E ect of size dependency on in-plane vibration of circular ...

Scientia Iranica B (2017) 24(4), 1996{2008

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

E�ect of size dependency on in-plane vibration ofcircular micro-rings

A. Karimzadeha, M.T. Ahmadianb;� and M. Rahaeifardc

a. School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran.b. Center of Excellence in Design, Robotics and Automation, Sharif University of Technology, Tehran, Iran.c. Department of Mechanical Engineering, Golpayegan University of Technology, Golpayegan, Iran.

Received 19 December 2015; accepted 17 April 2017

KEYWORDSMicro-ring;Modi�ed couple stresstheory;Resonant frequency.

Abstract. In this paper, based on the modi�ed couple stress theory, the size-dependentdynamic behavior of circular rings on elastic foundation is investigated. The ring is modeledby Euler-Bernoulli and Timoshenko beam theories, and Hamilton's principle is utilized toderive the equations of motion and boundary conditions. The formulation derived is ageneral form of the equation of motion of circular rings and can be reduced to the classicalform by eliminating the size-dependent terms. On this basis, the size-dependent naturalfrequencies of a circular ring are calculated based on the non-classical Euler-Bernoulliand Timoshenko beam theories. The �ndings are compared with classical beam theories.Response of the micro-ring under application of static and dynamic loads is investigatedand compared with the classical theories. Results show that when the thickness of the ringis in the order of the length scale of the ring material, the natural frequencies evaluatedusing the modi�ed couple stress are considerably more than those predicted based on theclassical beam theories, while the de ection and natural frequencies of the classical andnon-classical beam theories approach one another for the rings with thickness much largerthan the material length scale.

© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

Circular rings are widely used as micro-gyroscopes inmicro-electromechanical systems [1,2]. The key blockof micro-gyroscopes is a vibrating ring actuated byelectrostatic or electromagnetic forces [3,4]. By sensingthe lateral vibration of the ring, the angular velocityof the device is calculated. Many researchers havestudied the vibration and natural frequencies of circularrings [5,6]. For instance, Chang et al. [5] derived theequations of motion of a circular ring in polar coordi-

*. Corresponding author. Tel: +98 21 66165503;Fax: +98 21 66000021E-mail address: [email protected] (M.T. Ahmadian)

nate using energy method. They used their formulationto calculate the natural frequency and mode shapes ofa silicon micro-scale circular ring.

Wu and Parker [6] used perturbation method toderive an analytic expression for natural frequencyof circular rings on elastic foundations. Cooley andParker [7] investigated the vibration behavior of rotat-ing circular rings. They derived the nonlinear equa-tions of motion of the ring using Hamilton's principleand linearized these equations to obtain the naturalfrequency of the ring. Patel et al. [8] analyzed thevibration of isotropic and orthotropic circular rings.They considered the nonlinearity caused by the largedeformation of the ring and used �nite-element methodto study the mechanical behavior of the ring.

A. Karimzadeh et al./Scientia Iranica, Transactions B: Mechanical Engineering 24 (2017) 1996{2008 1997

Rings used in micro-gyroscopes have thicknessesin the order of microns. In the experiments carried outon the micro-scale structures, the size dependency isobserved in mechanical behavior of these systems. Theresponse of these components is signi�cantly di�erentfrom that predicted by the classical theory. Performedexperiments revealed that the normalized behavior ofmicro-mechanical structures is size-dependent, while,according to the classical theory, mechanical responseis independent of the structure size. For example, someexperiments are conducted to examine the mechani-cal behavior of micro-wires made of copper [9] andpolymeric micro-cantilevers [10]. Outcome of theseinvestigations a�rms the size-dependent behavior ofthese components, and shows that torsional sti�nessof micro-wires and bending sti�ness of polymeric can-tilevers are much more than the sti�ness predicted bythe classical theory. The size dependency in mechanicalbehavior of micro structures was also reported by otherresearchers [11-13].

In contrast to the classical theory which is not ableto justify the mechanical response of the micro-scalestructures, non-classical theories, such as couple stresstheory [14-16] and modi�ed couple stress theory [17-19], are appropriately able to explain this phenomenonand cover the gap between the experimental and clas-sical theory results. Couple stress theory as a theoryis capable of explaining the size-dependent behaviorof mechanisms proposed for the �rst time by Mindlinand Tiersten [14]. In this theory, new stresses areintroduced and considered acting on material element.These stresses were related to higher order derivativesof displacement �eld with two new constants. Thestresses are called higher order stresses, and the newmaterial parameters used in the relation are known aslength-scale parameters. A new equilibrium equation,named equilibrium equation of moment of couples, isconsidered and applied to the relations by Yang etal. [17]. Using this modi�cation, Yang et al. [17]presented the modi�ed couple stress theory.

