The New Boundary Condition Effect on The Free Vibration Analysis of Micro-beams Based on The Modified Couple Stress Theory

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1 International Research Journal of Applied and Basic Sciences 2015 Available online at ISSN X / Vol, 9 (3): Science Explorer Publications The New Boundary Condition Effect on The Free Vibration Analysis of Micro-beams Based on The Modified Couple Stress Theory Ahmad Ghanbari 1, Alireza Babaei 2 1. Department of Engineering-Emerging Technologies, University of Tabriz, Tabriz, Iran 2. Department of Mechanical Engineering, University of Tabriz, Tabriz, Iran Corresponding author babaeiar.mech.eng@gmail.com ABSTRACT: Based on the modified couple stress theory, the free vibration behavior of micro scale Bernoulli-Euler cantilever beam carrying an added mass is analytically investigated. In this study the nondimensional frequency equation for the clamped- free micro beam is obtained, using the variational methods and the most famous, Hamilton principle. The influence of the attached mass at the free end upon the first non-dimensional natural frequency is reported so that whit the increment of the mass attached to the micro beam, the non-dimensional natural frequency is decreasing. In the other case, the frequency shift for different values of the material mass ratio corresponded to the beam mass is investigated. It is remarkable that the different behavior of the micro beam is due to the weight- force and mass inertia of the attached mass during oscillation which is induced in a concentrate shape. Also comparison with the technical other article is done; showing the expected frequency shift. Key Words: attached mass, free vibration, fundamental frequency, Micro electromechanical systems, modified couple stress theory. INTRODUCTION With the advance of nanotechnology, micro beams, in which its thickness is generally on the order of microns and sub-microns, have been widely used in many applications of micro devices (i.e., atomic force microscopes (AFMs), the various applications of micro-electro-mechanical systems (MEMS)). Micro-electromechanical Systems or MEMS is a technology that in its most general form can be defined as miniaturized mechanical and electro-mechanical elements (i.e., devices and structures) that are made using the techniques of micro fabrication. While the functional elements of MEMS are miniaturized structures, sensors, actuators and microelectronics, the most notable (and perhaps the most interesting) elements are the micro sensors and micro actuators. However, it is experimentally proved that the size effect also becomes important for the mechanical behavior of microstructures, when the dimensions of structures become on the order of microns and sub-microns. As a result it is vital to consider the size effect in the analysis of mechanical behavior of microstructures. It is good to say that the size dependency does not appear in classical elasticity theories; for example, the experiments conducted by Lam et al. (2003) on micro - cantilever beams, indicate that tip deflection of a micro - cantilever beam varies with beam thickness even when the ratio of the beam length to the beam thickness is kept constant. This observation contradicts the prediction of the classical elasticity theory, which anticipates a tip deflection independent of the beam thickness as long as the beam length to beam thickness ratio is fixed. Experimental results reported by Andrew and Jonathan (2005) clearly illustrate that stiffness of micro - cantilever plates are at least four times greater than those calculated by the classical elasticity theory. Also as controlled experiments in micro scale is hard and costs a lot of money, the development of decent mathematical models for micro structures is the best way to consider approximate analysis of microstructures. Yang et al. (2002) has developed the modified couple stress theory which is derived from the general strain gradient theory. This modified theory has just one additional material length scale parameter, since for the non-classical elasticity theories, the most complicated part is to determine the additional material parameters, it can be supposed as an advantage for the modified couple stress theory and the other preference of the modified couple stress theory is due to the fact that the strain energy density function depends

