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Dynamics of functionally graded micro-cantilevers
International Journal of Engineering Science 115 (2017) 117–130
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International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Dynamics of functionally graded micro-cantilevers Hamed Farokhi a, Mergen H. Ghayesh b,∗, Alireza Gholipour b a b
Department of Mechanical Engineering, McGill University, Montreal, Quebec H3A 0C3, Canada School of Mechanical Engineering, University of Adelaide, South Australia 5005, Australia
a r t i c l e
i n f o
Article history: Received 18 January 2017 Accepted 21 January 2017
Keywords: Functionally graded micro-cantilever MCST Dynamics MEMS
a b s t r a c t The nonlinear size-dependent dynamics of a functionally graded micro-cantilever is investigated when subject to a base excitation resulting in large-amplitude oscillations. A geometric nonlinearities due to large changes in the curvature is taken into account. Employing the Mori–Tanaka homogenisation technique (for the material properties), the modified couple stress theory (MCST) is used to formulate the potential and kinetic energies of the system in terms of the transverse and axial motions. A dynamic energy balance is performed between the energy terms, yielding the continuous models for the axial and transverse displacements. The inextensibility condition results in the size-dependent model of the functionally graded micro-cantilever involving inertial and stiffness nonlinear terms. The resultant model is discretised based on a weighted-residual technique yielding a high-dimensional truncated model (required for accurate simulations). A parametercontinuation scheme together with a time integration method is introduced to the truncated model so as to determine the resonances with stable and unstable solution branches with special consideration to the effect of different system parameters, such as material gradient index and the length-scale effect on the nonlinear dynamics of the functionally graded micro-cantilever. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Cantilevers of microscale dimensions form the core components of many microelectromechanical systems (MEMS) (Asghari, Kahrobaiyan, & Ahmadian, 2010; Baghani, 2012; Ghayesh & Farokhi, 2015a, b; Ghayesh, Farokhi, & Amabili, 2013a, b; Zhang & Meng, 2007; Zheng, Dong, Lee, & Gao, 2009). Experiments showed that their deformation behaviour is highly size-dependent (Akgöz & Civalek, 2013a, b, Akgöz & Civalek, 2011; Dehrouyeh-Semnani, 2014; Dehrouyeh-Semnani, BehboodiJouybari, & Dehrouyeh, 2016; Farokhi, Ghayesh, Gholipour, & Hussain, 2017; Ghayesh, Amabili, & Farokhi, 2013a, b, Hosseini & Bahaadini, 2016; Kahrobaiyan, Rahaeifard, Tajalli, & Ahmadian, 2012; Karparvarfard, Asghari, & Vatankhah, 2015; Kong, Zhou, Nie, & Wang, 2008; Mojahedi & Rahaeifard, 2016; Shafiei, Kazemi, & Ghadiri, 2016a, b, Taati, 2016). Theoretically, only the advanced continuous theories such as strain gradient and MCST (Dai, Wang, & Wang, 2015; Dehrouyeh-Semnani et al., 2015; Farokhi & Ghayesh, 2015 b, Farokhi & Ghayesh, 2015 a, Farokhi, Ghayesh, & Amabili, 2013a, b, Ghayesh & Amabili, 2014; Ghayesh and Farokhi, 2015, Ghayesh, et al., 2013 a, b, Ghayesh, Farokhi, & Amabili, 2014; Ghayesh, Farokhi, & Hussain, 2016; Gholipour, Farokhi, & Ghayesh, 2015; Li & Pan, 2015; S¸ ims¸ ek, 2010; Tang, Ni, Wang, Luo, & Wang, 2014) theories are capable of incorporating size effects in the dynamical modelling of microscale elements; in other words, simulations based on the classical continuum mechanics may result in drastic errors (Farokhi, Ghayesh, & Amabili, 2013). ∗
Corresponding author. E-mail address: [email protected] (M.H. Ghayesh).
http://dx.doi.org/10.1016/j.ijengsci.2017.01.004 0020-7225/© 2017 Elsevier Ltd. All rights reserved.
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Fig. 1. Schematic representation of functionally graded micro-cantilever.
In recent years, microstructures made of functionally graded (FG) materials have gained high interests in MEMS industry due to their mechanical properties to achieve a better performance and sensitivity (Witvrouw, 2005; Huang, 2008; Lü, Lim, & Chen, 2009). Micro-cantilevers, as opposed to supported–supported (i.e. combination of clamps and pins) microbeams (Farokhi & Ghayesh, 2016), are subjected to large deformations when a transverse load is applied; this is due to the fact that one end of the microsystem possesses displacement-freedom. In other words, the effect of the curvature-related nonlinearities become dominant and should not be neglected, especially when coupled with the length-scale parameter. The literature on the dynamics/statics of supported-supported functionally graded microbeams, involving both linear and nonlinear analyses is large; for example, Akgöz and Civalek (2014) analysed the thermal variation influences on the buckling behaviour of a FG microbeam resting on an elastic foundation. Tajalli et al. (2013) formulated the motion behaviour of a FG Timoshenko microbeam. Aghazadeh, Cigeroglu, and Dag (2014) contributed to the field by developing a procedure of obtaining the linear dynamic/static characteristics of Timoshenko, Bernoulli–Euler, and shear deformable types of FG microbeams. Arbind and Reddy (2013) developed a geometrically nonlinear finite element model for a FG microbeam employing the Euler–Bernoulli and Timoshenko theories via use of MCST; the resultant model was used to obtain the buckling response of the microsystem. The literature on the dynamics/statics of functionally graded micro-cantilevers is not large. Asghari, Ahmadian, Kahrobaiyan, & Rahaeifard (2010) examined the size-dependent static and free dynamics of FG micro-cantilevers based on MCST. Shafiei et al. (2016a, b) examined the transverse vibrations of an axially FG micro-cantilever based on the Euler– Bernoulli and MCST. Akgöz & Civalek (2013a, b) employed MCST in order to obtain the dynamical response of non-uniform axially functionally graded micro-cantilevers. This paper is the first which examines the size-dependent nonlinear dynamics of a functionally graded micro-cantilever undergoing large oscillations via MCST incorporating curvature and inertial nonlinearities. The size-dependent potential energy of the functionally graded micro-cantilever is obtained as a continuous function of the system parameters employing the Mori–Tanaka technique as well as constitutive relations. A dynamic balance is introduced to the kinetic energy and the size-dependent potential energy giving the continuous equations of the transverse and axial displacements. The inextensibility condition leads to an integro-partial-differential equation of motion of the FG micro-cantilever. A system reduction is carried out on the basis of the Galerkin scheme. The resonances of the functionally graded micro-cantilever are obtained highlighting the influence of different microsystem parameters including the length-scale parameter as well as the material gradient index. 2. Microsystem model of functionally graded micro-cantilever Consider a functionally graded micro-cantilever of length L, thickness h, and cross-sectional area of A (Fig. 1). The left end of the functionally graded micro-cantilever is clamped while the right end is free to move. The clamped end is excited harmonically by b0 sin(ωt) where ω is the excitation frequency, t is time, and b0 denotes the excitation amplitude. The total displacement field is shown by the axial component u(x,t) and the transverse component w(x,t), where x is axial coordinate. The functionally graded micro-cantilever is made of metal and ceramic and the mechanical properties are Young’s modulus E, mass density ρ , Poisson’s ratio υ , and shear modulus μ. At z = h/2 (i.e. at the top surface) the micro-cantilever is ceramicrich and at z =-h/2 (the bottom surface) is metal-rich. The Mori–Tanaka homogenisation technique is utilised to determine the effective mechanical properties of the mixture. As such, the effective bulk modulus (shown by Ke )and the effective shear modulus (shown by μe ) are defined as functions of μc , μm , Kc , and Km (shear modulus of the ceramic, shear modulus of the metal, bulk modulus of the ceramic, and the bulk modulus of the metal, respectively) as well as vm and vc (the volume fractions of the metal and ceramic, respectively where vm + vc = 1); thus
Ke − Km vc = , Kc − Km 1 + vm (Kc − Km )/(Km + 4μm /3 )
(1)
H. Farokhi et al. / International Journal of Engineering Science 115 (2017) 117–130
−
μe − μm vc = . μm − μc 1 + vm (−μm + μc )/[μm + μm (8μm + 9Km )/(6(Km + 2μm ) )]
119
(2)
Different distributions can be obtained for the ceramic volume fraction by means of a power law function with the gradient index of n as follows
vc (z ) = (z/h + 1/2)n .