In recent years, many researchers utilized themodi�ed couple stress theory to analyze the staticand dynamic responses of mechanical components suchas micro-beams [20-23] and micro-plates [24,25]. Themodi�ed couple stress formulations of functionallygraded Euler-Bernoulli and Timoshenko beams weredeveloped by Asghari et al. [26,27]. The modi�edcouple stress-based formulation of laminated Timo-shenko beams was developed by Chen and Li [28].Fakhrabadi et al. [29] analyzed the nonlinear vibrationof carbon nanotubes based on modi�ed couple stresstheory. Tsiatas [30] utilized the modi�ed couplestress theory to develop the size-dependent formulationof Kirchho� plates. The formulations of laminatedReddy and Kirchho� micro plates derived based onthe modi�ed couple stress theory were presented by

Chen et al. [31,32]. Static and vibrational analyses ofnon-classical third-order shear deformation plate wereinvestigated by Gao et al. [33].

The mechanical behavior of uid conveying micro-tube was investigated based on the modi�ed couplestress theory by Wang [34]. He derived the naturalfrequency of the system and calculated the critical ow velocity. On the basis of modi�ed couple stresstheory, sensitivity analysis of atomic force microscopewas investigated by Kahrobaiyan et al. [35]. The size-dependent static and dynamic behaviors of electrostati-cally actuated micro-beams were studied by Rahaeifardet al. [36]. They showed that the modi�ed couple stresstheory can �ll the gap between the experimental andtheoretical results of the static pull-in of the micro-cantilevers.

In this paper, governing equations of motion andboundary conditions of a circular micro-ring on theelastic foundation are derived on the basis of modi�edcouple stress theory and Hamilton's principle. Euler-Bernoulli and Timoshenko beam theories are consid-ered to obtain two sets of micro-ring formulations. Asan example, the size-dependent natural frequencies ofcircular ring are calculated and compared with theresults obtained by the classical theory. Moreover,static and harmonic point forces are applied to themicro-ring and the response is obtained. Findingsindicate that for the rings with thickness of the orderof the material length-scale parameter, the di�erencebetween the non-classical and classical results is no-table. Furthermore, for the rings with the radius muchmore than the thickness, the results of the Timoshenkobeam theory converge to those predicted by the Euler-Bernoulli theory.

2. Modeling of circular rings

A simple model of the circular ring on elastic founda-tion (radial and tangential sti�ness) considered in thispaper and utilized in Euler-Bernoulli and Timoshenkotheory is illustrated in Figure 1.

2.1. Euler-Bernoulli ringsThe displacement �eld based on Euler-Bernoulli beamtheory in cylindrical coordinate system can be ex-pressed as follows:8><>:ur = w

u� = u+ rRu� r

R@w@�

uz = 0(1)

where w(�; t) is displacement in radial direction, u(�; t)is displacement in tangential direction, R is ring radius,and r denotes the distance of an arbitrary point fromthe neutral axis of the cross-section of the ring. Thein�nitesimal strains associated with the displacement

1998 A. Karimzadeh et al./Scientia Iranica, Transactions B: Mechanical Engineering 24 (2017) 1996{2008

Figure 1. A circular micro-ring: (a) Geometry and parameters, and (b) cross section of the ring.

�eld are computed as [37]:

"��=1R@u�@�

+urR

=1R

�w+

@u@�

+rR

�@u@�� @2w@�2

��;(2)

"rr = "r� = "rz = "�z = "zz = 0: (3)

Moreover, the components of the symmetric curvaturetensor can be expressed in terms of the displacementcomponents as [37]:

�sz� = �s�z =1

2R2

�@u@�� @2w@�2

�; (4)

�srr = �sr� = �srz = �s�� = �szz = 0: (5)

Classical and higher-order stresses �ij and msij are also

obtained as follows [37]:

�� =ER

�w +

@u@�

+rR

�@u@�� @2w@�2

��; (6)

�rr = �r� = �rz = ��z = �zz = 0; (7)

msz� = ms

�z =�l2

R2

�@u@�� @2w@�2

�; (8)

msrr = ms

r� = msrz = ms

�� = mszz = 0: (9)

The variation of potential energy due to a variation inthe body con�guration can be expressed as [17]:

�U =yV

(��"�� + 2mz��z�) dV

=Z 2�

0

xA

��R

��w + �

@u@�

+rR�@u@�� rR�@2w@�2

�+ 2mz�

�1

2R2

��@u@�� @2w

@�2

��+Krw�w

+K�u�u�RdAd�; (10)

where V and A represent the volume and cross-sectionof the micro-ring, and kr and k� are radial andtangential sti�ness of elastic foundation, respectively.To simplify the derivation of governing equations, stressresultants are de�ned as follows:

P =xA

��dA; (11)

M =xA

��rdA; (12)

Hz� =xA

mz�dA; (13)

which gives:

�U=Z 2�

0

�PR�w +

PR�@u@�

+MR2 �

@u@�� MR2 �

@2w@�2

+Hz�

R2 �@u@��Hz�

R2 �@2w@�2 +Krw�w+K�u�u

�Rd�:

(14)

Performing some mathematical manipulations yields:

�U =Z 2�

0

�PR�w � 1

R@P@�

�u� 1R2

@M@�

�u

� 1R2

@2M@�2 �w � 1

R2@Hz�

@��u

� 1R2

@2Hz�

@�2 �w�Rd� +

�PR�u+

MR2 �u

� MR2 �

�@w@�

�+

1R2

@M@�

�w +Hz�

R2 �u

� Hz�

R2 ��@w@�

�+

1R2

@Hz�

@��w +Krw�w

+K�u�u�2�

0: (15)

Total kinetic energy of the micro-ring is expressed as:


T =Z 2�

0

xA

12�

"�@ur@t

�2

+�@u�@t

�2#dV

=Z 2�

0

xA

12��

@w@t

�2

+�@u@t� rR@2w@t@�

+rR@u@t

�2�dV; (16)

in which � is the density of micro-ring material. Con-sidering the proposed displacement �eld and ignoringproduct of small components, the variation of thekinetic energy can be derived as follows:

@T = ��AZ 2�

0

�@2w@t2

�w +@2u@t2

�u�Rd�: (17)

The virtual work done by external forces on the ring iswritten in the following form:

�W =Z 2�

0[f(�; t)�w + p(�; t)�u]Rd�: (18)

Using Hamilton principle as:

�Z

(T +W � U)dt = 0; (19)

gives:Z T

0

Z 2�

0

��fR� P +

1R@2M@�2 +

1R@2Hz�

@�2

� �AR@2w@t2�Krw

��w +

�pR+

@P@�

+1R@M@�

+1R@Hz�

@�� AR@2u

@t2

�K�u��u�Rd�dt+

Z T

0

�� 1R@M@�

� 1R@Hz�

@�

��w +

��P � M

R� Hz�

R

��MR

� Hz�

R

��@w@�

��dt = 0: (20)

Performing some straightforward mathematical manip-ulations gives the governing equations of motion of theEuler-Bernoulli micro-ring as follows:

fR� P+1R@2M@�2 +

1R@2Hz�

@�2 �Krw=�AR@2w@t2

; (21)

pR+@P@�

+1R@M@�

+1R@2Hz�

@��K�u=�AR

@2u@t2

: (22)

The corresponding boundary conditions are:

1R@M@�

+1R@Hz�

@�=0 or

w=const.@�=0; 2�; (23)

P+MR

+Hz�

R=0 or

u=const.@�=0; 2�; (24)

MR

+Hz�

R= 0 or

@w@�

= const.@� = 0; 2�: (25)

Considering inextensionality of the centerline of therings, the following equation should be satis�ed [38]:

"�� = 0: (26)

Satisfying inextentionality condition of the centerlineand neglecting product of small quantities in Eqs. (2)gives the following expression for inextensional rings:

w = �@u@�: (27)

Eqs. (21) and (22) represent the governing equations ofEuler-Bernoulli micro-ring. Combining these equationsand considering inextensionality of the centerline of thering, a single di�erential governing equation is derivedas:�

1 +�Al2

EI

��@6u@�6 + 2

@4u@�4 +

@2u@�2

�+R4

EI

�@f@�

+ p�

+KrR3

EI@2u@�2 � k�R3

EIu

+�AR4

EI@2

@t2

�@2u@�2 � u

�= 0; (28)

where I is the area moment of inertia de�ned as I =Rr2dA. For convenience, the following dimensionless

parameters are introduced:

u =uR; (29)

f =f

(EIR3 ); p =

p(EIR3 )

; (30)

kr =KrR3

EI; k� =

K�R3

EI; (31)

t =�

EI�AR4

� 12

t: (32)


Using the introduced dimensionless parameters, thedimensionless governing equation can be written in thefollowing form:�

1 +�

E( �rl )2

��@6u@�6 + 2

@4u@�4 +

@2u@�2

�+

@f@�

+ p

!+�Kr

@2u@�2 � K�u

�+@2

@t2

�@2u@�2 � u

�= 0; (33)

where �r =pI=A is radius of gyration of the ring cross-

section.

