Vibration and Dynamic behavior of Multi-Layer Sandwich Composite Piezoelectric Micro Beam Using Higher-Order Elasticity Theory
الموضوعات : فصلنامه شبیه سازی و تحلیل تکنولوژی های نوین در مهندسی مکانیکMehrdad Soltani 1 , Javad Jafari 2 , Seyed Ali Galehdari 3 , Rasoul Tarkesh Esfahani 4 , Mohamad Shahgholi 5
1 - Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran
2 - Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran
3 - Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran
4 - Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran
5 - Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran
الکلمات المفتاحية: Multi-Layer Sandwich Composite, Micro beam, higher-order elasticity theory, vibration and dynamic analysis ,
ملخص المقالة :
Numerous scientists have examined the mechanical properties of materials at the micro- and nanoscale in recent years. Conversely, the remarkable advancements in micro and nanotechnology across several fields and industries, such as their extensive applications in micro- and nano-electro-mechanical systems (MEMS and NEMS), have piqued the curiosity of researchers. In this paper, vibration and dynamic behavior of multi-layer sandwich composite piezoelectric micro beam using higher-order elasticity theory by considering strain gradient and surface effects are investigated. The Hamilton’s principle is utilized to derive the sandwich micro beam model with Gurtin- Murdoch surface theory and generalized differential quadrature method is used to discretize and solve the differential equation Moreover, the effects of higher order materials and geometry of model on control respond and frequency behavior have been presented. In addition, the role of different theories based FFT results has been examined. The results show that the small material length scale considering in the higher order theories plays a key role in the dynamic respond of models. Also, the result of higher order theories should be considered in micro and nano scale and the classical theory could not predict the mechanical behavior very well. The study's conclusions imply that the effectiveness of controller scheme is within the piezoelectric voltage range in terms of vibration control and feedback damping factors.
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Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering 16 (3) (2024) 0027~0040 DOI 10.71939/jsme.2024.1127451
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Research article
Vibration and dynamic behavior of multi-layer sandwich composite piezoelectric micro beam using higher-order elasticity theory
Mehrdad Soltani1,2, Javad Jafari Fesharaki1,2, Seyed Ali Galehdari1,2, Rasoul Tarkesh Esfahani1,2 , Mohamad Shahgholi1,2
1Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran
2Modern Manufacturing Technologies Research center, Najafabad Branch, Islamic Azad University, Najafabad, Iran
*jjafari.f@gmail.com
(Manuscript Received --- 28 July 2024; Revised --- 29 Nov. 2024; Accepted --- 17 Dec. 2024)
Abstract
Numerous scientists have examined the mechanical properties of materials at the micro- and nanoscale in recent years. Conversely, the remarkable advancements in micro and nanotechnology across several fields and industries, such as their extensive applications in micro- and nano-electro-mechanical systems (MEMS and NEMS), have piqued the curiosity of researchers. In this paper, vibration and dynamic behavior of multi-layer sandwich composite piezoelectric micro beam using higher-order elasticity theory by considering strain gradient and surface effects are investigated. The Hamilton’s principle is utilized to derive the sandwich micro beam model with Gurtin- Murdoch surface theory and generalized differential quadrature method is used to discretize and solve the differential equation Moreover, the effects of higher order materials and geometry of model on control respond and frequency behavior have been presented. In addition, the role of different theories based FFT results has been examined. The results show that the small material length scale considering in the higher order theories plays a key role in the dynamic respond of models. Also, the result of higher order theories should be considered in micro and nano scale and the classical theory could not predict the mechanical behavior very well. The study's conclusions imply that the effectiveness of controller scheme is within the piezoelectric voltage range in terms of vibration control and feedback damping factors.
Keywords: Multi-Layer Sandwich Composite, Micro beam, higher-order elasticity theory, vibration and dynamic analysis
1- Introduction
The behavior of materials in the context of micro and nano electro-mechanical systems (MEMS and NEMS) has drawn the attention of numerous researchers in recent times. In many mechanical engineering applications, including atomic forces micro-scopes (AFMS), micro-switches, micro and nano sensors and actuators, micro rate gyros, and micro flexible joints, MEMS and NEMS play a significant role [1, 2]. Experimental studies have demonstrated that the size effect needs to be considered in the mechanical behavior of micro-scale structures when the length characteristic is on the order of microns [3]. Specifically, high longitudinal Young's modulus, strength to weight ratio, and high frequency operation are unique characteristics that have drawn a lot of study from many scientific fields. Sandwich composite microstructures are a well-known class of small-scale materials with a wide range of uses in contemporary industries. The results of the experiments showed that the behavior of micro and nanomaterials differs fundamentally and is dependent on the structure's dimensions [4-6]. In order to address the fundamental drawbacks of classical theory, higher order continuum theories that take material length scale factors into account have been extended. Recent mechanical testing and experimental observations have made it clear that the behavior of materials at the micro and nanoscales essentially varies from that of materials at the macroscale and is strongly reliant on size as well as structure dimension [7-12]. Researchers have determined that the size impact needs to be considered in mechanical behavior in order to overcome this inherent weakness of classical theory and get the precise prediction of behavior in small-scale dimensions [13, 14]. As a result, several higher order continuum theories