‘π‘Š(0,𝑠)=𝑑𝑑π‘₯π‘Š(0,𝑠)=2𝑑π‘₯2ξ‚‹π‘Š(𝐿,𝑠)=0(21) gives the Fourier sine transform of (18) as 𝑛4πœ‹4𝐿4𝑐1+𝑠1+𝑐24ξ‚‹π‘Šξ‚Άξ‚ΆπΉξ€·πœ†π‘›ξ€Έ+𝛽,𝑠1π’œ+𝛽2ℬ+𝛽3+𝛽4ξ€Έ4𝑠2ξ‚‹π‘ŠπΉξ€·πœ†π‘›ξ€Έ=𝛼,𝑠1π’œ+𝛼2ℬ𝑃4𝑠0πœ€ξ€·π‘›πœ‹1+(βˆ’1)𝑛+1ξ€Έ+𝑛3πœ‹3𝐿3(βˆ’1)𝑛+1ξ‚‹π‘Š(𝐿,𝑠)βˆ’π‘›πœ‹πΏξ‚‹π‘Šπ‘₯π‘₯+𝐢(0,𝑠)π‘Žπ‘‰π‘Žξ€·4π‘ π‘›πœ‹1+(βˆ’1)𝑛+1ξ€Έ.(22) Following the procedure outlined in Olunloyo et al. [21], the bending moment is computed as:ξ‚‹π‘Šπ‘₯π‘₯βŽ›βŽœβŽœβŽœβŽœβŽξ‚πΉ(0,𝑠)=12(𝑠)𝑏𝐸eq1β„Ž3eq1/π’œ+𝐸eq2β„Ž3eq2𝑐/ℬ+𝑠1𝐸eq1β„Ž3eq1/π’œ+𝑐2𝐸eq2β„Ž3eq2βˆ’ξƒ©/ℬ6πœ‡π‘ƒ0ξ€·β„Ž(1+(πœ€/2))eq1+β„Žeq2𝑠𝐸eq1β„Ž3eq1/π’œ+𝐸eq2β„Ž3eq2ξ€Έ/ℬ+𝑠2𝑐1𝐸eq1β„Ž3eq1/π’œ+𝑐2𝐸eq2β„Ž3eq2ξ€Έξƒͺ⎞⎟⎟⎟⎟⎠/ℬ𝐿.(23)This result clearly indicates that the value for expression (25) cannot be fully determined until the forcing function 𝐹(𝑑) is specified. Next, we limit the analysis to the case of harmonic loading function. For this case, the forcing function is 𝐹(𝑑)=𝐹0π‘’π‘–πœ”π‘‘ with the driving transform as 𝐹𝐹(𝑠)=0π‘ βˆ’π‘–πœ”,(24) where 𝐹0 represents the amplitude of the loading force and, πœ” is the maximum associated excitation frequency for the fluid-structure systems.