2.2. Timoshenko ringsThe displacement �eld of micro-ring with consideringthe �rst-order shear deformation in cylindrical coordi-nate can be stated as follows:8><>:ur = w

u� = u+ r�uz = 0

(34)

The in�nitesimal strains associated with this displace-ment �eld and symmetric curvature tensor can berepresented in terms of the displacement variables asfollows:

"�� =1R

�w +

@u@�

+ r@�@�

�; (35)

"r� = "�r =1

2R

�@w@�� u+R�

�; (36)

"rr = "rz = "zr = "�z = "z� = "zz = 0; (37)

�sz� = �s�z =1

4R2

�@u@�� @2w@�2 +R

@�@�

�; (38)

�szr = �srz =1

4R2

�@w@�� u+R�

�; (39)

�srr = �sr� = �s�r = �s�� = �szz = 0: (40)

Moreover, the non-zero components of correspondingclassical and higher order stresses are obtained asfollows:

�� =ER

�w +

@u@�

+ r@�@�

�; (41)

�r� = ��r =k�R

�@w@�� u+R�

�; (42)

msz� = ms

�z =�l2

2R2

�@u@�� @2w@�2 +R

@�@�

�; (43)

mszr = ms

rz =�l2

2R2

�@w@�� u+R�

�; (44)

where k is numerical factor that depends on the shapeof the cross-section of the ring. The variation of strainenergy can be calculated as follows:

�U=yV

��"��+2�r��"r�+2mz��sz�

+2mzr��szr�dV =

Z 2�

0

xA

��R

��w

+�@u@�

+r�@�@�

�+�r�R

��@w@��u+R��

�+mz�

2R2

��@u@�� @2w

@�2 +R�@�@�

�+mzr

2R2

��w��

��u+R��

+Krw�w+K�u�u�RdAd�: (45)

In addition to the aforementioned stress resultants, twomore stress resultants are de�ned for Timoshenko ringsas follows:

F =xA

�r�dA; (46)

Hzr =xA

mszrdA: (47)

Then, the variation of strain energy can be rewrittenas follows:

�U=Z 2�

0

��PR�w+

PR�@u@�

+MR�@�@�

�+�FR�@w@��FR�u+F��

�+�Hz�

2R2 �@u@��Hz�

2R2 �@2w@�2 +

Hz�

2R�@�@�

�+�Krw�w

+K�u�u�+�Hzr

2R2 ��w��Hzr

2R2 �u+Hzr

2R��RdAd�:

(48)

Using integration by part yields:

�U=Z 2�

0

��PR�w�1

R@P@�

�u� 1R@M@�

��+�� 1R@F@�

�w

�FR�u+FD��

�+�� 1

2R2@Hz�

@��u� 1

2R2@2Hz�

@�2 �w

� 12R

@Hz�

@��+�� 1

2R2@Hzr

@��w�Hzr

2R2 �u+Hzr

2R��

+[Krw�w+K�u�u]�Rd�+R

��PR�u+

MR��

+FR�w�

+�Hz�

2R2 �u�Hz�

2R2 �@w@�

+1

2R2@Hz�

@��w

+Hz�

2R��+�Hzr

2R2 �w��2�

0: (49)

Considering the proposed displacement �eld for Timo-


shenko ring and ignoring product of small components,the variation of kinetic energy is obtained as follows:

�T =�Z 2�

0

��A

@2w@t2

�w + �A@2u@t2

�u

+ �I@2�@t2

��Rd�: (50)

The variation of external work is written as follows:

�W =Z 2�

0[f(�; t)�w + p(�; t)�u]Rd�: (51)

Using Hamilton's principle gives:Z T

0

Z 2�

0

��fR� P +

@F@�

+1

2R@2Hz�

@�2 +1

2R@Hzr

@�

� �AR@2w@t2�Krw

��w +

�pR+ F +

@P@�

+1

2R@Hz�

@�+Hzr

2R� �AR@2u

@t2�K�u

��u

+�@M@�� FR+

12@Hz�

@�� Hzr

2

� �IR@2�@t2

��d�dt+

Z T

0

��P +

Hz�

2R

��u

��F +

12R

@Hz�

@�+Hzr

2R

��w �

�M +

Hz�

2

��

+Hz�

2R��@u@�

��dt2�0

�= 0:

(52)

The equation of motion for the Timoshenko circularring and the corresponding boundary conditions arederived as follows:

fR�P +@F@�

+1

2R@2Hz�

@�2 +1

2R@Hz�

@��Krw

= �AR@2w@t2

; (53)

pR+F +@P@�

+1

2R@Hz�

@�+Hzr

2R�K�u

= �AR@2u@t2

; (54)

@M@�� FR+

12@Hz�

@�� Hzr

2= �IR

@2�@t2

; (55)

P +Hz�

2R= 0 or

u = const.@� = 0; 2�; (56)

F +1

2R@Hz�

@�+Hzr

2R= 0 or

w=const.@�=0; 2�; (57)

M +Hz�

2= 0 or

� = const.@� = 0; 2�; (58)

Hz�

2R= 0 or

@w@�

= const.@� = 0; 2�: (59)

Using the governing equations of Timoshenko ring andconsidering inextensionality of the centerline of thering, a single di�erential governing equation for theTimoshenko micro-ring is derived as follows:

@8u@�8

�� l2

4kR2

�+@6u@�6

��Al2

EI

�1 +

l2

4kR2

�+�

1� l2

4kR2

��+@4u@�4

�2�Al2

EI

�1 +

l2

4kR2

�+�

2 +l2

4kR2

��+@2u@�2

��Al2

EI+ 1�

�1 +

l2

4kR2

��+

@8u@�6@t2

��l2

4kE

�+

@6u@�4@t2

��R2

E

�1� l2

4kR2 +Al2

4kI

�� R2

k�

�+

@4u@�2@t2

��AR4

EI

�1 +

l2

2kR2

+EI

kA�R2

�� R2

E

�2 +

l2

4kR2

��+

@6u@�2@t4

��2R4

kE�

�� @4u@t4

��2R4

kE�

�� @2u@t2

��AR4

EI+�R2

E

��1 +

l2

4kR2

��+KrR3

EI@2u@�2 � K�R3

EIu+

R4

EI

�1 +

l2

4kR2

��p+

@f@�

��l2R2

4kEI+

R2

kA�

�@2

@�2

�p+

@f@�

�+

�R4

kEA�@2

@t2

�p+

@f@�

�= 0: (60)


Employing introduced dimensionless parameters yields:

@8u@�8

�� 1

4k�Rl

�2 �+@6u@�6

��

E� �rl

�2�1 +1

4k�Rl

�2�+�

1� 1

4k�Rl

�2��+@4u@�4

�2�

E� �rl

�2�1+1

4k�Rl

�2�+�

2 +1

4k�Rl

�2��+@2u@�2

��

E� �rl

�2 + 1�

�1 +

1

4k�Rl

�2��+@8u

@�6@t2

�1

4k�Rl

�2 �R�r

�2 �+

@6u@�4@t2

�� 1�R

�r

�2�1� 1

4k�Rl

�2+

14k� �rl

�2 +Ek�

��+

@4u@�2@t2

��1

+1

2k�Rl

�2 +E

k��R

�r

�2�� 1�R

�r

�2�2 +1

4k�Rl

�2��+�Kr

@2u@�2 � K�u

�+

@6u@�2@t4

�E

k��R

�r

�4 �� @4u@t4

�E

k��R

�r

�4 �� @2u@t2

��1 +

1�R�r

�2��1 +1

4k�Rl

�2��+�

1 +1

4k�Rl

�2� p+@f@�

!��

1

4k�Rl

�2+

E

k��R

�r

�2� @2

@�2

p+

@f@�

!+

E

k��R

�r

�4 @2

@t2

p+

@f@�

!=0: (61)

3. Static case

In this section, the response of micro-ring under staticload is investigated. For this reason, as a case study,external point loads are exerted on the ring diagonallyin orientation �=3 from the horizontal direction asshown in Figure 2. In the static load case, the time-dependent derivatives are ignored and the radial force

acting on the ring is considered to be:

f(�) = F0��

3

�+ F0�

�� 4�

3

�; (62)

p(�) = 0; (63)

where F0 is amplitude of static load, and �=3 and4�=3 express the positions of the forces. The governingequation of Euler-Bernoulli micro-ring under staticload would be as follows:�

1 +�

E� �rl

�2��@6u@�6 + 2

@4u@�4 +

@2u@�2

�+ F0

@@�

��

3

�+ �

�� 4�

3

��

+�Kr

@2u@�2 � K�u

�= 0: (64)

In a similar procedure, the governing equation ofTimoshenko ring for static loading can be written asfollows:

@8u@�8

�� 1

4k�Rl

�2 �+@6u@�6

��

E� �rl

�2�1 +1

4k�Rl

�2�+�

1� 1

4k�Rl

�2��+@4u@�4

�2�

E� �rl

�2�1+1

4k�Rl

�2�+�

2 +1

4k�Rl

�2��+@2u@�2

��

E� �rl

�2 + 1��

1 +1

4k�Rl

�2��+�Kr

@2u@�2 � K�u

�+�

1 +1

4k�Rl

�2�F0

@@�

��

3

�+ �

�� 4�

3

��

1

4k�Rl

�2 +E

k��R

�r

�2�F0@3

@�3

��

3

�+ �

�� 4�

3

��= 0: (65)

The response of a structure under application of ex-ternal forces can be obtained using mode summationtechnique. Based on this method, the solution is ex-pressed as the summation of the mode shapes of thestructure. Therefore, the mode shapes of the ring are


Figure 2. Circular micro-ring loading.

needed. This mode which is satisfying the governingequations and boundary conditions can be mentionedas follows:Un = sin(n� + �) n = 2; 3; :::; (66)

where n is the mode number of the ring, noting thatn = 1 indicates the rigid body motion of the ring.Using the mode shapes, the response of the micro ringunder static loading can be expressed as follows:

u(�) =1Xn=2

Cn sin(n� + �): (67)

By inserting the above equation into the governingequations of Euler-Bernoulli and Timoshenko ring withsu�cient modes and using Galerkin approximation, onecan obtain the response of the ring.

4. Free vibration analysis

4.1. Natural frequency of Euler-Bernoulli ringTo �nd natural frequencies of the ring, external forcesare assumed to be zero and harmonic response isconsidered as follows:

u(�; t) = U(�)ei!t; (68)

where ! is dimensionless natural frequency of themicro-ring. Eq. (59) can be rewritten as follows:�

1 +�

E� �rl

�2��@6U@�6 + 2

@4U@�4 +

@2U@�2

�+�Kr

@2U@�2 � K�U

�� !2

�@2U@�2 � U

�= 0: (69)

As it is expressed in the static case section, the solution

of this equation can be proposed in the following form:

U(�) = c0 sin(n� + �) n = 2; 3; :::; (70)

in which c0 is a real constant. The natural frequency ofEuler-Bernoulli micro-ring, according to the modi�edcouple stress theory, is derived as follows:

!2 =�

1+�

E��rl

�2��n6�2n4+n2

n2+1

�+�Krn2+K�

n2 + 1

�: (71)

4.2. Natural frequency of Timoshenko ringAs performed in the previous section, external forcesare assumed to be equal to zero and the followingharmonic response is considered for the ring:

u(�; t) = c0 sin(n� + �)ei!t n = 2; 3; :::: (72)

Utilizing the proposed relation, the following algebraicequation is derived for natural frequency of the Timo-shenko micro-ring:

A!4 �B!2 + C = 0; (73)

in which:

A =�(n2 + 1)

�E

k��R

�r

�4��; (74)

B=�n6�

1

4k�Rl

�2�R�r

�2�+n4�

1�R�r

�2�1� 1

4k�Rl

�2+

14k��rl

�2 +Ek�

��+n2

��1+

1

2k�Rl

�2 +E

k��R

�r

�2�� 1�R

�r

�2�2+1

4k�Rl

�2��+�

1+1�R�r

�2��1

+1

4k�Rl

�2��; (75)

C=�n8�

1

4k�Rl

�2�+n6�

�

E�� r

l

�2��1+1

4k�Rl

�2�+�

1� 1

4k�Rl

�2��+ n4�

2�E� �rl

�2�1

+1

4k�Rl

�2�+�

2 +1

4k�Rl

�2��+ n2

��1 +

�E� �rl

�2��1 +1

4k�Rl

�2��+ (Krn2 + K�): (76)


5. Harmonic case

In the case of harmonic load, the radial force acting onthe ring is considered to have the form as follows:

f(�; t) =F0 sin(t)��

3

�+ F0 sin(t)�

�� 4�

3

�; (77)

where is the excitation frequency. The governingequation of Euler-Bernoulli micro-ring under harmonicloading would be as follows:�

1 +�

E� �rl

�2��@6u@�6 + 2

@4u@�4 +

@2u@�2

�+ F0 sin(t)

@@�

��

3

�+ �

�� 4�

3

��+�Kr

@2u@�2 � K�u

�+@2

@t2

�@2u@�2 � u

�= 0: (78)

