Designing distributed modal sensors for plate structures using finite element analysis

doi:10.1016/j.ymssp.2005.05.010

Mechanical Systems and Signal Processing

Volume 20, Issue 8, November 2006, Pages 2290-2304

https://doi.org/10.1016/j.ymssp.2005.05.010 Get rights and content

Abstract

Designing constant thickness, distributed modal transducers for plate structures is difficult. One option is to alter sensor effectiveness by changing the electrode pattern. This paper considers an alternative approach where the shape of the sensor is optimised, and parameterises the sensor boundary using a limited number of boundary points and spline interpolation. The modes are obtained from a finite element model of the plate structure, and used to create an objective function to design modal sensors. The approach is demonstrated on a plate structure with various boundary conditions.

Introduction

Modal sensors and actuators for beam and plate structures are intended to excite or measure a single mode of a structure [1], [2]. Clark et al. [3] gave a concise tutorial on modal transducers and applications. In active control, modal sensors reduce the problem of spillover, where high-frequency unmodelled modes affect the stability of the closed-loop system. The sensors and actuators may consist of a number of discrete transducers or alternatively distributed transducer material. For discrete transducers, the location and gain is determined for each discrete point transducer and established techniques for this are available [4], [5]. Distributed transducers essentially perform the signal processing by using the charge collecting phenomenon of the material to implement a modal transducer in the spatial domain, and this spatial filtering reduces the signal processing requirements. The transducers are usually based on piezoelectric material, and polyvinylidene fluoride (PVDF) film has been widely used because electrode pattern shaping is relatively easy and the film is flexible and has low mass. In the case of one-dimensional structures, the gain distribution can be easily implemented by varying the width of the PVDF film along the primary axis of the structure [1]. If the transducer covers the whole length of the beam then mode shape orthogonality may be used to design the transducer. Friswell [6] considered modal sensors that cover only part of the beam, and segmented modal sensors for multiple modes. Friswell [5] used finite element shape functions to design modal sensors and actuators for Euler–Bernoulli beams.

For two-dimensional structures, the approach used for beams may be implemented by varying the thickness of the PVDF, although this is very difficult to achieve in practice. Sun et al. [7] replaced an actuator layer with variable thickness by many small segments of uniform thickness. Kim et al. [8], [9] developed two design methods for distributed modal transducers for composite plates, the first using multi-layered PVDF films with optimised electrode pattern, lamination angle, and poling direction, and the second using PVDF film segments and an interface circuit. Preumont et al. [10] introduced the porous electrode concept, which allows the gains to be introduced by changing the local effectiveness of the electrodes.

The previous approaches to design modal sensors for plate structures have complicated electrode patterns, variable thickness or require extra interface circuits. The alternative is to design a distributed modal transducer by optimising the continuous shape of a constant thickness PVDF film by assuming a smooth boundary. In this paper the continuous boundary of a sensor is parameterised and the parameters optimised to produce the required modal output. To calculate the modal output of a sensor, the mode shape of the plate structure must be known, and for plate structures with complex boundary conditions it is difficult or impossible to obtain these mode shapes analytically. Finite element analysis is usually used, and this is the approach adopted in this paper. Thus distributed modal transducers for arbitrary plate structures may be designed.

Section snippets

Sensor output to modal response

Fig. 1 shows a plate with a piezoelectric film of uniform thickness T attached to the upper surface over the area $Ω$ . If the thickness of the piezoelectric film is negligible compared to the thickness of the plate, then the strain in the piezoelectric film is identical to the strain on the surface of the plate. The total charge generated in the piezoelectric film is [3], [8] $q (t) = - \frac{hT}{2} \iint_{Ω} χ (x, y) (e_{31} \frac{\partial^{2} w}{\partial x^{2}} + e_{32} \frac{\partial^{2} w}{\partial y^{2}} + 2 e_{36} \frac{\partial^{2} w}{\partial x \partial y}) d x d y,$ where $e_{31}$ , $e_{32}$ and $e_{36}$ are the piezoelectric strain/charge constants, h

