Sensors & TransducersSensors & Transducers Volume 103, Issue 4 April 2009 ISSN 1726-5479...

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Volume 103, Issue 4 April 2009 www.sensorsportal.com

ISSN 1726-5479

Editor-in-Chief: professor Sergey Y. Yurish, phone: +34 696067716, fax: +34 93 4011989, e-mail: [email protected]

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Abdul Rahim, Ruzairi, Universiti Teknologi, Malaysia Ahmad, Mohd Noor, Nothern University of Engineering, Malaysia Annamalai, Karthigeyan, National Institute of Advanced Industrial Science

and Technology, Japan Arcega, Francisco, University of Zaragoza, Spain Arguel, Philippe, CNRS, France Ahn, Jae-Pyoung, Korea Institute of Science and Technology, Korea Arndt, Michael, Robert Bosch GmbH, Germany Ascoli, Giorgio, George Mason University, USA Atalay, Selcuk, Inonu University, Turkey Atghiaee, Ahmad, University of Tehran, Iran Augutis, Vygantas, Kaunas University of Technology, Lithuania Avachit, Patil Lalchand, North Maharashtra University, India Ayesh, Aladdin, De Montfort University, UK Bahreyni, Behraad, University of Manitoba, Canada Baoxian, Ye, Zhengzhou University, China Barford, Lee, Agilent Laboratories, USA Barlingay, Ravindra, RF Arrays Systems, India Basu, Sukumar, Jadavpur University, India Beck, Stephen, University of Sheffield, UK Ben Bouzid, Sihem, Institut National de Recherche Scientifique, Tunisia Benachaiba, Chellali, Universitaire de Bechar, Algeria Binnie, T. David, Napier University, UK Bischoff, Gerlinde, Inst. Analytical Chemistry, Germany Bodas, Dhananjay, IMTEK, Germany Borges Carval, Nuno, Universidade de Aveiro, Portugal Bousbia-Salah, Mounir, University of Annaba, Algeria Bouvet, Marcel, CNRS – UPMC, France Brudzewski, Kazimierz, Warsaw University of Technology, Poland Cai, Chenxin, Nanjing Normal University, China Cai, Qingyun, Hunan University, China Campanella, Luigi, University La Sapienza, Italy Carvalho, Vitor, Minho University, Portugal Cecelja, Franjo, Brunel University, London, UK Cerda Belmonte, Judith, Imperial College London, UK Chakrabarty, Chandan Kumar, Universiti Tenaga Nasional, Malaysia Chakravorty, Dipankar, Association for the Cultivation of Science, India Changhai, Ru, Harbin Engineering University, China Chaudhari, Gajanan, Shri Shivaji Science College, India Chen, Jiming, Zhejiang University, China Chen, Rongshun, National Tsing Hua University, Taiwan Cheng, Kuo-Sheng, National Cheng Kung University, Taiwan Chiang, Jeffrey (Cheng-Ta), Industrial Technol. Research Institute, Taiwan Chiriac, Horia, National Institute of Research and Development, Romania Chowdhuri, Arijit, University of Delhi, India Chung, Wen-Yaw, Chung Yuan Christian University, Taiwan Corres, Jesus, Universidad Publica de Navarra, Spain Cortes, Camilo A., Universidad Nacional de Colombia, Colombia Courtois, Christian, Universite de Valenciennes, France Cusano, Andrea, University of Sannio, Italy D'Amico, Arnaldo, Università di Tor Vergata, Italy De Stefano, Luca, Institute for Microelectronics and Microsystem, Italy Deshmukh, Kiran, Shri Shivaji Mahavidyalaya, Barshi, India Dickert, Franz L., Vienna University, Austria Dieguez, Angel, University of Barcelona, Spain Dimitropoulos, Panos, University of Thessaly, Greece Ding, Jianning, Jiangsu Polytechnic University, China Djordjevich, Alexandar, City University of Hong Kong, Hong Kong

