Chapter 1 Mathematical Modeling of Piezoelectric Generators In recent years, the research on piezoelectric transducers of mechanical energy into electrical energy has been actively developed. This type of transducer is called a piezoelectric generator (PEG). Basic information about PEGs, as well as the prob- lems arising at the development stages of energy harvesting devices, was provided in the review papers [97, 95, 38, 73], as well as in the monographs [21, 60, 62]. PEGs are manufactured in two different configurations: stack and cantilever. PEGs have a broad application, e.g., for piezoelectric damping of vibrations [13, 15, 145, 143]. Most researchers have studied the characteristics of the cantilever-type PEGs. There are several ways of modeling PEGs: (i) mathematical model with lumped parameters, (ii) mathematical model with distributed parameters, and (iii) finite element model. Works [22, 59, 58, 3, 143, 144] are devoted to the construction of PEG models based on oscillations of a mechanical system with lumped parameters. The use of such systems is a convenient model approach, since researchers can obtain analytical dependencies between the output parameters of PEG (voltage, power, etc.) and electro-mechanical characteristics, and also electrical resistance of the external electric circuit. The modeling with lumped parameters provides initial representations on the problem, allowing one to use simple expressions for the description of the system. However, it is approximate and restricted by only one oscillation mode. This description does not take into account other important aspects of the system. Another type of modeling is distributed parameter modeling. Based on the Euler- Bernoulli hypothesis for beams, analytical solutions of the coupled problem have been obtained in [61, 53, 158, 157] for different configurations of cantilever-type PEGs. The authors obtained clear expressions for the output voltage on resistive electric load and for console displacements. Moreover, they studied, in detail, behaviors of PEGs with short-circuited and open-circuited electric circuits, and the influence of the piezoelectric couple effect and flexoelectric effect [53, 158]. © Springer International Publishing AG, part of Springer Nature 2018 S. N. Shevtsov et al., Piezoelectric Actuators and Generators for Energy Harvesting, Innovation and Discovery in Russian Science and Engineering, https://doi.org/10.1007/978-3-319-75629-5_1 1