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Coupled longitudinal-transverse behaviour of a geometrically imperfect microbeam
Composites: Part B 60 (2014) 371–377
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Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Coupled longitudinal-transverse behaviour of a geometrically imperfect microbeam Mergen H. Ghayesh ⇑, Marco Amabili Department of Mechanical Engineering, McGill University, Montreal, Quebec H3A 0C3, Canada
a r t i c l e
i n f o
Article history: Received 27 February 2013 Received in revised form 10 December 2013 Accepted 22 December 2013 Available online 2 January 2014 Keywords: B. Microstructures C. Micro-mechanics C. Numerical analysis
a b s t r a c t Based on the modified couple stress theory, the coupled longitudinal-transverse nonlinear behaviour of an imperfect microbeam is investigated numerically. The equations governing the longitudinal and transverse motions are obtained using Hamilton’s principle for the system with an initial geometric imperfection. The Galerkin scheme is employed to discretize the two partial differential equations of motion, yielding a set of second-order nonlinear ordinary differential equations with coupled terms. This set is cast into new set of first-order nonlinear ordinary differential equations and solved by means of the pseudo-arclength continuation technique. The nonlinear resonant response of the system along with bifurcations are presented via frequency–response curves. Moreover, the effect of different system parameter on the frequency–response curves is highlighted. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Investigations concerning the nonlinear behaviour of microstructures have been carried out for several years. This is because these systems arise in a large class of mechanical systems and modern machine components such as in micro-actuators, biosensors, micro-probes, atomic force microscopes, micro-switches, electrically excited micro-actuators, and vibration shock sensors. It is well known experimentally that the dynamical behaviour of microbeams is size-dependent. This motivated many investigators to develop new higher-order continuum theories, such as the strain gradient and modified couple stress, to predict the strange size-dependent behaviour of microbeams [1–5]. In early studies, the attention was focused on the linear aspects of the problem, aiming at determining natural frequencies and mode shapes. For example, Kong et al. [6] employed the modified couple stress theory in order to investigate the effect of the length scale parameter on the natural frequencies of a microbeam. The modified couple stress theory was also employed by Asghari and co-workers [7,8], who investigated the size-dependent dynamical behaviour of functionally graded microbeams. The same theory was employed by Ma et al. [9] for a Timoshenko microbeam. Wang et al. [10] employed the strain gradient elasticity theory so as to examine the free oscillations of a Timoshenko microbeam. An extensive number of efforts have been devoted by Akgöz and Civalek [11,12], who examined the buckling of an axially loaded microbeam employing both the strain gradient and modified ⇑ Corresponding author. E-mail address: [email protected] (M.H. Ghayesh). 1359-8368/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2013.12.030
couple stress theories. These studies were extended by Ke and Wang [13] for microbeams made of functionally graded materials. The strain gradient theory was employed by Ansari et al. [14] in order to study the free oscillations of a functionally graded Timoshenko microbeam. The effect of a moving microparticle on the oscillations of an embedded microbeam was investigated by S ß imsßek [15]. The size-dependent buckling analysis of functionally graded microbeams was carried out by Nateghi et al. [16]. When the amplitude of vibrations is large, the validity of linear theory diminishes; more sophisticated models including nonlinearity started appearing recently, notably by Asghari et al. [17], Moeenfard et al. [18], Ke et al. [19], Ramezani [20], and Ghayesh et al. [21–23], for example. In all of the aforementioned valuable studies, the longitudinal displacement or inertia is neglected in the solutions; the microbeam is assumed to be perfectly straight as well; however in reality, due to manufacturing imperfections, it is not the case. The main reason is that in the presence of the longitudinal displacement or the geometric imperfection, the quadratic nonlinear terms appear in the equations of motion which make them difficult to solve. The present study investigates the nonlinear coupled longitudinal-transverse behaviour of a microbeam taking into account a geometric imperfection (i.e. slightly curved microbeam, from physical perspective). A high-dimensional system is considered by employing a higher-mode Galerkin discretization. A linear analysis is performed to examine the dependence of the natural frequency of the system on different system parameters. The nonlinear resonant response (frequency–response curves) of the system is also examined by means of the pseudo-arclength continuation technique.
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2. Equations of motion and methods of solution The schematic representation of the system is depicted in Fig. 1, showing a geometrically imperfect microbeam of length L, flexural stiffness EI, and axial stiffness EA. The microbeam is hinged at both ends and subjected to an external distributed harmonic excitation force per unit length F(x)cos(xt). x and z represent the axial and transverse coordinates, respectively, and u(x, t) and w(x, t) denote the displacement in the longitudinal and transverse directions, respectively. The equations of motion are derived under the following assumptions: (1) the Euler–Bernoulli beam theory is used, neglecting the effect of shear deformation and rotary inertia; (2) the microbeam has an initial curvature denoted by w0(x); (3) the cross-section of the beam is constant over the entire length; (4) the nonlinearity is geometric and due to the stretching effect of the mid-plane of the microbeam. According to the modified couple stress theory, the strain energy of the system can be expressed as [24]
1 U¼ 2
Z
ðr : e þ m : vÞdv ;
ð1Þ
vxy ¼ vyx ¼
r ¼ ktrðeÞI þ 2le; 2
m ¼ 2l
lv;
ð2Þ ð3Þ
where k and l denote the Lamé constants, and l represents the material length scale parameter. The symmetric curvature tensor is related to the rotation vector h through the following relation
1 v ¼ ðrh þ ðrhÞT Þ; 2 1 h ¼ curlðuÞ; 2
ð5Þ
in which u is the displacement vector. The nonzero component of the strain tensor e for a geometrically imperfect and nonlinear Euler–Bernoulli beam is obtained as [26]
2 @u 1 @w @w dw0 @2w ¼ þ þ z 2 ; @x 2 @x @x dx @x
2
ð7Þ
@2w : @x2
ð8Þ
Inserting Eqs. (6)–(8) as well as Eq. (2) into Eq. (1) yields the strain energy of a microbeam as follows:
!2 @2w 1 ðEI þ lAl Þ dx þ EA @x2 2 0 #2 2 Z L" @u 1 @w @w dw0 þ þ dx: @x 2 @x @x dx 0
1 U¼ 2
Z
L
2
ð9Þ
The kinetic energy for a microbeam with coupled longitudinaltransverse dynamics can be formulated as
1 T ¼ qA 2
Z
L
0
" 2 # 2 @u @w dx: þ @t @t
ð10Þ
The variation of the work done by the distributed harmonic excitation force and the force due to viscous damping of the surrounding medium, can be written as
dW F ¼
Z
L
FðxÞ cosðxtÞdwdx;
ð11Þ
0
dW D ¼ cd
Z 0
L
@u @w du þ dw dx; @t @t
ð12Þ
where cd stands for the viscous damping coefficient. Eqs. (9)–(12) are substituted into Hamilton’s principle such that
Z
t2
d ð4Þ
1 @2w ; 2 @x2
mxy ¼ myx ¼ ll
V
in which r and e represent the stress and strain tensors, respectively, and m and v denote the deviatoric part of the couple stress tensor and the symmetric curvature tensor, respectively. The stress tensor and the deviatoric part of the couple stress tensor, for an isotropic linear elastic material, can be formulated as [25]
exx
Following the procedure defined by Kong et al. [6], one can obtain the non-zero components of the symmetric curvature tensor and the couple stress tensor for an Euler–Bernoulli beam as the following, respectively:
½TðtÞ UðtÞdt þ
t1
Z
t2
ðdW F þ dW D Þdt ¼ 0;
ð13Þ
t1
yielding the following nonlinear partial differential equations for the longitudinal and transverse motions of a geometrically imperfect microbeam, respectively:
" 2 # @2u @ @u @w dw0 1 @w @u þ cd qA 2 EA þ þ ¼ 0; @x @x @x 2 @x @t dx @t
ð14Þ
4 @2w 2 @ w þ ðEI þ lAl Þ 4 EA 2 @x @t " 2 !# @ @w dw0 @u @w dw0 1 @w FðxÞ þ þ þ @x @x @x @x dx 2 @x dx
qA ð6Þ
where exx is the axial strain at a generic point of the beam which is located at a distance z from the mid-plane surface, and w0(x) is the arbitrary initial curvature.
