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ISRN Mechanical Engineering
Volume 2012 (2012), Article ID 127238, 19 pages
http://dx.doi.org/10.5402/2012/127238
Research Article

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

1Department of Systems Engineering, Faculty of Engineering, University of Lagos, Akoka-Yaba, Lagos 23401, Nigeria
2Centre for Space Transport and Propulsion, National Space Research and Development Agency, Federal Ministry of Science and Technology FCT, Abuja, PMB 437, Nigeria

Received 10 January 2012; Accepted 12 March 2012

Academic Editors: C. F. Gao, R. Ohayon, and G.-J. Wang

Copyright © 2012 Vincent O. S. Olunloyo and Charles A. Osheku. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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].

127238.fig.001
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 equations2𝑃Φ=0,𝜌𝑤+𝜕Φ1𝜕𝑡+𝑔𝑧+2||||Φ2=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 𝜉0,0𝜂0<2𝜋,<𝑧0<.

fig2
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 1𝑐2sinh2𝜉+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, 1𝑐2cosh2𝜉cos2𝜂0𝜕2𝜙𝜉,𝜂0𝜕𝜉2+𝜕2𝜙𝜉,𝜂0𝜕𝜂02+𝑘2𝜙𝜉,𝜂0𝜕=0,(3)2𝑓𝑧0,𝑡𝜕𝑧2𝑘2𝑓𝑧0,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, 𝑑2𝐺𝜂0𝑑𝜂02+𝑎2𝑞cos2𝜂0𝐺𝜂0𝑑=0,(5)2𝐹(𝜉)𝑑𝜉2𝑎2𝑞cosh2𝜉𝐹(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=𝑆𝑃̂𝑒𝑧𝑑𝑆,(7) where 𝑃=𝜌𝑤𝜙𝜉,𝜂0𝑧𝜕𝑓0,𝑡1𝜕𝑡+𝑔𝑧+2𝑈2(8) and ̂𝑒𝑧 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. [2123] and can be written as 𝑓𝑧0𝑊𝑥,𝑠=𝑠0,𝑦0,𝑠𝑘sinh𝑘𝐷cosh𝑘𝑧0,(9) where 𝑊(𝑥0,𝑦0,𝑠) and 