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

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].

Figure 1 

Sandwich Plate System (SPS) technology. Source: Intelligent Engineering (IE), UK/Canada.

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 “out-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 equations𝑃∇²Φ=0,_ℎ𝜌_𝑤+𝜕Φ𝜕𝑡+𝑔𝑧+12||∇Φ||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 (𝑥₀,𝑦₀,𝑧₀) to elliptical cylindrical coordinates (𝜉,𝜂₀,𝑧₀) holds, namely, 𝑥₀=𝑐cosh𝜉cos𝜂₀,𝑦₀=𝑐sinh𝜉sin𝜂₀,𝑧₀=𝑧₀,(1b)where 𝜉≥0,0≤𝜂₀<2𝜋,−∞<𝑧₀<∞.

(a)

(b)

Figure 2 

(a) A mathematical model of ship/FPSO in ocean environment. (b) An elliptical model representation of a ship/FPSO bottom hull configuration.

Thereafter, the first part of (1a) takes the form 1𝑐²sinh²𝜉+sin²𝜂₀𝜕²Φ𝜕𝜉²+𝜕²Φ𝜕𝜂₀₂+𝜕²Φ𝜕𝑧₀₂=0.(2) The decomposition of Φ into Φ=𝜙(𝜉,𝜂₀)𝑓(𝑧₀,𝑡) via variable separation leads to a set of two equations namely, 1𝑐²cosh2𝜉−cos2𝜂₀𝜕²𝜙𝜉,𝜂₀𝜕𝜉²+𝜕²𝜙𝜉,𝜂₀𝜕𝜂₀₂+𝑘²𝜕𝜙𝜉,𝜂₀=0,(3)₂𝑓𝑧₀,𝑡𝜕𝑧²−𝑘²𝑓𝑧₀,𝑡=0,(4) where 𝑘² 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, 𝜙(𝜉,𝜂₀), which can be further decomposed as 𝜙(𝜉,𝜂₀)=𝐹(𝜉)𝐺, leads to the well-known Mathieu and the modified Mathieu differential equations, namely, 𝑑²𝐺𝜂₀𝑑𝜂₀₂𝑑+𝑎−2𝑞cos2𝜂₀𝐺𝜂₀=0,(5)₂𝐹(𝜉)𝑑𝜉²−𝑎−2𝑞cosh2𝜉𝐹(𝜉)=0,(6) here the separation constant 𝑎 is any characteristic eigenvalue while 𝑞=𝑘²𝑐²/4.

The solution to (6) and (7) is strongly influenced by the numerical value of 𝑞 and for deep and ultradeep water problems 𝑘 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(𝜉),𝐺(𝜂), and 𝑎 can all be expanded in perturbation series in terms of the parameter 𝑞.

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 𝐹_ℎ=_𝑆𝑃_ℎ𝑒_𝑧̂𝑑𝑆,(7) where 𝑃_ℎ=−𝜌_𝑤𝜙𝜉,𝜂⁰𝜕𝑓𝑧⁰,𝑡𝜕𝑡+𝑔𝑧+21𝑈²(8) and 𝑒_𝑧̂ is the outward normal unit vector for the element 𝑑𝑆 of the surface of the ship or floating structure.

Here, 𝑈 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,𝑡) in the Laplace domain has been reported by Olunloyo et al. [21–23] and can be written as 𝑓𝑧₀,𝑠=−𝑠𝑊𝑥⁰,𝑦₀,𝑠𝑘sinh𝑘ℎ_𝐷cosh𝑘𝑧⁰,(9) where 𝑊(𝑥₀,𝑦₀,𝑠) and ℎ_𝐷 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 𝑜(𝜀_𝑠2) yield 𝑃_ℎ=−𝜌_𝑤𝜙⁰×𝜕𝜉𝑓,𝜂₀+𝜀_𝑠𝜙₁𝜉,𝜂₀₀𝑧₀,𝑡+𝜀_𝑠𝑓₁𝑧₀,𝑡𝜕𝑡+𝑔𝑧+12𝑈²⁰+𝜀_𝑠𝑈²⁰(10) while 𝑓₀𝑧₀,𝑠=−𝑠𝑊⁰𝑥₀,𝑦₀,𝑠𝑘₀sinh𝑘₀ℎ_𝐷cosh𝑘⁰𝑓₁𝑧₀,𝑧₀,𝑠=−𝑠𝑊¹𝑥₀,𝑦₀,𝑠𝑘₁sinh𝑘₁ℎ_𝐷cosh𝑘¹𝑧₀,(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≥max⋅‖‖𝐹_ℎ‖‖.(12)

(a)

(b)

(c)

(d)

Figure 3 

(a) Preslip geometry for the composite sandwich structure under dynamic load. (b) Layering cross-section of composite structure. (c) Upper layer postslip geometry under dynamic load. (d) Lower layer postslip geometry under dynamic load.

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, ℎ_𝑎≪ℎ₁;ℎ_𝑠≪ℎ₂. 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 𝑊, same for each layer, satisfies the following PDE:𝑐¹+𝑐₂4𝜕⁵𝑊𝜕𝑡𝜕𝑥⁴+𝜕⁴𝑊𝜕𝑥⁴+𝛽¹𝒜+𝛽₂ℬ+𝛽₃+𝛽₄4𝜕²𝑊𝜕𝑡²=𝛼¹𝒜+𝛼₂ℬ4𝜕𝑃𝜕(𝑥𝑥,0)+𝐶4^𝑎𝜕²𝑉_𝑎𝜕𝑥²,(13) where 𝒜 denotes (1−𝜐²¹), ℬ denotes (1−𝜐²²) and the following parameters have been defined viz: 𝛼₁=6𝜇𝐸_eq1ℎ²_eq1,𝛽₁=𝜌¹𝑏ℎ₁𝐸₁𝐼₁𝛽₃,=𝜌^𝑎𝑏ℎ_𝑎𝐸_𝑎𝐼_𝑎1𝛼₂,ℎ_eq1=ℎ₁+ℎ_𝑎,=6𝜇𝐸_eq2ℎ²_eq2,𝛽₂=𝜌²𝑏ℎ₂𝐸₂𝐼₂𝛽₄,=𝜌^𝑠𝑏ℎ_𝑠𝐸_𝑠𝐼_𝑠𝐶_𝑎,ℎ_eq2=ℎ₂+ℎ_𝑠,=12𝐸^𝑎𝑑₃₁𝑏ℎ_𝑎+ℎ_𝑠,(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, (•)=^∞0(•)𝑒^−𝑠𝑡𝑑𝑡,(•)=12𝜋𝑖^{𝜂+𝑖∞𝜂−𝑖∞}(•)𝑒^𝑠𝑡𝑑𝑠(15) on (13) to obtain 1+𝑠𝑐¹+𝑐₂4𝑑⁴𝑊(𝑥,𝑠)𝑑𝑥⁴+𝛽¹𝒜+𝛽₂ℬ+𝛽₃+𝛽₄×𝑠₂4=𝛼𝑊(𝑥,𝑠)−𝑠𝑊(0)−𝑊̇(0)₁𝒜+𝛼₂ℬ4𝑠𝜕𝑃𝜕(𝑥𝑥,0)+𝐶4^𝑎𝑑²𝑉(𝑥,𝑠)_𝑎𝑑𝑥².(16) By limiting the analysis to the case of linear pressure variation along the laminate interface, namely, 𝑃(𝑥,0)=𝑃₀1+𝜀𝐿𝑥,(17) Equation (16) simplifies to the form 1+𝑠𝑐¹+𝑐₂4𝑑⁴𝑊(𝑥,𝑠)𝑑𝑥⁴+𝛽¹𝒜+𝛽₂ℬ+𝛽₃+𝛽₄×𝑠₂4=𝛼𝑊(𝑥,𝑠)−𝑠𝑊(0)−𝑊̇(0)₁𝒜+𝛼₂ℬ𝜀𝑃₀4𝑠𝐿+𝐶4^𝑎𝑑²𝑉_𝑎(𝑥,𝑠)_𝑎𝑑𝑥².(18) In (18), the spatial distribution of the actuating layer is considered to be uniform with respect to 𝑥. Under this assumption, the term 𝑑²𝑉_𝑎(𝑥,𝑠)_𝑎/𝑑𝑥² following Alberts et al. [11] becomes 𝑑²𝑉_𝑎(𝑥,𝑠)_𝑎𝑑𝑥²=𝛿^(𝑥−0)−𝛿^(𝑥−𝐿)𝑉_𝑎𝑠,(19) where 𝛿′(⋅) is the first derivative of the Dirac delta function.

