A two-dimensional vibration analysis of piezoelectrically actuated microbeam with nonideal boundary conditions

Author’s Accepted Manuscript A two-dimensional vibration analysis of piezoelectrically actuated microbeam with nonideal boundary conditions MP Rezaei, M Zamanian www.elsevier.com/locate/physe

PII: DOI: Reference:

S1386-9477(16)30375-7 http://dx.doi.org/10.1016/j.physe.2016.09.005 PHYSE12574

To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 10 May 2016 Revised date: 2 July 2016 Accepted date: 8 September 2016 Cite this article as: MP Rezaei and M Zamanian, A two-dimensional vibration analysis of piezoelectrically actuated microbeam with nonideal boundary conditions, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2016.09.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A two-dimensional vibration analysis of piezoelectrically actuated microbeam with nonideal boundary conditions M P Rezaeia , M Zamaniana,∗ a Department

of Mechanical engineering, Faculty of engineering, Kharazmi University, P.O.Box 15719-14911, Tehran, Iran

Abstract In this paper, the influences of nonideal boundary conditions (due to flexibility) on the primary resonant behavior of a piezoelectrically actuated microbeam have been studied, for the first time. The structure has been assumed to treat as an Euler-Bernoulli beam, considering the effects of geometric nonlinearity. In this work, the general nonideal supports have been modeled as a the combination of horizontal, vertical and rotational springs, simultaneously. Allocating particular values to the stiffness of these springs provides the mathematical models for the majority of boundary conditions. This consideration leads to use a two-dimensional analysis of the multiple scales method instead of previous works’ method (one-dimensional analysis). If one neglects the nonideal effects, then this paper would be an effort to solve the two-dimensional equations of motion without a need of a combination of these equations using the shortening or stretching effect. Letting the nonideal effects equal to zero and comparing their results with the results of previous approaches have been demonstrated the accuracy of the two-dimensional solutions. The results have been identified the unique effects of constraining and stiffening of boundaries in horizontal, vertical and rotational directions. This means that it is inaccurate to suppose the nonideality of supports only in one or two of these directions like as previous works. The findings are of vital importance as a better prediction of the frequency re∗ Corresponding

author Email addresses: [email protected] (M P Rezaei), [email protected] (M Zamanian )

Preprint submitted to Elsevier

September 14, 2016

sponse for the nonideal supports. Furthermore, the main findings of this effort can help to choose appropriate boundary conditions for desired systems. Keywords: two-dimensional, general boundary conditions, nonideal, direct method of multiple scales, piezoelectrically actuated, nonlinear microbeam

1. Introduction Nowadays, slender microbeams [1] are increasingly used as the main element of microelectro-mechanical systems (MEMS) because of their simple microfabrications, simple designs and high reliability. These components of MEMS devices can be used as sensors and/or actuators by laminating a piezoelectric layer on it. Acting and sensing with the help of piezoelectric materials are gotten more attention because of its excellent characteristics such as generating very accurate small motions [2]. For generating the motions, an AC voltage is applied between upper and lower sides of the piezoelectric layer. The mechanism of these systems depends sensitively on the forced-vibration response of the oscillated microbeam [3–17, 17–21]. The boundary conditions (BCs) have a major effect on the response of the systems which many works have been investigated this effect. The most important works on this subject are given below. Rezazadeh et al analyzed the linear responses of clamped-clamped (C-C) and clamped-free (C-F) microbeams [7]. They illustrated that applying the same voltage to the piezoelectric layer causes various responses for these two different BCs. Akg¨oz and Civalek obtained the linear response of pined-pined microbeam and compared it with the cantilever microbeam [8]. Zamanian et al analyzed the linear governing equation of C-C and C-F microbeams [9]. They showed that the C-C microbeam has bigger maximum deflection and linear resonant frequency. The nonlinear equation of motion may be obtained by combining the twodimensional equations of motion [10] in case the nonlinear geometric is considered (with the exception of some especial boundaries, e.g. an edge with the horizontal spring or or nonidal support). If one of the supports be horizon-