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(𝜀𝑠2) yield 𝑃=𝜌𝑤𝜙0𝜉,𝜂0+𝜀𝑠𝜙1𝜉,𝜂0×𝜕𝑓0𝑧0,𝑡+𝜀𝑠𝑓1𝑧0,𝑡1𝜕𝑡+𝑔𝑧+2𝑈20+𝜀𝑠𝑈20(10) while 𝑓0𝑧0𝑊,𝑠=𝑠0𝑥0,𝑦0,𝑠𝑘0𝑘sinh0𝐷𝑘cosh0𝑧0,𝑓1𝑧0𝑊,𝑠=𝑠1𝑥0,𝑦0,𝑠𝑘1𝑘sinh1𝐷𝑘cosh1𝑧0,(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 𝐹0𝐹max.(12)

fig3
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, 𝑎1;𝑠2. 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+𝛽2+𝛽3+𝛽44𝜕2𝑊𝜕𝑡2=𝛼1𝒜+𝛼24𝜕𝑃(𝑥,0)+𝐶𝜕𝑥𝑎4𝜕2𝑉𝑎𝜕𝑥2,(13) where 𝒜 denotes (1𝜐21), denotes (1𝜐22) and the following parameters have been defined viz: 𝛼1=6𝜇𝐸eq12eq1,𝛽1=𝜌1𝑏1𝐸1𝐼1,𝛽3=𝜌𝑎𝑏𝑎𝐸𝑎𝐼𝑎1,eq1=1+𝑎,𝛼2=6𝜇𝐸eq22eq2,𝛽2=𝜌2𝑏2𝐸2𝐼2,𝛽4=𝜌𝑠𝑏𝑠𝐸𝑠𝐼𝑠,eq2=2+𝑠,𝐶𝑎=12𝐸𝑎𝑑31𝑏𝑎+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()𝑒𝑠𝑡1𝑑𝑡,()=2𝜋𝑖𝜂+𝑖𝜂𝑖()𝑒𝑠𝑡𝑑𝑠(15) on (13) to obtain 𝑐1+𝑠1+𝑐24𝑑4𝑊(𝑥,𝑠)𝑑𝑥4+𝛽1𝒜+𝛽2+𝛽3+𝛽44×𝑠2̇=𝛼𝑊(𝑥,𝑠)𝑠𝑊(0)𝑊(0)1𝒜+𝛼24𝑠𝜕𝑃(𝑥,0)+𝐶𝜕𝑥𝑎4𝑑2𝑉(𝑥,𝑠)𝑎𝑑𝑥2.(16) By limiting the analysis to the case of linear pressure variation along the laminate interface, namely, 𝑃(𝑥,0)=𝑃0𝜀1+𝐿𝑥,(17) Equation (16) simplifies to the form 𝑐1+𝑠1+𝑐24𝑑4𝑊(𝑥,𝑠)𝑑𝑥4+𝛽1𝒜+𝛽2+𝛽3+𝛽44×𝑠2̇=𝛼𝑊(𝑥,𝑠)𝑠𝑊(0)𝑊(0)1𝒜+𝛼2𝜀𝑃0+𝐶4𝑠𝐿𝑎4𝑑2𝑉𝑎(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 𝑑2𝑉𝑎(𝑥,𝑠)𝑎/𝑑𝑥2 following Alberts et al. [11] becomes 𝑑2𝑉𝑎(𝑥,𝑠)𝑎𝑑𝑥2=𝛿(𝑥0)𝛿𝑉(𝑥𝐿)𝑎𝑠,(19) where 𝛿() is the first derivative of the Dirac delta function.

Introducing the Fourier finite sine transform []=𝐿0[]sin𝑛𝜋𝑥𝐿[]=2𝑑𝑥,𝐿𝑛=1[]sin𝑛𝜋𝑥𝐿(20) in conjunction with the following boundary conditions in the Laplace transform plane namely, 𝑑𝑊(0,𝑠)=𝑑𝑑𝑥𝑊(0,𝑠)=2𝑑𝑥2𝑊(𝐿,𝑠)=0(21) gives the Fourier sine transform of (18) as 𝑛4𝜋4𝐿4𝑐1+𝑠1+𝑐24𝑊𝐹𝜆𝑛+𝛽,𝑠1𝒜+𝛽2+𝛽3+𝛽44𝑠2𝑊𝐹𝜆𝑛=𝛼,𝑠1𝒜+𝛼2𝑃4𝑠0𝜀𝑛𝜋1+(1)𝑛+1+𝑛3𝜋3𝐿3(1)𝑛+1𝑊(𝐿,𝑠)𝑛𝜋𝐿𝑊𝑥𝑥+𝐶(0,𝑠)𝑎𝑉𝑎4𝑠𝑛𝜋1+(1)𝑛+1.