A theoretical study for the vibration of a cantilever microbeam as a free boundary problem

Accepted Manuscript

A theoretical study for the vibration of a cantilever microbeam as a free boundary problem Davoud Abdollahi, Samad Ahdiaghdam, Karim Ivaz, Rasool Shabani PII: DOI: Reference:

S0307-904X(15)00575-2 10.1016/j.apm.2015.09.041 APM 10740

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

23 February 2015 11 June 2015 22 September 2015

Please cite this article as: Davoud Abdollahi, Samad Ahdiaghdam, Karim Ivaz, Rasool Shabani, A theoretical study for the vibration of a cantilever microbeam as a free boundary problem, Applied Mathematical Modelling (2015), doi: 10.1016/j.apm.2015.09.041

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Highlights • A new method is proposed using boundary integral Eqs. with free boundary approach • The domains of the potential functions change in the reformed model

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• The obtained results are useful and validated with earlier results available

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A theoretical study for the vibration of a cantilever microbeam as a free boundary problem Davoud Abdollahia,∗, Samad Ahdiaghdama , Karim Ivaza , Rasool Shabanib a Faculty

of Mathematical Sciences, University of Tabriz, P. O. Box 51666-16471, 29 Bahman Blvd, Tabriz, Iran Engineering Department, Urmia University, P. O. Box 57561-51818, No. 165, Daneshgah Blvd, Urmia, Iran

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b Mechanical

Abstract

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Vibration of a beam has extensive applications in engineering and industry.The objective of this article is to provide a novel analytical model by free boundary approach to derive a more exact model for the small oscillations of a cantilever microbeam in contact with an incompressible bounded fluid in a cavity.First, a system of integral equations is proposed to solve and interpret the problem. Then, the existence of the solution is proved by the use of the Banach fixed point theorem At the end, an example is presented for verifying the proposed method and comparing the results with the fixed boundary approach. Keywords: Free vibration, Micro-beam, Free boundary equation, Added mass

1. Introduction

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Beam vibrations described by Timoshenko, have been applied over the years by many scientists in various majors. Micro-beams with different end conditions are used in many micro-devices such as micro-resonators [1-3], microswitches [4], Atomic Force microscopes [5], micro-actuators [6], micro-viscosity meters [7] and micro-biosensors [8]. Lam et al. [9] developed a modified set of second order deformation metrics, the dilatation gradient vector, the deviatoric stretch gradient tensor and the rotation gradient tensor, and derived the corresponding work-conjugate stress metrics as the basic strain and stress measures for a strain gradient theory for elasticity. McFarland et al. [10] studied the effect of micro-structure upon micro-cantilever sensors. Akgoz et al. [11] presented the size-dependent buckling analysis of embedded functionally graded (FG) microbeams in thermal environment based on trigonometric shear deformation beam and modified coupled stress theories. They determined effects of thickness to material length scale parameter ratio, material property gradient index, length to thickness ratio, temperature change and Winkler parameter on critical buckling loads of embedded FG microbeams. They also investigated the size-dependent buckling of axially loaded micro-scale beams and tubules by using the same theory [12, 13]. Recently, the modified couple stress theory (MCST) has been successfully utilized to predict the mechanical behavior of microbeams. Park et al. [14] showed that the bending rigidity predicted by the MCST is greater than that calculated by the classical theory (CT). Kong et al. [15] investigated the size effect on natural frequency of the Euler-Bernoulli microbeam. They introduced a non-classical model for the electrostatically actuated microbeam by using MCST with the Rayleigh-Ritz method. S. Kong [16] also showed that normalized pull-in voltage of the microbeam by the MCST is 3.1 times greater than that predicted by the CT when the microbeam thickness equals to the material length scale parameter. Askari et al. [17] studied the size-dependent dynamic pull-in analysis of clamped-clamped microbeams under mechanical shock based on MCST. They were able to predict dynamic pull-in voltage for a system under high shock accelerations. Wang et al. [18] presented a mathematical model and a numerical algorithm for the bending and post-buckling of a microbeam within the context of Euler-Bernolli beam theory involving geometric nonlinearity and MCST involving one internal scale parameter. Their proposed model was a size-dependent one able to interpret the small scale effect. Recently, ∗ Corresponding

author. Tel.: +989126305170. Email addresses: [email protected] (Davoud Abdollahi), [email protected] (Samad Ahdiaghdam), [email protected] (Karim Ivaz), [email protected] (Rasool Shabani) Preprint submitted to Elsevier

October 1, 2015

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some published papers investigated the added mass effects of the interacting fluids in micro-structures. In these cases, the operating microbeam may be surrounded by a liquid which exerts a reaction force characterized as the added mass in the dynamic analysis [19-21]. In the presented paper, free vibration of a cantilever microbeam surrounded by a bounded incompressible and inviscid fluid domain is studied. The outline of this study is organized as follows: In section 2, we briefly discuss the governing equations of the problem. In the third section, we are interested in convey the problem to a free boundary problem. In section 4, the existence of the solution is proved by using Banach fixed point theorem. In section 5, an example is illustrated for validating the proposed method. Finally, section 6 completes this article with a brief conclusion. 2. Mathematical model

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Shabani et al. [21] analyzed the vibration of a cantilever microbeam submerged in a fully contained rectangular cavity, where L and a are lengths of the microbeam and the cavity, respectively (See Fig. 1). The off-center position of the beam is specified by its distance from the lower and upper sides of the cavity, H1 and H2 . The microbeam and cavity are assumed to have an equal width b. The length of the cavity is longer than the length of microbeam making it possible for two fluid domains to interact.

