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Smart control of laminated plates using Murakami zig-zag functions

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Abstract

The study aims at vibration control of laminated plates by the active constraining layer damping (ACLD) treatment incorporating the Murakami zig-zag function (MZZF) while deriving the displacement fields. The control is achieved by activating the ACLD patches by supplying the control voltage. The ACLD patch has two component layers, namely 1–3 piezoelectric composite (PZC) constraining material layer and the viscoelastic constrained material layer. The complete ACLD plate system is modelled as a three layered structure using a MZZF in the displacement fields of the individual layers, and the FE model is obtained by the virtual work principle. A MATLAB FE code has been developed considering the closed loop feedback system for the control of the input parameters. The present method of modelling the ACLD system has proven to be accurate and cost effective for damping the vibrations of the composite plates. Also, the change in piezo fiber orientation angle on the control and performance of the ACLD patches has been effectively studied in detail.

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Authors and Affiliations

  1. Department of Mechanical Engineering, National Institute of Technology Raipur, Raipur, Chhattisgarh, 492010, India

    N. Mehadi Khan & Rajana Suresh Kumar

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  1. N. Mehadi Khan
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  2. Rajana Suresh Kumar
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Correspondence to Rajana Suresh Kumar.

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Appendix

Appendix

The matrices \(\left[{Z}_{1}\right], \left[{Z}_{2}\right], \left[{Z}_{3}\right], \left[{Z}_{4}\right], \left[{Z}_{5}\right],\mathrm{and} \left[{Z}_{6}\right]\) found in Eqs. (7) and (8) are given by

$$\left[ {Z_{1} } \right] = \left[ {\begin{array}{*{20}c} {\left[ {\overline{Z}_{1} } \right]} & {\left[ {\overline{M}_{Z1} } \right]} \\ \end{array} } \right], \left[ {Z_{2} } \right] = \left[ {\begin{array}{*{20}c} {\left[ {\overline{Z}_{2} } \right]} & {\left[ {\overline{M}_{Z2} } \right]} \\ \end{array} } \right], \left[ {Z_{3} } \right] = \left[ {\begin{array}{*{20}c} {\left[ {\overline{Z}_{3} } \right]} & {\left[ {\overline{M}_{Z3} } \right]} \\ \end{array} } \right]$$
$$\left[ {Z_{4} } \right] = \left[ {\begin{array}{*{20}c} {\overline{I}} & {\left[ {\overline{M}_{Z4}^{{\prime }} } \right]} & {\left[ {\overline{Z}_{4} } \right]} \\ \end{array} \left[ {\overline{M}_{Z4} } \right]} \right], \left[ {Z_{5} } \right] = \left[ {\begin{array}{*{20}c} {\overline{I}} & {\left[ {\overline{M}_{Z5}^{{\prime }} } \right]} & {\left[ {\overline{Z}_{5} } \right]} \\ \end{array} \left[ {\overline{M}_{Z5} } \right]} \right],$$
$$\left[ {Z_{6} } \right] = \left[ {\begin{array}{*{20}c} {\overline{I}} & {\left[ {\overline{M}z_{Z6}^{{\prime }} } \right]} & {\left[ {\overline{Z}_{6} } \right]} \\ \end{array} \left[ {\overline{M}_{Z6} } \right]} \right]$$

where \(\left[ {\overline{Z}_{1} } \right]_{k = 1 to N} = \left[ {\overline{Z}_{2} } \right]_{k = N + 1} = \left[ {\overline{Z}_{1} } \right]_{k = N + 2} = \left[ {\begin{array}{*{20}c} z & 0 & 0 & 0 \\ 0 & z & 0 & 0 \\ 0 & 0 & z & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right]\).

$$\left[ {\overline{M}_{Z1} } \right]_{k = 1 to N} = \left[ {\overline{M}_{Z1} } \right]_{k = N + 1} = \left[ {\overline{M}_{Z1} } \right]_{k = N + 2} = \left[ {\begin{array}{*{20}c} {\left( { - 1} \right)^{k} \zeta_{k} } & 0 & 0 & 0 \\ 0 & {\left( { - 1} \right)^{k} \zeta_{k} } & 0 & 0 \\ 0 & 0 & {\left( { - 1} \right)^{k} \zeta_{k} } & 0 \\ 0 & 0 & 0 & {2z} \\ \end{array} } \right]_{ }$$
$$\left[ {\overline{I}} \right] = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right],\left[ {\overline{Z}_{4} } \right]_{k = 1 to N} = \left[ {\overline{Z}_{5} } \right]_{k = N + 1} = \left[ {\overline{Z}_{6} } \right]_{k = N + 2} = \left[ {\begin{array}{*{20}c} z & 0 \\ 0 & z \\ \end{array} } \right]$$
$$\left[ {\overline{M}_{Z4} } \right]_{{k = 1{\text{ to}} N}} = \left[ {\overline{M}_{Z5} } \right]_{k = N + 1} = \left[ {\overline{M}_{Z6} } \right]_{k = N + 2} = \left[ {\begin{array}{*{20}c} {z^{2} } & 0 \\ 0 & {z^{2} } \\ \end{array} } \right]_{{}}$$
$$\left[ {\overline{M}_{Z4}^{{\prime }} } \right]_{k = 1 to N} = \left[ {\overline{M}_{Z4}^{{\prime }} } \right]_{k = N + 1} = \left[ {\overline{M}_{Z4}^{{\prime }} } \right]_{k = N + 2} = \left[ {\begin{array}{*{20}c} {\left( { - 1} \right)^{k} \frac{2}{{h^{k} }}} & 0 \\ 0 & {\left( { - 1} \right)^{k} \frac{2}{{h^{k} }}} \\ \end{array} } \right]_{{}}$$
$$\left[{B}_{tb}\right]=\left[{B}_{tb1 } {B}_{tb2 } \, \ldots \, {B}_{tb8 }\right], \, \left[{B}_{rb}\right]=\left[{B}_{rb1 } {B}_{rb2 } \, \ldots \, {B}_{rb8 }\right], \left[{B}_{ts}\right]=\left[{B}_{ts1 } {B}_{ts2 } \, \ldots \, {B}_{ts8 }\right] \mathrm{and} \, \left[{B}_{rs}\right]=\left[{B}_{rs1 } {B}_{rs2 } \, \ldots \, {B}_{rs8}\right]$$

