Deflection in Micro-Scale Thermoelastic Simply-Supported Beams due to Patch Loading

Ramandeep Kaur* and J N Sharma

Department of Mathematics, National Institute of Technology, Hamirpur (HP) 177005, India

*Corresponding Author: raman.nith@gmail.com, jnsnith@gmail.com

ABSTRACT:

In this paper, an analytical model is developed and investigated for the transverse deflection in a transversely isotropic, thermoelastic simply-supported beam resonator subjected to time harmonic patch loading. Closed form expressions based on Euler–Bernoulli beam theory are derived for the flexural vibrations in a transversely isotropic, thermoelastic micro-beam. A time harmonic patch load is assumed to act within the region of the beam. The Laplace transform technique has been used to find the transverse deflection in the transform domain due to patch loading on the beam. The deflection so obtained in a physical domain has been computed numerically with the help of MATLAB software and presented graphically for silicon material beams.

KEY WORDS: Anisotropy, MEMS, Beam Resonators, Euler-Bernoulli Equation, Laplace Transforms.

INTRODUCTION:

Micro-scale mechanical resonators are of great interest for a wide range of sensing , modulation and frequency filtration [1-3]. The dynamical responses of thermoelastic bodies under various external conditions have drawn significant attention and researches by many professional engineers and scientist. It is necessary to know how the parameters affect their physical properties and mechanical properties. Zener [4] explained the mechanism of thermoelastic damping and derived an analytical solution to relate the energy dissipation and the material properties of thin beam structures by assuming some mathematical and physical simplifications during his derivation. Lifshitz and Roukes [5] reported higher precision solution for beam systems. Sun et al. [6] presented 2-D analysis of frequency shifts by considering heat conduction along the beam thickness and beam span and taking sinusoidal temperature gradients across the thickness of the beam prior to the solution of coupled equation for flexural vibrations. Sharma [7] derived governing equations of flexural vibrations in a transversely isotropic beam in closed form based on Euler-Bernoulli theory and studied thermoelastic damping (TED) and frequency shift (FS) of vibrations in clamped and simply supported beam structures. Pustan et al. [8] describes the studies of the mechanical characteristics of flexible MEMS components including theoretical approach, finite element analysis and experimental investigations. Sharma and Ramandeep [9] modelled and analysed the transverse deflection of thermoelastic micro-cantilever beam due to concentrated point load.

However the analytical study for the transverse deflection in thermoelastic micro simply supported beam due to patch loading has not been reported yet.

In this paper, we present an analytical solution for transverse deflection in thermoelastic micro simply-supported beam due to patch load. The Laplace transform technique has been used to find the analytic solution of the model. The obtained analytic expression for deflection of simply supported beam has also been computed numerically by using MATLAB software. The computer simulated results have been presented graphically.

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Figure 1: Dimensionless deflection of pinned-pinned beam with dimensionless length at different times.

Figure 2: Dimensionless deflection of pinned-pinned beam with dimensionless length for different modes.

Fig. 1 demonstrates the transverse deflection of a micro-beam, which is simply-supported at both the ends. The deflection profiles have been noticed to be symmetrical about the mid-point of the beam. It is observed that the transverse deflection in a beam decreases with the increase in time. The reactions generated at the ends of the beam are clearly visible from the profiles.

Fig.2 shows the deflection of pinned-pinned beam for different modes for time . Here the profile of n=1 has been extrapolated in scale to have it on the same scale for discussion purpose. It is observed that deflection is maximum in case of fundamental mode as compared to that for second and higher modes.

CONCLUSION:

A closed form of mathematical model which governs the flexural vibrations in a transversely isotropic thermoelastic beam resonator subjected to time harmonic patch load has been developed under Euler-Bernoulli conditions. Laplace transform technique has been successfully used to obtain the analytical expressions for transverse deflection in a micro- beam under pinned-pinned conditions. The computer simulated results show that the deflection of fundamental mode is maximum at the mid-point of the beam and it decreases and decays with the increase in time. The deflection profiles are noticed to be symmetrical about the mid-point of beam. The deflection in case of fundamental mode of vibration is significantly large as compared to second and higher modes. The study may find applications in design and construction of beam type MEMS/ NEMS devices such as sensors, actuators and energy harvesters.

ACKNOWLEDGEMENT:

The authors gratefully acknowledge to Council of Scientific and Industrial Research (CSIR), New Delhi for providing financial assistance via Grant No. 09/918(0002)2010-EMR-I and 25(0184)/10/EMR-II to complete this work.

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Received on 28.01.2013 Accepted on 05.02.2013

Research J. Science and Tech 5(1): Jan.-Mar.2013 page 153-159