Recalling the only remaining unutilized boundary condition in (21), namely, ξ‚‹(𝑑/𝑑π‘₯)π‘Š(0,𝑠)=0 the Fourier inversion of (22) leads to the expressionξ‚‹βŽ§βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©π‘Š(π‘₯,𝑠)=2π‘ ξ‚‹π‘Š(𝐿,𝑠)π‘₯βˆ’2𝐿3βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽξ‚πΉ12𝑠(𝑠)𝑏𝐸eq1β„Ž3eq1/π’œ+𝐸eq2β„Ž3eq2𝑐/ℬ+𝑠1𝐸eq1β„Ž3eq1/π’œ+𝑐2𝐸eq2β„Ž3eq2/β„¬ξ€Έξ€Έβˆ’βŽ›βŽœβŽœβŽœβŽ6πœ‡π‘ƒ0ξ€·β„Ž(1+(πœ€/2))eq1+β„Žeq2𝐸eq1β„Ž3eq1/π’œ+𝐸eq2β„Ž3eq2𝑐/ℬ+𝑠1𝐸eq1β„Ž3eq1/π’œ+𝑐2𝐸eq2β„Ž3eq2ξ€Έ/β„¬βŽžβŽŸβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ Ξ›1+ξ€·3𝐿3/32ξ€Έξ€·π’œ/𝐸eq1β„Ž2eq1+ℬ/𝐸eq2β„Ž2eq2ξ€Έπœ‡π‘ƒ0πœ€Ξ›2+(1/16)πΆπ‘Žπ‘‰π‘ŽπΏ3Ξ›3⎫βŽͺβŽͺβŽͺ⎬βŽͺβŽͺβŽͺβŽ­π‘ ξ€·π‘ +πœ…1𝑠+πœ…2ξ€Έπœ”0βˆ’2,(25)where πœ…1=𝑐1+𝑐2ξ€Έπœ”208ξƒŽ+π‘–πœ”20βˆ’ξ‚΅π‘1+𝑐28ξ‚Ά2πœ”40,πœ…2=𝑐1+𝑐2ξ€Έπœ”208ξƒŽβˆ’π‘–πœ”20βˆ’ξ‚΅π‘1+𝑐28ξ‚Ά2πœ”40,πœ”20=πœ”20𝑐1βˆ’1+𝑐2ξ€Έ2πœ”3220ξƒͺ,πœ”20=𝑛4πœ‹4𝛽𝐿4.(26) Here, 𝛽𝛽=1π’œ+𝛽2ℬ+𝛽3+𝛽4ξ€Έ4,Ξ›1=ξ‚΅π‘₯6βˆ’π‘₯24+π‘₯3ξ‚Ά12,Ξ›2=ξ‚΅π‘₯βˆ’245π‘₯39+π‘₯43βˆ’2π‘₯5ξ‚Ά,Ξ›153=ξ‚΅π‘₯3βˆ’π‘₯22+π‘₯36ξ‚Ά.(27) Appropriate substitution for ξ‚‹π‘Š(𝐿,𝑠) into (25) and rearrangement givesξ‚‹π‘Šξ€·ξ€Έ=⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩π‘₯,π‘ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽ2𝑠𝐹0𝑏𝐸eq1β„Ž3eq1/π’œ+𝐸eq2β„Ž3eq2𝑐/ℬ+𝑠1𝐸eq1β„Ž3eq1/π’œ+𝑐2𝐸eq2β„Ž3eq2(/β„¬ξ‚ξ‚π‘ βˆ’π‘–πœ”)βˆ’πœ‡π‘ƒ0ξ€·β„Ž(1+(πœ€/2))eq1+β„Žeq2𝐸eq1β„Ž3eq1/π’œ+𝐸eq2β„Ž3eq2𝑐/ℬ+𝑠1𝐸eq1β„Ž3eq1/π’œ+𝑐2𝐸eq2β„Ž3eq2/β„¬βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ξ‚€3π‘₯2βˆ’π‘₯3+ξ‚€π’œ/𝐸eq1β„Ž2eq1+ℬ/𝐸eq2β„Ž2eq2ξ‚πœ‡π‘ƒ0πœ€ξ‚€βˆ’π‘₯3/48+π‘₯4/33βˆ’π‘₯5/90+πΆπ‘Žπ‘‰π‘Žξ‚€βˆ’π‘₯2/32+π‘₯3/96⎫βŽͺβŽͺβŽͺ⎬βŽͺβŽͺβŽͺβŽ­π‘ ξ€·π‘ +πœ…1𝑠+πœ…2ξ€Έπœ”0βˆ’2.