(22) Following the procedure outlined in Olunloyo et al. [21], the bending moment is computed as:𝑊𝑥𝑥𝐹(0,𝑠)=12(𝑠)𝑏𝐸eq13eq1/𝒜+𝐸eq23eq2𝑐/+𝑠1𝐸eq13eq1/𝒜+𝑐2𝐸eq23eq2/6𝜇𝑃0(1+(𝜀/2))eq1+eq2𝑠𝐸eq13eq1/𝒜+𝐸eq23eq2/+𝑠2𝑐1𝐸eq13eq1/𝒜+𝑐2𝐸eq23eq2/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)=𝐹0𝑒𝑖𝜔𝑡 with the driving transform as 𝐹𝐹(𝑠)=0𝑠𝑖𝜔,(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)𝑥2𝐿3𝐹12𝑠(𝑠)𝑏𝐸eq13eq1/𝒜+𝐸eq23eq2𝑐/+𝑠1𝐸eq13eq1/𝒜+𝑐2𝐸eq23eq2/6𝜇𝑃0(1+(𝜀/2))eq1+eq2𝐸eq13eq1/𝒜+𝐸eq23eq2𝑐/+𝑠1𝐸eq13eq1/𝒜+𝑐2𝐸eq23eq2/Λ1+3𝐿3/32𝒜/𝐸eq12eq1+/𝐸eq22eq2𝜇𝑃0𝜀Λ2+(1/16)𝐶𝑎𝑉𝑎𝐿3Λ3𝑠𝑠+𝜅1𝑠+𝜅2𝜔02,(25)where 𝜅1=𝑐1+𝑐2𝜔208+𝑖𝜔20𝑐1+𝑐282𝜔40,𝜅2=𝑐1+𝑐2𝜔208𝑖𝜔20𝑐1+𝑐282𝜔40,𝜔20=𝜔20𝑐11+𝑐22𝜔3220,𝜔20=𝑛4𝜋4𝛽𝐿4.(26) Here, 𝛽𝛽=1𝒜+𝛽2+𝛽3+𝛽44,Λ1=𝑥6𝑥24+𝑥312,Λ2=𝑥245𝑥39+𝑥432𝑥5,Λ153=𝑥3𝑥22+𝑥36.(27) Appropriate substitution for 𝑊(𝐿,𝑠) into (25) and rearrangement gives𝑊=𝑥,𝑠2𝑠𝐹0𝑏𝐸eq13eq1/𝒜+𝐸eq23eq2𝑐/+𝑠1𝐸eq13eq1/𝒜+𝑐2𝐸eq23eq2(/𝑠𝑖𝜔)𝜇𝑃0(1+(𝜀/2))eq1+eq2𝐸eq13eq1/𝒜+𝐸eq23eq2𝑐/+𝑠1𝐸eq13eq1/𝒜+𝑐2𝐸eq23eq2/3𝑥2𝑥3+𝒜/𝐸eq12eq1+/𝐸eq22eq2𝜇𝑃0𝜀𝑥3/48+𝑥4/33𝑥5/90+𝐶𝑎𝑉𝑎𝑥2/32+𝑥3/96𝑠𝑠+𝜅1𝑠+𝜅2𝜔02.(28)By employing the Laplace inversion we obtain the dynamic response as𝑊=𝑥,𝜏2𝐹2(𝜏)𝜇𝑃0𝒞𝐹13(𝜏)𝑥2𝑥3+𝜇𝑃03𝒞2𝐹1(𝜏)𝑥2+𝒞2𝐹1(𝜏)𝒜+𝛾1eqΨ2eq𝐹483𝑥(𝜏)3𝜀+𝐹3(𝜏)𝒜+𝛾1eqΨ2eq𝑥4𝑥335+90𝐶𝑎𝐹1(𝜏)𝑥2+32𝑥3,96(29)where 𝑡=2𝜋𝜏𝜔0𝜔,𝜂=𝜔0,(30) while 𝐹1(𝜏)=𝒜+𝛾eqΨ3eq𝒜+𝑒2𝜋𝜅1𝜏𝜅1𝜅2𝜅12𝑒2𝜋𝜅1𝜏𝜅4𝜅5𝜅12𝜔1𝑒2𝜋𝜂𝜅1𝜏𝜅13,𝐹2=𝑒(𝜏)2𝜋𝜅1𝜏𝜅2𝜅3+𝑒2𝜋𝜅4𝜏𝜅3𝜅5+𝑒𝑖2𝜋𝜂𝜏𝜅6𝜅7+𝜔1𝜅11𝑒𝑖2𝜋𝜂𝜏𝜅9𝜅10,𝐹3𝑒(𝜏)=1+2𝜋𝜅1𝜏𝜅1𝜅12+𝑒2𝜋𝜅4𝜏𝜅4𝜅12.