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 DOI: 10.1177/1077546311414600

2012 18: 696 originally published online 12 September 2011Journal of Vibration and ControlMohammad Hussein Kahrobaiyan, Mohsen Asghari, Masoud Hoore and Mohammad Taghi Ahmadian

continuum theoryNonlinear size-dependent forced vibrational behavior of microbeams based on a non-classical

  

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Article

Nonlinear size-dependent forcedvibrational behavior of microbeams basedon a non-classical continuum theory

Mohammad Hussein Kahrobaiyan1, Mohsen Asghari1,Masoud Hoore1 and Mohammad Taghi Ahmadian1,2

Abstract

In this paper, the nonlinear forced-vibration of Euler-Bernoulli beams with large deflections is investigated based on the

modified couple stress theory, a non-classical theory capable of capturing size effects. The classical theory is unable to

predict the size effects. In systems with the dimensions in order of microns and sub-microns the size effects are very

significant. For some specific beams subjected to a concentrated force at its middle as the harmonic exciter, the size-

dependent responses are investigated for primary, super-harmonic and sub-harmonic resonances. The results show that

the frequency-responses of the system are highly size-dependent.

Keywords

Microbeams, modified couple stress theory, non-classical continuum theory, nonlinear vibration, size effects

Received: 13 December 2010; accepted: 25 May 2011

1. Introduction

Microbeams are widely used in micro-and nano-electromechanical systems (MEMS and NEMS) suchas vibration shock sensors (Gao et al., 2005; Voglaet al., 2009), electro-statically excited micro actuators(Batra et al., 2008a,b; Chaterjee and Pohit, 2009; Hao,2008; Moghimi Zand and Ahmadian, 2009a,b;Mojahedi et al., 2010; Porfiri, 2008) and atomic forcemicroscopes (AFMs) (Mahdavi et al., 2008; Sinha,2005). The thickness of beams used in MEMS andNEMS is in the order of microns and sub-microns.The size-dependent static and vibration behaviors inmicro scales are experimentally validated. For examplein the micro-torsion test of thin copper wires, Flecket al. (1992) indicated that decrease of a wire’s diameterresults in a noteworthy enhancement of the torsionalhardening. Stolken and Evans (1998) reported a nota-ble increase of plastic work hardening caused by thedecrease of beam thickness in the micro bending testof thin nickel beams. Also, size-dependent behaviorsare detected in some kinds of polymers. For instance,during micro bending tests of beams made of epoxypolymers, Chong and Lam (1999) observed a significant

enhancement of bending rigidity caused by the beam’sthickness reduction.

McFarland and Colton (2005) detected a consider-able difference between the stiffness values predicted bythe classical beam theory and the stiffness valuesobtained during a bending test of polypropylenemicro-cantilever. These experiments reveal that size-dependent behavior is an inherent property of materialswhich appears for a beam when the characteristic size,such as thickness or diameter, is close to the internalmaterial length scale parameter (Kong et al., 2008).

The classical continuum mechanics theory is notcapable of appropriate prediction and explanation ofsize-dependent behaviors which occur in micron- and

1Department of Mechanical Engineering, Sharif University of Technology,

Tehran, Iran2Center of Excellence in Design, Robotics and Automation (CEDRA),

Sharif University of Technology, Tehran, Iran

Corresponding author:

M Asghari, Department of Mechanical Engineering, Sharif University of

Technology, Tehran, Iran

Email: asghari@sharif.edu

Journal of Vibration and Control

18(5) 696–711

! The Author(s) 2011

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DOI: 10.1177/1077546311414600

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sub-micron-scale structures. However, some non-classi-cal continuum theories such as higher-order gradienttheories and the couple stress theory have been devel-oped such that they are acceptably able to interpret thesize-dependencies.

In the 1960s some researchers such as Mindlin,Touplin and Koiter introduced the couple stress elastic-ity theory with the presence of two higher-order materiallength scale parameters in addition to the two classicalLame constants (Koiter, 1964; Mindlin and Tiersten,1962; Toupin, 1962) in the constitutive equations. Assome applications, Zhou and Li (2001) employed thistheory to investigate the static and dynamic torsion ofa micro-bar. Also, the size-effects in Timoshenko beamsmodeled on the basis of the couple stress theory havebeen investigated by Asghari et al. (2010b).

Yang et al. (2002) argued that in addition to the clas-sical equilibrium equations of forces and moments offorces, another equilibrium equation should be consid-ered for the material elements. This additional equationis the equilibrium of moments of couples. Then, they con-cluded that this additional equilibrium equation impliesthe symmetry of the couple stress tensor. Accordingly,they modified the constitutive equations of the couplestress theory and presented the new constitutive equa-tions with only one material length scale parameter.

In order to determine the length scale parameters fora specific material, some typical experiments such as themicro-bend test, micro-torsion test and specially micro/nano indentation test can be carried out (see: Begleyand Hutchinson, 1998; Fleck et al., 1992; Lam et al.,2003; McFarland and Colton, 2005; Nix and Gao,1998; Stolken and Evans, 1998).

The modified couple stress theory has been utilizedby some researchers to develop the size-dependent for-mulations for beams. For example, Park and Gao(2006) analyzed the static mechanical properties of anEuler-Bernoulli beam modeled on the basis of the mod-ified couple stress theory and interpreted the outcomesof the epoxy polymeric beam bending test. Also, Konget al. (2008) obtained the natural frequencies of thebeam based on the modified couple stress theory. Inaddition, the size-dependent natural frequencies offluid-conveying microtubes (Wang, 2010), the size-dependent buckling behavior of micro-tubules (Yimingand Zhang, 2010) and the size-dependent resonant fre-quencies and sensitivities of AFM microcantilevers(Kahrobaiyan et al., 2010) were investigated based onthe modified couple stress theory. A new Timoshenkobeam model based on the modified couple stress theorywas formulated by Ma et al. (2008). They assessed thesize-dependent static and free-vibration behavior of asimply-supported Timoshenko beam as a case study.

