Comparative investigation on nonlinear transient vibrations of laminated composite shallow shell with active constrained layer damping treatment using PFRC and AFC patch material

Here, the comparative investigation for the performance of active constrained layer damping treatment using PFRC and AFC patch material for controlling the nonlinear transient vibrations of laminated composite shallow shell is carried out. The patch material used in this case for making the constraining layer of the ACLD treatment is PFRC and AFC. For modelling of ACLD in the time domain Golla-Hughes-McTavish (GHM) method is used. The Von Kármán type non-linear strain displacement relations along with a simple first-order shear deformation theory are used for deriving this electromechanical coupled problem. A finite model in 3-dimensions have been developed for smart composite shallow shell integrated with the ACLD treated patches of PFRC and AFC. The results of both the cases have been compared and it has been found that the improved performance of AFC patch over the PFRC in subsiding the transient nonlinear vibrations of symmetric cross ply laminated composite shallow shell.


Introduction
Recent trend shows the use of lightweight flexible structures in many applications such as spacecrafts, aircrafts and so on. This demands the laminated composite structures in use. But lighter laminated flexible structures are weak to withstand the vibrations having wide-ranging decay time since they possess little internal damping. To increase the strength and damping, an alternative active control system is designed with the help of sensors and actuators and integrated with the host structure to make it a "smart structure" [1]. Different piezoelectric materials' distributed sensors and actuators are used in making of smart structures and many researchers have worked on this in recent years. Iman Fattahi and Hamid Reza Mirdamadi [2] have used piezoelectric smart structures in formulating the FE model based on equivalent single layer theory and all parameters were calculated for 3D active beam elements. M. Kerboua and et.al [3] have investigated the vibration control with the use of piezoelectric based smart materials and results showed that vibration reduction by using smart beam. Shaikh Tauseef and R. K. Agarwal [4] have studied the active vibration of cantilever beam. They used the lead zirconium titanate patches as actuators and by applying different voltages, amplitude of vibration IOP Conf. Series: Materials Science and Engineering 577 (2019) 012147 IOP Publishing doi:10.1088/1757-899X/577/1/012147 2 was observed. Ehsan Omidi and et al [5] have worked on vibration reduction using the velocity feedback control. The optimization problem was addressed by linear quadratic regulator (LQR). Ashok M H and et al [6,7] worked on actively damped laminated composite shallow shells and plates with AFC actuators. The surplus vibrations of the host structure could be minimized by using ACLD treatment with PFRC/AFC patches to it. Though the work related to PFRC and AFC patch materials is reported in the papers but comparative investigations were not made. In this paper, comparison of performance of ACLD with PFRC and AFC in controlling the nonlinear transient vibrations is presented.

Finite element model
In this section, the FE model is generated for performance evaluation of ACLD treatment. Figure 1 shown here gives details of N orthotropic layers' laminated composite shallow shell. The parameters like thickness, shallowness angle, length, circumferential width, and average radius of the shell are represented by symbols h, ϕ, a, s and R respectively. Figure 1illustrates the details of ACLD technique using PFRC/AFC patches applied on the top surface of the shell. The thickness of viscoelastic constrained layer used in ACLD treatment is taken as hv whereas hp is considered as thickness of PFRC/AFC patch material. To find out whole response of the shallow shell combined with ACLD treated PFRC/AFC patches, FSDT (First Order Shear Deformation) theory is used.
The generalized displacements are denoted by u, v, and w in different directions i.e. along x, y, and z directions at any point in any coat for entire panel respectively. These are given as: The governing equations of motion of a typical element of the overall panel with ACLD system can be obtained as follows: Modelling of viscoelastic material is done by GHM (Golla Hughes McTavish) method in time domain. The equations obtained are as follows.
Finally, the global equations of motion governing the open loop behaviour of the laminated composite shallow shells integrated with the PFRC/AFC patches of ACLD treatment are obtained as follows: The necessary control voltage required for the activation of the patches is found out using a simple velocity feedback control law. Such voltage of each patch is represented by following equation.
Where, control gain j d K is for the j th patch. The unit vector [] j U defines the location of sensing the velocity signal and this would be fed back to j th patch. By means of equation 9, the equation of motion for the dynamic response of the substrate plates treated by the patches of PFRC/AFC is given as follows:   Figure 3 shows the attenuation capabilities of AFC and PFRC patch material in active mode. It can be observed from figure 3 that, the amplitude of is less for AFC material compared to PFRC and we can also compare these responses under active mode with responses under passive mode. Figure 4 compares the control voltages requirement for AFC and PFRC for active mode which is nearly same. It means that, there is not much change in the control voltage required for activation of AFC or PFRC patches. Figure 5 gives the phase plot comparison for PFRC and AFC patch material.