Linear and non-linear vibration and frequency response analyses of microcantilevers subjected to tip–sample interaction

ARTICLE IN PRESS International Journal of Non-Linear Mechanics 45 (2010) 176–185

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Linear and non-linear vibration and frequency response analyses of microcantilevers subjected to tip–sample interaction Aidin Delnavaz a,1, S. Nima Mahmoodi b,2, Nader Jalili c,3, Hassan Zohoor a,,4 a

Center of Excellence in Design, Robotics and Automation, Sharif University of Technology, Tehran, Iran Center for Vehicle Systems and Safety, Virginia Tech, MC-0901 Blacksburg, VA 24061, USA c Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA b

a r t i c l e in f o

a b s t r a c t

Article history: Received 6 August 2008 Received in revised form 22 October 2009 Accepted 23 October 2009

Despite their simple structure and design, microcantilevers are receiving increased attention due to their unique sensing and actuation features in many MEMS and NEMS. Along this line, a non-linear distributed-parameters modeling of a microcantilever beam under the influence of a nanoparticle sample is studied in this paper. A long-range Van der Waals force model is utilized to describe the microcantilever–particle interaction along with an inextensibility condition for the microcantilever in order to derive the equations of motion in terms of only one generalized coordinate. Both of these considerations impose strong nonlinearities on the resultant integro-partial equations of motion. In order to provide an understanding of non-linear characteristics of combined microcantilever–particle system, a geometrical function is wisely chosen in such a way that natural frequency of the linear model exactly equates with that of non-linear model. It is shown that both approaches are reasonably comparable for the system considered here. Linear and non-linear equations of motion are then investigated extensively in both frequency and time domains. The simulation results demonstrate that the particle attraction region can be obtained through studying natural frequency of the system consisting of microcantilever and particle. The frequency analysis also proves that the influence of nonlinearities is amplified inside the particle attraction region through bending or shifting the frequency response curves. This is accompanied by sudden changes in the vibration amplitude estimated very closely by the non-linear model, while it cannot be predicted by the best linear model at all. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Microcantilever Tip–sample non-linear interaction Nanoparticle Non-linear modeling

1. Introduction Recently, microcantilevers have been emerged as versatile tools in many micro- and nano-electromechanical systems (MEMS and NEMS). In addition to their traditional utilization in atomic force microscopy (AFM) applications to sense the interaction forces between tip and substrate and produce topographical images of surfaces, they can also be used in various applications of sensing and actuation. For example, they can be used as chemical, thermal, physical and/or biological sensors for gases or vapors’ identification [1], thermal analysis at micromechanical calorimeters [2], nanotribological characterization of particles [3] and biologically induced surface stress sensing [4].

 Corresponding author. Tel.: + 98 912 103 6462.

E-mail address: [email protected] (H. Zohoor). Ph.D. candidate. Visiting Assistant Professor 3 Associate Professor. 4 Professor. 1 2

0020-7462/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2009.10.007

Microcantilevers have also been used extensively for nanomanipulation tasks from simple robotic arms for pushing/pulling nanoparticles [5], to indent surfaces or for use in lithography [6]. Microcantilever-based nanomanipulation systems can be operated in two different modes; (i) static mode (contact mode) in which the microcantilever deflects under the influence of sample without any vibrating motion and (ii) dynamic mode (non-contact or semi-contact modes) in which the microcantilever vibrates near its resonance frequency. Most microcantilever-based nanomanipulation systems use contact mode operation. In this mode of operation, the tip may push down the sample surface, and thus it will often destroy the surface of the sample. This drawback becomes more serious when the soft samples like biomaterials are to be manipulated. On the other hand, vibration-mode based nanomanipulation which has been developed for the past few years has many advantages over contact mode operation [7,8]. A system of a vibrating microcantilever and a particle which is discussed in this paper can be exemplified as a system of non-contact mode nanomanipulation. Although the microcantilever beam subjected to the tip– nanoparticle interaction has been widely investigated in contact mode [9,10], there are only a few comprehensive and reliable models

ARTICLE IN PRESS A. Delnavaz et al. / International Journal of Non-Linear Mechanics 45 (2010) 176–185

Nomenclature u v Kz K Jz Eb tb bb

rb Ab m l

m Q z t s rt rp Uvdw Fvdw

y

e0 T U L dW V M

d r

microcantilever longitudinal vibration microcantilever bending vibration microcantilever area moment of inertia microcantilever spring constant microcantilever mass moment of inertia microcantilever modulus of elasticity microcantilever thickness microcantilever width microcantilever density microcantilever cross section area mass per unit length of the microcantilever microcantilever length microcantilever damping coefficient quality factor support motion time position variable tip radius particle radius Van der Waals potential energy Van der Waals force relative angle Green’s strain at neutral axis kinetic energy potential energy Lagrangian function virtual work shear force bending moment tip–particle vertical distance before vibration total tip–particle distance during vibration