In beams used in MEMS and NEMS with twoimmovable supports, we face with the nonlinear

phenomena in the large amplitude deflections. The non-linearity is due to the induced mid-plane stretchingduring the transverse deflections. This nonlinearitycauses the static and vibration results to be changedsignificantly (Hamdan et al., 2010; Hassanpour et al.,2010; Hino et al., 1985; Khadem and Rezaee, 2002;Mojahedi et al., 2010; Mook et al., 1986). Hence, theabovementioned linear investigations on the couplestress beams are not appropriate in these conditions.Recently, a size-dependent nonlinear Euler-Bernoullibeam model has been presented by Xia et al. (2010)on the basis of the modified couple stress theory.They studied the nonlinear size-dependent static bend-ing, buckling and the free vibration of microbeams. Inaddition, the nonlinear free vibration of Timoshenkobeams has been investigated based on the modifiedcouple stress theory by Asghari et al. (2010a).Moreover, Ke et al. (2011) studied the nonlinear freevibration of functionally graded beams based on thistheory and using the Timoshenko displacement field.

In this paper, utilizing the modified couple stresstheory, the size-dependent nonlinear forced vibrationof Euler-Bernoulli beams is investigated. As an exam-ple, the size-dependent frequency and time responses ofa hinged-hinged beam subjected to a concentrated har-monic force at its middle are assessed for primary,super-harmonic and sub-harmonic resonances.

2. Governing equations

To have a self-contained work, in this section, the pro-cedures of obtaining the governing differential equa-tions are explained in a rigorous manner. In themodified couple stress theory, the strain energy density�u for a linearly elastic material in infinitesimal deforma-tions is written as (Yang et al., 2002)

�u ¼1

2�ij"ij þ

1

2mij�ij ði, j ¼ 1, 2, 3Þ, ð1Þ

where

�ij ¼ � tr eð Þ�ij þ 2�"ij, ð2Þ

"ij ¼1

2ðruÞij þ ruð Þ

Tij

� �, ð3Þ

mij ¼ 2l2��ij, ð4Þ

�ij ¼1

2ðrhÞij þ rhð ÞTij

� �, ð5Þ

in which �ij, "ij, mij and �ij denote the components of thesymmetric part of stress tensor r, the strain tensor e,the deviatoric part of the couple stress tensor m andthe symmetric part of the curvature tensor c, respec-tively. Also, u and h are the displacement vectorand the rotation vector. The two Lame constants and

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the material length scale parameter are represented by �,� and l, respectively. The Lame constants are written interms of the Young’s modulus and the Poisson’s ratio as� ¼ �E=ð1þ vÞð1� 2vÞ and � ¼ E=2ð1þ vÞ. The com-ponents of the rotation vector h are related to the com-ponents of the displacement vector field, ui according to

�i ¼1

2curl ðuÞð Þi, ð6Þ

the displacement vector can be represented as

u ¼ ux uy uz� �T

, ð7Þ

where ux, uy and uz refer to the displacement fieldsalong x, y and z axes respectively.

Consider a uniform homogeneous initially straightbeam along the x-direction with length L and alsowith top and bottom surfaces perpendicular to the z-direction.

The front and rear surfaces of the beam, which areplanes perpendicular to the y-direction, are assumed tobe force-traction free. The centroids of sections areassumed to lie on the plane z ¼ 0. Also, the line passingthrough the centroid of a section and parallel to the z-direction is assumed to be its principal axis of inertia.

Figure 1 illustrates the loading, kinematic parame-ters and the coordinate system for an Euler-Bernoullibeam. The axial and transverse components of the dis-placement field based on the Euler-Bernoulli modelafter loading are here considered as

ux ¼ u x, tð Þ þ z ðx, tÞ, uz ¼ wðx, tÞ, ð8Þ

with

x, tð Þ � �@w x, tð Þ

@x, ð9Þ

where u is the axial displacement of the centroid ofsections, w denotes the lateral deflection of the beamand stands for the angle of rotation (about y-axis) ofthe beam cross-sections. By this displacement field, thecross sections remain plane after deformation and theyare always perpendicular to the center line. We assumethat all parameters, except the displacement componentuy are independent of y-coordinate.

By assuming small slopes in the beam after deforma-tion but possible finite deflection w, the axial strain, i.e.the ratio of the elongation of a line element initially inthe axial direction to its initial length, can be approxi-mately expressed by the von-Karman strain as

"xx ¼@ux@xþ1

2

@w

@x

� �2

¼@u

@x� z

@2w

@x2þ1

2

@w

@x

� �2

: ð10Þ

It is noted the valid relation between the axial strain"xx and the displacement filed in the condition of largedeflections with small slopes of the beam is this equa-tion, and equation (3) should not be used for the cal-culation of "xx; equation (3) is valid only for theinfinitesimal deformations. Considering the definitionof the von Karman strain and the displacement fieldof equation (8), it is deduced that the other strain com-ponents are zero. By neglecting the derivatives of uywith respect to x and z, and using equations (6) to(9), it can be written

�y ¼ �@w x, tð Þ

@x�x ¼ �z ¼ 0: ð11Þ

It can be shown that for a geometrically nonlinearbeam, it is permissible to relate the rotation compo-nents to the displacement field as it is done in the infin-itesimal deformations.