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Microsystems, Italy Francis, Laurent, University Catholique de Louvain, Belgium Fu, Weiling, South-Western Hospital, Chongqing, China Gaura, Elena, Coventry University, UK Geng, Yanfeng, China University of Petroleum, China Gole, James, Georgia Institute of Technology, USA Gong, Hao, National University of Singapore, Singapore Gonzalez de la Rosa, Juan Jose, University of Cadiz, Spain Granel, Annette, Goteborg University, Sweden Graff, Mason, The University of Texas at Arlington, USA Guan, Shan, Eastman Kodak, USA Guillet, Bruno, University of Caen, France Guo, Zhen, New Jersey Institute of Technology, USA Gupta, Narendra Kumar, Napier University, UK Hadjiloucas, Sillas, The University of Reading, UK Hashsham, Syed, Michigan State University, USA Hasni, Abdelhafid, Bechar University, Algeria Hernandez, Alvaro, University of Alcala, Spain Hernandez, Wilmar, Universidad Politecnica de Madrid, Spain Homentcovschi, Dorel, SUNY Binghamton, USA Horstman, Tom, U.S. Automation Group, LLC, USA Hsiai, Tzung (John), University of Southern California, USA Huang, Jeng-Sheng, Chung Yuan Christian University, Taiwan Huang, Star, National Tsing Hua University, Taiwan Huang, Wei, PSG Design Center, USA Hui, David, University of New Orleans, USA Jaffrezic-Renault, Nicole, Ecole Centrale de Lyon, France Jaime Calvo-Galleg, Jaime, Universidad de Salamanca, Spain James, Daniel, Griffith University, Australia Janting, Jakob, DELTA Danish Electronics, Denmark Jiang, Liudi, University of Southampton, UK Jiang, Wei, University of Virginia, USA Jiao, Zheng, Shanghai University, China John, Joachim, IMEC, Belgium Kalach, Andrew, Voronezh Institute of Ministry of Interior, Russia Kang, Moonho, Sunmoon University, Korea South Kaniusas, Eugenijus, Vienna University of Technology, Austria Katake, Anup, Texas A&M University, USA Kausel, Wilfried, University of Music, Vienna, Austria Kavasoglu, Nese, Mugla University, Turkey Ke, Cathy, Tyndall National Institute, Ireland Khan, Asif, Aligarh Muslim University, Aligarh, India Kim, Min Young, Kyungpook National University, Korea South

Ko, Sang Choon, Electronics and Telecommunications Research Institute, Korea South Kockar, Hakan, Balikesir University, Turkey Kotulska, Malgorzata, Wroclaw University of Technology, Poland Kratz, Henrik, Uppsala University, Sweden Kumar, Arun, University of South Florida, USA Kumar, Subodh, National Physical Laboratory, India Kung, Chih-Hsien, Chang-Jung Christian University, Taiwan Lacnjevac, Caslav, University of Belgrade, Serbia Lay-Ekuakille, Aime, University of Lecce, Italy Lee, Jang Myung, Pusan National University, Korea South Lee, Jun Su, Amkor Technology, Inc. South Korea Lei, Hua, National Starch and Chemical Company, USA Li, Genxi, Nanjing University, China Li, Hui, Shanghai Jiaotong University, China Li, Xian-Fang, Central South University, China Liang, Yuanchang, University of Washington, USA Liawruangrath, Saisunee, Chiang Mai University, Thailand Liew, Kim Meow, City University of Hong Kong, Hong Kong Lin, Hermann, National Kaohsiung University, Taiwan Lin, Paul, Cleveland State University, USA Linderholm, Pontus, EPFL - Microsystems Laboratory, Switzerland Liu, Aihua, University of Oklahoma, USA Liu Changgeng, Louisiana State University, USA Liu, Cheng-Hsien, National Tsing Hua University, Taiwan Liu, Songqin, Southeast University, China Lodeiro, Carlos, Universidade NOVA de Lisboa, Portugal Lorenzo, Maria Encarnacio, Universidad Autonoma de Madrid, Spain Lukaszewicz, Jerzy Pawel, Nicholas Copernicus University, Poland Ma, Zhanfang, Northeast Normal University, China Majstorovic, Vidosav, University of Belgrade, Serbia Marquez, Alfredo, Centro de Investigacion en Materiales Avanzados, Mexico Matay, Ladislav, Slovak Academy of Sciences, Slovakia Mathur, Prafull, National Physical Laboratory, India Maurya, D.K., Institute of Materials Research and Engineering, Singapore Mekid, Samir, University of Manchester, UK Melnyk, Ivan, Photon Control Inc., Canada Mendes, Paulo, University of Minho, Portugal Mennell, Julie, Northumbria University, UK Mi, Bin, Boston Scientific Corporation, USA Minas, Graca, University of Minho, Portugal Moghavvemi, Mahmoud, University of Malaya, Malaysia Mohammadi, Mohammad-Reza, University of Cambridge, UK Molina Flores, Esteban, Benemérita Universidad Autónoma de Puebla,