(3)
Effective Poisson’s ratio and Young’s modulus are given, respectively, by
υ (z ) =
3Ke − 2μe . 6Ke + 2μe
(4)
E (z ) =
9μe Ke , μe + 3Ke
(5)
The effective mass density is given by
ρ ( z ) = ρm v m + ρc v c ,
(6)
where ρ m and ρ c are the mass densities of the metal and ceramic, respectively. Based on MCST, the potential energy of the functionally graded micro-cantilever of size-dependent strain-type can be formulated as
U=
1 2
(m : χ + σ : ε)dv,
V
(7)
where ε, χ, σ, and m represent the strain tensor, the symmetric curvature tensor, the stress tensor, and the deviatoric part of the symmetric couple stress tensor, respectively. Moreover,
m = 2 μe l 2 χ ,
(8)
where l represents the length-scale parameter. Furthermore,
σxx = E (z )εxx .
(9)
For an Euler–Bernoulli type of a functionally graded micro-cantilever, the strain at a point located at a distance z from the mid-plane of the functionally graded micro-cantilever can be determined as a function of u, w, and ψ (the angle of the centreline of the micro-cantilever with respect to the axial axis); as such
⎡
εxx = ⎣−1 + Moreover,
χxy
1 =− 4
∂w ∂x
2
+
1+
∂u ∂x
2
⎤
⎦ − z ∂ψ . ∂x
∂w ∂ 2 cos (ψ ) ∂ sin (ψ ) + +z = χyx , ∂x ∂x ∂ x2
χyz = χzy = −
1 ∂ cos (ψ ) . 4 ∂x
(10)
(11) (12)
Considering Eqs. (7)–(12), the elastic potential energy of the functionally graded micro-cantilever becomes
1 U = 2
⎡
L
0
⎛ ⎞ ⎛ ⎞ 2 2 2 2 2 ∂u ⎠ w ∂ u ∂w ∂ ⎢ ⎝ ⎠ ∂ψ + 1+ − 2B11 ⎝−1 + + 1+ ⎣A11 −1 + ∂x ∂x ∂x ∂x ∂x
2
2
2 A55 l 2 ∂ ∂ w B55 l 2 ∂ ∂ w ∂ψ ∂ cos (ψ ) + D11 + + sin (ψ ) + + sin (ψ ) ∂x 4 ∂x ∂x 2 ∂x ∂x ∂ x2 2 2 D55 l 2 ∂ 2 cos (ψ ) A55 l 2 ∂ cos (ψ ) + + dx, 2 4 4 ∂x ∂x with
A11 = A
E (z )dA,
(13)
(14a)
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B11 =
z E (z )dA,
(14b)
z2 E (z )dA,
(14c)
E (z ) dA, 2 (1 + υ (z ) )
(14d)
z E (z ) dA, 2 (1 + υ (z ) )
(14e)
z2 E (z ) dA. 2 (1 + υ (z ) )
(14f)
A
D11 = A
A55 = A
B55 = A
D55 = A
The kinetic energy of the functionally graded micro-cantilever is
1 T = 2 where
I1 =
L
I1
0
∂u ∂t
2
+ I1
∂w ∂t
2
dx,
(15)
ρ (z )dA.
(16)
A
Dynamically balancing the potential energy of the functionally graded micro-cantilever with the kinetic one gives
⎧⎡ ⎛ ⎞ ⎤ 2 2 ⎨ ∂ u ∂ ⎣ ⎝ ∂u ⎠ ∂ψ ⎦ ∂w I1 2 = A −1 + + 1+ − B11 cos (ψ ) ∂ x ⎩ 11 ∂x ∂x ∂x ∂t ⎡ ⎛ ⎞ ⎤ 2 2 ∂ ⎣ ⎝ ∂u ∂ψ ⎦ sin (ψ ) ∂w − B 1+ + − 1⎠ − D11 ∂ w 2 ∂ x 11 ∂x ∂x ∂x ∂u 2 2
1 ∂ ∂ + A55 l 2 sin (ψ ) + 4 ∂x ∂x
+
1 ∂ ∂ B55 l 2 sin (ψ ) + 4 ∂ x2 ∂x 2
1 ∂ ∂ cos (ψ ) − A55 l 2 4 ∂x ∂x
1 + ∂x
+ ∂x
1 + ∂x
+ ∂x
⎞ 2 ∂w ∂ cos ψ sin ψ cos ψ ( ) ( ) ( ) ⎝ ⎠ + B55 l 2 ∂ w 2 ∂x ∂ x2 ∂u 2 ⎛
⎛
∂w ∂ cos (ψ ) ⎝ + D55 l 2 ∂x ∂ x2 2
⎛ ⎝
sin (ψ ) 2
1 + ∂∂ ux
2
⎞⎫ ⎬ ⎠ ∂ w 2 ⎭
sin (ψ )
⎞
2
1 + ∂∂ ux
2
+ ∂∂wx
2
⎠
(17)
+ ∂x
⎧⎡ ⎛ ⎞ ⎤ 2 2 ⎨ ∂ w ∂ ⎣ ⎝ ∂u ∂ψ ⎦ ∂w I1 2 − A 1+ + − 1⎠ − B11 sin (ψ ) ∂ x ⎩ 11 ∂x ∂x ∂x ∂t ⎡ ⎛ ⎞ ⎤ 2 2 ∂ ⎣ ⎝ ∂u ⎠ ∂ψ ⎦ cos (ψ ) ∂w + B −1 + + 1+ − D11 ∂ x 11 ∂x ∂x ∂x ∂w 2 2
∂x
+ 1 + ∂∂ ux
⎛
1 ∂
2 ∂ ∂ w ∂ cos ψ ( ) ⎝1 + − l 2 A55 sin (ψ ) + + l 2 B55 4 ∂x ∂x ∂x ∂ x2
2
cos2 (ψ )
2 2 1 + ∂∂ ux + ∂∂wx
⎞ ⎠
H. Farokhi et al. / International Journal of Engineering Science 115 (2017) 117–130
121
Fig. 2. Frequency-response curves of the functionally graded micro-cantilever: (a, b) the first two generalised coordinates of the transverse motion, respectively. n = 2, ls = 0.45, b0 = 0.006, and ζ = 0.008.
1 ∂2 ∂ − B55 l 2 sin (ψ ) + 4 ∂ x2 ∂x
⎞ ⎛ 2 ∂w ∂ cos ψ sin ψ cos ψ ( ) ( ) ( ) ⎝ ⎠ + D55 l 2 ∂ w 2 ∂x ∂ x2 ∂u 2
⎞⎫
⎬ 1 ∂ ∂ cos (ψ ) ⎝ sin (ψ ) cos (ψ ) ⎠ + A55 l 2 ∂ w 2 ⎭ = 0 . 4 ∂x ∂x ∂u 2 ⎛
1 + ∂x
1 + ∂x
+ ∂x
(18)
+ ∂x
The inextensibility assumption is applied to Eqs. (17) and (18), while remembering that the axial force is zero at the right end of the functionally graded micro-cantilever. This operation leads to
! 2 2 x ∂ 2w ∂ ∂ w x ∂ 3w ∂ w ∂ w I1 2 + I1 + dx dx ∂x ∂x L ∂ x∂ t ∂t ∂ t 2∂ x ∂ x 0 2 2 ∂ ∂ 3w ∂ 3w ∂ w ∂ w ∂ 2w + D11 + + ∂ x ∂ x3 ∂ x ∂ x2 ∂ x3 ∂ x 2 2 2
3 3 3B55 l 2 ∂ ∂ 2 w ∂ 3 w ∂ w 1 ∂ w w 1 ∂ w ∂ ∂ ∂ w + A55 l 2 + + − ∂x 4 ∂ x3 ∂ x 4 ∂ x ∂ x2 2 ∂ x ∂ x2 ∂ x3 ∂ x3 2 2 1 ∂ 5w ∂ w ∂ 4w ∂ w ∂ 2w 3 ∂ w ∂ 3w ∂w 2 ∂ − D55 l + + +c = 0, ∂ x 4 ∂ x5 ∂ x ∂t ∂ x4 ∂ x ∂ x2 4 ∂ x ∂ x3
(19)
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Fig. 3. Force-response curves of the functionally graded micro-cantilever: (a, b) the first two generalised coordinates of the transverse motion, respectively. n = 2, ls = 0.45, Ω/ω1 = 0.98, and ζ = 0.008.