(31) Here,𝜅1=𝜇1+𝜇28𝜂𝑖1𝜇1+𝜇28𝜂2,𝜅2=+𝛾eqΨ3eq𝒜+𝜅1𝜇1/𝜂+𝜇2/𝜂𝒜𝛾eqΨ3eq(,𝒜)𝜅3=132𝜇1+𝜇2𝜂2+2𝜂+𝑖𝜇1+𝜇24𝜂1𝜇1+𝜇28𝜂2,2𝜅4=𝜇1+𝜇28𝜂+𝑖1𝜇1+𝜇28𝜂2,𝜅5=+𝛾eqΨ3eq𝒜+𝜅4𝜇1/𝜂+𝜇2/𝜂𝒜𝛾eqΨ3eq,(𝒜)𝜅6=1𝜂2𝑖𝜇1+𝜇2,4𝜂𝜅7=+𝛾eqΨ3eq𝒜+𝑖𝜇1+𝜇2𝒜𝛾eqΨ3eq,(𝒜)𝜅8=+𝛾eqΨ3eq𝒜𝜇1+𝜇2𝒜𝛾eqΨ3eq,𝜅9=𝜅82𝜂2+𝜅8𝜇1+𝜇2,4𝜂+1𝜅10=+𝛾eqΨ3eq𝒜+𝑖𝜇1+𝜇2𝒜𝛾eqΨ3eq,𝜅11=𝜇1+𝜇2𝒜𝛾eqΨ3eq,𝜅12=2𝑖1𝜇1+𝜇28𝜂2,𝜅13=𝜅8+𝜇1+𝜇24𝜅8+𝜅38,(32)where the following nondimensionalized parameters have also been introduced viz: 𝑊=𝑊𝑥,𝜏𝐸𝑥,𝜏eq1𝑏3eq1𝐿3𝐹0,𝑃0=𝑃0𝐹0/𝑏eq1.(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)𝑢2𝑥,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)=𝐸1eq1𝑥0𝜎𝑥1(𝜉,0,𝑡,0)𝑑𝜉𝐸1eq2𝑥0𝜎𝑥2(𝜉,0,𝑡,0+)𝑑𝜉(35) with the integral kernels as 𝜎𝑥1(𝜉,0,𝑡,0𝐸)=eq1eq1𝐶1(𝜕/𝜕𝑡)+121𝜐21×𝑑2𝑊(𝜉,𝑡)𝑑𝜉2+𝜇𝑃𝑎𝑣(𝑥𝐿)eq1,𝜎𝑥2𝜉,0,𝑡,0+𝐸=eq2eq2𝐶1(𝜕/𝜕𝑡)+121𝜐21×𝑑2𝑊(𝜉,𝑡)𝑑𝜉2𝜇𝑃𝑎𝑣(𝑥𝐿)eq2,(36) where eq1=1+𝑎,eq2=2+𝑠,𝐸eq1=𝐸11+𝛾𝑎,1,𝐸eq2=𝐸21+𝛾𝑠,2,𝛾eq=𝐸eq1𝐸eq2=𝐸11+𝛾𝑎,1𝐸21+𝛾𝑠,2,𝛾𝑎,1=𝐸𝑎𝐸1,𝛾𝑠,2=𝐸𝑠𝐸2,Ψ𝑎,1=𝑎1,Ψ𝑠,2=𝑠2,Ψ𝑎,1=211+Ψ𝑠,21+Ψ𝑎,1.(37) Equation (36) is integrated to give Δ1𝑢=2𝑐1𝒜+𝑐2Ψeq𝜕+𝜕𝜏Δ1𝜕𝑊𝜕𝑥+𝜇𝑃0Δ2𝑥22𝑥+𝜇𝑃0𝜀23𝑥3𝑥2,(38) where the following have been introduced: Δ1=𝑐1+𝑐2𝒜2Ψeq,2Δ2=1+Ψ2𝒜eq.2(39) Substitution for 𝑊 from (29) simplifies (38) asΔ𝑢=3𝜇𝑃0𝑐1𝒞+𝑐2𝒜2̇𝐹21𝑐(𝜏)31𝒜+𝑐2̇𝐹2+𝜇(𝜏)𝑃03-𝜆1𝐹1(𝜏)+-𝜆23-𝜆3𝐹2𝒵(𝜏)+𝜇𝑃0𝜀𝑐31𝒞𝒜+𝑐2̇𝐹1(𝜏)+-𝜆1𝐹1(𝜏)𝑥+𝑐1𝒞𝒜+𝑐2Ψ1eq̇𝐹1(𝜏)4𝑐1𝒞𝒜+𝑐2Ψ1eq-𝜆4̇𝐹321+(𝜏)-𝜆22𝐹1(𝜏)-𝜆4𝐹163(𝜏)1𝑥22-𝜆4𝑐331𝒜+𝑐2Ψ1eq̇𝐹34(𝜏)+33-𝜆4𝐹32(𝜏)+3𝑥35180-𝜆4𝑐1𝒜+𝑐2Ψ1eq̇𝐹35(𝜏)+90-𝜆4𝐹3(𝜏)𝑥4+𝐶𝑎𝑐1+𝑐2𝒜2Ψ1eq̇𝐹23(𝜏)+𝐹3(𝜏)116𝑥+𝑥232,(40)where 𝒵 denotes(𝑥22𝑥), 𝒞 denotes (1+Ψeq), and denotes (1+Ψ1eq)-𝜆1=(𝒞)Δ2,-𝜆2=Δ1Δ2,-𝜆3=2Δ2,-𝜆4=𝒜+𝛾1eqΨ2eq.(41) On setting 𝜀0 in (40), the dynamic slip at uniform pressure becomes Δ𝑢=3𝜇𝑃0𝑐1𝒞+𝑐2𝒜2𝑐231𝒜+𝑐2̇𝐹2+𝜇(𝜏)𝑃03-𝜆1𝐹1(𝜏)+-𝜆23-𝜆3𝐹2𝒵+(𝜏)𝐶𝑎𝑐1+𝑐2𝒜2Ψ1eq̇𝐹23(𝜏)+𝐹3(𝜏)116𝑥+𝑥2.32(42)

