fer

argctsobeialhe

d tos uresf thnt10.

I

r

ie

tr

c

a

ps

tha

v

gFing perturbation method 13. While, both shear deformation androtary inertia were considered, it was restricted to the pinnedboundary condition.

c

a

dppFm

ps

dtIa

w

x= + 2

p1v

JIn this paper, the large amplitude free vibration of a doublylamped Timoshenko beam, which includes shear deformationnd rotary inertia effects, is investigated. The governing partialifferential equation is obtained by the application of Hamiltonsrinciple. Then, the Galerkin technique is applied to convert theartial differential equation to an ordinary differential equation.inally, one of the most powerful perturbation methods, i.e., theethod of multiple scales is used to determine a second-order

erturbation solution. Then the developed theory is applied totudy a doubly clamped microbeam. The results give a basic un-erstanding of the influence of shear deformation and rotary iner-ia on the nonlinear frequencies of a doubly clamped microbeam.n this way, the method used by Foda 13 in macro-scale has beenpplied to another set of boundary conditions in micro-scale. To

where is the rotation of the beam cross section due to bendingand is the shear angle. The shear force is given by

V = kAG 3where A is the cross-sectional area of the beam, G is the shearmodulus and k is the shear correction factor that depends only onthe geometric properties of the cross section of the beam.

To apply Hamiltons principle, first the kinetic energy functionT and the strain energy function U of the beam should be ob-tained. They can be written as

T =120

L m wt2 + m u

t2 + mr2

t2dx

U =120

L EI x2 + kAG2 + EAx2dx 4

where I is the second moment of area of the cross section withrespect to the bending axis, L is the length of the beam, m is the

Contributed by the Technical Committee on Vibration and Sound of ASME forublication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April2, 2005; final manuscript received January 14, 2006. Assoc. Editor: Sotirios Natsia-as.

ournal of Vibration and Acoustics OCTOBER 2006, Vol. 128 / 611Copyright 2006 by ASMEAsghar RamezaniSchool of Mechanical Engineering,

Sharif University of Technology,P.O. Box 11365-9567, Tehran, Iran

e-mail: a.r@mech.sharif.edu

Aria AlastyCenter of Excellence in Design, Robotics,

and Automation (CEDRA),School of Mechanical Engineering,

Sharif University of Technology,Tehran, Iran

e-mail: aalasti@sharif.edu

Javad AkbariSchool of Mechanical Engineering,

Sharif University of Technology,Tehran, Iran

e-mail: akbari@sharif.edu

Effects oShear DFree VibIn this paper, the lconsidered. The effevibration of the micrin deriving the parttioned conditions. Tequation is converteof multiple scales iobtained ODE. Thenatural frequency oertia have significamicrobeams. DOI:

ntroductionThe Euler-Bernoulli formulation for a thin beam vibrating at

elatively low frequencies is sufficient. However, effects of rotarynertia and shear deformation are not negligible for thick beams orven thin beams that are vibrating at high frequencies. Due toheir dimensions, resonance frequencies of micro- and nano-scaleesonators are extremely high, namely in the range of kHz to GHz1. It has been observed experimentally that micro- and nanome-hanical resonators tend to behave nonlinearly at very modestmplitudes 25. This nonlinear behavior has already been ex-loited to achieve mechanical signal amplification and noisequeezing 6,7.

While there is an ample literature on the large amplitude vibra-ions of Euler-Bernoulli beams 811, only a few contributionsave considered the effects of shear deformation and rotary inertia12,13. Abramovich investigated the influence of compressivexial loads on the linear frequencies of Timoshenko beams havingarious boundary conditions 12. This formulation, however, ne-lected the joint action of rotary inertia and shear deformation.oda studied the nonlinear vibrations of a Timoshenko beam us-Rotary Inertia andformation on Nonlinearation of Microbeamse amplitude free vibration of a doubly clamped microbeam isof shear deformation and rotary inertia on the large amplitudeam are investigated. To this end, first Hamiltons principle is useddifferential equation of the microbeam response under the men-n, implementing the Galerkins method the partial differential

an ordinary nonlinear differential equation. Finally, the methodsed to determine a second-order perturbation solution for theults show that nonlinearity acts in the direction of increasing thee doubly clamped microbeam. Shear deformation and rotary in-effects on the large amplitude vibration of thick and short1115/1.2202167

the best knowledge of the authors, this is the first time that theeffects of rotary inertia and shear deformation on nonlinear vibra-tion of microbeams are investigated.

