# Resonance-Size Parameter Relationship and Dynamics of an AFM Subjected to Multimode Excitation and Based on the Modified Couple Stress Theory

Authors: Shueei-Muh Lin; Ching-Yao Chang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The mathematical model of AFM probe subjected to multimode excitation based on the modified couple stress theory is presented. The semianalytical solution of the system is proposed. The transient behavior and response spectrum of AFM probe subjected to multimode excitation are investigated. It is very helpful to predict the nanotopography and surface properties based on the response of multimodes excitation. The effects of the root excitation, size parameter, and interacting distance on the response spectrum and frequency shift are investigated. The resonant frequency relation of the two systems with different size parameters is discovered and expressed in a formula. The natural frequencies predicted via the formula and those determined by the semianalytical method are significantly consistent.


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## Body

## 1. Introduction

Atomic force microscopy (AFM) is a powerful device for scanning the atomic-scale topography and property of a sample’s surface [1].In convention, the literatures investigated the steady behavior of AFM probe [2–9]. Definitely, the investigation of the amplitude and phase behavior in the transient regime will increase significantly the speed of imaging and accuracy. Santos and Gaderlab [10] investigated the tip-sample force in transient behavior. Sahoo et al. [11] increased the speed of imaging and control of AFM by studying the transient motion of cantilever signal. Chang et al. [12] presented a new algorithm for high speed AFM imaging of biopolymers by investigating the transition from transient part to steady state regime. Payam [9] studied the transient behavior of tapping mode-AFM in the mass-spring-damper model.Some literatures [13–16] investigated the multifrequency excitation of FM-AFM systems in fast force spectroscopy and material characterization at the atomic scale as well. The topography and material properties of sample’s surface can be measured by using AFM subjected to the simultaneous multimodes excitations. The amplitude signal of the first mode is used to image the surface topography. The phase shift signal of the second mode is usually used to map changes in material properties of the sample’s surface [17–19].In convention, the structure’s size effect is not considered. Several literatures [20–23] found that the size-dependent effect on the dynamic behavior of the structures in the order of microns or submicrons is significant. Some nonclassical theories with size dependency are the couple stress theory [16, 24, 25], the strain gradient theory [26], the nonlocal theory [27], and the surface elasticity model [28]. Mindlin and Tiersten [24] and Toupin [25] presented the classical couple stress theory with the gradients of rotation and displacement. Yang et al. [29] modified the classical stress theory (MCST) by considering that the density of strain energy is a function of the strain and curvature tensors.Several literatures [30–34] investigated the size effect on the behavior of beam. Ansariet al. [35] investigated the size-dependent resonant frequency and flexural sensitivity of AFM based on the modified strain gradient theory by using the force gradient method. However, it is well known that the interpretation of frequency shift by using the force gradient method is unsatisfactory [4].So far, no literature is devoted to investigate the transient behavior of an AFM probe subjected to multimodes excitations based on the modified couple stress theory. In this study, the mathematical model of AFM probe subjected to two-mode excitation and the van der Waals force based on the modified couple stress theory is constructed. The semianalytical method is presented. The effects of several parameters on the transient behavior are investigated.


