Meshfree finite volume method for active vibration control of temperature-dependent piezoelectric laminated composite plates
Introduction
The stringent demands of dynamic systems in aerospace, mechanical, and civil engineering sectors have called for the development of smart structures containing bonded or embedded sensors/actuators, capable of actively altering the geometrical and physical properties of the system. These requirements involve system vibration control, shape restraint, and acoustic noise suppression, among others [1]. The dynamic nature of a smart structure in response to external forces makes it an excellent alternative in the vibration control problems. In this regard, there has been a considerable momentum on the advancement of piezoelectric materials in smart structures in terms of application and research interest, ever since Mason [2] published his article on the history and applications of piezoelectricity. The implementation of control systems in smart structures is considered as one of the effective notions in reducing their vibrations and maintaining structural and non-structural components [3]. Based on their use of energy resources, the control systems are categorized into four major groups of passive, semi-active, active, and hybrid control systems [4]. Passive control involves systems that use the structural response to develop the control forces without the need for an energy source to operate them. Active control, however, refers to systems that require a large source of energy to use stimuli to exert the control forces on the structure. In addition, semi-active control possesses a combination of active and inactive system features, while hybrid systems merge the capacities of active and inactive or inactive and semi-active control systems. Due to the efficiency of active systems in controlling the seismic response of structures, several studies have been carried out in this area [5], [6], [7], [8]. A key element in an active control system is, however, the controlling algorithm with which it operates. In other words, the design of an algorithm capable of generating the required control force at any given time based on the received feedback is of great significance [9]. The main difference between the active and passive controllers lies within their algorithms. In this regard, the most popular algorithms available in the field of active control include Linear Quadratic Regulator (LQR), Linear Quadratic Gaussian (LQG), Robust, Sliding Mode, Pole Assignment, and Probabilistic controllers, to name a few.
Piezoelectricity is, in particular, characterized by the linear electromechanical relation between the two electrical and mechanical states in dielectrics and crystals lacking inversion symmetry [10]. Indeed, piezoelectric materials generate electric charges when subjected to pressure or stress, termed as the direct piezoelectric effect. In reverse, these materials sustain mechanical strains when exposed to electric fields. The direct piezoelectric impact is employed in the production of sensors, and the reverse is utilized in the manufacturing of actuators. In recent years, piezoelectrics have been increasingly implemented as sensors, converters, and actuators [11]. Thus, a sensor converts information from a mechanical range to an electrical signal to be processed through the direct piezoelectric effect. However, the generated electrical signal decreases rapidly in power once the load is applied, nominating these sensors as suitable options for measuring loads exerted within short intervals of time. To date, three types of piezoelectric materials have known to be widely used in practice, including piezoelectric ceramics, composites, and polymers. Among the polymeric materials, polyvinylidene fluoride (PVDF) demonstrates considerably stronger piezoelectricity, in addition to relatively high tensile strength and stability [12]. The piezoelectric effect of this particular polymer is several times greater than that of quartz crystals, attributed to its high dielectric constant [13]. Besides, composite piezoelectrics have garnered much attention owing to their adaptive characteristics. On account of the promising benefits of composite materials in various engineering structures, a concerted effort has been made by several researchers in better characterizing the performance of these advanced materials respecting vibration, bending, and buckling [14]. The unique features of composites including low weight, high stiffness- and strength-to-weight ratios, electrical conductivity/nonconductivity, low thermal expansion, longer fatigue life, optimal design, and retaining properties at elevated temperatures [15, 16] has urged the researchers to develop various techniques to incorporate piezoelectric materials in composite laminates. As a class of composite materials, laminates are assembled by stacking individual orthotropic or transversely isotropic layers at different orientations, so they meet the required engineering properties such as strength and stiffness. Accordingly, different assumptions have been made to predict the response of laminated composites, the most straightforward of which is the classical laminated theory (CLT) that rests on the notion of displacement fields. The methodology is comparable to the isotropic plate theory, with the main difference that lies in the constitutive stress-strain relations of lamina [17]. The CLT is founded on the Kirchhoff postulations that offer a simplistic evaluation of the particularly thin plate structures. It, however, disregards the transverse shear strains, giving rise to the underestimation of deflections, and overestimation of natural frequencies [18]. Meanwhile, the first-order shear deformation theory (FSDT), proposed by Reissner [19, 20], is based on kinematics analysis to take into account the transverse shear deformation of plates and shells, which is adopted to assess the relatively thick plates by regarding the transverse shear strain effects and constant through-thickness shear stresses [21]. Since the transverse shear strain in FSDT is assumed to be constant through the thickness direction, it violates the zero shear stress condition on the top and bottom surfaces of the shell or plate. The shear correction factors are then required to accommodate the transverse shear stiffness, and as such, FSDT mainly relies on these factors for veracity [22, 23]. However, sophisticated theories are capable of specifying the response of thin plates without involving shear correction factors and predict the results more accurately as for thick plates; yet, a considerable computational attempt is required [24]. In this context, higher-order shear deformation theories (HSDTs) have been introduced [25], which postulate that the transverse shear stresses are non-linearly distributed perpendicular to the plate thickness. In addition, the use of numerical techniques in solving various engineering problems is highly advantageous since experimental and laboratory investigations are costly and time-consuming.