cosðxtÞ þ cd
@w @t
¼ 0;
ð15Þ
with the following immovable boundary conditions for a hingedhinged microbeam
ujx¼0 ¼ ujx¼L ¼ 0; wjx¼0 ¼ wjx¼L ¼ 0;
Fig. 1. The schematic representation of a geometrically imperfect micro-scale beam subjected to a transverse distributed harmonic excitation force.
ð16Þ @ 2 w @x2
x¼0
@ 2 w ¼ 2 @x
¼ 0:
ð17Þ
x¼L
In order to transform the equations of motion into the dimensionless form, one can introduce the following dimensionless parameters:
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x x ¼ ; L
g¼ cd
u ¼
lAl2
t ¼t
;
EI
cd L4 ¼ EI
u ; L
w ¼
w ; L
sffiffiffiffiffiffiffiffiffiffiffi EI
; qAL4
sffiffiffiffiffiffiffiffiffiffiffi EI : qAL4
w0 w0 ¼ ; L sffiffiffiffiffiffiffiffiffiffiffi qAL4 X¼x ; EI
sffiffiffiffiffiffiffiffiffiffi EAL2 b¼ ; EI F ¼
FL3 ; EI ð18Þ
Substituting these parameters into Eqs. (14) and (15), and disregarding the asterisk notation for brevity, results in the following dimensionless nonlinear partial differential equations of motion for the longitudinal and transverse displacements, respectively:
"
2 # @2u @u @w dw0 1 @w @u 2 @ b þ þ ¼ 0; þ cd @x @x @x dx 2 @x @t @t 2
@2w @4w @w dw0 2 @ þ ð1 þ g Þ b þ @x4 @x @x dx @t 2 2 !# @u @w dw0 1 @w @w FðxÞ cosðXtÞ þ cd þ þ ¼ 0: @x @x dx 2 @x @t
ð19Þ
1
" Z N X €j b2 /i /j dx p
0
j¼1
0
ð20Þ
0
N Z X
1
1
/i /j dx p_ j ¼ 0;
þ cd
ð21Þ
r¼1
/r ðxÞpr ðtÞ;
ð22Þ
r¼1
in which /r(x) denotes the rth eigenfunction for the transverse motion of a hinged-hinged linear microbeam while qr(t) and pr(t) represent the rth generalized coordinate for the transverse and longitudinal displacements, respectively.
i ¼ 1; 2; . . . ; N;
M Z X €j þ ð1 þ gÞ /i /j dx q
0
j¼1
# 0 00 /i /j /k dx qj qk
j¼1
M Z X
1
ð23Þ
0
0000 /i /j dx qj
N Z X /i /0j /00k dx qj pk þ k¼1
1
0
j¼1 k¼1
0
j¼1 k¼1
1
" M X N Z X /i /j dx q_ j b2
0
j¼1
M X N Z 1 X
N Z X þ
A0 /01 /i /00j dx qj
0
j¼1
M Z X
1
1
0
j¼1
j¼1 k¼1
þcd
/i /00j dx pj
M Z X A0 /001 /i /0j dx qj þ
þ
j¼1
1
0
j¼1
M Z 1 X
þ
M X wðx; tÞ ¼ /r ðxÞqr ðtÞ;
N X
N Z X
M X M Z X þ
In order to be able to solve the equations of motion numerically, the Galerkin scheme is employed to discretize them into a set of ordinary differential equations. To accomplish that, the eigenfunctions for the transverse motion of a linear hinged-hinged beam are selected as the basis functions for both the longitudinal and transverse motions. The transverse and longitudinal displacements are defined as the following approximate series expansions:
uðx; tÞ ¼
Substituting Eqs. (21) and (22) into Eqs. (19) and (20) and choosing the transverse external force as F(x) = f1/1(x) and the initial curvature as w0(x) = A0sin(px), the Galerkin method is applied by multiplying the resultant equations by the corresponding eigenfunction and integrating them with respect to x from 0 to 1; this procedure yields the following set of second-order nonlinear ordinary differential equations for the longitudinal and transverse motions, respectively:
0
1
/i /00j /0k dx qj pk
A0 /001 /i /0j dx pj
M X M X M Z 1 3X A0 /01 /i /00j dx pj þ /i /0j /0k /00l dx qj qk ql 2 j¼1 k¼1 l¼1 0 0 k¼1 Z M X M M Z 1 1 X 3X 2 þ ð A0 /001 /i /0j /0k dxÞqj qk þ ðA0 /01 Þ /i /00j dx qj 2 j¼1 k¼1 0 0 j¼1 # Z M M M 1 XX X Z 1 2 0 00 0 0 00 0 þ3 A0 /1 /i /j /k dx qj qk þ 2 ð A0 /1 /1 /i /j dxÞqj
Z
1
j¼1 k¼1
0
j¼1
0
1
f1 /1 /i dx cosðXtÞ ¼ 0;
i ¼ 1; 2; . . . ; M;
ð24Þ
0
20
Natural Frequency
18
16
η=1.0 14
η=0.5 12
η=0 10
8
0
0.0005
0.001
0.0015
0.002
A0 Fig. 2. The variation of the first linear dimensionless natural frequency of the transverse motion of the system with increasing A0 for different values of g.