Mexico Moradi, Majid, University of Kerman, Iran Morello, Rosario, University "Mediterranea" of Reggio Calabria, Italy Mounir, Ben Ali, University of Sousse, Tunisia Mulla, Imtiaz Sirajuddin, National Chemical Laboratory, Pune, India Neelamegam, Periasamy, Sastra Deemed University, India Neshkova, Milka, Bulgarian Academy of Sciences, Bulgaria Oberhammer, Joachim, Royal Institute of Technology, Sweden Ould Lahoucine, Cherif, University of Guelma, Algeria Pamidighanta, Sayanu, Bharat Electronics Limited (BEL), India Pan, Jisheng, Institute of Materials Research & Engineering, Singapore Park, Joon-Shik, Korea Electronics Technology Institute, Korea South Penza, Michele, ENEA C.R., Italy Pereira, Jose Miguel, Instituto Politecnico de Setebal, Portugal Petsev, Dimiter, University of New Mexico, USA Pogacnik, Lea, University of Ljubljana, Slovenia Post, Michael, National Research Council, Canada Prance, Robert, University of Sussex, UK Prasad, Ambika, Gulbarga University, India Prateepasen, Asa, Kingmoungut's University of Technology, Thailand Pullini, Daniele, Centro Ricerche FIAT, Italy Pumera, Martin, National Institute for Materials Science, Japan Radhakrishnan, S. National Chemical Laboratory, Pune, India Rajanna, K., Indian Institute of Science, India Ramadan, Qasem, Institute of Microelectronics, Singapore Rao, Basuthkar, Tata Inst. of Fundamental Research, India Raoof, Kosai, Joseph Fourier University of Grenoble, France Reig, Candid, University of Valencia, Spain Restivo, Maria Teresa, University of Porto, Portugal Robert, Michel, University Henri Poincare, France Rezazadeh, Ghader, Urmia University, Iran Royo, Santiago, Universitat Politecnica de Catalunya, Spain Rodriguez, Angel, Universidad Politecnica de Cataluna, Spain Rothberg, Steve, Loughborough University, UK Sadana, Ajit, University of Mississippi, USA Sadeghian Marnani, Hamed, TU Delft, The Netherlands Sandacci, Serghei, Sensor Technology Ltd., UK

Sapozhnikova, Ksenia, D.I.Mendeleyev Institute for Metrology, Russia Saxena, Vibha, Bhbha Atomic Research Centre, Mumbai, India Schneider, John K., Ultra-Scan Corporation, USA Seif, Selemani, Alabama A & M University, USA Seifter, Achim, Los Alamos National Laboratory, USA Sengupta, Deepak, Advance Bio-Photonics, India Shankar, B. Baliga, General Monitors Transnational, USA Shearwood, Christopher, Nanyang Technological University, Singapore Shin, Kyuho, Samsung Advanced Institute of Technology, Korea Shmaliy, Yuriy, Kharkiv National University of Radio Electronics,

Ukraine Silva Girao, Pedro, Technical University of Lisbon, Portugal Singh, V. R., National Physical Laboratory, India Slomovitz, Daniel, UTE, Uruguay Smith, Martin, Open University, UK Soleymanpour, Ahmad, Damghan Basic Science University, Iran Somani, Prakash R., Centre for Materials for Electronics Technol., India Srinivas, Talabattula, Indian Institute of Science, Bangalore, India Srivastava, Arvind K., Northwestern University, USA Stefan-van Staden, Raluca-Ioana, University of Pretoria, South Africa Sumriddetchka, Sarun, National Electronics and Computer Technology

Center, Thailand Sun, Chengliang, Polytechnic University, Hong-Kong Sun, Dongming, Jilin University, China Sun, Junhua, Beijing University of Aeronautics and Astronautics, China Sun, Zhiqiang, Central South University, China Suri, C. Raman, Institute of Microbial Technology, India Sysoev, Victor, Saratov State Technical University, Russia Szewczyk, Roman, Industrial Research Institute for Automation and