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Figure 1: Cantilever micro-beam submerged in a fully contained cavity [3]

The equations of the system were written as follows: ∂4 w ∂2 w + ρB 2 = b(P1 − P2 ), 4 ∂x ∂t

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EI

0 < x < L,

t>0

(1)

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where E is Young’s modulus, I = bh3 /12 is the moment of inertial of the microbeam, and the constants ρB ,h and b are the mass per unit length, thickness and width of the beam, respectively. For the length of the beam, we take the x coordinate of the fixed end, zero and the right end as L . Since the Eq. (1) involves a second order derivative with respect to time and a fourth order derivative with respect to x , four boundary conditions and two initial conditions are needed for finding a unique solution. The displacement w(x, t) and its slope are zero at the fixed end, while at the free end; the bending moment and shear force are zero. Usually, the values of transverse displacement and velocity are specified as f (x) and g(x) respectively. Thus, these statements are translated to the following set of boundary and initial conditions:  ∂w  w| x=0 = 0, t>0 (Fixed end) (a)   ∂x | x=0 = 0,   ∂2 w ∂3 w (2) | = 0, | = 0, t>0 (Free end) (b)   ∂x2 x=L ∂x3 x=L   ∂w  w|t=0 = f (x), 0
(3)

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where ϕ1 and ϕ2 are the velocity potential of the oscillating fluids (with density ρ f ). Now assuming incompressibility and inviscidity, the fluid movement induced by vibration of the microbeam in each domain (Fig. 1) is described using the associated velocity potential functions, which satisfy the following Laplace equations:

and

(a) (b) (c)

 2  ∇ ϕ2 (x, y2 , t) = 0, 0 < x < a, 0 < y 2 < H2 ,    ∂ϕ2   0 < y2 < H2 ,   ∂x | x=0,a = 0    ∂ϕ2 | = 0 0 < x < a, y =H  ∂y2 2 2     ∂w(x,t)     0
(a) (b) (c)

(5)

(d)

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(4)

(d)

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 2  ∇ ϕ1 (x, y1 , t) = 0, 0 < x < a, 0 < y1 < H1 ,    ∂ϕ1   | = 0 0 < y < H ,  x=0,a 1 1  ∂x    ∂ϕ1 | 0 < x < a, y =0 = 0  ∂y 1 1     ∂w(x,t)     0
The researchers in [21] investigated the free vibration of submerged cantilever microbeam using the FourierBessel series expansion and linear potential theory. Their analysis was based on the assumption of equal kinematic conditions on the two beam-fluid interfaces and one imaginary fluid-fluid interfaces in a fixed height (See Equations (4d) and (5d)), whereas it is clear that each point along the vibrating beam has not equal height along the x coordinate. Therefore, the real structure of the system should be reformulated as a free boundary problem.

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3. Motivation of the free boundary model

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In this section we sketch a derivation of the free boundary for the model (1-5). We should mention that Shabani et al. [21] derived the fluid pressure formulas in two regions as Eq. (3) by neglecting the elevation due to the microscale and ignoring the nonlinear term. Considering the structure of the system, the height of the off-center is not the same throughout the length of the microbeam. Therefore, in order to get more accurate model, we were motivated to reformulate the boundary conditions (4d) and (5d) with a change in the fluid pressure formulas (3) as follows:  ∂ϕ ∂w(x,t) 1   0
∂y2

∂y1

1 ,t) P1 = −ρ f ∂ϕ1 (x,y |y1 =H1 +w(x,t) , ∂t ∂ϕ2 (x,y2 ,t) P2 = −ρ f |y2 =w(x,t) , ∂t

0 < x < L, 0 < x < L,

t>0 t>0

(a) (b)

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for the following Laplace equations with free boundary domains: ( 0 < x < L, 0 < y1 < H1 + w(x, t) 2 ∇ ϕ1 (x, y1 , t) = 0, L < x < a, 0 < y1 < H1 ( 0 < x < L, w(x, t) < y2 < H2 2 ∇ ϕ2 (x, y2 , t) = 0, L < x < a, 0 < y2 < H2

(8)

(9) (10)

In the next section we will prove the existence of the solution for the reformulated structure of the problem by using Banach fixed point theorem.

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4. Existence of the Solution First, we begin by transforming the problem to a system of boundary integral equations. For this aim, we apply the following relation to solve Laplace equations (8) and (9): ! Z ∂ϕ(Q) 1 ∂ ϕ(P) = ln |P − Q| − ϕ(Q) ln |P − Q| dsQ , P ∈ ∂Ω (11) π ∂Ω ∂nQ ∂nQ

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where region Ω is not simply-connected and whose boundaries need not be smooth [22, Chap. 13]. Domains of the velocity potentials ϕ1 and ϕ2 are respectively as follows:

n o n o Ω1 = (x, y1 , t) | 0 ≤ x ≤ L, 0 ≤ y1 ≤ H1 + w(x, t), 0 ≤ t ≤ σ ∪ (x, y1 , t) | L ≤ x ≤ a, 0 ≤ y1 ≤ H1 , 0 ≤ t ≤ σ and

n o n o Ω2 = (x, y2 , t) | 0 ≤ x ≤ L, w(x, t) ≤ y2 ≤ H2 , 0 ≤ t ≤ σ ∪ (x, y2 , t) | L ≤ x ≤ a, 0 ≤ y2 ≤ H2 , 0 ≤ t ≤ σ .

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Considering the common boundary between the domains of ϕ1 and ϕ2 , there will be n o D1 = (x, y, t) | 0 ≤ x ≤ L, y = { y1 = H1 + w(x, t), y2 = w(x, t) }, 0 ≤ t ≤ σ n o D2 = (x, y, t) | L ≤ x ≤ a, y = { y1 = H1 , y2 = 0 }, 0 ≤ t ≤ σ

where D1 , D2 ⊂ Γ1 ∩ Γ2 . We should mention that Γ1 and Γ2 are the boundaries of Ω1 and Ω2 , respectively. Let us apply the formula (11) for ϕ1 to get

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! Z Z  Z ∂w(ξ, τ) ∂ ∂ϕ2 (ξ, η2 , τ) 1  ln |P − Q|dsQ + ϕ1 (Q) ln |P − Q|dsQ , (12) ϕ1 (P) = η =0 ln |P − Q|dsQ − π ∂τ ∂η2 ∂nQ 2 D1 Γ1 D2

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where P = (x, y1 , t) ∈ Γ1 and Q = (ξ, η1 , τ) ∈ Γ1 .