where a viscous damping force of c ∂ w/∂ t (with c being the coefficient) has been introduced. Defining v(x,t) as the transverse displacement relative to the base, Eq. (19) can be rewritten as
! x 2 2 ∂ 2 v ∂ ∂v x ∂v ∂ 3 v ∂ v I1 2 + I1 + dx dx ∂x ∂x L ∂ x∂ t ∂ x ∂ x∂ t 2 ∂t 0 2 2 2 ∂ ∂ 3 v ∂ 3 v ∂v ∂ v ∂v + D + + ∂ x 11 ∂ x3 ∂ x3 ∂ x ∂ x2 ∂ x 2 2
3B55 l 2 ∂ ∂ 3 v ∂ 2 v 1 ∂ 3 v ∂v 1 ∂ 2v ∂v ∂ 3v 2 ∂ + A55 l + + − ∂x 4 ∂ x3 ∂ x 4 ∂ x2 ∂x 2 ∂ x ∂ x3 ∂ x2 ∂ x3 2 2 1 ∂ 5 v ∂v ∂ 4 v ∂ 2 v ∂v 3 ∂ 3 v ∂v 2 ∂ − D55 l + + ∂ x 4 ∂ x5 ∂ x ∂ x4 ∂ x2 ∂ x 4 ∂ x3 ∂ x +c
∂v + cb0 ω cos (ω t ) − I1 b0 ω2 sin (ω t ) = 0. ∂t
(20)
Introducing the following dimensionless parameters
x v x = , ( v∗ ) = , L L ∗
η=
L l , ls = , h h
"
∗
c =c
L4 , I10 D110
∗
(I1 ) =
I 1
I10
"
∗
, t =t
{a55 , b55 , d55 } =
#
D110 , =ω I10 L4
A55 h2 B55 h D55 , , D110 D110 D110
$
"
I10 L4 b0 ∗ , b0 = , (d11 ) = D110 L
D 11
D110
,
(21)
H. Farokhi et al. / International Journal of Engineering Science 115 (2017) 117–130
123
Fig. 4. Frequency-response curves of the functionally graded micro-cantilever for different values of the material gradient index (ceramic, n = 2, n = 20, and metal): (a, b) the first two generalised coordinates of the transverse motion, respectively. ls = 0.45, b0 = 0.006, and ζ = 0.008.
the nonlinear integro-partial-differential equation of motion (Eq. (20)) can be rewritten as
! 2 2 x ∂ 2 v ∂ ∂v x ∂ 3 v ∂v ∂ v I1 2 + I + dx dx ∂ x ∂ x 1 1 0 ∂ x∂ t 2 ∂ x ∂ x∂ t ∂t 2 2 2 ∂ ∂ 3 v ∂ 3 v ∂v ∂ v ∂v + d11 + + ∂ x ∂ x3 ∂ x3 ∂ x ∂ x2 ∂ x 2 2
2 3b55 ls ∂ ∂ 3 v ∂ 2 v 1 ∂ 3 v ∂v 1 ∂ 2v ∂v ∂ 3v 2 ∂ + a55 ls + + − ∂x 4 ∂ x3 ∂ x 4 ∂ x2 ∂x 2η ∂ x ∂ x3 ∂ x2 ∂ x3 2 2 2 d55 ls ∂ 1 ∂ 5 v ∂v ∂ 4 v ∂ 2 v ∂v 3 ∂ 3 v ∂v − + + η 2 ∂ x 4 ∂ x5 ∂ x ∂ x4 ∂ x2 ∂ x 4 ∂ x3 ∂ x +c
∂v − I b 2 sin ( t ) + cb0 cos ( t ) = 0, ∂t 1 0
(22)
in which the asterisk notation is dropped for briefness. 3. Truncated model of functionally graded micro-cantilever The continuous nonlinear integro-partial-differential equation of motion of the functionally graded micro-cantilever given in Eq. (22) is recast into coupled nonlinear ordinary differential equations via use of the Galerkin technique. As such (Ghayesh & Amabili, 2012; Ghayesh, et al., 2013a, b)
v(x, t ) =
M % r=1
qr (t )ϕr (x ) ,
(23)
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H. Farokhi et al. / International Journal of Engineering Science 115 (2017) 117–130
Fig. 5. Force-response curves of the functionally graded micro-cantilever for different values of the material gradient index (ceramic, n = 2, n = 20, and metal): (a, b) the first two generalised coordinates of the transverse motion, respectively. ls = 0.45, Ω/ω1 = 0.98, and ζ = 0.008.
with ϕr (x ) being the rth eigenfunction for the transverse displacement of a linear non-functionally-graded fixed-free beam, and qr (x ) being the rth generalised coordinate. Substitution of Eq. (23) into Eq. (22) and application of the Galerkin scheme results in the following high-dimensional truncated model of the functionally graded microsystem
I1
M %
q¨ j
0
j=1
+
1
ϕi ϕ j dx
M % M % M % j=1 k=1 l=1
+
M % M % M % j=1 k=1 l=1
+
M % M % M % j=1 k=1 l=1
+
M % M % M %
#
q¨ j qk ql
#
0
q˙ j q˙ k ql
# q¨ j qk ql
q˙ j q˙ k ql
+ d11
M % j=1
+ d11 4
0
1
1
0
j=1 k=1 l=1
1
0
#
1
0
1
ϕi ϕi
x 1
x 1
ϕi I1
x
0
ϕi I1
x
0
$ ϕ l dx
ϕ
j ϕ k dx
ϕ
j ϕ k dx
1 0
$ ϕ l dx
M % M % M %
ϕi
ϕ
k
ϕ
l
ϕ
ϕi ϕ j ϕ k ϕ l dx q j qk ql
j
1 0
j=1 k=1 l=1
q j qk ql
$ ϕ j ϕ k dxdx ϕ l dx
0
ϕi ϕ j dx q j +
j=1 k=1 l=1
x
I1
$ ϕ j ϕ k dxdx ϕ l dx
0
M % M % M %
x
I1
dx
+
M % M % M % j=1 k=1 l=1
q j qk ql
1 0
ϕi
ϕ
k
ϕ
l ϕ j dx
H. Farokhi et al. / International Journal of Engineering Science 115 (2017) 117–130
125
Fig. 6. Frequency-response curves of the functionally graded micro-cantilever for different values of the length scale parameter (ls = 0.00, ls = 0.35, ls = 0.45, and ls = 0.55): (a, b) the first two generalised coordinates of the transverse motion, respectively. n = 2, b0 = 0.006, and ζ = 0.008.
+ a55 ls
M %
2
+ a55 ls
j=1
1 0
ϕi
ϕ
3b55 ls − 2η −
−
−
d55 ls
2
η2
d55 ls
2
η2
d55 ls
+c
η2
M % j=1
2
2
M % M % j=1 k=1
1 0
j=1 k=1 l=1
1 0
j
ϕ
k ϕ l dx
q j qk ql
j=1 k=1 l=1
M % M % M %
2
1 M M M 1 %%% dx q j + ϕi ϕ j 4 0
1 M M M 1 %%% ϕi ϕ j ϕ k ϕ l dx q j qk ql + ϕi ϕ j ϕ k ϕ l dx q j qk ql 4 0 j=1 k=1 l=1
ϕi
ϕ
j ϕ k dx
1 M M M 1 %%% ϕi ϕ j 4 0 j=1 k=1 l=1
q j qk +
1 0
j=1 k=1
ϕi
ϕ
j ϕ k dx
q j qk
1 M M M 3 %%% ϕ k ϕ l dx q j qk ql + ϕi ϕ j 2 0
1 M M M 5 %%% ϕi ϕ k ϕ l ϕ j 2 0 j=1 k=1 l=1
M % M %
j=1 k=1 l=1
dx q j qk ql +
1 M M M 3 %%% ϕi ϕ j ϕ k ϕ l dx q j qk ql 4 0
M % M % M % j=1 k=1 l=1
0
1
ϕ k ϕ l dx q j qk ql
ϕi
ϕ
j
ϕ
k ϕ l dx
q j qk ql
j=1 k=1 l=1
1 0
1 1 ϕi ϕ j dx q˙ j + ϕi cb0 cos ( t ) dx − ϕi I1 b0 2 sin ( t ) dx = 0. 0
(24)
0
As seen in Eq. (24), there are inertial nonlinearities present as well as geometric nonlinearities. The solution technique used for Eq. (24) is a coupled pseudo-arclength and the backward differentiation formula (BDF), by choosing M = 6
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(a) ls=0.00 ls=0.35
0.3
ls=0.45
max[q1]
ls=0.55
0.2
0.1
0 (b)
0
0.002
0.004
b0
0.006
0.008
0.004 ls=0.00 ls=0.35 ls=0.45 ls=0.55
max[q2]
0.003
0.002
0.001
0 0
0.002
0.004
b0
0.006
0.008
Fig. 7. Force-response curves of functionally graded micro-cantilever for different values of the length scale parameter (ls = 0.00, ls = 0.35, ls = 0.45, and ls = 0.55): (a, b) the first two generalised coordinates of the transverse motion, respectively. n = 2, Ω/ω 1 = 0.98, and ζ = 0.008.
