Present smart annular sandwich plate

Active control of vibration of annular plates using a new shear mode PFC actuator with cylindrically

6.2 Present smart annular sandwich plate

CHAPTER

6

Active control of vibration of annular plates using a

constructed by taking the material of the core as foam, and the patches of the shear mode PFC actuator are embedded within this foam layer. A schematic diagram of this smart annular sandwich plate is presented in Fig. 6.1. The inner/outer radius and the thickness of the overall annular plate are denoted by,

i / o

r r and h, respectively. The thicknesses of the top/bottom face layer and the core are indicated by, h_f and h_c, respectively. The thickness of the patches is the same as that is for the core.

Fig. 6.1 Schematic diagram of the smart annular sandwich plate.

For effective control of structural deformation/vibration using a piezoelectric actuator in the form of the patch, it is generally recommended to locate the patches within the domain of the overall structure in an appropriate manner. The present shear mode PFC actuator counteracts the mechanically induced transverse shear stress within the overall plate, and this transverse shear stress appears with its maximum value at the nodes of any bending mode shape of vibration of the plate.

So, the appropriate locations of the patches may be chosen as the nodes of the bending mode shape of vibration of the plate under study. But, the number and locations of the nodes change as the mode shape of the plate alters during its vibration within a range of operating frequency. So, it is a little difficult to decide

Chapter 6: Control capability of shear mode annular PFC actuator

appropriate locations of the patches over the plane of the annular plate. Presently, a strategy for the arrangement of the patches is proposed by taking the identical patches in the shape of annular-sector. The corresponding configuration of the patches is demonstrated in Fig. 6.1. The circumferential span of the annular plate is divided into a number (n_) of equal divisions. Similarly, the radial span (r_or_i) of the same plate is divided into a number (n_r) of equal radial divisions. These divisions yield identical annular-sectors over the plane of the plate with the radial and circumferential spans of,  r (r_o r_i) /n_r and   2 / n_, respectively. In every annular-sector, one patch is provided with the radial and circumferential lengths of, r_p and _p , respectively. The corresponding radial (r_F) and circumferential (

F

 ) gaps in every sector are filled by the foam (   ( _p _F),     r ( r_p r_F)) (Fig. 6.1). The gaps (r_F,_F) are very small in comparison to the in-plane dimensions of the patches (_p  _F,   r_p r_F). The main purpose of making this configuration of the core is to provide a sufficient number of patches around every node for any mode of vibration of the plate within an operating frequency-range of interest. If the operating frequency-range includes the modes of higher radial mode numbers, then the number (n_r) of divisions of the radial span (r_or_i) is to be increased. Similarly, the number (n_) of divisions of the circumferential span (2 ) is to be increased when the operating frequency-range includes the modes of higher circumferential mode numbers. For the increased number of radial/circumferential divisions, there would be no discrepancy in achieving effective actuation of the modes of lower radial/circumferential mode numbers. So, the number of radial/circumferential divisions is to be decided based on the maximum radial/circumferential mode number within the operating frequency-range of interest. Now, the mechanically induced transverse shear stress does not appear uniformly over the plane of the annular plate for any of its bending modes of vibration. So, it is required to activate the patches in an appropriate manner such that every patch can effectively counteract the mechanically induced transverse shear stress around its location. In order to achieve it according to the velocity feedback control strategy, every patch is equipped with a velocity sensor. For a typical patch, the velocity sensor is located on the top surface of the overall plate corresponding to the middle point of the plane of the patch. With this location of the velocity sensor, every patch is supposed to counteract the time-varying

transverse shear stress around its location by taking the feedback of local velocity.

According to this arrangement of the patches and velocity sensors, all the bending modes of vibration of the smart plate are expected to be attenuated effectively, and it is verified in the later section of this chapter. The control capability of the shear mode PFC actuator (Chapter 5) is investigated through this arrangement of its (PFC) patches where the PFC actuator is utilized with its two different geometric configurations separately.

For the first geometric configuration of the annular PFC actuator (as indicated by line 1/2 in Fig. 5.7), its inner and outer radii are equal to those of the host plate.

The patches are made by dividing the plane of this annular PFC actuator, and the material of the patches is denoted by PFC#1. For the second geometric configuration of the PFC actuator (as indicated by line 3/4 in Fig. 5.7), the patches within a radial division (S₁/S₂/S₃/S₄, Fig. 5.7) are made by dividing the plane of the corresponding part of the annular PFC actuator. In this second case, the material of the patches is denoted by PFC#2.