Derivation of Governing EquationsA doubly clamped beam with uniform cross section made of a

homogenous isotropic material with negligible damping is consid-ered. The case where the amplitude of vibration is of the sameorder as the thickness of the beam is presented. At this situation,due to large displacement of the beam centerline, the elementarylinear beam theory must be extended in order to include the non-linear effect of the stretching that occurs at the centerline. There-fore, the proper strain-displacement relations are

x =u

x+

12 wx

2x =

2w

x21

where u and w are the axial and transverse displacements, respec-tively, and x is the axial coordinate of the beam. Also, x and xare the normal strain and curvature at the centerline, respectively.

Considering the existence of shear deformation, the slope of thebeam centerline maybe written as

mass per unit length and r is the radius of gyration of the beamcross section i.e., r2= I /A. The mass moment of inertia is re-p 2

Leg

T

Ivm

r

N

w

TEtdrrt

N

d

w

iBt

4w

x4+

m2L4

EI2w

t2

mr22L2

EI 1 + EkG 4w

x2 t2+

m2r24L4

kAGEI4w

t4

6laced by mr .On the basis of the Hamiltons principle, the variations of the

agrangian t0t TUdt=0, where t is the time, provide the gov-

rning equations as well as the boundary conditions. The resultingoverning equations are

xEI

x + kAG w

x mr22

t2= 0 5

m2w

t2

xkAG w

x

xEA u

x+

12 wx

2 wx = 0

6

m2u

t2

xEA u

x+

12 wx

2 = 0 7he boundary conditions for a doubly clamped beam are

w = 0 and = 0 at x = 0,L 8n general, the axial inertia is small compared to both the trans-erse inertia and the rotary inertia 14. Neglecting the term2u /t2, Eq. 7 can be integrated with respect to x and the

esult substituted into Eqs. 5 and 6 to give

EI2

x2+ kAG w

x mr22

t2= 0 9

m2w

t2 kAG 2w

x2

x N2w

x2= 0 10

ow, the axial force N can be given by

N = N0 +EA2L0

L wx2dx 11

here N0 is the pre-tension in the beam, if any. Combining Eqs.9 and 10 results in

EI4w

x4+ m

2w

t2 mr2 + mEIkAG

4w

x2 t2+

m2r2

kAG4w

t4

N 2wx2

EIkAG

4w

x4+

mr2

kAG4w

x2 t2 = 0 12

he first and second terms in Eq. 12 correspond to classicaluler-Bernoulli beam model. The third term represents the correc-

ion for rotary inertia while the fourth term represents the sheareformation effect. The fifth term represents the joint effect ofotary inertia and shear deformation. The last bracketed terms rep-esent the effect of the axial force N that causes nonlinearity dueo large amplitudes.

ondimensionalization and Model ReductionIn general, it is more convenient and efficient to work with

imensionless quantities. Here, these can be introduced as

t = t x = x/L w = w/L 13

here

= 2EI/m 14s the linear frequency for a corresponding doubly clamped Euler-ernoulli beam. Substitution of Eq. 13 into Eq. 12 and use of

he chain rule of differentiation results in

12 / Vol. 128, OCTOBER 2006

NL2

EI 2w x2 EIkAGL2 4w x4 + mr22kAG 4w x2 t2 = 0 15where

N = N0 +EA2 0

1 w x2dx 16

Hereinafter the caret on all the variables is dropped for notationalconvenience.

Next, the solution for the nth mode is approximated as

wx,t = x qt 17where x is the mode shape of the corresponding doublyclamped Euler-Bernoulli beam and has a unit value at the mid-point of the beam i.e., 0.5=1. Substituting Eq. 17 into Eq.15, one obtains after some manipulations

. q +1

r24 kGEr2 kGEL21 + EkG N0EAL2

12L2

q20

1

2dxq + 1r48 kGEL4 N0kGE2Ar2L2+

N0EAL4

q 12r4801

2dx kGEr2L2

1L4q3 = 0 18

where the prime and dot denote derivation with respect to x and t,respectively. In order to reduce Eq. 18 to an ordinary differentialequation, averaging over the space variable Galerkins method isapplied. Multiplying both sides of Eq. 18 in and integratingover the interval of 0,1 results in

q + 1 + 2q2q + 3q + 4q3 = 0 19where the coefficients ii=1, . . . ,4 are

1 =1

f1r24E kGr2 f1 kGL2 1 + EkG f2 N0AL2 f22 =

12f1r24L2

f2f320

3 =1

f1r48E kGL4 f4 N0kGEIL2 f2 + N0AL4 f44 =

f32f1r48 kGEr2L2 f2 1L4 f4

where f is i=1, . . . ,4 are

f1 =0

1

2dx f2 =0

1

. dx f3 =0

1

2dx

21

f4 =0

1

. dx

Equation 19 represents a system with inertia and stiffness cubicnonlinearities. Four initial conditions are specified as

q0 = wmax/L q0 = q0 = q0 = 0 22where wmax is the midpoint amplitude of the beam.