## 2. Dynamic System Subjected to Multi-Mode Excitation

### 2.1. Governing Equation and Boundary Conditions

Consider the transient response of atomic force microscopy excited at the root measuring the interatomic van der Waals force, as shown in Figure1. The multimode harmonic excitation is presented, as shown in Figure 1. The governing equation is(1)EI+μAlc2∂4W∂x4+ρA∂2W∂t2=0,0<x<LThe boundary conditions are as follows.Figure 1
Geometry and coordinate system of AFM.At x = 0:(2)W=We1sin⁡Ωe1t+We2sin⁡Ωe2t≡F1t,(3)∂W∂x=0,At x = L:(4)∂2W∂x2=0,(5)EI+μAlc2∂3W∂x3-mt∂2W∂t2=Fvwhere A and I denote the cross-sectional area and the area moment of inertia, respectively. ρ is the mass density per unit volume, mt the tip mass, W the flexural displacement, E Young's modulus, x the coordinate along the beam, L the length of the beam, and lc the material length scale parameter. The van der Waals force [11] is(6)Fv=-AHR6D2,in which AH is the Hamaker constant, D the tip-surface distance, and R the tip radius.In terms of the following dimensionless quantities:(7)bξ=ExIxE0I0,cv=AHRL3E0I0L03,D-=DL0,fv=-cv6D-0-w2=-cv6D-2,m=ρxAxρ0A0,wξ,τ=Wx,tL0,we1=We1L0,we2=We2L0,m-t=mtρ0A0L,ξ=xL,τ=tL2E0I0ρ0A0,ωei=ΩeiL2ρAEIδ=μAlc2EIwhere L0 is the characteristic length. A small value of L0 is introduced to avoid the numerical transaction error. The dimensionless governing differential equation of the system is(8)1+δ∂4w∂ξ4+∂2w∂τ2=0The associated boundary conditions are as follows.Atξ = 0:(9)w=we1sin⁡ωe1τ+we2sin⁡ωe2τ≡f1τ(10)∂w∂ξ=0⁡Atξ = 1:(11)∂2w∂ξ2=0,(12)1+δ∂3w∂ξ3-m-t∂2w∂τ2=fvw1,τThe initial conditions are(13)wξ,0=0,∂wξ,0∂τ=0


### 2.2. Solution Method

Because the system composed of (8)-(13) is nonlinear and the boundary condition is time-dependent, it is difficult to directly solve. The semianalytical method is presented here.


#### 2.2.1. Linearization of System

Due to its complexity, the nonlinear general system is approximated by infinite linearized subsystems. At first, the time variableτ is divided into infinite sections and the dynamic performance of the system is derived step by step. The methodology is described here. The tip displacement is presented by the Taylor series as(14)w~1,τ=w1,τi+∂w1,τi∂ττ-τi+12∂2w1,τi∂τ2τ-τi2+Δτi+1,∂w1,τ∂τ≈∂w1,τi∂τ+∂2w1,τi∂τ2τ-τi∂2w1,τ∂τ2≈∂2w1,τi∂τ2,forτi<τ<τi+1where Δτi+1 is an unknown transaction difference to be determined. Based on this relation and the following methods one can determine the displacement w(ξ,τ). Moreover, the error of the tip displacement must approach zero:(15)Errorτi+1=w1,τi+1-w~1,τi+1<ε→0Substituting (14) into the nonlinear nonhomogeneous boundary condition (12), the linearized time-dependent boundary condition is obtained:(16)∂3w∂ξ3≈11+δm-t∂2w1,τi∂τ2+fvw~1,τ≜f2τ,Δ,forτi<τ<τi+1So far, the nonlinear system is linearized in each time domain. The approximated system can be exactly solved by the solution method presented in the following sections. Therefore, the overall displacement can be derived step by step.So far, the linearized subsystem in the time domainτi,τi+1 can be expressed as follows:(17)1+δ∂4w∂ξ4+∂2w∂τ2=0,forτi<τ<τi+1The associated time-dependent boundary conditions are as follows.Atξ = 0:(18)w=we1sin⁡ωe1τ+we2sin⁡ωe2τ≡f1τ(19)∂w∂ξ=0Atξ = 1:(20)∂2w∂ξ2=0,(21)∂3w∂ξ3=f2τ,Δ,forτi<τ<τi+1