Similarly, we conclude that by assuming P = (x, y2 , t) ∈ Γ2 and Q = (ξ, η2 , τ) ∈ Γ2 , the velocity potential ϕ2 will be as follows: ! Z Z  Z 1  ∂w(ξ, τ) ∂ϕ1 (ξ, η1 , τ) ∂ ln |P − Q|dsQ + ln |P − ln |P − Q|ds − ϕ (Q) Q|ds η =H 2 Q Q . (13) π ∂τ ∂η1 ∂nQ 1 1 D1 D2 Γ2

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ϕ2 (P) =

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The fundamental solution of Eq. (1) is given in [23] as ∂ ∂t

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w(x, t) =

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f (ξ)G(x, ξ, t) dξ +

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g(ξ)G(x, ξ, t) dξ +

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Z tZ 0

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b(P1 − P2 )G(x, ξ, t − τ) dξ dτ,

(14)

which the Green’s functions can be evaluated from the formula

where α =

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G(x, ξ, t) =

∞ X ψn (x)ψn (ξ) sin(λ2n αt), 2 2 λ αkψ k n 2 n n=1 L (0,L)

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and

      ψn (x) = sinh(λn L) + sin(λn L) cosh(λn x) − cos(λn x) − cosh(λn L) + cos(λn L) sinh(λn x) − sin(λn x) , 5

(16)

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which λn are positive roots of the transcendental equation cosh(λL)cos(λL) = −1 . The numerical values of the roots can be calculated from λn = µLn , where µ1 = 1.875104,

µ2 = 4.694091,

 1 µn = n − π. 2

n≥3

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The values of λn L are related to the natural frequencies of the dry beam (ωn ) by the following [21]: s EI 2 ωn = (λn L) ρ B L4

(17)

(18)

Therefore, we rewrite the boundary integral equations (12-14) in terms of ϕ1 ,ϕ2 and w such as

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 !  R ∂w(ξ,τ)  R R ∂ϕ (ξ,η ,τ)   1 ∂ 2 2   ϕ (P) = ln |P − Q| ds + ln |P − Q| ds − ϕ (Q) ln |P − Q| ds (a)  1 Q Q Q  π ∂τ ∂η2 ∂nQ D1 D2 Γ1 1  η2 =0   !   R  R R 1 ,τ)   ϕ2 (P) = π1 D ∂w(ξ,τ) ln |P − Q| dsQ + D ∂ϕ1 (ξ,η ln |P − Q| ds − Γ ϕ2 (Q) ∂n∂Q ln |P − Q| dsQ (b)  Q  ∂τ ∂η 1 1 2 2  η1 =H1     R R R R     w(x, t) = L f (ξ)Gt (x, ξ, t) dξ + L g(ξ)G(x, ξ, t) dξ + bρ f t L ∂ϕ2 (ξ,w(ξ,τ),τ) − ∂ϕ1 (ξ,H1 +w(ξ,τ),τ) G(x, ξ, t − τ) dξ dτ. (c) ρB 0 0 ∂τ ∂τ 0 0 (19) In the next section, by using Banach Fixed Point Theorem, we construct some conditions which would give us the existence and uniqueness of the solution in the above system (19).

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4.1. Iteration Scheme The system of integral equations (19) will be solved iteratively. Given Ψ = (ϕ1 , ϕ2 , w) , we define the operator e) through the three-stages procedure as follows: T : (ϕ1 , ϕ2 , w) −→ (ϕe1 , ϕe2 , w   ϕe = T 1 Ψ,    1 e2 = T 1 Ψ, ϕ (20)     w e=T Ψ 3

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where the nonlinear operators T 1 ,T 2 and T 3 are defined as Eq. (19). We will show that T has a fixed point which obviously provides a solution to the original integral equations (19). Since, Eqs. (19a) and (19b) are the representation of Laplace equations as boundary integral equations [22] and Eq. (19c) is the fundamental solution of Eq. (1), so the fixed point of the system of integral equations (19) would satisfy the model (1-5) that had been reformulated as a free boundary problem.

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4.2. Invariant Ball n o In the space of χ = Ψ = (ϕ1 , ϕ2 , w) | ϕ1 , ϕ2 , w ∈ C 1 (Γ), Γ = Γ1 ∪ Γ2 , we define the norm n o kΨkχ = max kϕ1 kC 1 (Γ) , kϕ2 kC 1 (Γ) , kwkC 1 (Γ) , that makes χ a Banach space. The fixed point will be sought in the closed ball n o Bσ,M,a = Ψ = (ϕ1 , ϕ2 , w) | kΨkχ ≤ M ⊂ χ,

(21)

(22)

with M, σ and a to be determined. Referring to [24], the operators T 1 , T 2 : C 1 (Γ) −→ C 1 (Γ) are well defined. The green function G(x, ξ, t) in the operator T 3 is clearly in C ∞ (Γ). Then, by assuming f (x) and g(x) of (2c) in C 1 (Γ) , the third operator T 3 : C 1 (Γ) → C 1 (Γ) is well defined as well. Hence, T : C 1 (Γ) → C 1 (Γ) is valid. Now, we need to prove that for suitable M, σ and a , the operator T : B → B is well defined. This is equivalent of demonstrating kT i kC 1 (Γ) ≤ M, i = 1, 2, 3. To do this, let us introduce the following notion for | α |≤ 2 as 6

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λα = max P∈Γ

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Γ

| Dα (ln |P − Q|) | dsQ ,

(23)

similar to Hilbert-Schmidt norms [25]. For the first and second integrals of Eq.(19a), we will have

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and for the third integral of Eq. (19a), the upper bound is

