## 1 Introduction  
Nowadays, energy harvesting and, in general, the research for alternative sources of energy represents a topic of great interest. In fact, the continuous and growing development of new electronic technologies is leading to the spread of an enormous number of new generation portable devices that denote an important change and improvement in the wellness of the humankind. In addition to the well-known smartphones and tablets, examples of new generation portable devices are linked to the Internet of Things and the possibility of making smart remote controlling and monitoring equipment for different applications like home automation [1, 2], or the new wearable devices for monitoring biomedical parameters or sport activities [3, 4], or wireless sensor networks (WSNs) and autonomous sensors for environmental monitoring [5, 6, 7, 8, 9, 10, 11] or for improving the supply chain and the efficiency of industrial warehouses. Although the quality of life has greatly improved in this sense, this incessant diffusion of portable electron devices leads to an increase in the need for energy sources capable of sustaining their functionality. Batteries have always been considered the primary source of portable energy and therefore many technological advances have been made in order to improve their performance, in terms of capacity per unit of volume and reliability. However, there are many negative aspects linked to such massive use of batteries for portable applications. First of all, disposal and environmental impact. Although eco-sustainable technologies have recently been developed [12], the vast majority of batteries currently in circulation constitute a danger to human health and the environment, as they consist of toxic materials that are difficult to dispose of [13], and an increase in their diffusion would intensify this problem. Secondly, the batteries have a limited duration in time, so they need maintenance for recharging or for replacing them, constituting a cost as well as a limitation for portable devices that benefit from it, especially those sensory autonomous devices used, for example, environmental monitoring, that is dislocated in hostile places and of difficult access. Therefore, a possible solution to this problem, currently of great scientific interest, is represented by energy harvesting from low-cost alternative sources [14; 15; 16; 17; 18; 19], with zero environmental impact, that can directly supply ideally infinite energy to portable electronic devices, or at least assist the use of the batteries, prolonging their life and therefore reducing their disposal over time. There are many alternative energy sources and the related conversion techniques used to date for this purpose. Solar energy, for example, has always been considered one of the most important alternative fonts for both low-power and high-power applications. In this sense, the development of new materials and new technologies for recovering solar energy through photovoltaic panels is constantly evolving, to further improve conversion efficiency and reduce costs [20; 21; 22]. Another promising energy source is vibrational energy, generated for example by natural oscillations of structures such as bridges, frames of industrial equipment, but also by the very movement of the human body [23; 24; 25]. The most adopted conversion techniques of this energy are cantilevers with electromagnetic induction or the use of piezoelectric materials. The latter exploit a particular property for which by mechanically altering the crystal lattice of the material itself, a potential difference develops at its ends which is transformed into electric current if closed on a load or, vice versa, it is possible to compress or expand the piezoelectric material subjecting it to a proper voltage. This last effect was the first to be exploited, in the realization of actuators as speakers, but recently is being studied for the possibility of applying piezoelectric materials in contact with vibration sources to recover energy from them.