Parameterisation of the sensor shape and the calculation of modal sensitivity

The calculation of the sensitivity of the charge output to the ith modal coordinate, $P_{i}$ , is more conveniently computed by integrating over the boundary rather than the sensor area. Green's theorem in the plane, $\oint_{\partial Ω} f (x, y) d x + g (x, y) d y = \iint_{Ω} (\frac{\partial g (x, y)}{\partial x} - \frac{\partial f (x, y)}{\partial y}) d x d y,$ may be used, where $\partial Ω$ is the simple closed boundary curve of the continuous area $Ω$ . Applying Green's theorem to Eq. (4) gives $P_{i} = e \oint_{\partial Ω} χ (x, y) (\frac{\partial φ_{i} (x, y)}{\partial x} d y - \frac{\partial φ_{i} (x, y)}{\partial y} d x) .$ For small deformations, $θ_{i} (x, y) = \frac{\partial φ_{i} (x, y)}{\partial y}, ψ_{i} (x, y) = - \frac{\partial φ_{i} (x, y)}{\partial x},$ where $θ_{i} (x, y)$

The design of distributed modal sensors

Eq. (3) shows that a perfect modal sensor for the kth mode, requires that the sensor output is insensitive to the other modes. Thus, $P_{i} = \{\begin{matrix} \neq 0 & for i = k, \\ = 0 & otherwise . \end{matrix}$ These properties have to be met for the modes of interest by shaping the piezoelectric film. The first step is to choose the number of points on the boundary curve of the sensor $\partial Ω$ , shown in Fig. 1. The positions of these points are the unknown parameters that must be identified. The previous section demonstrated how the modal sensitivities,

Examples

The proposed approach will be demonstrated by designing modal sensors for a rectangular plate with different boundary conditions. The plate has length $a = 200 mm$ , width $b = 100 mm$ and thickness $h = 1 mm$ , and is made of steel, with Young's modulus $E = 200 GPa$ , Poisson's ratio $ν = 0.3$ and mass density $ρ = 7800 kg / m^{3}$ . We assume that $e_{31} = e_{32} = e$ and $e_{36} = 0$ . It is clear that the piezoelectric strain/charge constants do not affect the shape of modal sensors, and so we set $e = 1$ in the calculation of modal sensitivity. In

Conclusion

This paper has presented a method to design distributed modal sensors for plate structures with arbitrary boundary conditions by using the finite element method. The approach assumes the thickness of the sensor is constant and optimises the shape of the sensor boundary. This allows for sensor patches with smooth boundaries. The use of the finite element method allows distributed modal sensors to be designed for arbitrary plate structures. Although the paper has demonstrated the trade off

Acknowledgements

Kailin Jian gratefully acknowledge the financial support to this work from Visiting Scholar Foundation of State Key Laboratory of Mechanical Transmission of Chongqing University. Michael Friswell gratefully acknowledges the support of the Royal Society through a Royal Society-Wolfson Research Merit Award.

References (14)

H. Irschik
A review on static and dynamic shape control of structures by piezoelectric actuation
Engineering Structures
(2002)
W. Gawronski
Modal actuators and sensors
Journal of Sound and Vibration
(2000)
M.I. Friswell
On the design of modal actuators and sensors
Journal of Sound and Vibration
(2001)
J. Kim et al.
Optimal gain distribution for two-dimensional modal transducer and its implementation using multi-layered PVDF films
Journal of Sound and Vibration
(2002)
A. Preumont et al.
Spatial filters in structural control
Journal of Sound and Vibration
(2003)
C.-K. Lee et al.
Modal sensors/actuators
ASME Journal of Applied Mechanics
(1990)
R.L. Clark et al.
Adaptive Structures: Dynamics and Control
(1998)

There are more references available in the full text version of this article.

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Designing distributed modal sensors for plate structures using finite element analysis

Abstract

Introduction

Section snippets

Sensor output to modal response

Parameterisation of the sensor shape and the calculation of modal sensitivity

The design of distributed modal sensors

Examples

Conclusion

Acknowledgements

Engineering Structures

Journal of Sound and Vibration

Journal of Sound and Vibration

Journal of Sound and Vibration

Journal of Sound and Vibration

Modal sensors/actuators

ASME Journal of Applied Mechanics

Adaptive Structures: Dynamics and Control