#### 2.2.2. Solution Method of Linearized Subsystem

(A) Transform of Variable. The linearized subsystem includes two nonhomogeneous time-dependent boundary conditions. Solving this subsystem, the following transformation of variable is considered:(22)wξ,τ=vξ,τ+∑i=12giξfiτ,forτi<τ<τi+1Substituting relation (22) into (17)-(21), two following subsystems are obtained. The first subsystem is expressed in terms of the transformed variable v(ξ,τ) as follows: the transformed governing equation is(23)1+δ∂4v∂ξ4+∂2v∂τ2=-∑i=12giξd2fidτ2,forτi<τ<τi+1The associated boundary conditions are as follows.Atξ = 0:(24)v0,τ=0,(25)∂v0,τ∂ξ=0Atξ = 1:(26)∂2v1,τ∂ξ2=0,(27)∂3v1,τ∂ξ3=0The corresponding initial conditions are(28)vξ,0=-∑i=12giξfi0(29)∂vξ,0∂τ=-∑i=12giξdfi0dτThe second subsystem is expressed in terms of the shifting function g1(ξ). The second transformed governing equation is(30)d4g1dξ4=0,0<ξ<1At ξ = 0:(31)g10=1,(32)dg10dξ=0At ξ = 1:(33)d2g11dξ2=0,(34)d3g11dξ3=0The third subsystem is expressed in terms of the shifting function g2(ξ). The third transformed governing equation is(35)d4g2dξ4=0,0<ξ<1At ξ = 0:(36)g20=0,(37)dg20dξ=0At ξ = 1:(38)d2g21dξ2=0,(39)d3g21dξ3=1The solutions of the two shifting functions {g1,g2} are easily discovered as follows:(40)g1ξ=1,(41)g2ξ=-12ξ2+16ξ3(B) Solution of the First Subsystem(B-1) Orthogonality of Eigenfunctions. The frequency equation of the first subsystem composed of (23)-(27) is(42)cosh⁡ω-cos⁡ω-+1=0where ω-=ω/1+δ. The nth mode shapes are (43)yn=sinh⁡ω-n+sin⁡ω-ncosh⁡ω-nξ-cos⁡ω-nξ-cosh⁡ω-n+cos⁡ω-nsinh⁡ω-nξ-sin⁡ω-nξwhich is the same as that given by Rao [36]. The orthogonality of the eigenfunctions is presented as follows:(44)∫01ynykdξ=0,n≠kδn,n=k(B-2) Modes Superposition Method. Based on the orthogonality conditions (44), the mode superposition method is used to derive the solution of the first subsystem composed of (23)-(27). The transformed variable is expressed in terms of the eigenfunctions(45)vξ,τ=∑n=1∞ynξTnτSubstituting (45) into (23) and multiplying it by yn and integrating it from 0 to 1, one obtains (46)d2Tmτdτ2+ωm2Tmτ=1δm∫01ftξ,τymdξ≡F~mτ,forτi<τ<τi+1;m=1,2,…where(47a)ftξ,t=ω12w1sin⁡ω1τ+ω22w2sin⁡ω2τ--12ξ2+16ξ311+δd2f~tsw1,τdτ2,F~nτ=1δnαnω12w1sin⁡ω1τ+ω22w2sin⁡ω2τ-βn1+δd2fvdτ2in which(47b)αn=-2ω-n1+cosh⁡ω-ncos⁡ω-n-cosh⁡ω-n-cos⁡ω-nβn=γn1+γn2sinh⁡ω-n+sin⁡ω-n-γn3+γn4cosh⁡ω-n+cos⁡ω-n,γn1=-12ω-n3/2ω-n+2sinh⁡ω-n-2ωncosh⁡ω-n-2ω-ncos⁡ω-n+ω-n-2sin⁡ω-n,γn2=16ω-n2ω-n3/2+6ω-nsinh⁡ω-n-3ω-n+6cosh⁡ω-n+6-3ω-n-6cos⁡ω-n+ω-n3/2-6ω-nsin⁡ω-n+6,γn3=-12ω-n3/2ω-n+2cosh⁡ω-n-2ω-nsinh⁡ω-n-2-2ω-nsin⁡ω-n-ω-n-2cos⁡ω-n-2γn4=16ω-n2ω-n3/2+6ω-ncosh⁡ω-n-3ω-n+6sinh⁡ω-n-3ω-n-6sin⁡ω-n-ω-n3/2-6ω-ncos⁡ω-n.The solution of (40) is(48a)Tnτ=cos⁡ωnτ-τiTnτi+1ωnsin⁡ωnτ-τidTnτidτ+1ωn∫τiτsin⁡ωnτ-χF~nχdχ(48b)dTmτdτ=-ωmsin⁡ωmτ-τiTmτi+cos⁡ωmτ-τidTmτidτ+∫τiτcos⁡ωmτ-χF~mχdχSubstituting (48a) and (48b) back into (22) results in the displacement of the cantilever. It should be noted that the tip displacement must satisfy condition (15).


