Equivalent pin-forces or equivalent moments for the modelling of piezoelectric patches: a parametric study

Abstract

Abstract Many piezoelectric (PZ) actuation or sensing systems consist of PZ patches bonded on elastic beams or blades. In order to optimise the design of such systems, Finite Element Analysis (FEA) can be used. However, this option is relatively time consuming and not necessarily appropriate to the first steps of the design process. The replacement of FEA by simple analytical tools is desirable in the early design stages in order to explore the optimal configurations for the device (beam dimensions, patch position and properties). Two main modelling approaches can be found in the literature, based on the Solid Mechanics beam theory. The first approach consists in replacing the PZ patch by two opposite forces positioned at the surface of the beam, each at one extremity of the PZ patch. The second approach consists in replacing the PZ patch by two opposite moments positioned at the neutral axis of the beam, each at one extremity of the PZ patch. The object of this paper is to detail these options, and to evaluate their range of validity. For this purpose, a parametric study is conducted on a cantilever beam structure to compare the different approaches for standard dimensions and material properties. The results of corresponding FEA simulations are taken as reference. It is shown that the validity of analytical models is restricted to a narrow range of material properties and dimensions. This range is chosen when the error between the normalised displacements obtained analytically and those obtained by numerical calculation does not exceed 6%. Within this range, the two-moment model is revealed a more precise choice than two pin-forces. As a consequence, its validity range is larger compared to other analytical approaches. This is due to the introduction of the flexural stiffness of the PZ patch and of a realistic strain profile across the section of the structure. These results can be used to obtain analytical expressions of stress and strains in PZ actuation and sensing devices.