Vibration analysis of multiple-layer microbeams based on the modified couple stress theory: analytical approach
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T E C H N I C A L N OT E S
Abbas Rahi
Vibration analysis of multiple-layer microbeams based on the modified couple stress theory: analytical approach
Received: 16 July 2020 / Accepted: 17 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The modified couple stress theory (MCST) is used to capture size effect on dynamic response in multiple-layer microbeams in the present article. Governing equations of the system are obtained based on the MCST and using Hamilton’s principle. The natural frequencies of the multiple-layer microbeam are calculated using the analytical method. Then, the results of the natural frequencies are presented with respect to different values of the system parameters such as the geometric layers and also the dimensionless material length-scale parameter. The results show that the material length-scale parameter values and also the length, width, and thickness of each layer are extremely effective on the vibration characteristic of the multiple-layer microbeams. Keywords Multiple-layer microbeams · Modified couple stress theory · Size dependency · Free vibration · MEMS
1 Introduction The vibration response study of microelectromechanical system (MEMS) devices is very important to the design and optimization of a small component of the equipment. Today, manufacturing of small size devices in the field of the MEMS is possible by the development of new knowledge and technologies. Microbeams are one of the most common elements which are used in the field of the MEMS sensors [1–9]. Several of the investigators studied the vibration response of the microcomponents in the field of MEMS based on the classical continuum mechanics theories or using the finite element method (FEM) [6,10,11]. In recent years, it has been observed that the classical continuum mechanics theories are unable to predict and explain the static and dynamic behaviors of the materials in small sizes such as microbeams [12–16]. In other words, capturing the size effect is a significant challenge in the study of the vibration response of the small size structures. Therefore, several of the non-classical continuum theories such as nonlocal elasticity theory, strain gradient theory, and couple stress theory have been offered to capture the size effect in dynamic response for microcomponents [12,17–19]. The couple stress theory was presented by Mindlin and Tiersten [18] as a non-classical continuum theory to consider the size dependency effect using two material length-scale parameters. Then, Yang et al. [19] proposed the modified couple stress theory (MCST) based on using one material length-scale parameter to capture the size effect in microcomponents. The MCST has been used by some researchers to study the size dependency of the static and dynamic behaviors of materials in micro components. Liang et al. [20]; Park and Gao [21]; Dai et al. [22]; Ma et al. [23]; Ghiasi [24]; and Asghari et al. [25] investigated static or dynamic behaviors of micro-/nanobeams based on the MCST to capture the size d
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