## 1 Introduction  
Smart electronic devices used over the years mostly consist of printed circuit boards (PCBs) and micro-electromechanical systems (MEMS). These boards are mainly made of thin copper films bounded to fiber epoxy laminates, such as FR4. PCB board sizes can vary from about 2 cm to 50 cm and most commonly have a rectangular shape in the electronic industry [1]. When attached to a machine structure, circuit boards can undergo various vibrating forces. Statistics show that about 20% of airborne electronic equipment failures occur due to vibration factors [2]. Damage, or even total failure, can occur when shock and vibration produce high stress on substrate, components, and joints. The type of damage is dependent on the frequency and amplitude of bending moments and inertial loads imposed on the PCB structure [3]. Since the dynamic loading is very critical for electronic equipment, studies to analyze and isolate vibrations in these systems are very important. However, market pressure and shortened development stages have rendered reliability analysis gradually unaffordable and, hence, simulations based on failure physics have been proposed to test the reliability of circuit boards [4]. Modal analysis is the study of the dynamic characteristics of a system by evaluating natural frequencies and damping factors, among other dynamic parameters. This analysis is used to formulate a mathematical model that explains the dynamic behavior of the studied system and is an important tool to optimize the dynamic characteristics of structures and materials [5]. In the case of PCB boardsand integrated circuits, their reliability depends, among other factors, on the possibility to endure vibration loads and impacts that can affect their performance. Modal analysis can thus be a very useful tool for engineers to reveal vulnerable areas and avoid premature failure [6; 7; 8]. Different studies have addressed the problem of determining the dynamic properties of PCB boards, using either experimental or analytical methods. In an early work, [PERSON] [9] determined the factors that play a critical role in the dynamic behavior of a PCB having three components of similar mass density. This author found that the first vibration mode should account for more than 90% of the mass of the structure and minor rearrangements of the components on the board cause major redistributions of the dynamic response towards higher modes. [PERSON] and [PERSON] [10] suggested an analytical model of a simply supported PCB with a component and studied the vibrating responses of the critical elements on a PCB for different design alternates. They found that a two-degree-of-freedom spring mass model can accurate predict the response to a random vibration profile and could be used in preliminary design stages. [PERSON] et al. [11] performed modal analysis on a PCB, by the finite element method and successfully predicted the changes in natural frequencies associated to changing the board thicknesses. [PERSON] et al. [12] determined the mechanical properties of a multi-layered PCB and validated their results using Ansys(r) and [PERSON](r) (Canonsburg, PA, USA; Gwynedd, PA, USA) computer software. They observed frequency changes by changing material properties such as the out-of-plane modulus (E\({}_{x}\)), Poisson's ratio, and shear modulus. On the other hand, [PERSON] [13] described various methods for reducing the destructive resonant amplitude of PCBs in electronic systems. He compared the effectiveness of isolation, extensional damping, and shear damping techniques with a standard printed circuit board. [PERSON] [14] solved vibration isolation problems in electronic boxes, by using a two-degree-of-freedom mass, spring, and damper system in order to minimize the dynamic response of internal sensitive components of the electronic platform. Recently, [PERSON] [15] has introduced the presence of connector, tray, and ribs as a boundary condition in a finite element model of a PCB.