Substitution of equation (11) into (5) yields

�xy ¼ �1

2

@2w x, tð Þ

@x2�xx ¼ �yy ¼ �zz ¼ �xz ¼ �yz ¼ 0:

ð12Þ

Although due to the possible distributed loading onthe top surface of the beam, the normal stress �zz doesnot vanish, we neglect it with respect to �xx. Now,recalling the considered assumptions (including frontand rear traction-free surfaces condition, the indepen-dence of y-coordinate for stress and strain components,and the negligibility of @uy=@x and @uy=@z), then usingequations (2), (3), (8), (9), the stress component �xx asthe only non-zero component of r can be written as

�xx ¼ E"xx ¼ E@u

@x� z

@2w

@x2þ1

2

@w

@x

� �2 !

: ð13ÞFigure 1. An Euler-Bernoulli beam, loading, kinematic parame-

ters and coordinate system.

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Also substituting equation (12) into (4), the onlynon-zero components of the deviatoric part of thecouple stress tensor m are obtained as

mxy ¼ myx ¼ ��l2 @

2w x, tð Þ

@x2: ð14Þ

To obtain the governing equations, the kineticenergy of the beam T, the beam strain energy due tobending and also due to the change of the stretch withrespect to the initial configuration Ubs, and the increasein the stored energy with respect to the initial configu-ration due to the existence of initially axial load Uis areconsidered as follows

T ¼1

2

Z L

0

ZA

@u

@t� z

@2w

@t@x

� �2

þ@w

@t

� �2" #

dAdx, ð15Þ

Ubs ¼

ZV

�u dV ¼1

2

Z L

0

ZA

�ij"ij þmij�ij�

dAdx

¼1

2

Z L

0

ZA

E@u

@x� z

@2w

@x2þ1

2

@w

@x

� �2 !2

8<:

þ�l2@2w

@x2

� �2)dAdx, ð16Þ

Uis ¼

Z L

0

N0

A

ZA

@u

@x� z

@

@xþ1

2

@w

@x

� �2 !

dA dx, ð17Þ

where V, A, L and respectively denote the volume, thecross-section area, length and density of the beam,noting that the term in the bracket presented by equa-tion (15) represents the squared magnitude of thevelocity of particles by neglecting the velocity in they-direction. It is noted that the reference configurationis the one coincident with the status of the beam whencarrying the axial load N0 without transverse deflection.As a result, the component �xx is not the total stressacting on beam elements. The total normal axial stressis �xx þN0=A.

SinceRA zdA ¼ 0, the kinetic energy T and the total

potential energy, U ¼ Ubs þUis can be rewritten as

T¼1

2

Z L

0

A@u

@t

� �2

þI@2w

@t@x

� �2

þA@w

@t

� �2" #

dx, ð18Þ

U¼1

2

Z L

0

EA@u

@xþ1

2

@w

@x

� �2 !2

þN0 2@u

@xþ

@w

@x

� �2 !8<

:þ EIþ�Al2� @2w

@x2

� �2)dx, ð19Þ

where I ¼RA z2dA represents the beam area moment of

inertia. The virtual work done by the external forcesand couples during a virtual change in the beam con-figuration can be written as

�W ¼

Z L

0

F x, tð Þ � B@w

@t

� ��wdxþ

Z L

0

G x, tð Þ�udx

þ �M�@w

@x

� �þ �V�wþ �N�u

� �x¼0,L

, ð20Þ

where F x, tð Þ and G x, tð Þ, respectively, refer to the lat-eral and axial body forces (see Figure 1). Also, Bdenotes the viscous damping coefficient of the viscousfluid possibly existent around the beam, and the termincluding it represents the lost work done by its damp-ing force (Banks and Inman, 1991). Moreover, theexternal transverse force, the axial force and the totalmoment acting on the beam end sections are denotedby �V, �N and �M, respectively. The Hamilton principle isnow used as Z t2

t1

�Tþ �U� �Wð Þdt ¼ 0: ð21Þ

Substituting equations (18) to (20) into (21), the size-dependent nonlinear governing equation of motion andcorresponding boundary conditions of Euler-Bernoullibeams are obtained as

@

@xN0 þ EA

@u

@xþ1

2

@w

@x

� �2 !" #

þ G x, tð Þ ¼ A@2u

@t2,

ð22Þ

EIþ�Al2� @4w

@x4�@

@xN0þEA

@u

@xþ1

2

@w

@x

� �2 !" #

@w

@x

( ),

þA@2w

@t2�I

@4w

@t2@x2þB

@w

@t¼F x,tð Þ ð23Þ

N0 þ EA@u

@xþ1

2

@w

@x

� �2 !

� �N

!x¼0,L

¼ 0

or �ujx¼0,L¼ 0, ð24Þ

N0 þ EA@u

@xþ1

2

@w

@x

� �2 !" #

@w

@x

� EIþ �Al2� @3w

@x3� �Vþ I

@3w

@t2@x

�x¼0,L

¼ 0

or �wjx¼0,L¼ 0, ð25Þ

EIþ �Al2� @2w

@x2� �M

� �x¼0,L

¼ 0 or �@w

@x

� �x¼0,L

¼ 0:

ð26Þ

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The axial and lateral governing equations of non-linear size-dependent beams are presented in equations(22) and (23), respectively. Now, the presence of axialbody forces is neglected, and as a consequence N0 isindependent of x. Moreover, if the longitudinal inertiaA@2u=@t2 is neglected, equation (22) can be double-integrated with respect to x as

EA@u

@xþ1

2

@w

@x

� �2 !