2

tally moveable without any external loads acting along the horizontal direction, free or horizontal guided edge, the shortening effect is assumed and the one-dimensional governing equation can be obtained [11, 12]. Otherwise, the stretching effect can help to reduce two-dimensional governing equations into one [13–16]. C-C, pined-pined, clamped-pined, clamped-vertically guided and pined-vertically guided BCs are the most common of these kinds of BCs which ¨ Ozkaya et al studied in [22]. In the real systems, these BCs do not occur entirely, in fact the imperfection of fabrication leads to cause the nonideality of BCs because of its flexibilities which may be a result of residual stresses [23]. There are several approaches for modeling of the nonideal BCs. Mullen et al modeled the flexibility of the clamped edge of step-up C-C and C-F microbeams by assuming horizontal and rotational springs [17]. They obtained the fundamental frequency via finite element method and indicated that this frequency is considerably shifted because of the non-ideality of BCs. The deflection of nonideal and ideal step-up C-C microbeam was obtained by Kobrinsky et al. He showed that the nonideal results are in good agreement with the experimental data[24]. Pakdemirli and Boyaci investigated the natural frequencies and also the time history of nonideal pined-pined and pined-vertically guided beams[25]. H. K. KIM and M. S. KIM suggested a method to find vibration frequencies of beams with general BCs which is restrained by vertical and rotational springs [26]. Boyac considered the nonideality in the vertical and rotational directions for C-C BCs and found that it caused a shift in frequency response [27]. A pined-pined beam which had rotational springs on each pined edge was solved by HassanpourAsl et al [28]. They considered various values for the stiffness of these springs and studied the influence of their stiffness on the natural frequencies. Boyaci assumed the vertical and rotational nonideality for a pined-pined beam[29]. Rinaldi et al modeled the non-ideal microcantilever by considering the rotational and vertical springs [18] and they also analyzed this model via finite element methods [19]. Ekici and Boyaci considered only the springs for nonideal C-C and demonstrated shifting in the frequency response curve of microbeam [30]. Eigoli and Ahma3

dian indicated that the nonideality of pined-pined beam could directly affect the nonlinear features [31]. The value of rotational and vertical spring coefficients of the nonideal modeled BCs can be estimated by tuning their stiffness with experimental according to [20, 21]. Moreover, the method of obtaining the value of the stiffness of the horizontal spring of nonideal modeled BCs was presented by Claeys et al [32]. They obtained this coefficient by comparing the results of numerical simulations with the measured results of experimental data. The literature states that many works have been carried out the vibration analysis of beams with the nonideal supports, whereas all of these works did not consider the nonideal BCs for a microbeam with the piezoelectric actuation. As this structure is the main element of many of MEMS devices such as microresonatores, investigation of primary resonance of these systems is of major significant and here is studied. In this work, the full flexible clamped edge is assumed as a general model of the nonideal BCs which is modeled as an edge with the combination of horizontal, vertical and rotational springs. Allocating particular values to the stiffness of these springs provides the mathematical models for all kinds of the nonideal BCs [33]. For example, the nonideal model of pined edge is the same model as this model when the stiffness of the rotational spring is equated to zero [34]. Another example, the possible nonideal model of C-F microbeam (which is flexible due to residual stresses) is same as this model when the stiffness of the horizontal and rotational springs is assumed infinity [35]. This assumption and also considering the geometric nonlinearity leads to use a two-dimensional approach of the multiple scales method [36]. The reason why, here, the two-dimensional analysis should be used is that the two-dimensinal governing equations of motion cannot be combined into one equation for some BCs. As previously mentioned, the combination of these equations is possible by assuming shortening or stretching effect relying on the kind of BCs. In the stretching case [13–16], the axial inertia term in longitudinal equation of motion is dropped due to a slender beam (in the two-dimensional equations this term is considered). Afterward, the outcome is integrated twice 4

with respect to longitudinal position. Hence, the longitudinal displacement may be obtained according to the transverse displacement by imposing BCs. However, there are some BCs (such as an edge with horizontal spring or nonideal support) that cannot be imposed to this twice integrated equation. This is the main reason that the two-dimensional equations of motion should be considered. If one neglects these nonideal effects, then this paper would be an effort to solve the two-dimensional equations of motion without a need of combination of these equations using the shortening or stretching effect. The two-dimensional method will be verified by comparing the results of previous approaches (onedimensional analysis) with results achieved by letting the nonideal effects equal to zero. To investigate the influences of the nonideality of BCs in each direction, the stiffness of every springs are given various values depending upon the direction and their results are presented. The main findings of this paper are of vital importance as a better prediction of the frequency response for the nonideal supports. Furthermore, the findings can help to choose appropriate BCs for desired systems. Eventually, the most important findings are summarized and categorized.

2. Problem formulation Here, the system’s model is a micro-beam coated with a piezoelectric layer for different sets of BCs(Fig. 1).

5

Fig. 1: The model of system

The left and right ends of this piezoelectric layer are at distance l1 and l2 from the left edge of the micro-beam, respectively. The piezoelectric layer is actuated by applying an AC voltage which equals to VAC cos(Ωτ ). VAC and Ω are respectively the amplitude and the frequency of AC voltage. It should be noted that here the electrical displacement in the piezoelectric layer is assumed to be one-dimensional [37, 38]. Furthermore, it is assumed that the neutral axis of bending is stretched when the microbeam is deflected. Using Newton’s second law, applying the Taylor series, keeping the terms up to third order and considering both the longitudinal and transverse displacements U (s, τ ) and W (s, τ ), give the governing equation of motion as [16]: ¨ − (Cζ (s)U  ) = [Cζ (s)( 1 W 2 − U  W 2 )] + {W  [Cη (s)(W  − m(s)U 2 U  W  ) ] − 2U  W  (Cη (s)W  ) } − [Cα (s)VAC cos(Ωτ )(1 − 12 W 2 − U  W 2 )] +[Cγ (s)VAC cos(Ωτ )(W  − 2W  U  + 2U 2 W  − W 3 − U 3 W  )]

