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Active constrained layer damping of geometrically nonlinear vibration of rotating composite beams using 1-3 piezoelectric composite

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Abstract

In this paper, an analysis for active constrained layer damping (ACLD) of rotating composite beams undergoing geometrically non linear vibrations has been carried out. Commercially available vertically/obliquely reinforced 1-3 piezoelectric composite (PZC) material has been used as the material of the constraining layer of the ACLD treatment. A finite element (FE) model has been derived to carry out the analysis. The substrate beam is considered thin and hence, first order shear deformation theory (FSDT) and von-Karman type nonlinear strain–displacement relations are used to derive the coupled electromechanical nonlinear FE model. The rotary effect has been suitably modelled by incorporating extensional strain energy due to centrifugal force. The Golla–Hughes–McTavish method has been employed to model the constrained viscoelastic layer of the ACLD treatment in the time domain. The numerical responses revealed that the ACLD treatment with 1-3 PZC constraining layer efficiently performs the task of active damping of geometrically nonlinear vibrations of the rotating composite beams. The effects of the fibre orientation angles of the angle-ply substrate beams and the 1-3 PZC constraining layer on the ACLD of the geometrically nonlinear vibrations have been investigated. Also, the effect of the thickness variations of the 1-3 PZC layer and the viscoelastic constrained layer on the damping characteristics of the overall rotating composite beams has been studied.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, 721302, India

    D. Biswas & M. C. Ray

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  1. D. Biswas
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  2. M. C. Ray
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Correspondence to M. C. Ray.

Appendices

Appendix 1

$$ \begin{aligned} \left[ {Z_{b1} } \right] & = \left[ {\begin{array}{*{20}c} z & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \begin{array}{*{20}c} 0 & 0 & 0 \\ 1 & 0 & 0 \\ \end{array} } \right] ,\quad \quad \quad \left[ {Z_{b2} } \right] = \left[ {\begin{array}{*{20}c} \frac{h}{2} & {z - \frac{h}{2}} & 0 \\ 0 & 0 & 0 \\ \end{array} \begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 1 & 0 \\ \end{array} } \right] , \\ \left[ {Z_{b3} } \right] & = \left[ {\begin{array}{*{20}c} \frac{h}{2} & {h_{v} } & {z - \frac{h}{2} - h_{v} } \\ 0 & 0 & 0 \\ \end{array} \begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] ,\quad \left[ {Z_{s1} } \right] = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ \end{array} \begin{array}{*{20}c} { z } & 0 & 0 \\ \end{array} } \right] , \\ \left[ {Z_{s2} } \right] & = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 \\ \end{array} \begin{array}{*{20}c} \frac{h}{2} & z & 0 \\ \end{array} } \right] \quad {\text{and }}\quad \quad \quad \left[ {Z_{s3} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 & 1 \\ \end{array} \begin{array}{*{20}c} \frac{h}{2} & {h_{v} } & {z - \frac{h}{2}} \\ \end{array} - h_{v} } \right] \\ \end{aligned} $$

Appendix 2

$$ \begin{aligned} \left[ {B_{tbi} } \right] & = \left[ {\begin{array}{*{20}c} {\frac{{\partial n_{i} }}{\partial x}} & 0 \\ \end{array} } \right], \quad \quad \quad \quad \quad \left[ {B_{tsi} } \right] = \left[ {\begin{array}{*{20}c} 0 & {\frac{{\partial n_{i} }}{\partial x}} \\ \end{array} } \right], \quad \quad \quad \quad \quad \left[ {B_{nti} } \right] = \left[ { \begin{array}{*{20}c} {\frac{{\partial n_{i} }}{\partial x}} & 0 \\ 0 & {\frac{{\partial n_{i} }}{\partial x}} \\ \end{array} } \right] , \\ \left[ {B_{rbi} } \right] & = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial n_{i} }}{\partial x}} & 0 & 0 \\ 0 & {\frac{{\partial n_{i} }}{\partial x}} & 0 \\ 0 & 0 & {\frac{{\partial n_{i} }}{\partial x}} \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} {n_{i} } & 0 & 0 \\ 0 & {n_{i} } & 0 \\ 0 & 0 & {n_{i} } \\ \end{array} } \\ \end{array} } \right] \;{\text{and}}\quad \left[ {B_{rsi} } \right] = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {n_{i} } & 0 & 0 \\ 0 & {n_{i} } & 0 \\ 0 & 0 & {n_{i} } \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} {\frac{{\partial n_{i} }}{\partial x}} & 0 & 0 \\ 0 & {\frac{{\partial n_{i} }}{\partial x}} & 0 \\ 0 & 0 & {\frac{{\partial n_{i} }}{\partial x}} \\ \end{array} } \\ \end{array} } \right] \\ \end{aligned} $$

