non-linear vibrations and frequency response analysis of piezoelectrically driven microcantilevers

International Journal of Non-Linear Mechanics 42 (2007) 577–587www.elsevier.com/locate/nlm

Non-linear vibrations and frequency response analysis of piezoelectricallydriven microcantilevers

S. Nima Mahmoodi, Nader Jalili∗

Smart Structures and Nanoelectromechanical Systems Laboratory, Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA

Received 15 June 2006; received in revised form 25 January 2007; accepted 31 January 2007

Abstract

Microcantilevers have recently received widespread attentions due to their extreme applicability and versatility in both biological and non-biological applications. Along this line, this paper undertakes the non-linear vibrations of a piezoelectrically driven microcantilever beam as acommon configuration in many scanning probe microscopy and nanomechanical cantilever biosensor systems. A part of the microcantileverbeam surface is covered by a piezoelectric layer (typically ZnO), which acts both as an actuator and sensor. The bending vibrations of themicrocantilever beam are studied considering the inextensibility condition and the coupling between electrical and mechanical properties inthe piezoelectric materials. The non-linear terms appear in the form of quadratic expression due to presence of piezoelectric layer, and cubicform due to geometrical non-linearities. The Galerkin approximation is then utilized to discretize the equations of motion. In addition, themethod of multiple scales is applied to arrive at the closed form solution for the fundamental natural frequency of the system. An experimentalsetup consisting of a commercial piezoelectric microcantilever attached on the stand of a state-of-the-art microsystem analyzer for non-contactvibration measurement is utilized to verify the theoretical developments. It is found that the experimental results and theoretical findingsare in good agreement, which demonstrates that the non-linear modeling framework could provide a better dynamic representation of themicrocantilever than the previous linear models. Due to microscale nature of the system, excitation amplitude plays an important role sinceeven a small change in the amplitude of excitation can lead to significant vibrations and frequency shift.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Non-linear vibrations; Piezoelectrically driven microcantilevers; Non-linear frequency response analysis; Method of multiple scales

1. Introduction

Recently, microcantilevers have been widely utilized in noveltechnologies such as scanning force microscopy (SFM) andsmall mass and biochemical sensors [1,2]. Due to small scaledisplacements and motions, non-linearity of the system be-comes important when more accurate measurement is needed.Because of small scale nature of microcantilevers, they mayhave large deformation in response to even a small appliedforce. The microcantilever beam studied here consists of ametallic beam (usually Si alloys) with a layer of piezoelec-tric material (ZnO) on it as an actuator. The applied forceis the moment due to the applied voltage to the piezoelec-tric layer [3–5]. Even in the absence of large deformation, the

∗ Corresponding author. Tel.: +1 864 656 5642.E-mail address: [email protected] (N. Jalili).

0020-7462/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijnonlinmec.2007.01.019

effect of non-linear terms is not negligible when compared withlinear terms, and hence, cannot be neglected. From a modelingpoint of view, the actuation force comes through piezoelectriclayer and hence the stress–strain relations are related to actua-tion voltage and electrical properties of the piezoelectric layer.Therefore, the electrical properties appear in actuation momentand come into equations of motion eventually [4,5].

The linear dynamics of cantilever beams with piezo-electric layer attachment considering the electromechanicalstress–strain relations have been studied [4,5]. A generic non-linear finite element formulation for vibration sensing andcontrol analysis of laminated electro/elastic non-linear shellstructures has also been derived based on the virtual workprinciple [6]. The non-linear bending behavior of laminatedpiezoelectric actuators as a function of the amplitude andfrequency of large electric fields has been both analytically (viafinite element method) and experimentally investigated [7].

http://www.elsevier.com/locate/nlm

mailto:[email protected]

578 S. Nima Mahmoodi, N. Jalili / International Journal of Non-Linear Mechanics 42 (2007) 577–587

Nomenclature

� impermittivity coefficients (∈ R3×3), m/F� strain vector (∈ R6), dimensionless� curvature of the beam, m−1

�b volumetric mass density of the beam, Kg/m3

�p volumetric mass density of the piezoelectriclayer, Kg/m3

� Poisson ratio, dimensionless� stress vector (∈ R6), Pa� bending angle of the beam, rad� angular velocity of the beam, rad/sN non-linear frequency of the microcantilever,

rad/s excitation frequency, rad/san0 constant primary amplitude, mA(T1) amplitude, mb beam width, mD electrical displacement vector (∈ R3), dimen-

sionlessE elastic stiffness coefficients matrix (∈ R6×6), PaEb beam modulus of elasticity, Pa

Ep piezoelectric modulus of elasticity, Pah coupling coefficients matrix (∈ R6×3), V/mJ mass moment of inertia of microcantilever

beam, Kg/m2

l beam length, ml2 − l1 piezoelectric layer length, mm mass per unit length of the beam with piezoelec-

tric layer, Kg/mPe(t) applied voltage to the piezoelectric layer, VQ electrical field vector (∈ R3), V/mT0 fast time scale, sT1 slow time scale, st time, stb beam thickness, mtp piezoelectric layer thickness, mu(s, t) beam longitudinal vibration, mv(s, t) beam bending vibration, mwp piezoelectric layer width, mwt tip mass width, m

The non-linear bending vibration of a metallic cantileverbeam with non-linear terms in inertia and stiffness due to ge-ometry had been studied earlier [8–12]. Although the obtainedgoverning equations of motion have been solved for theclamped-free boundary conditions under forced vibration, theequations were presented in general form, and hence, one canutilize analytical methods such as method of multiple scalesto solve for any conventional boundary conditions [13,14].Since there is longitudinal deformation in addition to bendingdeformation in planar vibration, inextensibility condition canbe utilized to relate the longitudinal displacement to bendingvibration which, in turn, introduces more non-linear terms inthe equations of motion [15–17]. In this study, the beam isconsidered to be inextensible with only planar vibration, asstudied earlier in many other works [17–20]. In addition, thequadratic and cubic non-linearities [21] are considered.