Over the last two decades, significant innovations and developments in numerical methods have emerged. Among the most renowned approaches are the finite elements (FE), the finite difference (FD), and the finite volumes (FV), each of which bears its own merits and limitations. However, the presence of a predefined mesh to connect nodes remains an essential cornerstone in the formulation of all these methods. Nevertheless, finite volume method (FVM) has been practiced in a wide range of problems, such as elasto-statics [26], beams [19], plates [27], and crack analyses [28, 29]. In this technique, the mesh can be readily generated, though there is still a need for trial and effort to achieve an optimal one. In addition to developing the previous methods, research is running in order to optimize the three factors of time and accuracy, as also cost to obtain appropriate results through this method. In this respect, numerical analyses have been conducted on composite plates to obtain approximate results since no exact solution can be established on account of different geometries and boundary conditions of such plates. Based on the existing studies on the different methods, meshfree finite volume (MFV) method remains a promising technique to solve various governing differential equations of the plates numerically. Hence, the efficacy of MFV has been explored in numerous investigations [30], [31], [32], [33], [34], [35], [36], [37], [38]. The MVF-based approaches are advantageous for simulations with configuration complexity, cracks, discontinuities, moving geometries, and large-deformation analyses. They also enjoy simplicity, stability, and shear-locking-free quality in the developing of stiffness matrix [39]. Among the earliest meshless techniques is widely known to be the smooth particle hydrodynamics (SPH) [40], which incorporated an interpolation method based on kernel estimation initially applied in astrophysical modeling. Ever since, multiple studies have further improved the SPH, and thus, considerable progress has been made [41], [42], [43]. In this respect, Chen et al. [44] proposed correction functions in the development of reproducing kernel particle method (RKPM). Belytschko et al. [45, 46] introduced the element-free Galerkin (EFG) method and applied it to the analyses of thin plates and shells [47, 48]. Regarding the piezoelectric laminates, Crawley and de Luis [49] developed analytical models to obtain the dynamic response of a cantilevered laminated composite using bonded or embedded piezoelectric actuators. Reddy [50] implemented Navier solutions in the evaluation of laminated composite plates using integrated sensors and actuators. Chen et al. [51] investigated the exact solution of orthotropic shells coupled with piezoelectric layers. Liew et al. [52] developed an element-free method based on FSDT for the shape control and vibration suppression of the piezolaminated composite plates. Recently, the vibration control of shells with distributed piezoelectric sensor and actuator layers have been investigated by Jamshidi and Jafari [53]. Also, the active vibration control of laminated composite beams with piezoelectric layers using third-order shear deformation plate theory (TSDT) and the LQR control scheme has been studied by Tian et al. [54].
The layout of the presented paper is as follows: the Moving Least Square Approximation is shortly explained in Section 2. The constitutive equation of the orthotropic composite plate, the strain-displacement relations, finite volume discretization of governing equations based on FSDT, the various plate boundary conditions, space-State Modeling, modal analysis and LQR controller are defined in Section 3. The numerical results and discussion are presented in Section 4, and finally, conclusions are given in Section 5.
Section snippets
Moving least square approximation
The moving least square (MLS) approximation was first innovated by the mathematicians for surface fitting and reconstruction purposes [14, 55]. So far, it has paved the way for developing various meshfree methods since it is capable of providing a continuous approximation for the field functions across the entire domain [56, 57]. In the mesh-free methods, support domain or support radius is used to express the domain by which the solution field variables are approximated.
The MLS shape
Problem formulation and solution procedure
Assume a thin rectangular orthotropic composite plate with length a, width b, and height h. According to Fig. 2, two piezoelectric layers, which completely cover the plate, are attached to the top and bottom of the laminate so as to control the vibration.
Verification
In order to verify the derived relationships via the MFV approach proposed in this study, mass and stiffness matrices, and thus, frequency need to be verified. According to Fig. 9, a three-ply laminated square composite plate with stacking orientations [0°/90°/0°] and boundary conditions CFFF was considered, conforming to that of Liew et al. [68].
Table 1 reports the frequency parameter results obtained in the present study with αs = 2.6, a/b = 1, t/b = 0.01, and with 21 × 21 control volumes
Conclusions
In this paper, the first-order shear deformation speculations coupled with the MFV approach were triggered in the active vibration control of piezoelectric composite laminates with different stacking schemes under a point load. The MLS shape functions were employed to increase the accuracy of the problem-solving. In this regard, a continuous-time LQR loop was used to damp the vibration of the laminate excited by a step force by adjusting the weighting matrix of the objective function. The
Declaration of Competing Interest
None.
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