### 2.3. Numerical Results of Transient Response and Discussion

The influences of the excitation frequencyfe1, the excitation amplitude We1, and tip-sample distance D0 on the tip transient response are investigated and shown in Figures 2(a) and 2(b). Figure 2(a) demonstrates that the interacting distance is D0=4 nm. The initial displacement and velocity are zero and are listed in (13). One has the natural frequency of the cantilever without the interacting force,f1 = 53525.9 Hz. When the frequency of root excitation is far from the natural frequency such as 48000 and 60000 Hz the small vibration of the cantilever is harmonically excited. When the frequency of excitation such as 53000 Hz approaches the natural frequencyf1, the vibration response is significantly increased. It is obvious that the frequency shift occurs due to the interacting force. Figure 2(b) demonstrates that the interacting distance is D0=2 nm. When the frequency of excitation is 52000 Hz, the vibration response is significantly increased. It is obvious from Figures 2(a) and 2(b) that the smaller the interacting distance D0 is, the greater the frequency shift is.Figure 2
The influences of the excitation frequencyfe1, the excitation amplitude We1, and tip-sample distance D0 on the tip transient response [A0=10nm,B = 45 μm, H = 2.50 μm,E = 70.3×109Pa,ρ = 2.5×103 kg/m3, mt=0.06723×10-13kg,L = 200 μm,k = 1.5447 N/m, andf1 = 53525.9 Hz,].
(a)
W e 1 = 0.05  n m , W e 2 = 0 , D 0 = 4  n m (b)
W e 1 = 0.01  n m , W e 2 = 0 , D 0 = 2  n mFigure3 demonstrates the influence of the distance between the tip and a sample surface on the tip response spectrum. It is shown that the smaller the interacting distance is, the larger the resonance frequency shift is.Figure 3
The influence of the distance between the tip and a sample surface on the tip response spectrum [A0=10nm,B = 45 μm, H = 3.50 μm,E = 70.3×109Pa,ρ = 2.5×103 kg/m3, mt=0.06723×10-13kg,L = 200 μm,k = 4.2386 N/m,f1 = 74941.4 Hz, We1=0.01nm, We2=0, and d0=3, 5, 10 nm].Figures4(a) and 4(b) demonstrate the influence of the size parameter δ on the tip response spectrum forD0= 3, 5 nm. It is observed that the larger the size parameter δ is, the greater the frequency shift is. Moreover, the smaller the interacting distance D0 is, the greater the frequency shift is.Figure 4
The influence of the size parameterδ on the tip response spectrum [A0=10nm,B = 45 μm, H = 2.50 μm,E = 70.3×109Pa,ρ = 2.5×103 kg/m3, mt=0.06723×10-13kg,L = 200 μm,k = 1.5447 N/m,f1 = 53525.9 Hz, W1e=0.01 nm, and W2b=0.0 nm].
(a) (b)Further, the influence of two excitation modes on the tip response spectrum is investigated; Figure5 demonstrates that if the second frequency of excitation fe2 = 76500 Hz is far from the natural frequencyf1 = 53525.9 Hz, the response spectrum is as shown Figures 3 and 4 with one-mode excitation. Moreover, the influence of the interacting distanceD0 on the response spectrum is slight. However, if the second frequency of excitation fe2 = 56500 Hz is close to the natural frequencyf1 = 53525.9 Hz, several resonant responses occur. Moreover, the influence of the interacting distanceD0 on the response spectrum is significant. It is concluded that the response spectrum of two-mode excitations is significantly different to that of one-mode excitation.Figure 5
The influence of the two-mode excitation on the tip response [δ=0.1, A0=10nm,B = 45 μm, H = 2.50 μm,E = 70.3×109Pa,ρ = 2.5×103 kg/m3, mt=0.06723×10-13kg,L = 200 μm,k = 1.5447 N/m,f1 = 53525.9 Hz, and W1e= W2e=0.01 nm].