(28)By employing the Laplace inversion we obtain the dynamic response asπ‘Šξ€·ξ€Έ=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽξ‚€π‘₯,𝜏2𝐹2(𝜏)βˆ’πœ‡π‘ƒ0π’žπΉ13(𝜏)π‘₯2βˆ’π‘₯3+πœ‡π‘ƒ0ξƒ¬βˆ’3π’ž2𝐹1(𝜏)π‘₯2+ξƒ©ξ‚€π’ž2𝐹1(𝜏)βˆ’π’œ+β„¬π›Ύβˆ’1eqΞ¨βˆ’2eqξƒͺ𝐹483ξƒͺπ‘₯(𝜏)3ξƒ­πœ€+𝐹3ξ‚Έξ€·(𝜏)π’œ+β„¬π›Ύβˆ’1eqΞ¨βˆ’2eqξ€Έξ‚΅π‘₯4βˆ’π‘₯335+90ξ‚Άξ‚ΉπΆπ‘ŽπΉ1ξ‚΅βˆ’(𝜏)π‘₯2+32π‘₯3ξ‚ΆβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,96(29)where 𝑑=2πœ‹πœπœ”0πœ”,πœ‚=πœ”0,(30) while 𝐹1(𝜏)=ξƒ¬ξƒ©π’œβ„¬β„¬+𝛾eqΞ¨3eqπ’œξƒͺ+π‘’βˆ’2πœ‹πœ…1πœπœ…1πœ…2πœ…12βˆ’π‘’βˆ’2πœ‹πœ…1πœπœ…4πœ…5πœ…12βˆ’πœ”βˆ’1π‘’βˆ’2πœ‹πœ‚πœ…1πœπœ…13ξ‚Ή,𝐹2=𝑒(𝜏)βˆ’2πœ‹πœ…1πœπœ…2πœ…3+π‘’βˆ’2πœ‹πœ…4πœπœ…3πœ…5+𝑒𝑖2πœ‹πœ‚πœπœ…6πœ…7+πœ”βˆ’1πœ…11𝑒𝑖2πœ‹πœ‚πœπœ…9πœ…10ξ‚Ή,𝐹3𝑒(𝜏)=1+βˆ’2πœ‹πœ…1πœπœ…1πœ…12+π‘’βˆ’2πœ‹πœ…4πœπœ…4πœ…12ξ‚Ή.(31) Here,πœ…1=ξ‚΅πœ‡1+πœ‡2ξ‚ΆξƒŽ8πœ‚βˆ’π‘–ξ‚΅1βˆ’πœ‡1+πœ‡2ξ‚Ά8πœ‚2,πœ…2=ℬ+𝛾eqΞ¨3eqπ’œξ€Έ+πœ…1ξ€·ξ€·πœ‡1ξ€Έξ€·/πœ‚β„¬+πœ‡2ξ€Έ/πœ‚π’œπ›ΎeqΞ¨3eqξ€Έ(ξƒ­,π’œβ„¬)πœ…3=⎑⎒⎒⎣1ξ‚΅32πœ‡1+πœ‡2πœ‚ξ‚Ά2+ξ‚΅ξ‚΅2πœ‚+π‘–πœ‡1+πœ‡2ξ„Άξ„΅ξ„΅βŽ·4πœ‚ξ‚Άξ‚Άξƒ©ξ‚΅1βˆ’πœ‡1+πœ‡2ξ‚Ά8πœ‚2ξƒͺ⎀βŽ₯βŽ₯⎦,βˆ’2πœ…4=ξ‚΅πœ‡1+πœ‡2ξ‚ΆξƒŽ8πœ‚+𝑖1βˆ’πœ‡1+πœ‡2ξ‚Ά8πœ‚2,πœ…5=ℬ+𝛾eqΞ¨3eqπ’œξ€Έ+πœ…4ξ€·ξ€·πœ‡1ξ€Έξ€·/πœ‚β„¬+πœ‡2ξ€Έ/πœ‚π’œπ›ΎeqΞ¨3eqξ€Έξƒ­,(π’œβ„¬)πœ…6=ξ‚Έξ€·1βˆ’πœ‚2ξ€Έξ‚΅βˆ’π‘–πœ‡1+πœ‡2,4πœ‚ξ‚Άξ‚Ήπœ…7=ℬ+𝛾eqΞ¨3eqπ’œξ€Έξ€·+π‘–πœ‡1ℬ+πœ‡2π’œπ›ΎeqΞ¨3eqξ€Έξƒ­,(π’œβ„¬)πœ…8=ℬ+𝛾eqΞ¨3eqπ’œξ€Έξ€·πœ‡1ℬ+πœ‡2π’œπ›ΎeqΞ¨3eqξ€Έξƒ­,πœ…9=ξ‚Έπœ…82πœ‚2+πœ…8ξ‚΅πœ‡1+πœ‡2ξ‚Άξ‚Ή,4πœ‚+1πœ…10=βˆ’ξξ€·β„¬+𝛾eqΞ¨3eqπ’œξ€Έξ€·+π‘–πœ‡1ℬ+πœ‡2π’œπ›ΎeqΞ¨3eq,ξ€Έξžπœ…11=βˆ’ξξ€·πœ‡1ℬ+πœ‡2π’œπ›ΎeqΞ¨3eq,ξ€Έξžπœ…12ξ„Άξ„΅ξ„΅βŽ·=2𝑖1βˆ’πœ‡1+πœ‡2ξ‚Ά8πœ‚2ξƒͺ,πœ…13ξ‚Έ=βˆ’πœ…8+ξ‚΅πœ‡1+πœ‡24ξ‚Άπœ…8+πœ…38ξ‚Ή,(32)where the following nondimensionalized parameters have also been introduced viz: π‘Šξ€·ξ€Έ=π‘Šξ€·π‘₯,πœξ€ΈπΈπ‘₯,𝜏eq1π‘β„Ž3eq1𝐿3𝐹0,𝑃0=𝑃0𝐹0/π‘β„Žeq1ξ€Έ.(33)