in non-contact modes. Refs. [5] and [11] proposed a lumpedparameters model of microcantilever under the influence of nanoparticle for nanomanipulation tasks. Moreover, a lumpedparameters model that is interacting with a non-linear force field is thoroughly investigated in [12–14] for tapping-mode AFM. A linear distributed-parameters model of the microcantilever subject to tip– sample interaction has also been studied in [15] to identify the interaction forces. Therefore, almost all recently developed microcantilever–sample models use either lumped-parameters or linear distributed-parameters modeling of the microcantilever. These models are not sufficiently reliable for designing accurate and effective microcantilever-based nanoscale sensors and manipulators. Such reduced-order and/or linear models do not reflect profound understanding of the dynamic characteristics and vibration behavior of the microcantilever while its tip is influenced by nanoparticle. In this paper, a non-linear distributed-parameters model of the microcantilever beam subjected to a tip–nanoparticle interaction is proposed. The remainder of the paper is organized as follows. The mathematical model of the system is developed in Section 2. In Section 3, the frequency response equation and non-linear natural frequency are derived and the stability analysis is discussed. Numerical simulations are provided in Section 4, followed by a conclusion in Section 5.

2. Dynamic modeling Linear and non-linear distributed-parameters models for microcantilever vibrations subject to the tip–particle interaction

l AH H

on oNn s O

e g K^ i K~ i

ai b^ i b~ i

g^ in g~ in ln

mi Cn qn qin Ti An an

fn cn

177

Lagrangian multiplier factor Hamaker constant Heaviside function natural frequency non-linear natural frequency detuning parameter frequency of excitation perturbation parameter geometrical function Taylor series expansion coefficients of vdw force in linear system Taylor series expansion coefficients of vdw potential in non-linear system microcantilever mechanical properties in non-dimensional form K^ i in non-dimensional form K~ i in non-dimensional form coefficients of differential equation of motion in linear system coefficients of differential equation of motion in nonlinear system roots of the frequency equation roots of the characteristic equation mode shape function for nth mode of the vibration generalized time-dependent coordinate for nth mode of the vibration generalized coordinate expansion coefficients as an order of e time expansion coefficients as an order of e complex amplitude of vibration amplitude of vibration phase angle of vibration transformed phase angle of vibration

are presented in this section. In the linear model, the amplitude of vibrations is considered to be comparably small, and hence, the bending motion of the microcantilever is only in the vertical direction. However, in non-linear analysis, the microcantilever deforms in both longitudinal and vertical directions due to its large amplitude of vibrations. This not only causes geometrical nonlinearity in the microcantilever model, but also the distance between the tip of the microcantilever and the particle can no longer be considered as linear vertical. Due to change in the distance of microcantilever tip and particle, the Van der Waals energy terms and consequently the interaction force will change. The comparison between linear and non-linear models will provide a comprehensive physical understanding of the system. 2.1. Non-linear modeling To mathematically investigate the non-linear vibration of the microcantilever beam, the Hamilton’s approach is adopted and used here. As shown in Fig. 1, an Euler–Bernoulli beam is taken into account, where element ds being the distance s from its support at time t for which v(s,t) and u(s,t) are the bending and longitudinal displacements of the microcantilever, respectively. {x,y} and {x,Z} are respectively defined as a fixed and moving coordinate system with the following relative angle:  0  v ð1Þ y ¼ arctan 1 þ u0 The kinetic and potential energies of the individual microcantilever element are constructed and integrated over the

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Fig. 1. Schematic representation of microcantilever/nanoparticle system for non-linear modeling.

microcantilever length l  Z l 2 1 1 T¼ mðu_ 2 þ v_ 2 Þ þ Jz y_ ds 2 0 2 U¼

 Z l 1 1 2 Kz y0 þ Eb Ab e20 ds 2 0 2

ð2Þ

0

þ K~ 4 u2 Hðs  lÞ þ K~ 5 zðtÞ þ K~ 6 vHðs  lÞ þ K~ 7 Þ ds

y_ ¼ v_ 0  v_ 0 u0  v0 2 v_ 0  v0 u_ 0

ð4Þ

y0 ¼ v00  v00 u0  v0 2 v00  v0 u00

ð5Þ

Also, e0 is associated with the Green’s strain at neutral axis given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e0 ¼ ð1 þu0 Þ2 þ v0 2  1 ð6Þ Since microcantilever neutral axis remains inextensible during vibration [16], i.e., e0 = 0, the longitudinal and bending vibrations are related to each other by non-elongation constraint 1 02 ð7Þ v ¼0 2 This relationship reduces the number of independent generalized coordinates to only 1. The influence of the nanoparticle on the microcantilever vibration is also taken into account by the Van der Waals (vdw) potential formulation which can be extracted from continuum mechanics solution of interaction problem for the system of cantilever tip and spherical particle [5], i.e., u0 þ

AH rt rp 1 3 ðr þr þ Þðr  r þ Þ

ð10Þ

ð3Þ

where y_ and y0 denote temporal and spatial derivatives of y respectively, which can be estimated by applying the Taylor series expansion as