Measurement, Poland Tan, Ooi Kiang, Nanyang Technological University, Singapore, Tang, Dianping, Southwest University, China Tang, Jaw-Luen, National Chung Cheng University, Taiwan Teker, Kasif, Frostburg State University, USA Thumbavanam Pad, Kartik, Carnegie Mellon University, USA Tian, Gui Yun, University of Newcastle, UK Tsiantos, Vassilios, Technological Educational Institute of Kaval, Greece Tsigara, Anna, National Hellenic Research Foundation, Greece Twomey, Karen, University College Cork, Ireland Valente, Antonio, University, Vila Real, - U.T.A.D., Portugal Vaseashta, Ashok, Marshall University, USA Vazquez, Carmen, Carlos III University in Madrid, Spain Vieira, Manuela, Instituto Superior de Engenharia de Lisboa, Portugal Vigna, Benedetto, STMicroelectronics, Italy Vrba, Radimir, Brno University of Technology, Czech Republic Wandelt, Barbara, Technical University of Lodz, Poland Wang, Jiangping, Xi'an Shiyou University, China Wang, Kedong, Beihang University, China Wang, Liang, Advanced Micro Devices, USA Wang, Mi, University of Leeds, UK Wang, Shinn-Fwu, Ching Yun University, Taiwan Wang, Wei-Chih, University of Washington, USA Wang, Wensheng, University of Pennsylvania, USA Watson, Steven, Center for NanoSpace Technologies Inc., USA Weiping, Yan, Dalian University of Technology, China Wells, Stephen, Southern Company Services, USA Wolkenberg, Andrzej, Institute of Electron Technology, Poland Woods, R. Clive, Louisiana State University, USA Wu, DerHo, National Pingtung University of Science and Technology,

Taiwan Wu, Zhaoyang, Hunan University, China Xiu Tao, Ge, Chuzhou University, China Xu, Lisheng, The Chinese University of Hong Kong, Hong Kong Xu, Tao, University of California, Irvine, USA Yang, Dongfang, National Research Council, Canada Yang, Wuqiang, The University of Manchester, UK Yaping Dan, Harvard University, USA Ymeti, Aurel, University of Twente, Netherland Yong Zhao, Northeastern University, China Yu, Haihu, Wuhan University of Technology, China Yuan, Yong, Massey University, New Zealand Yufera Garcia, Alberto, Seville University, Spain Zagnoni, Michele, University of Southampton, UK Zeni, Luigi, Second University of Naples, Italy Zhong, Haoxiang, Henan Normal University, China Zhang, Minglong, Shanghai University, China Zhang, Qintao, University of California at Berkeley, USA Zhang, Weiping, Shanghai Jiao Tong University, China Zhang, Wenming, Shanghai Jiao Tong University, China Zhou, Zhi-Gang, Tsinghua University, China Zorzano, Luis, Universidad de La Rioja, Spain Zourob, Mohammed, University of Cambridge, UK

Sensors & Transducers Journal (ISSN 1726-5479) is a peer review international journal published monthly online by International Frequency Sensor Association (IFSA). Available in electronic and on CD. Copyright © 2009 by International Frequency Sensor Association. All rights reserved.

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Volume 103 Issue 4 April 2009

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Research Articles

Frontiers of Nanosensor Technology Vinod Kumar Khanna ......................................................................................................................... 1 Dual Comb Unit High-g Accelerometer Based on CMOS-MEMS Technology Mehrdad Mottaghi, Farzan Ghalichi, Habib B. Ghavifekr................................................................... 17 Modeling of Micromachined Thermopiles Powered from the Human Body for Energy Harvesting in Wearable Devices Vladimir Leonov, Ziyang Wang, Paolo Fiorini and Chris Van Hoof .................................................... 29 Design and Development of Polysilicon-based Microhotplate for Gas Sensing Application Mahanth Prasad, V. K. Khanna and Ram Gopal................................................................................ 44 Design of a Capacitive SOI Micromachined Accelerometer Wenjing Zhao, Limei Xu. .................................................................................................................... 52 Characteristic Features of RF MEMS Switches and its Various Applications B. Mishra, Z. C. Alex........................................................................................................................... 65 Study on the Effects of Added Mass on Mechanical Behavior of a Microbeam Mohammad Fathalilou, Ghader Rezazadeh, Yashar Alizadeh, Soheil Talebian ............................... 73 Titanium Hydride Formation in Current-Biased Titanium Microbolometer and Nanobolometer Devices S. F. Gilmartin, K. Arshak, D. Collins, B. Lane, D. Bain, S. B. Newcomb, B. McCarthy, A. Arshak. . 83 Squeeze-Film Damping Effect on Dynamic Pull-in Voltage of an Electrostatically-Actuated Microbeam Hadi Yagubizade, Mohammad Fathalilou, Ghader Rezazadeh, Soheil Talebian .............................. 96 Porous Silicon Hydrogen Sensor at Room Temperature: the Effect of Surface Modification and Noble Metal Contacts Jayita Kanungo, Hiranmay Saha, Sukumar Basu .............................................................................. 102 Design and Analyses of Electromagnetic Microgenerator Nibras Awaja, Dinesh Sood, Thurai Vinay. ........................................................................................ 109 Dynamic Pull-in Phenomenon in the Fully Clamped Electrostatically Actuated Rectangular Microplates Considering Damping Effects Ghader Rezazadeh, Soheil Talebian, Mohammad Fathalilou............................................................ 122 Finite Element Analysis of Static and Dynamic Pull-In Instability of a Fixed-Fixed Micro Beam Considering Damping Effects Mohammad Reza Ghazavi, Ghader Rezazadeh, Saber Azizi ........................................................... 132

