Vibration damping in sandwich panels

Abstract

Currently, there is incomplete knowledge of the damping level and its sources in satellite structures and a suitable method to model it constitutes a necessary step for reliable dynamic predictions. As a first step of a damping characterization, the damping of honeycomb structural panels, which is identified as a main contributor to global damping, has been considered by ALCATEL SPACE. In this work, the inherent vibration damping mechanism in sandwich panels, including those with both aluminium and carbon fibre-reinforced plastic (CFRP) skins, is considered. It is first shown how the theoretical modal properties of the sandwich panel can be predicted from the stiffness and damping properties of its constituent components using the basic laminate theory, a first-order shear deformation theory and a simple discretization method. Next, a finite-element transcription of this approach is presented. It is shown to what extent this method can be implemented using a finite-element software package to predict the overall damping value of a sandwich honeycomb panel for each specific mode. Few of the many theoretical models used to predict natural frequencies of plates are supported by experimental data and even fewer for damping values. Therefore, in a second, experimental part, the Rayleigh–Ritz method and NASTRAN (finite-element software used by ALCATEL SPACE) predicted modal characteristics (frequency and damping) are compared with the experimentally obtained values for two specimens of typical aluminium core honeycomb panels (aluminium and CFRP skins) used by ALCATEL SPACE as structural panels. It is shown through these results that the method (theoretical and finite element) is satisfactory and promising.

Appendix

Components of the Rayleigh–Ritz stiffness matrix

$$ C_{11} =\sum_{i=1}^M {\sum_{j=1}^N \left\{A_{44} \int_a {w_i w_m {\rm d}x} \;\int_b {w_i^{\prime} w_n^{\prime} {\rm d}y} + A_{55} \int_a {w_i^{\prime} w_m^{\prime} {\rm d}x} \;\int_b {w_j w_n {\rm d}y} +A_{45}\left[\int_a {w_i^{\prime} w_m {\rm d}x} \;\int\nolimits_b {w_j w_n^{\prime} {\rm d}y} + \int\nolimits_a {w_m^{\prime} w_i {\rm d}x} \;\int\nolimits_b {w_n w_j^{\prime} {\rm d}y} \right]\;\right\}} $$

$$ C_{12}=\sum_{i=1}^M {\sum_{j=1}^N { \left\{A_{55} \int\nolimits_a {\psi_m w_i^{\prime} {\rm d}x} \;\int\nolimits_b {w_n w_j {\rm d}y} + A_{45} \int\nolimits_a {\psi_m w_i {\rm d}x} \;\int\nolimits_b {w_n w_j^{\prime} {\rm d}y} \right\}} } $$

$$ C_{13} =\sum_{i=1}^M {\sum_{j=1}^N { \left\{A_{44} \int\nolimits_a {w_m w_i {\rm d}x} \;\int\nolimits_b {\psi_n w_j^{\prime} {\rm d}y} + A_{45} \int\nolimits_a {w_m w_i^{\prime} {\rm d}x} \;\int\nolimits_b {\psi_n w_j {\rm d}y}\right\}} } $$

$$ C_{21} =\sum_{i=1}^M {\sum_{j=1}^N { \left \{\;A_{55} \int\nolimits_a {\psi_i w_m^{\prime} {\rm d}x} \;\int\nolimits_b {w_j w_n {\rm d}y} + A_{45} \int\nolimits_a {\psi_i w_m {\rm d}x} \;\int\nolimits_b {w_j w_n^{\prime} {\rm d}y} \right \}} } $$

$$ C_{22} =\sum_{i=1}^M {\sum_{j=1}^N \left\{\;D_{11} \int\nolimits_a {\psi_i^{\prime} \psi_m^{\prime} {\rm d}x} \;\int\nolimits_b {w_j w_n {\rm d}y} + D_{16} \left[\int\nolimits_a {\psi_i^{\prime} \psi_m {\rm d}x} \;\int\nolimits_b {w_j w_n^{\prime} {\rm d}y} +\int\nolimits_a {\psi_m^{\prime} \psi_i {\rm d}x} \;\int\nolimits_b {w_n w_j^{\prime} {\rm d}y}\right] + D_{66} \int\nolimits_a {\psi_i \psi_m {\rm d}x} \;\int\nolimits_b {w_j^{\prime} w_n^{\prime} {\rm d}y} + A_{55} \int\nolimits_a {\psi_i \psi_m {\rm d}x} \;\int\nolimits_b {w_j w_n {\rm d}y}\right\}} $$