Utilizing the mode summation method, the de ectionof the ring can be considered as follows:

U(�; t) =1Xn=2

�(t) sin(n� + �): (79)

Inserting the above equation into Eq. (78) gives thefollowing di�erential equation for �(t) which can beeasily solved:

(n2 + 1)@2�(t)@t2

+�

1 +�

E� �rl

�2��n6 � 2n4

+ n2��(t) + (Krn2 + K�)�(t) = N(t); (80)

where:

N(t) =1�

Z 2p

0

�F0 sin(t)

@@�

��

3

��+F0 sin(t)

@@�

�� 4�

3

��sin(n�+�)d�:

(81)

After the time part of de ection (�(t)) is solved, theresponse of the Euler-Bernoulli micro ring can beobtained from the mode summation technique by mul-tiplying the time part solution to corresponding mode.A similar procedure is performed on Timoshenko ringand the di�erential equation for time part of responseis derived as follows:

A@4�(t)@t4

+B@2�(t)@t2

+ C�(t) = N(t); (82)

whereA, B, and C coe�cients are the same as Eqs. (74)to (76), and N(t) can be achieved from Eq. (81).By solving the time part solution of Timoshenko ringand multiplying them to the corresponding modes, theresponse of Timoshenko ring can be obtained.

6. Results and discussion

Ring with rectangular cross-section is considered as acase study in this section. Let parameters h and bdenote the height and width of the ring cross-section,respectively. These parameters' radius of gyrationof the cross-section of the ring may be replaced by(1=2p

3)h.The �rst dimensionless natural frequency of

Euler-Bernoulli micro-ring with respect to the ratio ofring thickness to the material length scale is illustratedin Figure 3. The dimensionless coe�cients of elasticfoundation (Kr; K�) are assumed to be zero. The sizedependency of the normalized natural frequency of theEuler-Bernoulli micro-ring can be seen clearly in this�gure. It can be easily concluded that when the ratio ofh=l is less than 10, the di�erence between the naturalfrequencies given by the modi�ed couple stress andclassical theories is considerable. As is well-known inthe literature and can be understood from this �gure,the modi�ed couple stress theory predicts the sti�nessof the micro-scale structures higher than the classicaltheory. Hence, the natural frequency predicted bythe modi�ed couple stress theory is higher than thatevaluated by that classical theory.

The �rst dimensionless natural frequencies ofTimoshenko ring are plotted in Figure 4. The di-mensionless coe�cients of elastic foundation (Kr; K�)are assumed to be zero. To calculate the resultspresented in this �gure, the ratio of the ring radiusto the thickness of the cross-section is considered asR=h = 10. The results indicate that when the thicknessof the cross-section is in the order of the material lengthscale, the di�erence between classical and non-classicaltheories is considerable. As an example for h=l = 3,this di�erence is 25%. This di�erence means that sizee�ect should be considered for the structures in whichthe constitutive length (e.g., the thickness of the beam-

Figure 3. First dimensionless natural frequency ofEuler-Bernoulli ring.


Figure 4. First dimensionless natural frequency ofTimoshenko ring.

Figure 5. First dimensionless natural frequency ofEuler-Bernoulli ring on elastic foundation.

type structure) is in the order of material length-scaleparameter.

In Figure 5, the �rst dimensionless natural fre-quencies of Euler-Bernoulli micro-ring in terms of sti�-ness coe�cient are presented. Results indicated thatnatural frequency increases as radial and tangentialsti�ness coe�cient enhances. The same result can beobtained from Figure 6 in which the �rst dimensionlessnatural frequencies of Timoshenko micro-ring in termsof sti�ness coe�cient are investigated. The dimension-less coe�cients of elastic foundation are assumed to bebKr = bK� = 10.

The ratios of the 1st, 5th, and 10th naturalfrequencies of Timoshenko ring to those of the Euler-Bernoulli are presented in Figure 7 for h=l = 3. Basedon this �gure, it can be realized that the di�erencebetween the natural frequencies given by the Timo-shenko and Euler-Bernoulli theories would be largeras the number of the mode increases. Furthermore,as the ratio of ring radius to ring thickness increases

Figure 6. First dimensionless natural frequency ofTimoshenko ring on elastic foundation.

Figure 7. Natural frequency of Timoshenko ring overthat of Euler ring for �rst, �fth and tenth frequencies.

the di�erence between the natural frequencies of theTimoshenko ring and that of the Euler-Bernoulli ringtend to each other. However, for the values largerthan 10, the natural frequencies of micro-rings forTimoshenko and Euler beams are assumed to be thesame.