3.1. Analysis of Dynamic Slip

Recalling the description in Olunloyo et al. [19], the relative slip at the interface of the sandwich smart structure is given by Δ𝑒(π‘₯,𝑑)=𝑒1(π‘₯,𝑑,0βˆ’)βˆ’π‘’2ξ€·π‘₯,𝑑,0+ξ€Έ,(34) where πœ‰ is a dummy axial spatial variable of integration across the interface, 0+ and 0βˆ’ denote the origin of the transverse spatial variable for each layer, and 𝑑 is the time (state) variable.

Following Goodman and Klumpp [13], this can also be written as Δ𝑒(π‘₯,𝑑)=πΈβˆ’1eq1ξ€œπ‘₯0πœŽξ€½ξ€·π‘₯ξ€Έ1ξ€Ύ(πœ‰,0,𝑑,0βˆ’)π‘‘πœ‰βˆ’πΈβˆ’1eq2ξ€œπ‘₯0πœŽξ€½ξ€·π‘₯ξ€Έ2ξ€Ύ(πœ‰,0,𝑑,0+)π‘‘πœ‰(35) with the integral kernels as ξ€·πœŽπ‘₯ξ€Έ1(πœ‰,0,𝑑,0βˆ’πΈ)=eq1β„Žeq1𝐢1ξ€Έ(πœ•/πœ•π‘‘)+12ξ€·1βˆ’πœ21×𝑑2π‘Š(πœ‰,𝑑)π‘‘πœ‰2+πœ‡π‘ƒπ‘Žπ‘£(π‘₯βˆ’πΏ)β„Žeq1,ξ€·πœŽπ‘₯ξ€Έ2ξ€·πœ‰,0,𝑑,0+𝐸=βˆ’eq2β„Žeq2𝐢1ξ€Έ(πœ•/πœ•π‘‘)+12ξ€·1βˆ’πœ21×𝑑2π‘Š(πœ‰,𝑑)π‘‘πœ‰2βˆ’πœ‡π‘ƒπ‘Žπ‘£(π‘₯βˆ’πΏ)β„Žeq2,(36) where β„Žeq1=ξ€·β„Ž1+β„Žπ‘Žξ€Έ,β„Žeq2=ξ€·β„Ž2+β„Žπ‘ ξ€Έ,𝐸eq1=𝐸1ξ€·1+π›Ύπ‘Ž,1ξ€Έ,𝐸eq2=𝐸2ξ€·1+𝛾𝑠,2ξ€Έ,𝛾eq=𝐸eq1𝐸eq2=𝐸1ξ€·1+π›Ύπ‘Ž,1𝐸2ξ€·1+𝛾𝑠,2ξ€Έ,π›Ύπ‘Ž,1=πΈπ‘ŽπΈ1,𝛾𝑠,2=𝐸𝑠𝐸2,Ξ¨π‘Ž,1=β„Žπ‘Žβ„Ž1,Ψ𝑠,2=β„Žπ‘ β„Ž2,Ξ¨π‘Ž,1=β„Ž2β„Ž1ξ‚΅1+Ψ𝑠,21+Ξ¨π‘Ž,1ξ‚Ά.(37) Equation (36) is integrated to give Ξ”1𝑒=2𝑐1π’œ+𝑐2Ξ¨eqβ„¬ξ‚Άπœ•+πœ•πœΞ”1ξ‚Άπœ•π‘Šπœ•π‘₯+πœ‡π‘ƒ0Ξ”2ξ‚€π‘₯2βˆ’2π‘₯+πœ‡π‘ƒ0πœ€ξ‚€23π‘₯3βˆ’π‘₯2,(38) where the following have been introduced: Ξ”1=𝑐1+𝑐2π’œ2Ξ¨eqξ‚Ά,2ℬΔ2=ξ‚΅1+Ξ¨2π’œeqξ‚Ά.2ℬ(39) Substitution for π‘Š from (29) simplifies (38) asΞ”βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽœβŽξ‚΅π‘’=3πœ‡π‘ƒ0𝑐1π’ž+𝑐2π’œ2ℰ̇𝐹2ℬ1𝑐(𝜏)βˆ’31π’œ+𝑐2ℰℬ̇𝐹2ξ‚Ά+ξ‚ƒπœ‡(𝜏)𝑃0ξ€Ί3-πœ†1𝐹1(𝜏)+-πœ†2ξ€»βˆ’3-πœ†3𝐹2ξ‚„βŽžβŽŸβŽŸβŽŸβŽ π’΅(𝜏)+πœ‡π‘ƒ0πœ€βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘βˆ’3ξ‚΅ξ‚΅1π’žπ’œ+𝑐2ℰℬ̇𝐹1(𝜏)+-πœ†1𝐹1ξ‚Ά(𝜏)π‘₯+βŽ›βŽœβŽœβŽœβŽξƒ©π‘1π’žπ’œ+𝑐2Ξ¨βˆ’1eqℬξƒͺ̇𝐹1(𝜏)4βˆ’ξƒ©π‘1π’žπ’œ+𝑐2Ξ¨βˆ’1eqℬξƒͺ-πœ†4̇𝐹321+(𝜏)-πœ†22𝐹1(𝜏)βˆ’-πœ†4𝐹163⎞⎟⎟⎟⎠(𝜏)βˆ’1π‘₯22-πœ†4𝑐331π’œ+𝑐2Ξ¨βˆ’1eqℬξƒͺ̇𝐹3ξ‚€4(𝜏)+33-πœ†4𝐹32(𝜏)+3ξƒͺπ‘₯3βˆ’ξƒ©5180-πœ†4𝑐1π’œ+𝑐2Ξ¨βˆ’1eqℬξƒͺ̇𝐹35(𝜏)+90-πœ†4𝐹3ξƒͺ(𝜏)π‘₯4+πΆπ‘Žπ‘ξƒ©ξƒ©1+𝑐2π’œ2Ξ¨βˆ’1eqξƒͺ̇𝐹2ℬ3(𝜏)+𝐹3ξƒͺξ‚΅(𝜏)βˆ’116π‘₯+π‘₯2ξ‚ΆβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦32,(40)where 𝒡 denotes(π‘₯2βˆ’2π‘₯), π’ž denotes (1+Ξ¨eq), and β„° denotes (1+Ξ¨βˆ’1eq)-πœ†1ξ‚€=(π’ž)β‹…Ξ”2,-πœ†2=ξ‚€Ξ”1⋅Δ2,-πœ†3ξ‚€=2β‹…Ξ”2,-πœ†4=ξ€·π’œ+β„¬π›Ύβˆ’1eqΞ¨βˆ’2eqξ€Έ.(41) On setting πœ€β†’0 in (40), the dynamic slip at uniform pressure becomes Δ𝑒=⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©βŽ›βŽœβŽœβŽœβŽξ‚΅3πœ‡π‘ƒ0𝑐1π’ž+𝑐2π’œ2ℰ𝑐2β„¬βˆ’31π’œ+𝑐2ℰℬ̇𝐹2ξ‚Ά+ξ‚ƒπœ‡(𝜏)𝑃0ξ€Ί3-πœ†1𝐹1(𝜏)+-πœ†2ξ€»βˆ’3-πœ†3𝐹2ξ‚„βŽžβŽŸβŽŸβŽŸβŽ π’΅+(𝜏)πΆπ‘Žπ‘ξƒ©ξƒ©1+𝑐2π’œ2Ξ¨βˆ’1eqξƒͺ̇𝐹2ℬ3(𝜏)+𝐹3ξƒͺξ‚΅(𝜏)βˆ’116π‘₯+π‘₯2ξ‚ΆβŽ«βŽͺβŽͺβŽͺ⎬βŽͺβŽͺβŽͺ⎭.32(42)

