## 1 Introduction  
The negative effects of vibration in machine tools, civil infrastructures, automotive, and aerospace are huge and enormous. For instance, in the machine tool industry, mechanical vibration reduces both production rate and end-product quality, and shortens the service life of machine components, as assessed by [1, 2]. Various traditional methods, such as passive insulators and dampers, have been used to dampen mechanical vibrations in various fields of engineering [3]. However, due to advances in digital signal processing (DSP), sensors, and actuators technologies, and the limitations in the effectiveness of traditional vibration damping mechanisms [4], a great emphasis has been placed on active vibration control over the past two decades. Today, active vibration control is a vast research area that synergistically integrates interdisciplinary technologies such as mechanical and electrical components. The fundamental components of active vibration control are the actuator, sensor, controller, and mechanical structure influenced by the disturbance. The main function of the actuators in active vibration control is to counteract the impact of the disturbance on the structure (mechanical body) by utilizing control signals generated intelligently by the controllers using input signals from sensors that sense the nature and magnitude of the disturbance. This way, the control system can reduce and/or cancel the disturbance by the principle of destructive interference. While there are different types of actuators such as electric, hydraulic, pneumatic, and piezo, to name a few, the only type of actuator focused on in this thesis is the piezoelectric actuator. Before implementing and manufacturing any active vibration control systems, it is important to ensure the feasibility and reliability of the control system in the specific area. According to [5], one of the determinants of the feasibility and reliability of active vibration control is the dynamics of the actuator (piezoelectric actuator in this article) that dampen the unwanted vibration on the mechanical body with the help of commands from the controller. In turn, the performance of any piezoelectric actuator depends primarily on the quality of the actuator material [2] and its design approach [6], i.e., single-layer or multi-layer actuators, stacks, benders, or amplified actuators. Not only for piezoelectric actuators, but also in any engineering design, material selection is a fundamental and crucial step, since proper material selection provides better quality, reliability, extended product life, and economic feasibility. Material selection in most engineering design processes, including piezoelectric actuators, is a multi-criteria decision making (MCDM) problem, as there are many constraints and several conflicting requirements to be met [7]. In addition to this, the availability of a number of potential materials for a specific area of application, such as piezoelectric actuators, makes the material selection a daunting task and requires a precise and efficient selection method. In this regard, different techniques and algorithms have been applied in the material selection problem and they are generally categorized into multi-objective decision making (MODM) and multi-attribute decision making (MADM) [7]. In this work, two approaches were applied in order to select the optimum material for the piezoelectric actuator. The first method was based on the figures of merit (FOM) approach under MODM, and the second was the technique for order of preference by similarity to the ideal solutions (TOPSIS), which is the most common material selection algorithm under MADM [7]. Different approaches to piezoelectric actuator design have been used to develop high-performance actuators with relatively large displacement and high blocking force. Monolayer, multi-layer, stack, bending, or amplified actuators are the common approaches [8].