Introducing the Fourier finite sine transform [•]=^𝐿0[•]sin𝑛𝜋𝑥𝐿𝑑𝑥,[•]=𝐿2^∞_𝑛=1[•]sin𝑛𝜋𝑥𝐿(20) in conjunction with the following boundary conditions in the Laplace transform plane namely, 𝑊(0,𝑠)=𝑑𝑑𝑥𝑊(0,𝑠)=𝑑𝑑𝑥²²𝑊(𝐿,𝑠)=0(21) gives the Fourier sine transform of (18) as 𝑛⁴𝜋⁴𝐿⁴1+𝑠𝑐¹+𝑐₂4𝑊^𝐹𝜆_𝑛+𝛽,𝑠₁𝒜+𝛽₂ℬ+𝛽₃+𝛽₄4𝑠²=𝛼𝑊^𝐹𝜆_𝑛,𝑠₁𝒜+𝛼₂ℬ4𝑠𝑃⁰𝜀+𝑛₃𝑛𝜋1+(−1)^𝑛+1𝜋³𝐿³(−1)^𝑛+1𝑊(𝐿,𝑠)−𝑛𝜋𝐿𝑊^𝑥𝑥+𝐶(0,𝑠)_𝑎𝑉_𝑎4𝑠𝑛𝜋1+(−1)^𝑛+1.(22) Following the procedure outlined in Olunloyo et al. [21], the bending moment is computed as:𝑊_𝑥𝑥(0,𝑠)=⎛⎜⎜⎜⎜⎝12𝐹(𝑠)−𝑏𝐸_eq1ℎ³_eq1/𝒜+𝐸_eq26ℎ𝜇𝑃³_eq20/ℬ+𝑠𝑐₁𝐸_eq1ℎ³_eq1/𝒜+𝑐₂𝐸_eq2ℎ³_eq2/ℬ(1+(𝜀/2))ℎ_eq1+ℎ_eq2𝑠𝐸_eq1ℎ³_eq1/𝒜+𝐸_eq2ℎ³_eq2/ℬ+𝑠²𝑐₁𝐸_eq1ℎ³_eq1/𝒜+𝑐₂𝐸_eq2ℎ³_eq2/ℬ⎞⎟⎟⎟⎟⎠𝐿.(23)This result clearly indicates that the value for expression (25) cannot be fully determined until the forcing function 𝐹(𝑡) is specified. Next, we limit the analysis to the case of harmonic loading function. For this case, the forcing function is 𝐹(𝑡)=𝐹₀𝑒^𝑖𝜔𝑡 with the driving transform as 𝐹(𝑠)=𝐹⁰𝑠−𝑖𝜔,(24) where 𝐹₀ represents the amplitude of the loading force and, 𝜔 is the maximum associated excitation frequency for the fluid-structure systems.