Uvdw ¼

rewritten as Z l ðK~ 1 zðtÞvHðs  lÞ þ K~ 2 zðtÞ2 þ K~ 3 v2 Hðs  lÞ Uvdw ðtÞ ¼

ð8Þ

Where rt is the microcantilever tip radius, rp is the particle radius and r + =rt + rp. Moreover, AH is the Hamaker constant and r is the total tip–nanoparticle distance given by: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð9Þ r ¼ ½d þ zðtÞ þ vHðs  lÞ2 þ½uHðs  lÞ2 In expression (9), z(t) is a support excitation and d represents the vertical distance between the tip and nanoparticle while the tip stands exactly over the particle. H is the Heaviside function to concentrate the tip–nanoparticle potential at the microcantilever free end where the tip is commonly laid. By applying Taylor series expansion and integrating over the microcantilever length, the Van der Waals potential can be

where 2

2 AH rt rp ð3d þ r 2þ Þ K~ 1 ¼ 3 ðd þ r þ Þ3 ðd  r þ Þ3

ð11aÞ

2

1 AH rt rp ð3d þ r 2þ Þ K~ 2 ¼ K~ 3 ¼ 3 ðd þ r þ Þ3 ðd  r þ Þ3

ð11bÞ

1 A H rt rp K~ 4 ¼ 3 ðd þ r þ Þ2 ðd  r þ Þ2

ð11cÞ

2 A H rt rp d K~ 5 ¼ K~ 6 ¼ 3 ðd þ r þ Þ2 ðd  r þ Þ2

ð11dÞ

1 AH rt rp d K~ 7 ¼ 3 ðd2  r 2þ Þ

ð11eÞ

In the Hamiltonian approach adopted here, the boundary input force mz€ ðtÞ and damping force mv_ are all collected in the following virtual work expression Z l dW ¼ ðmz€ ðtÞdv  mv_ dvÞ ds ð12Þ 0

For convenience in computing and understanding the followup equations, the following non-dimensional variables are introduced and used. rffiffiffiffiffiffiffiffi Kz s u v s ¼ u ¼ v ¼ t ¼ t l l l ml4 rffiffiffiffiffiffiffiffi J E A l2 ml4 Kz a1 ¼ z 2 a2 ¼ b b a3 ¼ ð13Þ Kz Kz ml4 ml It should be noted that the mechanical properties of the microcantilever beam are all represented through a0 s coefficients while the tip–particle interactions are related by the following b~ 0 s coefficients. 8 > < 4 i ¼ 1; 2; 3; 4 j~ l K i i ¼ 5; 6 b~ i ¼ ; j¼ 3 ð14Þ > Kz : 2 i¼7

ARTICLE IN PRESS A. Delnavaz et al. / International Journal of Non-Linear Mechanics 45 (2010) 176–185

Now, the total non-dimensional Lagrangian function and the non-dimensional virtual work L ¼ ðl2 =Kz ÞL dW  ¼ ðl2 =Kz ÞdW can be obtained as:

179

In which ln are the roots of the frequency equation 1þ cos ln cosh ln ¼ 0

ð22Þ

  1 1 2 1 2 1 _ þ u_ þv00 2 u0 þv00 v0 u00  v00 2 þ v0 2 v00 2 v C B 2 2 2 B    C C B 1 1 1 1 Z 1B _ 0  v0 2 v_ 0 2  v_ 0 2 u0 þ v_ 0 2 þ a2  u0 v0 2  v0 4  u0 2 C C B þ a1 v_ 0 vu C ds B 2 2 8 2 L¼ C B 0B C 2 b~ 1 zðtÞvHðs  1Þ  b~ 2 zðtÞ  b~ 3 v2 Hðs  1Þ  b~ 4 u2 Hðs  1Þ C B C B A @ 1 2 b~ 5 zðtÞ  b~ 6 vHðs  1Þ  b~ 7 þ lðu0 þ v0 Þ 2 0

dW ¼

Z

1

ðz€ ðtÞdv  a3 v_ dvÞ ds

ð16Þ

0

where asterisk has been dropped for convenience and l is the Lagrangian multiplier to handle non-elongation constraint. Rt By applying the Hamilton’s extended principle, 0 ðdL þ dWÞ dt ¼ 0, the governing equations of motions are obtained as:     1 u€  ðv0 v00 Þ00 þ ða1 v0 v_ 0 Þ0 þ a1 v_ 0 2  a2 u0 þ v0 2 þ v00 2 0 2 þ 2b~ uHðs  1Þ  l ¼ 0 ð17aÞ 4

ð15Þ

Substituting Eq. (20) into Eq. (18) and considering the orthogonality of comparison functions by taking the inner product of the resulting equation with Cn(s) results in the following ordinary differential equation of motion

g~ 1n q€ n þ g~ 2n qn þ g~ 3n q_ n þ g~ 4n q3n þ g~ 5n q2n q€ n þ g~ 6n qn q_ 2n þ g~ 7n z€ ðtÞ þ g~ 8n zðtÞ þ g~ 9n ¼ 0