## 2.1. Governing Equation and Boundary Conditions

Consider the transient response of atomic force microscopy excited at the root measuring the interatomic van der Waals force, as shown in Figure1. The multimode harmonic excitation is presented, as shown in Figure 1. The governing equation is(1)EI+μAlc2∂4W∂x4+ρA∂2W∂t2=0,0<x<LThe boundary conditions are as follows.Figure 1
Geometry and coordinate system of AFM.At x = 0:(2)W=We1sin⁡Ωe1t+We2sin⁡Ωe2t≡F1t,(3)∂W∂x=0,At x = L:(4)∂2W∂x2=0,(5)EI+μAlc2∂3W∂x3-mt∂2W∂t2=Fvwhere A and I denote the cross-sectional area and the area moment of inertia, respectively. ρ is the mass density per unit volume, mt the tip mass, W the flexural displacement, E Young's modulus, x the coordinate along the beam, L the length of the beam, and lc the material length scale parameter. The van der Waals force [11] is(6)Fv=-AHR6D2,in which AH is the Hamaker constant, D the tip-surface distance, and R the tip radius.In terms of the following dimensionless quantities:(7)bξ=ExIxE0I0,cv=AHRL3E0I0L03,D-=DL0,fv=-cv6D-0-w2=-cv6D-2,m=ρxAxρ0A0,wξ,τ=Wx,tL0,we1=We1L0,we2=We2L0,m-t=mtρ0A0L,ξ=xL,τ=tL2E0I0ρ0A0,ωei=ΩeiL2ρAEIδ=μAlc2EIwhere L0 is the characteristic length. A small value of L0 is introduced to avoid the numerical transaction error. The dimensionless governing differential equation of the system is(8)1+δ∂4w∂ξ4+∂2w∂τ2=0The associated boundary conditions are as follows.Atξ = 0:(9)w=we1sin⁡ωe1τ+we2sin⁡ωe2τ≡f1τ(10)∂w∂ξ=0⁡Atξ = 1:(11)∂2w∂ξ2=0,(12)1+δ∂3w∂ξ3-m-t∂2w∂τ2=fvw1,τThe initial conditions are(13)wξ,0=0,∂wξ,0∂τ=0


## 2.2. Solution Method

Because the system composed of (8)-(13) is nonlinear and the boundary condition is time-dependent, it is difficult to directly solve. The semianalytical method is presented here.


### 2.2.1. Linearization of System

Due to its complexity, the nonlinear general system is approximated by infinite linearized subsystems. At first, the time variableτ is divided into infinite sections and the dynamic performance of the system is derived step by step. The methodology is described here. The tip displacement is presented by the Taylor series as(14)w~1,τ=w1,τi+∂w1,τi∂ττ-τi+12∂2w1,τi∂τ2τ-τi2+Δτi+1,∂w1,τ∂τ≈∂w1,τi∂τ+∂2w1,τi∂τ2τ-τi∂2w1,τ∂τ2≈∂2w1,τi∂τ2,forτi<τ<τi+1where Δτi+1 is an unknown transaction difference to be determined. Based on this relation and the following methods one can determine the displacement w(ξ,τ). Moreover, the error of the tip displacement must approach zero:(15)Errorτi+1=w1,τi+1-w~1,τi+1<ε→0Substituting (14) into the nonlinear nonhomogeneous boundary condition (12), the linearized time-dependent boundary condition is obtained:(16)∂3w∂ξ3≈11+δm-t∂2w1,τi∂τ2+fvw~1,τ≜f2τ,Δ,forτi<τ<τi+1So far, the nonlinear system is linearized in each time domain. The approximated system can be exactly solved by the solution method presented in the following sections. Therefore, the overall displacement can be derived step by step.So far, the linearized subsystem in the time domainτi,τi+1 can be expressed as follows:(17)1+δ∂4w∂ξ4+∂2w∂τ2=0,forτi<τ<τi+1The associated time-dependent boundary conditions are as follows.Atξ = 0:(18)w=we1sin⁡ωe1τ+we2sin⁡ωe2τ≡f1τ(19)∂w∂ξ=0Atξ = 1:(20)∂2w∂ξ2=0,(21)∂3w∂ξ3=f2τ,Δ,forτi<τ<τi+1