which ensures reliable and converged results for the large-amplitude nonlinear oscillations of the functionally graded microcantilever; the linear natural frequencies are obtained using an eigenfrequency method (Ghayesh & Amabili, 2013; Kazemirad et al., 2013). 4. Frequency-responses and force-responses Based on the solution procedure described in Section 3, the frequency-responses and force-responses in the resonant region of the functionally graded micro-cantilever are constructed in this section. The microsystem considered is of h = 10 μm, L/h = 100, and b/h = 4; the two constituents are aluminium with EAl = 69 GPa, ρAl = 2700 kg/m3 , and υAl = 0.33 and ceramic (SiC) with ESiC = 427 GPa, ρSiC = 3100 kg/m3 , and υSiC = 0.17. The frequency-response diagrams of the functionally graded micro-cantilever are depicted in Fig. 2, displaying a weakly softening-type behaviour; the microsystem parameters are set to n = 2, ls = 0.45, b0 = 0.006, and ζ = 0.008 (where ζ is the modal damping ratio). This set of parameters results in the fundamental natural frequency of ω1 = 7.1621 for the oscillation behaviour of the functionally graded micro-cantilever. These are two saddle-node bifurcations present at A and B (Ω/ω 1 = 0.9809 and 0.9764, respectively). The weak softening-type nonlinearity in the oscillation behaviour of the functionally graded micro-cantilever is the balance between different sources of nonlinearity involving inertial and geometric. Experimentally, the saddle-node bifurcations are responsible for jumps in the oscillation amplitude. The force-response diagrams of the functionally graded micro-cantilever are shown in Fig. 3 for n = 2, ls = 0.45, Ω/ω1 = 0.98, and ζ = 0.008. There are two saddle-node bifurcation points present at the amplitude of the base excitation of b0 = 0.0064 and 0.0054 (at points A and B, respectively); at point A the oscillation amplitude of the functionally graded micro-cantilever jumps to a larger value; however, at point B a reduction occurs in the motion amplitude. 5. Effect of material gradient index, length-scale parameter, and amplitude of base excitation This section analyses the effect of the parameters of the functionally graded micro-cantilever, such as the length-scale parameter, material gradient index, and the base-excitation, on the size-dependent oscillation behaviour of the functionally
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Fig. 8. Frequency-response curves of the functionally graded micro-cantilever: (a, b) the first two generalised coordinates of the transverse motion, respectively. n = 2, ls = 0.45, b0 = 0.006, and ζ = 0.008.
graded microsystem. The length, width, and thickness of the micro-cantilever as well as its constituents and their properties are kept the same as of Section 4. The frequency-response diagrams of the functionally graded micro-cantilever is highlighted in Fig. 4 for several material gradient indices; other microsystem parameters are set to ls = 0.45, b0 = 0.006, and ζ = 0.008. The largest peak-amplitude belongs to the ceramic micro-cantilever. For all the cases of the functionally graded micro-cantilever the nonlinearity is of a weak softening-type with two saddle-node bifurcations. The influence of the material gradient index on the force-response curves of the functionally graded micro-cantilever is illustrated in Fig. 5; ls = 0.45, Ω/ω1 = 0.98, and ζ = 0.008. As seen, the smallest base excitation amplitude corresponding to the second saddle-node bifurcation belongs to the metal case, the largest b0 for the first saddle-node belongs to the case with n = 20, and the overall motion amplitude of the case with n = 2 is the largest. Fig. 6 shows that how the length-scale parameter affects the vibrations of the functionally graded micro-cantilever; n = 2, b0 = 0.006, and ζ = 0.008. It can be seen that accounting for the effect of the small size of the functionally graded microcantilever (shown by the length-scale parameter) reduces the overall oscillation amplitude slightly. The fundamental natural frequency is larger for larger values of ls . For all the functionally graded cases, the type of nonlinearity is softening and the largest peak-amplitude belongs to the case with ls = 0, as the representative of the classical continuum mechanics. It is worth noting that the softening nonlinearity becomes stronger with increasing the length-scale parameter. Fig. 7 plots the force-response diagrams of the functionally graded micro-cantilever for various length-scale parameters; n = 2, Ω/ω1 = 0.98, and ζ = 0.008. It can be seen that for the functionally graded micro-cantilever modelled on the basis of classical continuum mechanics, there is no jump occurring, while, for the all other cases there are two jump-points present. It is seen that, for the range of b0 studied, the overall amplitude of the case with ls = 0.35 is the largest. For the microsystem with n = 2, ls = 0.45, and ζ = 0.008, as base-excitation amplitude is increased, as shown in Fig. 8, the oscillation amplitude of the functionally graded micro-cantilever becomes larger, which is characterised by the occurrence of the peak amplitudes at smaller base excitation frequencies.
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Fig. 9. Comparison of frequency-responses of the functionally graded micro-cantilever (current study) and silicon micro-cantilever (from Ref. (Farokhi et al., 2016)): (a, b) the first two generalised coordinates of the transverse motion, respectively. ls = 0.35, b0 = 0.0065, and ζ = 0.008.
6. Validation of nonlinear equation of motion The equation of motion of the functionally graded micro-cantilever derived in this paper (Eq. (20)) is validated by comparing it to the equation of motion of a non-functionally-graded micro-cantilever of Ref. Farokhi, Ghayesh, and Hussain (2016). To this end, when Eq. (14) and (16) are considered, one obtains
A11 = EA,
(25a)
B11 = 0,
(25b)
D11 = EI,
(25c)
A55 = μA.
(25d)
B55 = 0,
(25e)
D55 = μI,
(25f)
I1 = ρ A,
(25g)
which results in
! x x 2 2 ∂ 2v ∂ ∂v ∂ 3 v ∂v ∂ v ρA 2 + ρA + dx dx ∂x ∂ x∂ t ∂ x ∂ x∂ t 2 ∂x ∂t L 0 2 2 2 ∂ ∂ 3 v ∂ 3 v ∂v ∂ v ∂v + EI + + ∂ x ∂ x3 ∂ x3 ∂ x ∂ x2 ∂ x
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129
2 2 ∂ 3v ∂ 1 ∂ 3 v ∂v 1 ∂ 2v ∂v + μA l + + ∂x 4 ∂ x3 ∂ x 4 ∂ x2 ∂x ∂ x3 2 2 1 ∂ 5 v ∂v ∂ 4 v ∂ 2 v ∂v 3 ∂ 3 v ∂v 2 ∂ − μI l + + ∂ x 4 ∂ x5 ∂ x ∂ x4 ∂ x2 ∂ x 4 ∂ x3 ∂ x 2
+c
∂v + cb0 ω cos (ω t ) − I1 b0 ω2 sin (ω t ) = 0, ∂t
(26)
which is exactly the same as Eq. (10) of Ref. Farokhi et al. (2016). The computer codes have also been validated by obtaining the results of Ref. Farokhi et al. (2016) while our codes are simplified; the comparison shown in Fig. 9 displays excellent agreement. 7. Conclusions This paper was the first which modelled and simulated the large-amplitude nonlinear oscillations of functionally graded micro-cantilevers incorporating curvature-related nonlinearities and size-dependence via MCST. Based on the Mori–Tanaka homogenisation technique, using the constitutive relations, the potential energy term of the functionally graded microcantilever was dynamically balanced with the kinetic energy term and the work of the base excitation, resulting in a nonlinear integro-partial-differential equation of motion of the microsystem when inextensibility was considered. A highdimensional truncated model was obtained using the Galerkin method and the resultant model of the functionally graded micro-cantilever was solved by means of the backward differentiation formula (BDF) together with a continuation technique. The numerical modelling and simulations showed that: (i) when both the curvature and inertial nonlinearities are considered, the nonlinear dynamics of the functionally graded micro-cantilever shows a weakly softening behaviour; (ii) for most of the cases studied, the frequency-response curves of the functionally graded micro-cantilever shows two saddlenode bifurcations; (iii) among the cases studied, the purely ceramic micro-cantilever displays the largest peak-amplitude in the frequency response curves; (iv) in the force-response diagrams, the smallest base excitation amplitude corresponding to the second saddle-node bifurcation belongs to the metal case and the largest base-excitation amplitude for the first saddlenode belongs to the case with the material gradient index of 20; (v) as the value for length-scale parameter is increased, the fundamental frequency of free oscillations increases, and the softening nonlinearity becomes stronger. Acknowledgements The financial support to this research by the start-up grant of the University of Adelaide is gratefully acknowledged. References Aghazadeh, R., Cigeroglu, E., & Dag, S. (2014). 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, 1–11. Akgöz, B., & Civalek, Ö. (2011). Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. International Journal of Engineering Science, 49, 1268–1280. Akgöz, B., & Civalek, Ö. (2013a). Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory. Composite Structures, 98, 314–322. Akgöz, B., & Civalek, Ö. (2013b). A size-dependent shear deformation beam model based on the strain gradient elasticity theory. International Journal of Engineering Science, 70, 1–14. 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Composites Part B: Engineering, 60, 371–377. Ghayesh, M. H., Amabili, M., & Farokhi, H. (2013a). Coupled global dynamics of an axially moving viscoelastic beam. International Journal of Non-Linear Mechanics, 51, 54–74. Ghayesh, M. H., Amabili, M., & Farokhi, H. (2013b). Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. International Journal of Engineering Science, 63, 52–60. Ghayesh, M. H., & Farokhi, H. (2015a). Nonlinear dynamics of microplates. International Journal of Engineering Science, 86, 60–73. Ghayesh, M. H., & Farokhi, H. (2015b). Chaotic motion of a parametrically excited microbeam. International Journal of Engineering Science, 96, 34–45. Ghayesh, M. H., Farokhi, H., & Amabili, M. (2013b). Nonlinear behaviour of electrically actuated MEMS resonators. International Journal of Engineering Science, 71, 137–155. Ghayesh, M. H., Farokhi, H., & Amabili, M. (2014). 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International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Dynamics of functionally graded micro-cantilevers Hamed Farokhi a, Mergen H. Ghayesh b,∗, Alireza Gholipour b a b