ð23Þ

where Z 1 g~ 1n ¼ Cn2 ds

ð24aÞ

0

v€  ð2v0 2 v00 þ v0 u00  v00 þ 2v00 u0 Þ00  ða1 ð2v_ 0 u0  2v0 2 v_ 0 þ v_ 0  v0 u_ 0 ÞÞ0     1 þ a1 ð2v0 v_ 0 2  v_ 0 u_ 0 Þ þ a2 u0 v0  v0 3 þ 2v0 v00 2 þ v00 u00 0 2 Z s ~ ~ ~ þ a3 v_ þ b 1 zðtÞ þ 2b 3 vHðs  1Þ þ b 6  z€ ðtÞ  ðv0 l dsÞ0 ¼ 0

g~ 2n ¼

Z

1 0

g~ 3n ¼ a3

  0000 Cn ð2b~ 3 Cn ð1ÞÞ þ C n ds

Z

0

1

ð17bÞ

g~ 4n ¼ The beam mass moment of inertia is neglected in comparison with its translational inertia and flexural stiffness, i.e., a1 =0. Furthermore, l is eliminated from Eq. (17a and 17b). The non-elongation constraint is eventually applied to obtain the following non-linear equation governing the microcantilever bending vibration:  Z s Z s  0000 ð v0 v_ 0 dsÞ ds 0 v€ þ v þ ðv0 ðv0 v00 Þ0 Þ0 þ v0 1

0

þ a3 v_ þ b~ 1 zðtÞ þ 2b~ 3 vHðs  1Þ  Z s  1 02 þ 2b~ 4 v0 v ds Hðs  1Þ þ b~ 6  z€ ðtÞ ¼ 0 02

ð18Þ

Z

1

1 0

ð24bÞ

Cn2 ds 0

Cn @

ð24cÞ

1 Rs 0000 000 C 0 2n Cn þ 3C 0 n Cn00 Cn þC 0 3n þ C 0 n b~ 4 0 C 0 2n dsjs ¼ 1 A ds R R 000 0000 s s þ Cn00 1 ðCn00 Cn þ Cn0 Cn þ b~ 4 0 C 0 2n dsjs ¼ 1 Þ ds ð24dÞ

Z 1  Z sZ s Z g~ 5n ¼ g~ 6n ¼ Cn C 00 n C 0 2n ds ds þ Cn0 0

g~ 7n ¼  g~ 8n ¼ b~ 1

Z

1

0

 s dy 0 2 C n ds ds 0 dx

ð24eÞ

1

Cn ds

ð24fÞ

Cn ds

ð24gÞ

0

Z

1 0

The comparison functions are orthonormal and consequently, with the following corresponding boundary conditions: vð0; tÞ ¼ 0;

v0 ð0; tÞ ¼ 0;

v00 ð1; tÞ ¼ 0;

000

v ð1; tÞ ¼ 0

ð19Þ

In order to further investigate Eq. (18), the Galerkin’s approximation is used to separate the solution into the following separable form vðs; tÞ ¼

1 X

Cn ðsÞqn ðtÞ

g~ 1n ¼ 0. In Eqs. (23) and (24), g~ 2n ¼ o2n represents the linear natural frequency of vibration, g~ 3n is associated with the viscous damping of the microcantilever and g~ 4n is the coefficient of the cubic nonlinearity due to the microcantilever geometry and tip– particle interaction. Also, g~ 5n and g~ 6n are the same and originate from working with non-elongation constraint. g~ 7n and g~ 8n are also related to the support motion and g~ 9n is a constant.

ð20Þ

n¼1

where qn (t) is the generalized time-dependant coordinate for nth mode of the microcantilever vibration, and Cn(s) is a comparison function satisfying the boundary conditions (at least geometrical ones). For the cantilever with homogenized boundary conditions, the comparison function can be written in the non-dimensional form as: Cn ðsÞ ¼ cosh ln s  cos ln s þðsin ln s  sinh ln sÞ

cosh ln þcos ln sinh ln þsin ln

ð21Þ

2.2. Linear modeling To obtain the linear differential equation for the bending vibration of the microcantilever beam, the forces and moments acting on an element ds of the microcantilever are considered as shown in Fig. 2, with V and M being shear and bending moments respectively. Using the Newton’s second law and summing the forces in the y direction and moments about any point on the right face of the element as well as ignoring higher order terms yields [17]:

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dimensional form. 0000

v€ þ v þ a3 v_  z€ ðtÞ ¼ 0

ð35Þ

v0 ð0; tÞ ¼ 0;

vð0; tÞ ¼ 0;

v00 ð1; tÞ ¼ 0;

000

v ð1; tÞ þ Fvdw ðtÞ ¼ 0 ð36Þ

It should be noted that Fvdw (t) can also be brought into nondimensional form by Fvdw ðtÞ ¼ b^ 1 zðtÞ þ b^ 2 vHðs  1Þ þ b^ 3

ð37Þ

where

b^ i ¼

Fig. 2. Schematic representation of microcantilever/nanoparticle system for linear modeling.

lj K^ i ; K^ z

j¼



4

i ¼ 1; 2

3

i¼3

ð38Þ

The boundary conditions (36) need to be homogenized for further analysis. Therefore, an appropriate geometrical function g(s) can be selected to perform the following change of variables: v ðs; tÞ ¼ vðs; tÞ þFvdw ðtÞgðsÞ

ð39Þ

where g(s) should satisfy the following conditions dV  mv_ dsþ mz€ ðtÞ ds ¼ mv€ ds dM  V ds ¼ 0

gð0Þ ¼ 0;

ð26Þ

Substituting Eq. (39) into Eqs. (35) and (36) produces the nondimensional governing equations of motion with homogenized boundary conditions as:

Combining Eqs. (25) and (26) results in: d2 M ¼  ðmv€ þ mv_  mz€ ðtÞÞ ð27Þ ds2 Also the bending moment can be related to the curvature by the following flexure equation [17] M ¼ Kz v00

ð28Þ

The linear governing equation of motion is derived by substituting Eq. (28) into Eq. (27) 0000

mv€ þKz v þ mv_  mz€ ðtÞ ¼ 0

ð29Þ

with the boundary conditions vð0; tÞ ¼ 0;

v0 ð0; tÞ ¼ 0;

v00 ðl; tÞ ¼ 0;

000

where Fvdw ðtÞ is an attraction force between the tip and particle at the microcantilever free end and can be calculated by taking gradient from previously studied the Van der Waals potential (Eq. (8)):

g^ 2n ¼

Z

1 0

0000 0000 Cn ðb^ 2 g ð1ÞCn ð1Þ þ Cn Þ ds

Z

Z

1 0

ð31Þ 1 0

g^ 9n ¼ b^ 3

Z

1

v ð1; tÞ ¼ 0

ð34bÞ

The equation of motion and its corresponding boundary conditions are multiplied by l3 =kx to transform them into non-

ð41Þ ð42Þ

ð43Þ

ð44aÞ

0000

Cn g ds

ð44bÞ

ð44cÞ

ð44dÞ

ð44eÞ

0

Z

1

0000

Cn g ds

ð44fÞ

0

g^ 10n ¼ a3 b^ 1 ð34aÞ

Cn ðb^ 2 gð1ÞCn ð1Þ þ Cn Þ ds

Cn ð1 þg b^ 1 Þ ds

ð33Þ

where

AH rt rp d 2 K^ 3 ¼  3 ðd þr þ Þ2 ðd  r þ Þ2

000

ð40Þ

0

Taylor series expansion is utilized to linearize the interaction force as:

2 AH rt rp ð3d K^ 1 ¼ K^ 2 ¼  3 ðd þr þ Þ3 ðd

v00 ð1; tÞ ¼ 0;

where Z 1 g^ 1n ¼ Cn ðgð1ÞCn ð1Þ þ Cn Þ ds

ð32Þ

 r 2þ Þ  r þ Þ3

g ð1Þ ¼ 0

where the double asterisk for v has been dropped for brevity. The Galerkin’s approximation method is now used to discretize the linear governing equation of motion with respect to the position and time. The comparison functions (21) are used as displacement functions and, eventually, the linear differential equation of the generalized time-dependant coordinate for nth mode of the microcantilever vibration is obtained as:

g^ 8n ¼ b^ 1

2

v0 ð0; tÞ ¼ 0;

vð0; tÞ ¼ 0;

g^ 3n ¼ a3

Moreover, the total tip–nanoparticle distance reduces to the vertical direction according to the following expression:

Fvdw ðtÞ ¼ K^ 1 zðtÞ þ K^ 2 vHðs  lÞ þ K^ 3

g 00 ð1Þ ¼ 0;

0000 0000 v€ þ v þ a3 v_  z€ ðtÞ þg Fvdw ðtÞ þ a3 g F_ vdw ðtÞ þ g F€ vdw ðtÞ ¼ 0

g^ 7n ¼

r ¼ d þzðtÞ þ vHðs  lÞ

g 0 ð0Þ ¼ 0;

g^ 1n q€ n þ g^ 2n qn þ g^ 3n q_ n þ g^ 7n z€ ðtÞ þ g^ 8n zðtÞ þ g^ 9n þ g^ 10n z_ ðtÞ ¼ 0

Kz v ðl; tÞ ¼  Fvdw ðtÞ ð30Þ

Fvdw ðtÞ ¼  rUvdw ðtÞ d U ðtÞ ¼  dr vdw AH rrt rp 2 ¼  3 ðr  r þ Þ2 ðr þr þ Þ2

000

ð25Þ

Z

1

Cn g ds

ð44gÞ

0

The indices used for g^ s in Eq. (44) are selected in a way that they are in a complete similarity in succession with their counterparts in a non-linear system. Since the natural frequency ðo2n ¼ g2 =g1 Þ of both linear and non-linear systems are equal, by comparing g~ 1 and g~ 2 with their corresponding coefficients in linear system, g^ 1 and g^ 2 , and considering b^ 2 ¼  2b^ 3 leads to the following conditions for g(s) to guarantee the equal natural

ARTICLE IN PRESS A. Delnavaz et al. / International Journal of Non-Linear Mechanics 45 (2010) 176–185

frequency expression for both linear and non-linear modeling 0000

g ð1Þ ¼  1

gð1Þ ¼ 0;

ð45Þ

Eq. (45) also imposes an important relation between the bending vibration at the microcantilever free end before and after changing the variable by: 

v ð1; tÞ ¼ vð1; tÞ

5 5 29 4 25 3 13 2 s  s þ s  s 96 96 48 48

ð47Þ

Now that both linear and non-linear models have been obtained, the frequency response of the system needs to be analyzed as described next.