Recalling the only remaining unutilized boundary condition in (21), namely, (𝑑/𝑑𝑥)𝑊(0,𝑠)=0 the Fourier inversion of (22) leads to the expression𝑊(𝑥,𝑠)=⎧⎪⎪⎪⎨⎪⎪⎪⎩^2𝑠𝑊_{(𝐿,𝑠)𝑥−2𝐿}^{3⎛⎜⎜⎜⎜⎜⎜⎜⎝}12𝑠𝐹(𝑠)𝑏𝐸_eq1ℎ³_eq1/𝒜+𝐸_eq2ℎ³_eq2/ℬ+𝑠𝑐₁𝐸_eq1ℎ³_eq1/𝒜+𝑐₂𝐸_eq2ℎ³_eq2/ℬ_{−⎛⎜⎜⎜⎝}6𝜇𝑃₀(1+(𝜀/2))ℎ_eq1+ℎ_eq2𝐸_eq1ℎ³_eq1/𝒜+𝐸_eq2ℎ³_eq2/ℬ+𝑠𝑐₁𝐸_eq1ℎ³_eq1/𝒜+𝑐₂𝐸_eq2ℎ³_eq2/ℬ^{⎞⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎟⎠Λ1}_{+3𝐿3/32𝒜/𝐸eq1ℎ2eq1+ℬ/𝐸eq2ℎ2eq2𝜇𝑃0𝜀Λ2+(1/16)𝐶𝑎𝑉𝑎𝐿3Λ3}⎫⎪⎪⎪⎬⎪⎪⎪⎭𝑠𝑠+𝜅₁𝑠+𝜅₂𝜔₀₋₂(2,5)where 𝜅₁=𝑐¹+𝑐₂𝜔²⁰8+𝑖𝜔²⁰−𝑐¹+𝑐₂8²𝜅₂𝜔⁴⁰,=𝑐¹+𝑐₂𝜔²⁰8−𝑖𝜔²⁰−𝑐¹+𝑐₂8²𝜔₂₀𝜔⁴⁰,=𝜔²⁰1−𝑐¹+𝑐₂²32𝜔²⁰,𝜔²⁰=𝑛⁴𝜋⁴𝛽𝐿⁴.(26) Here, 𝛽=𝛽¹𝒜+𝛽₂ℬ+𝛽₃+𝛽₄Λ₁4,=𝑥6−𝑥²4+𝑥³12,Λ²=𝑥45−2𝑥³9+𝑥⁴3−2𝑥⁵Λ15₃,=𝑥3−𝑥²2+𝑥³6.(27) Appropriate substitution for 𝑊(𝐿,𝑠) into (25) and rearrangement gives𝑊𝑥,𝑠=⎧⎪⎪⎪⎨⎪⎪⎪⎩^{⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝}2𝑠𝐹₀𝑏𝐸_eq1ℎ³_eq1/𝒜+𝐸_eq2ℎ³_eq2/ℬ+𝑠𝑐₁𝐸_eq1ℎ³_eq1/𝒜+𝑐₂𝐸_eq2ℎ³_eq2/ℬ(𝑠−𝑖𝜔)₋𝜇𝑃₀(1+(𝜀/2))ℎ_eq1+ℎ_eq2𝐸^eq1ℎ³_eq1/𝒜+𝐸_eq2ℎ³_eq2/ℬ+𝑠𝑐₁𝐸_eq1ℎ³_eq1/𝒜+𝑐₂𝐸_eq2ℎ³_eq2/ℬ^{⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠3𝑥2−𝑥3+𝒜/𝐸}_{eq1ℎ2eq1+ℬ/𝐸eq2ℎ2eq2𝜇𝑃0𝜀−𝑥3/48+𝑥4/33−𝑥5/90+𝐶𝑎𝑉𝑎−𝑥2/32+𝑥3/96}⎫⎪⎪⎪⎬⎪⎪⎪⎭𝑠𝑠+𝜅₁𝑠+𝜅₂𝜔₀₋₂(28.)By employing the Laplace inversion we obtain the dynamic response as𝑊=⎛⎜⎜⎜⎜⎜⎜⎝𝑥2,𝜏𝐹₂+𝜇(𝜏)−𝜇𝑃₀𝒞𝐹₁(𝜏)3𝑥²−𝑥³𝑃₀−32𝒞𝐹¹(𝜏)𝑥²+𝒞2𝐹¹(𝜏)−𝒜+ℬ𝛾^−1eqΨ⁻²_eq48𝐹³(𝜏)𝑥³+𝐹₃𝜀(𝜏)𝒜+ℬ𝛾⁻¹_eqΨ⁻²_eq𝑥⁴33−9𝑥0⁵+𝐶_𝑎𝐹₁(𝜏)−𝑥²32+𝑥³96⎞⎟⎟⎟⎟⎟⎟⎠,(29)where 𝑡=2𝜋𝜏𝜔₀,𝜂=𝜔𝜔₀,(30) while 𝐹₁(𝜏)=𝒜ℬ+ℬ𝑒+𝛾_{eq−2𝜋𝜅1𝜏}Ψ³_eq𝒜𝜅₁𝜅₂𝜅₁₂−𝑒^{−2𝜋𝜅1𝜏}𝜅₄𝜅₅𝜅₁₂−𝜔⁻¹𝑒^{−2𝜋𝜂𝜅1𝜏}𝐹₂𝜅₁₃,(𝜏)=𝑒^{−2𝜋𝜅1𝜏}𝜅₂𝜅₃+𝑒^{−2𝜋𝜅4𝜏}𝜅₃𝜅₅+𝑒^{𝑖2𝜋𝜂𝜏}𝜅₆𝜅₇+𝜔⁻¹𝜅₁₁𝑒^{𝑖2𝜋𝜂𝜏}𝐹₃𝜅₉𝜅₁₀,(𝜏)=1+𝑒^{−2𝜋𝜅1𝜏}𝜅₁𝜅₁₂+𝑒^{−2𝜋𝜅4𝜏}𝜅₄𝜅₁₂.(31) Here,𝜅₁=𝜇₁+𝜇₂8𝜂−𝑖1−𝜇₁+𝜇₂8𝜂²,𝜅₂=ℬ+𝛾^eqΨ³_eq𝒜+𝜅₁𝜇₁/𝜂ℬ+𝜇₂/𝜂𝒜𝛾_eqΨ³_eq(𝒜ℬ),𝜅₃=⎡⎢⎢⎣132𝜇₁+𝜇₂𝜂²+2𝜂+𝑖𝜇₁+𝜇₂4𝜂⎷1−𝜇₁+𝜇₂8𝜂²−2⎤⎥⎥⎦,𝜅₄=𝜇₁+𝜇₂8𝜂+𝑖1−𝜇₁+𝜇₂8𝜂²,𝜅₅=ℬ+𝛾^eqΨ³_eq𝒜+𝜅₄𝜇₁/𝜂ℬ+𝜇₂/𝜂𝒜𝛾_eqΨ³_eq(𝒜ℬ),𝜅₆=1−𝜂²−𝑖𝜇₁+𝜇₂4𝜂,𝜅₇=ℬ+𝛾^eqΨ³_eq𝒜+𝑖𝜇₁ℬ+𝜇₂𝒜𝛾_eqΨ³_eq(𝒜ℬ),𝜅₈=ℬ+𝛾^eqΨ³_eq𝒜𝜇₁ℬ+𝜇₂𝒜𝛾_eqΨ³_eq,𝜅₉=𝜅₈₂𝜂²+𝜅₈𝜇₁+𝜇₂4𝜂+1,𝜅₁₀=−ℬ+𝛾_eqΨ³_eq𝒜+𝑖𝜇₁ℬ+𝜇₂𝒜𝛾_eqΨ³_eq,𝜅₁₁=−𝜇₁ℬ+𝜇₂𝒜𝛾_eqΨ³_eq,𝜅₁₂=2𝑖⎷1−𝜇₁+𝜇₂8𝜂²,𝜅₁₃=−𝜅₈+𝜇₁+𝜇₂4𝜅₈+𝜅³⁸,(32)where the following nondimensionalized parameters have also been introduced viz: 𝑊𝑥,𝜏=𝑊𝑥,𝜏𝐸_eq1𝑏ℎ³_eq1𝐿³𝐹₀,𝑃₀=𝑃⁰𝐹₀/𝑏ℎ_eq1.(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 Δ𝑢(𝑥,𝑡)=𝑢₁(𝑥,𝑡,0⁻)−𝑢₂𝑥,𝑡,0⁺,(34) where 𝜉 is a dummy axial spatial variable of integration across the interface, 0⁺ and 0⁻ denote the origin of the transverse spatial variable for each layer, and 𝑡 is the time (state) variable.