in which the dot notation denotes the differentiation with respect to dimensionless time while the prime notation represents the differentiation with respect to the dimensionless axial coordinate. Eqs. (23) and (24) (which form a set of M + N nonlinear ordinary differential equations) are transformed into a set of first-order nonlinear ordinary differential equations via a standard change of variables by means of xi ¼ p_ i ði ¼ 1; 2; . . . ; NÞ and yi ¼ q_ i ði ¼ 1; 2; . . . ; MÞ; this operation yields a set of 2(M + N) first-order nonlinear ordinary differential equations with coupled terms. In the present study, a 15-degree-of-freedom system is considered (M = 7 and N = 8) which results in 30 first-order nonlinear ordinary differential equations with coupled nonlinear terms. Both linear and nonlinear analyses are performed upon this set of equations. The linear analysis is employed to obtain the linear natural frequencies of the system and examine the effect of geometric imperfection and small scale parameter on the linear natural frequencies of the system. The nonlinear analysis is performed by means of the pseudo-arclength continuation technique [27] which is capable of continuing both stable and unstable solution branches as well as determining different types of bifurcations.
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24
0.008
(a)
A
22
0.006
A0=0.002 18
Max (q1)
Natural Frequency
20
16
0.004
A0=0.001
14
0.002
12
B
A0=0 10 8
0 0.9 0
0.5
1
η
1.5
0.95
1
1.05
1.1
1.15
1.2
Ω/ω1
2 0
Fig. 3. The variation of the first linear dimensionless natural frequency of the transverse motion of the system with increasing g for different values of A0.
(b)
B
-0.002
3. Linear natural frequencies of the system
4. Frequency–response curves of the system The frequency–response curves are presented in this section so as to examine the nonlinear resonant response of the system. The frequency of the external excitation force is chosen near the first linear natural frequency of the system, i.e. x1, in order to investigate the resonant response of the system. The microbeam is made of epoxy material, the mechanical properties of which can be found in [5,9,28]. Fig. 4 shows the frequency–response curves for (a and b) the maximum and minimum of the first generalized coordinates of the transverse motion, respectively. The numerical calculations are carried out using the following dimensionless parameters: b = 415.692, g = 4.347, cd = 0.04, A0 = 0.002, and f1 = 0.010. As shown in Fig. 4(a), theoretically, the maximum amplitude of the first generalized coordinate of the transverse motion of the system increases with increasing external excitation frequency; the first limit point bifurcation occurs at point A (X = 1.1822x1) leading the system to have an unstable solution. As the excitation frequency is decreased, the unstable solution between points A and B is followed where the amplitude of the unstable response decreases until point B (X = 1.01020x1) is hit; at this point, the system regains the stability via a second limit point bifurcation. Fig. 4(b) shows the frequency–response curve of the system for the minimum amplitude of the first generalized coordinate. It
-0.004
Min (q1)
In this section the effect of the amplitude of the initial imperfection (A0) and small scale parameter (g) on the first linear natural frequency of the transverse motion is examined. Fig. 2 shows the variation of the first linear dimensionless natural frequency of the transverse motion of the system versus A0, for different values of g. As shown in the figure, the first natural frequency of the transverse motion of the system increases with increasing A0 (for the system with A0 = 0 (i.e. perfect system) the natural frequency is the smallest). The variation of the first linear natural frequency of the system with g is depicted in Fig. 3, for different values of A0. It can be concluded that as the value of g is increased, the first linear natural frequency of the transverse motion of the system increases accordingly.
-0.006
-0.008
A
-0.01 0.9
0.95
1
1.05
1.1
1.15
1.2
Ω/ω1 Fig. 4. Frequency–response curve of the system: (a and b) the maximum and minimum amplitudes of the first generalized coordinate of the transverse motion, respectively; b = 415.692, g = 4.347, A0 = 0.002, cd = 0.04, and f1 = 0.010.
can be observed that, as a result of the geometric imperfection, the maximum and minimum amplitudes of the first generalized coordinate are not the same; in fact, the maximum amplitude of q1 is 0.00828 while the minimum amplitude of q1 is 0.00985. From Eq. (20), it can be concluded that both quadratic and cubic nonlinearities are present in the equations of motion. Since w0 is associated with quadratic nonlinear terms, it tends to cause softening behaviour in the system. However, since the value of A0 is small, the effect of the cubic nonlinear terms are dominant and the system displays a hardening behaviour. Increasing the value of A0 to 0.005, from 0.002 in the system of Fig. 4, a new frequency–response curve is generated and plotted in Fig. 5. As shown in this figure, due to increased value of A0, the effect of quadratic nonlinear terms increase and the system displays both softening and hardening behaviours. Specifically, Fig. 5(a) shows that, theoretically, the maximum amplitude of q1 is increased with increasing excitation frequency until point A (X = 0.993504x1) is reached, where the stability is lost by means of a limit point bifurcation; this early limit point bifurcation is due to increased amplitude of imperfection which causes an initial softening behaviour. As the excitation frequency is decreased, the
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0.008
0.002
(a)
0.003
0.008
C
0.004 0.005 0.006
0.006
B
0.004
D
q1
Max (q1)
0.006
0.004
0.002
0.002
A
0 0.96
0.98
1
1.02
0
1.04
0.95
1
1.05
1.1
1.2
Fig. 6. The effect of A0 on the frequency–response curve of the system for the first generalized coordinate of the transverse motion. The values of A0 are denoted on the curves; b = 415.692, g = 4.347, cd = 0.04, and f1 = 0.010.
0
(b) A
-0.002
Classical
-0.004
Min (q1)
1.15
Ω/ω1
Ω/ω1
B -0.006
Modified couple
0.003
D
-0.008
q1
0.002
-0.01
C 0.96
0.98
1
1.02
1.04
0.001
Ω/ω1 Fig. 5. Frequency–response curve of the system: (a and b) the maximum and minimum amplitudes of the first generalized coordinate of the transverse motion, respectively; b = 415.692, g = 4.347, A0 = 0.005, cd = 0.04, and f1 = 0.010.