3. Frequency response analysis In order to utilize multiple time scales [18,19], and ultimately solve Eq. (23), e is introduced as a perturbation parameter to separate the ordinary differential equation of motion into the following form q€ n þ o2n qn þ eg~ 3n q_ n þ eg~ 4n q3n þ eg~ 5n ðq2n q€ n þ qn q_ 2n Þ þ eg~ 7n z€ ðtÞ þ eg~ 8n zðtÞ þ eg~ 9n ¼ 0

as T1

e

ð3g~ 4n  2g~ 5n ÞA2n A n þ ðg~ 7n o2n þ g~ 8n ÞzeisT1   dAn ¼0 þ i g~ 3n on An þ2on dT1

ð56Þ

þ ...

ð49Þ

qn ðt; eÞ ¼ q0n ðT0 ; T1 Þ þ eq1n ðT0 ; T1 Þ þ . . .

ð50Þ

where T0 is associated with the time scale in which motions occur at the linear natural frequency on and T1 is the defining shift in the natural frequency due to nonlinearities. Substituting expressions (49) and (50) into (48) and equating the terms with the same order of e yields d2 q0n þ o2n q0n ¼ 0 dT02

ð51Þ

1 an eifn 2

ð57Þ

The frequency response analysis is commonly carried out in two different forms: (i) undamped free vibration and (ii) damped forced vibration. The former leads to the non-linear natural frequency while the latter results in the frequency response equation. To determine the non-linear natural frequency, the undamped, free vibration version of the equation of motion is considered here by taking zðtÞ ¼ 0 and g~ 3n ¼ 0 in Eq. (56). Hence, the polar form is substituted and the real and imaginary parts are separated to obtain 8 > < an ¼ an0 a2n0 d ð58Þ 2 > : dT fðT1 Þ ¼ 8o ð2g~ 5n on  3g~ 4n Þ n 1 The non-linear natural frequency (oNn) can be expressed as:

oNn ¼ on þ e

a2 d fðT1 Þ ¼ on þ e n0 ð2g~ 5n o2n  3g~ 4n Þ 8on dT1

ð59Þ

The same procedure is now repeated for the damped and forced vibration of Eq. (56). Consequently, the modulation equations of frequency and amplitude can be extracted as   8 d 1 g~ > > ~ 3n þ g~ 7n on  8n z sin cn a ðT Þ ¼  g a n n > 1 < dT1 on 2   2   ~ d 3 g a g~ z > 4n n > > cn ðT1 Þ ¼ s þ 2g~ 5n on  cos cn þ g~ 7n on  8n : on 8 on an dT1 ð60Þ in which cn ¼  fn þ sT1 . To relate the excitation frequency or its representative s to the vibration amplitude, cn is eliminated to generate the following frequency response equation:

ð61Þ

ð52Þ

The obtained frequency response equation obviously indicates that there may be more than one amplitude for specific detuning parameter, as will be discussed in the next section.Moreover, the stability of the response will be insured by the stability of Table 1 System parameters.

The solution of Eq. (51) is assumed to be in the form: q0n ðT0 ; T1 Þ ¼ An ðT1 Þeion T0 þcc

An ¼

16a2n g~ 23n o2n þ a2n ð8son  3g~ 4n a2n þ2g~ 5n o2n a2n Þ2 ¼ 64ðg~ 7n o2n  g~ 8n Þ2 z2

d2 2d2 d q1n þ o2n q1n ¼  q0n  g~ 3n q0n  g~ 4n q30n 2 dT0 dT1 dT0 dT0  2 ! d2 d g~ 5n q20n 2 q0n þ q0n q0n dT0 d T0 g~ 7n z€ ðtÞ þ g~ 8n zðtÞ þ g~ 9n ¼ 0

To solve Eq. (56) for the complex amplitude, it must be written in its polar form as

ð48Þ

Time and the steady-state solution are expanded by order of e

t ¼ T0 þ

Substituting Eqs. (53) and (55) into Eq. (52) gives the following secular terms as coefficients of eionT0 which produce timeincreasing solutions and ought to be eliminated as a consequence

ð46Þ

It is now possible to find an appropriate geometrical function by considering conditions (40) and (45) as gðsÞ ¼

181

ð53Þ

Sym.

Value

Unit

zðtÞ ¼ zeiOt ¼ zeiðon T0 þ sT1 Þ

AH

460 2 50 0.1 97 2.3  10  7 30 10 100 1 30 6  10  20

mm mm mm

in which s is a detuning parameter representing the deviation from natural frequency. z(t) can now be written as a harmonic function as:

l t b K Eb m Q rt rp z

where An is a complex amplitude and cc stands for the complex conjugate of the preceding terms. Also, the frequency of excitation is assumed to remain near the natural frequency by the following expression

O ¼ on þ es

ð54Þ

ð55Þ

s

N/m GPa kg/m nm nm nm Hz nm

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Fig. 5. Time response for d Z 400 nm.

Fig. 3. Time response for d o 150 nm.

Fig. 6. Variation of natural frequency.