Effect of Polyimide Variation and its Curing Temperature on CMOS Based Capacitive Humidity Sensor and Characterization of Integrated Heater B. N. Baliga, D. N. Tiwari,Kamaljeet Singh, Sanjay Verma,K. Nagachenchaiah ............................... 144 Sputtered Silicon as a Potential Masking Material for Glass Micromachining – A Feasibility Study Abhay B. Joshi, Dhananjay Bodas, S. A. Gangal............................................................................... 155 Thermo-Mechanical Behavior of a Bilayer Microbeam Subjected to Nonlinear Electrostatic Pressure Maliheh Pashapour, Seyed-Mehdi Pesteii, Ghader Rezazadeh, Shahriyar Kouravand.................... 161 Hydrogen and Methane Response of Pd Gate MOS Sensor Preeti Pandey, J. K. Srivastava, V. N. Mishra and R. Dwivedi........................................................... 171

Authors are encouraged to submit article in MS Word (doc) and Acrobat (pdf) formats by e-mail: [email protected] Please visit journal’s webpage with preparation instructions: http://www.sensorsportal.com/HTML/DIGEST/Submition.htm

International Frequency Sensor Association (IFSA).

Sensors & Transducers Journal, Vol. 103, Issue 4, April 2009, pp. 73-82

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Study on the Effects of Added Mass on Mechanical Behavior of a Microbeam

1Mohammad FATHALILOU, 1Ghader REZAZADEH, 2Yashar ALIZADEH,

1Soheil TALEBIAN 1Mech. Eng. Dept. Urmia University, Urmia, Iran

Tel.: +98-914-145-1407 2Sama Organization (Affiliated with Islamic Azad University), Khoy Branch, Iran

Tel.: +98-914-363-6714 E-mail: [email protected]

Received: 29 November 2008 /Accepted: 20 April 2009 /Published: 27 April 2009 Abstract: This paper presents a numerical study on the effects of added mass on the microbeams oscillating in an incompressible inviscid stationary fluid. By applying “Slender Wing” and “Three dimensional aerodynamic” theories the inertial effects of fluid on the microbeam dynamics has been modeled as a mass added to microbeam mass. The results have showed that the added mass for clamped-clamped beams are higher than cantilever beams for first mode and lower for other three modes. Galerkin-based step by step linearization method (SSLM) and Galerkin-based reduced order model have been applied to solve the nonlinear static and dynamic governing equations, respectively. Water by neglecting viscidity effects, has been considered as a surrounding fluid and the frequency response of the microbeam has been compared with that of vacuum conditions. It has been shown that because of the added mass effects in water environment, the natural frequencies of the microbeam decrease. Copyright © 2009 IFSA. Keywords: MEMS, Microbeam, Added mass, Frequency response 1. Introduction It is known that Microelectromechanical systems (MEMS) play an important role in modern technologies. The ability of MEMS to miniaturize, reduce the cost and energy consumption makes them to be interesting object for researches. These micro devises are commonly seen in various structures such as pressure sensors, micro-mirrors, micro-pumps, accelerometers and etc [1-4]. Many