amplitude of the unstable response (the unstable solution between points A and B) increases until reaching the next limit point bifurcation at point B (X = 0.987855x1). By further increasing the excitation frequency, the stable solution branch between points B and C is followed, where the system displays a hardening behaviour. The third limit point bifurcation occurs at point C (X = 1.028668x1) where the motion of the system becomes unstable once again. As the excitation frequency is decreased, the unstable solution branch lasts until point D (X = 0.990201x1) is hit corresponding to the fourth limit point bifurcation. Beyond that point, the system displays a stable response with decreasing amplitude. From Fig. 5(a and b), it can be observed that the maximum and minimum amplitudes of the first generalized coordinate are 0.007202 and 0.01066, respectively. Comparing Figs. 4 and 5, it can be concluded that due to increased amplitude of initial imperfection, the difference between the maximum and minimum amplitudes of the q1 motion increases, the number of limit point bifurcations increases to four, and the system displays both hardening and softening behaviours. Fig. 6 shows the frequency–response curves of the system for different values of the initial curvature amplitude A0. The other
0 0.9
0.95
1
1.05
1.1
Ω/ω1 Fig. 7. Comparison between the frequency–response curves of the system for the first generalized coordinate of the transverse motion, obtained via modified couple and classical theories; b = 346.41, cd = 0.03, A0 = 0.001, and f1 = 0.0012; g = 0.4830 for the modified couple stress theory and g = 0 for the classical theory.
dimensionless parameters are selected as: b = 415.692, g = 4.347, cd = 0.04, and f1 = 0.10. This figure shows that, as the amplitude of the initial imperfection increases, the hardening behaviour of the system decreases; moreover, for higher values of A0 both hardening and softening behaviours are observed. As shown in the figure, for A0 = 0.006, the maximum amplitude of the system occurs at a frequency lower that the first linear natural frequency of the transverse motion. A comparison between the classical and modified couple stress theories is illustrated in Fig. 7. In particular, the frequency– response curve of the system is plotted in the absence (classical) and presence (modified couple) of the length scale parameter. As shown in the figure, the hardening behaviour of the system decreases when the modified couple stress theory is employed. In particular, the modified couple stress theory predicts lower
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From Eqs. (19) and (20), it can be inferred that the contribution of both quadratic and cubic nonlinear terms increases with increasing b, which is verified in Fig. 9. As shown in this figure, due to increased value of b, both the initial softening and the latter hardening behaviours of the system increase; in other words, the curves bend more to the left at first and then bend more to the right.
0.010 0.008
0.006
q1
0.006
0.004
5. Conclusions
0.004
0.002
0 0.96
0.98
1
1.02
1.04
Ω/ω1 Fig. 8. The effect of the forcing amplitude, f1, on the frequency–response curve of the system for the first generalized coordinate of the transverse motion. The values of f1 are denoted on the curves; b = 415.692, g = 4.347, A0 = 0.005, and cd = 0.04.
0.009 300 350
400
0.006
q1
450
0.003
0
0.96
1
1.04
1.08
Ω/ω1 Fig. 9. The effect of b on the frequency–response curves of the system for the first generalized coordinate of the transverse motion. The values of b are denoted on the curves; g = 4.347, A0 = 0.005, and cd = 0.04.and f1 = 0.0011.
nonlinear resonance frequency and hence lower amplitude of oscillations. This is mainly due to the reason that the presence of g in Eq. (20) increases the flexural stiffness of the system. The effect of the forcing amplitude on the frequency–response curves of the system is depicted in Fig. 8. Four different forcing amplitudes are considered, the values of which are denoted on the curve. As shown in Fig. 8, for lower forcing amplitudes (e.g. f1 = 0.004), the system displays only a softening behaviour; however, for higher forcing amplitudes, the initial softening behaviour is accompanied by a latter hardening behaviour which becomes more visible as the forcing amplitude is increased. It is also observed that due to increased forcing amplitude, the whole response region becomes wider and the maximum amplitude of the oscillations increases. Finally, the frequency–response curves of the system for different values of the dimensionless parameter b are depicted in Fig. 9.
The nonlinear resonant behaviour of a geometrically imperfect microbeam has been examined numerically, taking into account the longitudinal and transverse displacements. The nonlinear partial differential equations governing the motion of the microbeam in the longitudinal and transverse directions were derived employing Hamilton’s principle on the basis of the modified couple stress theory. The equations of motion were discretized into a set of second-order nonlinear ordinary differential equations by means of the Galerkin technique. A new set of first-order nonlinear ordinary differential equations was obtained by introducing a standard change of variables to the previous set. A linear analysis was performed upon this set of equations so as to investigate the effect the amplitude of the initial curvature and small scale parameter on the linear natural frequencies of the system. The pseudo-arclength continuation technique was employed in order to examine the nonlinear resonant response of the system and to construct the frequency–response curves. The results of the linear analysis showed that the first natural frequency of the transverse motion of the system increases with increasing both the amplitude of the initial imperfection and g (the size-dependent factor). Examining the nonlinear resonant response of the system revealed that the hardening behaviour of the system decreases with increasing the amplitude of the initial imperfection and the system displays both hardening and softening behaviours for higher values of the initial imperfection; examining the effect of the amplitude of the initial imperfection on the frequency–response curves revealed that the nonlinear behaviour of the system can be qualitatively changed due to the presence of a small geometric imperfection. It was shown that the modified couple stress theory predicts lower nonlinear resonances and hence weaker nonlinear behaviours, compared to the classical theory. Examining the effect of the forcing amplitude on the frequency–response curves of the system showed that the initial softening behaviour is continued by a latter hardening behaviour for higher values of the forcing amplitude.
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M.H. Ghayesh, M. Amabili / Composites: Part B 60 (2014) 371–377 [10] Wang B, Zhao J, Zhou S. A micro-scale Timoshenko beam model based on strain gradient elasticity theory. Eur J Mech A Solids 2010;29(4):591–9. [11] Akgöz B, Civalek Ö. Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory. Arch Appl Mech 2012;82(3):423–43. [12] Akgöz B, Civalek Ö. Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. Int J Eng Sci 2011;49(11):1268–80. [13] Ke L-L, Wang Y-S. Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Compos Struct 2011;93(2):342–50. [14] Ansari R, Gholami R, Sahmani S. Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Compos Struct 2011;94(1):221–8. [15] Sßimsßek M. Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory. Int J Eng Sci 2010;48(12):1721–32. [16] Nateghi A, Salamat-talab M, Rezapour J, Daneshian B. Size dependent buckling analysis of functionally graded micro-beams based on modified couple stress theory. Appl Math Model 2012;36(10):4971–87. [17] Asghari M, Kahrobaiyan M, Nikfar M, Ahmadian M. A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory. Acta Mech 2012;223(6):1233–49. [18] Moeenfard H, Mojahedi M, Ahmadian M. A homotopy perturbation analysis of nonlinear free vibration of Timoshenko microbeams. J Mech Sci Technol 2011;25(3):557–65.