(1)

and ¨ + cˆ W ˙ = [Cζ (s)(U  W  − U 2 W  + 1 W 3 )] + (Cη (s)W  ) + m(s)W 2 {[Cη (s)(U  W  ) ] (1 − U  ) + (Cη (s)W  ) (U  − U 2 + W 2 ) − [Cη (s)(U 2 W  − 13 W 3 ) ] } − [Cα (s)VAC cos(Ωτ )(W  − U  W  + U 2 W  − 12 W 3 )] −[Cγ (s)VAC cos(Ωτ )(1 − U  + U 2 − W 2 + 3U  W 2 )] 6

(2)

where: m(s) = ρb Ab + (Hl1 − Hl2 )ρp Ap ,

Cα (s) = Ep d31 wb (Hl1 − Hl2 ),

Cζ (s) = (1 − Hl1 )Eb Ab + (Hl1 − Hl2 )(Eb Ab + Ep Ap ) + Hl2 Eb Ab , Cη (s) = (1 − Hl1 )Eb Ib + (Hl1 − Hl2 )(Eb I¯b + Ep Ip ) + Hl2 Eb Ib , Cγ (s) = (Hl1 − Hl2 ) Ib = A¯p =

wb 2 2 (hp

Ep d31 Ap , hp

3 1 12 wb hb ,

+ hp hb − 2hp z¯n ),

Ap = wb hp ,

I¯b =

3 1 12 wb hb

Ab = wb hb ,

+ hb wb z¯n2 ,

Ip = wb [ 13 (hp 3 + 32 hb hp 2 + 34 hp hb 2 ) +

zn ], hp z¯n2 − (hp 2 + hp hb )¯

z¯n =

Ep hp (hp +hb ) 2(Eb hb +Ep hp) .

(3)

In the above equations, the prime and dot signs indicate the differentiation with respect to longitudinal position (s) and time (τ ), respectively. Also, cˆ is the external viscous damping per unit length of the microbeam and d31 is the transversal piezoelectric strain coefficient. ρb and ρp show respectively the mass densities of the microbeam and the piezoelectric layer, moreover, Eb and Ep are the modulus of elasticity of the microbeam and the piezoelectric layer, respectively. As the Fig. 1 illustrates the geometrical properties, wb is the width of the microbeam and the piezoelectric layer, l is the microbeam’s length, hb and hp are the thickness of the microbeam and the piezoelectric layer, respectively. In these equations, the expression of Hli is Heaviside function which is defined in accordance with the longitudinal position as:

⎧ ⎨ 1 Hli = Heaviside function(s − li ) = ⎩ 0

s ≥ li

(4)

s < li

For convenience, the non-dimensional variables will be introduced as follows:  ρb h b w b l 4 τ W s U t= , T = , w= , x= , . (5) u= hb hb l T Eb I b Now, by substituting Eq. (5) into Eq. (1) and Eq. (2), and eliminated trivial coefficients according to the order of magnitude of microsystem and their mechanical properties [16], the dimensionless form of equations of motion will be: 2

∂ u α1u m(x)( ˆ ∂t2 ) −

∂ ∂u ∂x (Hζ (x) ∂x )

2

dHP (x) ∂ u − α2u ∂x (Hζ (x)( ∂w = 0 (6) ∂x ) ) − α3 P (t) dx

7

and 2

∂2 ∂2w ∂x2 (Hη (x) ∂x2 )

∂ w m(x) ˆ ∂t2 +

∂ ∂w w + Cˆ ∂w ∂t + α1 P (t) ∂x (HP (x) ∂x ) 3

∂w ∂ ∂u w ∂ −α2w ∂x (Hζ (x) ∂w ∂x ∂x ) − α3 ∂x (Hζ (x)( ∂x ) ) − 2

∂ w ∂ + ∂u ∂x ∂x (Hη (x) ∂x2 ) +

hb ∂ ∂ ∂ ∂u ∂w l ∂x { ∂x [Hη (x) ∂x ( ∂x ∂x )]

hb ∂ ∂ 1 ∂w 3 l ∂x [Hη (x) ∂x ( 3 ∂x )]}

+ α4w P (t) d

2

HP (x) dx2

=0

(7)

