## 1 Introduction  
Piezoelectric actuators are transducers that convert electrical energy into mechanical displacement or stress. Owing to their ease of use and ability to generate huge forces and accurate displacements, they have found usage in a variety of sectors, including ultra-precision positioning and the creation and management of forces or pressures in static and dynamic domains. Piezoelectric actuators have a wide frequency range and may be employed in a variety of setups [1, 2]. All of these characteristics make piezoelectric actuators very attractive in a wide range of applications; for instance, their use as prevention systems for ice formation in aircraft structures represents a potential solution that has been studied in the literature in recent years [3, 4, 5, 6]. The formation of ice on aircraft structures has been considered a dangerous phenomenon posing a significant risk to the safety of the flight. Wings, blades, and aerodynamic surfaces in general suffer from this issue, which leads to losses in performance and safety for the aircraft [7, 8, 9]. The mechanism of the ice formation on the airframe is quite simple. It is known from the literature that liquid water can exist in an unstable condition at temperatures below zero degrees Celsius [10]. If the temperatures of the clouds range from 0 to \(-20\)\({}^{\circ}\)C, super-cooled water can exist, even in many g/m\({}^{3}\). Under these conditions, when the droplets impact the surfaces of the aircraft structures they immediately freeze on the skin, thereby promoting ice formation and growth [11, 12, 13]. Icing reduces the aircraft's efficiency by increasing its weight, reducing lift, decreasing thrust, and increasing drag. The growth of the ice produces an increase in the roughness of the wing surface that results in airfoil distortion and boundary layer transition. This means that both the friction and the pressure drag increase, with additional fuel consumption,while the lift and critical angle of attack decrease due to ice formation. Furthermore, theicing on the horizontal tail seriously damages the stability and manipulation of airplanes. Several accidents had occurred before authorities figured out the importance of icing formation as a main cause of them [9]. To preserve aircrafts from the dangerous effects of ice formation, various anti- or de-icing systems have been developed over the years. The state of the art of these technologies has been in continuous development since the 1980s when the authorities started to implement new regulations concerning on-ground and on-flight procedures. The more traditional systems include electro-thermal and hot bleed air de-icing or anti-icing methods, fluid anti-icing techniques, hydrophobic coating anti-icing methods, and mechanical de-icing methods (electro-vibratory, microwave, shape memory alloy, pneumatic boots, and electro-impulsive) [14; 15; 16; 17]. These methods are currently used but are energy-hungry and expensive to make and operate. The weight is of primary concern, as are the power consumption and effectiveness. The most common de-icing system currently in use for fixed-wing aircrafts is fluid de-icing, which presents great environmental concerns and only provides temporary ice protection. In this scenario, the research into low-power, high-frequency de-icing and anti-icing systems has been ongoing. In particular, the relatively new piezo-based actuation concept exhibits significant development prospects for improved energy efficiency. It relies on the capability of piezoelectric transducers bonded on the structure to induce mechanical waves (e.g., Lamb waves) causing shear stresses that produce ice delamination and fracture effects [18; 19; 20; 21; 22]. It is, thus, worthwhile investigating in depth the phenomena of resonance between the piezoelectric actuator and the beam to which it is attached.