Appendix 3

$$ \left[ {D_{ttb} } \right]_{c} = \sum\limits_{k = 1}^{N} \int\limits_{{h_{k} }}^{{h_{k + 1} }} \left[ {\bar{C}_{b}^{k} } \right]dz, \quad \quad \quad \quad \quad \quad \left[ {D_{trb} } \right]_{c} = \sum\limits_{k = 1}^{N} \int\limits_{{h_{k} }}^{{h_{k + 1} }} \left[ {\bar{C}_{b}^{k} } \right]\left[ {Z_{b1} } \right]dz, $$
$$ \left[ {D_{rrb} } \right]_{c} = \sum\limits_{k = 1}^{N} \int\limits_{{h_{k} }}^{{h_{k + 1} }} \left[ {Z_{b1} } \right]^{T} \left[ {\bar{C}_{b}^{k} } \right]\left[ {Z_{b1} } \right]dz,\quad \quad \quad \quad \quad \quad \left[ {D_{tts} } \right]_{c} = \sum\limits_{k = 1}^{N} \int\limits_{{h_{k} }}^{{h_{k + 1} }} \left[ {\bar{C}_{s}^{k} } \right]dz, $$
$$ \left[ {D_{trs} } \right]_{c} = \sum\limits_{k = 1}^{N} \int\limits_{{h_{k} }}^{{h_{k + 1} }} \left[ {\bar{C}_{s}^{k} } \right]\left[ {Z_{s1} } \right]dz , \quad \left[ {D_{rrb} } \right]_{c} = \sum\limits_{k = 1}^{N} \int\limits_{{h_{k} }}^{{h_{k + 1} }} \left[ {Z_{b1} } \right]^{T} \left[ {\bar{C}_{s}^{k} } \right]\left[ {Z_{s1} } \right]dz, $$
$$ \left[ {D_{tts} } \right]_{c} = \sum\limits_{k = 1}^{N + 2} \int\limits_{{h_{k} }}^{{h_{k + 1} }} \left[ {\bar{C}_{s}^{k} } \right]dz, \quad \left[ {D_{trs} } \right]_{c} = \sum\limits_{k = 1}^{N} \int\limits_{{h_{k} }}^{{h_{k + 1} }} \left[ {\bar{C}_{s}^{k} } \right]\left[ {Z_{s1} } \right]dz, $$
$$ \\ \left[ {D_{rrb} } \right]_{c} = \sum\limits_{k = 1}^{N} \int\limits_{{h_{k} }}^{{h_{k + 1} }} \left[ {Z_{b1} } \right]^{T} \left[ {\bar{C}_{s}^{k} } \right]\left[ {Z_{s1} } \right]dz ,\quad \quad \quad \quad \quad \quad \left[ {D_{rtb} } \right]_{c} = \left[ {D_{trb} } \right]_{c}^{T} $$
$$ \left[ {D_{ttbs} } \right]_{v} = \int\limits_{{h_{N + 1} }}^{{h_{N + 2} }} dz = h_{v} ,\quad \quad \quad \quad \quad \quad \left[ {D_{trbs} } \right]_{v} = \int\limits_{{h_{N + 1} }}^{{h_{N + 2} }} \left[ {Z_{s2} } \right]dz , $$
$$ \left[ {D_{rtbs} } \right]_{v} = \left[ {D_{trb} } \right]_{v}^{T} ,\quad \quad \quad \quad \quad \left[ {D_{trbs} } \right]_{v} = \int\limits_{{h_{N + 1} }}^{{h_{N + 2} }} \left[ {Z_{s2} } \right]^{T} \left[ {Z_{s2} } \right]dz, $$
$$ \left[ {D_{ttb} } \right]_{p} = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {\bar{C}_{pb1} } \right]dz, \quad \quad \quad \quad \left[ {D_{trb} } \right]_{p} = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {\bar{C}_{pb1} } \right]\left[ {Z_{b3} } \right]dz, $$
$$ \left[ {D_{rrb} } \right]_{p} = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {Z_{b3} } \right]^{T} \left[ {\bar{C}_{pb1} } \right]\left[ {Z_{b3} } \right]dz,\quad \quad \quad \quad \left[ {D_{rtb} } \right]_{p} = \left[ {D_{trb} } \right]_{p}^{T} , $$
$$ \left[ {D_{tts} } \right]_{p} = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \bar{C}_{55} dz , \quad \quad \quad \quad \left[ {D_{trs} } \right]_{p} = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \bar{C}_{55} \left[ {Z_{s3} } \right]dz, $$