¼ k1 tð Þ

)

REA uþ

Z x

0

1

2

@w

@x

� �2

dx

!¼ xk1 tð Þ þ k2 tð Þ, ð27Þ

where k1 and k2 are some functions. In the following,we focus on beams with two immovable supports toobtain their specific formulation from the derived gen-erally valid equations presented in the above. In beamswith immovable supports (e.g. hinged-hinged, clamped-clamped, hinged-clamped, etc), are u 0, tð Þ ¼ u L, tð Þ ¼ 0,and in these conditions, equation (27) results in

k1 tð Þ ¼EA

2L

Z L

0

@w

@x

� �2

dx, k2 ¼ 0: ð28Þ

Therefore, the following is obtained

EA@u

@xþ1

2

@w

@x

� �2 !

¼ k1 tð Þ ¼EA

2L

Z L

0

@w

@x

� �2

dx : ¼ N:

ð29Þ

Indeed, N is the variation of the axial force in thebeam with respect to the initial configuration due to theextension caused by the transverse deflection, notingthat it is a constant value all over the beam. It iscalled the mid-plane stretching and is the source ofnonlinearity in the considered problem. Substitutionof equation (29) into (23) and (25) leads to

EIþ �Al2� @4w

@x4�N

@2w

@x2þ A

@2w

@t2þ B

@w

@t¼ F x, tð Þ,

ð30Þ

N@w

@x� EIþ �Al2� @3w

@x3� �V

� �x¼L

x¼0

¼ 0 or �wjx¼Lx¼0¼ 0,

ð31Þ

where

N ¼ N0 þ N ¼ N0 þEA

2L

Z L

0

@w

@x

� �2

dx: ð32Þ

It should be noted that for obtaining equation (30),the rotating inertia of cross-sections, I@4w=@t2@x2 hasbeen neglected.

Letting l ¼ 0, equations (30) to (32) reduce to thegoverning equation and corresponding boundary con-ditions of a nonlinear Euler-Bernoulli beam modeledon the basis of the classical continuum theory(Khadem and Rezaee, 2002). Similarly, letting N ¼ 0,equations (30) to (32) reduce to the governing equationand corresponding boundary conditions of a linearEuler-Bernoulli beam modeled on the basis of the mod-ified couple stress theory (Kong et al., 2008).

3. Examples

In this section as a specific case, a beam with a uniformrectangular cross-section subjected to a concentratedharmonic transverse force at its middle with twoimmovable end supports is considered (see Figure 2).The most prevalent kinds of immovable end supportsincluding hinged-hinged (H-H), clamped-clamped (C-C) and hinged-clamped (H-C) are investigated. Forthese three types of supports, are

w 0, tð Þ ¼ w L, tð Þ ¼@2w 0, tð Þ

@x2¼@2w L, tð Þ

@x2¼ 0, ð33Þ

w 0, tð Þ ¼ w L, tð Þ ¼@w 0, tð Þ

@x¼@w L, tð Þ

@x¼ 0, ð34Þ

w 0, tð Þ ¼ w L, tð Þ ¼@w 0, tð Þ

@x¼@2w L, tð Þ

@x2¼ 0: ð35Þ

At a hinge support, the external moment �M is con-sidered equal to zero in this work.

The concentrated transverse harmonic external forceF can be written as

F x, tð Þ ¼ P0� x�L

2

� �cos�t, ð36Þ

where � represents the Dirac delta function; also, � andP0 respectively denote the frequency and the amplitudeof excitation. It is helpful to normalize the governingequation and boundary conditions and work with

Figure 2. A beam with immovable supports having a rectangu-

lar cross-section subjected to a harmonic concentrated force at

its middle.

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non-dimensional parameters. In order to do so, somedimensionless variables are introduced as

~w ¼w

h, ~x ¼

x

L, ¼ t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEIþ �Al2

A

s�2

L2,

~� ¼ �L2

�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA

EIþ �Al2

s, ð37Þ

where the dimensionless forms of the lateral deflection,axial coordinate, time and frequency of excitation aredenoted by ~w, ~x, and ~�, respectively. Also, the valueof parameter � is taken equal to � ¼ �, � ¼ 4:73 and� ¼ 3:92, respectively, for the cases of H-H, C-C andH-C supports (Rao, 2007). Utilizing these dimensionlessvariables, the normalized forms of the governing equa-tion (23) and also equations (33) to (35) are written as

@2 ~w ~x, ð Þ

@2þ

1

�4@4 ~w ~x, ð Þ

@ ~x4� ~N

@2 ~w ~x, ð Þ

@ ~x2

þ ~B@ ~w ~x, ð Þ

@¼ SF ~x, ð Þ, ð38Þ

~w 0, ð Þ ¼ ~w 1, ð Þ ¼@2 ~w 0, ð Þ

@ ~x2¼@2 ~w 1, ð Þ

@ ~x2¼ 0, ð39Þ

~w 0, ð Þ ¼ ~w 1, ð Þ ¼@ ~w 0, ð Þ

@ ~x¼@ ~w 1, ð Þ

@ ~x¼ 0, ð40Þ

~w 0, ð Þ ¼ ~w 1, ð Þ ¼@ ~w 0, ð Þ

@ ~x¼@2 ~w 1, ð Þ

@ ~x2¼ 0, ð41Þ

where

~N ¼ ~N0 þ6

�4 1þ 61þ�

lh

� 2� � Z 1

0

@ ~w

@ ~x

� �2

d ~x, ð42Þ

~N0 ¼N0

EI�4 1þ 61þ�

lh

� 2� � , ð43Þ

~B ¼

ffiffiffiffiffi12p

B Lh

� 2�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE 1þ 6

1þ�lh

� 2� �r , ð44Þ

S ¼12 L

h

� 4b�4E 1þ 6

1þ�lh

� 2� � , ð45Þ

F ~x, ð Þ ¼ P0 � ~x�1

2

� �cos ~�: ð46Þ

In order to derive the governing ordinary differentialequation of motion from the partial one mentioned in

