Active Vibration Control of Piezoelectric Sandwich Plates

. 2022 May 31;15(11):3907. doi: 10.3390/ma15113907

M_{b} = ρ_{b} h_{b} \int_{0}^{a} \int_{0}^{b} W W^{T} d x d y

M_{e} = ρ_{e} h_{e} \int_{0}^{a} \int_{0}^{b} W W^{T} d x d y

K_{e} = K_{e 1} + K_{e 2} + K_{e 3} + K_{e 4} + K_{e 5} + K_{e 6}

K_{e 1} = \frac{Q_{11}^{k}}{3} [{(\frac{h}{2} + h_{p})}^{3} - {(\frac{h}{2})}^{3}] \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial x^{2}} \frac{\partial^{2} W}{\partial x^{2}} d x d y

K_{e 2} = \frac{2 Q_{12}^{k}}{3} [{(\frac{h}{2} + h_{p})}^{3} - {(\frac{h}{2})}^{3}] \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial x^{2}} \frac{\partial^{2} W}{\partial y^{2}} d x d y

K_{e 3} = \frac{Q_{22}^{k}}{3} [{(\frac{h}{2} + h_{p})}^{3} - {(\frac{h}{2})}^{3}] \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial y^{2}} \frac{\partial^{2} W^{T}}{\partial y^{2}} d x d y

K_{e 4} = \frac{4 Q_{16}^{k}}{3} [{(\frac{h}{2} + h_{p})}^{3} - {(\frac{h}{2})}^{3}] \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial x \partial y} \frac{\partial^{2} W^{T}}{\partial x^{2}} d x d y

K_{e 5} = \frac{4 Q_{26}^{k}}{3} [{(\frac{h}{2} + h_{p})}^{3} - {(\frac{h}{2})}^{3}] \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial x \partial y} \frac{\partial^{2} W^{T}}{\partial y^{2}} d x d y

K_{e 6} = \frac{4 Q_{66}^{k}}{3} [{(\frac{h}{2} + h_{p})}^{3} - {(\frac{h}{2})}^{3}] \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial x \partial y} \frac{\partial^{2} W^{T}}{\partial x \partial y} d x d y

K_{d} = \frac{∍_{66}}{h_{p}} \int_{0}^{a} \int_{0}^{b} d x d y

F = \int_{0}^{a} \int_{0}^{b} q W^{T} d x d y

F_{e} = F_{e 1} + F_{e 2} + F_{e 3}

F_{e 1} = e_{31} (\frac{h_{b}}{2} + h_{e}) \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial x^{2}} d x d y

F_{e 2} = e_{32} (\frac{h_{b}}{2} + h_{e}) \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial y^{2}} d x d y

F_{e 3} = 2 e_{36} (\frac{h_{b}}{2} + h_{e}) \int_{0}^{a} \int_{0}^{b} \frac{\partial W}{\partial x \partial y} d x d y

K_{b} = K_{b 1} + K_{b 2} + K_{b 3} + K_{b 4} + K_{b 5} + K_{b 6}

K_{b 1} = \frac{Q_{11}^{k} \sum_{1}^{k} (h_{k}^{3} - h_{k - 1}^{3})}{3} \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial x^{2}} \frac{\partial^{2} W^{T}}{\partial x^{2}} d x d y

K_{b 2} = \frac{2 Q_{12}^{k} \sum_{1}^{k} (h_{k}^{3} - h_{k - 1}^{3})}{3} \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial x^{2}} \frac{\partial^{2} W^{T}}{\partial y^{2}} d x d y

K_{b 3} = \frac{Q_{22}^{k} \sum_{1}^{k} (h_{k}^{3} - h_{k - 1}^{3})}{3} \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial y^{2}} \frac{\partial^{2} W^{T}}{\partial y^{2}} d x d y

K_{b 4} = \frac{4 Q_{16}^{k} \sum_{1}^{k} (h_{k}^{3} - h_{k - 1}^{3})}{3} \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial x \partial y} \frac{\partial^{2} W^{T}}{\partial x^{2}} d x d y

K_{b 5} = \frac{4 Q_{26}^{k} \sum_{1}^{k} (h_{k}^{3} - h_{k - 1}^{3})}{3} \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial x \partial y} \frac{\partial^{2} W^{T}}{\partial y^{2}} d x d y

K_{b 4} = \frac{4 Q_{66}^{k} \sum_{1}^{k} (h_{k}^{3} - h_{k - 1}^{3})}{3} \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial x \partial y} \frac{\partial^{2} W^{T}}{\partial x \partial y} d x d y

K_{c} = K_{c 1} + K_{c 2} + K_{c 3}

K_{c 1} = \frac{e_{31}}{2 h_{b}} [{(\frac{h}{2} + h_{p})}^{2} - {(\frac{h}{2})}^{2}] \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial x^{2}} d x d y

K_{c 2} = \frac{e_{32}}{2 h_{b}} [{(\frac{h}{2} + h_{p})}^{2} - {(\frac{h}{2})}^{2}] \int_{0}^{a} \int_{0}^{b} \frac{\partial^{2} W}{\partial y^{2}} d x d y

K_{c 3} = \frac{e_{36}}{2 h_{b}} [{(\frac{h}{2} + h_{p})}^{2} - {(\frac{h}{2})}^{2}] \int_{0}^{a} \int_{0}^{b} \frac{\partial W}{\partial y \partial x} d x d y