Department of Mechanical Engineering, McGill University, Montreal, Quebec H3A 0C3, Canada School of Mechanical Engineering, University of Adelaide, South Australia 5005, Australia
a r t i c l e
i n f o
Article history: Received 18 January 2017 Accepted 21 January 2017
Keywords: Functionally graded micro-cantilever MCST Dynamics MEMS
a b s t r a c t The nonlinear size-dependent dynamics of a functionally graded micro-cantilever is investigated when subject to a base excitation resulting in large-amplitude oscillations. A geometric nonlinearities due to large changes in the curvature is taken into account. Employing the Mori–Tanaka homogenisation technique (for the material properties), the modified couple stress theory (MCST) is used to formulate the potential and kinetic energies of the system in terms of the transverse and axial motions. A dynamic energy balance is performed between the energy terms, yielding the continuous models for the axial and transverse displacements. The inextensibility condition results in the size-dependent model of the functionally graded micro-cantilever involving inertial and stiffness nonlinear terms. The resultant model is discretised based on a weighted-residual technique yielding a high-dimensional truncated model (required for accurate simulations). A parametercontinuation scheme together with a time integration method is introduced to the truncated model so as to determine the resonances with stable and unstable solution branches with special consideration to the effect of different system parameters, such as material gradient index and the length-scale effect on the nonlinear dynamics of the functionally graded micro-cantilever. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Cantilevers of microscale dimensions form the core components of many microelectromechanical systems (MEMS) (Asghari, Kahrobaiyan, & Ahmadian, 2010; Baghani, 2012; Ghayesh & Farokhi, 2015a, b; Ghayesh, Farokhi, & Amabili, 2013a, b; Zhang & Meng, 2007; Zheng, Dong, Lee, & Gao, 2009). Experiments showed that their deformation behaviour is highly size-dependent (Akgöz & Civalek, 2013a, b, Akgöz & Civalek, 2011; Dehrouyeh-Semnani, 2014; Dehrouyeh-Semnani, BehboodiJouybari, & Dehrouyeh, 2016; Farokhi, Ghayesh, Gholipour, & Hussain, 2017; Ghayesh, Amabili, & Farokhi, 2013a, b, Hosseini & Bahaadini, 2016; Kahrobaiyan, Rahaeifard, Tajalli, & Ahmadian, 2012; Karparvarfard, Asghari, & Vatankhah, 2015; Kong, Zhou, Nie, & Wang, 2008; Mojahedi & Rahaeifard, 2016; Shafiei, Kazemi, & Ghadiri, 2016a, b, Taati, 2016). Theoretically, only the advanced continuous theories such as strain gradient and MCST (Dai, Wang, & Wang, 2015; Dehrouyeh-Semnani et al., 2015; Farokhi & Ghayesh, 2015 b, Farokhi & Ghayesh, 2015 a, Farokhi, Ghayesh, & Amabili, 2013a, b, Ghayesh & Amabili, 2014; Ghayesh and Farokhi, 2015, Ghayesh, et al., 2013 a, b, Ghayesh, Farokhi, & Amabili, 2014; Ghayesh, Farokhi, & Hussain, 2016; Gholipour, Farokhi, & Ghayesh, 2015; Li & Pan, 2015; S¸ ims¸ ek, 2010; Tang, Ni, Wang, Luo, & Wang, 2014) theories are capable of incorporating size effects in the dynamical modelling of microscale elements; in other words, simulations based on the classical continuum mechanics may result in drastic errors (Farokhi, Ghayesh, & Amabili, 2013). ∗
Corresponding author. E-mail address: [email protected] (M.H. Ghayesh).
http://dx.doi.org/10.1016/j.ijengsci.2017.01.004 0020-7225/© 2017 Elsevier Ltd. All rights reserved.
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Fig. 1. Schematic representation of functionally graded micro-cantilever.
In recent years, microstructures made of functionally graded (FG) materials have gained high interests in MEMS industry due to their mechanical properties to achieve a better performance and sensitivity (Witvrouw, 2005; Huang, 2008; Lü, Lim, & Chen, 2009). Micro-cantilevers, as opposed to supported–supported (i.e. combination of clamps and pins) microbeams (Farokhi & Ghayesh, 2016), are subjected to large deformations when a transverse load is applied; this is due to the fact that one end of the microsystem possesses displacement-freedom. In other words, the effect of the curvature-related nonlinearities become dominant and should not be neglected, especially when coupled with the length-scale parameter. The literature on the dynamics/statics of supported-supported functionally graded microbeams, involving both linear and nonlinear analyses is large; for example, Akgöz and Civalek (2014) analysed the thermal variation influences on the buckling behaviour of a FG microbeam resting on an elastic foundation. Tajalli et al. (2013) formulated the motion behaviour of a FG Timoshenko microbeam. Aghazadeh, Cigeroglu, and Dag (2014) contributed to the field by developing a procedure of obtaining the linear dynamic/static characteristics of Timoshenko, Bernoulli–Euler, and shear deformable types of FG microbeams. Arbind and Reddy (2013) developed a geometrically nonlinear finite element model for a FG microbeam employing the Euler–Bernoulli and Timoshenko theories via use of MCST; the resultant model was used to obtain the buckling response of the microsystem. The literature on the dynamics/statics of functionally graded micro-cantilevers is not large. Asghari, Ahmadian, Kahrobaiyan, & Rahaeifard (2010) examined the size-dependent static and free dynamics of FG micro-cantilevers based on MCST. Shafiei et al. (2016a, b) examined the transverse vibrations of an axially FG micro-cantilever based on the Euler– Bernoulli and MCST. Akgöz & Civalek (2013a, b) employed MCST in order to obtain the dynamical response of non-uniform axially functionally graded micro-cantilevers. This paper is the first which examines the size-dependent nonlinear dynamics of a functionally graded micro-cantilever undergoing large oscillations via MCST incorporating curvature and inertial nonlinearities. The size-dependent potential energy of the functionally graded micro-cantilever is obtained as a continuous function of the system parameters employing the Mori–Tanaka technique as well as constitutive relations. A dynamic balance is introduced to the kinetic energy and the size-dependent potential energy giving the continuous equations of the transverse and axial displacements. The inextensibility condition leads to an integro-partial-differential equation of motion of the FG micro-cantilever. A system reduction is carried out on the basis of the Galerkin scheme. The resonances of the functionally graded micro-cantilever are obtained highlighting the influence of different microsystem parameters including the length-scale parameter as well as the material gradient index. 2. Microsystem model of functionally graded micro-cantilever Consider a functionally graded micro-cantilever of length L, thickness h, and cross-sectional area of A (Fig. 1). The left end of the functionally graded micro-cantilever is clamped while the right end is free to move. The clamped end is excited harmonically by b0 sin(ωt) where ω is the excitation frequency, t is time, and b0 denotes the excitation amplitude. The total displacement field is shown by the axial component u(x,t) and the transverse component w(x,t), where x is axial coordinate. The functionally graded micro-cantilever is made of metal and ceramic and the mechanical properties are Young’s modulus E, mass density ρ , Poisson’s ratio υ , and shear modulus μ. At z = h/2 (i.e. at the top surface) the micro-cantilever is ceramicrich and at z =-h/2 (the bottom surface) is metal-rich. The Mori–Tanaka homogenisation technique is utilised to determine the effective mechanical properties of the mixture. As such, the effective bulk modulus (shown by Ke )and the effective shear modulus (shown by μe ) are defined as functions of μc , μm , Kc , and Km (shear modulus of the ceramic, shear modulus of the metal, bulk modulus of the ceramic, and the bulk modulus of the metal, respectively) as well as vm and vc (the volume fractions of the metal and ceramic, respectively where vm + vc = 1); thus
Ke − Km vc = , Kc − Km 1 + vm (Kc − Km )/(Km + 4μm /3 )
(1)
H. Farokhi et al. / International Journal of Engineering Science 115 (2017) 117–130
−
μe − μm vc = . μm − μc 1 + vm (−μm + μc )/[μm + μm (8μm + 9Km )/(6(Km + 2μm ) )]
119
(2)
Different distributions can be obtained for the ceramic volume fraction by means of a power law function with the gradient index of n as follows
vc (z ) = (z/h + 1/2)n .