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Z Z ∂w(ξ, τ) ∂ϕ2 (ξ, η2 , τ) = ln |P − Q| dsQ + η =0 ln |P − Q| dsQ ∂τ ∂η2 2 D1 D2 C(Γ) Z Z ∂w(ξ, τ) ∂ϕ2 (ξ, η2 , τ) max ln |P − Q| dsQ + (24) η =0 ln |P − Q| dsQ ≤ P∈Γ D1 ∂τ ∂η2 2 D2 Z Z ∂ϕ2 (ξ, η2 , τ) ∂w(ξ, τ) max ln |P − Q| dsQ + max ln |P − Q| dsQ ≤ P∈Γ D2 η2 =0 P∈Γ D1 ∂τ ∂η2 Z Z ∂w(ξ, τ) ∂ϕ2 (ξ, η2 , τ) ln |P − Q| dsQ + ln |P − Q| dsQ ≤ 2Mλ0 , max max P∈Γ P∈Γ ∂τ ∂η2 D1 D2 C(Γ) C(Γ) Z Z ∂ ∂ ϕ (Q) = max ϕ (Q) ln |P − Q| ds ln |P − Q| ds 1 1 Q Q P∈Γ ∂n ∂n Q Q Γ1 Γ1 C(Γ) Z ∂ ≤ kϕ1 (Q)kC(Γ) max ln |P − Q| dsQ ≤ Mλ1 P∈Γ ∂n Q Γ1

So, the relations (24) and (25) gives

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 M  1 T 1 Ψ(P) ≤ 2Mλ0 + Mλ1 ≤ 2λ0 + λ1 , C(Γ) π π

and by the

(25)

(26)

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! Z Z  Z ∂ 1  ∂w(ξ, τ) ∂ ∂2 ∂ϕ2 (ξ, η2 , τ) ln |P−Q| dsQ + ln |P−Q| dsQ − ln |P−Q| dsQ DP (T 1 Ψ(P)) = ϕ1 (Q) η =0 π ∂τ ∂nP ∂η2 ∂nP ∂nP ∂nQ 2 D1 D2 Γ1 (27) We can write for the first and second integrals of the relation (27),

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Z Z ∂w(ξ, τ) ∂ ∂ϕ2 (ξ, η2 , τ) ∂ ln |P − Q| ds + ln |P − Q| ds = η =0 Q Q C(Γ) ∂τ ∂n ∂η ∂n 2 P 2 P D1 D2 Z Z ∂w(ξ, τ) ∂ ∂ϕ2 (ξ, η2 , τ) ∂ max ln |P − Q| dsQ + ln |P − Q| dsQ ≤ η2 =0 ∂nP P∈Γ D1 ∂τ ∂nP ∂η2 D2 Z Z ∂w(ξ, τ) ∂ ∂ϕ2 (ξ, η2 , τ) ∂ max ln |P − Q| dsQ + max ln |P − Q| dsQ ≤ η =0 P∈Γ D2 P∈Γ D1 ∂τ ∂nP ∂η2 ∂nP 2 Z ∂w(ξ, τ) ∂ ln |P − Q| dsQ max P∈Γ ∂τ D1 ∂nP C(Γ) Z ∂ϕ2 (ξ, η2 , τ) ∂ + max ln |P − Q| dsQ ≤ 2Mλ1 , (28) C(Γ) P∈Γ D2 ∂nP ∂η2

and for the third integral of the relation (27), we have

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Z Z ∂2 ∂2 ϕ1 (Q) ϕ1 (Q) = max ln |P − Q| dsQ ln |P − Q| dsQ P∈Γ ∂nP ∂nQ ∂nP ∂nQ Γ1 Γ1 C(Γ) Z ∂2 ≤ kϕ1 (Q)kC(Γ) max ln |P − Q| dsQ ≤ Mλ2 P∈Γ Γ1 ∂nP ∂nQ

Then,

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  1 M DP (T 1 Ψ(P)) ≤ 2Mλ1 + Mλ2 ≤ 2λ1 + λ2 , C(Γ) π π So, from the relations (26) and (30), we get

n o n o M T 1 Ψ(P) = max kT 1 Ψ(P)kC(Γ) , kDP (T 1 Ψ(P))kC(Γ) ≤ max 2λ0 + λ1 , 2λ1 + λ2 . C 1 (Γ) π

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Therefore, for adjusting kT 1 kC 1 (Γ) ≤ M, we need to have n o max 2λ0 + λ1 , 2λ1 + λ2 ≤ π.

(29)

(30)

(31)

(32)

In a similar fashion, we get the condition (32) by adjusting kT 2 kC 1 (Γ) ≤ M for ϕ2 as well.

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Now, we just need to show that, T 3 is from B into itself. First, the following notions are defined:  RL   max 0 | DαG(x, ξ, t) | dξ (a)   µα = (x,t)∈D  1 RtRL     | DαG(x, ξ, t − τ) | dξ dτ (b)  να = max (x,t)∈D1 0

(33)

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which |α| ≤ 2. Then, we can write from Eq. (19c) that

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Z L Z L k T 3 kC(Γ) = max f (ξ)Gt (x, ξ, t) dξ + g(ξ)G(x, ξ, t) dξ P∈Γ 0 0 Z tZ L bρ f ∂ϕ2 (ξ, w(ξ, τ), τ) ∂ϕ1 (ξ, H1 + w(ξ, τ), τ)  − G(x, ξ, t − τ) dξ dτ + ρB 0 0 ∂τ ∂τ Z L Z L | Gt (x, ξ, t) | dξ + k g kC(Γ) max | G(x, ξ, t) | dξ ≤ k f kC(Γ) max P∈Γ P∈Γ 0 0 Z tZ L bρ f ∂ϕ2 ∂ϕ1 + k − kC(Γ) max | G(x, ξ, t − τ) | dξ dτ P∈Γ ρB ∂τ ∂τ 0 0 bρ f ≤ k f kC(Γ) µ1 + k g kC(Γ) µ0 + 2Mν0 , (34) ρB