Following Goodman and Klumpp [13], this can also be written as Δ𝑢(𝑥,𝑡)=𝐸⁻¹_eq1^𝑥0𝜎_𝑥₁−𝐸_−1eq2(𝜉,0,𝑡,0−)𝑑𝜉^𝑥0𝜎_𝑥₂(𝜉,0,𝑡,0+)𝑑𝜉(35) with the integral kernels as 𝜎_𝑥₁(𝜉,0,𝑡,0⁻)=𝐸^eq1ℎ_eq1𝐶₁(𝜕/𝜕𝑡)+1×𝑑₂21−𝜐²¹𝑊(𝜉,𝑡)𝑑𝜉²+𝜇𝑃^𝑎𝑣(𝑥−𝐿)ℎ_eq1𝜎_𝑥,₂𝜉,0,𝑡,0⁺=−𝐸^eq2ℎ_eq2𝐶₁(𝜕/𝜕𝑡)+1×𝑑₂21−𝜐²¹𝑊(𝜉,𝑡)𝑑𝜉²−𝜇𝑃^𝑎𝑣(𝑥−𝐿)ℎ_eq2,(36) where 𝐸ℎ_eq1eq1=ℎ₁+ℎ_𝑎,ℎ_eq2=ℎ₂+ℎ_𝑠,𝛾_eq=𝐸₁1+𝛾_𝑎,1,𝐸_eq2=𝐸₂1+𝛾_𝑠,2,=𝐸^eq1𝐸_eq2=𝐸¹1+𝛾_𝑎,1𝐸₂1+𝛾_𝑠,2,𝛾^𝑎,1=𝐸^𝑎𝐸₁𝛾_𝑠,2,=𝐸^𝑠𝐸₂,Ψ_𝑎,1=ℎ^𝑎ℎ₁Ψ_𝑠,2,=ℎ^𝑠ℎ₂,Ψ_𝑎,1=ℎ²ℎ₁11++ΨΨ^𝑠,2_𝑎,1.(37) Equation (36) is integrated to give Δ𝑢=12𝒜𝑐¹+𝑐²Ψ_eqℬ𝜕𝜕𝜏+Δ₁𝜕𝑊+𝜇𝜕𝑥𝑃₀Δ₂𝑥²−2𝑥+𝜇𝑃₀𝜀23𝑥³−𝑥²,(38) where the following have been introduced: Δ₁=𝑐¹2𝒜+𝑐²Ψ_eq2ℬ,Δ₂=12𝒜+2Ψℬ^eq.(39) Substitution for 𝑊 from (29) simplifies (38) asΔ𝑢=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎛⎜⎜⎜⎝3𝜇𝑃₀𝑐¹𝒞2𝒜+𝑐²ℰ2ℬ𝐹̇¹(𝜏)−3𝑐¹𝒜+𝑐²ℰℬ𝐹̇²+𝜇(𝜏)+𝜇𝑃₀3𝜆^-₁𝐹₁(𝜏)+𝜆^-₂−3𝜆^-₃𝐹₂(𝜏)⎞⎟⎟⎟⎠𝒵𝑃₀𝜀⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−3𝑐¹𝒞𝒜+𝑐²ℰℬ𝐹̇¹+⎛⎜⎜⎜⎝𝑐₁(𝜏)+𝜆^-₁𝐹₁(𝜏)𝑥𝒞𝒜+𝑐²Ψ⁻¹_eqℬ𝐹̇¹(𝜏)4−𝑐¹𝒞𝒜+𝑐²Ψ⁻¹_eqℬ^-𝜆₄32𝐹̇¹+(𝜏)_-𝜆₂2𝐹¹(𝜏)−^-𝜆₄16𝐹³(𝜏)−1⎞⎟⎟⎟⎠𝑥²2^-𝜆₄33𝒜𝑐¹+𝑐²Ψ⁻¹_eqℬ𝐹̇³(𝜏)+433^-𝜆₄𝐹₃(𝜏)+23𝑥³−5180^-𝜆₄𝒜𝑐¹+𝑐²Ψ⁻¹_eqℬ𝐹̇³(𝜏)+590^-𝜆₄𝐹₃(𝜏)𝑥⁴+𝐶_𝑎2𝑐𝒜¹+𝑐²Ψ⁻¹_eq2ℬ𝐹̇³(𝜏)+𝐹₃(𝜏)−116𝑥+𝑥²32⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(40)where 𝒵 denotes(𝑥²−2𝑥), 𝒞 denotes (1+Ψ_eq), and ℰ denotes (1+Ψ⁻¹_eq)𝜆^-₁=(𝒞)⋅Δ₂,_-𝜆₂=Δ₁⋅Δ₂,𝜆^-₃=2⋅Δ₂,_-𝜆₄=𝒜+ℬ𝛾⁻¹_eqΨ⁻²_eq.(41) On setting 𝜀→0 in (40), the dynamic slip at uniform pressure becomes Δ=𝑢⎧⎪⎪⎪⎨⎪⎪⎪⎩⎛⎜⎜⎜⎝3𝜇𝑃₀𝑐¹𝒞2𝒜+𝑐²ℰ2ℬ−3𝒜𝑐¹+𝑐²ℰℬ𝐹̇²+𝜇(𝜏)+𝑃₀3𝜆^-₁𝐹₁(𝜏)+𝜆^-₂−3𝜆^-₃𝐹₂(𝜏)⎞⎟⎟⎟⎠𝒵𝐶_𝑎2𝑐𝒜¹+𝑐²Ψ⁻¹_eq2ℬ𝐹̇³(𝜏)+𝐹₃(𝜏)−116𝑥+𝑥²32⎫⎪⎪⎪⎬⎪⎪⎪⎭.(42)

4. Energy Dissipation

Following Goodman and Klumpp [13], the energy dissipated can be computed from the relation 𝐷=4𝜇𝑏^0𝜋/2𝜔^𝐿0𝑃(𝑥)Δ𝑢(𝑥,𝑡)𝑑𝑥𝑑𝑡(43) and nondimensionalised as 𝐷=4𝜇^01/4¹⁰𝑃_𝑎𝑣Δ𝑢𝑑𝑥𝑑𝜏,(44) where 𝐷=𝐷(𝑥,𝜏)𝐸𝑏ℎ³/𝐿³𝐹²⁰; substitution for Δ𝑢 from (40) gives the closed form solution as 𝐷=𝐷₁+𝐷₂+𝐷₃+𝐷₄,(45) where𝐷₁=⎡⎢⎢⎢⎢⎢⎢⎣⎛⎜⎜⎜⎝8Δ⁵(𝜂)𝜆^-₃+Δ₃(𝜂)𝑐¹𝒜+𝑐²ℰ−𝑐₁ℬ𝒜+𝑐²ℰℬΔ⁷(𝜂)+Δ₆(𝜂)𝐶_𝑎24⎞⎟⎟⎟⎠𝜇𝑃₀−𝑐¹𝒜+𝑐²ℰℬ+8^-𝜆₁Δ₄(𝜂)+23^-𝜆₂𝜇²𝑃²⁰⎤⎥⎥⎥⎥⎥⎥⎦,(46a)𝐷₂=𝜀⎡⎢⎢⎢⎢⎢⎢⎣⎛⎜⎜⎜⎝4Δ⁵(𝜂)𝜆^-₃+Δ₅(𝜂)𝑐¹𝒜+𝑐²ℰ−𝑐₁ℬ𝒜+𝑐²ℰℬΔ⁷(𝜂)+Δ₆(𝜂)𝐶_𝑎48⎞⎟⎟⎟⎠𝜇𝑃₀−𝑐¹2𝒜+𝑐²ℰ2ℬ+4^-𝜆₁Δ₄(𝜂)+13^-𝜆₂𝜇²𝑃²⁰⎤⎥⎥⎥⎥⎥⎥⎦(46b)for all negative values of 𝐶_𝑎.Similarly,𝐷₃=⎡⎢⎢⎢⎣⎛⎜⎜⎜⎝𝜀𝑐¹𝒜+𝑐²ℰℬ2𝒴4Δ⁸(𝜂)+41495^-𝜆₄Δ₇(𝜂)𝜇²𝑃²⁰+𝜀2^-𝜆₂+6𝜆^-₁3Δ⁴(𝜂)+1631980^-𝜆₄Δ₆(𝜂)−16𝜇²𝑃²⁰⎞⎟⎟⎟⎠⎤⎥⎥⎥⎦,(46c)𝐷₄=⎡⎢⎢⎢⎣⎛⎜⎜⎜⎝𝜀²𝒜𝑐¹+𝑐²ℰℬ4𝒴8Δ⁸(𝜂)+41990^-𝜆₄Δ₇(𝜂)𝜇²𝑃²⁰+𝜀²^-𝜆₂+3𝜆^-₁3Δ⁴(𝜂)+1633960^-𝜆₄Δ₆(𝜂)−112𝜇²𝑃²⁰⎞⎟⎟⎟⎠⎤⎥⎥⎥⎦,(46d)where Δ₃(𝜂)=𝑒^{−𝜋𝜅1/2}𝜅₂𝜅₃+𝑒^{−𝜋𝜅4/2}𝜅₃𝜅₅+𝑒^{𝑖𝜋𝜂/2}𝜅₆𝜅₇+𝜔⁻¹𝜅₁₁𝑒^{𝑖𝜋𝜂/2}Δ₄𝜅₉𝜅₁₀,(𝜂)=(𝒜ℬ)−4ℬ𝑒+𝛾_{eq−𝜋𝜅1/2}Ψ³_eq𝒜2𝜋𝜅²¹𝜅₂𝜅₁₂+𝑒^{−𝜋𝜅1/2}2𝜋𝜅₁𝜅₄𝜅₅𝜅₁₂+𝜔⁻¹𝑒^{−𝜋𝜂𝜅1/2}Δ₅2𝜋𝜅₁𝜅₁₃,(𝜂)=𝑒^{−𝜋𝜅1/2}2𝜋𝜅₁𝜅₂𝜅₃+𝑒^{−𝜋𝜅4/2}2𝜋𝜅₃𝜅₄𝜅₅−𝑖𝑒^{𝑖𝜋𝜂/2}2𝜋𝜂𝜅₆𝜅₇−𝑖𝜔⁻¹𝜅₁₁𝑒^{𝑖𝜋𝜂/2}Δ₆2𝜋𝜂𝜅₉𝜅₁₀,(𝜂)=14−2𝜋𝑒𝜅₁₂𝜅^{−𝜋𝜅1/2}₁₂−𝑒^{−𝜋𝜅4/2}Δ₇2𝜋𝜅₄₂𝜅₁₂,(𝜂)=1+𝑒^{−𝜋𝜅1/2}𝜅₁𝜅₁₂+𝑒^{−𝜋𝜅4/2}Δ₈𝜅₄𝜅₁₂,(𝜂)=(𝒜ℬ)+ℬ𝑒+𝛾_{eq−𝜋𝜅1/2}Ψ³_eq𝒜𝜅₁𝜅₂𝜅₁₂−𝑒^{−𝜋𝜅1/2}𝜅₄𝜅₅𝜅₁₂−𝜔⁻¹𝑒^{−𝜋𝜂𝜅1/2}𝜅₁₃.(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 𝛿=12ln𝐸𝐸^𝑛_𝑛+1=12ln𝐸^ne+𝐸_loss𝐸_𝑛𝐸_loss,(48) where 𝐸_ne is the strain energy of the laminate material, whereas 𝐸_loss is the energy loss per cycle, and 𝛿 is the Logarithmic damping decrement.