The main objective of this paper is to derive the non-linearequations of motion of a non-homogenous piezoelectrically ac-tuated microcantilever beam. The new non-linear terms appeardue to presence of piezoelectric material and the coupling be-tween electrical and mechanical fields in stress–strain relation.The non-linear response of the system in presence of thesenew non-linear terms is analytically studied in order to inves-tigate the effect of electromechanical coupling on the systemresponse. The non-linear natural frequency of the beam is alsosimulated and experimentally verified.

2. Problem statement and motivation

Microelectromechanical systems (MEMS) have been exten-sively utilized in novel technologies with their applications

progressing rapidly. Their small scale size gives them the abil-ity to be used in small scale measurements. Recently, MEMShave been widely used in daily-utilized applications such asinkjet printer heads. In addition, MEMS have been used inchemical/biological micro sensor/actuators, SFM and wirelesscommunications, see Fig. 1.

Fig. 2 depicts a dynamic micro actuated silicon probe(DMASP) microcantilever beam manufactured by Veeco In-struments, which has been used in this paper for experimentalverification of the non-linear vibration analysis. The beam canbe excited by applying a voltage to the piezoelectric layer onthe beam.

The advantages of using microcantilever beam for sensingcan be briefly summarized as follows [23]. (i) Four parame-ters of resonance frequency, amplitude, deflection and qualityfactor (which is a measure of the resonance peak bandwidth)can be measured simultaneously; (ii) the sensor can be oper-ated in vacuum, air or liquid; (iii) it offers an improvement inthe dynamic response, increase in precision and reliability aswell as decrease in the overall dimensions; (iv) it is the sim-plest MEMS that can be mass produced; (v) it can easily be in-corporated on integrated circuits with readout techniques, andfinally; (vi) it can be heated and cooled within microseconds,which is advantageous when utilized in reversal of molecularadsorption in rapid detection techniques.

It is observed that the piezoelectrically driven microcan-tilever can also be utilized as a sensing probe which exhibitshigh sensitivity suitable for use in sensing a variety of physi-cal parameters such as mass, temperature, humidity, and vari-ous chemical and biological agents. The operating principle isbased on the transduction of physical, chemical or biologicalprocess into a microcantilever motion response. One of the ap-

S. Nima Mahmoodi, N. Jalili / International Journal of Non-Linear Mechanics 42 (2007) 577–587 579

Fig. 1. MEMS applications [22]: (a) micro scanner; (b) RF MEMS switch; and (c) MEMS mass sensor.

Fig. 2. Comparison of the Veeco DMASP microcantilever beam size with aUS penny.

plications of non-linear analysis of cantilever beam is in atomicforce microscopy (AFM) in which a microcantilever is usedand its bending vibration is utilized for force microscopy. Fordynamical modeling of a microcantilever beam with applica-tion to surface stress and distributed mass sensing, the effectof distributed mass and surface stress should be considered inbending vibration of the beam, as discussed next.

3. Mathematical modeling

3.1. Preliminaries

In order to develop a model for the microcantilever beam,a uniform flexible beam with a piezoelectric layer on its topsurface is considered as shown in Fig. 3(a). It is assumed thatthe piezoelectric width is the same as the beam width. The beamis initially straight and it is clamped at one end and free at theother end. In addition, the beam follows the Euler–Bernoullibeam theory, where shear deformation and rotary inertia termsare negligible.

Fig. 3(b) shows a beam segment of length s with x–y and �–�axes being the inertial and the principal axes of the beam crosssection, respectively. The bending angle between x-axis and �-axis is �. Using Fig. 3(b), angle � for an element of length dscan be obtained as

� = tan−1 v′

1 + u′ , (1)

where over prime denotes derivative with respect to position s.The transformation of the coordinates can be represented in thefollowing matrix form:

{x

y

z

}=[ cos(�) sin(�) 0

− sin(�) cos(�) 00 0 1

]{��

}. (2)

y

x

s+u(s,t)

v(s,t)

s

l2

Piezoelectric

Layer

l

l1

θξ ψ

Fig. 3. (a) Schematic of the microcantilever beam, and (b) its coordinatesystems.

Therefore, using Eqs. (1) and (2) and Taylor’s series expan-sion, the curvature and angular velocity of the beam can be,respectively, obtained as

� = v′′ − v′′u′ − v′u′′ − v′′v′2, (3)

� = v′ − v′u′ − v′u′ − v′v′2, (4)

where over dot indicates the derivative with respect to time t.The Green’s strain associated with the material located atneutral axis is given by [16]

�0 =√

(1 + u′)2 + v′2 − 1, (5)

which is utilized to relate longitudinal and bending vibrationsthrough the inextensibility condition.