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[19] Ke L-L, Wang Y-S, Yang J, Kitipornchai S. Nonlinear free vibration of sizedependent functionally graded microbeams. Int J Eng Sci 2012;50(1):256–67. [20] Ramezani S. A micro-scale geometrically non-linear Timoshenko beam model based on strain gradient elasticity theory. Int J Non-Linear Mech 2012;47(8):863–73. [21] Ghayesh MH, Amabili M, Farokhi H. Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. Int J Eng Sci 2013;63:52–60. [22] Ghayesh MH, Farokhi H, Amabili M. Nonlinear dynamics of a micro-scale beam based on the modified couple stress theory. Compos B Eng. [23] Ghayesh MH, Farokhi H, Amabili M. Nonlinear behaviour of electrically actuated MEMS resonators. Int J Eng Sci 2013;71:137–55. [24] Yang F, Chong ACM, Lam DCC, Tong P. Couple stress based strain gradient theory for elasticity. Int J Solids Struct 2002;39(10):2731–43. [25] Reddy JN, Kim J. A nonlinear modified couple stress-based third-order theory of functionally graded plates. Compos Struct 2012;94(3):1128–43. [26] Öz HR, Pakdemirli M. Two-to-one internal resonances in a shallow curved beam resting on an elastic foundation. Acta Mech 2006;185(3–4):245–60. [27] Doedel EJ, Champneys AR, Fairgrieve TF, Kuznetsov YA, Sandstede B, Wang X. AUTO 97: Continuation and bifurcation software for ordinary differential equations (with HomCont). Montreal, Canada: Concordia University; 1998. [28] Maneschy C, Miyano Y, Shimbo M, Woo T. Residual-stress analysis of an epoxy plate subjected to rapid cooling on both surfaces. Exp Mech 1986;26(4):306–12.
Contents lists available at ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Coupled longitudinal-transverse behaviour of a geometrically imperfect microbeam Mergen H. Ghayesh ⇑, Marco Amabili Department of Mechanical Engineering, McGill University, Montreal, Quebec H3A 0C3, Canada
a r t i c l e
i n f o
Article history: Received 27 February 2013 Received in revised form 10 December 2013 Accepted 22 December 2013 Available online 2 January 2014 Keywords: B. Microstructures C. Micro-mechanics C. Numerical analysis
a b s t r a c t Based on the modified couple stress theory, the coupled longitudinal-transverse nonlinear behaviour of an imperfect microbeam is investigated numerically. The equations governing the longitudinal and transverse motions are obtained using Hamilton’s principle for the system with an initial geometric imperfection. The Galerkin scheme is employed to discretize the two partial differential equations of motion, yielding a set of second-order nonlinear ordinary differential equations with coupled terms. This set is cast into new set of first-order nonlinear ordinary differential equations and solved by means of the pseudo-arclength continuation technique. The nonlinear resonant response of the system along with bifurcations are presented via frequency–response curves. Moreover, the effect of different system parameter on the frequency–response curves is highlighted. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Investigations concerning the nonlinear behaviour of microstructures have been carried out for several years. This is because these systems arise in a large class of mechanical systems and modern machine components such as in micro-actuators, biosensors, micro-probes, atomic force microscopes, micro-switches, electrically excited micro-actuators, and vibration shock sensors. It is well known experimentally that the dynamical behaviour of microbeams is size-dependent. This motivated many investigators to develop new higher-order continuum theories, such as the strain gradient and modified couple stress, to predict the strange size-dependent behaviour of microbeams [1–5]. In early studies, the attention was focused on the linear aspects of the problem, aiming at determining natural frequencies and mode shapes. For example, Kong et al. [6] employed the modified couple stress theory in order to investigate the effect of the length scale parameter on the natural frequencies of a microbeam. The modified couple stress theory was also employed by Asghari and co-workers [7,8], who investigated the size-dependent dynamical behaviour of functionally graded microbeams. The same theory was employed by Ma et al. [9] for a Timoshenko microbeam. Wang et al. [10] employed the strain gradient elasticity theory so as to examine the free oscillations of a Timoshenko microbeam. An extensive number of efforts have been devoted by Akgöz and Civalek [11,12], who examined the buckling of an axially loaded microbeam employing both the strain gradient and modified ⇑ Corresponding author. E-mail address: [email protected] (M.H. Ghayesh). 1359-8368/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2013.12.030
couple stress theories. These studies were extended by Ke and Wang [13] for microbeams made of functionally graded materials. The strain gradient theory was employed by Ansari et al. [14] in order to study the free oscillations of a functionally graded Timoshenko microbeam. The effect of a moving microparticle on the oscillations of an embedded microbeam was investigated by S ß imsßek [15]. The size-dependent buckling analysis of functionally graded microbeams was carried out by Nateghi et al. [16]. When the amplitude of vibrations is large, the validity of linear theory diminishes; more sophisticated models including nonlinearity started appearing recently, notably by Asghari et al. [17], Moeenfard et al. [18], Ke et al. [19], Ramezani [20], and Ghayesh et al. [21–23], for example. In all of the aforementioned valuable studies, the longitudinal displacement or inertia is neglected in the solutions; the microbeam is assumed to be perfectly straight as well; however in reality, due to manufacturing imperfections, it is not the case. The main reason is that in the presence of the longitudinal displacement or the geometric imperfection, the quadratic nonlinear terms appear in the equations of motion which make them difficult to solve. The present study investigates the nonlinear coupled longitudinal-transverse behaviour of a microbeam taking into account a geometric imperfection (i.e. slightly curved microbeam, from physical perspective). A high-dimensional system is considered by employing a higher-mode Galerkin discretization. A linear analysis is performed to examine the dependence of the natural frequency of the system on different system parameters. The nonlinear resonant response (frequency–response curves) of the system is also examined by means of the pseudo-arclength continuation technique.