in which: m(x) ˆ = (1 − Hl1 /l ) + αm ˆ (Hl1 /l − Hl2 /l ) + Hl2 /l ,

αm ˆ =1+

ρp h p ρb hb ,

Hζ (x) = (1 − Hl1 /l ) + αζ (Hl1 /l − Hl2 /l ) + Hl2 /l ,

αζ = 1 +

E P Ap E b Ab ,

HP (x) = Hl1 /l − Hl2 /l , α1u =

Ib Ab l 2 ,

α2u =

P (t) = VAC cos(Ωt),

1 hb 2 l ,

l d31 Ep Ap h b h p E b Ab ,

α3u =

Hη (x) = (1 − Hl1 /l ) + αη (Hl1 /l − Hl2 /l ) + Hl2 /l , α1w =

l2 Ep d31 Ap E b Ib h p ,

α4w =

α2w =

¯p l2 Ep d31 A E b Ib h b h p ,

h b Ab l Ib ,

Cˆ =

αη =

α3w = l4 cˆ E b Ib T

.

Eb I¯b +Ep Ip , E b Ib

h b 2 Ab 2Ib ,

(8)

Eq. (6) and Eq. (7) are respectively partial differential equation of order four and two in x, hence these require six BCs which can be determined from the geometrical features of the deflection and slope and also, from the forces and moments balance on the beam edges [35, 39]. Therefore the general nonideal BCs which is modeled as an edge with the combination of horizontal, vertical and rotational springs are: ∂u Kh l = ±αh u, αh = ∂x E b Ab ∂3w Kv l3 = ±αv w, αv = 3 ∂x Eb I b ∂w ∂2w Kr l , αr = = ±αr . 2 ∂x ∂x Eb I b

(9a) (9b) (9c)

where Kh , Kv and Kr are the horizontal, vertical and rotational spring’s constants and also the sign ± is positive or negative depending on whether the spring is at the left or the right end of the microbeam.

8

3. Two-dimensional analysis In this work, the multiple scales method are used for solving the equations and the case of primary resonance is considered [40, 41]. For two-dimensional analysis via this method, it is of major significance to consider the accurate order between the longitudinal motion and the transverse motion. The parameter that  I which can be helpful to understand this order is the slenderness ratio (r = A I and A are, respectively, the moment of inertia and the cross section) which is extremely small for this system. Hence, in Eq. (6) the longitudinal inertia is small compared with the restoring force, and it follows that u = O(w2 ) ([42] in page 450). For this reason, the solutions of Eq. (6) and Eq. (7) in terms of different time scales are expressed as: u(x, t) = ε2 u2 (x, T0 , T1 , T2 ) + ε4 u4 (x, T0 , T1 , T2 ) + . . . ,

(10)

w(x, t) = εw1 (x, T0 , T1 , T2 ) + ε2 w2 (x, T0 , T1 , T2 ) + . . .

(11)

where ε is a small non-dimensional bookkeeping parameter, T0 = t, T1 = εt and T2 = ε2 t. Now, in order to balance the nonlinear terms with the terms of air damping Cˆ and excitation VAC , these are considered as order ε2 and ε3 terms, respectively. Substituting Eq. (10) and Eq. (11) into the dimensionless equations of motion Eq. (6) and Eq. (7) and also equating coefficients of the same powers of ε to zero, gives order ε: 2

∂ w1 ˆ + L(w1 ) = m(x) ∂T0 2

∂ 2 w1 ∂2 ∂x2 (Hη (x) ∂x2 )

=0

(12)

order ε2 : 2

∂ u2 ˆ − α1u m(x) ∂T0 2

∂u2 ∂ ∂x (Hζ (x) ∂x )

2

∂ 1 = α2u ∂x (Hζ (x)( ∂w ∂x ) )

(13)

2

∂ w1 L(w2 ) = −2m(x) ˆ ∂T1 ∂T0

(14)

order ε3 : 2

2

2

∂ w1 ∂ w1 ∂ w2 ˆ ∂w1 ˆ − 2m(x) ˆ ˆ L(w3 ) = −m(x) ∂T2 ∂T0 − 2m(x) ∂T1 ∂T0 − C ∂T0 + ∂T1 2 ∂ ∂ ∂ ∂u2 ∂w1 ∂x { ∂x [Hη (x) ∂x ( ∂x ∂x )]

+

∂u2 ∂ ∂ 2 w1 ∂x ∂x (Hη (x) ∂x2 ) 3

+

×

hb ∂ ∂ 1 ∂w1 3 l ∂x [Hη (x) ∂x ( 3 ∂x )]}

d ∂w1 1 w ∂ w ∂ 1 ∂u2 (Hξ (s) ∂w +α2w ∂x ∂x ∂x ) + α3 ∂x (Hζ (x) ∂x ) − 2 α4 P (t)

9

hb l

2

HP (x) iΩT0 e . dx2

(15)