The piezoelectric layer is not attached to the entire lengthof the beam, and hence, the neutral surface changes for eachsection of the beam. For s < l1 or s > l2 where piezoelectricis not attached, the neutral surface is the geometric center ofthe beam (y = 0). For the section where piezoelectric layer isattached, the neutral surface, yn, is calculated as

yn = Eptp(tp + tb)

2(Eptp + Ebtb). (6)

As the last note before deriving the equations of motion, the“plane strain” configuration of the beam must be carefully con-sidered here. For such microscale size beam, the thickness ofthe beam is typically much smaller than its width, and hence,the modulus of elasticity must be corrected in the form of [2,24]

E = E∗

(1 − �2). (7)

3.2. The governing equations of motion

An energy method is used here in order to derive the equa-tions of motion. Using the obtained angular velocity, the total


kinetic energy of the system can be expressed as

T = 1

2

∫ l

0{m(s)(u2 + v2) + J (s)(v′2 − 2v′2u′ − 2v′u′v′

− 2v′2v′2)} ds, (8)

where

m(s) = b(�btb + (Hl1 − Hl2)�ptp), (9)

Hl1 = H(s − l1), Hl2 = H(s − l2), (10)

and H(s) is the Heaviside function. The stress–strain relationsare needed to obtain the potential energy of the system. Thefundamental relations for the piezoelectric materials are givenas [4]

� = E� − hD, (11)

Q = −hT� + �D. (12)

For the microcantilever beam considered here, the electricalfield is one-dimensional, therefore, the electrical displacementvector has one non-zero component which is D2. The electricaldisplacement vector can now be defined as

D1 = D3 = 0, D2(s, t). (13)

Therefore, the coupling of stress and electrical field for piezo-electric material can be related as follows:

�p11 = Ep�

p11 − h12D2, (14)

Q2 = −h12�11 + �22D2. (15)

The total potential energy of the system can now be written as[4]

U = 1

2

∫ l1

0

∫ ∫A

�b11�

b11 dA ds + 1

2

∫ l2

l1

∫ ∫A

�b11�

b11 dA ds

+ 1

2

∫ l2

l1

∫ ∫A

(�p11�

p11 + Q2D2) dA ds

+ 1

2

∫ l

l2

∫ ∫A

�b11�

b11 dA ds

+ 1

2

∫ l

0EA(s)

(u′2 + u′v′2 + 1

4v′4)

ds. (16)

Using Eqs. (3)–(16), the Lagrangian of the system can beexpressed as

L = 1

2

∫ l

0{m(s)(u2 + v2) + J (s)

× (v′2 − 2v′2u′ − 2v′u′v′ − 2v′2v′2)− C (s)(v

′′2 − 2v′′2v′2 − 2v′′2u′ − 2v′v′′u′′)

+ EA(s)

(u′2 + u′v′2 + 1

4v′4)

− 2Cd(s)D2

× (v′′ − v′′u′ − v′u′′ − v′′v′2) + C�(s)D22} ds, (17)

where

C (s) = (H0 − Hl1)EbIb + (Hl1 − Hl2)Eb(Ib + btby2n )

+ (Hl1 − Hl2)EpIp + (Hl2 − Hl)EbIb, (18)

Cd(s) = (Hl1 − Hl2)h12Id , (19)

C�(s) = (Hl1 − Hl2)btb�22, (20)

Ib = bt3b

12;

Ip = b(tpy2n + (t2

p + tbtp)yn + 13 (t3

p + 32 tbtp + 3

4 t2b tp));

Id = b

2(tbtp + t2

p − 2tpyn). (21)

The beam is considered to be inextensible here, in which theinextensibility condition demands no relative elongation of theneutral axis (i.e., �0 = 0). Therefore, using Eq. (5) it yields

(1 + u′)2 + v′2 = 1. (22)

Using Eq. (22), variable u can now be coupled to v. This re-duces the number of independent variables to two, i.e., v (thebeam bending vibration) and D2 (the non-zero component ofdielectric displacement vector). Hence, the Lagrangian expres-sion derived in Eq. (17) will lead to two equations, one for v

and one for D2. Considering the Euler–Bernoulli beam theory,ignoring the rotary inertia effect, while substituting Eq. (22)into Eq. (17), the governing equations of motion of the systemusing extended Hamilton Principle can be obtained as

�

�s

(v′(C (s)v

′′v′)′ − v′(Cd(s)D2v′)′)

− �2

�s2

[C (s)(v

′′) + Cd(s)D2

(1 − 1

2v′2)]

,

− m(s)v − �

�s

[v′∫ s

l

m(s)

∫ s

0(vv′ + v′2) ds ds

]= 0

(23a)

Cd(s)(v′′ + 12v′′v′2) + C�(s)D2 = bP e(t), (23b)

v(0, t) = v′(0, t) = v′′(l, t) = v′′′(l, t) = 0. (23c)

Eqs. (23a) and (23b) are coupled partial differential equationsfor v and D2. Eq. (23b) is derived from the constitutiverelationship for piezoelectric materials given in Eqs. (11)–(15).One can combine Eqs. (23a) and (23b) to eliminate variableD2. Hence, the equation of motion of the system reduces to itsfinal form as

m(s)v + �2

�s2

(C (s)v

′′)+ ��s

[v′∫ s

lm(s)