Transactions of the ASME

Application of the Method of Multiple ScalesAt this step a second-order uniform expansion for the solution

o

wa

=

d

I

w

So

T

wtc

wffltbd

E

w

A1 = A1T2 and A2 = A2T2 35Consequently, the right hand side of Eq. 28 is zero and

Jf Eq. 19 is to be obtained using the method of multiple scales14,15 in the form

qt = . q1T0,T1,T2 + 2q2T0,T1,T2 + 3q3T0,T1,T223

here the small dimensionless parameter is the order of themplitude of vibration. Expansion of qt is performed around q0 which is the linear solution of 19. Also, the time scales areefined as

T0 = t T1 = t T2 = 2t 24t follows that the derivatives with respect to t become

ddt

= D0 + D1 + 2D2 +

d2

dt2= D0

2 + 2D0D1 + 2D12 + 2D0D2 +

d4

dt4= D0

4 + 4D03D1 + 26D02D12 + 4D03D2 + 25

here

Dj =

Tjj = 0,1,2 26

ubstituting Eqs. 23 and 25 into 19 and equating coefficientsf like powers of , one obtains

D04q1 + 1D0

2q1 + 3q1 = 0 27

D04q2 + 1D0

2q2 + 3q2 = 4D03D1q1 21D0D1q1 28

D04q3 + 1D0

2q3 + 3q3 = 6D02D12 + 4D03D2q1 1D12

+ 2D0D2q1 2q12D0

2q1 4D03D1q2

21D0D1q2 4q13 29

he general solution of Eq. 27 is

q1T0,T1,T2 = A1T1,T2ei1T0 + A2T1,T2ei2T0 + c . c. 30here c.c. stands for the complex conjugate of the preceding

erms. The linear frequencies 1 and 2, i.e., the natural frequen-ies of q +1q+3q=0 are

12

=

12

124

3 31

22

=

12

+124

3 32

hich indicate that there are two sinusoidal modes of differentrequencies corresponding to the same spatial mode. The smallerrequency 1 is associated with bending deformation, and thearger one 2 is associated with shear deformation. The exis-ence of these frequencies, which is the most important differenceetween a Timoshenko and an Euler-Bernoulli beam, has beenemonstrated experimentally 16.

Substituting Eq. 30 into 28 yields

D04q2 + 1D0

2q2 + 3q2 = 2i1212

1D1A1ei1T0 + 2i2222

1D1A2ei2T0 + cc 33limination of secular terms implies that

D1A1 = 0 and D1A2 = 0 34hich gives

ournal of Vibration and Acousticsq2 = 0 36Substituting Eqs. 30 and 36 into Eq. 29 yields

D04q3 + 1D0

2q3 + 3q3 = 2i1212

1D2A1 + 3212

4A12A 1 + 221

2 + 222

34A1A2A 2ei1T0 + 2i2222

1D2A2 + 3222

4A22A 2 + 222

2

+ 212 34A1A 1A2ei2T0 + 21

2

4A13e3i1T0 + 22

2 4A2

3e3i2T0

+ 212 + 22

2 34A1A22ei1+22T0

+ A1A 22ei122T0 + 22

2 + 212

34 A12A2ei21+2T0

+ A12A 2ei212T0 + c . c. 37

where A j is the complex conjugate of Aj.Elimination of secular terms in q3 implies that

2i1212

1D2A1 + 3212

4A12A 1 + 221

2 + 222

34A1A2A 2 = 0 38

2i2222

1D2A2 + 3222

4A22A 2 + 222

2 + 212

34A1A 1A2 = 0 39Now A1 and A2 are expressed in polar forms, i.e.,

A1 =12a1T2e

i1T2 A2 =12a2T2e

i2T2 40where aj and j are real.