equation (38), the Galerkin method is employed. Sincein beams, the first mode is dominant, only the firstmode shape of the beam is considered here, and it iswritten

w ~x, ð Þ ¼ q ð Þ’ ~xð Þ: ð47Þ

where ’ ~xð Þ stands for the first mode shape of the beam.By inserting equation (47) into (38), then multiplyingboth sides of the result by ’ ~xð Þ and integrating alongthe beam length, the governing ordinary differentialequation of motion is achieved as

€qþ 2 _qþ qþ �q3 ¼ K cos ~� ð48Þ

in which a dot denotes the derivation with respect to thedimensionless time , and

¼~B

2, ð49Þ

� ¼ �6R 10 ’0ð Þ2d ~x

� � R 10 ’’

00d ~x� �

�4 1þ 61þ�

lh

� 2� � R 10 ’

2d ~x� � , ð50Þ

K ¼SP0 ’ 1=2ð ÞR 1

0 ’2d ~x

: ð51Þ

In deriving equation (48), it has been assumed thatthere is no pre-tension in the beam, i.e. N0 ¼ 0. In thecases of H-H, C-C and C-H beams, the function ’ iswritten, respectively, as follows (Rao, 2007)

’ ~xð Þ ¼ sin � ~xð Þ, ð52Þ

’ ~xð Þ ¼ cosh � ~xð Þ � cos � ~xð Þ

�cosh �ð Þ � cos �ð Þ

sinh �ð Þ � sin �ð Þsinh � ~xð Þ � sin � ~xð Þð Þ, ð53Þ

’ ~xð Þ ¼ cosh � ~xð Þ � cos � ~xð Þ

�cosh �ð Þ þ cos �ð Þ

sinh �ð Þ þ sin �ð Þsinh � ~xð Þ � sin � ~xð Þð Þ: ð54Þ

It is noted that in each of the previous equations, thementioned values of � corresponding to the type of thesupports should be used, i.e. � ¼ � in equation (52),� ¼ 4:73 in equation (53) and � ¼ 3:92 in equation (54).

As an example, the parameters involved in equation(48) for a hinged-hinged beam are expressed as follows:

� ¼3

1þ 61þ�

lh

� 2� � , ð55Þ

K ¼ 2S P0 ¼ 212 L

h

� 4b�4E 1þ 6

1þ�lh

� 2� �P0: ð56Þ

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Equation (48) is a Duffing equation with cubic non-linearity. It is noted that the coefficients of the dampingterm , the nonlinear term � and also the amplitude ofthe excitation K are size-dependent, i.e. there are func-tions of the ratio of the beam characteristic length (inthis example, the thickness) to the material length scaleparameter h=l. Hence, it is predictable that the non-linear behavior of the beam should be size-dependent.To obtain some numerical estimation, it is assumedthat the beams have the geometrical parameters men-tioned in Table 1 and also they are made of epoxywith the mechanical properties expressed in Table 2.Also, the damping coefficient is considered asB ¼ 0:1 Ns=m2 (like the value considered by Banksand Inman, 1991). For the hinged-hinged boundarycondition, the ratios = Classic, �=�Classic and K=KClassic

are presented in Table 3 for different values of h=l,where Classic, �Classic and KClassic denote the values ofthe coefficients when using the classical theory, i.e.when assuming l ¼ 0.

3.1. The Primary Resonance

In this section, the primary resonance of the hinged-hinged (H-H), clamped-clamped (C-C) and clamped-hinged (C-H) beams with aforementioned geometricaland material specifications is investigated. First, the

following assumptions should be considered (Nayfeh,1979)

� ¼ " ~�, ð57Þ

¼ " ~ , ð58Þ

K ¼ " ~K, ð59Þ

where " is a small dimensionless parameter with a valuein the order of the amplitude of the deflection. Bysubstituting equations (57) to (59) into (48), the follow-ing equation is obtained

€qþ 2" ~ _qþ qþ " ~�q3 ¼ " ~K cos ~�: ð60Þ

Since in primary resonance ~� � 1, the followingequation can be written

~� ¼ 1þ "�, ð61Þ

where � denotes the detuning parameter. Following thewell-known Method of Multiple Scales (MMS)(Nayfeh, 1979) for the governing nonlinear equation(60), the solution is obtained as q ¼ a cos ~� � �

� þ

O "ð Þ where a is the non-dimensional amplitude ofresponse and � denotes the phase of response. In addi-tion, the frequency response of the system can beobtained as

~ 2 þ � �3

8~�a2

� �2" #

a2 ¼~K2

4: ð62Þ

Equation (62) implicitly relates the amplitude ofresponse a, the detuning parameter �, and the ampli-tude of excitation ~K.