∫ s

0

(vv′ + v′2) ds ds

]

+[v′ �2

�s2

(C (s)v

′′v′)+ v′′ ��s

(C (s)v

′′v′)]

+ v′ ��s

(C2

d(s)

C�(s)v′′v′ − bCd(s)

C�(s)Pe(t)v

′)

− �2

�s2

[bCd(s)

C�(s)Pe(t)

(1 − 1

2v′2)

+ C2d(s)

C�(s)(v′′)

]= 0 (24)


with the boundary conditions (23c). As expected, the cubicnon-linear inertia and stiffness terms appear in the equationsof motion, but due to coupling of electrical and mechanicalfields there emerges quadratic and cubic non-linear terms dueto piezoelectric effect. The linear terms due to presence ofpiezoelectric layer have been obtained before [4]. It can be,however, observed that new non-linearities due to piezoelectricterms appear in the equations of motion.

4. Non-linear natural frequency development

In order to solve the original partial differential equation andobtain the resonance frequency and beam mode shape, the timeand position functions must be separated. For this, the Galerkinmethod is utilized to discretize the system as

v(s, t) =∞∑

n=1

vn(s, t) =∞∑

n=1

�n(s)qn(t), (25)

where �n are the companion functions (i.e., only satisfyingboundary conditions and not necessarily the equation of mo-tion) of the microcantilever beam and qn are the generalizedtime-dependent coordinates. Substituting Eq. (25) into Eq. (24)and using the Lagrangian approach, the equation of motion canbe obtained as

g1nqn + g2nqn + g3nPe(t)q2n + g4nq

3n

+ g5n(q2nqn + qnq

2n) + g6nPe(t) = 0, (26a)

where

g1n =∫ l

0m(s)[�n(s)]2 ds, (26b)

g2n =∫ l

0C (s)�

′′n(s) ds − h12Id(�′

n(l2) − �′n(l1))

btp�22(l2 − l1)

×∫ l

0Cd(s)�′′

n(s) ds, (26c)

g3n = 1

tp�22

∫ l

0Cd(s)�′′

n(s)[�′n(s)]2 ds, (26d)

g4n = 2∫ l

0C (s)[�′′

n(s)]2[�n(s)]2 ds

− h12Id([�′n(l2)]3 − [�′

n(l1)]3)

2btp�22(l2 − l1)

×∫ l

0Cd(s)�′′

n(s) ds

− h12Id(�′n(l2) − �′

n(l1))

btp�22(l2 − l1)

×∫ l

0Cd(s)�′′

n(s)[�′n(s)]2 ds, (26e)

g5n = 2∫ l

0�n(s)

[m(s)�′

n(s)

∫ s

l

∫ s

02[�′

n(s)]2 ds ds

]′ds,

(26f)

g6n = 1

tp�22

∫ l

0Cd(s)�′′

n(s) ds. (26g)

Since the boundary conditions of the beam are clamped-free,the linear mode shapes are considered here to be the followingcompanion functions:

�n(x) = cosh(znx) − cos(znx) + [sin(znx)

− sinh(znx)]cosh(zn) + cos(zn)

sin(zn) + sinh(zn), (27)

where zn are the roots of the frequency equation

1 + cos(zn) cosh(zn) = 0. (28)

In order to utilize the method of multiple scales and ultimatelysolve Eq. (26a), this equation is rewritten in the following form:

qn + ��qn + 2nqn + �2k1nf (t)q2

n + �k2nq3n

+ �k3n(q2nqn + qnqn

2) + �k4nf (t) = 0, (29)

where � is perturbation parameter (in order to use the method ofmultiple scales [25,26]), and 2

n = g2n/g1n, k1n = �−1g3n/g1n,k2n=�−1g4n/g1n, k3n=�−1g5n/g1n, k4n=g6n/g1n and fn(t)=�−1Pe(t). Since there exist a damping term (even very small) inreal system, a damping term is added to the equation, in whichı` is damping coefficient. qn(t) can now be expanded by orderof � as:

qn(t; �) = q0n(T0, T1) + �q1n(T0, T1) + · · · , (30)

where T0 and T1 are the time scales. T1 = �t is defining shift inthe natural frequencies because of the non-linearity, while T0=t

demonstrates motions occurring at the natural frequencies, n.The time derivative becomes

d

dt= D0 + �D1 + · · · + �nDn, (31)

where Dn =�/�Tn. Substituting expression (30) into the differ-ential equations of motion (29) and boundary conditions (23c)and separating terms at orders of �, yields

(�0): D20q0n + 2

nq0n = 0, (32)

(�1): D20q1n + 2

nq1n = − �D0q0n − 2D0D1q0n − k2nq30n

− k3n(q20nq0n + q0nq

20n)

− k4nf (t). (33)

The solution of linear equation (32) can be expressed as

q0n = An(T1)einT0 + cc, (34)

where An can be found by applying the solvability conditionsto the problem as will be discussed later in this section and ccstands for complex conjugate of the preceding terms. f (t) is


now taken to be f (t) = f eit which is a harmonic excitationvoltage. Substituting Eq. (34) into (33) yields

D20r2n + 2

nr2n = − [�inAn(T1) + 2inA′n(T1)

+ 3k2nAn(T1)An(T1)]einT0

− [k2n − 2k3n2n]A3

n(T1)einT0

+ k4nf eiT0 . (35)

The secular terms which should be equal to zero become

�inAn(T1) + 2inA′n(T1) + 3k2nA

2n(T1)An(T1)

− k4nf eiT0 = 0. (36)

Eq. (36) can be utilized for both case of stability analysis andfinding non-linear natural frequency. The stability analysis willbe extensively discussed in the next section. For obtaining thenon-linear natural frequency of the system, it is considered thatthe system is going under free vibration (no external excitation)and the damping is zero.