Substituting Eq. 40 into Eqs. 38 and 39, separating realand imaginary parts and solving the resulting differential equa-tions, it follows that a1 and a2 are constants and

1 = 1T2 + 10 2 = 2T2 + 20 41where 10 and 20 are constants and

1 =321

2 4a1

2 + 2212 + 22

2 34a22

81212

1

2 =322

2 4a2

2 + 2222 + 21

2 34a12

82222

142

Returning to 40, one finds

A1 =12a1 expi

21t + i10 A2 =12a2 expi

22t + i20 43where T2=2t is used. Solving for q3 with retaining only the par-ticular solutions and combining the resulting expressions for q1,q2 and q3 gives

q = a1 cos1t + 10 + a2 cos2t + 20 + 3C1a13 cos31t

+ 310 + C2a23 cos32t + 320 + C3a1a2

2 cos1 + 22t

+ 10 + 220 + C4a1a22 cos1 22t + 10 220

+ C5a12a2 cos21 + 2t + 210 + 20 + C6a1

2a2 cos21 2t + 210 20 44

where the nonlinear frequencies are

j = j + 2 j + O3 j = 1,2 45

and the coefficients Cj are

OCTOBER 2006, Vol. 128 / 613

C1 =21

2 4

48114

9112 + 3

T

ltg

wt

Ai

EAa

wi

wt

T

w

Table 1 The parameters 02 and in Eq. 48 for Euler-Bernoulli, Rayleigh and shear beams

6C2 =22

2 4

48124

9122 + 3

C3 =21

2 + 222 34

41 + 224 11 + 222 + 3

C4 =21

2 + 222 34

41 224 11 222 + 3

C5 =22

2 + 212 34

421 + 24 121 + 22 + 3

C6 =22

2 21

2 34421 24 121 22 + 3

46

he satisfaction of the initial conditions 22 requires that

a1 =2

2wmax

22

12L

a2 =1

2wmax

12

22L

10 = 20 = 0 47

Now, we consider the special cases of Euler-Bernoulli, Ray-eigh no shear deformation and shear beams. In all of these caseshere is only one linear natural frequency 0 and Eq. 19 de-enerates into the general form:

q + 02q + q3 = 0 48

here 02 and are according to the Table 1. The initial condi-

ions are

q0 = wmax/L q0 = 0 49pplication of the method of multiple scales results in the follow-

ng solvability conditions

D1A = 0 50

2i0D2A + 3A2A = 0 51quation 50 implies A=AT2. Expressing A in polar form, i.e.,=

12aT2e

iT2, substituting it into Eq. 51 and separating realnd imaginary parts results in

0a = 0

0 +38 a

2= 0 52

here the primes indicate differentiation with respect to T2. Solv-ng Eq. 52 gives

a = a0

=3a2

80T2 + 0 53

here a0 and 0 are constants determined from the initial condi-ions as

0 = 0

a0 = wmax/L 54herefore,

q = a0 cost + 0 + O3 55here, the nonlinear natural frequency is

= 0 +32a2

8056

14 / Vol. 128, OCTOBER 2006Vibration of a MicrobeamA doubly clamped microbeam with rectangular cross section,

made of silicon undergoing flexural vibration is considered. Themicrobeam geometric and material properties are given in Table 2.

Since the linear Euler-Bernoulli beam theory is widely used byresearchers, for comparison the ratio of the nonlinear frequency,, of different beams to the linear frequency, 0, of Euler-Bernoulli beam has been computed and plotted for different slen-derness ratios and maximum amplitude-thickness ratios. In all ofthe figures, the frequencies correspond to the first spatial mode ofvibration. In addition, for Timoshenko beam the lower fundamen-tal frequency 1 is used in the figures.

In Fig. 1, the slenderness ratio, L /r, is 10, which corresponds to

Table 2 Microbeam geometric and material properties

Fig. 1 Nonlinear frequency of free vibration of a doublyclamped microbeam for L /r=10

Transactions of the ASME

Lfientnctmi

es

fiqbib

tcwe

C

mfmlnsttttmc

Fc

J/h=2.8867, where h is the thickness of the microbeam. Thegure shows that the effects of shear deformation and rotary in-rtia are significant for thick microbeams. Therefore, the nonlinearatural frequency which is predicted from Euler-Bernoulli beamheory deviates considerably from the actual one. Comparing theonlinear frequencies of shear and Rayleigh beams, it can be con-luded that the effect of shear deformation is more significant thanhe effect of rotary inertia on the large amplitude vibration of

icrobeams. However, as the midpoint amplitude-thickness rationcreases, the effect of rotary inertia becomes more pronounced.

In Fig. 2, the slenderness ratio is 20 i.e., L /h=5.7735. Theffects of shear deformation and rotary inertia are significant formall midpoint amplitude-thickness ratios.