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of researchers are trying to develop modelling and simulating the various environmental conditions on MEMS structures. One of these conditions can be due to quality of fluid which surrounds the structure. It can be shown that structural component, when submerged in a fluid exhibits decrease in natural frequencies. In general, the surrounding fluid exerts a reaction force on the structure which is represented as an added mass [5, 6]. In the dynamical analysis of micro-pumps and micro-densitometers where a microstructure submerged in a fluid, its inertial effects can be modeled as a mass added to microstructure mass. Some researchers have presented theories that treat the concept of added mass and fluid loading on the structures. For example three-dimensional linear aerodynamic theory [7], the slender wing theory [8], or the traveling wave solution [9, 10] are popular theories that can be applied to investigate the mechanical behavior of the structures submerged in a fluid. Minami [11] has applied the thin airfoil theory to calculate the added mass on a membrane with fixed ends. He has shown that the frequency and the amplitude of the oscillations dos not affect the added mass and the added mass only depends on the fluid density and the membrane dimensions. Meyerhoff [12] has calculated the added mass of thin rectangular plates in infinite fluid, and used the dipole singularities to describe the potential flow around a flat rectangular plate. Ergin et al. [13] have calculated wet natural frequencies and associated modes of partially submerged cantilever plates by the method of images and boundary-integral equation method. They have concluded that the proposed method is suitable for relatively high frequency vibrations of partially submerged elastic structures. Sinha et al. [5] have suggested a formula for added mass of the vibrating perforated plate-type structures submerged in fluid based on the experimental and analytical studies. As they mentioned this could be of considerable importance to designers for structural dynamic evaluation of such components without the need to conduct a modal test. Muthuveerappan et al [14] and Rao et al. [15] have used the finite-element method to solve the fluid–structure interaction problems for completely submerged elastic plates. Yadykin et al. [6] have presented a review of different theories that determine the fluid loading on plates which is required to calculate the added mass. But, it is mentionable that all of mentioned researches are done in macro-scale structures. The purpose of this paper is investigations of dynamic specifications and forced response of an electrostatically-actuated micro clamped-clamped beam by considering added mass effects, where this study can be useful to the analysis of micropumps and densitometers behavior. It is assumed that the microbeam is immersed in an incompressible inviscid stationary fluid. The fluid equivalent mass obtained using “Slender Wing” and “Three dimensional aerodynamic” theories, is added to the microbeam mass. Galerkin-based step by step linearization method (SSLM) is applied to calculate the static deflection of the microbeam. Then nonlinear governing dynamic equation is linearized about the static equilibrium position and Galerkin-based reduced order model is used to obtain the forced response of the microbeam vibrations. 2. Model Description 2.1. Nonlinear Distributed Electromechanical Coupled Model Electrostatically-actuated microbeams with clamped-clamped boundary conditions are studied here (see Fig. 1) to show the concept of the added mass on oscillating micro structures. The work is easily extendable to cantilever and simply-supported microbeams. The devices consist of a deformable elastic beam that is electrostatically actuated by an applied voltage. When a voltage, V(t) is applied on the upper and lower electrodes, the upper deformable beam moves transversely due to the time-dependent electrical force.


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Fig. 1. Schematic of an electrostatically-actuated microbeam. 2.2. Mathematical Modeling The governing equation which introduces the transverse displacement of the beam, ( )txw , actuated by an electrical load of voltage )(tV is written as [16]:

2

2

2

4

4

),()(

2~

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=∂∂

+∂∂

+∂∂

txwdtVb

twc

twbh

xwIE κερ , (1)

where E~ is dependent on the beam width, b and thickness, h . A beam is considered wide when

hb 5≥ . Wide beams exhibit plane-strain conditions, and therefore, E~ becomes the plate modulus, )1( 2υ−E where E and υ are the Young’s modulus and Poisson’s ratio, respectively. A beam is

considered narrow when hb 5< . In this case, E~ becomes the Young’s modulus, E. I is the effective moment of inertia of the cross-section and is 123bh which is wide relative to thickness and width. ρ is density of beam. d, κ and ε are initial gap, dielectric coefficient of the gap medium and dielectric constant, respectively. The microbeam is subjected to a structural damping [17]. This effect is approximated by an equivalent damping coefficient, c per unit length [16]. The clamped-clamped beam’s boundary conditions are given by:

0),(),0(),(),0(

=∂∂

=∂∂

==tLt x

wxwtLwtw (2)