4. Energy Dissipation

Following Goodman and Klumpp [13], the energy dissipated can be computed from the relation 𝐷=4𝜇𝑏0𝜋/2𝜔𝐿0𝑃(𝑥)Δ𝑢(𝑥,𝑡)𝑑𝑥𝑑𝑡(43) and nondimensionalised as 𝐷=4𝜇01/410𝑃𝑎𝑣Δ𝑢𝑑𝑥𝑑𝜏,(44) where 𝐷=𝐷(𝑥,𝜏)𝐸𝑏3/𝐿3𝐹20; substitution for Δ𝑢 from (40) gives the closed form solution as 𝐷=𝐷1+𝐷2+𝐷3+𝐷4,(45) where𝐷1=8Δ5(𝜂)-𝜆3+Δ3𝑐(𝜂)1𝒜+𝑐2𝑐1𝒜+𝑐2Δ7(𝜂)+Δ6(𝜂)𝐶𝑎𝜇24𝑃0𝑐1𝒜+𝑐2+8-𝜆1Δ42(𝜂)+3-𝜆2𝜇2𝑃20,(46a)𝐷24Δ=𝜀5(𝜂)-𝜆3+Δ5𝑐(𝜂)1𝒜+𝑐2𝑐1𝒜+𝑐2Δ7(𝜂)+Δ6(𝜂)𝐶𝑎𝜇48𝑃0𝑐1+𝑐2𝒜2+42-𝜆1Δ41(𝜂)+3-𝜆2𝜇2𝑃20(46b)for all negative values of 𝐶𝑎.Similarly,𝐷3=𝜀𝑐1𝒜+𝑐2𝒴Δ248(𝜂)+41495-𝜆4Δ7𝜇(𝜂)2𝑃202+𝜀-𝜆2+6-𝜆13Δ4(𝜂)+1631980-𝜆4Δ61(𝜂)6𝜇2𝑃20,(46c)𝐷4=𝜀2𝑐1𝒜+𝑐2𝒴Δ488(𝜂)+41990-𝜆4Δ7𝜇(𝜂)2𝑃20+𝜀2-𝜆2+3-𝜆13Δ4(𝜂)+1633960-𝜆4Δ61(𝜂)𝜇122𝑃20,(46d)where Δ3𝑒(𝜂)=𝜋𝜅1/2𝜅2𝜅3+𝑒𝜋𝜅4/2𝜅3𝜅5+𝑒𝑖𝜋𝜂/2𝜅6𝜅7+𝜔1𝜅11𝑒𝑖𝜋𝜂/2𝜅9𝜅10,Δ4(𝜂)=(𝒜)4+𝛾eqΨ3eq𝒜𝑒𝜋𝜅1/22𝜋𝜅21𝜅2𝜅12+𝑒𝜋𝜅1/22𝜋𝜅1𝜅4𝜅5𝜅12+𝜔1𝑒𝜋𝜂𝜅1/22𝜋𝜅1𝜅13,Δ5𝑒(𝜂)=𝜋𝜅1/22𝜋𝜅1𝜅2𝜅3+𝑒𝜋𝜅4/22𝜋𝜅3𝜅4𝜅5𝑒𝑖𝑖𝜋𝜂/22𝜋𝜂𝜅6𝜅7𝜔𝑖1𝜅11𝑒𝑖𝜋𝜂/22𝜋𝜂𝜅9𝜅10,Δ61(𝜂)=4𝑒𝜋𝜅1/22𝜋𝜅12𝜅12𝑒𝜋𝜅4/22𝜋𝜅42𝜅12,Δ7(𝑒𝜂)=1+𝜋𝜅1/2𝜅1𝜅12+𝑒𝜋𝜅4/2𝜅4𝜅12,Δ8(𝜂)=(𝒜)+𝛾eqΨ3eq𝒜+𝑒𝜋𝜅1/2𝜅1𝜅2𝜅12𝑒𝜋𝜅1/2𝜅4𝜅5𝜅12𝜔1𝑒𝜋𝜂𝜅1/2𝜅13.(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=2𝐸ln𝑛𝐸𝑛+1=12𝐸lnne+𝐸loss𝐸𝑛𝐸loss,(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, 𝑈1=𝐿0𝑀22𝐸eq𝐼eq𝑑𝑥,(49) whilst the energy stored at the free end follows from the theory of strength of materials namely, 𝑈2=32𝐸eq𝐿3𝑊2𝐿.(50) here, 𝑀 is the bending moment, 𝐸eq the modulus of rigidity of the material, 𝐼eq=(2/3)𝑏3eq the moment of inertia of the clamped sandwich composite smart structure, and 𝑊𝐿 the deflection at the free end.