(3)
Effective Poisson’s ratio and Young’s modulus are given, respectively, by
υ (z ) =
3Ke − 2μe . 6Ke + 2μe
(4)
E (z ) =
9μe Ke , μe + 3Ke
(5)
The effective mass density is given by
ρ ( z ) = ρm v m + ρc v c ,
(6)
where ρ m and ρ c are the mass densities of the metal and ceramic, respectively. Based on MCST, the potential energy of the functionally graded micro-cantilever of size-dependent strain-type can be formulated as
U=
1 2
(m : χ + σ : ε)dv,
V
(7)
where ε, χ, σ, and m represent the strain tensor, the symmetric curvature tensor, the stress tensor, and the deviatoric part of the symmetric couple stress tensor, respectively. Moreover,
m = 2 μe l 2 χ ,
(8)
where l represents the length-scale parameter. Furthermore,
σxx = E (z )εxx .
(9)
For an Euler–Bernoulli type of a functionally graded micro-cantilever, the strain at a point located at a distance z from the mid-plane of the functionally graded micro-cantilever can be determined as a function of u, w, and ψ (the angle of the centreline of the micro-cantilever with respect to the axial axis); as such
⎡
εxx = ⎣−1 + Moreover,
χxy
1 =− 4
∂w ∂x
2
+
1+
∂u ∂x
2
⎤
⎦ − z ∂ψ . ∂x
∂w ∂ 2 cos (ψ ) ∂ sin (ψ ) + +z = χyx , ∂x ∂x ∂ x2
χyz = χzy = −
1 ∂ cos (ψ ) . 4 ∂x
(10)
(11) (12)
Considering Eqs. (7)–(12), the elastic potential energy of the functionally graded micro-cantilever becomes
1 U = 2
⎡
L
0
⎛ ⎞ ⎛ ⎞ 2 2 2 2 2 ∂u ⎠ w ∂ u ∂w ∂ ⎢ ⎝ ⎠ ∂ψ + 1+ − 2B11 ⎝−1 + + 1+ ⎣A11 −1 + ∂x ∂x ∂x ∂x ∂x
2
2
2 A55 l 2 ∂ ∂ w B55 l 2 ∂ ∂ w ∂ψ ∂ cos (ψ ) + D11 + + sin (ψ ) + + sin (ψ ) ∂x 4 ∂x ∂x 2 ∂x ∂x ∂ x2 2 2 D55 l 2 ∂ 2 cos (ψ ) A55 l 2 ∂ cos (ψ ) + + dx, 2 4 4 ∂x ∂x with
A11 = A
E (z )dA,
(13)
(14a)
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B11 =
z E (z )dA,
(14b)
z2 E (z )dA,
(14c)
E (z ) dA, 2 (1 + υ (z ) )
(14d)
z E (z ) dA, 2 (1 + υ (z ) )
(14e)
z2 E (z ) dA. 2 (1 + υ (z ) )
(14f)
A
D11 = A
A55 = A
B55 = A
D55 = A
The kinetic energy of the functionally graded micro-cantilever is
1 T = 2 where
I1 =
L
I1
0
∂u ∂t
2
+ I1
∂w ∂t
2
dx,
(15)
ρ (z )dA.
(16)
A
Dynamically balancing the potential energy of the functionally graded micro-cantilever with the kinetic one gives
⎧⎡ ⎛ ⎞ ⎤ 2 2 ⎨ ∂ u ∂ ⎣ ⎝ ∂u ⎠ ∂ψ ⎦ ∂w I1 2 = A −1 + + 1+ − B11 cos (ψ ) ∂ x ⎩ 11 ∂x ∂x ∂x ∂t ⎡ ⎛ ⎞ ⎤ 2 2 ∂ ⎣ ⎝ ∂u ∂ψ ⎦ sin (ψ ) ∂w − B 1+ + − 1⎠ − D11 ∂ w 2 ∂ x 11 ∂x ∂x ∂x ∂u 2 2
1 ∂ ∂ + A55 l 2 sin (ψ ) + 4 ∂x ∂x
+
1 ∂ ∂ B55 l 2 sin (ψ ) + 4 ∂ x2 ∂x 2
1 ∂ ∂ cos (ψ ) − A55 l 2 4 ∂x ∂x
1 + ∂x
+ ∂x
1 + ∂x
+ ∂x
⎞ 2 ∂w ∂ cos ψ sin ψ cos ψ ( ) ( ) ( ) ⎝ ⎠ + B55 l 2 ∂ w 2 ∂x ∂ x2 ∂u 2 ⎛
⎛
∂w ∂ cos (ψ ) ⎝ + D55 l 2 ∂x ∂ x2 2
⎛ ⎝
sin (ψ ) 2
1 + ∂∂ ux
2
⎞⎫ ⎬ ⎠ ∂ w 2 ⎭
sin (ψ )
⎞
2
1 + ∂∂ ux
2
+ ∂∂wx
2
⎠
(17)
+ ∂x
⎧⎡ ⎛ ⎞ ⎤ 2 2 ⎨ ∂ w ∂ ⎣ ⎝ ∂u ∂ψ ⎦ ∂w I1 2 − A 1+ + − 1⎠ − B11 sin (ψ ) ∂ x ⎩ 11 ∂x ∂x ∂x ∂t ⎡ ⎛ ⎞ ⎤ 2 2 ∂ ⎣ ⎝ ∂u ⎠ ∂ψ ⎦ cos (ψ ) ∂w + B −1 + + 1+ − D11 ∂ x 11 ∂x ∂x ∂x ∂w 2 2
∂x
+ 1 + ∂∂ ux
⎛
1 ∂
2 ∂ ∂ w ∂ cos ψ ( ) ⎝1 + − l 2 A55 sin (ψ ) + + l 2 B55 4 ∂x ∂x ∂x ∂ x2
2
cos2 (ψ )
2 2 1 + ∂∂ ux + ∂∂wx
⎞ ⎠
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121
Fig. 2. Frequency-response curves of the functionally graded micro-cantilever: (a, b) the first two generalised coordinates of the transverse motion, respectively. n = 2, ls = 0.45, b0 = 0.006, and ζ = 0.008.
1 ∂2 ∂ − B55 l 2 sin (ψ ) + 4 ∂ x2 ∂x
⎞ ⎛ 2 ∂w ∂ cos ψ sin ψ cos ψ ( ) ( ) ( ) ⎝ ⎠ + D55 l 2 ∂ w 2 ∂x ∂ x2 ∂u 2
⎞⎫
⎬ 1 ∂ ∂ cos (ψ ) ⎝ sin (ψ ) cos (ψ ) ⎠ + A55 l 2 ∂ w 2 ⎭ = 0 . 4 ∂x ∂x ∂u 2 ⎛
1 + ∂x
1 + ∂x
+ ∂x
(18)
+ ∂x
The inextensibility assumption is applied to Eqs. (17) and (18), while remembering that the axial force is zero at the right end of the functionally graded micro-cantilever. This operation leads to
! 2 2 x ∂ 2w ∂ ∂ w x ∂ 3w ∂ w ∂ w I1 2 + I1 + dx dx ∂x ∂x L ∂ x∂ t ∂t ∂ t 2∂ x ∂ x 0 2 2 ∂ ∂ 3w ∂ 3w ∂ w ∂ w ∂ 2w + D11 + + ∂ x ∂ x3 ∂ x ∂ x2 ∂ x3 ∂ x 2 2 2
3 3 3B55 l 2 ∂ ∂ 2 w ∂ 3 w ∂ w 1 ∂ w w 1 ∂ w ∂ ∂ ∂ w + A55 l 2 + + − ∂x 4 ∂ x3 ∂ x 4 ∂ x ∂ x2 2 ∂ x ∂ x2 ∂ x3 ∂ x3 2 2 1 ∂ 5w ∂ w ∂ 4w ∂ w ∂ 2w 3 ∂ w ∂ 3w ∂w 2 ∂ − D55 l + + +c = 0, ∂ x 4 ∂ x5 ∂ x ∂t ∂ x4 ∂ x ∂ x2 4 ∂ x ∂ x3
(19)
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Fig. 3. Force-response curves of the functionally graded micro-cantilever: (a, b) the first two generalised coordinates of the transverse motion, respectively. n = 2, ls = 0.45, Ω/ω1 = 0.98, and ζ = 0.008.