The amplitude An is written in polar form as

An = 12anei�n . (37)

Substituting Eq. (37) into (36) and separating the real and imag-inary parts, the modulation equations can be expressed as

a′n = 0, (38)

kan�′n = 3k2n

8a3n. (39)

Using the modulation equations and considering free vibrationsof the system, the amplitude and non-linear natural frequencycan be presented as{

an = an0,

Nn = n + �3k2n

8n

a2n0,

(40)

where an0 is a constant. Eq. (40) indicates that the non-linearnatural frequency of the system is related to square of ampli-tude of vibration. The constant coefficients � and kn are depen-dent on beam and piezoelectric layer mechanical and electricalproperties; as discussed earlier.

5. Frequency response and stability analysis

The frequency response function of the system for the lastcase in the preceding section is studied here. For this case, theexcitation frequency can be written as

= n(1 + ��), (41)

where � is a detuning parameter. In this case, Eq. (36) changesto

�inAn(T1) + 2inA′n(T1) + 3k2nA

2n(T1)An(T1)

− k4nf ei�nT1 = 0. (42)

Substituting Eq. (37) into (42), and separating the real andimaginary parts, the modulation equations of frequency and

amplitude can be written as

na′n = − 1

2 �nan + k4nf sin(n�T1 − �n), (43)

nan�′n = 3

8 k2na3n − k4nf cos(n�T1 − �n). (44)

Now, introducing �n = n�T1 − �n, and substituting it inEqs. (43) and (44) yields

na′n = − 1

2 �nan + k4nf sin(�n), (45)

nan�′n = 2

n�an − 38 k2na

3n + k4nf cos(�n). (46)

Using modulation equations (45) and (46), the frequencyresponse of the system can be expressed as

( 12 �nan)

2 + ( 38 k2na

3n − 2

n�an)2 = k2

4nf2. (47)

The stability of the frequency response equation (47) can bedetermined through linearizing the system and finding theeigenvalues of the Jacobian matrix of equations (45) and (46)given by

Ja =⎡⎢⎣ −1

2�n −2

n�an + 3

8k2na

3n[

2n� − 9

8k2na

2n

]1

an

−1

2�n

⎤⎥⎦ . (48)

The corresponding eigenvalues �n of the Jacobian matrix arethe roots of the following characteristic equation

�2n + �n�n + 1

4�22n + (2

n� − 98k2na

2n)(

2n� − 3

8k2na2n) = 0.

(49)

As clearly seen from Eq. (49), the sum of the eigenvalues is−�n, and since � and n are both positive this summation isnegative. This indicates that one of the eigenvalues has alwaysnegative real part. On the other hand, one of the eigenvalues iszero when

14�22

n + (2n� − 9

8k2na2n)(

2n� − 3

8k2na2n) = 0. (50)

In this case, the tangent bifurcation occurs which leads to twoturning points in the response curve of the non-linear system.These points create discontinuity in the frequency responsecurve and a jump will occur in frequency response when itarrives at these points. These features will be demonstratedthrough numerical simulations later in the text (see Section 7).

For special case of linearized steady state system, the externalforce and higher order terms of amplitude are put zero and theeigenvalues of matrix (48) are negative only if

� > 0 (51)

and since � is positive, the linearized system is stable.

6. Experimental setup and methods

The experimental setup for non-linear vibrations of the mi-crocantilever beam is presented here. The objective of the ex-periment is to find the fundamental natural frequency of the


Fig. 4. (a) The MSA-400 setup at Clemson SSNEMS Laboratory, and (b) tip vibration measurement.

wt

tptb

Piezoelectric layer

Si beam

l2 l

wpb

Fig. 5. Geometry of the microcantilever beam.

microcantilever beam (DMASP manufactured by Veeco Instru-ments) shown in Fig. 2 and consequently compare the resultswith the resonance frequency derived from theoretical devel-opment. A state-of-the-art microsystem analyzer, the MSA-400manufactured by Polytec Inc. [22], is utilized as the testing de-vice (see Fig. 4(a)). The MSA-400 utilizes laser-based inter-ferometry for non-contact measurement of three-dimensionalmotions in microstructures.

The laser light measures the vibration of the microcantilevertip. Using the MSA-400, one can precisely adjust the laser lighton the beam tip, as shown in Fig. 4. As seen in Fig. 4(b), thepiezoelectric layer does not cover the entire beam and the tip hasa smaller width than the rest of the beam. The piezoelectric layerconsists of a 3.5 �m ZnO stack and two layers of 0.25 �m Ti/Auon top and beneath the ZnO at the base of each microcantileverbeam acting as electrodes. This layer together with the siliconcantilever act as a bimorph configuration to control the verticaldisplacement of the tip. The beam geometrical and the physicalproperties are provided in Fig. 5 and Table 1, respectively.

