A comparative study on the smart damping capabilities of cylindrically orthotropic
4.6 Present strategy for optimal configuration of actuator-patches
Chapter 4: A comparative study… in vibration control of annular plates
locations of its patches for a mode of vibration under study. A mode of vibration of the overall plate is presently denoted by the radial and circumferential mode numbers as, m and n , respectively. Generally, two kinds of mode-shapes appear in the natural vibration of an annular plate with regard to its circular symmetry. One is the symmetric mode-shape (n 0) and other one is the asymmetric mode-shape (n0). For every asymmetric mode-shape, there is a similar second mode-shape that is orthogonal to the first one with an angular shift of / 2n. The couple of mode-shapes appear at the same natural frequency of every asymmetric mode due to the exchange of sine and cosine functions in the circumferential coordinate () (Amabili, 2008). For these two similar mode- shapes, the only difference between the corresponding optimal configurations of the actuator-patches would be an angular shift of / 2n . So, the optimal configuration of actuator-patches can be decided by taking any one of the couple of mode-shapes in an asymmetric mode. In the following section, two typical symmetric (Fig. 4.5(a)) and asymmetric (Fig. 4.5(b)) mode-shapes of the smart annular plate are considered, and a methodology in deciding optimal size and locations of actuator-patches for each of the mode-shapes is demonstrated.
4.6.1 Initial configuration of the smart annular plate
It may be observed from Fig. 4.5 that every mode-shape is comprised of certain numbers of positive and negative half sine-waves along each of the radial and circumferential directions. These positive and negative half sine-waves are of uniform dimensions along radial/circumferential direction, and they also appear alternatively in both the orthogonal directions. Following the radial and circumferential boundaries of the half sine-waves, the top surface of the host annular plate is divided into equal number of sectors as shown in Figs. 4.5(a) and 4.5(b) by the dash-lines. Basically, the sectors of half sine-waves are created following the nodes of the mode-shape. The PFC actuator is first taken in layer- form over the top surface of the substrate plate and then divided in the form of patches following the boundaries (dash-lines) of the sectors. These patches are considered to be uniformly separated in a little gap along both the radial and circumferential directions. For a flexure mode of deformation of an annular plate, generally, the mechanically induced in-plane normal stresses appear with significant magnitudes at the locations of anti-nodes. The extension mode PFC actuators provide electrically induced in-plane normal forces against these
mechanical stresses at the host plate-surface on which they (PFCs) are attached.
So, the initial configuration of the smart plate is taken by providing actuator- material around the anti-nodes or by separating the actuator-layer following the nodes. The mechanical stress at a point of a sector is in opposite phase to that
Fig. 4.5 Typical (a) symmetric and (b) asymmetric mode-shapes of the annular plate along with the separated sectors (by dash-lines) of half sine-waves.
(phase) of the same stress at similar points within the consecutive sectors. So, for effective use of the PFC patches, every patch is supposed to act against the time-varying mechanical stresses around its location by taking the feedback of
Chapter 4: A comparative study… in vibration control of annular plates
local velocity at the anti-node of the corresponding sector. As the velocity in any sector/PFC patch appears with its maximum value at the corresponding anti- node, this point/node is chosen for the feedback of local velocity. However, this initial configuration of the actuator-patches is to be modified for optimal size and location of every actuator-patch within the corresponding sector. It should be noted here that all the sectors are made uniformly, and the actuator-patches act in a uniform manner against the mechanical stresses within the corresponding sectors. So, any one of the sectors can be taken as the representative sector to study the actuation characteristics of the actuator- patches. Also, the optimal size and locations of actuator-patches within the corresponding sectors can be decided based on this representative sector that is presented in the next section.
4.6.2 Optimal configuration of a typical/representative sector
According to the aforesaid initial configuration of the smart annular plate, every PFC patch almost covers the area (in r plane) of the corresponding sector. So, the initial PFC patch within a sector is presently denoted as PFC sector-layer.
The optimal size and location of the actuator-patch within a sector are decided on the basis of the importance of the actuator-material at every point over its (sector) r plane. The corresponding tests over a set of points on a PFC sector- layer (representative sector-layer) are carried out without alteration of reaming sector-layers. A typical test-point is taken over the top surface of the PFC sector- layer, and a differential area on the same surface around the test-point is selected such that its (differential area) edges are in parallel to those of the sector-layer. The material of the PFC actuator over this differential area is then removed in such a manner that the PFC sector-layer has a differential hole throughout its thickness around the test-point. Keeping the radial and circumferential spans of the differential area as constant parameters, the test- point is moved at different points/locations over the r plane of the sector- layer. At any location of the test-point along with the differential hole, the remaining area of the PFC sector-layer is a constant area. So, the volume of the PFC sector-layer does not change when the test-point is located at different points. For every location of the test-point, the modal loss factor of the overall plate corresponding to its (plate) mode of vibration is evaluated, and the magnitude of the loss factor is noted along with the location/coordinate (r, ) of
the test-point. The evaluated magnitudes of modal loss factor are then plotted as altitudes against the corresponding locations (r,) of the test-point over the r plane of the sector-layer. These altitudes constitute a surface of modal loss factor over the r plane of the sector-layer.
Physically, the removal of differential PFC material from an important location/point significantly affects the control-capability of the PFC sector-layer, and it is implied by the decrease of the corresponding magnitude of modal loss factor. Similarly, if the differential PFC material is removed from an unimportant point/location of a PFC sector-layer, then there would be no significant change in the corresponding magnitude of the modal loss factor. According to this analogy, the projected points on the r plane from the troughs of the aforesaid surface of loss factor signify the important locations of the PFC material.
Consequently, the area around a projected point on the r plane is the important zone for the PFC material. After determination of important and unimportant zones over the plane of the sector-layer, the optimal size of the important zone can be decided based on the rate of change of loss factor in every space coordinate (r, ). Within an important or unimportant zone, loss factor changes in insignificant rate along any of the space coordinates. But, the magnitude of the same parameter significantly differs between these important and unimportant zones. So, there would be a steep rate of change of loss factor at the transition zone which indicates the separation of important zone from the whole area of the sector-layer. The size of the PFC sector-layer can then be reduced according to this separated zone, and it implies optimal size and location of the PFC patch within a sector.