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2. Equations of motion and methods of solution The schematic representation of the system is depicted in Fig. 1, showing a geometrically imperfect microbeam of length L, flexural stiffness EI, and axial stiffness EA. The microbeam is hinged at both ends and subjected to an external distributed harmonic excitation force per unit length F(x)cos(xt). x and z represent the axial and transverse coordinates, respectively, and u(x, t) and w(x, t) denote the displacement in the longitudinal and transverse directions, respectively. The equations of motion are derived under the following assumptions: (1) the Euler–Bernoulli beam theory is used, neglecting the effect of shear deformation and rotary inertia; (2) the microbeam has an initial curvature denoted by w0(x); (3) the cross-section of the beam is constant over the entire length; (4) the nonlinearity is geometric and due to the stretching effect of the mid-plane of the microbeam. According to the modified couple stress theory, the strain energy of the system can be expressed as [24]
1 U¼ 2
Z
ðr : e þ m : vÞdv ;
ð1Þ
vxy ¼ vyx ¼
r ¼ ktrðeÞI þ 2le; 2
m ¼ 2l
lv;
ð2Þ ð3Þ
where k and l denote the Lamé constants, and l represents the material length scale parameter. The symmetric curvature tensor is related to the rotation vector h through the following relation
1 v ¼ ðrh þ ðrhÞT Þ; 2 1 h ¼ curlðuÞ; 2
ð5Þ
in which u is the displacement vector. The nonzero component of the strain tensor e for a geometrically imperfect and nonlinear Euler–Bernoulli beam is obtained as [26]
2 @u 1 @w @w dw0 @2w ¼ þ þ z 2 ; @x 2 @x @x dx @x
2
ð7Þ
@2w : @x2
ð8Þ
Inserting Eqs. (6)–(8) as well as Eq. (2) into Eq. (1) yields the strain energy of a microbeam as follows:
!2 @2w 1 ðEI þ lAl Þ dx þ EA @x2 2 0 #2 2 Z L" @u 1 @w @w dw0 þ þ dx: @x 2 @x @x dx 0
1 U¼ 2
Z
L
2
ð9Þ
The kinetic energy for a microbeam with coupled longitudinaltransverse dynamics can be formulated as
1 T ¼ qA 2
Z
L
0
" 2 # 2 @u @w dx: þ @t @t
ð10Þ
The variation of the work done by the distributed harmonic excitation force and the force due to viscous damping of the surrounding medium, can be written as
dW F ¼
Z
L
FðxÞ cosðxtÞdwdx;
ð11Þ
0
dW D ¼ cd
Z 0
L
@u @w du þ dw dx; @t @t
ð12Þ
where cd stands for the viscous damping coefficient. Eqs. (9)–(12) are substituted into Hamilton’s principle such that
Z
t2
d ð4Þ
1 @2w ; 2 @x2
mxy ¼ myx ¼ ll
V
in which r and e represent the stress and strain tensors, respectively, and m and v denote the deviatoric part of the couple stress tensor and the symmetric curvature tensor, respectively. The stress tensor and the deviatoric part of the couple stress tensor, for an isotropic linear elastic material, can be formulated as [25]
exx
Following the procedure defined by Kong et al. [6], one can obtain the non-zero components of the symmetric curvature tensor and the couple stress tensor for an Euler–Bernoulli beam as the following, respectively:
½TðtÞ UðtÞdt þ
t1
Z
t2
ðdW F þ dW D Þdt ¼ 0;
ð13Þ
t1
yielding the following nonlinear partial differential equations for the longitudinal and transverse motions of a geometrically imperfect microbeam, respectively:
" 2 # @2u @ @u @w dw0 1 @w @u þ cd qA 2 EA þ þ ¼ 0; @x @x @x 2 @x @t dx @t
ð14Þ
4 @2w 2 @ w þ ðEI þ lAl Þ 4 EA 2 @x @t " 2 !# @ @w dw0 @u @w dw0 1 @w FðxÞ þ þ þ @x @x @x @x dx 2 @x dx
qA ð6Þ
where exx is the axial strain at a generic point of the beam which is located at a distance z from the mid-plane surface, and w0(x) is the arbitrary initial curvature.
cosðxtÞ þ cd
@w @t
¼ 0;
ð15Þ
with the following immovable boundary conditions for a hingedhinged microbeam
ujx¼0 ¼ ujx¼L ¼ 0; wjx¼0 ¼ wjx¼L ¼ 0;
Fig. 1. The schematic representation of a geometrically imperfect micro-scale beam subjected to a transverse distributed harmonic excitation force.
ð16Þ @ 2 w @x2
x¼0
@ 2 w ¼ 2 @x
¼ 0:
ð17Þ
x¼L
In order to transform the equations of motion into the dimensionless form, one can introduce the following dimensionless parameters:
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x x ¼ ; L
g¼ cd
u ¼
lAl2
t ¼t
;
EI
cd L4 ¼ EI
u ; L
w ¼
w ; L
sffiffiffiffiffiffiffiffiffiffiffi EI
; qAL4
sffiffiffiffiffiffiffiffiffiffiffi EI : qAL4
w0 w0 ¼ ; L sffiffiffiffiffiffiffiffiffiffiffi qAL4 X¼x ; EI
sffiffiffiffiffiffiffiffiffiffi EAL2 b¼ ; EI F ¼
FL3 ; EI ð18Þ
Substituting these parameters into Eqs. (14) and (15), and disregarding the asterisk notation for brevity, results in the following dimensionless nonlinear partial differential equations of motion for the longitudinal and transverse displacements, respectively:
"
2 # @2u @u @w dw0 1 @w @u 2 @ b þ þ ¼ 0; þ cd @x @x @x dx 2 @x @t @t 2
@2w @4w @w dw0 2 @ þ ð1 þ g Þ b þ @x4 @x @x dx @t 2 2 !# @u @w dw0 1 @w @w FðxÞ cosðXtÞ þ cd þ þ ¼ 0: @x @x dx 2 @x @t
ð19Þ
1
" Z N X €j b2 /i /j dx p
0
j¼1
0
ð20Þ
0
N Z X
1
1
/i /j dx p_ j ¼ 0;
þ cd
ð21Þ
r¼1
/r ðxÞpr ðtÞ;
ð22Þ
r¼1
in which /r(x) denotes the rth eigenfunction for the transverse motion of a hinged-hinged linear microbeam while qr(t) and pr(t) represent the rth generalized coordinate for the transverse and longitudinal displacements, respectively.