Since only the excited mode identify the feature of the system, the mode-shape that influences on the response is the first mode-shape and the other modeshapes do not contribute. Therefore the general solution of Eq. (12) may be expressed as: ¯ 1 , T2 )e−iωT0 φ(x) w1 = A(T1 , T2 )eiωT0 φ(x) + A(T ∂ 3 φ(x) ∂φ(x) ∂ 2 φ(x) at x = 0 , 1. = ±α φ(x) and = ±αr v ∂x3 ∂x2 ∂x

(16)

Where A(T1 , T2 ) is an undetermined complex function which is determined by eliminating the secular terms at third order and furthermore, the over-bar represents the complex conjugate. Moreover, φ(x) and ω denote the first normalized 1 2 ˆ dx = 1) and the fundamental natural fremode shape of the system ( 0 m(x)φ quency of the system, respectively (φ(x) = φ1 (x), ω = ω1 ). The mode shapes are yielded by solving the linear homogenous part of Eq. (7). As the microbeam includes three segments, the mode shapes of system may be written as follows [9]: III φi (x) = (1 − Hl1 /l )φIi (x) + (Hl1 /l − Hl2 /l )φII i (x) + (Hl2 /l )φi (x)

(17)

where φji (x) represents the ith linear mode shape in jth part of microbeam depicted in Fig. 1. By applying Eq. 17 into the linear part of Eq. 7, three equations are obtained for each sub-domain which their solutions may be yielded as: φji (x) = Cj1 cosh(βji x) + Cj2 sinh(βji x) + Cj3 cos(βji x) + Cj4 sin(βji x), j = 1..3,

(18)

where β1i = β3i =



√ ωi ,

β2i =

αm 2 ˆ αη ωi

14

.

(19)

In which ωi is the ith natural frequency and Cjk , k = 1..4 are twelve unknown coefficients which are determined by applying the four related BCs and the eight continuity conditions (balancing the shear forces, the moments, the displacement 10

and the slop at the two continuity points [43]). These coefficients are computed in a way that the mode shapes be normalized. Accordingly, by substituting the solution of the first-order Eq. (16) in the second-order of transverse direction Eq. (14) would be: ∂A(T1 ,T2 ) iωT0 ˆ e φ + cc, L(w2 ) = −2m(x)iω ∂T1

(20)

where cc refers to the complex conjugate of the preceding terms. If A(T1 , T2 ) = A(T2 ), the secular term dose not arise in Eq. (20). Consequently: w2 (x, T0 , T1 , T2 ) = 0.

(21)

Ergo, considering A only depends on T2 and Eq. (16) the second-order equation in axial direction will be: α1u m(x) ˆ

1 u d ∂ 2 u2 ∂ ∂u2 dφ 2 2 2iωT0 ¯ + cc + AA) 2 − ∂x (Hξ (x) ∂x ) = 2 α2 dx (Hξ (x) dx )(A e ∂T0

∂u2 = ±αh u2 at x = 0 , 1. ∂x

(22)

So, its particular solution is given by: ¯ 2 )ψ1 (x) + cc. u2 = A2 e2iωT0 ψ2 (x) + AA(T

(23)

As a result, the governing equations of the functions ψ1 (x) and ψ2 (x) are respectively: dψ1 d dx (Hζ (x) dx )

2

d = −α2u dx (Hξ (x)( dφ dx ) )

(24)

and ˆ 4α1u ω 2 m(x)ψ 2+

dψ2 d dx (Hζ (x) dx )

2

d = −α2u dx (Hζ (x)( dφ dx ) ),

(25)

The ψ1 can simply be given by integrating Eq. (24) twice and imposing the related BCs. For obtaining ψ2 , it should be considered in three segments as follows: ψ2 (x) = (1 − Hl1 /l )ψ2I (x) + (Hl1 /l − Hl2 /l )ψ2II (x) + (Hl2 /l )ψ2III (x) 11

(26)

where the I, II and III superscripts stand for the first, second and third segments of microbeam depicted in Fig. 1. This assumption leads to these following equations: 4α1u ω 2 ψ2I +

d2 ψ2I dx2

2 II 4α1u αm ˆ ω ψ2 + αζ

4α1u ω 2 ψ2III +

2

= −2α2u

d ψ2II dx2

2

d ψ2III dx2

d2 φI1 dφI1 dx2 dx

= −2α2u αζ = −2α2u

2

for

d φII 1 dx2

2

d φIII 1 dx2

dφII 1

for

dx

dφIII 1

0 < s ≥ l1 ,

for

dx

l 1 < s ≥ l2 ,

l2 < s ≥ 1.

(27a) (27b) (27c)

These equations can simply be solved by using mathematical software. The solved ψ2 includes six undetermined coefficients. Imposing the two related BCs and the four continuity conditions (balancing the axial forces and the displacement at the two continuity points [43]) determine these six undetermined coefficients. Now, the obtained w1 , w2 and u2 are substituted in the third-order problem Eq. (15). Then in order to express the primary resonance, a detuning parameter σ is introduced and consequently the excited frequency is defined as Ω = ω + ε2 σ. Accordingly, the secular term will be obtained as: dA 2 ¯ iσT2 iωT0 ˆ ˆ L(w3 ) = [−2iω m(x) ˆ ]e dT2 φ − iω CA + χ(x)A A + F (x)e

+cc + N ST.