2.3. Definition of the Added Mass As mentioned, concept of the added mass is widely used in the cases that a solid structure undergoes fluid loading especially when it is immersed in a fluid. When a beam oscillates in a fluid, the inertial effects of fluid on the beam dynamics is modeled as a mass added to beam mass. In this section magnitude of the added mass will be calculated and added to the beam mass in governing equation. For an incompressible inviscid stationary fluid, the rate of change in time of kinetic energy of any portion of the fluid is equal to the work done by the pressures on its surface [18]:

∫∫ ∆−=S

zk PdSV

dtdE

, (3)

where Ek is the fluid kinetic energy, zV is the vertical velocity of the fluid particle, S indicates the fluid-solid interaction surface and P∆ denotes the pressure exerted from outside an element dS of the boundary and is doing work. Assuming the beam to be fluid–solid interface [19], and considering small vibrations the integral in equation 3 can be rewritten as:


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∫ ∫∫∫ ∆∂∂

=∆b L

Sz Pdxdyt

wPdSV0 0

, (4)

where tw ∂∂ / is the vertical velocity of the beam. As reported by Minami [11] the rate of change of kinetic energy in time of the oscillating beam having a mass per unit area equivalent to the added mass, M is written as:

∫ ∫ ∂∂

∂∂

=b L

k dxdytw

twM

dtdE

0 02

2

(5)

So, with attention to Eqs. 3, 4 and 5 it can be obtained an expression for the added mass as:

1

0 02

2

0 0

−

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∆−= ∫ ∫∫ ∫b Lb L

dxdytw

twdxdy

twPM (6)

As found out from equation 6, the fluid loading, P∆ has a direct effect on the magnitude of added mass. Different formulations for P∆ have been reported by some researchers which have been reviewed by Yadykin et al [6]. For a thin narrow plate "Slender-Wing" theory [8] can be applied as following:

),(4

),(2

txwxL

Ut

AL

txP F ⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

=∆πρ (7)

where Fρ is the fluid density, U is the fluid constant velocity parallel to the plate and LbA /= is the aspect ratio of the plate. Here, for a stationary fluid the added mass is:

ALM F4πρ

= (8)

But the generalized fluid loading on the flexible plate is obtained using "Three dimensional aerodynamic" theory [7]. Using this theory for stationary fluid, P∆ can be expressed as:

(9)

where ζ and η are dummy variables. and are nondimensional coordinates which are related with x and y by following relations

(10)

and

(11)


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Now, the calculated added mass must be entered into the Eq. 1 as:

( )2

2

2

4

4

),()(

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛−

=∂∂

+∂∂

++∂∂

txwdtVb

twc

twMbbh

xwEI κερ (12)

For convenience, the nondimensional added mass, µ and variables are introduced as:

(13)

Time is non-dimensionalized by a characteristic period of the dry system according to:

(14)

By substituting these nondimensional parameters into Eq. 12, this equation is obtained as following:

, (15)

where

(16)

3. Numerical Solution It is considered that the microbeam is actuated by a small amplitude AC voltage. So, in Eq. 15, V(t) is

equal to . To solve Eq. 15, the forcing term is linearized about the undeflected position of the beam and the linearized equation is written as following:

, (17)

where is nondimensional excitation frequency related with Ω by following equation:

(18)

In order to solve Eq. 17, a reduced-order model can be used [20]. So, can be expressed as:


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, (19)

where is the jth shape function that satisfies the boundary conditions. The unknown , is approximated by truncating the summation series to a finite number, N :

(20)

In this paper, is the jth linear undamped mode shape of the straight microbeam.

By substituting the Eq. 20 into Eq. 17, and multiplying by as a weight function in Galerkin method and then integrating the outcome from , the Galerkin based reduced-order model is generated:

, (21)

where C M , , mechK , elecK are mass, damping, mechanical and electrical stiffness matrixes, respectively and F introduces the force vector as following:

∫=1

0

dxK ivjimechij φφ ,

, ∫=

1

0

202 dxVF ii φα i,j=1,…,N (22)

In order to determine the natural frequencies of the microbeam, free vibration of the microbeam around its undeflected position can be considered. To this end the AC voltage and damping term in Eq. 21 must be equaled to zero. So, this equation is rewritten as:

(23)

The ith nondimensional natural frequency of the beam is determined as:

ii

mechii

i MK

=ω (24)

By solving the governing equation, also frequency response of the system is investigated with and without considering added mass effect. 4. Results and Discussion To show the numerical results of the analysis presented in previous section, a clamped-clamped microbeam which vibrates in a single natural mode, is considered by characteristics introduced in


79

Table 1. It must be mentioned that width of the beam will change in various aspect ratios. The structural damping is estimated to be 0.5% of critical damping of undeflected beam [17]. It is considered that the proposed model oscillates in both water and vacuum where κ is 78 and 1, respectively.