The microcantilever is installed on the stand of MSA-400(see Fig. 6) in order to utilize the laser light for measuringthe beam vibration. The microcantilever beam velocity is thenmeasured in response to a 1V AC voltage applied to the piezo-electric layer as the excitation source. The frequency responseof the velocity signal is depicted in Fig. 7.

First strip from left side in FFT plot of Fig. 7 locates thefirst resonance of the beam. The software provides the abilityto identify the beam first natural frequency which is 55,560 Hzin this case.

Table 1Physical properties of the microcantilever

Microcantilever beam ZnO piezoelectric layer

Symbol Value Symbol Value

Eb 105 GPa Ep 104 GPaL 500 �m l2 375 �mb 250 �m �22 45.5 Mm/Fwt 55 �m h12 500 MV/m�b 2330 kg/m3 �p 6390 Kg/mtb 4 �m tp 4 �mvb 0.28 wp 130 �m

vp 0.25

Fig. 6. Experimental setup for the measurement of the microcantilever tipvibrations.

7. Numerical and experimental results comparison

The non-linear equations of motion for vibrations of a piezo-electrically driven microcantilever beam have been theoreti-cally derived, and experimental frequency response has beenobtained in the preceding sections. The frequency response of


Fig. 7. Microcantilever fundamental frequency response for 1V piezoelectricexcitation.

45 50 55 60 650

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Frequency Response

Frequency (KHz)

v(l

,t)

(Mic

rom

ete

r)

Fig. 8. Fast Fourier transform of the microcantilever tip deflection, experi-mental.

the experimental data for the microcantilever beam with proper-ties listed in Table 1 is depicted in Fig. 8 which clearly presentsthe first natural frequency of the microcantilever beam.

The frequency response of model (26) is illustrated in Fig. 9.When comparing Figs. 8 and 9, it is observed that the resonancefrequency of the system matches the experimental results veryclosely (i.e., 55,560 Hz in experiment and 55,579 Hz in theory).

In order to compare the results with linear methods, themethod used in [27,28] is utilized here to obtain the resonancefrequency of the system. This is a linear method which usesthe following formulations [27]:

n = 3.52

2�l2

√EI

�btb + �ptp, (52)

45 50 55 60 650

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Frequency Response

Frequency (KHz)

v(l

,t)

(Mic

rom

ete

r)

Fig. 9. Fast Fourier transform of the microcantilever tip deflection, simulations.

where

EI = b(E2p t4

p + E2b t4

b + EptpEbtb(4t2p + 4t2

b + 6tptb))

12(Ebtb + Eptp). (53)

Eqs. (52) and (53) are the same expressions appeared as Eqs.(2) and (3) of [27]. In this case, there is a significant differencebetween experimental and linear analytical results. Using Eqs.(52) and (53) the frequency is calculated to be 32,385 Hz whichindicates significant error where compared with experimentalresults.

For investigating the influence of correction in modulus ofelasticity presented in Eq. (7), the resonance frequency of thesystem has been obtained without considering the correctionin Eq. (7). The obtained frequency is 53,598 Hz which showsa 3.55% error in result when compared with the frequencyobtained with corrected modulus of elasticity as in Eq. (7).Therefore, it is concluded that the correction in modulus ofelasticity must be considered.

One of the important results that can be concluded is thenoticeable sensitivity of the amplitude of microcantilever vi-bration to input excitation. In Fig. 10, the excitation voltage isexperimentally increased from 0.5V in Fig. 10(a) to 1.5V inFig. 10(b). This results in significant increase in vibrationamplitude as shown in Fig. 10(b). The non-linear modelingproposed here can predict this phenomenon very closely asdepicted in Fig. 11.

There are some small discrepancies between amplitude ofexperimental results and the numerical ones which is expectedsince the numerical parameters (provided by the manufacturer)used in simulations may not exactly match the real parameters(microcantilever’s parameters). These parameters are typicallyobtained and measured by manufacturer in a specific conditionwhich may be different from the condition of present laboratoryexperiment. These differences might also be due to the dampingin real experiment which has not been modeled here.


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-4

-3

-2

-1

0

1

2

3

4 Time Response

Time (Milisecond)

v(l

,t)

(Mic

rom

ete

r)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

-4

-3

-2

-1

0

1

2

3

Experimental TimeResponse

Time (Milisecond)

v(l,t) (M

icro

mete

r)

Fig. 10. Experimental results for amplitude of tip vibration: (a) 0.5V excita-tion, and (b) 1.5V excitation.

At this stage, the frequency response curves of Eq. (47) areplotted for two different piezoelectric excitation amplitudes of0.5V and 1.5V in Figs. 12 and 13, respectively. It is clearfrom both Figs. 12 and 13 that the linear and non-linear naturalfrequencies are different. The zero point on � axis in Fig. 12is the linear natural frequency point and the peak point of thecurve shows the place of non-linear natural frequency.