The frequency-response curves of beams with H-H,C-C and C-H supports in primary resonance are respec-tively depicted in Figures 3 to 5. The amplitude ofresponse a and the phase of response � versus thedetuning parameter � are illustrated in (a) and (b)parts, respectively, for various values of the ratio ofthe beam thickness to the material length scale param-eter h=l. In the figures, the case h=l ¼ 1 refers to thefrequency-response curves predicted by the classicalbeam theory. Due to the nonlinear nature of thesystem, the jump phenomenon, i.e. a sudden changein the amplitude and phase of oscillations caused by aslight change in the frequency of the excitation, can beobserved in this figure. It is observed that as h=ldecreases, i.e. approaching the conditions belongingto the MEMS and NEMS, the deviation of the fre-quency-response curves from the straight line � ¼ 0diminishes.

Table 1. Geometrical parameters of the microbeams in the

case studies

Beam length, L Beam width, b

L ¼ 20h b ¼ 2h

Table 2. Mechanical properties of epoxy (Ma et al., 2008;

Park and Gao, 2006)

Elastic

modulus,

E ( Gpa)

Shear

modulus,

� (Mpa)

Poisson

ratio, v

Density,

(kg m�3)

Length scale

parameter,

l (�m)

1:44 521:7 0:38 1220 17:6

Table 3. The ratio of the coefficients appeared in the governing

ODE evaluated by the modified couple stress theory to those

predicted by the classical theory for different values of h=l

h=l ¼ 2 h=l ¼ 4 h=l ¼ 6 h=l ¼ 8 h=l ¼ 10

= classic 0.6922 0.8867 0.9446 0.9677 0.9789

�=�classic 0.4792 0.7863 0.8922 0.9364 0.9583

K=Kclassic 0.4792 0.7863 0.8922 0.9364 0.9583

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Furthermore, the height of the jump decreases. Thedimensionless amplitude of the response versus thedimensionless amplitude of the excitation for beamswith H-H, C-C and C-H supports is depicted, respec-tively, in Figures 6 to 8 for the primary resonance. Thefigures show that for a fixed value of the amplitude ofthe excitation K, the amplitude of the response oscilla-tions increases as h=l decreases.

Moreover in the following, the effects of the simul-taneous variation of the aspect ratio L=h and also h/l onthe response amplitude a have been depicted, for each

–20 –15 –10 –5 0 5 10 15 20 25 30

0.2

0.4

0.6

0.8

1

1.2a

σ

(a)

3

2.5

2

1.5

–20 –15 –10 –5 0 5 10 15 20 25 300

0.5

1

()

rad

γ

σ

(b)

Figure 3. The frequency-response of the hinged-hinged beam

for primary resonance: (a) amplitude of response (b) phase of

response.

1.11

0.90.80.7

–30 –20 –10 0 10 20 30

0.10.20.30.40.50.6a

σ

(a)

3

2.5

2

–30 –20 –10 0 10 20 300

0.5

1

1.5()

rad

γ

σ

(b)

Figure 4. The frequency-response of the clamped-clamped

beam for primary resonance: (a) amplitude of response (b)

phase of response.

1

–30

–30

–20 –10 0 10 20 30

0.10.20.30.40.50.60.70.80.9

a

σ

(a)

3

2.5

2

1.5

–20 –10 0 10 20 300

0.5

1

()

rad

γ

σ

(b)

Figure 5. The frequency-response of the clamped-hinged beam

for primary resonance: (a) amplitude of response (b) phase of

response.

0 2 4 6 8 10 12 14 16 18 20

0.5

1

1.5

2

2.5

a

K

Figure 6. The amplitude of response of the hinged-hinged

beam versus the amplitude of excitation for primary resonance.

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of hinged-hinged, clamped-clamped and clamped-hinged supports, separately.

Figures 9 to 11 show these effects, respectively, forthese support conditions. In these figures, the responseamplitude of the beams is evaluated for � ¼ 0, i.e.~� ¼ 1. According to the obtained results, the responseamplitude of the beams increase as L=h becomesgreater, for all values of h/l. This behavior can be

interpreted as the decrease in the beam stiffness dueto an increase in L=h.

3.2. The Super-harmonic Resonance

Due to the condition of the super-harmonic resonance,~� � 1=3, one can get (Nayfeh, 1979)

3 ~� ¼ 1þ "�, ð63Þ

Utilizing the method of multiple scales (Nayfeh,1979), the response of the system is obtained asq ¼ a cos 3 ~� � �

� þ K=1� ~�2�

cos ~� þO "ð Þ inwhich a denotes the non-dimensional amplitude ofresponse and � refers to the phase of response.Moreover, the frequency response of the system withgoverning nonlinear differential equation (60) can beobtained by eliminating the secular terms of the resul-tant equations as follows:

2 þ � � 3��2 �3

8�a2

� �2" #

a2 ¼ �2�6: ð64Þ

where

� ¼1

2

K

1� ~�2: ð65Þ

K0 5 10 15 20 25 30

0.5

1

1.5

2

2.5

3

3.5

a

Figure 7. The amplitude of response of the clamped-clamped

beam versus the amplitude of excitation for primary resonance.

K

0 2 4 6 8 10 12 14 16 18 20

0.5

1

1.5

2

2.5

a

Figure 8. The amplitude of response of the clamped-hinged

beam versus the amplitude of excitation for primary resonance.

Figure 9. Effect of L=h and h=l on the response amplitude of the

hinged-hinged beam.

Figure 10. Effect of L=h and h=l on the response amplitude of

the clamped-clamped beam.

Figure 11. Effect of L=h and h=l on the response amplitude of

the clamped-hinged beam.