In particular, 𝐸_loss, the slip energy, is computed from (45)-(46a), (46b), (46c), and (46d), that is 𝐷, whereas 𝐸_ne 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, 𝑈₁=^𝐿0𝑀²2𝐸_eq𝐼_eq𝑑𝑥,(49) whilst the energy stored at the free end follows from the theory of strength of materials namely, 𝑈₂=32𝐸𝐿^eq3𝑊^2𝐿.(50) here, 𝑀 is the bending moment, 𝐸_eq the modulus of rigidity of the material, 𝐼_eq=(2/3)𝑏ℎ³_eq the moment of inertia of the clamped sandwich composite smart structure, and 𝑊_𝐿 the deflection at the free end.

For this case, the computed nondimensionalized strain energy components are𝑈₁=27⎧⎪⎪⎨⎪⎪⎩Θ¹𝑒^{𝑖4𝜋𝜂}−Θ₂𝑒^{2𝜋𝜂(𝑖−𝜒)}+𝜔⁻²Θ₃𝑒^{−4𝜋𝜒𝜂}−2𝜇𝑃₀1+𝜀2𝒞Θ⁴+𝜇₂𝑒^{𝑖2𝜋𝜂}−Θ₅𝜔⁻¹𝑒^{2𝜋𝜂(𝑖−𝜒)}−Θ₆𝜔⁻¹𝑒^{−2𝜋𝜒𝜂}+Θ₇𝜔⁻²𝑒^{−4𝜋𝜒𝜂}𝑃²⁰1+𝜀+𝜀4²𝒞²Θ⁸−2Θ₉𝜔⁻¹𝑒^{−2𝜋𝜒𝜂}+Θ₁₀𝜔⁻²𝑒^{−4𝜋𝜒𝜂}⎫⎪⎪⎬⎪⎪⎭,(51)whereΘ₁=2𝒜ℬℬ1+𝜇₁+𝒜1+𝜇₂𝛾_eqΨ_eq²Θ₂,=4𝒜ℬΘℬ+𝒜𝛾_eqΨ³_eq−𝑖𝜇₁ℬ+𝜇₂ℬ𝛾_eqΨ³_eq₃,=𝜇¹ℬ+𝜇₂𝒜𝛾_eqΨ³_eqℬ+𝒜𝛾_eqΨ³_eq−𝑖𝜇₁ℬ+𝜇₂ℬ𝛾_eqΨ³_eq²Θ₄,=(𝒜ℬ)²ℬ1+𝜇₁+𝒜1+𝜇₂𝛾_eqΨ³_eqℬ+𝒜𝛾_eqΨ³_eq²Θ₅,=𝒜ℬΘ₆ℬ+𝒜𝛾_eqΨ³_eq,=𝒜ℬ×ℬ+𝒜𝛾_eqΨ³_eq𝜇₁ℬ+𝜇₂𝒜𝛾_eqΨ³_eqΘℬ+𝒜𝛾_eqΨ³_eq−𝑖𝜇₁ℬ+𝜇₂ℬ𝛾_eqΨ³_eq₇,=⎛⎜⎜⎝𝜇₁ℬ+𝜇₂𝒜𝛾_eqΨ³_eq²Θℬ+𝒜𝛾_eqΨ³_eqℬ+𝒜𝛾_eqΨ³_eq−𝑖𝜇₁ℬ+𝜇₂ℬ𝛾_eqΨ³_eq₈⎞⎟⎟⎠,=𝒜ℬℬ+𝒜𝛾_eqΨ³_eq²Θ₉,=⎛⎜⎜⎝1−𝜐¹1−𝜐₂𝜇₁ℬ+𝜇₂𝒜𝛾_eqΨ³_eqℬ+𝒜𝛾_eqΨ³_eq²Θ₁₀⎞⎟⎟⎠,=𝜇₁ℬ+𝜇₂𝒜𝛾_eqΨ³_eqℬ+𝒜𝛾_eqΨ³_eq²𝜒=,𝜇₁ℬ+𝜇₂𝒜𝛾_eqΨ³_eqℬ+𝒜𝛾_eqΨ³_eq,𝑈₂=⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩Θ¹𝑒^{𝑖4𝜋𝜂}−Θ₂𝑒^{2𝜋𝜂(𝑖−𝜒)}+𝜔⁻²Θ₃𝑒^{−4𝜋𝜒𝜂}−2𝜇𝑃₀1+𝜀2𝒞Θ⁴+𝜇₂𝑒^{𝑖2𝜋𝜂}−Θ₅𝜔⁻¹𝑒^{2𝜋𝜂(𝑖−𝜒)}−Θ₆𝜔⁻¹𝑒^{−2𝜋𝜒𝜂}+Θ₇𝜔⁻²𝑒^{−4𝜋𝜒𝜂}𝑃²⁰1+𝜀+𝜀4²𝒞²Θ⁸−⎡⎢⎢⎢⎢⎢⎣Θ₁₁−2Θ₉𝜔⁻¹𝑒^{−2𝜋𝜒𝜂}+Θ₁₀𝜔⁻²𝑒^{−4𝜋𝜒𝜂}480𝜇𝑃₀𝜀+Θ¹⁵48𝐶_𝑎𝑒^{𝑖2𝜋𝜂}−𝜔⁻¹4Θ80¹²𝜇𝑃₀𝜀+Θ¹⁶48𝐶_𝑎𝑒^{−2𝜋𝜒𝜂}−Θ¹³480𝜇²𝑃²⁰𝜀+𝜀2²𝒞+𝜔⁻¹Θ₁₄480𝜇²𝑃²⁰𝜀+𝜀2²𝒞𝑒^{−2𝜋𝜒𝜂}−196𝜇𝑃₀1+𝜀2𝒞+Θ₁₉𝐶_𝑎Θ₁₇−𝜔⁻¹Θ₁₈𝑒^{−2𝜋𝜒𝜂}⎤⎥⎥⎥⎥⎥⎦960²𝜇²𝑃²⁰𝜀²+Θ²⁰96²𝜇𝑃₀𝜀𝐶_𝑎+196²𝐶^2𝑎⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭,(52)whileΘ₁₁=2𝒜ℬ𝒜+ℬ𝛾^−1eqΨ⁻²_eqΘ₁₂ℬ1+𝜇₁+𝒜1+𝜇₂𝛾_eqΨ³_eq,=2𝒜+ℬ𝛾^−1eqΨ⁻²_eq𝜇₁ℬ+𝜇₂𝒜𝛾_eqΨ³_eqΘℬ+𝒜𝛾_eqΨ³_eqℬ+𝒜𝛾_eqΨ³_eq−𝑖𝜇₁ℬ+𝜇₂ℬ𝛾_eqΨ³_eq₁₃,=𝒜ℬ𝒜+ℬ𝛾^−1eqΨ⁻²_eqΘ₁₄ℬ+𝒜𝛾_eqΨ³_eq,=ℬ𝜇₁+𝒜𝜇₂𝛾_eqΨ³_eq𝒜ℬ𝛾⁻¹_eqΨ⁻²_eqΘ₁₅ℬ+𝒜𝛾_eqΨ³_eq,=𝒜ℬΘ₁₆ℬ1+𝜇₁+𝒜1+𝜇₂𝛾_eqΨ³_eq,=𝜇₁ℬ+𝜇₂𝒜𝛾_eqΨ³_eqΘℬ+𝒜𝛾_eqΨ³_eq−𝑖𝜇₁ℬ+𝜇₂ℬ𝛾_eqΨ³_eq₁₇,=2𝒜ℬΘ₁₈ℬ+𝒜𝛾_eqΨ³_eq,=2ℬ𝜇₁+𝒜𝜇₂𝛾_eqΨ³_eqΘ₁₉ℬ+𝒜𝛾_eqΨ³_eq,Θ₂₀=𝒜+ℬ𝛾⁻¹_eqΨ⁻²_eq²,=𝒜+ℬ𝛾^−1eqΨ⁻²_eq²5.(53)Here, we have introduced the nondimensionalization 𝑈=𝑈^𝑛𝐸𝑏ℎ³𝐹²⁰𝐿³,𝑛=1,2,(54) to write the logarithmic damping decrement as 𝛿=12ln1+D𝑈₁+𝑈₂.(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) as𝑊𝑥,𝑠𝐹(𝑠)=𝑊𝑥,𝑠𝐿³𝐸_eq1𝑏ℎ³_eq1=𝑇𝐹₁=𝐿³⎛⎜⎜⎜⎜⎜⎜⎝_2𝐹_{2(𝑠)−𝜇𝑃0𝒞}𝐹_{13(𝑠)𝑥}²_−𝑥³_{+𝜇𝑃0⎡⎢⎢⎢⎣−}3𝒞2𝐹^{1(𝑠)𝑥2+⎛⎜⎜⎜⎝}𝒞2^𝐹_{1⎛⎜⎜⎜⎝(𝑠)−}𝒜+ℬ𝛾⁻¹_eqΨ⁻²_eq48^{⎞⎟⎟⎟⎠}₃𝐹_{⎞⎟⎟⎟⎠𝑥(𝑠)}^{3⎤⎥⎥⎥⎦𝜀+}𝐹_{3⎡⎢⎢⎣(𝑠)𝒜+ℬ𝛾}⁻¹_eqΨ⁻²_{eq⎛⎜⎜⎝}𝑥⁴33⁻𝑥⁵90^{⎞⎟⎟⎠⎤⎥⎥⎦+𝐶𝑎}𝐹_{1⎛⎜⎜⎝(𝑠)}−𝑥²32⁺𝑥³96^⎞⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎠𝐸_eq1𝑏ℎ³_eq1,(56)while the corresponding transfer function for the active control of dynamic interfacial slip can be derived or computed from (40) asΔ𝑢̃𝑥,𝑠𝐹(𝑠)=Δ𝑢𝑥,𝑠𝐿²𝐸_eq1𝑏ℎ²_eq1=𝑇𝐹₂=𝐿²⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩^{⎛⎜⎜⎜⎜⎝⎛⎜⎜⎝3𝜇}_𝑃0^⎛⎜⎜⎝𝑐₁𝒞2𝒜⁺𝑐₂ℰ2ℬ^{⎞⎟⎟⎠𝑠−1⎛⎜⎜⎝−3}𝑐₁𝒜⁺𝑐₂ℰℬ^{⎞⎟⎟⎠𝑠}_{2⎞⎟⎟⎠+𝜇(𝑠}𝐹_{)𝑃03-𝜆1}𝐹_1(𝑠)+^-_𝜆2𝑠⁻¹_−3^-_𝜆3𝐹_{2⎞⎟⎟⎟⎟⎠𝒵(𝑠)+𝜇𝑃0𝜀⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎛⎜⎜⎝⎛⎜⎜⎝−3}𝑐₁𝒞𝒜⁺𝑐₂ℰℬ^{⎞⎟⎟⎠𝑠}𝐹_1(𝑠)+^-_𝜆1𝐹_{1⎞⎟⎟⎠(𝑠)𝑥+⎛⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎝}𝑐₁𝒞𝒜⁺𝑐₂Ψ⁻¹_eqℬ^{⎞⎟⎟⎟⎠}𝑠𝐹₁(𝑠)4^{−⎛⎜⎜⎜⎝}𝑐₁𝒞𝒜⁺𝑐₂Ψ⁻¹_eqℬ^{⎞⎟⎟⎟⎠-}𝜆₄32^𝑠₁₊𝐹_(𝑠)-𝜆₂2𝐹^1(𝑠)−-𝜆₄16^{3(𝑠)−𝑠−1⎞⎟⎟⎟⎟⎟⎟⎠𝑥2}_{⎛⎜⎜⎜⎝}𝐹2𝜆^-₄33^{⎛⎜⎜⎜⎝}𝑐₁𝒜⁺𝑐₂Ψ⁻¹_eqℬ^{⎞⎟⎟⎟⎠𝑠}𝐹_3(𝑠)+433^-𝜆4𝐹_3(𝑠)+2^{𝑠−1⎞⎟}3^{⎟⎟⎠𝑥3}_{−⎛⎜⎜⎜⎝}5180^{-𝜆4⎛⎜⎜⎜⎝}𝑐₁𝒜⁺𝑐₂Ψ⁻¹_eqℬ^{⎞⎟⎟⎟⎠𝑠}𝐹_3(𝑠)+590^-𝜆4_{3⎞⎟⎟⎟⎠(𝑠}𝐹_)𝑥^{4⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+}_{𝐶𝑎⎛⎜⎜⎜⎝}c₁2𝒜⁺c₂Ψ⁻¹_eq2ℬ^{⎞⎟⎟⎟⎠𝑠}𝐹_3(𝑠)+𝐹_{3⎛⎜⎜⎝(𝑠)}−116^𝑥+𝑥²32^⎞⎟⎟⎠⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭𝐸_eq1𝑏ℎ²_eq1.(57)Within the context of robust active vibration and noise control designs, the common denominator admits the form𝑊=⎛⎜⎜⎜⎜⎜⎜⎝𝑥,𝑠2𝐹₂+𝜇(𝑠)−𝜇𝑃₀𝒞𝐹₁(𝑠)3𝑥²−𝑥³𝑃₀−32𝒞𝐹¹(𝑠)𝑥²+𝒞2𝐹¹(𝑠)−𝒜+ℬ𝛾^−1eqΨ⁻²_eq48𝐹³(𝑠)𝑥³+𝐹₃𝜀(𝑠)𝒜+ℬ𝛾⁻¹_eqΨ⁻²_eq𝑥⁴33−9𝑥0⁵+𝐶_𝑎𝐹₁(𝑠)−𝑥²32+𝑥³96⎞⎟⎟⎟⎟⎟⎟⎠,(58)where 𝐹₁(𝑠), 𝐹₂(𝑠), and 𝐹₃(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 𝜀=0 and irrespective of the value of the piezoelectric parameter 𝐶_𝑎, 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 as𝜇𝑃_opt=(2+𝜀)Γ⁻¹4Δ⁵(𝜂)𝜆^-₃+Δ₃(𝜂)𝑐¹𝒜+𝑐²ℰ−𝑐₁ℬ𝒜+𝑐²ℰℬΔ⁷(𝜂)+Δ₆(𝜂)𝐶_𝑎48,(59)whereΓ=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑐¹𝒜+𝑐²ℰℬΔ⁷(𝜂)+Δ₆(𝜂)𝐶_𝑎+𝜀⎛⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎝𝑐₁242𝒜+𝑐²ℰ+⎛⎜⎜⎜⎝𝑐₁2ℬ𝒜+𝑐²ℰℬ2𝒴4Δ⁸(𝜂)+41495^-+2_-𝜆₄Δ₇(𝜂)𝜆₂+6𝜆^-₁3Δ⁴(𝜂)+1631980^-𝜆₄Δ₆(𝜂)−1+𝜀₂6⎞⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎠⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭⎛⎜⎜⎜⎝−𝑐¹𝒜+𝑐²ℰℬ4𝒴8Δ⁸(𝜂)+41990^-+_-𝜆₄Δ₇(𝜂)𝜆₂+3𝜆^-₁3Δ⁴(𝜂)+1633960^-𝜆₄Δ₆(𝜂)−112⎞⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=𝐽^𝑑𝜓,(60)where 𝒴 denotes −144−7𝜆^-𝐽_𝑑=(2+𝜀)Γ⁻¹,(61)𝜓=⎛⎜⎜⎜⎝4Δ⁵(𝜂)𝜆^-₃+Δ₃(𝜂)𝑐¹𝒜+𝑐²ℰ−𝑐₁ℬ𝒜+𝑐²ℰℬΔ⁷(𝜂)+Δ₆(𝜂)𝐶_𝑎48⎞⎟⎟⎟⎠.(62) The use of (59) gives the optimal slip energy as 𝐷_opt=⁴_𝑛=1𝐷_𝑛opt,(63) where𝐷_1opt=𝐽_𝑑2𝜓²−𝐽_𝑑𝜓𝑐¹𝒜+𝑐²ℰℬ+8^-𝜆₁Δ₄(𝜂)+23^-𝜆₂𝐷_2opt=𝐽_𝑑𝜀𝜓²−𝐽_𝑑𝜓𝑐¹2𝒜+𝑐²ℰ2ℬ+4^-𝜆₁Δ₄(𝜂)+13^-𝜆₂,𝐷_3opt=𝐽^2𝑑𝜓²⎡⎢⎢⎢⎣⎛⎜⎜⎜⎝𝜀⎛⎜⎜⎜⎝𝑐₁𝒜+𝑐²ℰℬ⎛⎜⎜⎜⎝𝒴24Δ⁸+41(𝜂)495^-𝜆₄Δ₇(𝜂)⎞⎟⎟⎟⎠⎞⎟⎟⎟⎠+𝜀⎛⎜⎜⎜⎝2^-𝜆₂+6𝜆^-₁3Δ⁴+163(𝜂)1980^-𝜆₄Δ₆(𝜂)−16⎞⎟⎟⎟⎠⎞⎟⎟⎟⎠⎤⎥⎥⎥⎦,𝐷_4opt=𝐽_𝑑2𝜓²⎡⎢⎢⎢⎣⎛⎜⎜⎜⎝𝜀₂𝒜𝑐¹+𝑐²ℰℬ4𝒴8Δ⁸(𝜂)+41990^-+𝜀₂𝜆₄Δ₇(𝜂)⎛⎜⎜⎜⎝𝜆^-₂+3𝜆^-₁3Δ⁴+163(𝜂)3960^-𝜆₄Δ₆(𝜂)−112⎞⎟⎟⎟⎠⎞⎟⎟⎟⎠⎤⎥⎥⎥⎦.(64)This result highlights the effects of the piezoelectric parameter 𝐶_𝑎 and the associated frequency ratio 𝜅 as embedded in the parameter Δ(𝜅) with selected values of 𝛾_eq,Ψ_eq, and Poisson’s ratios 𝜐₁,𝜐₂ of the composite elastic beam-plate layers (1 and 2). Relation (59) also shows that in the absence of interfacial pressure gradient, the excitation frequency has a significant role to play in defining the optimal pressure for slip damping. In fact a comparison of Figure 4 with Figure 6 or Figure 5 with Figure 7 illustrates that higher frequency ratios support higher energy dissipation as expected with viscoelastic structures. In all cases, the level of dissipation increases with the value of the piezoelectric parameter 𝐶_𝑎 thereby confirming the efficacy of the introduction of smart structures into the damping mechanism.