4. Energy Dissipation

Following Goodman and Klumpp [13], the energy dissipated can be computed from the relation ξ€œπ·=4πœ‡π‘0πœ‹/2πœ”ξ€œπΏ0𝑃(π‘₯)Δ𝑒(π‘₯,𝑑)𝑑π‘₯𝑑𝑑(43) and nondimensionalised as ξ€œπ·=4πœ‡01/4ξ€œ10π‘ƒπ‘Žπ‘£Ξ”π‘’π‘‘π‘₯π‘‘πœ,(44) where 𝐷=𝐷(π‘₯,𝜏)πΈπ‘β„Ž3/𝐿3𝐹20; substitution for Δ𝑒 from (40) gives the closed form solution as 𝐷=𝐷1+𝐷2+𝐷3+𝐷4,(45) where𝐷1=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽœβŽ8ξ‚΅Ξ”5(πœ‚)-πœ†3+Ξ”3𝑐(πœ‚)1π’œ+𝑐2β„°β„¬βˆ’π‘ξ‚Άξ‚Άξ‚΅ξ‚΅1π’œ+𝑐2ℰℬΔ7(πœ‚)+Ξ”6(ξ‚Άπœ‚)πΆπ‘ŽβŽžβŽŸβŽŸβŽŸβŽ πœ‡24𝑃0βˆ’π‘ξ‚΅ξ‚΅1π’œ+𝑐2ℰℬ+ξ‚€8-πœ†1Ξ”42(πœ‚)+3-πœ†2ξ‚ξ‚Άπœ‡2𝑃20⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,(46a)𝐷2βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽœβŽ4ξ‚΅Ξ”=πœ€5(πœ‚)-πœ†3+Ξ”5𝑐(πœ‚)1π’œ+𝑐2β„°β„¬βˆ’π‘ξ‚Άξ‚Άξ‚΅ξ‚΅1π’œ+𝑐2ℰℬΔ7(πœ‚)+Ξ”6(ξ‚Άπœ‚)πΆπ‘ŽβŽžβŽŸβŽŸβŽŸβŽ πœ‡48𝑃0βˆ’π‘ξ‚΅ξ‚΅1+𝑐2π’œ2β„°ξ‚Ά+ξ‚€42ℬ-πœ†1Ξ”41(πœ‚)+3-πœ†2ξ‚ξ‚Άπœ‡2𝑃20⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦(46b)for all negative values of πΆπ‘Ž.Similarly,𝐷3=βŽ‘βŽ’βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽœβŽπœ€π‘ξ‚΅ξ‚΅1π’œ+𝑐2ℰℬ𝒴Δ248(πœ‚)+41495-πœ†4Ξ”7πœ‡(πœ‚)ξ‚Άξ‚Ά2𝑃202+πœ€ξ‚΅ξ‚΅-πœ†2+6-πœ†13ξ‚ΆΞ”4(πœ‚)+1631980-πœ†4Ξ”61(πœ‚)βˆ’6ξ‚Άπœ‡2𝑃20⎞⎟⎟⎟⎠⎀βŽ₯βŽ₯βŽ₯⎦,(46c)𝐷4=βŽ‘βŽ’βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽœβŽπœ€2𝑐1π’œ+𝑐2ℰℬ𝒴Δ488(πœ‚)+41990-πœ†4Ξ”7πœ‡(πœ‚)ξ‚Άξ‚Ά2𝑃20+πœ€2ξ‚΅ξ‚΅-πœ†2+3-πœ†13ξ‚ΆΞ”4(πœ‚)+1633960-πœ†4Ξ”61(πœ‚)βˆ’ξ‚Άπœ‡122𝑃20⎞⎟⎟⎟⎠⎀βŽ₯βŽ₯βŽ₯⎦,(46d)where Ξ”3𝑒(πœ‚)=βˆ’πœ‹πœ…1/2πœ…2πœ…3+π‘’βˆ’πœ‹πœ…4/2πœ…3πœ…5+π‘’π‘–πœ‹πœ‚/2πœ…6πœ…7+πœ”βˆ’1πœ…11π‘’π‘–πœ‹πœ‚/2πœ…9πœ…10ξ‚Ή,Ξ”4(πœ‚)=(π’œβ„¬)4ℬ+𝛾eqΞ¨3eqπ’œξƒͺβˆ’π‘’βˆ’πœ‹πœ…1/22πœ‹πœ…21πœ…2πœ…12+π‘’βˆ’πœ‹πœ…1/22πœ‹πœ…1πœ…4πœ…5πœ…12+πœ”βˆ’1π‘’βˆ’πœ‹πœ‚πœ…1/22πœ‹πœ…1πœ…13ξƒ­,Ξ”5𝑒(πœ‚)=βˆ’πœ‹πœ…1/22πœ‹πœ…1πœ…2πœ…3+π‘’βˆ’πœ‹πœ…4/22πœ‹πœ…3πœ…4πœ…5π‘’βˆ’π‘–π‘–πœ‹πœ‚/22πœ‹πœ‚πœ…6πœ…7πœ”βˆ’π‘–βˆ’1πœ…11π‘’π‘–πœ‹πœ‚/22πœ‹πœ‚πœ…9πœ…10ξ‚Ή,Ξ”61(πœ‚)=4βˆ’π‘’βˆ’πœ‹πœ…1/22πœ‹πœ…12πœ…12βˆ’π‘’βˆ’πœ‹πœ…4/22πœ‹πœ…42πœ…12ξƒ­,Ξ”7(ξ‚Έπ‘’πœ‚)=1+βˆ’πœ‹πœ…1/2πœ…1πœ…12+π‘’βˆ’πœ‹πœ…4/2πœ…4πœ…12ξ‚Ή,Ξ”8(πœ‚)=(π’œβ„¬)ℬ+𝛾eqΞ¨3eqπ’œξƒͺ+π‘’βˆ’πœ‹πœ…1/2πœ…1πœ…2πœ…12βˆ’π‘’βˆ’πœ‹πœ…1/2πœ…4πœ…5πœ…12βˆ’πœ”βˆ’1π‘’βˆ’πœ‹πœ‚πœ…1/2πœ…13ξ‚Ή.(47)

5. Analysis of Logarithmic Damping Decrement

For this problem the logarithmic damping coefficient is used as a measure of the damping capacity of the structure under consideration. Following Masuko et al. [24], the relationship between the energy dissipation from two consecutive cycles and the associated logarithmic damping decrement satisfies the form 1𝛿=2𝐸ln𝑛𝐸𝑛+1ξ‚Ά=12𝐸lnne+𝐸loss𝐸𝑛𝐸lossξ‚Ά,(48) where 𝐸ne is the strain energy of the laminate material, whereas 𝐸loss is the energy loss per cycle, and 𝛿 is the Logarithmic damping decrement.

In particular, 𝐸loss, the slip energy, is computed from (45)-(46a), (46b), (46c), and (46d), that is 𝐷, whereas 𝐸ne is presently analyzed.

5.1. Analysis of Strain Energy of the Clamped Smart Structure

The total strain energy of the smart structure is a combination of the energy introduced by the bending moment as well and that stored from the deflection of the free end. The energy from the bending moment can be evaluated from Castigliano’s theorem, namely, π‘ˆ1=ξ€œπΏ0𝑀22𝐸eq𝐼eq𝑑π‘₯,(49) whilst the energy stored at the free end follows from the theory of strength of materials namely, π‘ˆ2=32𝐸eq𝐿3π‘Š2𝐿.(50) here, 𝑀 is the bending moment, 𝐸eq the modulus of rigidity of the material, 𝐼eq=(2/3)π‘β„Ž3eq the moment of inertia of the clamped sandwich composite smart structure, and π‘ŠπΏ the deflection at the free end.