(28)

where N ST demonstrates the all terms which cannot lead to secular terms and 3

2

3

hb 2 d d dφ d χ(x) = 3α3w dx (Hζ (x)( dφ dx ) ) + ( l ) dx2 [Hη (x) dx ( dx )] dψ1 dφ d hb d d 2 + dx { l dx [Hη (x) dx (( dψ dx + 2 dx ) dx )] dψ1 w dφ 2 +( dψ dx + 2 dx )[α2 dx Hζ (x) +

d2 φ hb d l dx (Hη (x) dx2 )]}

Fˆ (x) = − 12 α4w VAC d

2

HP (x) . dx2

(29)

The left-hand side of Eq. (29) is self-adjoint, hence the adjoint solution is quite similar to the solution of Eq. (12). The non-homogeneous of Eq. (29) has a solution only in case the right-hand side of this equation is orthogonal to every solution of the self-adjoint homogeneous equation, i.e. φ(x)eiωTo . Thus, multiplying the right-hand of Eq. (29) by φ(x)e−iωTo and integrating the result from x = 0 to 1 leads to the solvability condition as: dA 2iω(m ˜ dT + 2

μA 2 )

+ SA2 A¯ + F˜ eiσT2 = 0 12

(30)

where m ˜ =

1

μ = Cˆ

2 m(x)φ ˆ dx = 1,

0

1 0

F˜ = −

1 0

1 S = − φχ(x)dx,

φ2 dx,

φFˆ (x)dx = − 21 α4w VAC

 dφ  dx 

l1 /l

  − dφ dx 

0



l2 /l

.

(31)

It should be pointed out that the calculation of these integrations by mathematical software (e.g. maple) makes an enormous calculated error due to exist the Heaviside function. Hence, it would be sensible to eliminating this function by using the integration by part first. Subsequently, by introducing polar notation A =

1 iβ 2 ae

with amplitude a and phase β into Eq. (30) and assuming

γ = σT2 − β, it results in: da dT2 dγ dT2

= − μ2 a −

=σ−

S 2 8ω a

F˜ ω

−

sin(γ),

(32a)

F˜ aω

(32b)

cos(γ),

and also substituting Eq. (16) into Eq. (11) and equating ε to one, yields the transverse displacement as: w(x, t) = a cos(Ωt − γ)φ(x). Due to a and γ are invariant in the equilibrium state, the

(33) da dT2

and

dγ dT2

are

equated to zero in Eq. (32a), as a consequent:

2 a2 ( μ2 ) + (σ −

S 2 2 8ω a )

 =

 2 F˜ ω

(34)

This equation is often referred to as the frequency-response equation (function) which the amplitude a is a maximum if the expression inside parentheses be zero. Hence: a=

2F˜ ωμ ,

σ=

S 2 8ω a

(35)

Also, to consider σ = Ω − ε2 ω ,it can be found the nonlinear resonant frequency as: Ω=ω+ 13

S F˜ 2 2ω 3 μ2 .

(36)

Table 1: Geometrical and mechanical properties of considered micro-system

l(μm)

l1

l2

hb (μm)

hp

Cˆ

200

0.25l

0.75l

1

0.25hb

0.08

Eb (Gpa)

Ep (Gpa)

ρb (kg m−3 )

ρp (kg m−3 )

d31 (C N−1 )

VAC (Volt)

160

67

2300

7700

−1.75 10−10

0.002

4. Verification by one-dimensional analysis The MEMS literature review shows that the governing equations of motion Eq. (6) and Eq. (7) could be combined into one equation on some occasions. This is possible for some BCs by assuming shortening or stretching effect relying on the kind of BCs. In these assumptions the axial inertia term is neglected due to a small size [10]. Nevertheless, this term is considered in the two-dimensional analysis (which most probably have not any influence). In order to verify and check the two-dimensional approach both in shortening and stretching cases, the result of a microsystem via this method are compared with the result of the one-dimensional approach (previous works’ method). To achieve this goal, the most used and popular of microbeams, cantilever and C-C, are analysed. The microsystem is considered a microbeam and a laminated layer with the properties of silicon and PZT5A. These properties and the geometry of the considered system are given in Table 1 [44]. 4.1. Shortening effect (C-F) If one of the supports be moveable in axial direction without any external load acting along horizontal direction, the shortening effect will be assumed and thus the dependent variables can be reduced from two to one by doing some stages in accordance with [10]. In this situation, the one-dimensional equation