The submatrices \(\left[{B}_{tbj}\right]\), \(\left[{B}_{rbj}\right]\), \(\left[{B}_{tsj}\right]\) and \(\left[{B}_{rsj}\right]\) found in Eq. (22) are as follows:

$$\left[{B}_{tbj}\right]=\left[\begin{array}{ccc}\frac{\partial {n}_{j}}{\partial x}& 0& 0\\ 0& \frac{\partial {n}_{j}}{\partial y}& 0\\ \frac{\partial {n}_{j}}{\partial x}& \frac{\partial {n}_{j}}{\partial x}& 0\\ 0& 0& 0\end{array}\right], \left[{B}_{tsj}\right]=\left[\begin{array}{ccc}0& 0& \frac{\partial {n}_{j}}{\partial x}\\ 0& 0& \frac{\partial {n}_{j}}{\partial y}\end{array}\right], {\left[{B}_{tbj}\right]}^{*}=\left[\begin{array}{ccc}\frac{\partial {n}_{j}}{\partial x}& 0& 0\\ 0& \frac{\partial {n}_{j}}{\partial y}& 0\\ \frac{\partial {n}_{j}}{\partial x}& \frac{\partial {n}_{j}}{\partial x}& 0\\ 0& 0& 1\end{array}\right],$$
$$\left[{B}_{rbj}\right]=\left[\begin{array}{cc}{\left[{B}_{tbj}\right]}^{*}& {0}^{1*}\\ {0}^{1*}& {\left[{B}_{tbj}\right]}^{*}\end{array}\right], \left[{B}_{rsj}\right]=\left[\begin{array}{c}\begin{array}{cc}{\mathrm{I}}^{2}& {0}^{2*}\\ {0}^{2*}& {\mathrm{I}}^{2}\\ \left[{B}_{tsj}\right]& {0}^{2*}\end{array}\\ \begin{array}{cc}{0}^{2*}& \left[{B}_{tsj}\right]\end{array}\end{array}\right]$$

While, the rigidity matrices, coupling electro-elastic rigidity vectors pertaining to 1–3 PZC material found in Eq. (29) are explicitly given as