Figure 8 shows the e�ect of sti�ness coe�cientof elastic foundation on size-dependent mechanicalbehavior of micro-ring. In this �gure, the ratio ofsize-dependent natural frequency to classic naturalfrequency with respect to radial and tangential sti�nesscoe�cients is plotted for R=h > 20. This �gure revealsthat as the sti�ness coe�cient of the system increases,the size-dependent parameter would be less e�ective innatural frequency of the system, and the ratio of non-classic to classic natural frequencies tends to 1.

Utilizing the mode summation technique and thederived equations, the response of the micro ring understatic load for R=h > 20 and h=l = 3 is presented in


Figure 8. Ratio of non-classic to classic naturalfrequency with respect to elastic foundation sti�ness.

Figure 9. Maximum de ection of micro-ring under staticload h=l = 3.

Figure 9. It should be noted that since the ratio ofring radius to ring thickness (R=h) is assumed to behigher than 20, the results of static loading based onEuler-Bernoulli and Timoshenko theories match each

other. As it is expected from the free vibration analysis,micro ring based on non-classical theory is sti�er thanclassical theory and de ection is smaller compared toclassical theory.

Using mode summation technique and considering4 modes of the ring, the e�ect of size dependency on themaximum de ection of the micro ring at di�erent ratiosof h=l under harmonic loading is plotted in Figure 10.The excitation frequency () is assumed to be closeto the �rst natural frequency of the ring. Findingsindicate that as h=l increases, the classical and non-classical results tend to one another (for h=l = 3, thereis a meaningful gap between classic and non-classicresponses, while the responses completely match eachother h=l > 100). Comparing the natural frequenciesand de ection of the micro ring under external forcebased on classical and non-classical theories revealsthat for the vibrating ring gyroscope which operatesbased on natural frequency of the micro ring, the sizedependency must be considered in the analysis of thesesystems.

7. Conclusion

In this paper, based on the Euler-Bernoulli and Tim-oshenko beam theories, size-dependent formulationsare developed for micro-scaled circular rings on elasticfoundation. A displacement �eld is proposed for circu-lar rings and Hamilton's principle is utilized to derivethe governing equations and boundary conditions ofmicro-rings. As a case study, the dimensionless size-dependent natural frequencies of a circular micro-ring with rectangular cross-section are obtained andcompared with those evaluated based on the classicaltheory. The response of the micro-ring under applica-tion of static and harmonic point loads is investigated,and the �ndings are compared with classical ones.Results reveal that the normalized natural frequencyof the micro ring and the de ection of micro ringunder static and harmonic forces are size-dependent,

Figure 10. Maximum de ection of micro-ring under harmonic load.


while according to the classical theory, the normalizedbehavior of a structure is independent of its size.As the ring thickness increases, the results of themodi�ed couple stress theory converge to those ofthe classical theory. Furthermore, as the radius ofthe ring increases, the natural frequency given by theTimoshenko beam theory converges to that predictedby the Euler-Bernoulli beam theory.

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Biographies

Ali Karimzadeh was born in 1988. He receivedthe BS degree in Mechanical Engineering from theUniversity of Shiraz, Iran, in 2010, and the MS degreein Mechanical Engineering from the Sharif University ofTechnology in 2012. Now he is a PhD candidate in theSchool of Mechanical Engineering in Sharif Universityof Technology. His research interests are in the areas ofmicro electromechanical systems, strength of material,and advanced �nite-element method.

Mohammd Taghi Ahmadian received a PhDdegree from the Department of Mechanical Engineeringat the University of Kansas, USA, in 1986. From 1984to 1985, he was an Assistant Professor at the Univer-sity of Missouri, and from 1985 to 1988, he was anAssistant Professor in the Department of MechanicalEngineering at the University of Kansas at Lawrence.He started working as an Assistant Professor at SharifUniversity of Technology in 1989; at present, he is aProfessor in the School of Mechanical Engineering atSharif University of Technology, Tehran, Iran. Hisresearch interests include cellular biomechanics, softtissue modeling, strength of material, and advanced�nite-element method.

Masoud Rahaeifard was born in 1985. He re-ceived BS degree in Mechanical Engineering from theAmirkabir University of Technology, Iran, in 2007,and MS degree in Mechanical Engineering from theSharif University of Technology in 2009. He receiveda PhD degree from the Department of MechanicalEngineering at the Sharif University of Technology in2013. He has started working as an Assistant Professorat Golpayegan University of Technology from 2013. Hisresearch interests are in the areas of micro electrome-chanical systems, nonclassical continuum theories.

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