14

of motion would be [11, 12]: 2

∂ w m(x) ˆ ∂t2 +

∂2 ∂2w ∂x2 (Hη (x)

∂x2 ) ∂ +( hlb )2 ∂x

∂w ∂x

2

hb 2 ∂ ∂w ∂ ∂w ∂ w + Cˆ ∂w η (x) ∂x ∂x2 )] ∂t + ( l ) ∂x [ ∂x ∂x (H x x ∂ 2 1 ∂w 2 m(x)[ ˆ ∂t2 ( 2 ∂x )dx]dx 1

0

2 ∂ d + 12 α4w P (t)( hlb )2 ∂x [ dx HP (x)( ∂w ∂x ) ]

2

d + α4w P (t) dx 2 (HP (x)) = 0

(37)

and u=0 ∂u ∂x

=

w=

2 −α2u ∂w ∂x [45],

∂w ∂x 2

∂ w ∂x2

= 0 at x = 0 =

∂3w ∂x3

= 0 at x = 1.

(38)

In the previous works of MEMS application, the one-dimensional equation has been solved many times for the microcantilever (C-F). In these articles, Eq. (11) has been replaced in the perturbed equation and then the frequency response has been achieved using the multiple scales method. Here, the frequency responses of one-dimensional (black dotted line) and two-dimensional (red solid line) analysis are displayed in Fig. 2. As there is no difference between the two results, the two-dimensional method is confirmed in this situation.

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Fig. 2: The comparison between the one-dimensional and two-dimensional analysis for cantilever microbeam

4.2. Stretching effect (C-C) In this case, the axial inertia term in Eq. (6) is dropped due to a slender beam. Afterward, the outcome is integrated twice with respect to x. At the end, the axial displacement u(x, t) may be obtained according to the transverse displacement w(x, t) by imposing the BCs. It should be emphasised that applying this stage cannot be possible for certain BCs (such as an edge with horizontal spring (∂u/∂x = ±αh u)). This is the main reason that the uncombined (two-dimensional) equations are considered in this paper. However, the one-dimensional equation of motion will be [15, 16]: 2

∂ w m(x) ˆ ∂t2 +

∂2 ∂2w ∂x2 (Hη (x) ∂x2 )

∂2w w + Cˆ ∂w ∂t − [αΓ Γ(w, w) + α5 P (t)] ∂x2 2

d +α4w P (t) dx 2 (HP (x)) = 0,

16

(39)

where αΓ = 6(

αζ l −l αζ (1− 2 l 1

2

6(

l2 −l1

)

Ep

l Eb α5w = 2lhbd331 ( l −l l −l ), αζ (1− 2 l 1 )+ 2 l 1  1 ∂f ∂f Γ(f, f ) = 0 ∂x ∂x dx.

l −l )+ 2 l 1

),

(40)

and u=0

w=

∂w ∂x

= 0 at x = 0

u=0

w=

∂w ∂x

= 0 at x = 1.

(41)

Again, the result of one-dimensional method is obtained in similar process which is shown in Fig. 3. As it can be realized, it gives just the same result as the twodimensional analysis. So, again on this occasion the two-dimensional method is verified.

Fig. 3: The comparison between one-dimensional and two-dimensional analysis for C-C microbeam

Ergo, in general, there is no difference between the results of the onedimensional and two-dimensional approaches, but the second approaches have 17

two significant advantages. Firstly, this method can be applied for all BCs, while the other analysis cannot. Secondly, there is no need to use shortening or stretching effect depending on kind of BCs.

5. Results and discussions BCs can directly affect the fundamental frequency (=the linear resonant frequency), on the nonlinear shift of resonant frequency and on the maximum deflection (wmax = aφ(xmax ) which the xmax are obtained by setting

dφ(x) dx

to zero). Hence, in this section the frequency response curve are plotted and discussed. The general BCs includes the most familiar BCs especially ideal BCs, but in the real systems, these BCs do not occur entirely. For an illustration, it may be inaccurate to suppose clamped edge without any change of its slope and position. This change appears because of the flexibility of the support which may be a result of residual stresses. The full flexible clamped edge can be modeled by supposing horizontal, vertical and rotational springs or any combination of these (Fig. 1 and equations Eq. (9a), Eq. (9b) and Eq. (9c)). The majority of other supports are a special case of this full flexible clamped edge. For instance, the flexible pined edge is the same model as the full flexible clamped edge in case αr = 0. For this reason, the general BCs are modeled exactly like as the full flexible clamped edge. For a better study of the influence of general nonideal BCs, the results will be separately obtained for each direction. To fulfill this ambition, the two values of αh , αv and αr coefficients will be assumed infinity and the other one will be gotten different value. 5.1. Horizontally nonideal clamped edge: This means the nonideal BCs which vertically and rotationally be rigid (αv , αr = ∞). Here, the results are given for different value of αh in Fig. 4 which are for Horizontally flexible C-C.