Table 1. The values of design variables. Design Variable L h d E

ρ ε V0

Value mµ400 mµ5.1 mµ0.2 Gpa169 3/2331 mkg pF/m 85.8 V01.0 In Fig. 2, nondimensional added mass for the microbeam calculated using both "Slender-Wing" theory and "Three-dimensional aerodynamic" theory is shown. Gained from Eq. 8 and Fig. 2 the added mass calculated by "Slender-Wing" theory dos not depends on the order of the mode number. But in "Three-dimensional aerodynamic" theory the added mass have a different value for different mode numbers. Also it is shown that increasing the order of natural mode number decreases the added mass. Comparison between the values calculated by the three-dimensional aerodynamic theory and those calculated by the slender-wing theory shows that there is a considerable discrepancy especially for low aspect ratio and low modes.

Fig. 2. The variation of the nondimensional added mass versus aspect ratio; "Slender-Wing" theory (dotted line) and "Three-dimensional aerodynamic" theory (continues lines).

It can be obtained from results of Yadykin et al. [6] that for first four modes the values of added mass calculated for cantilever plates are higher than those for the simply supported plates except the added mass of first mode. It may be interesting to find out the difference of the values of added mass between cantilever and clamped-clamped beams. So, the values of added mass for clamped-clamped microbeams with the same characteristics of the cantilever beams are calculated. In Fig. 3 nondimensional added mass for undeflected clamped-clamped and cantilever microbeams are shown and compared. It is clearly seen that the added mass for clamped-clamped beams are higher than


80

cantilever beams for first mode and lower for other three modes. Also it is seen that the difference becomes smaller when the beams have low aspect ratio or oscillate in higher mode orders.

Fig. 3. The comparison of the nondimensional added mass between clamped-clamped (dotted lines) and cantilever (continues lines) microbeams for various aspect ratios and natural mode orders: first mode (circles);

second mode (squares); third mode (triangles); fourth mode (stars). Fig. 4 shows the calculated non-dimensional natural frequencies of the clamped-clamped microbeam for different fluids. The fluid densities are selected between 3/556 mkg (Butane) to 3/1584 mkg (Carbon Tetrachloride) where the densities of many of industrial fluids are in this range. It is clearly seen that because of the direct effects of the fluid density on the added mass, increasing the fluid density decreases the natural frequencies of the microbeam.

Fig. 4. The variation of the nondimensional natural frequencies of the microbeam with fluid density for aspect ratio=0.1.


81

Fig. 5 shows the frequency response of the microbeam when oscillates in both of water and vacuum environments. Gained from equation 21, the electrical stiffness is a time dependent parameter and so this problem can be introduced as a parametric excitation but as shown in Fig. 5, due to small V0 value the electrical stiffness term has no considerable effect on the frequency response of the beam. It can be seen that the amplitude of the vibrations of first mode of the model increases when oscillates in water in comparison with vacuum. This is due to decrease of the stiffness. Also it is shown that oscillation in water decreases the natural frequency of the model due to the added mass effects. This problem can be more important for some devices such as densitometers, micro-pumps and etc. where it strongly is required to find out the resonance frequency of the system. It is mentionable that although the micropumps are in full-clamped microplate form and our model is not a proper model for micropumps, but it is expected that similar results are obtained for microplates with full-clamped boundary conditions.

Fig. 5. The frequency response of the microbeam for aspect ratio=0.1. 5. Conclusion In the presented work, the effects of added mass on the mechanical behavior of microbeams were studied. Comparison between the values calculated by the "three-dimensional aerodynamic" theory and those calculated by the "slender-wing" theory showed that there is a noticeable discrepancy especially for low aspect ratio and low modes. The results showed that the added mass for clamped-clamped beams are higher than cantilever beams for first mode and lower for other three modes. The analysis showed that increasing the fluid density or increasing the aspect ratio increases the added mass and decreases the natural frequencies of oscillating beam. Investigated frequency response illustrated that due to higher dielectric coefficient, amplitude of oscillation in water environment increases compared with vacuum. Also because of small AC voltage, the time dependent electrical stiffness has no considerable effects on dynamic behavior of the microbeam. The applied proposed model can be used as a fluid densitometer.


82

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