$$\left[ {D_{tb} } \right] = \mathop \sum \limits_{k = 1}^{N} \mathop \int \nolimits_{{h_{k} }}^{{h_{k + 1} }} \left[ {\overline{C}_{b}^{k} } \right] dz, \left[ {D_{trb} } \right] = \mathop \sum \limits_{k = 1}^{N} \mathop \int \nolimits_{{h_{k} }}^{{h_{k + 1} }} \left[ {\overline{C}_{b}^{k} } \right] \left[ {Z_{1} } \right] dz,$$
$$\left[ {D_{rrb} } \right] = \mathop \sum \limits_{k = 1}^{N} \mathop \int \nolimits_{{h_{k} }}^{{h_{k + 1} }} \left[ {Z_{1} } \right]^{T} \left[ {\overline{C}_{b}^{k} } \right]\left[ {Z_{1} } \right] dz, \left[ {D_{ts} } \right] = \mathop \sum \limits_{k = 1}^{N} \mathop \int \nolimits_{{h_{k} }}^{{h_{k + 1} }} \left[ {\overline{C}_{s}^{k} } \right] dz$$
$$\left[ {D_{trs} } \right] = \mathop \sum \limits_{k = 1}^{N} \mathop \int \nolimits_{{h_{k} }}^{{h_{k + 1} }} \left[ {\overline{C}_{b}^{k} } \right] \left[ {Z_{4} } \right] dz, \left[ {D_{rrs} } \right] = \mathop \sum \limits_{k = 1}^{N} \mathop \int \nolimits_{{h_{k} }}^{{h_{k + 1} }} \left[ {Z_{4} } \right]^{T} \left[ {\overline{C}_{b}^{k} } \right]\left[ {Z_{4} } \right] dz$$
$$\left[ {D_{tb} } \right]_{v} = h_{v} \left[ {\overline{C}_{b}^{N + 1} } \right],\left[ {D_{trb} } \right]_{v} = \mathop \int \nolimits_{{h_{N + 1} }}^{{h_{N + 2} }} \left[ {\overline{C}_{b}^{N + 1} } \right] \left[ {Z_{2} } \right]{\text{ dz}}$$
$$\left[ {D_{rrb} } \right]_{v} = \mathop \int \nolimits_{{h_{N + 1} }}^{{h_{N + 2} }} \left[ {Z_{2} } \right]^{T} \left[ {\overline{C}_{b}^{N + 1} } \right] \left[ {Z_{2} } \right]{\text{ dz, }}\left[ {D_{ts} } \right]_{v} = h_{v} \left[ {\overline{C}_{s}^{N + 1} } \right]$$
$$\left[ {D_{trs} } \right]_{v} = \mathop \int \nolimits_{{h_{N + 1} }}^{{h_{N + 2} }} \left[ {\overline{C}_{s}^{N + 1} } \right] \left[ {Z_{5} } \right]{\text{ dz}},\left[ {D_{rrs} } \right]_{v} = \mathop \int \nolimits_{{h_{N + 1} }}^{{h_{N + 2} }} \left[ {Z_{5} } \right]^{T} \left[ {\overline{C}_{b}^{N + 1} } \right] \left[ {Z_{5} } \right]{\text{ dz}}$$
$$\left[ {D_{tb} } \right]_{p} = h_{p} \left[ {\overline{C}_{b}^{N + 2} } \right],\left[ {D_{trb} } \right]_{p} = \mathop \int \nolimits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {\overline{C}_{b}^{N + 2} } \right] \left[ {Z_{3} } \right]{\text{ dz}}$$
$$\left[ {D_{rrb} } \right]_{p} = \mathop \int \nolimits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {Z_{3} } \right]^{T} \left[ {\overline{C}_{b}^{N + 2} } \right] \left[ {Z_{3} } \right]{\text{ dz, }}\left[ {D_{ts} } \right]_{p} = h_{p} \left[ {\overline{C}_{s}^{N + 2} } \right]$$
$$\left[ {D_{trs} } \right]_{p} = \mathop \int \nolimits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {\overline{C}_{s}^{N + 2} } \right] \left[ {Z_{6} } \right]{\text{ dz, }}\left[ {D_{rrs} } \right]_{p} = \mathop \int \nolimits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {Z_{6} } \right]^{T} \left[ {\overline{C}_{b}^{N + 2} } \right] \left[ {Z_{6} } \right]{\text{ dz}}$$
$$\left[ {D_{tbs} } \right]_{p} = \mathop \int \nolimits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {\overline{C}_{bs}^{N + 2} } \right]{\text{ dz, }}\left[ {D_{trbs} } \right]_{p} = \mathop \int \nolimits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {\overline{C}_{bs}^{N + 2} } \right] \left[ {Z_{6} } \right]{\text{ dz}}$$
$$\left[ {D_{rtbs} } \right]_{p} = \mathop \int \nolimits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {Z_{3} } \right]^{T} \left[ {\overline{C}_{bs}^{N + 2} } \right]{\text{ dz, }}\left[ {D_{rrbs} } \right]_{p} = \mathop \int \nolimits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {Z_{3} } \right]^{T} \left[ {\overline{C}_{bs}^{N + 3} } \right]\left[ {Z_{6} } \right]{\text{ dz}}$$
$$\left\{ {D_{tb} } \right\}_{p} = \mathop \int \nolimits_{{h_{N + 2} }}^{{h_{N + 3} }} - \frac{{\left\{ {\overline{e}_{b} } \right\}}}{{h_{p} }}{\text{ dz}}, \left\{ {D_{rb} } \right\}_{p} = \mathop \int \nolimits_{{h_{N + 2} }}^{{h_{N + 3} }} - \left[ {Z_{3} } \right]^{T} \frac{{\left\{ {\overline{e}_{b} } \right\}}}{{h_{p} }}{\text{ dz}}$$
$$\left\{ {D_{ts} } \right\}_{p} = \mathop \int \nolimits_{{h_{N + 2} }}^{{h_{N + 3} }} - \frac{{\left\{ {\overline{e}_{s} } \right\}}}{{h_{p} }}{\text{ dz, }}\left\{ {D_{rs} } \right\}_{p} = \mathop \int \nolimits_{{h_{N + 2} }}^{{h_{N + 3} }} - \left[ {Z_{6} } \right]^{T} \frac{{\left\{ {\overline{e}_{s} } \right\}}}{{h_{p} }}{\text{ dz}}$$

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Khan, N.M., Suresh Kumar, R. Smart control of laminated plates using Murakami zig-zag functions. Int J Mech Mater Des 17, 463–487 (2021). https://doi.org/10.1007/s10999-021-09542-0

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