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Equation (64) is the frequency-response equation ofthe system for the super-harmonic resonance which canbe used for determining the response amplitude a.

In order to evaluate the numerical results, here it isassumed that P0 ¼ 7:5�N.

The frequency-response curves of beams with H-H,C-C and C-H supports are depicted, respectively, inFigures 12 to 14 for the super-harmonic resonance.The figures show that as h=l decreases, the height ofthe jump decreases too, and also the deviation of thefrequency-response curves from the line � ¼ 0 reduces.Also, the dimensionless amplitude of the responseversus the dimensionless amplitude of the excitation isdepicted in Figures 15 to 17, respectively, for the threetypes of the supports under consideration. It isobserved that similar to the primary resonance, for agiven value of the amplitude of the excitation �, theamplitude of the sub-harmonic response oscillationsincreases as h=l decreases.

3.3. The Sub-harmonic Resonance

In the sub-harmonic resonance, one can consider ~� � 3(Nayfeh, 1979); so that we have ~� ¼ 3þ "�. Similarto what followed for obtaining the primary andsuper-harmonic resonances, elimination of the secularterms obtained from governing nonlinear differential

3

2.5

2

–5 0 5 10 15 20 25 30

0.5

1

1.5

a

σ

(a)

3

2.5

2

–5 0 5 10 15 20 25 300

0.5

1

1.5()

rad

γ

σ

(b)

Figure 12. The frequency-response of the hinged-hinged beam

for super-harmonic resonance: (a) amplitude of response (b)

phase of response.

1

–10 –5 0 5 10 15 20

0.10.20.30.40.50.60.70.80.9

a

σ

(a)

3

2.5

2

1.5

–10 –5 0 5 10 15 200

0.5

1

()

rad

γ

σ

(b)

Figure 13. The frequency-response of the clamped-clamped

beam for super-harmonic resonance: (a) amplitude of response

(b) phase of response.

1.4

–10 –5 0 5 10 15 20

0.2

0.4

0.6

0.8

1

1.2

a

σ

(a)

3

2.5

2

1.5

–10 –5 0 5 10 15 200

0.5

1

()

rad

γ

σ

(b)

Figure 14. The frequency-response of the clamped-hinged

beam for super-harmonic resonance: (a) amplitude of response

(b) phase of response.

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equation (60), the sub-harmonic frequency response isdetermined as (Nayfeh, 1979)

9 2 þ � � 9��2 �9

8�a2

� �2

¼81

16�2�2a2, ð66Þ

where a represents the non-dimensional amplitude ofthe response. Equation (66) can be rewritten as

a2 ¼ g1 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig21 � g2

q, ð67Þ

where

g1 ¼8�

9�� 6�2, ð68Þ

g2 ¼64

81�29 2 þ � � 9��2

� 2h i: ð69Þ

The nontrivial response occurs only if g1 4 0 andg21 � g2 (Nayfeh, 1979). Meeting these conditions,there should be

�

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

2� 63

s�

63��2

4 ��

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

2� 63

s: ð70Þ

Based on the mechanical properties of the epoxymentioned in Table 2 and also the specifications ofthe beam geometry tabulated in Table 1, the numericalresults for the sub-harmonic resonance are presented inthe following. Here in order to obtain some numericalresults, it is assumed that P0 ¼ 50�N. In Figures 18 to20 the amplitude a of the beam response has beendepicted, respectively, for H-H, C-C and H-C supportsversus the detuning parameter � for some differentvalues of h=l. It is observed that unlike the primaryand super-harmonic resonances, the sub-harmonic res-onance does not involve the jump phenomenon for allvalues of h/l. The figure also shows that a decrease inh=l leads to an increase in a. Also, the dimensionlessamplitude of the response versus the dimensionlessamplitude of the excitation is depicted in Figures 21to 23, respectively, for the three types of the supports

Λ0 0.5 1 1.5 2 2.5 3 3.5

0

1

2

3

4

5

6

a

Figure 16. The amplitude of the response of the clamped-

clamped beam versus the amplitude of excitation for super-

harmonic resonance.

Λ0 0.5 1 1.5 2 2.5 3 3.5

0

0.5

1

1.5

2

2.5

3

3.5

4

a

Figure 17. The amplitude of the response of the clamped-

hinged beam versus the amplitude of excitation for super-

harmonic resonance.

σ0 10 20 30 40 50 60 70 80 90 100

0123456789

10

a

Figure 18. The frequency-response of the hinged-hinged beam

for sub-harmonic resonance.

a

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

4

Λ

Figure 15. The amplitude of the response of the hinged-hinged

beam versus the amplitude of excitation for super-harmonic

resonance.

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under consideration. The figure indicates that like theprimary and super harmonic resonances, for a givenvalue of the amplitude of the excitation, the amplitudeof the sub-harmonic response oscillations increases ash=l decreases. Moreover, the boundaries of the regionwhere the nontrivial response exists are depicted in

Figures 24 to 26, respectively, for the three types ofthe supports. The sub-harmonic nontrivial responsescan exist only in the right side of the depicted curves.From the figures, it can be inferred that as h=ldecreases, the region of the possible responses alsodecreases.

20

0 50 100 150 200 25002468

1012141618

a

σ

Figure 19. The frequency-response of the clamped-clamped

beam for sub-harmonic resonance.

14

0 50 100 1500

2

4

6

8

10

12

a

σ

Figure 20. The frequency-response of the clamped-hinged

beam for sub-harmonic resonance.