Figure 4 

Energy dissipation profile for the case 𝜀=0;𝜐1=0.35,𝜐2=0.25,𝛾eq=0.1,Ψeq=0.001,𝜇1=0.4,𝜇2=0.2, and frequency ratio 𝜂=0.005 and 𝜔=8KHz.

Figure 5 

Energy dissipation profile for the case 𝜀=0;𝜐1=0.35,𝜐2=0.25,𝛾eq=0.1,Ψeq=0.001,𝜇1=0.4,𝜇2=0.2, and frequency ratio 𝜂=0.005 and 𝜔=8KHz.

Figure 6 

Energy dissipation profile for the case 𝜀=0;𝜐1=0.35,𝜐2=0.25,𝛾eq=0.1,Ψeq=0.001,𝜇1=0.4,𝜇1=0.2, and frequency ratio 𝜂=0.005 and 𝜔=10KHz.

Figure 7 

Energy dissipation profile for the case 𝜀=0;𝜐1=0.35,𝜐2=0.25,𝛾eq=0.1,Ψeq=0.001,𝜇1=0.4,𝜇1=0.2, and frequency ratio 𝜂=0.005 and 𝜔=100KHz.

Also of interest here is to ascertain to what extent can we influence the amount of energy dissipated either by our choice of different materials for the upper and lower laminates or by varying the thicknesses of the upper and lower laminates? These effects are best simulated by varying the values of 𝛾_eq and Ψ_eq, these being the relative ratios of Young’s moduli and laminate thicknesses, respectively, 𝛾^eq=𝐸²𝐸₁11++𝛾𝛾^𝑠,2_𝑎,1,𝛾^𝑠,2=𝐸^𝑠𝐸₂,𝛾_𝑎,1=𝐸^𝑎𝐸₁Ψ_eq,=ℎ²ℎ₁11++ΨΨ^𝑠,2_𝑎,1,Ψ^𝑠,2=ℎ^𝑠ℎ₂,Ψ_𝑎,1=ℎ^𝑎ℎ₁.(65) As stated above, we note that the value of 𝜅 increases with decrement in energy dissipation for a prescribed value of the normalised pressure 𝜇𝑃, part of the essence of this exercise is to know which is more efficacious between the ratio of laminate thicknesses Ψ_eq and modulus of rigidity of laminate materials 𝛾_eq for dissipation of vibration energy. The effect of the ratio of modulus of rigidity of laminate materials 𝛾_eq is shown in Figures 6 and 7 for the same values of 𝜅 and Ψ_eq, while the effect of laminate thicknesses Ψ_eq is shown in Figures 9–11. Results indicate that more energy is dissipated as the value of either 𝛾_eq or Ψ_eq increases. In particular, for values higher than 1, dissipation is higher than the results earlier reported in Olunloyo et al. [19].

It is clear in comparison that more energy can be dissipated by varying Ψ_eq as against 𝛾_eq as seen in Figures 8 and 10. By carefully selecting both Ψ_eq and 𝛾_eq higher energy dissipation can be achieved than in any of the cases discussed above as seen from Figure 13, where Ψ_eq=𝛾_eq=4/3.