In Figs. 12 and 13, points B and C are the turning pointsobtained from Eq. (50). The region between points B and C(the dashed lines) is unstable. If a frequency sweep is per-formed from lower frequencies to higher frequencies of res-onance (from points A to D), the amplitude increases until itreaches the turning point B, then the amplitude drops and movesalong curve CD with the same � picked up at point B (pointB′ in Fig. 13). If the frequency sweep is done the other way

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-4

-3

-2

-1

0

1

2

3

4

Time (Milisecond)

v(l,t)

(Mic

rom

ete

r)

Time Response

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-4

-3

-2

-1

0

1

2

3

4

Time (Milisecond)

v(l,t)

(Mic

rom

ete

r)Time Response

Fig. 11. Simulation results for amplitude of tip vibration: (a) 0.5V excitation,and (b) 1.5V excitation.

around (i.e., from points D to A), the amplitude can be obtainedfrom curve DC until it reaches turning point C, then it jumpson curve BA with the same � of point C and continues to reachpoint A. This hysteresis phenomenon is clearly observable inanalytical results of Fig. 13 but not in the simulation results.The reason might be due to relatively small excitation ampli-tudes selected here (f = 0.5 and 1.5V). It is expected that forhigher excitation, the jump in simulation results will be morevisible.

By studying Eq. (40) numerically, it is observed that thenon-linear frequency for large amount of an (in this case largerthat 10−9) can become very considerable. In addition, it isdemonstrated in Fig. 12 that for forced vibration there is also ashift in frequency due to non-linearity and applied excitation. Asseen from Fig. 13, the response amplitude obviously increases


-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

0

0.5

1

1.5

2

2.5

3

x 10-9

Vib

ration a

mplit

ude,a

n f

rom

Eq.

(47)

B

D A

C

Fig. 12. Analytical and numerical results for frequency response of the systemwith 0.5V excitation with points B and C representing the turning points(the “dots” represent the corresponding numerical results obtained by directlysolving Eq. (26)).

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

5

6

x 10-9

Vib

ratio

n a

mp

litu

de

,an

fro

mE

q.

(47)

A

B

C

D

B

Fig. 13. Analytical and numerical results for frequency response of the systemwith 1.5V excitation with points B and C representing the turning points(the “dots” represent the corresponding numerical results obtained by directlysolving Eq. (26)).

when higher excitation voltage is applied compared to Fig. 12.In addition, increasing the voltage of excitation shifts the non-linear natural frequency further more to the right side of theplot which shows that non-linear frequency increases by higherexcitation voltages.

In order to better compare the numerical and analyticalresults, the vibration amplitude an given in Eq. (47) and plot-ted in Figs. 12 and 13 is compared with the beam tip deflectionv(l, t) given by Eqs. (25)–(26) and plotted in Fig. 11. To relatean (the amplitude of generalized coordinate q(t)) to v(l, t),Eqs. (34) and (37) and modulation equations of (43) and (44)

can be used to arrive at

q(t) = an cos(�nt + �0). (54)

Hence, an = |v(l, t)|/|�(l)|. The “dots” in Figs. 12 and 13represent the corresponding simulation results for an usingEq. (26) for different excitation frequencies (equivalently dif-ferent �). While Fig. 12 shows a good match between nu-merical and analytical results, such agreement between ana-lytical and numerical approaches begins to diminish in Fig.13 as the excitation frequency passes the resonance. However,it can be argued that these discrepancies are still acceptableand a good match between analytical and simulation results isobserved.

8. Conclusions

The non-linear equations of motion of a non-homogenousnon-linear microcantilever beam have been derived. The non-linear terms due to presence of piezoelectric material and thecoupling of electrical and mechanical fields in stress–strainrelation of the system have been identified. The quadratic andcubic non-linearities were observed due to piezoelectric layerand geometry of beam. The method of multiple scales was ap-plied to the resultant equations of motion in order to ascertainthe non-linear natural frequency in closed form. The experimen-tal results matched the theoretical ones with only a small error.It was demonstrated that the non-linear approach presented inthis paper is much more precise than linear method reportedin literature in capturing the fundamental resonance frequencyand its changes. This precise prediction of frequency is of im-portance in many nanomechanical sensors utilizing microcan-tilevers. In addition, the influence of considering correction inmodulus of elasticity was studied. It was concluded that in mi-croscale beams, a small change in amplitude of excitation couldincrease the amplitude of vibration considerably. This is espe-cially important in atomic force microscopy applications. Forvery small amplitude of the microcantilever beam, a compre-hensive linear model can be utilized instead of non-linear oneonly when studying the natural frequency of the system. How-ever, in microscale, the amplitude can easily grow and bringthe system into non-linear regime, and therefore it is highlyrecommended to utilize the non-linear analysis for studying thefrequency response. It was demonstrated that in the microscale,the instability of the system can be easily spread or influencedover a large range of frequencies.

Acknowledgments

The authors would like to thank Dr. M. Daqaq for his use-ful comments and suggestions on the frequency response andstability analysis sections.

References

[1] N. Jalili, M. Dadfarnia, D.M. Dawson, A fresh insight in to themicrocantilever-sample interaction problem in non-contact atomic forcemicroscopy, ASME J. Dyn. Syst. Meas. Control 126 (2004) 327–335.


[2] C. Ziegler, Cantilever-based biosensors, Anal. Bioanal. Chem. 379 (2004)946–959.

[3] W.C. Xie, H.P. Lee, S.P. Lim, Nonlinear dynamic analysis of MEMSswitches by nonlinear modal analysis, Nonlinear Dynam. 31 (2003)243–256.

