Identification of the shock load on an electroelastic bimorph disk

Abstract

A numerical-analytic method for the identification of the axisymmetric mechanical shock load on a disk-shaped metal-piezoceramic bimorph transducer is proposed. A problem is formulated based on the theory of thin two-layer plates. The solution is found using the Laplace transform. By recovering the original function analytically, the problem is reduced to a system of Volterra equations, solved numerically using Tikhonov's regularization algorithm. The finite-element solution of the direct problem is used as input data (potential difference between the electrodes of the piezoceramic layer). The results are analyzed Keywords: electroelasticity, asymmetric bimorph disk, nonstationary process, load identification, Laplace transformIntroduction. Electroelastic bimorph transducers with thin metal and piezoceramic layers are widely used nowadays. In [1,5,6], the equations of vibrations of solid and stepped-layer bimorph disks are derived and solved assuming that the dynamic process is time-periodic. Nonstationary problems of electroelasticity are important because piezoelectric transducers have enhanced functionality in transient modes. The unsteady vibrations of a two-layer beam and strip under electric or mechanical loads are studied in [2,3]. The papers [9, 10] propose ways to identify a signal waveform that keeps a bimorph beam in a nearly undeformed state under a given shock load (control problem) and to approximate the form of external mechanical load, given the potential difference between the electrodes of the piezoceramic layer (identification problem).We will solve the inverse problem, i.e., identify the shock load on an electroelastic two-layer disk in the mode of direct piezoelectric effect.1. Problem Formulation. Consider a two-layer (metal + piezoceramic) circular plate (asymmetric bimorph disk). The layers are perfectly bonded, have radius R and thickness h m and h p (Fig. 1). We will hereafter use the indices "m" and "p" to refer to the metal and piezoceramic layers, respectively. At time t = 0, the outside surface of the elastic layer starts experiencing a nonstationary load P uniformly distributed inside a circle of radius r 0 : P(r, t) = p(t)H(t)H(r 0 -r), where H is the Heaviside function, r is the radial coordinate. The piezoelectric element is polarized throughout the thickness and has solid conductive coatings whose mass and stiffness can be neglected. The inside electrode z = z 0 (z is the axial coordinate) is grounded. A datum plane for the plate (z = 0) will be selected below. We assume, without loss of generality, that the disk is hinged at the edge (r = R).Let the structural element under consideration be thin and the Kirchhoff-Love hypotheses can be used to describe its motion. Moreover, we assume that the normal component of the electric-flux density is constant within the piezoceramic layer and that Poisson's ratios of the constitutent materials of the bimorph are equal (n n n

Dynamic Force Identification Problem Based on a Novel Improved Tikhonov Regularization Method

“…Reference [31] demonstrated that the Tikhonov regularization method can solve the inverse problem in a stable manner despite the presence of noisy data. Reference [32] identified the shock load on an electroelastic bimorph disk using the Tikhonov regularization method. However, the Tikhonov regularization method is not completely perfect; there are numerous unavoidable limitations and disadvantages, which is described as follows: (a) the approximate solution provided by Tikhonov regularization method is too smooth; (b) the approximate solution provided by Tikhonov regularization method may lack some details that the desired real solution might possess.…”