Figure 8 

Energy dissipation profile for the case 𝜀=0;𝜐1=0.35,𝜐2=0.25,𝛾eq=0.001,Ψeq=0.1,𝜇1=0.4,𝜇2=0.2, and frequency ratio 𝜂=0.5 and 𝜔=8KHz.

Figure 9 

Energy dissipation profile for the case 𝐶𝑎=−1000;𝜐1=0.35,𝜐2=0.25,𝛾eq=1,Ψeq=1,𝜇1=0.4,𝜇2=0.2, and frequency ratio 𝜂=0.5 and 𝜔=8KHz.

Figure 10 

Energy dissipation profile for the case 𝐶𝑎=−1000;𝜐1=0.25,𝜐2=0.5,𝛾eq=4/3,Ψeq=1,𝜇1=0.4,𝜇2=0.2, and frequency ratio 𝜂=0.5 and 𝜔=8KHz.

Figure 11 

Energy dissipation profile for the case 𝐶𝑎=−1000;𝜐1=0.25,𝜐2=0.5,𝛾eq=1,Ψeq=4/3,𝜇1=0.4,𝜇2=0.2, and frequency ratio 𝜂=0.5 and𝜔=8KHz.

From engineering and economic points of view, it is much easier to have a hold on Ψ than 𝛾 since the former requires the preparation of laminates of certain thickness ratios. This is a straightforward matter, whereas it is sometimes a formidable problem to find appropriate laminate materials that will give a prescribed 𝛾_eq ratio knowing that there might be other engineering, economic, or environmental requirements to be met in the use of such materials.

The pattern of results for energy dissipation also suggests that for effective energy dissipation there is a preferred arrangement of the laminates. The computation of Ψ_eq shows that it is a composite laminate, which implies that the thickness of the upper laminate ℎ₁ should always be less than that of the lower laminate ℎ₂. Similarly, if the ratio of Young’s moduli of the laminates 𝛾_eq=𝐸_eq2/𝐸_eq1 has to be greater than 1, the softer laminate is to be placed on top of the harder one. Thus when dealing with composite laminates, for optimal energy dissipation, it is imperative that the softer material (i.e., with Young’s Modulus 𝐸₁) be placed on top of the harder laminate.

The general profile of logarithmic damping decrement curves for low frequency ratios clearly displays (cf. Figures 12 and 13) the pattern earlier reported by Masuko et al. 1973 [24]. This pattern is partly modified by the smart layers as observed from Figures 14 and 15.

Figure 12 

Energy dissipation profile for the case 𝜀=0;𝜐1=0.35,𝜐2=0.25,𝛾eq=4/3,Ψeq=4/3,𝜇1=0.4,𝜇2=0.2, and frequency ratio 𝜂=0.5 and 𝜔=8KHz.

Figure 13 

Logarithmic damping decrement profile for the case 𝐶𝑎=−1000;𝜐1=0.35,𝜐2=0.25,𝛾eq=100,Ψeq=0.1,𝜇1=0.4,𝜇2=0.1, and frequency ratio 𝜂=0.05 and 𝜔=8KHz.

Figure 14 

Logarithmic damping decrement profile for the case 𝐶𝑎=−1000;𝜐1=0.35,𝜐2=0.25,𝛾eq=100,Ψeq=0.1,𝜇1=0.4,𝜇2=0.2, and frequency ratio 𝜂=0.85 and𝜔=8KHz.

Figure 15 

Logarithmic damping decrement profile for the case 𝐶𝑎=−5;𝜐1=0.25,𝜐2=0.5,𝛾eq=500,Ψeq=0.1,and frequency ratio 𝜂=0.85 and 𝜔=8KHz.

The modulating effects of playing with the values of 𝛾_eq and Ψ_eq for moderate values of 𝜂 is also noted in Figures 16 and 17. Furthermore, as the value of the frequency ratio 𝜂 gets large and beyond resonance as in Figures 16 and 17, the nature of the logarithmic damping decrement profile transits from the results of Masuko et al. [24] to those of Nanda and Behera [15]. Irrespective of the value of 𝜀, the logarithmic damping decrement 𝛿 decreases as the value of 𝜇𝑃 increases reflecting a higher normal force at the interface of the laminates. Thus, the effect of the smart layers is to moderate the logarithmic damping profile. The significance of these results depends upon the practical relevance of the regime of frequency ratio of the situation under consideration.

Figure 16 

Logarithmic damping decrement profile for the case 𝐶𝑎=−1000;𝜐1=0.25,𝜐2=0.5,𝛾eq=0.5,Ψeq=1,and frequency ratio 𝜂=2.5 and 𝜔=8KHz.

Figure 17 

Logarithmic damping decrement profile for the case 𝐶𝑎=−1000;𝜐1=0.35,𝜐2=0.25,𝛾eq=0.5,Ψeq=1, and frequency ratio 𝜂=3.5 and 𝜔=8KHz.

8. Summary and Conclusion

In this paper, vibration and noise dissipation in ships FPS and FPSO structures using layered smart systems is discussed. In the formulated boundary value problem, the physics derives from energy dissipation through press fit joints or interface in an environment of linear pressure distribution. By employing the integral transform approach,(a)explicit solutions for the response, slip, optimal pressure, and dissipated energy (i.e., 𝑊,Δ𝑢,𝑃_opt and 𝐷, resp.) under periodic forcing functions were computed,(b)the amount of slip damping as influenced by the linear pressure distribution across the laminate interface, the relative thicknesses of the laminates members, and their elasticity moduli was evaluated,(c)the ordering of the laminates is equally significant, as the pattern of results for energy dissipation indicates that there is a preferred sandwich geometry,(d)in particular, for optimal energy dissipation the harder material (i.e., with Young’s Modulus 𝐸₂) is placed under the softer laminate,(e)similarly, more energy is dissipated when the harder laminate is thicker than the softer one,(f)with respect to the logarithmic damping decrement, results show that for small values of the frequency ratio 𝜂 there is a close agreement between these results and those reported earlier in Masuko et al. [24], while, for high values of 𝜂, the nature of profiles highlighted results in Olunloyo et al. [19], thus, behavioural pattern predicted by earlier workers can be recovered in appropriate limits of the present work,(g)the results also show that by incorporating smart structures, the level of damping can be significantly increased.

Nomenclature

𝑑/𝑑𝑥:	Differential operator
𝑥:	Space coordinate along the beam-plate interface
𝑧:	Space coordinate perpendicular to the beam-plate interface
𝐸:	Modulus of rigidity
𝐼:	Moment of inertia
𝑊:	Transverse dynamic response
𝑏:	Width of laminated beam plate
ℎ:	Depth of laminated beam plate
𝑡:	Time coordinate
𝑃:	Clamping pressure at the interface of the laminates
𝐹(𝑡):	Applied end force
𝐿:	Length of laminated beam plate
𝑢1:	Longitudinal displacement of the upper laminate at interface
𝑢2:	Longitudinal displacement of the lower laminate at interface
𝑊𝐹:	Transverse response in Fourier plane.

Greek Letters

𝜇:	Coefficient of friction at the interface of laminated viscoelastic laminates
𝜇:	Hysteretic damping constant
𝛾:	Structural damping coefficient of viscoelastic laminates
𝜌:	Density of laminate material
𝜏𝑥𝑧:	Shear stress at the interface of the laminates
(𝜎𝑥)1:	Bending stress at the upper half of the laminates
(𝜎𝑥)2:	Bending stress at the lower half of the laminates
𝜉:	Dummy variable
𝜐:	Poisson’s ratio.