Pressure Measurement Sensor for Jointed Structures

Abstract

The talk will deal with measurements of normal loads in assembled structures. A new sensor has been developed to measure the distribution of the normal force in a bolted joint. This sensor is piezoelectric and is associated to an electronic device for carrying out a static measurement. Moreover, the electrode of the sensor is shaped in order to make possible the measurement of a detailed normal stress field. Furthermore, preliminary results of the study of the vibrations are presented.

Appendix: Piezoelectric Materials

This section summarizes the datas provided by PI and the datas that are used for finite element simulation.

41.1.1 Manufacturer Datas

The manufacturers provide incomplete and inhomogeneous datas. The following summarizes the datas given by PI :

• Densityρ = 7, 800 kg/m³

• Strain piezoelectric constants$\begin{array}{ccc} d_{31} = -180 \cdot { 10}^{-12}\;\text{C/N}&d_{33} = 400 \cdot { 10}^{-12}\;\text{C/N}&d_{15} = 550 \cdot { 10}^{-12}\;\text{C/N}\\ \end{array}$

• Dielectric constants$\begin{array}{ccc} \in _{33}^{T}\text{=1750}\in _{0}& \in _{11}^{T}\text{=1650}\in _{0}&\in _{0}\text{=8}.854 \cdot 1{0}^{12}\\ \end{array} \;\text{F/m}$

• Elastic compliance at constant electric field$\begin{array}{cc} S_{11}^{E} = 16.1 \cdot { 10}^{-12}\;{\text{m}}^{2}\text{/N}&S_{33}^{E} = 20.7 \cdot { 10}^{-12}\;{\text{m}}^{2}\text{/N}\\ \end{array}$

• Material coupling factor (IEEE definition, see [11])$\begin{array}{*{20}{c}} k_{31} = 0.35&k_{33} = 0.69 \\ k_{15} = 0.66& k_{p} = 0.62\end{array}$

41.1.2 ABAQUS Datas

Parameters are given for an assumed p thickness polarized piezoceramic material in the coordinate frame R ₀.

• Densityρ = 7, 800 kg/m³

• Strain piezoelectric constants $d = \left [\begin{array}{cccccc} 0 & 0 & 0 &0&d_{15} & 0 \\ 0 & 0 & 0 &0& 0 &d_{15} \\ d_{31} & d_{31} & d_{33} & 0& 0 & 0\\ \end{array} \right ]$ where all the parameters are given by PI

• Elastic compliance at constant electric field ${S}^{E} = \left [\begin{array}{cc} \begin{array}{ccc} S_{11}^{E}&S_{12}^{E}&S_{12}^{E} \\ S_{12}^{E}&S_{11}^{E}&S_{13}^{E} \\ S_{12}^{E}&S_{13}^{E}&S_{33}^{E}\\ \end{array} & 0 \\ 0 &\begin{array}{ccc} S_{55}^{E}& & \\ &S_{55}^{E}& \\ & &S_{66}^{E}\\ \end{array}\\ \end{array} \right ]$ with $\begin{array}{*{35}{l}} S_{55}^{E} = \frac{d_{15}^{2}} {\in _{11}^{T}k_{15}^{2}} \\ S_{12}^{E} = -S_{11}^{E} + 2 \frac{d_{31}^{2}} {\in _{33}^{T}k_{p}^{2}} \\ S_{13}^{E} = -\nu _{13}^{E}S_{11}^{E} \\ S_{66}^{E} = 2\left (S_{11}^{E} - S_{12}^{E}\right )\\ \end{array}$

• Engineering constants at constant electric field $E_{i} = 1/S_{pp}^{E}$ for i = { 1, 2, 3} and p = { 1, 2, 3}$G_{ij} = 1/S_{pp}^{E}$ for i = { 2, 1, 1},j = { 3, 3, 2} and p = { 4, 5, 6}

• Dielectric constants $e = d{\left ({S}^{E}\right )}^{-1}$ and ${\in }^{S} ={ \in }^{T} - d{e}^{t}$ $\begin{array}{cc} \in _{11}^{S}\text{=0}.8245 \cdot { 10}^{-9}\;\text{m/F}& \in _{33}^{S}\text{=0}.7122 \cdot { 10}^{-9}\;\text{m/F}\\ \end{array}$