and

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Z L Z L f (ξ)DPGt (x, ξ, t) dξ + g(ξ)DPG(x, ξ, t) dξ k DP T 3 kC(Γ) = max P∈Γ 0 0 Z tZ L bρ f ∂ϕ2 (ξ, w(ξ, τ), τ) ∂ϕ1 (ξ, H1 + w(ξ, τ), τ)  + − DPG(x, ξ, t − τ) dξ dτ ρB 0 0 ∂τ ∂τ Z L Z L ≤ k f kC(Γ) max | DPGt (x, ξ, t) | dξ + k g kC(Γ) max | DPG(x, ξ, t) | dξ P∈Γ P∈Γ 0 0 Z tZ L bρ f ∂ϕ2 ∂ϕ1 + − kC(Γ) max | DPG(x, ξ, t − τ) | dξ dτ k P∈Γ ρB ∂τ ∂τ 0 0 bρ f ≤ k f kC(Γ) µ2 + k g kC(Γ) µ1 + 2Mν1 , (35) ρB Therefore, by combining the estimations (34) and (35), we get

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o n bρ f bρ f 2Mν0 , k f kC(Γ) µ2 + k g kC(Γ) µ1 + 2Mν1 . kT 3 kC 1 (Γ) = max k f kC(Γ) µ1 + k g kC(Γ) µ0 + ρB ρB

For adjusting kT 3 kC 1 (Γ) ≤ M , we can write ( ) k f kC(Γ) µ1 + k g kC(Γ) µ0 k f kC(Γ) µ2 + k g kC(Γ) µ1 max , ≤ M. bρ bρ 1 − ρBf 2ν0 1 − ρBf 2ν1

(36)

(37)

The above condition is insured by choosing suitable σ and L ≤ a for the relation (33) in which n o 2bρ f max ν0 , ν1 < 1. ρB

(38)

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0 ≤

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The condition (38) would also give us the contraction of T 3 , which will be proved later on. Thus, M depends explicitly on k f kC(Γ) and k g kC(Γ) . It means that for suitable σ and a , the relations (32) and (37) guarantees T : B→ B. Finally, we will prove that, T : B → B is a contraction by the selected values for σ , a and M in the conditions (32) and (37). Let ϕ1 = T 1 Ψ and ϕ01 = T 1 Ψ0 . For the ϕ1 − componet , the estimations are as follows:

where

T 1 Ψ(P) − T 1 Ψ0 (P) =

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 R  ∂ϕ (ξ,η ,τ) R  ∂w(ξ,τ) ∂w0 (ξ,τ)   2 2  −   κ1 = D1 ∂τ − ∂τ ln |P − Q| dsQ + D2 ∂η2   R   ∂ 0   κ2 = ϕ1 (Q) − ϕ1 (Q) ∂nQ ln |P − Q| dsQ , Γ 1

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thus,

 1 κ1 − κ2 , π

(39)



∂ϕ02 (ξ,η2 ,τ)  η =0 ∂η2 2

ln |P − Q| dsQ ,

(40)

Z Z   ∂w(ξ, τ) ∂w0 (ξ, τ)  ∂ϕ2 (ξ, η2 , τ) ∂ϕ02 (ξ, η2 , τ)  k κ1 kC(Γ) = max − − ln |P − Q| dsQ + η =0 ln |P − Q| dsQ P∈Γ ∂τ ∂τ ∂η2 ∂η2 2 D1 D 2 Z Z 0 0 ∂ϕ (ξ, η , τ) ∂ϕ2 (ξ, η2 , τ) ∂w(ξ, τ) ∂w (ξ, τ) 2 max ln |P − Q| dsQ + max ln |P − Q| dsQ ≤ − − 2 ∂τ P∈Γ Γ ∂τ P∈Γ Γ ∂η2 ∂η2 Z (41) ≤ 2 k Ψ − Ψ 0 k max ln |P − Q| dsQ = 2λ0 k Ψ − Ψ 0 k, P∈Γ

Γ

and

9

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Z   ∂ 0 ϕ1 (Q) − ϕ1 (Q) k κ2 kC(Γ) = max ln |P − Q| dsQ P∈Γ Γ1 ∂nQ ≤ k ϕ1 (Q) −

ϕ01 (Q) kC(Γ)

Z ∂ ln |P − Q| dsQ ≤ λ1 k Ψ − Ψ 0 k max P∈Γ Γ ∂nQ

(42)

Therefore, by combining the estimations (41) and (42), we get

0≤

2λ0 + λ1 < 1 π

Similarly, for the derivative of T 1 , we have

 1 κ3 − κ4 , π

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DP T 1 Ψ(P) − DP T 1 Ψ0 (P) =

CR IP T

 1 2λ0 + λ1 T 1 Ψ(P) − T 1 Ψ0 (P) = k κ1 kC(Γ) + k κ2 kC(Γ) ≤ kΨ−Ψ0 k C(Γ) π π In order to make the T 1 a contraction in the Ψ− component, the following condition is received:

where

 R ∂w(ξ,τ) ∂w0 (ξ,τ) ∂ R  ∂ϕ (ξ,η ,τ)   2 2  −   κ3 = D1 ( ∂τ − ∂τ ) ∂nP ln |P − Q| dsQ + D2 ∂η2   R  2   κ4 = (ϕ1 (Q) − ϕ0 (Q)) ∂ ln |P − Q| dsQ , 1 ∂nP ∂nQ Γ 1



∂ϕ02 (ξ,η2 ,τ)  ∂η2

η2 =0

∂ ∂nP

ln |P − Q| dsQ ,

(43)

(44)

(45)

(46)