2 only on the strain tensor and the symmetric part of the curvature tensor. For more explanation it is noticeable that, in the modified couple stress theory, the effects of the dilatation gradient and the deviatoric stretch gradient are neglected, thus only a single material length scale parameter is required, which relates the couple stress to the symmetric part of the rotational gradient. The model considered in this study can be explained for many applications in microbiological devices and micromechanical systems. This model has a sharp sensitivity to environment mass and inertia changes, therefor many researchers explored similar models for various investigations especially in corporation to resonators. Rodolfo Sánchez - Fraga et al. (2014) studied the mass detection using vertical and horizontal designs and showed that the vertical has more sensitivity but less accuracy. Dai et al. (2009) studied the nano mechanical mass detection using nonlinear oscillators based on continuum elastic model and obtained that nonlinear oscillation leads to the unique resonant frequency shift due to mass adsorption, quite different from that in harmonic oscillation. R. Aghazadeh et al. (2014) utilized the modified couple stress theory for the static and free vibration analysis of functionally graded micro scale beams and stating the influences of material length scale parameter upon the first fundamental natural frequency when it is not constant. M. Şimşek (2014) investigated the free vibration behavior of micro beams, founded on the nonlinear foundation using the modified couple stress theory and expressing the foundation parameters stiffness influences upon natural frequency. H. Rokni et al. (2014) did research for the free vibration behavior of functionally graded micro beams based on the same modified theory and since the mechanical properties can vary continuously through the axial coordinate, they tried to describe the condition for which the maximum frequency can be obtained. with respect to the researches done, it is clear that the influences of the attached mass for pure metal structure upon the fundamental frequency of a clamped- free micro beam is not studied, although H. Lee et al studied the similar problem but they used the nonlocal elasticity theory. The main objective of the present study is to examine the clamped- free micro beam supporting an attached mass using the modified couple stress elasticity theory and comparing the results with the case of free end; so that the influence of the attached mass upon the vibrational response can be obtained and a proper simulation for the practical cases in industry and bio mechanics is studied. The methodology is based on the Euler-Bernoulli beam theory. Governing partial differential equations are derived by following the variational approach and using Hamilton s principle. Governing equations are solved analytically. Mathematical Formulation In order to use the modified couple stress theory, Consider a flat, thin micro cantilever beam of length L, width w and uniform thickness h, as depicted in Fig. 1. According to the modified couple stress theory expressed by Yang et al. (2002) the stored strain energy U s of an elastic linear isotropic Euler- Bernoulli micro-beam is expressed in terms of the macro-scale strain tensor and the symmetric curvature tensor, which is related to the micro-scale rotation of material particles. Figure 1 U s = 1 2 (σ ij: ε ij + m ij : χ ij ) dv V (1) where V is volume, σ ij designates the Cauchy stress tensor, ε ij is the strain tensor, m ij stands for the deviatoric part of the couple stress tensor, and χ ij represents the symmetric curvature tensor. The tensors ε ij and χ ij are defined by: ε ij = 1 2 (u ij + u ji ) (2) 275

3 χ ij = 1 2 (e ipqε qj,p + e jpq ε qi,p ) (3) u in Eq. (2) is the displacement vector; e ipq in Eq. (3) denotes the alternating tensor; and a comma stands for differentiation. Constitutive relations regarding the Cauchy stress tensor and the deviatoric part of the couple stress tensor can be written in the following form: σ ij = 2με ij + λδ ij ε kk (4) m ij = 2μl 2 χ ij (5) Where λ and μ are Lame, s parameters. Also ν is the Poisson s ratio and E is the modulus of elasticity. By means of two constraint equations, the number of elastic parameters needed to describe the mechanical properties of a structure will reduce to two parameters, the equations mentioned above are λ = Eν (1+ν)(1 2ν), μ = E 2(1+ν) (6) l in Eq. (5) is the material length scale parameter characterizing the effect of the couple stress. In the modified couple stress theory, the length scale parameter is also defined as a quantity whose square is equal to the ratio of the modulus of curvature to the modulus of rigidity (Mindlin, 1963; Park and Gao, 2006). Therefore, introduction of the strain gradient into the formulation results in the definition of an intrinsic length scale parameter, which can be simply defined as a property of the elastic medium similar to modulus of elasticity and rigidity and Poisson s ratio. The displacement components in an Euler-Bernoulli beam can be represented by (Park and Gao, 2006): u 1 (x 1, x 3, t) = x 3 w x 1 (7) u 2 (x 1, x 3, t) = 0 (8) u 3 (x 1, x 3, t) = w(x 1, t) (9) In the above equations u i, (i = 1,2,3) are the general displacement components in x 1, x 2, x 3 directions. By utilizing this displacement fields and Eqs. (2) - (5), one can derive the elements of ε ij, χ ij, σ ij and m ij as follows: ε 11 = x 3 ( w ) (10) x 1 x 1 χ 12 = χ 21 = 1 ( w ) (11) 2 x 1 x 1 σ 11 = Ex 3 ( w ) (12) x 1 x 1 m 12 = m 21 = μl 2 ( w ) (13) x 1 x 1 The other components are zero. By substitution of the Eqs. (10)-(13) into Eq. (1) gives 2 U s = 1 {E ( ( w )) (x 2 2 x 1 x 3 + l2 1 2 )} dv (14) V Total kinetic energy of the Euler- Bernoulli beam carrying an attached mass can be expressed as 276