For this case, the computed nondimensionalized strain energy components are𝑈1Θ=271𝑒𝑖4𝜋𝜂Θ2𝑒2𝜋𝜂(𝑖𝜒)+𝜔2Θ3𝑒4𝜋𝜒𝜂2𝜇𝑃0𝜀1+2𝒞Θ4𝑒𝑖2𝜋𝜂Θ5𝜔1𝑒2𝜋𝜂(𝑖𝜒)Θ6𝜔1𝑒2𝜋𝜒𝜂+Θ7𝜔2𝑒4𝜋𝜒𝜂+𝜇2𝑃20𝜀1+𝜀+24𝒞2Θ82Θ9𝜔1𝑒2𝜋𝜒𝜂+Θ10𝜔2𝑒4𝜋𝜒𝜂,(51)whereΘ1=2𝒜1+𝜇1+𝒜1+𝜇2𝛾eqΨeq2,Θ2=4𝒜+𝒜𝛾eqΨ3eq𝜇𝑖1+𝜇2𝛾eqΨ3eq,Θ3=𝜇1+𝜇2𝒜𝛾eqΨ3eq+𝒜𝛾eqΨ3eq𝜇𝑖1+𝜇2𝛾eqΨ3eq2,Θ4=(𝒜)21+𝜇1+𝒜1+𝜇2𝛾eqΨ3eq+𝒜𝛾eqΨ3eq2,Θ5=𝒜+𝒜𝛾eqΨ3eq,Θ6=𝒜+𝒜𝛾eqΨ3eq×𝜇1+𝜇2𝒜𝛾eqΨ3eq+𝒜𝛾eqΨ3eq𝑖𝜇1+𝜇2𝛾eqΨ3eq,Θ7=𝜇1+𝜇2𝒜𝛾eqΨ3eq2+𝒜𝛾eqΨ3eq+𝒜𝛾eqΨ3eq𝑖𝜇1+𝜇2𝛾eqΨ3eq,Θ8=𝒜+𝒜𝛾eqΨ3eq2,Θ9=1𝜐11𝜐2𝜇1+𝜇2𝒜𝛾eqΨ3eq+𝒜𝛾eqΨ3eq2,Θ10=𝜇1+𝜇2𝒜𝛾eqΨ3eq+𝒜𝛾eqΨ3eq2,𝜒=𝜇1+𝜇2𝒜𝛾eqΨ3eq+𝒜𝛾eqΨ3eq,𝑈2=Θ1𝑒𝑖4𝜋𝜂Θ2𝑒2𝜋𝜂(𝑖𝜒)+𝜔2Θ3𝑒4𝜋𝜒𝜂2𝜇𝑃0𝜀1+2𝒞Θ4𝑒𝑖2𝜋𝜂Θ5𝜔1𝑒2𝜋𝜂(𝑖𝜒)Θ6𝜔1𝑒2𝜋𝜒𝜂+Θ7𝜔2𝑒4𝜋𝜒𝜂+𝜇2𝑃20𝜀1+𝜀+24𝒞2Θ82Θ9𝜔1𝑒2𝜋𝜒𝜂+Θ10𝜔2𝑒4𝜋𝜒𝜂Θ11𝜇480𝑃0Θ𝜀+1548𝐶𝑎𝑒𝑖2𝜋𝜂𝜔1Θ12𝜇480𝑃0Θ𝜀+1648𝐶𝑎𝑒2𝜋𝜒𝜂Θ13𝜇4802𝑃20𝜀𝜀+22𝒞+𝜔1Θ14𝜇4802𝑃20𝜀𝜀+22𝒞𝑒2𝜋𝜒𝜂1𝜇96𝑃0𝜀1+2𝒞𝐶𝑎Θ17𝜔1Θ18𝑒2𝜋𝜒𝜂+Θ199602𝜇2𝑃20𝜀2+Θ20962𝜇𝑃0𝜀𝐶𝑎+1962𝐶2𝑎,(52)whileΘ11=2𝒜𝒜+𝛾1eqΨ2eq1+𝜇1+𝒜1+𝜇2𝛾eqΨ3eq,Θ12=2𝒜+𝛾1eqΨ2eq𝜇1+𝜇2𝒜𝛾eqΨ3eq+𝒜𝛾eqΨ3eq+𝒜𝛾eqΨ3eq𝑖𝜇1+𝜇2𝛾eqΨ3eq,Θ13=𝒜𝒜+𝛾1eqΨ2eq+𝒜𝛾eqΨ3eq,Θ14=𝜇1+𝒜𝜇2𝛾eqΨ3eq𝒜𝛾1eqΨ2eq+𝒜𝛾eqΨ3eq,Θ15=𝒜1+𝜇1+𝒜1+𝜇2𝛾eqΨ3eq,Θ16=𝜇1+𝜇2𝒜𝛾eqΨ3eq+𝒜𝛾eqΨ3eq𝑖𝜇1+𝜇2𝛾eqΨ3eq,Θ17=2𝒜+𝒜𝛾eqΨ3eq,Θ18=2𝜇1+𝒜𝜇2𝛾eqΨ3eq+𝒜𝛾eqΨ3eq,Θ19=𝒜+𝛾1eqΨ2eq2,Θ20=𝒜+𝛾1eqΨ2eq25.