different fiber orientation angles ( with z-axis) of the patches of obliquely reinforced 1-3 PFC are presented in Fig. 3.9. Also, the total actuation of the overall annular plate through all the coefficients (e3j 0, j1,2,…6) is presented in the same figure (Fig. 3.9) by the legend as “Obliquely reinforced 1-3 PFC”.
3.6.2.1 Extension mode actuation of the annular sandwich plate
Since the stiffness matrix of the patches of obliquely reinforced 1-3 PFC is fully populated (in the reference cylindrical coordinate system), any of the piezoelectric coefficients (e3j, j1,2,…6) provides all the electrically induced extensional and shear stress components/actuation forces in the patches for the aforesaid applied transverse electric field. Among these actuation forces, the in- plane extensional ones through any coefficient (e3j) are not effective to cause bending deformation of the overall annular plate since the actuator patches are located at the core of the symmetric annular sandwich plate. However, induced transverse extensional stress/actuation force through any coefficient (e3j) causes thickness mode actuation that is usually not so effective in causing bending deformation of thin plates. So, rest of the actuation forces, i.e. shear actuation forces in the patches of obliquely reinforced 1-3 PFC mainly cause the bending deformation of the overall annular plate (Fig. 3.9) as it is described in the next section.
3.6.2.2 Shear mode actuation of the annular sandwich plate
The electrically induced shear stress components/shear actuation forces in the patches due to the coefficient (e31/e32) mainly appear through the shear- extension coupling stiffness coefficients as C14/C24, C15/C25 and C16/C26. But these shear actuation forces through the coefficient (e31/e32) are not so effective to cause the bending deformation of the overall annular plate as it is observed from the curves in Fig. 3.9 for e310(e3j 0, j1) and e320(e3j 0, j2).
The other coefficients like e34 and e36 directly induce the shear stresses/actuation forces in the
z and r
planes of the patches of 1-3 PFC,Chapter 3: Shear actuation capability of obliquely-reinforced 1-3 PFC respectively. But, these coefficients (e34, e36) appear with very small magnitudes since the angle (Fig. 3.5) at any point over the plane of a patch does not arise with indicative value for its (patch) small circumferential span (p, Fig. 3.2). So, these coefficients (e36and e34) do not have indicative effect on the bending deformation of the overall annular plate as it is observed from the curves in Fig.
3.9 for e34 0(e3j 0, j4) or e360(e3j 0, j6).
Fig. 3.9 Variation of maximum transverse deflection (wmax /h) of the annular sandwich plate with the fiber orientation angle ( with the
z
-axis) of the patches of obliquely reinforced 1-3 PFC.The shear actuation forces through the coefficient e33 appear due to the shear-extension coupling stiffness coefficients as C34, C35 and C36. Among these coefficients, C34 and C36 appear with the small magnitudes because of the small value of (Fig. 3.5) at any point in a patch, while the other stiffness coefficient C35 appears with indicative magnitude because of the alignment of the obliquely oriented fibers of 1-3 PFC in the xz-plane of its (1-3 PFC) Cartesian material coordinate system. So, the shear actuation of the overall annular plate through the coefficient (e33) appears mainly due to the corresponding shear actuation force in the rz-plane, and it occurs indicatively as shown in Fig. 3.9 by the curve for e330(e3j 0, j3).
The coefficient e35directly induces the transverse shear stress in the rz- plane of the patches although the other induced shear stresses in the
z andChapter 3: Shear actuation capability of obliquely-reinforced 1-3 PFC
r
planes also appear through the same coefficient (e35) due to the shear-shear coupling stiffness coefficients as C54and C56, respectively. But, the magnitudes of these stiffness coefficients (C54and C56) are very small in comparison to the magnitude of the stiffness coefficient C55 mainly because of the small value of at any point in an actuator patch (Fig. 3.5). So, the shear actuation of the overall annular plate through e35 mainly arises due to the electrically induced transverse shear stress (rz), and the corresponding maximum bending deflections of the overall annular plate for different fiber orientation angles of the patches of 1-3 PFC are shown in Fig. 3.9 by the curve for e350 (e3j0, j5).
From the results in Fig. 3.9, it is clear that the actuation of the annular sandwich plate by the patches of obliquely reinforced 1-3 PFC mainly appears by means of shear actuation forces in the rz-plane through the coefficients e33 and e35. But, the shear actuation force in the rz-plane through the coefficient
e33 is opposite to that through the coefficient e35 (Fig. 3.9), and thus the total shear actuation of the annular sandwich plate decreases as it is shown in Fig.
3.9 by the curve for “Obliquely reinforced 1-3 PFC ”. Despite this fact, it may be observed from Fig. 3.9 that the patches of obliquely reinforced 1-3 PFC cause an indicative shear actuated bending deformation of the annular sandwich plate.
3.6.3 Shear-based active control of vibration of the annular sandwich plate
The capability of the obliquely reinforced 1-3 PFC in shear-based attenuation of vibration of the annular sandwich plate is assessed in this section by means of evaluating the controlled frequency responses of the plate under a transversely distributed harmonic mechanical load (Eq. 2.46) at its bottom surface. Under this harmonic mechanical load, the frequency responses of the overall annular plate are evaluated within an operating frequency-range that includes its (plate) first three bending modes of vibration. All these bending modes of vibration are of fundamental radial mode member (m1) while those are separated by the circumferential-mode numbers (n) as 0, 1 and 2.
The shear actuator patches of the 1-3 PFC are taken with a fiber orientation angle () of 35oon the basis of the results in Fig. 3.9, and they (actuator patches) are activated according to the aforesaid shear-based active
Chapter 3: Shear actuation capability of obliquely-reinforced 1-3 PFC
control strategy where the feedback control gains (kds, s1, 2,....np) are taken with a uniform value (kds kdfor s1, 2,....np). In the following results for frequency responses of the annular sandwich plate, the maximum nodal transverse displacement-amplitude is presented at every operating frequency ( ) , and it is denoted in the dimensionless form as, wmax/h. Besides, the required control voltage at every operating frequency is presented as the maximum one (Vmax) among the control voltages over the actuator patches.
Fig. 3.10 (a) Variation of the maximum nodal transverse displacement- amplitude (wmax /h) of the annular sandwich plate with the operating frequency (), (b) the corresponding variations of the maximum control voltage ( po1.6 N/m2,nr 2, n 8).
Figure 3.10(a) illustrates the variation of the maximum nodal transverse displacement-amplitude (wmax/h) of the annular sandwich plate with the operating frequency () for three different values of the control gain (kd), where the bending mode at a resonant frequency is denoted by (m n, ). The
Chapter 3: Shear actuation capability of obliquely-reinforced 1-3 PFC