4 L T = 1 2 ρa( w 0 t )2 dx t) M( w(l, ) 2 (15) t Based on Hamilton's principle (Ginsberg, 2001), the governing equation of the beam along with initial conditions and boundary conditions can be determined by using the following variational equation: t 2 δ [ (T U s ) dt] = 0 (16) t 1 Substituting the Eqs.(14),(15) into the Eq.(16) and after some mathematical process, one can obtain the governing equation as following: Q 4 w x 4 + ρa 2 w 1 t 2 = 0 (17) Since it is the case of free vibration, it seems reasonable to expect that transverse displacement varies harmonically with respect to time variable. Also due to homogeneity of the partial differential equation obtained, it is acceptable to use the method of separation of variables like: w(x 1, t) = W(x 1 )T(t) (18) Substituting of the Eq.(18) into the Eq.(17) yields the following ordinary differential equation: Q d4 W d 4 ρaω 2 W = 0 x 1 (19) Q = EI x2 x 2 + GAl 2 (20) Where ω denotes the natural frequency of vibration, W is the displacement amplitude at x 1 = 0 and I x2 x 2 is the second moment of mass inertia about the x 2 direction. By introducing the following parameters x = x 1 L (21), β 4 = (ρaω2 L 4 ) Q (22) And substituting into the Eq.(19), one can obtain d 4 W d 4 x β4 W = 0, 0 x 1 (23) And the corresponding boundary conditions are: W(0) = 0 (24) dw (0) = 0 dx (25) d 2 W (1) = 0 dx2 (26) d 3 W dx (1) 3 + (M Q ω2 ) W(1) = 0 (27) The general solution of the Eq.(23) can be supposed as : W(x) = C 1 cosh βx + C 2 sinh βx + C 3 cos βx + C 4 sin βx (28) By exerting the boundary conditions into the general solution of Eq.(23), the following matrix form can be obtained: [A]{C} = 0 (29) Which the components of [A] are as follows: A(1,1) = 1, A(1,2) = 0, A(1,3) = 1, A(1,4) = 0 (30) A(2,1) = 0, A(2,2) = 1, A(2,3) = 0, A(2,4) = 1 (31) A(3,1) = β 2 cosh βl, A(3,2) = β 2 sinh βl, A(3,3) = cos βl (32) A(3,4) = sin βl (33) A(4,1) = β 3 sinh βl + ( M Q ω2 ) cosh βl, A(4,2) = β 3 cosh βl + ( M Q ω2 ) sinh βl, A(4,3) = β 3 sin βl + ( M Q ω2 ) cos βl, 277

5 A(4,4) = β 3 cos βl + ( M Q ω2 ) sin βl (34) And the coefficient vector matrix including C i, (i = 1,2,3,4). As this is the homogenous system of equations, according to linear algebra it is expected to consider the determinant of the coefficient matrix equal to zero. which introduces the frequency equation as fallows: Rβ(tanh βl tan βl) = 0 (35) (cosh βl)(cos βl) In the above non transcendental equation R is the mass ratio defined as: R = M (36) ρal RESULTS AND DISCUSSIONS The model presented in this study shows the frequency shift due to the presence of the attached mass at the free end of the micro beam. Based on the modified couple stress theory, we have derived the frequency equation corresponded to the Euler- Bernoulli beam to analyze the effects of non-classical parameter, also renowned as the couple stress parameter along with the fixed attached mass on the frequency shift of micro beambased mass sensor. According to the equation, the relationship and comparison between the dimensionless frequency parameter and carrying the added mass on the cantilever mass- sensor for the first mode of vibration is shown and compared in Table1. Table 1. Comparison of the dimensionless natural frequency, free end and carrying attached mass. R Rokni et al. (2014) ωl 2 ρa f shows the frequency shift, the second column is the reference for comparison Q f CONCLUSION The results show the important shifts of frequency for the new model to use it as a decent micro beambased mass sensor model for mass detection in practical cases. It is clear that the definition of this new problem changes the frequency equation remarkably, since for the fourth boundary condition, we face a new dynamical boundary condition derived from the force equilibrium of the oscillating added mass, which causes strong dependency of the dynamical response of micro beam to the inertia of the added mass. Also we conclude that since this is the micro structure (the boom of the bending is negligent), it is good design for supporting the large mass ratios. REFERENCES Aghazadeh R, Cigeroglu E, Dag S Static and free vibration analyses of small-scale functionally graded beams possessing a variable length scale parameter using different beam theories. European Journal of Mechanics-A/Solids, 46, Dai MD, Eom K, Kim CW Nanomechanical mass detection using nonlinear oscillations. Applied Physics Letters, 95(20), Ginsberg JH Mechanical and Structural Vibrations: Theory and Applications. John Wiley & Sons. Ke LL, Yang J, Kitipornchai S Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams. Composite Structures. 92, Rao SS Vibration of continuous systems. John Wiley & Sons. Reddy JN Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45(2), Rokni H, Milani AS, Seethaler RJ Size-dependent vibration behavior of functionally graded CNT-Reinforced polymer microcantilevers: Modeling and optimization. European Journal of Mechanics-A/Solids, 49,

6 Şimşek M Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He s variational method. Composite Structures, 112, Vel SS, Batra RC Exact solution for thermoelastic deformations of functionally graded thick rectangular plates. AIAA journal, 40(7), Yang F, Chong ACM, Lam DCC, et al Couple stress based strain gradient theory for elasticity.int. J. Solids Structures. 39 (10), 2731e