i ¼ 1; 2; . . . ; N;
M Z X €j þ ð1 þ gÞ /i /j dx q
0
j¼1
# 0 00 /i /j /k dx qj qk
j¼1
M Z X
1
ð23Þ
0
0000 /i /j dx qj
N Z X /i /0j /00k dx qj pk þ k¼1
1
0
j¼1 k¼1
0
j¼1 k¼1
1
" M X N Z X /i /j dx q_ j b2
0
j¼1
M X N Z 1 X
N Z X þ
A0 /01 /i /00j dx qj
0
j¼1
M Z X
1
1
0
j¼1
j¼1 k¼1
þcd
/i /00j dx pj
M Z X A0 /001 /i /0j dx qj þ
þ
j¼1
1
0
j¼1
M Z 1 X
þ
M X wðx; tÞ ¼ /r ðxÞqr ðtÞ;
N X
N Z X
M X M Z X þ
In order to be able to solve the equations of motion numerically, the Galerkin scheme is employed to discretize them into a set of ordinary differential equations. To accomplish that, the eigenfunctions for the transverse motion of a linear hinged-hinged beam are selected as the basis functions for both the longitudinal and transverse motions. The transverse and longitudinal displacements are defined as the following approximate series expansions:
uðx; tÞ ¼
Substituting Eqs. (21) and (22) into Eqs. (19) and (20) and choosing the transverse external force as F(x) = f1/1(x) and the initial curvature as w0(x) = A0sin(px), the Galerkin method is applied by multiplying the resultant equations by the corresponding eigenfunction and integrating them with respect to x from 0 to 1; this procedure yields the following set of second-order nonlinear ordinary differential equations for the longitudinal and transverse motions, respectively:
0
1
/i /00j /0k dx qj pk
A0 /001 /i /0j dx pj
M X M X M Z 1 3X A0 /01 /i /00j dx pj þ /i /0j /0k /00l dx qj qk ql 2 j¼1 k¼1 l¼1 0 0 k¼1 Z M X M M Z 1 1 X 3X 2 þ ð A0 /001 /i /0j /0k dxÞqj qk þ ðA0 /01 Þ /i /00j dx qj 2 j¼1 k¼1 0 0 j¼1 # Z M M M 1 XX X Z 1 2 0 00 0 0 00 0 þ3 A0 /1 /i /j /k dx qj qk þ 2 ð A0 /1 /1 /i /j dxÞqj
Z
1
j¼1 k¼1
0
j¼1
0
1
f1 /1 /i dx cosðXtÞ ¼ 0;
i ¼ 1; 2; . . . ; M;
ð24Þ
0
20
Natural Frequency
18
16
η=1.0 14
η=0.5 12
η=0 10
8
0
0.0005
0.001
0.0015
0.002
A0 Fig. 2. The variation of the first linear dimensionless natural frequency of the transverse motion of the system with increasing A0 for different values of g.
in which the dot notation denotes the differentiation with respect to dimensionless time while the prime notation represents the differentiation with respect to the dimensionless axial coordinate. Eqs. (23) and (24) (which form a set of M + N nonlinear ordinary differential equations) are transformed into a set of first-order nonlinear ordinary differential equations via a standard change of variables by means of xi ¼ p_ i ði ¼ 1; 2; . . . ; NÞ and yi ¼ q_ i ði ¼ 1; 2; . . . ; MÞ; this operation yields a set of 2(M + N) first-order nonlinear ordinary differential equations with coupled terms. In the present study, a 15-degree-of-freedom system is considered (M = 7 and N = 8) which results in 30 first-order nonlinear ordinary differential equations with coupled nonlinear terms. Both linear and nonlinear analyses are performed upon this set of equations. The linear analysis is employed to obtain the linear natural frequencies of the system and examine the effect of geometric imperfection and small scale parameter on the linear natural frequencies of the system. The nonlinear analysis is performed by means of the pseudo-arclength continuation technique [27] which is capable of continuing both stable and unstable solution branches as well as determining different types of bifurcations.
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24
0.008
(a)
A
22
0.006
A0=0.002 18
Max (q1)
Natural Frequency
20
16
0.004
A0=0.001
14
0.002
12
B
A0=0 10 8
0 0.9 0
0.5
1
η
1.5
0.95
1
1.05
1.1
1.15
1.2
Ω/ω1
2 0
Fig. 3. The variation of the first linear dimensionless natural frequency of the transverse motion of the system with increasing g for different values of A0.
(b)
B
-0.002
3. Linear natural frequencies of the system
4. Frequency–response curves of the system The frequency–response curves are presented in this section so as to examine the nonlinear resonant response of the system. The frequency of the external excitation force is chosen near the first linear natural frequency of the system, i.e. x1, in order to investigate the resonant response of the system. The microbeam is made of epoxy material, the mechanical properties of which can be found in [5,9,28]. Fig. 4 shows the frequency–response curves for (a and b) the maximum and minimum of the first generalized coordinates of the transverse motion, respectively. The numerical calculations are carried out using the following dimensionless parameters: b = 415.692, g = 4.347, cd = 0.04, A0 = 0.002, and f1 = 0.010. As shown in Fig. 4(a), theoretically, the maximum amplitude of the first generalized coordinate of the transverse motion of the system increases with increasing external excitation frequency; the first limit point bifurcation occurs at point A (X = 1.1822x1) leading the system to have an unstable solution. As the excitation frequency is decreased, the unstable solution between points A and B is followed where the amplitude of the unstable response decreases until point B (X = 1.01020x1) is hit; at this point, the system regains the stability via a second limit point bifurcation. Fig. 4(b) shows the frequency–response curve of the system for the minimum amplitude of the first generalized coordinate. It
-0.004
Min (q1)
In this section the effect of the amplitude of the initial imperfection (A0) and small scale parameter (g) on the first linear natural frequency of the transverse motion is examined. Fig. 2 shows the variation of the first linear dimensionless natural frequency of the transverse motion of the system versus A0, for different values of g. As shown in the figure, the first natural frequency of the transverse motion of the system increases with increasing A0 (for the system with A0 = 0 (i.e. perfect system) the natural frequency is the smallest). The variation of the first linear natural frequency of the system with g is depicted in Fig. 3, for different values of A0. It can be concluded that as the value of g is increased, the first linear natural frequency of the transverse motion of the system increases accordingly.
-0.006
-0.008
A
-0.01 0.9
0.95
1
1.05
1.1
1.15
1.2
Ω/ω1 Fig. 4. Frequency–response curve of the system: (a and b) the maximum and minimum amplitudes of the first generalized coordinate of the transverse motion, respectively; b = 415.692, g = 4.347, A0 = 0.002, cd = 0.04, and f1 = 0.010.
can be observed that, as a result of the geometric imperfection, the maximum and minimum amplitudes of the first generalized coordinate are not the same; in fact, the maximum amplitude of q1 is 0.00828 while the minimum amplitude of q1 is 0.00985. From Eq. (20), it can be concluded that both quadratic and cubic nonlinearities are present in the equations of motion. Since w0 is associated with quadratic nonlinear terms, it tends to cause softening behaviour in the system. However, since the value of A0 is small, the effect of the cubic nonlinear terms are dominant and the system displays a hardening behaviour. Increasing the value of A0 to 0.005, from 0.002 in the system of Fig. 4, a new frequency–response curve is generated and plotted in Fig. 5. As shown in this figure, due to increased value of A0, the effect of quadratic nonlinear terms increase and the system displays both softening and hardening behaviours. Specifically, Fig. 5(a) shows that, theoretically, the maximum amplitude of q1 is increased with increasing excitation frequency until point A (X = 0.993504x1) is reached, where the stability is lost by means of a limit point bifurcation; this early limit point bifurcation is due to increased amplitude of imperfection which causes an initial softening behaviour. As the excitation frequency is decreased, the
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0.008
0.002
(a)
0.003
0.008
C
0.004 0.005 0.006
0.006
B
0.004
D
q1
Max (q1)
0.006
0.004
0.002
0.002
A
0 0.96
0.98
1
1.02
0
1.04
0.95
1
1.05
1.1
1.2
Fig. 6. The effect of A0 on the frequency–response curve of the system for the first generalized coordinate of the transverse motion. The values of A0 are denoted on the curves; b = 415.692, g = 4.347, cd = 0.04, and f1 = 0.010.