For this case, the computed nondimensionalized strain energy components areπ‘ˆ1⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩Θ=271𝑒𝑖4πœ‹πœ‚βˆ’Ξ˜2𝑒2πœ‹πœ‚(π‘–βˆ’πœ’)+πœ”βˆ’2Θ3π‘’βˆ’4πœ‹πœ’πœ‚βˆ’2πœ‡π‘ƒ0πœ€ξ‚€ξ‚€1+2ξ‚π’žΞ˜ξ‚ξ‚€4𝑒𝑖2πœ‹πœ‚βˆ’Ξ˜5πœ”βˆ’1𝑒2πœ‹πœ‚(π‘–βˆ’πœ’)βˆ’Ξ˜6πœ”βˆ’1π‘’βˆ’2πœ‹πœ’πœ‚+Θ7πœ”βˆ’2π‘’βˆ’4πœ‹πœ’πœ‚ξ‚+πœ‡2𝑃20πœ€ξ‚΅ξ‚΅1+πœ€+24ξ‚Άπ’ž2ξ‚Άξ€·Ξ˜8βˆ’2Θ9πœ”βˆ’1π‘’βˆ’2πœ‹πœ’πœ‚+Θ10πœ”βˆ’2π‘’βˆ’4πœ‹πœ’πœ‚ξ€ΈβŽ«βŽͺβŽͺ⎬βŽͺβŽͺ⎭,(51)whereΘ1=2π’œβ„¬ξ€·β„¬ξ€·1+πœ‡1ξ€Έξ€·+π’œ1+πœ‡2𝛾eqΞ¨eqξ€Έξƒͺ2,Θ2=4π’œβ„¬ξ€·β„¬+π’œπ›ΎeqΞ¨3eqξ€·πœ‡βˆ’π‘–1ℬ+πœ‡2ℬ𝛾eqΞ¨3eqξƒͺ,Ξ˜ξ€Έξ€Έ3=ξƒ©πœ‡1ℬ+πœ‡2π’œπ›ΎeqΞ¨3eqℬ+π’œπ›ΎeqΞ¨3eqξ€·πœ‡βˆ’π‘–1ℬ+πœ‡2ℬ𝛾eqΞ¨3eqξƒͺξ€Έξ€Έ2,Θ4=(π’œβ„¬)2ℬ1+πœ‡1ξ€Έξ€·+π’œ1+πœ‡2𝛾eqΞ¨3eqℬ+π’œπ›ΎeqΞ¨3eqξ€Έξƒͺ2,Θ5=ξƒ©π’œβ„¬ξ€·β„¬+π’œπ›ΎeqΞ¨3eqξ€Έξƒͺ,Θ6=ξƒ©ξƒ©π’œβ„¬ξ€·β„¬+π’œπ›ΎeqΞ¨3eqξ€ΈξƒͺΓ—ξƒ©πœ‡1ℬ+πœ‡2π’œπ›ΎeqΞ¨3eqℬ+π’œπ›ΎeqΞ¨3eqξ€·βˆ’π‘–πœ‡1ℬ+πœ‡2ℬ𝛾eqΞ¨3eq,Ξ˜ξ€Έξ€Έξƒͺξƒͺ7=βŽ›βŽœβŽœβŽξ€·πœ‡1ℬ+πœ‡2π’œπ›ΎeqΞ¨3eqξ€Έ2ℬ+π’œπ›ΎeqΞ¨3eqℬ+π’œπ›ΎeqΞ¨3eqξ€·βˆ’π‘–πœ‡1ℬ+πœ‡2ℬ𝛾eqΞ¨3eq⎞⎟⎟⎠,Ξ˜ξ€Έξ€Έ8=ξƒ©π’œβ„¬ξ€·β„¬+π’œπ›ΎeqΞ¨3eqξ€Έξƒͺ2,Θ9=βŽ›βŽœβŽœβŽξ€·ξ€·1βˆ’πœ1ξ€Έξ€·1βˆ’πœ2ξ€Έξ€Έξ€·πœ‡1ℬ+πœ‡2π’œπ›ΎeqΞ¨3eqℬ+π’œπ›ΎeqΞ¨3eqξ€Έ2⎞⎟⎟⎠,Θ10=ξƒ©πœ‡1ℬ+πœ‡2π’œπ›ΎeqΞ¨3eqℬ+π’œπ›ΎeqΞ¨3eqξ€Έξƒͺ2,ξƒ©πœ’=πœ‡1ℬ+πœ‡2π’œπ›ΎeqΞ¨3eqℬ+π’œπ›ΎeqΞ¨3eqξ€Έξƒͺ,π‘ˆ2=⎧βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎩Θ1𝑒𝑖4πœ‹πœ‚βˆ’Ξ˜2𝑒2πœ‹πœ‚(π‘–βˆ’πœ’)+πœ”βˆ’2Θ3π‘’βˆ’4πœ‹πœ’πœ‚βˆ’2πœ‡π‘ƒ0πœ€ξ‚€ξ‚€1+2ξ‚π’žΞ˜ξ‚ξ‚€4𝑒𝑖2πœ‹πœ‚βˆ’Ξ˜5πœ”βˆ’1𝑒2πœ‹πœ‚(π‘–βˆ’πœ’)βˆ’Ξ˜6πœ”βˆ’1π‘’βˆ’2πœ‹πœ’πœ‚