$$ \left[ {D_{rrb} } \right]_{p} = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {Z_{s3} } \right]^{T} \bar{C}_{55} \left[ {Z_{s3} } \right]dz,\quad \quad \quad \quad \left[ {D_{rts} } \right]_{p} = \left[ {D_{trs} } \right]_{p}^{T} , $$
$$ \left[ {D_{ttbs1} } \right]_{p} = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {\bar{C}_{pb2} } \right]dz,\quad \quad \quad \quad \left[ {D_{trbs1} } \right]_{p} = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {\bar{C}_{pb2} } \right]\left[ {Z_{s3} } \right]dz , $$
$$ \left[ {D_{rtbs1} } \right]_{p} = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {Z_{b3} } \right]^{T} \left[ {\bar{C}_{pb2} } \right]dz, \quad \quad \quad \quad \left[ {D_{rrbs1} } \right]_{p} = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {Z_{b3} } \right]^{T} \left[ {\bar{C}_{pb2} } \right]\left[ {Z_{s3} } \right]dz, $$
$$ \left[ {D_{ttsb2} } \right]_{p} = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {\bar{C}_{ps1} } \right]dz , \quad \quad \quad \quad \left[ {D_{trsb2} } \right]_{p} = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {\bar{C}_{ps1} } \right]\left[ {Z_{b3} } \right]dz, $$
$$ \left[ {D_{rtsb2} } \right]_{p} = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {Z_{s3} } \right]^{T} \left[ {\bar{C}_{ps1} } \right]dz,\quad \quad \quad \quad \left[ {D_{rrsb2} } \right]_{p} = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {Z_{s3} } \right]^{T} \left[ {\bar{C}_{ps1} } \right]\left[ {Z_{b3} } \right]dz, $$
$$ \left[ {D_{ttsb1} } \right]_{p} = \left[ {D_{ttbs1} } \right]_{p}^{T} ,\;\left[ {D_{trsb1} } \right]_{p} = \left[ {D_{rtbs1} } \right]_{p}^{T} , $$
$$ \left[ {D_{rtsb1} } \right]_{p} = \left[ {D_{trbs1} } \right]_{p}^{T} , \;\left[ {D_{rrsb1} } \right]_{p} = \left[ {D_{rrbs1} } \right]_{p}^{T} , $$
$$ \left[ {D_{ttbs2} } \right]_{p} = \left[ {D_{ttsb2} } \right]_{p}^{T} , \;\left[ {D_{trbs2} } \right]_{p} = \left[ {D_{rtsb2} } \right]_{p}^{T} , $$
$$ \left[ {D_{rtbs2} } \right]_{p} = \left[ {D_{trsb2} } \right]_{p}^{T} ,\; \left[ {D_{rrbs2} } \right]_{p} = \left[ {D_{rrsb2} } \right]_{p}^{T} , $$
$$ \left[ {D_{tbp} } \right] = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left\{ {\bar{e}_{b} } \right\}\frac{1}{{h_{p} }}dz , \quad \quad \quad \quad \left[ {D_{rbp} } \right] = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {Z_{b3} } \right]^{T} \left\{ {\bar{e}_{b} } \right\}\frac{1}{{h_{p} }}dz, $$
$$ \left[ {D_{tsp} } \right] = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \bar{e}_{35} \frac{1}{{h_{p} }}dz,\quad \quad \quad \quad \left[ {D_{rsp} } \right] = \int\limits_{{h_{N + 2} }}^{{h_{N + 3} }} \left[ {Z_{s3} } \right]^{T} \bar{e}_{35} \frac{1}{{h_{p} }}dz, $$

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Biswas, D., Ray, M.C. Active constrained layer damping of geometrically nonlinear vibration of rotating composite beams using 1-3 piezoelectric composite. Int J Mech Mater Des 9, 83–104 (2013). https://doi.org/10.1007/s10999-012-9207-5

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