M

thus, for the norm of the relations (46), we can write

PT

ED

Z Z   ∂w(ξ, τ) ∂w0 (ξ, τ)  ∂ ∂ ∂ϕ2 (ξ, η2 , τ) ∂ϕ02 (ξ, η2 , τ)  k κ3 kC(Γ) = max − ln |P − Q| dsQ + − ln |P − Q| dsQ P∈Γ D1 ∂τ ∂τ ∂nP ∂η2 ∂η2 ∂n P D2 η2 =0 Z Z 0 0 ∂w(ξ, τ) ∂w (ξ, τ) ∂ ∂ ∂ϕ2 (ξ, η2 , τ) ∂ϕ2 (ξ, η2 , τ) ≤ − ln |P − Q| dsQ + − ln |P − Q| dsQ max max ∂τ P∈Γ P∈Γ ∂τ ∂η2 ∂η2 Γ ∂nP Γ ∂nP Z ∂ ≤ 2 k Ψ − Ψ 0 k max ln |P − Q| dsQ = 2λ1 k Ψ − Ψ 0 k, (47) P∈Γ Γ ∂nP

CE

and

AC

Z   ∂2 0 ln |P − Q| dsQ k κ4 kC(Γ) = max ϕ1 (Q) − ϕ1 (Q) P∈Γ Γ1 ∂nP ∂nQ Z ≤ k ϕ1 (Q) − ϕ01 (Q) kC(Γ) max P∈Γ

These estimations give us

Γ

∂2 ln |P − Q| dsQ ≤ λ2 k Ψ − Ψ 0 k ∂nP ∂nQ

 1 2λ1 + λ2 DP T 1 Ψ(P) − DP T 1 Ψ0 (P) = k κ3 kC(Γ) + k κ4 kC(Γ) ≤ kΨ−Ψ0 k C(Γ) π π So, the following condition give the contraction of the derivative of T 1 : 0≤

2λ1 + λ2 < 1 π 10

(48)

(49)

(50)

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which by combining the relations (44) and (50), the same condition as (32) will be obtained. Similarly, this condition gives the contraction of T 2 as well. For the contraction of w−component of T 3 , there are Z t Z L  ∂ϕ2 (ξ, w(ξ, τ), τ) ∂ϕ1 (ξ, H1 + w(ξ, τ), τ)  bρ f = max − C(Γ) ρB P∈Γ 0 0 ∂τ ∂τ 0 0 0 0  ∂ϕ2 (ξ, w (ξ, τ), τ) ∂ϕ1 (ξ, H1 + w (ξ, τ), τ) ! − G(x, ξ, t − τ) dξ dτ − ∂τ ∂τ Z t Z L bρ f ∂ϕ2 (ξ, w(ξ, τ), τ) ∂ϕ1 (ξ, H1 + w0 (ξ, τ), τ) ≤ max − ρB P∈Γ 0 0 ∂τ ∂τ 0 ! ∂ϕ (ξ, w0 (ξ, τ), τ) ∂ϕ01 (ξ, H1 + w0 (ξ, τ), τ) G(x, ξ, t − τ) dξ dτ + 2 − ∂τ ∂τ Z tZ L 2bρ f 2bρ f ≤ k Ψ − Ψ 0 k max ν0 k Ψ − Ψ 0 k, G(x, ξ, t − τ) dξ dτ = P∈Γ ρB ρB 0 0

(51)

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and for the derivation of T 3 , it can be bounded as follows:

CR IP T

T 3 Ψ(P) − T 3 Ψ0 (P)

ED

M

Z t Z L  ∂ϕ2 (ξ, w(ξ, τ), τ) ∂ϕ1 (ξ, H1 + w(ξ, τ), τ)  bρ f DP T 3 Ψ(P) − DP T 3 Ψ0 (P) = max − C(Γ) ρB P∈Γ 0 0 ∂τ ∂τ  ∂ϕ02 (ξ, w0 (ξ, τ), τ) ∂ϕ01 (ξ, H1 + w0 (ξ, τ), τ) ! − − DPG(x, ξ, t − τ) dξ dτ ∂τ ∂τ Z t Z L 0 bρ f ∂ϕ2 (ξ, w(ξ, τ), τ) ∂ϕ1 (ξ, H1 + w (ξ, τ), τ) ≤ max − ρB P∈Γ 0 0 ∂τ ∂τ 0 ! ∂ϕ (ξ, w0 (ξ, τ), τ) ∂ϕ01 (ξ, H1 + w0 (ξ, τ), τ) + 2 − DPG(x, ξ, t − τ) dξ dτ ∂τ ∂τ Z tZ L 2bρ f 2bρ f ≤ k Ψ − Ψ 0 k max ν1 k Ψ − Ψ 0 k, DPG(x, ξ, t − τ) dξ dτ = P∈Γ ρB ρ B 0 0 (52)

PT

So, the proceeding estimations yield

T 3 Ψ(P) − T 3 Ψ0 (P)

C 1 (Γ)

=

n o 2bρ f max ν0 , ν1 k Ψ − Ψ 0 k ρB

(53)

CE

Therefore, for having the contraction of T , we put the following condition: 0 ≤

n o 2bρ f max ν0 , ν1 k Ψ − Ψ 0 k < 1 ρB

(54)

AC

which is the same as the condition (38). This completes the contraction of the operator , so the system of integral equations (19) has a unique solution that produces the solution for the model (1-5) with free boundary approach. 5. An example

Let us give an illustrated example for the results obtained as above for validating the proposed method. The geometrical dimensions and fluid density on the natural frequencies of the example are listed in Table 1. We also consider f (x) = −0.05x and g(x) = 0 in the initial conditions (2c). The width of the microbeam meets b ≥ 5h. Consequently, plane strain conditions should be taken into consideration. Therefore, E is replaced by E/(1 − v2 ), where v is Poisson’s ratio. The results are calculated taking various aspect ratio ( Lb ) and thickness ratios ( hb ) for the medium type of water. Table 2 represents the comparison of the fundamental frequencies obtained by Liang et al. [25], experimental 11

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Table 1: The data used in the calculations Value 350 µm 500 µm 250 µm 50 µm 3 µm 200 µm 300 µm 169 Gpa 3.4965× 10−7 kg/m 1000 kg/m3 0.06 4 4 0.2 S ec.