have been put forth that include factors related to the material length scale, such as strain gradient, Eringen, micropolar elasticity, couple stress, and modified couple stress. One of these advanced theories that many researchers have effectively used is the modified pair stress theory. Alashti et al. studied at the size-dependent behavior of microbeams in order to investigate static and free vibration problems utilizing the pair stress theory. The findings demonstrate that the deflection estimated by the previously indicated technique is less than that of the classical hypothesis [15]. Nikpourian and colleagues [16] report size-dependent nonlinear dynamic resonance of a piezoelectrically laminated MEMS. For both rectangular and circular micro-plates, Jomehzadeh et al. [17] provided a micro-mechanical study based on the modified couple stress for examining a wide range of length scale parameters, distinct aspect ratios, and alternative boundary conditions. Dynamic stability of manufactured double-sided NEMS with finite conductivity, surface energy, and nonlocal effect was achieved by Sedighi et al. [18]. Ding et al. [19] assessed the impact of the thickness, beam width, and electrode gap on the frequency response using a modified couple stress theory of electrostatically actuated micro-beam with von Kármán geometric nonlinearity. The dynamic behavior of geometrical imperfect micro-beam was studied by Farokhi et al. [20] taking into account the modified pair stress theory. To discretize the equation of motion, they used the Galerkin method. Sahmani and Bahrami [21] noted that the dynamic stability of micro-beams driven by piezoelectric voltage is depending on their size. A narrow elastic beam's bending and buckling were proven by Lazopoulos & Lazopoulos [22]. They came to the conclusion that when beam thickness decreased, so did the cross-section area's dependence, based on the strain gradient hypothesis. The increase in thin beam stiffness is responsible for this impact. Trindade and Benjeddou's finite element model [23] was created to assess the vibrational response of a traditional sandwich beam with a dynamic piezoelectric actuator and sensor. Ghaznavi et al. [24] examined the stability of transvers motion for a cantilever micro-beam integrated with piezoelectric layers for the actuator and sensor. Using strain gradient theory, Sahmani et al. [21] clarified the dynamic stability of a micro-beam driven by piezoelectric voltage. Additionally, they contrasted the critical piezoelectric voltages predicted by the classical theory with those that included a range of length scale parameter values.
Researchers have been paying close attention to MEMS vibration abatement and control. Therefore, vibration control has made efficient use of a variety of control strategies. Based on the Strain Gradient nonlinear theory, Vatankhah et al. [25] examined a closed-loop control approach to reduce the vibration of a tiny Euler-Bernoulli cantilever beam utilizing linear piezoelectric actuation. Using the Galerkin projection approach, they transformed the controlling partial differential equation into a few ordinary differential equations, and a reliable linear controller was created for this model. Ansari et al.'s study included a computer analysis of Timoshenko nano-beam vibrations at various end conditions that took surface stress effects into account. They demonstrated how raising the dimensionless fundamental frequency is a direct result of raising the residual surface stress. The global dynamics and integrity of a micro-plate pressure sensor were expanded by Belarinelli et al. [26]. Gurtin-Murdoch elasticity theory was utilized by Mohammadimehr et al. [27] to examine the influence of surface effects on the free vibration of an isotropic piezoelectric Timoshenko micro-beam. Their research demonstrated how the natural frequency responses are impacted by surface stress effects. The Gurtin–Murdoch surface stress and strain gradient theory was utilized by Mirkalantari et al. [28] to forecast the pull-in instability behavior of nano-plates. They employed the GDQM to solve governing equations in their research.
To the best of the author's knowledge, no research has been published on the best wy to control a microbeam using piezoelectric layers based on modified couple stress theory. The majority of related studies in micro and nano mechanical behavior that take into account higher order continuum theories concentrated on size dependent effect, stability, and vibration analyses of the model. Furthermore, sandwich structures with piezoelectric layers in micro scale that take surface stress elasticity into account have received much too little attention.
Surface stress elasticity and higher order elasticity theory are used to model the structure of micro cantilever beam model in the current study. Based on Hamilton's principle, the governing equations of motion and boundary conditions for a multilayer piezoelectric and micro-beam are constructed, and GDQM is then used to discretize the data. The LQR with output feedback is utilized to assess the active vibration control capability.
2- Governing equations of motion
Composite cantilever sandwich microbeam with two active silicon material qualities with PZT-4 piezoelectric layers linked to a thick porous core constitutes the fundamental idea of the problem. This sophisticated, one-of-a-kind structure was developed using surface stress and strain gradient theories.
A schematic of a micro cantilever beam integrated with piezoelectric layers is presented in Figure 1. The length and width of the microbeam in this model are L and b, respectively. Furthermore, two thickness parameters related to the bulk and piezoelectric thickness, respectively, are indicated by 
and 
[29].
Fig. 1 Schematic of micro cantilever beam with piezoelectric layers[30]
The displacement field of an Euler-Bernoulli beam is stated as [31]:
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Piezoelectric layer | Bulk | Parameters |
3 | 3 | Thickness(µm) |
450 | 450 | Length(µm) |
50 | 50 | Width(µm) |
64 | 210 | Young’s modulus (Gpa) |
7500 | 2331 | Mass density |
0.27 | 0.24 | Poisson’s ratio |
-10 | - |
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1.0275×10-8 | - |
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17.6 | 17.6 |
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(31)
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