[4] M. Dadfarnia, N. Jalili, Z. Liu, D.M. Dawson, An observer-basedpiezoelectric control of flexible Cartesian robot arms: theory andexperiment, Control Eng. Pract. 12 (2004) 1041–1053.

[5] M. Dadfarnia, N. Jalili, B. Xian, D.M. Dawson, A Lyapunov-basedpiezoelectric controller for flexible cartesian robot manipulators, ASMEJ. Dyn. Syst. Meas. Control 126 (2004) 347–358.

[6] D.W. Wang, H.S. Tzou, H.J. Lee, Control of nonlinear electro/elasticbeam and plate systems (finite element formulation and analysis), ASMEJ. Vib. Acoust. 126 (2004) 63–70.

[7] F. Narita, Y. Shindo, M. Mikami, Analytical and experimental study ofnonlinear bending response and domain wall motion in piezoelectriclaminated actuators under AC electric fields, Acta Mater. 53 (2005)4523–4529.

[8] M.R.M. Crespo da Silva, C.C. Glynn, Nonlinear flexural–flexural-torsional dynamics of inextensional beams. I. Equations of motion,J. Struct. Mech. 6 (1978) 437–448.

[9] M.R.M. Crespo da Silva, C.C. Glynn, Nonlinear flexural–flexural-torsional dynamics of inextensional beams. II. Forced motions, J. Struct.Mech. 6 (1978) 449–461.

[10] M.R.M. Crespo da Silva, Nonlinear flexural–flexural-torsional-extensional dynamics of beams—I formulation, Int. J. Solid Struct. 24(1988) 1225–1234.

[11] M.R.M. Crespo da Silva, Nonlinear flexural–flexural-torsional-extensional dynamics of beams—II. Response analysis, Int. J. SolidStruct. 24 (1988) 1235–1242.

[12] A.H. Nayfeh, C. Chin, A.S. Nayfeh, Nonlinear normal modes of acantilever beam, ASME J. Vib. Acoust. 117 (1995) 477–481.

[13] A.H. Nayfeh, S.A. Nayfeh, On non-linear modes of continuous systems,ASME J. Vib. Acoust. 116 (1994) 129–136.

[14] S.N. Mahmoodi, S.E. Khadem, M. Rezaee, Analysis of nonlinear modeshapes and natural frequencies of continuous damped systems, J. SoundVib. 275 (2004) 283–298.

[15] H.N. Arafat, A.H. Nayfeh, C. Chin, Nonlinear nonplanar dynamics ofparametrically excited cantilever beams, Nonlinear Dynam. 15 (1998)31–61.

[16] S. Hsieh, S.W. Shaw, C. Pierre, Normal modes for large amplitudevibration of a cantilever beam, Int. J. Solids Struct. 31 (1994)1981–2014.

[17] E. Esmailzadeh, N. Jalili, Parametric response of cantilever Timoshenkobeams with tip mass under harmonic support motion, Int. J. NonlinearMech. 33 (1998) 765–781.

[18] N. Jalili, E. Esmailzadeh, A nonlinear double-winged adaptive neutralizerfor optimum structural vibration suppression, J. Commun. Nonlinear Sci.Numer. Simulation 8 (2003) 113–134.

[19] S.N. Mahmoodi, N. Jalili, S.E. Khadem, Passive nonlinear vibrationsof a directly excited nanotube-reinforced composite cantilever beam,in: Proceedings of 2005 ASME International Mechanical EngineeringCongress & Exposition, Symposium on Vibration and Noise Control,Orlando, FL, November, 5–11, 2005.

[20] S.N. Mahmoodi, S.E. Khadem, N. Jalili, Theoretical development andclosed-form solution of nonlinear vibrations of a directly excitednanotube-reinforced composite cantilever beam, Arch. Appl. Mech. 75(2006) 153–163.

[21] A.H. Nayfeh, J.F. Nayfeh, D.T. Mook, On methods for continuoussystems with quadratic and cubic nonlinearities, Nonlinear Dynam. 3(1992) 145–162.

[22] Micro System Analyzer Manual MSA-400, Polytec Inc., www.polytec.com.

[23] P.G. Datskos, T. Thundat, N.V. Lavrik, Micro and nanocantilever sensors,Encyclopedia Nanosci. Nanotechnol. X (2004) 1–10.

[24] D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, P. Tong, Experimentsand theory in strain gradient elasticity, J. Mech. Phys. Solids 51 (2003)1477–1508.

[25] A.H. Nayfeh, Perturbation Methods, Wiley, New York, 1973.[26] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley, New York, 1979.[27] J.H. Lee, K.H. Yoon, K.S. Hwang, J. Park, S. Ahn, T.S. Kim, Label free

novel electrical detection using micromachined PZT monolithic thin filmcantilever for the detection of C-reactive protein, Biosens. Bioelectron.20 (2004) 269–275.

[28] J.H. Lee, K.S. Hwang, J. Park, K.H. Yoon, D.S. Yoon, T.S.Kim, Immunoassay of prostate-specific antigen (PSA) using resonantfrequency shift of piezoelectric nanomechanical microcantilever, Biosens.Bioelectron. 20 (2005) 2157–2162.

http://www.polytec.com

http://www.polytec.com

non-linear vibrations and frequency response analysis of piezoelectrically driven microcantilevers

Documents