Λ

a

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

1.61.41.210.80.60.40.20–0.20

Figure 21. The amplitude of the response of the hinged-hinged

beam versus the amplitude of excitation for sub-harmonic

resonance.

Λ

a

7

6

5

4

3

2

1

021.510.50

Figure 22. The amplitude of the response of the clamped-

clamped beam versus the amplitude of excitation for sub-harmo-

nic resonance.

Λ–0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

a

Figure 23. The amplitude of the response of the clamped-

hinged beam versus the amplitude of excitation for sub-harmonic

resonance.

60

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

263

4α ηΛ

ση

Figure 24. The region where the nontrivial response of the

hinged-hinged beam in sub-harmonic resonance exists.

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It is noted there is no couple-stress based simulationof the nonlinear forced vibration of microbeams in theliterature and this work is the first attempt in investi-gating this subject. However, since the couple stresstheory is degenerated into the classical continuumtheory in the special case of l ¼ 0, here the results ofour formulation are compared with an available simu-lation in the literature in this special case. Bauchau andBottasso (1994) used the space-time perturbationmodes to investigate some aspects of the nonlinearforced vibration of beams based on the classical con-tinuum theory. As a numerical result, they reported thefrequency-response of a clamped-clamped nonlinearbeam subjected to an external harmonic force in theabsence of viscous damping. We have considered thesame case with geometrical and material specificationsin accordance with what considered by Bauchau andBottasso (1994). In Figure 27, the numerical results ofthe formulations of our work with l ¼ 0 is comparedwith those of Bauchau and Bottasso (1994) for the con-sidered case. It is observed that there is a good agree-ment with the results of this work and that of Bauchauand Bottasso (1994).

4. Conclusion

In this paper, the modified couple stress theory isemployed to study the size-dependent nonlinearforced oscillations of Euler-Bernoulli microbeams.The considered nonlinearity is due to the large deflec-tions and the consequent mid-plane stretching. Themotivation for utilizing the couple stress theory istaking into account the small scale effects in the behav-ior of microbeams. First, the nonlinear governing par-tial differential equations have been obtained. Then, theGalerkin method has been used to derive the associatednonlinear ordinary differential equation for beams withthree different end-support conditions, includinghinged-hinged, clamped-clamped, and hinged-clampedcases. The derived ordinary differential equation is aDuffing type with cubic nonlinearity. The beams havebeen assumed to be under a harmonic concentratedtransverse force at the middle span. Numerical valueshave been obtained for different parameters involved inthe primary and secondary resonances.

The results show that in the cases of very thin micro-beams, there are significant differences between the pre-dictions of the couple stress theory and the classicalcontinuum theory for the nonlinear forced vibrationresponse parameters.

Notation

List of English symbols

A Cross section areaa Amplitude of responseB Viscous damping coefficientb Beam widthE Young’s modulus

F x, tð Þ Lateral force per unit length exerted to the

beam

80

0 5 10 15 20 250

10

20

30

40

50

60

70

263

4α ηΛ

ση

Figure 25. The region where the nontrivial response of the

clamped-clamped beam in sub-harmonic resonance exists.

80

0 5 10 15 20 250

10

20

30

40

50

60

70

263

4α ηΛ

ση

Figure 26. The region where the nontrivial response of the

clamped-hinged beam in sub-harmonic resonance exists.

2.5E–3

2E–3

1.5E–3

–0.10 –0.075 –0.05 –0.025 0 0.025 0.05 0.075 0.100

0.5E–3

1E–3

a

σ

Figure 27. The frequency-response of the clamped-clamped

beam subjected to an external harmonic force in the absence

of damping.

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G x, tð Þ Axial force per unit length exerted to the

beamh Beam thicknessI Area moment of inertia of beam cross

sectionK Dimensionless amplitude of excitationL Beam lengthl Length scale parameter�M Total moment acting on the beam end

sectionsm Deviatoric part of couple stress tensor�N Axial force acting on the beam end sectionsN Variation of the axial force in the beam with

respect to the initial configuration due to the

extension caused by the transverse deflectionN0 Axial load exerted to the beam~N0 Dimensionless axial load exerted to the beamP0 Amplitude of the concentrated transverse

external harmonic forceq Time part of the beam dimensionless lateral

deflectionT Kinetic energy of the beamt TimeU Total strain energy of the beam

Ubs Beam strain energy due to bending and the

change of the stretch with respect to the ini-

tial configurationUis Stored energy of the beam with respect to

the initial configuration due to the existence

of initially axial loadu Displacement field vectoru Axial displacement of the centroid of beam

sections�u Strain energy density�V Transverse force acting on the beam end

sectionsW Work done by external forces and couplesw Lateral deflection of the beam~w Dimensionless lateral deflection of the beamx Longitude coordinate~x Dimensionless longitude coordinatez Lateral coordinate

List of Greek symbols

� Dimensionless coefficient of the nonlinear

term of the beam governing ordinary differ-

ential equation� Dirac delta function�ij Kronecker deltae Strain tensor" A small dimensionless parameter with a

value in the order of the amplitude of the

motion

Dimensionless damping coefficienth Rotation vector

� Dimensionless amplitude of excitation forsecondary resonances

� Lame constant� Shear modulusv Poisson ratio Material densityr Stress tensor� Detuning parameter Dimensionless time� First mode shape of the beamc Symmetric part of curvature tensor Angle of rotation of the beam cross-sections� Frequency of excitation~� Dimensionless frequency of excitation

Funding

This research received no specific grant from any fundingagency in the public, commercial, or not-for-profit sectors.

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