Modelling of Energy Harvesting Beams Using Dynamic Stiffness Method (DSM)

The overall aims of the chapter are twofold: (i) to present two alternatives modelling techniques for the uniform-section beam systems in Fig. 4.1 and (ii) to use these techniques in a theoretical study of a bimorph. The chapter presents the use of DSM for the solution of such a problem and illustrates its applicability to more complex problems. The DSM formulae derived are compared with the AMAM originally developed by Erturk and Inman [1] and used in the previous chapter. It is important to note that none of the works in [1] and Chap. 3, of this book, verify their modelling against an alternative method. As in [1] and Chap. 3, the Euler-Bernoulli model with piezoelectric coupling is used. However, the electrical load is a generic linear impedance, rather than just a resistor used in [1] and Chap. 3. Moreover, certain damping-related details that are unexplained or neglected in [1] and Chap. 3 are considered. In order to enable a direct comparison with the DSM, the AMAM of Erturk and Inman [1] (and Chap. 3) is reformulated to allow for a unified modelling of all three systems in Fig. 4.1. This reformulation of AMAM is a contribution in itself since the modelling in [1] is fragmented to the extent that different ad hoc formulae, notation and variable definitions are used for the three systems in Fig. 4.1. Each reformulated AMAM formula presented here covers all cases in [1]. The specialised DSM formulae similarly encompass all three systems in Fig. 4.1. Moreover, the dynamic stiffness matrix of the beam itself could be used in the modelling of beams with different boundary conditions or an assembly of uniform-section beams [2]. This is a major advantage over the AMAM approach, which is restricted to the systems in Fig. 4.1. Additionally, since DSM is based on an exact solution, less elements are required than the finite element method for an assembly of uniform-section beams, making it more accurate than finite element techniques for high-frequency applications [4]. The analysis performed in this research yields findings relating to the following issues: (i) tuning range of the energy harvester (with approximate formulae derived for the open-circuit resonances); (ii) the effect of the type of the external electrical impedance (resistive or capacitive), as well as the effect of series and parallel connection of the piezoelectric layers; (iii) the effect of damping-related assumptions; (iv) the neutralising effect of a tuned harvester on the vibration at its base; and (v) the application of boundary constraints and segmented electrodes for increased power generation [3]. The modelling procedure is described from first principles in Sect. 4.2. This is followed by a theoretical study of a cantilevered bimorph in Sect. 4.3. Section 4.4 illustrates the extension of DSM to more complex systems through an example and discusses the limitations of DSM.