(53)Here, we have introduced the nondimensionalization 𝑈𝑈=𝑛𝐸𝑏3𝐹20𝐿3,𝑛=1,2,(54) to write the logarithmic damping decrement as 1𝛿=2ln1+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, 2530]. 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) as𝑊𝑥,𝑠=𝐹(𝑠)𝑊𝐿𝑥,𝑠3𝐸eq1𝑏3eq1=𝑇𝐹1=𝐿32𝐹2(𝑠)𝜇𝑃0𝒞𝐹13(𝑠)𝑥2𝑥3+𝜇𝑃03𝒞2𝐹1(𝑠)𝑥2+𝒞2𝐹1(𝑠)𝒜+𝛾1eqΨ2eq48𝐹3𝑥(𝑠)3𝜀+𝐹3(𝑠)𝒜+𝛾1eqΨ2eq𝑥433𝑥590+𝐶𝑎𝐹1(𝑠)𝑥232+𝑥396𝐸eq1𝑏3eq1,(56)while the corresponding transfer function for the active control of dynamic interfacial slip can be derived or computed from (40) asΔ̃𝑢𝑥,𝑠=Δ𝐹(𝑠)𝑢𝐿𝑥,𝑠2𝐸eq1𝑏2eq1=𝑇𝐹2=𝐿23𝜇𝑃0𝑐1𝒞2𝒜+𝑐22𝑠13𝑐1𝒜+𝑐2𝑠𝐹2+𝜇(𝑠)𝑃03-𝜆1𝐹1(𝑠)+-𝜆2𝑠13-𝜆3𝐹2𝒵(𝑠)+𝜇𝑃0𝜀3𝑐1𝒞𝒜+𝑐2𝑠𝐹1(𝑠)+-𝜆1𝐹1(𝑠)𝑥+𝑐1𝒞𝒜+𝑐2Ψ1eq𝑠𝐹1(𝑠)4𝑐1𝒞𝒜+𝑐2Ψ1eq-𝜆432𝑠𝐹1+(𝑠)-𝜆22𝐹1(𝑠)-𝜆4𝐹163(𝑠)𝑠1𝑥22-𝜆433𝑐1𝒜+𝑐2Ψ1eq𝑠𝐹3(𝑠)+433-𝜆4𝐹3(𝑠)+23𝑠1𝑥35180-𝜆4𝑐1𝒜+𝑐2Ψ1eq𝑠𝐹3(𝑠)+590-𝜆4𝐹3(𝑠)𝑥4+𝐶𝑎c12𝒜+c2Ψ1eq2𝑠𝐹3(𝑠)+𝐹3(𝑠)116𝑥+𝑥232𝐸eq1𝑏2eq1.(57)Within the context of robust active vibration and noise control designs, the common denominator admits the form𝑊=2𝐹𝑥,𝑠2(𝑠)𝜇𝑃0𝒞𝐹1(3𝑠)𝑥2𝑥3+𝜇𝑃03𝒞2𝐹1(𝑠)𝑥2+𝒞2𝐹1(𝑠)𝒜+𝛾1eqΨ2eq𝐹483𝑥(𝑠)3𝜀+𝐹3(𝑠)𝒜+𝛾1eqΨ2eq𝑥4𝑥335+90𝐶𝑎𝐹1(𝑠)𝑥2+32𝑥3,96(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 𝜀=0 and irrespective of the value of the piezoelectric parameter 𝐶𝑎, 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 