Distributed Sensing and Control of Elastic Shells

Abstract

Distributed sensing and control of a generic distributed parameter system (DPS) or a generic smart structronic shell system, i.e., a deep elastic shell laminated with distributed piezoelectric sensor and actuator layers, was proposed and corresponding generic theories derived. Based on the direct piezoelectric effect, the distributed sensor can be used to monitor shell oscillations; the converse effect enables the distributed actuators to manipulate structural behaviors and to suppress structural vibrations. Two generic sensor/actuator design principles, i.e., the segmentation technique and the shaping technique, were also presented.

Appendix

4.1.1 Piezoelectricity Theory

Two fundamental equations are used in the derivation of distributed sensor theory:

$$\left\{ \text{T} \right\} = \left[ {\text{c}^{\text{D} } } \right]\left\{ \text{S} \right\} - \left[ \text{h} \right]^{\text{t} } \left\{ \text{D} \right\},$$

(4.9.1)

$$\left\{ \text{E} \right\} = \left[ {\upbeta^{\text{S} } } \right]\left\{ D \right\} - \left[ \text{h} \right]\left\{ \text{S} \right\},$$

(4.9.2)

where {T} is the stress vector (i.e., {T} = {T₁₁ T₂₂ T₃₃ T₂₃ T₃₁ T₁₂}^t); [c^D] is the elasticity matrix evaluated at constant dielectric displacement; {S} is the strain vector (i.e., {S} = {S₁₁ S₂₂ S₃₃ S₂₃ S₃₁ S₁₂}^t); [h] is the piezoelectric constant matrix ; {D} is the electric displacement vector; [.]^t indicates the matrix transpose; {E} is the electric field vector; [β^S] is the dielectric impermeability matrix evaluated at constant strain.

4.1.2 Piezoelectric Matrix of Polyvinylidene Fluoride (PVDF)

Polymeric polyvinylidene fluoride (PVDF) has a mm2 structure (Kawai, 1969). The piezoelectric matrix [d] of a PVDF polymer can be expressed as (Sessler, 1981)

$$\left[ {\text{d}_{{\text{ij}}} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & {\text{d}_{15} } & 0 \\ 0 & 0 & 0 & {\text{d}_{24} } & 0 & 0 \\ {\text{d}_{31} } & {\text{d}_{32} } & {\text{d}_{33} } & 0 & 0 & 0 \\ \end{array} } \right].$$

(4.9.3)

Note that the piezoelectric coefficient d₂₄ is equal to d₁₅ for a PVDF electrically polarized and not mechanical stretched.