where a viscous damping force of c ∂ w/∂ t (with c being the coefficient) has been introduced. Defining v(x,t) as the transverse displacement relative to the base, Eq. (19) can be rewritten as
! x 2 2 ∂ 2 v ∂ ∂v x ∂v ∂ 3 v ∂ v I1 2 + I1 + dx dx ∂x ∂x L ∂ x∂ t ∂ x ∂ x∂ t 2 ∂t 0 2 2 2 ∂ ∂ 3 v ∂ 3 v ∂v ∂ v ∂v + D + + ∂ x 11 ∂ x3 ∂ x3 ∂ x ∂ x2 ∂ x 2 2
3B55 l 2 ∂ ∂ 3 v ∂ 2 v 1 ∂ 3 v ∂v 1 ∂ 2v ∂v ∂ 3v 2 ∂ + A55 l + + − ∂x 4 ∂ x3 ∂ x 4 ∂ x2 ∂x 2 ∂ x ∂ x3 ∂ x2 ∂ x3 2 2 1 ∂ 5 v ∂v ∂ 4 v ∂ 2 v ∂v 3 ∂ 3 v ∂v 2 ∂ − D55 l + + ∂ x 4 ∂ x5 ∂ x ∂ x4 ∂ x2 ∂ x 4 ∂ x3 ∂ x +c
∂v + cb0 ω cos (ω t ) − I1 b0 ω2 sin (ω t ) = 0. ∂t
(20)
Introducing the following dimensionless parameters
x v x = , ( v∗ ) = , L L ∗
η=
L l , ls = , h h
"
∗
c =c
L4 , I10 D110
∗
(I1 ) =
I 1
I10
"
∗
, t =t
{a55 , b55 , d55 } =
#
D110 , =ω I10 L4
A55 h2 B55 h D55 , , D110 D110 D110
$
"
I10 L4 b0 ∗ , b0 = , (d11 ) = D110 L
D 11
D110
,
(21)
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123
Fig. 4. Frequency-response curves of the functionally graded micro-cantilever for different values of the material gradient index (ceramic, n = 2, n = 20, and metal): (a, b) the first two generalised coordinates of the transverse motion, respectively. ls = 0.45, b0 = 0.006, and ζ = 0.008.
the nonlinear integro-partial-differential equation of motion (Eq. (20)) can be rewritten as
! 2 2 x ∂ 2 v ∂ ∂v x ∂ 3 v ∂v ∂ v I1 2 + I + dx dx ∂ x ∂ x 1 1 0 ∂ x∂ t 2 ∂ x ∂ x∂ t ∂t 2 2 2 ∂ ∂ 3 v ∂ 3 v ∂v ∂ v ∂v + d11 + + ∂ x ∂ x3 ∂ x3 ∂ x ∂ x2 ∂ x 2 2
2 3b55 ls ∂ ∂ 3 v ∂ 2 v 1 ∂ 3 v ∂v 1 ∂ 2v ∂v ∂ 3v 2 ∂ + a55 ls + + − ∂x 4 ∂ x3 ∂ x 4 ∂ x2 ∂x 2η ∂ x ∂ x3 ∂ x2 ∂ x3 2 2 2 d55 ls ∂ 1 ∂ 5 v ∂v ∂ 4 v ∂ 2 v ∂v 3 ∂ 3 v ∂v − + + η 2 ∂ x 4 ∂ x5 ∂ x ∂ x4 ∂ x2 ∂ x 4 ∂ x3 ∂ x +c
∂v − I b 2 sin ( t ) + cb0 cos ( t ) = 0, ∂t 1 0
(22)
in which the asterisk notation is dropped for briefness. 3. Truncated model of functionally graded micro-cantilever The continuous nonlinear integro-partial-differential equation of motion of the functionally graded micro-cantilever given in Eq. (22) is recast into coupled nonlinear ordinary differential equations via use of the Galerkin technique. As such (Ghayesh & Amabili, 2012; Ghayesh, et al., 2013a, b)
v(x, t ) =
M % r=1
qr (t )ϕr (x ) ,
(23)
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Fig. 5. Force-response curves of the functionally graded micro-cantilever for different values of the material gradient index (ceramic, n = 2, n = 20, and metal): (a, b) the first two generalised coordinates of the transverse motion, respectively. ls = 0.45, Ω/ω1 = 0.98, and ζ = 0.008.
with ϕr (x ) being the rth eigenfunction for the transverse displacement of a linear non-functionally-graded fixed-free beam, and qr (x ) being the rth generalised coordinate. Substitution of Eq. (23) into Eq. (22) and application of the Galerkin scheme results in the following high-dimensional truncated model of the functionally graded microsystem
I1
M %
q¨ j
0
j=1
+
1
ϕi ϕ j dx
M % M % M % j=1 k=1 l=1
+
M % M % M % j=1 k=1 l=1
+
M % M % M % j=1 k=1 l=1
+
M % M % M %
#
q¨ j qk ql
#
0
q˙ j q˙ k ql
# q¨ j qk ql
q˙ j q˙ k ql
+ d11
M % j=1
+ d11 4
0
1
1
0
j=1 k=1 l=1
1
0
#
1
0
1
ϕi ϕi
x 1
x 1
ϕi I1
x
0
ϕi I1
x
0
$ ϕ l dx
ϕ
j ϕ k dx
ϕ
j ϕ k dx
1 0
$ ϕ l dx
M % M % M %
ϕi
ϕ
k
ϕ
l
ϕ
ϕi ϕ j ϕ k ϕ l dx q j qk ql
j
1 0
j=1 k=1 l=1
q j qk ql
$ ϕ j ϕ k dxdx ϕ l dx
0
ϕi ϕ j dx q j +
j=1 k=1 l=1
x
I1
$ ϕ j ϕ k dxdx ϕ l dx
0
M % M % M %
x
I1
dx
+
M % M % M % j=1 k=1 l=1
q j qk ql
1 0
ϕi
ϕ
k
ϕ
l ϕ j dx
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Fig. 6. Frequency-response curves of the functionally graded micro-cantilever for different values of the length scale parameter (ls = 0.00, ls = 0.35, ls = 0.45, and ls = 0.55): (a, b) the first two generalised coordinates of the transverse motion, respectively. n = 2, b0 = 0.006, and ζ = 0.008.
+ a55 ls
M %
2
+ a55 ls
j=1
1 0
ϕi
ϕ
3b55 ls − 2η −
−
−
d55 ls
2
η2
d55 ls
2
η2
d55 ls
+c
η2
M % j=1
2
2
M % M % j=1 k=1
1 0
j=1 k=1 l=1
1 0
j
ϕ
k ϕ l dx
q j qk ql
j=1 k=1 l=1
M % M % M %
2
1 M M M 1 %%% dx q j + ϕi ϕ j 4 0
1 M M M 1 %%% ϕi ϕ j ϕ k ϕ l dx q j qk ql + ϕi ϕ j ϕ k ϕ l dx q j qk ql 4 0 j=1 k=1 l=1
ϕi
ϕ
j ϕ k dx
1 M M M 1 %%% ϕi ϕ j 4 0 j=1 k=1 l=1
q j qk +
1 0
j=1 k=1
ϕi
ϕ
j ϕ k dx
q j qk
1 M M M 3 %%% ϕ k ϕ l dx q j qk ql + ϕi ϕ j 2 0
1 M M M 5 %%% ϕi ϕ k ϕ l ϕ j 2 0 j=1 k=1 l=1
M % M %
j=1 k=1 l=1
dx q j qk ql +
1 M M M 3 %%% ϕi ϕ j ϕ k ϕ l dx q j qk ql 4 0
M % M % M % j=1 k=1 l=1
0
1
ϕ k ϕ l dx q j qk ql
ϕi
ϕ
j
ϕ
k ϕ l dx
q j qk ql
j=1 k=1 l=1
1 0
1 1 ϕi ϕ j dx q˙ j + ϕi cb0 cos ( t ) dx − ϕi I1 b0 2 sin ( t ) dx = 0. 0
(24)
0
As seen in Eq. (24), there are inertial nonlinearities present as well as geometric nonlinearities. The solution technique used for Eq. (24) is a coupled pseudo-arclength and the backward differentiation formula (BDF), by choosing M = 6
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(a) ls=0.00 ls=0.35
0.3
ls=0.45
max[q1]
ls=0.55
0.2
0.1
0 (b)
0
0.002
0.004
b0
0.006
0.008
0.004 ls=0.00 ls=0.35 ls=0.45 ls=0.55
max[q2]
0.003
0.002
0.001
0 0
0.002
0.004
b0
0.006
0.008
Fig. 7. Force-response curves of functionally graded micro-cantilever for different values of the length scale parameter (ls = 0.00, ls = 0.35, ls = 0.45, and ls = 0.55): (a, b) the first two generalised coordinates of the transverse motion, respectively. n = 2, Ω/ω 1 = 0.98, and ζ = 0.008.