### 2.2.2. Solution Method of Linearized Subsystem

(A) Transform of Variable. The linearized subsystem includes two nonhomogeneous time-dependent boundary conditions. Solving this subsystem, the following transformation of variable is considered:(22)wξ,τ=vξ,τ+∑i=12giξfiτ,forτi<τ<τi+1Substituting relation (22) into (17)-(21), two following subsystems are obtained. The first subsystem is expressed in terms of the transformed variable v(ξ,τ) as follows: the transformed governing equation is(23)1+δ∂4v∂ξ4+∂2v∂τ2=-∑i=12giξd2fidτ2,forτi<τ<τi+1The associated boundary conditions are as follows.Atξ = 0:(24)v0,τ=0,(25)∂v0,τ∂ξ=0Atξ = 1:(26)∂2v1,τ∂ξ2=0,(27)∂3v1,τ∂ξ3=0The corresponding initial conditions are(28)vξ,0=-∑i=12giξfi0(29)∂vξ,0∂τ=-∑i=12giξdfi0dτThe second subsystem is expressed in terms of the shifting function g1(ξ). The second transformed governing equation is(30)d4g1dξ4=0,0<ξ<1At ξ = 0:(31)g10=1,(32)dg10dξ=0At ξ = 1:(33)d2g11dξ2=0,(34)d3g11dξ3=0The third subsystem is expressed in terms of the shifting function g2(ξ). The third transformed governing equation is(35)d4g2dξ4=0,0<ξ<1At ξ = 0:(36)g20=0,(37)dg20dξ=0At ξ = 1:(38)d2g21dξ2=0,(39)d3g21dξ3=1The solutions of the two shifting functions {g1,g2} are easily discovered as follows:(40)g1ξ=1,(41)g2ξ=-12ξ2+16ξ3(B) Solution of the First Subsystem(B-1) Orthogonality of Eigenfunctions. The frequency equation of the first subsystem composed of (23)-(27) is(42)cosh⁡ω-cos⁡ω-+1=0where ω-=ω/1+δ. The nth mode shapes are (43)yn=sinh⁡ω-n+sin⁡ω-ncosh⁡ω-nξ-cos⁡ω-nξ-cosh⁡ω-n+cos⁡ω-nsinh⁡ω-nξ-sin⁡ω-nξwhich is the same as that given by Rao [36]. The orthogonality of the eigenfunctions is presented as follows:(44)∫01ynykdξ=0,n≠kδn,n=k(B-2) Modes Superposition Method. Based on the orthogonality conditions (44), the mode superposition method is used to derive the solution of the first subsystem composed of (23)-(27). The transformed variable is expressed in terms of the eigenfunctions(45)vξ,τ=∑n=1∞ynξTnτSubstituting (45) into (23) and multiplying it by yn and integrating it from 0 to 1, one obtains (46)d2Tmτdτ2+ωm2Tmτ=1δm∫01ftξ,τymdξ≡F~mτ,forτi<τ<τi+1;m=1,2,…where(47a)ftξ,t=ω12w1sin⁡ω1τ+ω22w2sin⁡ω2τ--12ξ2+16ξ311+δd2f~tsw1,τdτ2,F~nτ=1δnαnω12w1sin⁡ω1τ+ω22w2sin⁡ω2τ-βn1+δd2fvdτ2in which(47b)αn=-2ω-n1+cosh⁡ω-ncos⁡ω-n-cosh⁡ω-n-cos⁡ω-nβn=γn1+γn2sinh⁡ω-n+sin⁡ω-n-γn3+γn4cosh⁡ω-n+cos⁡ω-n,γn1=-12ω-n3/2ω-n+2sinh⁡ω-n-2ωncosh⁡ω-n-2ω-ncos⁡ω-n+ω-n-2sin⁡ω-n,γn2=16ω-n2ω-n3/2+6ω-nsinh⁡ω-n-3ω-n+6cosh⁡ω-n+6-3ω-n-6cos⁡ω-n+ω-n3/2-6ω-nsin⁡ω-n+6,γn3=-12ω-n3/2ω-n+2cosh⁡ω-n-2ω-nsinh⁡ω-n-2-2ω-nsin⁡ω-n-ω-n-2cos⁡ω-n-2γn4=16ω-n2ω-n3/2+6ω-ncosh⁡ω-n-3ω-n+6sinh⁡ω-n-3ω-n-6sin⁡ω-n-ω-n3/2-6ω-ncos⁡ω-n.The solution of (40) is(48a)Tnτ=cos⁡ωnτ-τiTnτi+1ωnsin⁡ωnτ-τidTnτidτ+1ωn∫τiτsin⁡ωnτ-χF~nχdχ(48b)dTmτdτ=-ωmsin⁡ωmτ-τiTmτi+cos⁡ωmτ-τidTmτidτ+∫τiτcos⁡ωmτ-χF~mχdχSubstituting (48a) and (48b) back into (22) results in the displacement of the cantilever. It should be noted that the tip displacement must satisfy condition (15).