Fig. 4. Time response for d o200 nm.

the trajectories of modulation equation about its stationary points where ðd=dT1 Þan ðT1 Þ ¼ 0 and ðd=dT1 Þcn ðT1 Þ ¼ 0. Hence, the modulation equations are linearized and the Jacobian matrix can be constructed as 2 6 6 Ja ¼ 6 4



1 g~ 2 3n

1 ð6g~ 5n o2n a2n þ 8son  9g~ 4n a2n Þ 8on an



3 an ð8son  3g~ 4n a2n þ 2g~ 5n o2n a2n Þ 7 8on 7 7 1 5  g~ 3n 2

ð62Þ The corresponding eigenvalues mn of the Jacobian matrix are the roots of the following characteristics equation 1 4

m2n þ g~ 3n mn þ g~ 23n þ 3

   1 1 3g~ 4n a2n 1 3g~ 4n a2n s þ g~ 5n on a2n  s þ g~ 5n on a2n  ¼0 8on 8on 3 4 4

ð63Þ

As clearly seen from Eq. (63), the sum of the eigenvalues can be written as rffiffiffiffiffiffiffiffi Z 1 ml4 Kz m1 þ m2 ¼  g~ 3n ¼  a3 Cn2 ds ¼  o0 ð64Þ Kz ml4 0 This indicates that one of the eigenvalues has always a negative real part which guarantees the stability of at least one of the stationary points. In order to investigate the linear natural frequency, the linear equation of motion is written in the following standard form of the vibration equations q€ n þ o2n qn þ g^ 3n q_ n þ g^ 7n z€ ðtÞ þ g^ 8n zðtÞ þ g^ 9n þ g^ 10n z_ ðtÞ ¼ 0

ð65Þ

Complex quantities method [17] is utilized for frequency response analysis of the linear system. Hence, the generalized coordinate and input motion are assumed to be harmonic complex quantities as qn ðtÞ ¼ An eiOt

ð66Þ

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Fig. 7. Difference between linear and non-linear natural frequency.

183

Fig. 9. Non-linear frequency response curve for d =150 nm and jump phenomenon region.

listed in Table 1. Microcantilever is chosen from the MikroMaschs series 17 with the low spring constant suitable for manipulation purposes and true imaging of soft samples in non-contact mode. The natural frequency of the microcantilever is 12 kHz. These microcantilevers possess a very sharp tip that protects soft samples from being damaged and are usually used in applications with lower resonance frequencies [20]. The smaller radius of the tip they have, the less possibility of sticking exists between the tip and sample. Nanoparticle is also chosen from 3200A series of ThermoScientifics and is made of polystyrene. At first, time responses of the tip vibration are simulated for different values of tip–nanoparticle separation values (d). For this, three types of motions with respect to d are considered:

Fig. 8. Amplitude of vibration for both linear and non-linear modeling.

zðtÞ ¼ zeiOt

ð67Þ

Substituting Eqs. (66) and (67) into Eq. (65) yields the following relation ðo2n  O2 þ ig^ 3n OÞAn þðg^ 8n  g^ 7n O2 þig^ 10n OÞz ¼ 0

ð68Þ

For primary resonance as stated in Eq. (54), and substituting Eq. (57) into Eq. (68), linear frequency response equation of the system is obtained as: 1 ½s2 ð2on þ sÞ2 4 2

 g^ 3n ðon þ sÞ2 a2n ¼ ½ðg^ 8n  g^ 7n ðon þ sÞ2 Þ2 þ g^ 10n o2n z2

ð69Þ

Unlike non-linear modeling, the linear system has always one stable solution. This important difference will be effectively utilized and elaborately discussed in the next section.

4. Numerical simulation and results In order to numerically investigate the obtained ordinary differential equation, a typical system of microcantilever–nanoparticle, as depicted in Fig. 1, is considered with the parameters

(1) For d o150 nm there is no harmonic motion and the attraction forces are dominant which force the tip to accelerate towards the particle without any vibration. (2) In the region where 150r150 nm, the microcantilever tip vibrates under the influence of nanoparticle. Such a motion is demonstrated in Figs. 3 and 4. It is obvious that the microcantilever bends downward and vibrates around the point lower than its static equilibrium due to the attraction forces between the tip and particle. Also, it vibrates with a beating type motion as a consequence of keeping the frequency of excitation very close to the natural frequency of vibration. By increasing the separation, the vibration amplitude increases until the distance reaches 400 nm. Despite these similarities, there is a difference in time responses between linear and non-linear models in either DC position of the tip and/or amplitude value. (3) For d Z400 nm, microcantilever vibration may not be affected by the tip–nanoparticle separation. In the other words, the tip escapes from the particle attraction region for d Z400 nm and microcantilever vibrates around its static equilibrium. The vibration of single microcantilever beam far away from the particle is shown in Fig. 5. The time response of linear model almost fits the non-linear model response when the microcantilever vibrates around its initial straight position. To better understand the influence of tip–particle interaction on microcantilever vibration, the resonance frequency expression (24b) is considered again. The resonance frequency (natural

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Fig. 10. Non-linear frequency response curve for three different separation values.