$$ C_{23}= \sum_{i=1}^M {\sum_{j=1}^N \left\{\;D_{12} \int\nolimits_a {\psi_i^{\prime} w_m {\rm d}x} \;\int\nolimits_b {\psi_n^{\prime} w_j {\rm d}y} + D_{16} \int\nolimits_a {\psi_i^{\prime} w_m^{\prime} {\rm d}x} \;\int\nolimits_b {w_j \psi_n {\rm d}y}+D_{26} \int\nolimits_a {w_m \psi_i {\rm d}x} \int\nolimits_b {\psi_n^{\prime} w_j^{\prime} {\rm d}y} + D_{66} \int\nolimits_a {\psi_i w_m^{\prime} {\rm d}x} \;\int\nolimits_b {w_j^{\prime} \psi_n {\rm d}y} +A_{45} \int\nolimits_a {\psi_i w_m {\rm d}x} \;\int\nolimits_b {w_j \psi_n {\rm d}y}\right\}} $$

$$ C_{31} =\sum_{i=1}^M {\sum_{j=1}^N { \left\{\;A_{44} \int\nolimits_a {w_i w_m {\rm d}x} \;\int\nolimits_b {\psi_j w_n^{\prime} {\rm d}y} + A_{45} \int\nolimits_a {w_i w_m^{\prime} {\rm d}x} \;\int\nolimits_b {\psi_j w_n {\rm d}y} \right\}} } $$

$$ C_{32} = \sum_{i=1}^M {\sum_{j=1}^N \left\{D_{12} \int\nolimits_a {\psi_m^{\prime} w_i {\rm d}x} \;\int\nolimits_b {\psi_j^{\prime} w_n {\rm d}y} + D_{16} \int\nolimits_a {\psi_m^{\prime} w_i^{\prime} {\rm d}x} \;\int\nolimits_b {w_n \psi_j {\rm d}y} + D_{26} \int\nolimits_a {w_i \psi_m {\rm d}x} \;\int\nolimits_b {\psi_j^{\prime} w_n^{\prime} {\rm d}y} + D_{66} \int\nolimits_a {\psi_m w_i^{\prime} {\rm d}x} \;\int\nolimits_b {w_n^{\prime} \psi_j {\rm d}y} + A_{45} \int\nolimits_a {\psi_m w_i {\rm d}x} \;\int\nolimits_b {w_n \psi_j {\rm d}y}\right\}}$$

$$ C_{33} = \sum_{i=1}^M {\sum_{j=1}^N { \left\{\;D_{22} \int\nolimits_a {w_i w_m {\rm d}x} \;\int\nolimits_b {\psi_j^{\prime} \psi_n^{\prime} {\rm d}y} + D_{26} \left[\int\nolimits_a {w_i w_m^{\prime} {\rm d}x} \;\int\nolimits_b {\psi_j^{\prime} \psi_n {\rm d}y} +\int\nolimits_a {w_m w_i^{\prime} {\rm d}x} \;\int\nolimits_b {\psi_n^{\prime} \psi_j {\rm d}y}\right] + D_{66} \int\nolimits_a {w_i^{\prime} w_m^{\prime} {\rm d}x} \;\int\nolimits_b {\psi_j \psi_n {\rm d}y} + A_{44} \int\nolimits_a {w_i w_m {\rm d}x} \;\int\nolimits_b {\psi_j \psi_n {\rm d}y}\right\}}} \quad\quad\quad\quad\quad\quad(m=1,\;2,\;\ldots \; M;\;\;n=1,\;2,\;\ldots \; N) $$

Components of the Rayleigh–Ritz mass matrix

$$ \begin{aligned} m_{11}&=\sum_{i=1}^M {\sum_{j=1}^N {\rho h \left[\int\nolimits_a {w_i w_m {\rm d}x} \;\int\nolimits_b {w_j w_n {\rm d}y} \right]} } \\ m_{22}&=\sum_{i=1}^M {\sum_{j=1}^N {\rho \frac{h^{3}}{12} \left[\int\nolimits_a {\psi_i \psi_m {\rm d}x} \;\int\nolimits_b {w_j w_n {\rm d}y} \right]} } \\ m_{33}&=\sum_{i=1}^M {\sum_{j=1}^N {\rho \frac{h^{3}}{12} \left[\int\nolimits_a {w_i w_m {\rm d}x} \;\int\nolimits_b {\psi_j \psi_n {\rm d}y} \right]} } \\ &\quad\quad\quad\quad\quad\quad(m=1,\;2,\;\ldots\;M;\;\;n=1,\;2,\;\ldots\;N) \\ \end{aligned} $$