## 2.2.1. Linearization of System

Due to its complexity, the nonlinear general system is approximated by infinite linearized subsystems. At first, the time variableτ is divided into infinite sections and the dynamic performance of the system is derived step by step. The methodology is described here. The tip displacement is presented by the Taylor series as(14)w~1,τ=w1,τi+∂w1,τi∂ττ-τi+12∂2w1,τi∂τ2τ-τi2+Δτi+1,∂w1,τ∂τ≈∂w1,τi∂τ+∂2w1,τi∂τ2τ-τi∂2w1,τ∂τ2≈∂2w1,τi∂τ2,forτi<τ<τi+1where Δτi+1 is an unknown transaction difference to be determined. Based on this relation and the following methods one can determine the displacement w(ξ,τ). Moreover, the error of the tip displacement must approach zero:(15)Errorτi+1=w1,τi+1-w~1,τi+1<ε→0Substituting (14) into the nonlinear nonhomogeneous boundary condition (12), the linearized time-dependent boundary condition is obtained:(16)∂3w∂ξ3≈11+δm-t∂2w1,τi∂τ2+fvw~1,τ≜f2τ,Δ,forτi<τ<τi+1So far, the nonlinear system is linearized in each time domain. The approximated system can be exactly solved by the solution method presented in the following sections. Therefore, the overall displacement can be derived step by step.So far, the linearized subsystem in the time domainτi,τi+1 can be expressed as follows:(17)1+δ∂4w∂ξ4+∂2w∂τ2=0,forτi<τ<τi+1The associated time-dependent boundary conditions are as follows.Atξ = 0:(18)w=we1sin⁡ωe1τ+we2sin⁡ωe2τ≡f1τ(19)∂w∂ξ=0Atξ = 1:(20)∂2w∂ξ2=0,(21)∂3w∂ξ3=f2τ,Δ,forτi<τ<τi+1