0
(b) A
-0.002
Classical
-0.004
Min (q1)
1.15
Ω/ω1
Ω/ω1
B -0.006
Modified couple
0.003
D
-0.008
q1
0.002
-0.01
C 0.96
0.98
1
1.02
1.04
0.001
Ω/ω1 Fig. 5. Frequency–response curve of the system: (a and b) the maximum and minimum amplitudes of the first generalized coordinate of the transverse motion, respectively; b = 415.692, g = 4.347, A0 = 0.005, cd = 0.04, and f1 = 0.010.
amplitude of the unstable response (the unstable solution between points A and B) increases until reaching the next limit point bifurcation at point B (X = 0.987855x1). By further increasing the excitation frequency, the stable solution branch between points B and C is followed, where the system displays a hardening behaviour. The third limit point bifurcation occurs at point C (X = 1.028668x1) where the motion of the system becomes unstable once again. As the excitation frequency is decreased, the unstable solution branch lasts until point D (X = 0.990201x1) is hit corresponding to the fourth limit point bifurcation. Beyond that point, the system displays a stable response with decreasing amplitude. From Fig. 5(a and b), it can be observed that the maximum and minimum amplitudes of the first generalized coordinate are 0.007202 and 0.01066, respectively. Comparing Figs. 4 and 5, it can be concluded that due to increased amplitude of initial imperfection, the difference between the maximum and minimum amplitudes of the q1 motion increases, the number of limit point bifurcations increases to four, and the system displays both hardening and softening behaviours. Fig. 6 shows the frequency–response curves of the system for different values of the initial curvature amplitude A0. The other
0 0.9
0.95
1
1.05
1.1
Ω/ω1 Fig. 7. Comparison between the frequency–response curves of the system for the first generalized coordinate of the transverse motion, obtained via modified couple and classical theories; b = 346.41, cd = 0.03, A0 = 0.001, and f1 = 0.0012; g = 0.4830 for the modified couple stress theory and g = 0 for the classical theory.
dimensionless parameters are selected as: b = 415.692, g = 4.347, cd = 0.04, and f1 = 0.10. This figure shows that, as the amplitude of the initial imperfection increases, the hardening behaviour of the system decreases; moreover, for higher values of A0 both hardening and softening behaviours are observed. As shown in the figure, for A0 = 0.006, the maximum amplitude of the system occurs at a frequency lower that the first linear natural frequency of the transverse motion. A comparison between the classical and modified couple stress theories is illustrated in Fig. 7. In particular, the frequency– response curve of the system is plotted in the absence (classical) and presence (modified couple) of the length scale parameter. As shown in the figure, the hardening behaviour of the system decreases when the modified couple stress theory is employed. In particular, the modified couple stress theory predicts lower
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From Eqs. (19) and (20), it can be inferred that the contribution of both quadratic and cubic nonlinear terms increases with increasing b, which is verified in Fig. 9. As shown in this figure, due to increased value of b, both the initial softening and the latter hardening behaviours of the system increase; in other words, the curves bend more to the left at first and then bend more to the right.
0.010 0.008
0.006
q1
0.006
0.004
5. Conclusions
0.004
0.002
0 0.96
0.98
1
1.02
1.04
Ω/ω1 Fig. 8. The effect of the forcing amplitude, f1, on the frequency–response curve of the system for the first generalized coordinate of the transverse motion. The values of f1 are denoted on the curves; b = 415.692, g = 4.347, A0 = 0.005, and cd = 0.04.
0.009 300 350
400
0.006
q1
450
0.003
0
0.96
1
1.04
1.08
Ω/ω1 Fig. 9. The effect of b on the frequency–response curves of the system for the first generalized coordinate of the transverse motion. The values of b are denoted on the curves; g = 4.347, A0 = 0.005, and cd = 0.04.and f1 = 0.0011.
nonlinear resonance frequency and hence lower amplitude of oscillations. This is mainly due to the reason that the presence of g in Eq. (20) increases the flexural stiffness of the system. The effect of the forcing amplitude on the frequency–response curves of the system is depicted in Fig. 8. Four different forcing amplitudes are considered, the values of which are denoted on the curve. As shown in Fig. 8, for lower forcing amplitudes (e.g. f1 = 0.004), the system displays only a softening behaviour; however, for higher forcing amplitudes, the initial softening behaviour is accompanied by a latter hardening behaviour which becomes more visible as the forcing amplitude is increased. It is also observed that due to increased forcing amplitude, the whole response region becomes wider and the maximum amplitude of the oscillations increases. Finally, the frequency–response curves of the system for different values of the dimensionless parameter b are depicted in Fig. 9.
The nonlinear resonant behaviour of a geometrically imperfect microbeam has been examined numerically, taking into account the longitudinal and transverse displacements. The nonlinear partial differential equations governing the motion of the microbeam in the longitudinal and transverse directions were derived employing Hamilton’s principle on the basis of the modified couple stress theory. The equations of motion were discretized into a set of second-order nonlinear ordinary differential equations by means of the Galerkin technique. A new set of first-order nonlinear ordinary differential equations was obtained by introducing a standard change of variables to the previous set. A linear analysis was performed upon this set of equations so as to investigate the effect the amplitude of the initial curvature and small scale parameter on the linear natural frequencies of the system. The pseudo-arclength continuation technique was employed in order to examine the nonlinear resonant response of the system and to construct the frequency–response curves. The results of the linear analysis showed that the first natural frequency of the transverse motion of the system increases with increasing both the amplitude of the initial imperfection and g (the size-dependent factor). Examining the nonlinear resonant response of the system revealed that the hardening behaviour of the system decreases with increasing the amplitude of the initial imperfection and the system displays both hardening and softening behaviours for higher values of the initial imperfection; examining the effect of the amplitude of the initial imperfection on the frequency–response curves revealed that the nonlinear behaviour of the system can be qualitatively changed due to the presence of a small geometric imperfection. It was shown that the modified couple stress theory predicts lower nonlinear resonances and hence weaker nonlinear behaviours, compared to the classical theory. Examining the effect of the forcing amplitude on the frequency–response curves of the system showed that the initial softening behaviour is continued by a latter hardening behaviour for higher values of the forcing amplitude.
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