+Θ7πœ”βˆ’2π‘’βˆ’4πœ‹πœ’πœ‚ξ‚+πœ‡2𝑃20πœ€ξ‚΅ξ‚΅1+πœ€+24ξ‚Άπ’ž2ξ‚Άξ€·Ξ˜8βˆ’2Θ9πœ”βˆ’1π‘’βˆ’2πœ‹πœ’πœ‚+Θ10πœ”βˆ’2π‘’βˆ’4πœ‹πœ’πœ‚ξ€Έβˆ’βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£ξ‚΅Ξ˜11πœ‡480𝑃0Ξ˜πœ€+1548πΆπ‘Žξ‚Άπ‘’π‘–2πœ‹πœ‚βˆ’πœ”βˆ’1ξ‚΅Ξ˜12πœ‡480𝑃0Ξ˜πœ€+1648πΆπ‘Žξ‚Άπ‘’βˆ’2πœ‹πœ’πœ‚βˆ’Ξ˜13πœ‡4802𝑃20ξ‚΅πœ€πœ€+22ξ‚Άπ’ž+πœ”βˆ’1Θ14πœ‡4802𝑃20ξ‚΅πœ€πœ€+22ξ‚Άπ’žπ‘’βˆ’2πœ‹πœ’πœ‚βˆ’1πœ‡96𝑃0ξ‚€πœ€1+2ξ‚π’žπΆπ‘Žξ€·Ξ˜17βˆ’πœ”βˆ’1Θ18π‘’βˆ’2πœ‹πœ’πœ‚ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦+ξ‚ΈΞ˜199602πœ‡2𝑃20πœ€2+Θ20962πœ‡π‘ƒ0πœ€πΆπ‘Ž+1962𝐢2π‘Žξ‚ΉβŽ«βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎬βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎭,(52)whileΘ11=ξ€·2π’œβ„¬π’œ+β„¬π›Ύβˆ’1eqΞ¨βˆ’2eqℬ1+πœ‡1ξ€Έξ€·+π’œ1+πœ‡2𝛾eqΞ¨3eqξ€Έ,Θ12=2ξ€·π’œ+β„¬π›Ύβˆ’1eqΞ¨βˆ’2eqξ€Έξ€·πœ‡1ℬ+πœ‡2π’œπ›ΎeqΞ¨3eqℬ+π’œπ›ΎeqΞ¨3eqℬ+π’œπ›ΎeqΞ¨3eqξ€·βˆ’π‘–πœ‡1ℬ+πœ‡2ℬ𝛾eqΞ¨3eqξƒͺ,Ξ˜ξ€Έξ€Έ13=ξ€·π’œβ„¬π’œ+β„¬π›Ύβˆ’1eqΞ¨βˆ’2eqℬ+π’œπ›ΎeqΞ¨3eqξ€Έ,Θ14=ξ€·β„¬πœ‡1+π’œπœ‡2𝛾eqΞ¨3eqξ€Έξ€·π’œβ„¬π›Ύβˆ’1eqΞ¨βˆ’2eqℬ+π’œπ›ΎeqΞ¨3eqξ€Έ,Θ15=π’œβ„¬ξ€·β„¬ξ€·1+πœ‡1ξ€Έξ€·+π’œ1+πœ‡2𝛾eqΞ¨3eqξ€Έ,Θ16=ξƒ©ξ€·πœ‡1ℬ+πœ‡2π’œπ›ΎeqΞ¨3eqℬ+π’œπ›ΎeqΞ¨3eqξ€·βˆ’π‘–πœ‡1ℬ+πœ‡2ℬ𝛾eqΞ¨3eqξƒͺ,Ξ˜ξ€Έξ€Έ17=2π’œβ„¬ξ€·β„¬+π’œπ›ΎeqΞ¨3eqξ€Έ,Θ18=2ξ€·β„¬πœ‡1+π’œπœ‡2𝛾eqΞ¨3eqℬ+π’œπ›ΎeqΞ¨3eqξ€Έ,Θ19=ξ€·π’œ+β„¬π›Ύβˆ’1eqΞ¨βˆ’2eqξ€Έ2,Θ20=ξ€·π’œ+β„¬π›Ύβˆ’1eqΞ¨βˆ’2eqξ€Έ25.(53)Here, we have introduced the nondimensionalization π‘ˆπ‘ˆ=π‘›πΈπ‘β„Ž3𝐹20𝐿3,𝑛=1,2,(54) to write the logarithmic damping decrement as 1𝛿=2ln1+Dπ‘ˆ1+π‘ˆ2ξƒͺ.(55)