Fig. 11. Linear frequency response curve for three different separation values.

frequency) has two different parts according to the g~ 2n expression as given also below (please note that g~ 1n ¼ 1). Z 1 0000 g~ 2n ¼ o2n ¼ ð2b~ 3 Cn ð1ÞCn þ C n Cn Þds 0

The first part having b~ 3 is associated with the tip–particle 0000 interaction, while the second part including Cn is related to the microcantilever elasticity properties. In the first region, the interaction forces are very large and consequently g~ 2n o 0 in this region. Hence, there is no harmonic motion for d o150 nm. The microcantilever elasticity and the interaction forces reach to the same order of strength in region 2. Thus, the microcantilever vibrates and the nanoparticle influences the microcantilever vibration. For larger separation, the interaction force influences are gradually diminishing until they completely vanish in comparison with the microcantilever elasticity forces for d Z400 nm and the microcantilever vibrates as if there is no particle in its neighborhood. The variations of the linear natural frequency with respect to the tip–nanoparticle separation are illustrated in Fig. 6. The linear natural frequency for the combined microcantilever–nanoparticle

system is lower than the single microcantilever beam vibrating far away from the nanoparticle. The farther the tip gets away from the particle, the closer the natural frequency approaches the single microcantilever natural frequency, and this behavior is perfectly non-linear. For d Z400 nm, two curves coincide and it is this point where the tip escapes from the particle influence region. In addition to linear natural frequency, the non-linear estimation of the natural frequency of the microcantilever (Eq. (59)) is deeply affected by the particle attraction force. Fig. 7 shows the difference between linear and non-linear estimations of natural frequency with respect to separation values. It is obvious that for the smaller separation values, the difference of two estimations becomes more significant. In other words, the presence of nonlinearities becomes much important near the particle. To effectively investigate the difference between linear and non-linear models of microcantilever under the influence of nanoparticle, the amplitudes of vibration predicted by both models for different separation values are illustrated in Fig. 8. There is a sharp difference between the amplitude of vibration inside the particle attraction domain, which originates from the difference between modeling of attraction force in the linear and non-linear systems. It should be noted that in the linear modeling, the linear terms of the Taylor’s series expansion are adopted, while in the non-linear modeling, more terms are chosen to estimate the interaction force. Both curves finally approach each other, but they never completely merge. This is due to the difference between the linear and non-linear models of the microcantilever. The plot of the vibration amplitude an as a function of s in the frequency response equation (Eq. (38)), for given parameters in Table 1, is depicted in Fig. 9 for d = 150 nm. The bending of the curves to the right leads to the multivalued amplitudes in the region of ABCD in this figure. In other words, inside the specified region, the amplitude of vibration has more than one solution in the frequency response equation and may take each of the possible values. This behavior is accompanied with the sudden jump in the amplitude of vibration for the specific frequency of excitation. The observed jumps are the consequence of the multivaluedness of the frequency response curves, which in turn is a consequence of the nonlinearity. In Fig. 9, there are three steady-state solutions inside the region of ABCD. The middle one is a saddle point; hence the response corresponding to it is unstable. The other two are stable foci. The initial conditions determine which of the possible responses actually develops. The amplitude of the second stable foci is much larger than the tip–sample separation and it may contaminate or damage the particle or soft samples if it is run. Consequently, this phenomenon cannot be ignored at all for precise and sensitive manipulation purposes. The non-linear frequency response curves for three different values of separation are demonstrated in Fig. 10. It shows that the amplitude of vibration decreases when the tip approaches the particle. On the other hand, the bending of the frequency response curve originating from nonlinearities is amplified in the presence of particle as depicted in Fig. 10. Comparing linear and non-linear modeling, frequency response curves are different. The frequency response curve of linear model for three different separation values is also shown in Fig. 11. In both Figs. 10 and 11, by increasing the separation, the curves show the larger amplitude for greater distance. However, in a non-linear model, the curves also bend to right. The bending of the curves to the right means that for specific range of detuning parameters in the right half-plane of non-linear modeling there

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are more than one solution for amplitude in frequency response equation.

5. Conclusion The linear and non-linear vibration analyses of microcantilever beam subject to the tip–nanoparticle interaction were investigated and numerically simulated for a typical system of microcantilever and particle. Depending on the tip–nanoparticle separation, the time responses show that the tip may: (i) be absorbed by the particle, (ii) vibrates under the influence of the particle, (iii) vibrates as if there is no particle in its vicinity. The attraction region of the nanoparticle was profoundly inspected by considering linear natural frequency expression. This expression revealed that the tip–particle distance raises the natural frequency of vibration. If the separation distance exceeds a specific value, the natural frequency remains unchanged and demonstrates the farthest point of the particle attraction region. Moreover, the frequency response curve for different separation values showed that the amplitude of vibration is gradually decreasing by approaching the particle. In addition, there was a bending in the curves which was signified near the particle and led to the multivaluedness of amplitude. Consequently, the amplitude of vibration may abruptly grow by jumping from one stable focus to another stable one. This phenomenon originates from nonlinearities and cannot be predicted by linear or lumpedparameters modeling. In lumped system analysis, the exact form of the non-linear terms cannot be found and it was assumed that they are in quadratic or cubic forms based on other researches or experiments. In this paper, the terms that confirm and complete those researches have been analytically found. Overall comparison of linear and non-linear responses shows when the interatomic distance is quite large, weak differences are seen between linear and non-linear approaches. However, as the tip gradually approaches the particle, the non-linear terms are dominated and the differences in amplitude and frequency become significant. Hence, it can be concluded that the influences of nonlinearities are amplified near the particle, and consequently, the non-linear model should become the first priority for studying the system of microcantilever beam and particle.

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