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Fig. 4: The influence of horizontal nonideality of C-C microbeam on the nonlinear behavior

The figure demonstrates that the growth of the horizontal stiffness αh has no influence on the fundamental frequency and the maximum deflection whereas it leads to a logarithmic rise in the nonlinear shift of resonant frequency. As it can be observed, changing horizontal flexibility of BCs has enormously effect on the nonlinear shift of resonant frequency. Therefore it may not be true to ignore the horizontal spring for modeling of the nonideal BCs. It should be commented that the two-dimensional analysis can be applied to a BCs which includes the horizontal spring, while the previous works’ method cannot be applied. It can be added that, in this case, the nonideal clamped edge changes to the ideal one, when αh becomes infinity. It should be announced that this result cannot be obtained via previous approach and the two-dimensional approach should be used, because on this occasion the two-dimensinal governing equations of motion cannot be combined into one equation.

19

5.2. Vertically nonideal clamped edge. Here, αh , αr = ∞ while αv is gotten different values. In accordance with the literature, it is clear that the fundamental frequency is gone up by increasing vertical stiffness αv which leads to shift of the frequency response curve. Hence, here, for determining the nonlinear shift of resonant frequency and the maximum deflection, the maximum deflection of vertical nonideal C-C and C-F with respect to σ are respectively displayed in Fig. 5 and Fig. 6.

Fig. 5: The influences of vertical nonideality of C-C microbeam on the nonlinear behavior

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Fig. 6: The influences of vertical nonideality of cantilever microbeam on the nonlinear behavior

These figures reveal that a rise in the vertical stiffness αv leads to a logarithmic increase in the nonlinear shift of resonant frequency and the maximum deflection both for vertically nonideal C-C and C-F. As it can be realized, in this situation, the nonideal clamped support becomes ideal by increasing αv to infinity. 5.3. Rotationally nonideal clamped edge: If the nonideal clamped edge be horizontally and vertically rigid (αh , αv = ∞), the rotational nonideality of clamped edge will be happened. Again, it is evident that the fundamental frequency is gone up by rising αr . The Fig. 7 and Fig. 8 provide the variations of wmax with respect to σ for the rotationally nonideal C-C and C-F, respectively.

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Fig. 7: The influences of rotational nonideality of C-C microbeam on the nonlinear behavior

22

Fig. 8: The influence of rotational nonideality of cantilever microbeam on the nonlinear behavior

These figures reflect that the nonlinear shift of resonant frequency are exponentially declined by rising the rotational stiffness αr . However, the maximum deflection is logarithmically gone up and exponentially dropped for rotational flexible C-F and C-C, respectively. On this occasion, the nonideality of clamped BCs is removed when αr becomes infinity. It should be commented that the rotational nonideal C-F may be a studious model of micro-cantilever system which is flexible because of residual stresses ([35] in page section 3 of chapter 6).

6. Summary and conclusion In this paper, the influences of nonideal (due to flexibility) boundary conditions on the primary resonant behaviour of a piezoelectrically actuated microbeam have been studied. The structure has been assumed to treat as an Euler-Bernoulli beam, considering the effects of geometric nonlinearity. In this work, the general nonideal BCs have been modeled by assume simultaneously 23

all of the horizontal, vertical and rotational springs which in the MEMS literature have not been considered. This assumption leads to use a two-dimensional analysis of the multiple scales method. If one neglects the nonideal effects, then this paper would be an effort to solve the two-dimensional equations of motion without a need of a combination of these equations using the shortening or stretching effect. Letting the nonideal effects equal to zero and comparing the results of it with the results of previous approaches (one-dimensional analysis) have been demonstrated the accuracy of the two-dimensional solutions. The results of this paper has briefly been revealed that: 1. Constraining and stiffening of the boundaries in horizontal direction increase the nonlinear shift of resonant frequency, while it has not any impact on the fundamental frequency and the maximum deflection. 2. Constraining and stiffening of the boundaries in vertical direction rise the nonlinear shift of resonant frequency, the fundamental frequency and the maximum deflection. 3. Constraining and stiffening of the boundaries in rotational direction grow the fundamental frequency and also it decays the nonlinear shift of resonant frequency. Hence, As the constraining and stiffening of boundaries in each directions have unique effects, it is inaccurate to suppose the nonideality of supports only in one or two of these directions like as previous works. Furthermore, the findings indicate that the type of a support is of major significance to dynamic behavior of microbeam and also it is of major significance to consider the nonideal of the support in all of directions. [1] L. M. Roylance, J. B. Angell, A batch-fabricated silicon accelerometer, Electron Devices, IEEE Transactions on 26 (12) (1979) 1911–1917. [2] U. Simu, S. Johansson, Fabrication of monolithic piezoelectric drive units for a miniature robot, 2002 j, Micromech. Microeng 12 582.

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