A comparative study on the smart damping capabilities of cylindrically orthotropic
4.1 Introduction
The literature review in Chapter 1 shows a good number of studies on the actuation capabilities of horizontally and vertically reinforced 1-3 PFC actuators (Mallik and Ray, 2003; Smith and Auld, 1991). Of these 1-3 PFCs, the horizontally reinforced 1-3 PFC can provide in-plane actuation force, and thus it is mostly utilized for counteraction of the bending mode of deformation of thin- walled and flexible structures. This 1-3 PFC is presently utilized for control of bending deformation of thin annular/circular plates. For efficient utilization of this 1-3 PFC (Mallik and Ray, 2003), its construction is converted from rectangular to cylindrical coordinates, and the laminate is made in the cylindrical coordinate frame as shown in Fig. 4.1(a). Every layer of this laminate is a cylindrically orthotropic 1-3 PFC lamina that is comprised of transversely poled piezoelectric fibers. For control activity of every 1-3 PFC layer, the external electric field would act in the transverse direction, and it is achieved by means of supplying external voltage across the top and bottom fully electrode-surfaces (Fig. 4.1(a)). The poling direction (p) of a 1-3 PFC layer within the laminate is in opposite to that of its consecutive layers so that the major electrically induced actuation forces from all the layers appear in the same (positive/negative) radial direction. For achieving the major in-plane actuation force along the circumferential direction, a similar laminate can also be formed by altering the orientation of the fibers in every layer from radial to the circumferential direction. This laminate is also shown in Fig. 4.1(b). The actuation capabilities of these PFC laminates (Figs. 4.1(a) and 4.1(b)) are quantified by the piezoelectric coefficient (e31) of the fibers. As the piezoelectric coefficient (e33) of the fibers has a greater magnitude than that of the coefficient (e31) of the same fibers, the 1-3 PFC based on the piezoelectric coefficient (e33) may possess greater actuation
capability. So, it was implemented in the design of 2-2 PFC (AFC/MFC), and presently the same has been performed for the design of CPFC or 1-3 PFC in cylindrical coordinates (Chapter 2) particularly for having the in-plane actuation
force based on the piezoelectric coefficient of maximum magnitude. The present SPFC (Chapter 2) is a modification of the CPFC mainly for having greater flexibility and conformability. However, similar to the laminates in Figs. 4.1(a) Fig. 4.1 Schematic diagrams of cylindrically orthotropic PFC laminates with (a) radially or (b) circumferentially reinforced fibers with transverse poling direction (
: electric potential, p: poling direction of fibers).Fig. 4.2 Schematic diagrams of cylindrically orthotropic PFC laminates with (a) radially or (b) circumferentially reinforced fibers with longitudinal poling direction (
: electric potential, p: poling direction of fibers).Chapter 4: A comparative study… in vibration control of annular plates
and 4.1(b), the laminates of present SPFC/CPFC (Fig. 2.8) can also be formed for achieving the in-plane actuation force either in radial or in the circumferential direction as shown in Figs. 4.2(a) or 4.2(b), respectively. These PFC laminates act as the extension mode PFC actuators.
Generally, an extension mode piezoelectric actuator in cylindrical coordinates is attached to the top/bottom surface of an annular or a circular plate. The actuator provides the in-plane actuation forces along both the radial and circumferential directions against the mechanically induced in-plane normal stresses in the overall plate due to its bending mode of deformation. As the requirement of actuation force usually appears in both the radial and circumferential directions in control of flexural vibration of an annular/a circular plate, any of the aforesaid PFC laminates (Figs. 4.1 and 4.2) can be utilized. But, it is not possible to achieve identical control-performance of all the PFC laminates as per their electro-elastic properties. So, a quantitative assessment of their (PFC laminates in Figs. 4.1 and 4.2) control-capabilities or a quantitative comparative study on their (PFC laminates) control-capabilities is needed to identify appropriate one for efficient control of vibration of an annular or a circular plate. It is carried out in this chapter to address appropriate one among the available 1-3 PFCs (Figs. 4.1 and 4.2) for efficient control of an annular plate (Kumar et al., 2017). It should be noted here that this comparison study is performed considering the PFC laminates with continuous piezoelectric fibers. Every PFC actuator/laminate is presently utilized in the form of patches which are optimally configured over the host plate-surface through the proposition of a new numerical methodology. With the aid of these optimal configurations of the PFC actuators, they are considered to act in the form of smart dampers, and their actuation/damping capabilities are quantified for a study on their relative control-capabilities. The overall study is presented in the following manner.
First, a configuration of an annular plate integrated with PFC actuator- patches is presented for effective utilization of the actuator-patches by means of the feedback of their local velocities. Next, the geometrical and material properties of the PFC actuators (Figs. 4.1 and 4.2) are outlined for handling all these actuators in a uniform manner especially for a comparative study on their control-capabilities. On the basis of these properties of PFC actuators, the functional relations between the driving electric field and applied voltage are
evaluated for all the actuators. These relations are subsequently utilized in the derivation of an FE model of the smart annular plate. With the aid of this FE model, a new numerical methodology in deciding the optimal configuration of actuator-patches corresponding to a mode of vibration of the smart annular plate is presented. Through this numerical methodology, the optimal configurations of the smart annular plate for different PFC actuators are decided for each of its (plate) fundamental symmetric and asymmetric modes of vibration. Based on these optimal configurations of the smart annular plate, the modal damping-capabilities of the PFC actuators are evaluated. These evaluated results for each of the fundamental symmetric and asymmetric modes of vibration are then analyzed for the study on their (PFC actuators) relative damping-capabilities. The numerical results not only address the control/damping-capabilities of the actuators in a common reference frame of the annular plate but also recommend the potential one among the aforesaid PFC actuators in control of similar plane structures.