which ensures reliable and converged results for the large-amplitude nonlinear oscillations of the functionally graded microcantilever; the linear natural frequencies are obtained using an eigenfrequency method (Ghayesh & Amabili, 2013; Kazemirad et al., 2013). 4. Frequency-responses and force-responses Based on the solution procedure described in Section 3, the frequency-responses and force-responses in the resonant region of the functionally graded micro-cantilever are constructed in this section. The microsystem considered is of h = 10 μm, L/h = 100, and b/h = 4; the two constituents are aluminium with EAl = 69 GPa, ρAl = 2700 kg/m3 , and υAl = 0.33 and ceramic (SiC) with ESiC = 427 GPa, ρSiC = 3100 kg/m3 , and υSiC = 0.17. The frequency-response diagrams of the functionally graded micro-cantilever are depicted in Fig. 2, displaying a weakly softening-type behaviour; the microsystem parameters are set to n = 2, ls = 0.45, b0 = 0.006, and ζ = 0.008 (where ζ is the modal damping ratio). This set of parameters results in the fundamental natural frequency of ω1 = 7.1621 for the oscillation behaviour of the functionally graded micro-cantilever. These are two saddle-node bifurcations present at A and B (Ω/ω 1 = 0.9809 and 0.9764, respectively). The weak softening-type nonlinearity in the oscillation behaviour of the functionally graded micro-cantilever is the balance between different sources of nonlinearity involving inertial and geometric. Experimentally, the saddle-node bifurcations are responsible for jumps in the oscillation amplitude. The force-response diagrams of the functionally graded micro-cantilever are shown in Fig. 3 for n = 2, ls = 0.45, Ω/ω1 = 0.98, and ζ = 0.008. There are two saddle-node bifurcation points present at the amplitude of the base excitation of b0 = 0.0064 and 0.0054 (at points A and B, respectively); at point A the oscillation amplitude of the functionally graded micro-cantilever jumps to a larger value; however, at point B a reduction occurs in the motion amplitude. 5. Effect of material gradient index, length-scale parameter, and amplitude of base excitation This section analyses the effect of the parameters of the functionally graded micro-cantilever, such as the length-scale parameter, material gradient index, and the base-excitation, on the size-dependent oscillation behaviour of the functionally
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Fig. 8. Frequency-response curves of the functionally graded micro-cantilever: (a, b) the first two generalised coordinates of the transverse motion, respectively. n = 2, ls = 0.45, b0 = 0.006, and ζ = 0.008.
graded microsystem. The length, width, and thickness of the micro-cantilever as well as its constituents and their properties are kept the same as of Section 4. The frequency-response diagrams of the functionally graded micro-cantilever is highlighted in Fig. 4 for several material gradient indices; other microsystem parameters are set to ls = 0.45, b0 = 0.006, and ζ = 0.008. The largest peak-amplitude belongs to the ceramic micro-cantilever. For all the cases of the functionally graded micro-cantilever the nonlinearity is of a weak softening-type with two saddle-node bifurcations. The influence of the material gradient index on the force-response curves of the functionally graded micro-cantilever is illustrated in Fig. 5; ls = 0.45, Ω/ω1 = 0.98, and ζ = 0.008. As seen, the smallest base excitation amplitude corresponding to the second saddle-node bifurcation belongs to the metal case, the largest b0 for the first saddle-node belongs to the case with n = 20, and the overall motion amplitude of the case with n = 2 is the largest. Fig. 6 shows that how the length-scale parameter affects the vibrations of the functionally graded micro-cantilever; n = 2, b0 = 0.006, and ζ = 0.008. It can be seen that accounting for the effect of the small size of the functionally graded microcantilever (shown by the length-scale parameter) reduces the overall oscillation amplitude slightly. The fundamental natural frequency is larger for larger values of ls . For all the functionally graded cases, the type of nonlinearity is softening and the largest peak-amplitude belongs to the case with ls = 0, as the representative of the classical continuum mechanics. It is worth noting that the softening nonlinearity becomes stronger with increasing the length-scale parameter. Fig. 7 plots the force-response diagrams of the functionally graded micro-cantilever for various length-scale parameters; n = 2, Ω/ω1 = 0.98, and ζ = 0.008. It can be seen that for the functionally graded micro-cantilever modelled on the basis of classical continuum mechanics, there is no jump occurring, while, for the all other cases there are two jump-points present. It is seen that, for the range of b0 studied, the overall amplitude of the case with ls = 0.35 is the largest. For the microsystem with n = 2, ls = 0.45, and ζ = 0.008, as base-excitation amplitude is increased, as shown in Fig. 8, the oscillation amplitude of the functionally graded micro-cantilever becomes larger, which is characterised by the occurrence of the peak amplitudes at smaller base excitation frequencies.
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Fig. 9. Comparison of frequency-responses of the functionally graded micro-cantilever (current study) and silicon micro-cantilever (from Ref. (Farokhi et al., 2016)): (a, b) the first two generalised coordinates of the transverse motion, respectively. ls = 0.35, b0 = 0.0065, and ζ = 0.008.
6. Validation of nonlinear equation of motion The equation of motion of the functionally graded micro-cantilever derived in this paper (Eq. (20)) is validated by comparing it to the equation of motion of a non-functionally-graded micro-cantilever of Ref. Farokhi, Ghayesh, and Hussain (2016). To this end, when Eq. (14) and (16) are considered, one obtains
A11 = EA,
(25a)
B11 = 0,
(25b)
D11 = EI,
(25c)
A55 = μA.
(25d)
B55 = 0,
(25e)
D55 = μI,
(25f)
I1 = ρ A,
(25g)
which results in
! x x 2 2 ∂ 2v ∂ ∂v ∂ 3 v ∂v ∂ v ρA 2 + ρA + dx dx ∂x ∂ x∂ t ∂ x ∂ x∂ t 2 ∂x ∂t L 0 2 2 2 ∂ ∂ 3 v ∂ 3 v ∂v ∂ v ∂v + EI + + ∂ x ∂ x3 ∂ x3 ∂ x ∂ x2 ∂ x
H. Farokhi et al. / International Journal of Engineering Science 115 (2017) 117–130
129
2 2 ∂ 3v ∂ 1 ∂ 3 v ∂v 1 ∂ 2v ∂v + μA l + + ∂x 4 ∂ x3 ∂ x 4 ∂ x2 ∂x ∂ x3 2 2 1 ∂ 5 v ∂v ∂ 4 v ∂ 2 v ∂v 3 ∂ 3 v ∂v 2 ∂ − μI l + + ∂ x 4 ∂ x5 ∂ x ∂ x4 ∂ x2 ∂ x 4 ∂ x3 ∂ x 2
+c
∂v + cb0 ω cos (ω t ) − I1 b0 ω2 sin (ω t ) = 0, ∂t
(26)
which is exactly the same as Eq. (10) of Ref. Farokhi et al. (2016). The computer codes have also been validated by obtaining the results of Ref. Farokhi et al. (2016) while our codes are simplified; the comparison shown in Fig. 9 displays excellent agreement. 7. Conclusions This paper was the first which modelled and simulated the large-amplitude nonlinear oscillations of functionally graded micro-cantilevers incorporating curvature-related nonlinearities and size-dependence via MCST. Based on the Mori–Tanaka homogenisation technique, using the constitutive relations, the potential energy term of the functionally graded microcantilever was dynamically balanced with the kinetic energy term and the work of the base excitation, resulting in a nonlinear integro-partial-differential equation of motion of the microsystem when inextensibility was considered. A highdimensional truncated model was obtained using the Galerkin method and the resultant model of the functionally graded micro-cantilever was solved by means of the backward differentiation formula (BDF) together with a continuation technique. The numerical modelling and simulations showed that: (i) when both the curvature and inertial nonlinearities are considered, the nonlinear dynamics of the functionally graded micro-cantilever shows a weakly softening behaviour; (ii) for most of the cases studied, the frequency-response curves of the functionally graded micro-cantilever shows two saddlenode bifurcations; (iii) among the cases studied, the purely ceramic micro-cantilever displays the largest peak-amplitude in the frequency response curves; (iv) in the force-response diagrams, the smallest base excitation amplitude corresponding to the second saddle-node bifurcation belongs to the metal case and the largest base-excitation amplitude for the first saddlenode belongs to the case with the material gradient index of 20; (v) as the value for length-scale parameter is increased, the fundamental frequency of free oscillations increases, and the softening nonlinearity becomes stronger. 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