In Fig. 3, the slenderness ratio is 50 i.e., L /h=14.4338. Thegure presents that for slender microbeams, the nonlinear fre-uency of Rayleigh, Euler-Bernoulli, Shear, and Timoshenkoeams are almost equal. Hence, as expected, the effects of rotarynertia and shear deformation can be neglected for slender micro-eams.

Finally, all of the three figures show that increasing the vibra-ion amplitude intensifies the stretching effect which leads to in-rease of the nonlinear frequency of vibration for the microbeamsith different slenderness ratios. Therefore, application of the lin-

ar theory in these cases results in erroneous predictions.

onclusionThe large amplitude vibration corresponding to the first spatialode of a doubly clamped microbeam, which includes shear de-

ormation and rotary inertia effects, was studied. The method ofultiple scales was used to find an approximate perturbation so-

ution for the resulting nonlinear equation. It was observed that aonlinear model results in higher natural frequencies for the con-idered doubly clamped microbeam. The effects of shear deforma-ion and rotary inertia are significant and cannot be neglected inhe case of thick and short microbeams undergoing large ampli-ude vibrations. Therefore, when the theory of beams is used forhe study of micro- and nanomechanical structures, shear defor-

ation and rotary inertia effects should be considered for an ac-urate dynamic analysis.

ig. 2 Nonlinear frequency of free vibration of a doublylamped microbeam for L /r=20

ournal of Vibration and AcousticsReferences1 Huang, X. M. H., Zorman, C. A., Mehregany, M., and Roukes, M. L., 2003,

Nanodevice Motion at Microwave Frequencies, Nature London, 421, pp.496496.

2 Turner, K. L., Miller, S. A., Hartwell, P. G., MacDonald, N. C., Strogatz, S. H.,and Adams, S. G., 1998, Five Parametric Resonances in a Microelectrome-chanical System, Nature London, 396, pp. 149152.

3 Craighead, H. G., 2000, Nanoelectromechanical systems, Science, 290, pp.15321535.

4 Scheible, D. V., Erbe, A., and Blick, R. H., 2002, Evidence of a Nanome-chanical Resonator Being Driven into Chaotic Response via the Ruelle-TakensRoute, Appl. Phys. Lett., 81, pp. 18841886.

5 Zhang, W., Baskaran, R., and Turner, K. L., 2002, Effect of Cubic Nonlin-earity on Autoparametrically Amplifed Resonant MEMS Mass Sensor, Sens.Actuators, A, 102, pp. 139150.

6 Rugar, D., and Grtter, P., 1991, Mechanical Parametric Amplification andThermomechanical Noise Squeezing, Phys. Rev. Lett., 67, pp. 699702.

7 Carr, D. W., Evoy, S., Sekaric, L., Craighead, H. G., and Parpia, J. M., 2000,Parametric Amplification in a Torsional Microresonator, Appl. Phys. Lett.,77, pp. 15451547.

8 Sarma, B. S., Varadn, T. K., and Parathap, G., 1988, Various Formulations ofLarge Amplitude Free Vibrations of Beams, Comput. Struct., 29, pp. 959966.

9 Singh, G., Sharma, A. K., and Rao, G. V., 1990, Large-Amplitude Free Vi-brations of Beams-a Discussion on Various Formulations and Assumptions, J.Sound Vib., 142, pp. 7785.

10 Ozkaya, E., and Pakdemirli, M., 1997, Nonlinear Vibrations of a Beam-MassSystem under Different Boundary Conditions, J. Sound Vib., 199, pp. 679696.

11 Rao, B., 1992, Large-Amplitude Vibrations of Simply Supported Beams withImmovable Ends, J. Sound Vib., 155, pp. 523527.

12 Abramovich, H., 1992, Natural Frequencies of Timoshenko Beams underCompressive Axial Loads, J. Sound Vib., 157, pp. 183189.

13 Foda, M. A., 1999, Influence of Shear Deformation and Rotary Inertia onNonlinear Free Vibration of a Beam with Pinned Ends, Comput. Struct., 71,pp. 663670.

14 Neyfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillation Wiley Inter-science, New York.

15 Neyfeh, A. H., 1981, Introduction to Perturbation Techniques Wiley Inter-science, New York.

16 Abramovich, H., and Elishakoff, I., 1987, Application of the Kreins Methodfor Determination of Natural Frequencies of Periodically Supported BeamBased on Simplified Bresse-Timoshenko Equations, Acta Mech., 66, pp. 3559.

Fig. 3 Nonlinear frequency of free vibration of a doublyclamped microbeam for L /r=50

OCTOBER 2006, Vol. 128 / 615