CR IP T

Parameter Container length, a Container height, H Microbeam length, L Microbeam width, b Microbeam thickness, h Liquid depth of domain 1, H1 Liquid depth of domain 2, H2 Young’s modulus, E Microbeam mass, ρB Fluid density, ρ f Poisson’s ratio, υ Number of beam vibration modes, n Number of fluid oscillation modes, m Upper bound of the time, σ

AN US

results of Linholm et al. [26], and Shabani et al. [21] to the presented method. Fig. 2 demonstrates the mode shapes of wet beam. The fluid movement patterns within the upper and lower domains are illustrated in Figs. 3 and 4 by using local velocity vectors, which conforms the mode shapes presented in Fig. 2. Table 2: Comparison of fundamental frequencies in water (ρ=1000 kg/m3 )

M

Aspect ratio, L/b Thickness ratio, h/b ω1 (Hz) Analytical (Liang et al. [25]) Experimental (Lindholm et al. [26]) Fixed boundary (Shabani et al. [21]) Proposed method

First mode

−5

x 10

−0.5

0

1

CE

−1.5

PT

−1

−6

1

x 10

3 0.061

2 0.061

1 0.024

15.63 14.60 15.62 13.09

18.30 17.80 18.82 17.89

42.30 40.30 46.80 40.24

51.93 51.40 57.90 59.33

Second mode

−6

ED

0

5 0.124

2 0 −2

2

x Third mode

x 10

−4

3

0

1

x 10

−7

5

2 x Fourth mode

−4

x 10

3 −4

x 10

0.5

AC

0

0

−0.5 −1

0

1

2 x

−5

3

0

1

2 x

−4

x 10

Figure 2: The microbeam modes

12

3 −4

x 10

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6. Conclusions and future works

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A new method with free boundary approach was introduced to analyze the free vibration frequencies of a microbeam in contact with incompressible bounded fluid in a cavity. Shabani et al. [21] and other researchers like Liang [25] and Lindholm [26] investigated the presented system of Fig. 1 as a fixed boundary problem, whereas the height through y coordinate from y1 = 0 up to the microbeam is not of equal height along the x coordinate. It means that the range of y1 is from zero up to H1 + w(x, t) instead of H1 and so, w(x, t) ≤ y2 ≤ H2 while 0 ≤ x ≤ L . Since, the deflection w(x, t) is unknown, therefore the problem is considered as a free boundary problem. First, the system of the integral equations (19) was established to solve the problem (1-5) as a free boundary problem by using the reformed conditions (6-10). Then, the existence of the solution was proved by the use of Banach fixed point theorem. Finally, we illustrated the validation of our new method by presenting an example from [21]. Figs. 3 and 4 depict that the shapes of the fluid movement patterns change around the microbeam. This was expected, because the domains of the Laplace equations had changed in our proposed method. Fig. 5 shows that by applying the proposed method, the domains of the microbeam would be changed. It means that, as the time proceeds, the differences between the domains of the modes resulted from the proposed method and the method in [21] are increased. In order to investigate the effects of the microbeam location on its vibration properties, the microbeam is fixed along different locations in the cavity. Fig. 6 shows the dimensionless natural frequencies of the microbeam as a function of its off-center position. The graphs depict that the natural frequencies decrease when the microbeam is close to the bottom or top surfaces. Nevertheless, the frequencies are not affected much for different off-center location of the microbeam. Therefore, by reforming the model of the microbeam via free boundary approach, we get different results in comparison with [21]. It was the reason why the domains of the ϕ1 and ϕ2 changed to the free boundary. Thus, this affected the previous results. As a future work, we will derive added mass and frequencies for asymmetric free vibration of coupled system including clamped circular plate in contact with incompressible bounded fluid by free boundary approach.

M

Acknowledgments

AC

CE

PT

ED

The authors gratefully acknowledge the many helpful suggestions of Prof. M. Y. Rahimi Ardabili and all three referees for carefully reading the paper and for comments and suggestions which have improved this work.

13

x 10

Fluid domain 1

−4

Fluid domain 1

−4

3

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3

x 10

First mode at t=0.15

Second mode at t=0.15

2.5

2

2

y1

y1

1.5

1

0.5

0

0

−0.5 −0.5

0

0.5

1

1.5

2

2.5

3

3.5

x

4 −4

x 10

(a) First mode

0.5

1

1.5

2

2.5

3

3.5

x

4 −4

x 10

Fluid domain 1

x 10

Third mode at t=0.15 2.5

Fourth mode at t=0.15 2.5 2

ED

2

1

PT

1

1.5 y1

y1

1.5

0.5

CE

0 −0.5 −0.5

0

−4

M

x 10

−0.5

(b) Second mode

Fluid domain 1

−4

3

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1

0

0.5

1

1.5

2 x

2.5

0.5 0

3

3.5

−0.5 −0.5

4

0

0.5

x 10

(c) Third mode

AC

1

1.5

2 x

−4

(d) Fourth mode Figure 3: The fluid movement patterns in domain 1

14

2.5

3

3.5

4 −4

x 10

Fluid domain 2

−4

x 10

3.5

First mode at t=0.15

3

2.5

2.5

2

2

Second mode at t=0.15

y2

3

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

x

−0.5

−4

x 10

(a) First mode Third mode at t=0.15

ED

2.5

0.5 0 0

0.5

1

1.5

CE

−0.5

PT

1

2

x

1.5

2

2.5

3

3.5

x

−4

x 10

Fluid domain 2

2.5

3.5

Fourth mode at t=0.15 3 2.5 2 y2

1.5

1

−4

3

2

0.5

x 10

M

3.5

0

(b) Second mode

Fluid domain 2

−4

x 10

y2

Fluid domain 2

−4

x 10

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y2

3.5

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ACCEPTED MANUSCRIPT

1.5 1 0.5 0

3

3.5

−0.5

0

0.5

1

1.5

2 x

−4

x 10

(c) Third mode

(d) Fourth mode

AC

Figure 4: The fluid movement patterns in domain 2

15

2.5

3

3.5

4 −4

x 10

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ACCEPTED MANUSCRIPT

(b) Second mode differences

AC

CE

PT

ED

M

(a) First mode differences

(c) Third mode differences

(d) Fourth mode differences

Figure 5: The differences of the microbeam modes obtained by the fixed and the free boundary methods