6. Analysis of Spatial Transfer Functions for Active Vibration and Noise Control Designs

Several authors in the field of active noise and vibration control have derived useful rational expressions with experimental verification for transfer functions with piezoelectric beams or smart structures systems such as in [12, 25–30]. These transfer functions are characterized by a common denominator that is strongly influenced by the parameters of the piezoelectric beams or smart structures. For this paper, the focus is to derive closed-form analytic rational expressions in the Laplace domain for both the active control of the dynamic deflection and interfacial slip at the interface of the layered sandwich composite elastic beam-plate smart structure. We shall now proceed to address these issues.

6.1. Computation of Spatial Transfer Functions for Active Control of Dynamic Deflection and Interfacial Slip

The transfer function for the active control of dynamic deflection can be derived or computed from (28) asξ‚‹π‘Šξ€·ξ€Έπ‘₯,𝑠=𝐹(𝑠)π‘Šξ€·ξ€ΈπΏπ‘₯,𝑠3𝐸eq1π‘β„Ž3eq1=𝑇𝐹1=𝐿3βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽξ‚΅2𝐹2(𝑠)βˆ’πœ‡π‘ƒ0π’žξ‚πΉ1ξ‚Άξ‚€3(𝑠)π‘₯2βˆ’π‘₯3+πœ‡π‘ƒ0βŽ‘βŽ’βŽ’βŽ’βŽ£βˆ’3π’ž2𝐹1(𝑠)π‘₯2+βŽ›βŽœβŽœβŽœβŽξƒ©π’ž2ξƒͺ𝐹1βŽ›βŽœβŽœβŽœβŽ(𝑠)βˆ’π’œ+β„¬π›Ύβˆ’1eqΞ¨βˆ’2eq48βŽžβŽŸβŽŸβŽŸβŽ ξ‚πΉ3⎞⎟⎟⎟⎠π‘₯(𝑠)3⎀βŽ₯βŽ₯βŽ₯βŽ¦πœ€+𝐹3βŽ‘βŽ’βŽ’βŽ£ξ€·(𝑠)π’œ+β„¬π›Ύβˆ’1eqΞ¨βˆ’2eqξ€ΈβŽ›βŽœβŽœβŽπ‘₯433βˆ’π‘₯590⎞⎟⎟⎠⎀βŽ₯βŽ₯⎦+πΆπ‘Žξ‚πΉ1βŽ›βŽœβŽœβŽ(𝑠)βˆ’π‘₯232+π‘₯396⎞⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎠𝐸eq1π‘β„Ž3eq1,(56)while the corresponding transfer function for the active control of dynamic interfacial slip can be derived or computed from (40) asΔ̃𝑒π‘₯,𝑠=Δ𝐹(𝑠)𝑒𝐿π‘₯,𝑠2𝐸eq1π‘β„Ž2eq1=𝑇𝐹2=𝐿2⎧βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽ©βŽ›βŽœβŽœβŽœβŽœβŽβŽ›βŽœβŽœβŽ3πœ‡π‘ƒ0βŽ›βŽœβŽœβŽπ‘1π’ž2π’œ+𝑐2β„°2β„¬βŽžβŽŸβŽŸβŽ π‘ βˆ’1βŽ›βŽœβŽœβŽβˆ’3𝑐1π’œ+𝑐2β„°β„¬βŽžβŽŸβŽŸβŽ π‘ ξ‚πΉ2⎞⎟⎟⎠+ξ‚Έπœ‡(𝑠)𝑃0ξ‚Έ3-πœ†1𝐹1(𝑠)+-πœ†2π‘ βˆ’1ξ‚Ήβˆ’3-πœ†3𝐹2ξ‚ΉβŽžβŽŸβŽŸβŽŸβŽŸβŽ π’΅(𝑠)+πœ‡π‘ƒ0πœ€βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽβŽ›βŽœβŽœβŽβˆ’3𝑐1π’žπ’œ+𝑐2β„°β„¬βŽžβŽŸβŽŸβŽ π‘ ξ‚πΉ1(𝑠)+-πœ†1𝐹1⎞⎟⎟⎠(𝑠)π‘₯+βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽβŽ›βŽœβŽœβŽœβŽπ‘1π’žπ’œ+𝑐2Ξ¨βˆ’1eqβ„¬βŽžβŽŸβŽŸβŽŸβŽ π‘ ξ‚πΉ1(𝑠)4βˆ’βŽ›βŽœβŽœβŽœβŽπ‘1π’žπ’œ+𝑐2Ξ¨βˆ’1eqβ„¬βŽžβŽŸβŽŸβŽŸβŽ -πœ†432𝑠𝐹1+(𝑠)-πœ†22𝐹1(𝑠)βˆ’-πœ†4𝐹163(𝑠)βˆ’π‘ βˆ’1⎞⎟⎟⎟⎟⎟⎟⎠π‘₯2βŽ›βŽœβŽœβŽœβŽ2-πœ†433βŽ›βŽœβŽœβŽœβŽπ‘1π’œ+𝑐2Ξ¨βˆ’1eqβ„¬βŽžβŽŸβŽŸβŽŸβŽ π‘ ξ‚πΉ3(𝑠)+433-πœ†4𝐹3(𝑠)+23π‘ βˆ’1ξƒͺ⎞⎟⎟⎟⎠π‘₯3βˆ’βŽ›βŽœβŽœβŽœβŽ5180-πœ†4βŽ›βŽœβŽœβŽœβŽπ‘1π’œ+𝑐2Ξ¨βˆ’1eqβ„¬βŽžβŽŸβŽŸβŽŸβŽ π‘ ξ‚πΉ3(𝑠)+590-πœ†4𝐹3⎞⎟⎟⎟⎠(𝑠)π‘₯4⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦+πΆπ‘ŽβŽ›βŽœβŽœβŽœβŽc12π’œ+c2Ξ¨βˆ’1eq2β„¬βŽžβŽŸβŽŸβŽŸβŽ π‘ ξ‚πΉ3(𝑠)+𝐹3βŽ›βŽœβŽœβŽ(𝑠)βˆ’116π‘₯+π‘₯232⎞⎟⎟⎠⎫βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎬βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎭𝐸eq1π‘β„Ž2eq1.(57)Within the context of robust active vibration and noise control designs, the common denominator admits the formξ‚‹π‘Šξ€·ξ€Έ=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽξ‚€2𝐹π‘₯,𝑠2(𝑠)βˆ’πœ‡π‘ƒ0π’žξ‚πΉ1(3𝑠)π‘₯2βˆ’π‘₯3+πœ‡π‘ƒ0ξƒ¬βˆ’3π’ž2𝐹1(𝑠)π‘₯2+ξƒ©ξ‚€π’ž2𝐹1(𝑠)βˆ’π’œ+β„¬π›Ύβˆ’1eqΞ¨βˆ’2eqξƒͺ𝐹483ξƒͺπ‘₯(𝑠)3ξƒ­πœ€+𝐹3ξ‚Έξ€·(𝑠)π’œ+β„¬π›Ύβˆ’1eqΞ¨βˆ’2eqξ€Έξ‚΅π‘₯4βˆ’π‘₯335+90ξ‚Άξ‚ΉπΆπ‘Žξ‚πΉ1ξ‚΅βˆ’(𝑠)π‘₯2+32π‘₯3ξ‚ΆβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,96(58)where 𝐹1(𝑠), 𝐹2(𝑠), and 𝐹3(𝑠) are the Laplace transform of 𝐹1(𝜏),𝐹2(𝜏) and 𝐹3(𝜏) as previously defined, subject to zero initial conditions, namely, 𝐹1(0), 𝐹2(0), and 𝐹3(0).

7. Analysis of Results

By studying Figure 2, it is significant to note that for this case πœ€=0 and irrespective of the value of the piezoelectric parameter πΆπ‘Ž, the energy dissipation assumes the same parabolic profile reported in earlier studies such as Damisa et al. [15, 16] and Olunloyo et al. [19, 20]. This is a direct consequence of (45) and ((46a)–(46d)) from which we deduce the condition for optimal energy dissipation asπœ‡π‘ƒopt=(2+πœ€)Ξ“βˆ’1ξ‚΅4ξ‚΅Ξ”5(πœ‚)-πœ†3+Ξ”3𝑐(πœ‚)1π’œ+𝑐2β„°β„¬βˆ’π‘ξ‚Άξ‚Άξ‚΅ξ‚΅1π’œ+𝑐2ℰℬΔ7(πœ‚)+Ξ”6ξ‚Ά(πœ‚)πΆπ‘Žξ‚Ά,48(59)whereβŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβŽ§βŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽ©π‘Ξ“=