as𝜇𝑃opt=(2+𝜀)Γ14Δ5(𝜂)-𝜆3+Δ3𝑐(𝜂)1𝒜+𝑐2𝑐1𝒜+𝑐2Δ7(𝜂)+Δ6(𝜂)𝐶𝑎,48(59)where𝑐Γ=1𝒜+𝑐2Δ7(𝜂)+Δ6(𝜂)𝐶𝑎𝑐24+𝜀1+𝑐2𝒜2+𝑐21𝒜+𝑐2𝒴Δ248(𝜂)+41495-𝜆4Δ7+2(𝜂)-𝜆2+6-𝜆13Δ4(𝜂)+1631980-𝜆4Δ61(𝜂)6+𝜀2𝑐1𝒜+𝑐2𝒴Δ488(𝜂)+41990-𝜆4Δ7+(𝜂)-𝜆2+3-𝜆13Δ4(𝜂)+1633960-𝜆4Δ6(1𝜂)12=𝐽𝑑𝜓,(60)where 𝒴 denotes 1447-𝜆𝐽𝑑=(2+𝜀)Γ1,(61)4Δ𝜓=5(𝜂)-𝜆3+Δ3𝑐(𝜂)1𝒜+𝑐2𝑐1𝒜+𝑐2Δ7(𝜂)+Δ6(𝜂)𝐶𝑎48.(62) The use of (59) gives the optimal slip energy as 𝐷opt=4𝑛=1𝐷𝑛opt,(63) where𝐷1opt=𝐽𝑑2𝜓2𝐽𝑑𝜓𝑐1𝒜+𝑐2+8-𝜆1Δ42(𝜂)+3-𝜆2𝐷2opt=𝐽𝑑𝜀𝜓2𝐽𝑑𝜓𝑐1+𝑐2𝒜2+42-𝜆1Δ41(𝜂)+3-𝜆2,𝐷3opt=𝐽2𝑑𝜓2𝜀𝑐1𝒜+𝑐2𝒴Δ248+(𝜂)41495-𝜆4Δ72(𝜂)+𝜀-𝜆2+6-𝜆13Δ4+(𝜂)1631980-𝜆4Δ61(𝜂)6,𝐷4opt=𝐽𝑑2𝜓2𝜀2𝑐1𝒜+𝑐2𝒴Δ488(𝜂)+41990-𝜆4Δ7(𝜂)+𝜀2-𝜆2+3-𝜆13Δ4+(𝜂)1633960-𝜆4Δ61(𝜂).12(64)This result highlights the effects of the piezoelectric parameter 𝐶𝑎 and the associated frequency ratio 𝜅 as embedded in the parameter Δ(𝜅) with selected values of 𝛾eq,Ψeq, 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 𝐶𝑎 thereby confirming the efficacy of the introduction of smart structures into the damping mechanism.

127238.fig.004
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.
127238.fig.005
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.
127238.fig.006
Figure 6: Energy dissipation profile for the case 𝜀=0;𝜐1=0.35