16

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References

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PT

ED

M

AN US

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[1] J. D. Zook, D. W. Burns, H. Guckel, J. J. Sniegowski, R. L. Engelstad, Z. Feng, Characteristics of polysilicon resonant microbeams, Sensors and Actuators A Physical, 35 (1992) 51-59. [2] C.L.A. Berli, A. Cardona, On the calculation of viscous damping of microbeam resonators in air, J. Sound Vib. 327 (2009) 249-253. [3] V. Ostasevicius, R. Dauksevicius, R. Gaidys, A. Palevicius, Numerical analysis of fluid-structure interaction effects on vibrations of cantilever microstructure, J. Sound Vib. 308 (2007) 660-673. [4] D. Acquaviva, A. Arun, R. Smajda, D. Grogg, A. Magrez, T. Skotnicki, Micro-electro-mechanical switch based on suspended horizontal dense mat of cants by fib nanomanipulation. Procedia Chemistry, 1 (2009) 1411-1414. [5] A. Torii, M. Sasaki, K. Hane, S. Okuma, Adhesive force distribution on microstructures investigated by an atomic force microscope, Sensors and Actuators A Physical, 44 (1994) 153-158. [6] E. S. Hung, S. D. Senturia, Extending the travel range of analog-tuned electrostatic actuators, Journal of Microelectromechanical Systems, 8 (1999) 497-505. [7] P.I. Oden, G. Y. Chen, R. A. Steele, R. J. Warmack, T. Thundat, Viscous drag measurement utilizing microfabricated cantilevers, Appl. Phys. Lett. 68 (1996) 3814-3816. [8] R. Raiteri, M. Grattarola, H. J. Butt, P. Skladal, Micromechanical cantilever-based biosensors, Sens. Actuators B, 79 (2001) 115-126. [9] D. C. C. Lam, F. Yang, A. C. M. Chong, J. Wang, P. Tong. Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids 51 (2003) 1477-1508. [10] A.W. McFarland, J.S. Colton, Role of material microstructure in plate stiffness with relevance to microcantilever sensors, Journal of Micromechanics and Microengineering 15 (2005) 1060-1067. [11] B. Akgoz, O. Civalek. Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium, International Journal of Engineering Science 85 (2014) 90-104. [12] B. Akgoz, O. Civalek, Application of strain gradient elasticity theory for buckling analysis of protein microtubules, Curr. Appl. Phys. 11 (2011) 1133-1138. [13] B. Akgoz, O. Civalek, Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams, Int. J. Eng. Sci. 49 (2011) 1268-1280. [14] S.K. Park, X.L. Gao, Bernoulli-Euler beam model based on a modified couple stress theory, J. Micromech. Microeng. 16 (2006) 2355-2359. [15] S. Kong, S. Zhou, Z. Nie, K. Wang, The size-dependent natural frequency of Bernoulli-Euler micro-beams, Int. J. Eng. Sci. 46 (2008) 427-437. [16] S. Kong, Size effect on pull-in behavior of electrostatically actuated microbeams based on a modified couple stress theory, Applied Mathematical Modelling 37 (2013) 7481-7488. [17] A.R. Askari, M. Tahani. Size-dependent dynamic pull-in analysis of beam-type MEMS under mechanical shock based on the modified couple stress theory, Applied Mathematical Modelling 39 (2015) 934-946. [18] Y. G. Wang, W. H. Lin, N. Liu. Nonlinear bending and post-buckling of extensible microscale beams based on modified couple stress theory, Applied Mathematical Modelling 39 (2015) 117-127. [19] J.E. Sader, Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope, J. Appl. Phys. 84(1) (1998) 64-76. [20] M. Esmailzadeh, A. A. Lakis, M. Thomas, L. Marcouiller, Three-dimensional modeling of curved structures containing and/or submerged in fluid, Finite Elem. Anal. Des. 44 (2008) 334-345. [21] R. Shabani, H. Hatami, F. G. Golzar, S. Tariverdilo, Coupled vibration of a cantilever micro-beam submerged in a bounded incompressible fluid domain, Acta Mechanica, Volume 224, 4 (2013) 841-850. [22] K. Atkinson,W. Han, Theoretical Numerical Analysis, Vol. 39, Springer, Berlin (2005). [23] Polyanin,D. Andrei, Handbook of linear partial differential equations for engineers and scientists, CRC Press (2010). [24] Kendal E. Atkinson, Graeme Chandler: Boundary integral equation methods for solving Laplace equation with nonlinear boundary conditions: the smooth boundary case, mathematics of computation, vol. 55, no. 192 (1990) 451-472. [25] C.C. Liang, C.C. Liao,Y.S. Tai,W.H. Lai, The free vibration analysis of submerged cantilever plates, Ocean Eng. 28(9) (2001) 1225-1245. [26] U.S. Lindholm, D.D. Kana,W.H. Chu,H.N. Abramson, Elastic vibration characteristics of cantilever plates in water, J. Ship Res. 9 (1965) 11-12.

6

6

5 4

ω1

3

ω2

Frequency (rad/s)

Frequency (rad/s)

5

x 10

AC 6

2 1 0

0

0.1

0.2 0.3 H1/(H1+H2)

0.4

5 ω3

4

ω4

3 2 1

0.5

x 10

0

0.1

0.2 0.3 H1/(H1+H2)

Figure 6: The effect of vertical position on natural frequencies

17

0.4

0.5