## 2.2.2. Solution Method of Linearized Subsystem

(A) Transform of Variable. The linearized subsystem includes two nonhomogeneous time-dependent boundary conditions. Solving this subsystem, the following transformation of variable is considered:(22)wξ,τ=vξ,τ+∑i=12giξfiτ,forτi<τ<τi+1Substituting relation (22) into (17)-(21), two following subsystems are obtained. The first subsystem is expressed in terms of the transformed variable v(ξ,τ) as follows: the transformed governing equation is(23)1+δ∂4v∂ξ4+∂2v∂τ2=-∑i=12giξd2fidτ2,forτi<τ<τi+1The associated boundary conditions are as follows.Atξ = 0:(24)v0,τ=0,(25)∂v0,τ∂ξ=0Atξ = 1:(26)∂2v1,τ∂ξ2=0,(27)∂3v1,τ∂ξ3=0The corresponding initial conditions are(28)vξ,0=-∑i=12giξfi0(29)∂vξ,0∂τ=-∑i=12giξdfi0dτThe second subsystem is expressed in terms of the shifting function g1(ξ). The second transformed governing equation is(30)d4g1dξ4=0,0<ξ<1At ξ = 0:(31)g10=1,(32)dg10dξ=0At ξ = 1:(33)d2g11dξ2=0,(34)d3g11dξ3=0The third subsystem is expressed in terms of the shifting function g2(ξ). The third transformed governing equation is(35)d4g2dξ4=0,0<ξ<1At ξ = 0:(36)g20=0,(37)dg20dξ=0At ξ = 1:(38)d2g21dξ2=0,(39)d3g21dξ3=1The solutions of the two shifting functions {g1,g2} are easily discovered as follows:(40)g1ξ=1,(41)g2ξ=-12ξ2+16ξ3(B) Solution of the First Subsystem(B-1) Orthogonality of Eigenfunctions. The frequency equation of the first subsystem composed of (23)-(27) is(42)cosh⁡ω-cos⁡ω-+1=0where ω-=ω/1+δ. The nth mode shapes are (43)yn=sinh⁡ω-n+sin⁡ω-ncosh⁡ω-nξ-cos⁡ω-nξ-cosh⁡ω-n+cos⁡ω-nsinh⁡ω-nξ-sin⁡ω-nξwhich is the same as that given by Rao [36]. The orthogonality of the eigenfunctions is presented as follows:(44)∫01ynykdξ=0,n≠kδn,n=k(B-2) Modes Superposition Method. Based on the orthogonality conditions (44), the mode superposition method is used to derive the solution of the first subsystem composed of (23)-(27). The transformed variable is expressed in terms of the eigenfunctions(45)vξ,τ=∑n=1∞ynξTnτSubstituting (45) into (23) and multiplying it by yn and integrating it from 0 to 1, one obtains (46)d2Tmτdτ2+ωm2Tmτ=1δm∫01ftξ,τymdξ≡F~mτ,forτi<τ<τi+1;m=1,2,…where(47a)ftξ,t=ω12w1sin⁡ω1τ+ω22w2sin⁡ω2τ--12ξ2+16ξ311+δd2f~tsw1,τdτ2,F~nτ=1δnαnω12w1sin⁡ω1τ+ω22w2sin⁡ω2τ-βn1+δd2fvdτ2in which(47b)αn=-2ω-n1+cosh⁡ω-ncos⁡ω-n-cosh⁡ω-n-cos⁡ω-nβn=γn1+γn2sinh⁡ω-n+sin⁡ω-n-γn3+γn4cosh⁡ω-n+cos⁡ω-n,γn1=-12ω-n3/2ω-n+2sinh⁡ω-n-2ωncosh⁡ω-n-2ω-ncos⁡ω-n+ω-n-2sin⁡ω-n,γn2=16ω-n2ω-n3/2+6ω-nsinh⁡ω-n-3ω-n+6cosh⁡ω-n+6-3ω-n-6cos⁡ω-n+ω-n3/2-6ω-nsin⁡ω-n+6,γn3=-12ω-n3/2ω-n+2cosh⁡ω-n-2ω-nsinh⁡ω-n-2-2ω-nsin⁡ω-n-ω-n-2cos⁡ω-n-2γn4=16ω-n2ω-n3/2+6ω-ncosh⁡ω-n-3ω-n+6sinh⁡ω-n-3ω-n-6sin⁡ω-n-ω-n3/2-6ω-ncos⁡ω-n.The solution of (40) is(48a)Tnτ=cos⁡ωnτ-τiTnτi+1ωnsin⁡ωnτ-τidTnτidτ+1ωn∫τiτsin⁡ωnτ-χF~nχdχ(48b)dTmτdτ=-ωmsin⁡ωmτ-τiTmτi+cos⁡ωmτ-τidTmτidτ+∫τiτcos⁡ωmτ-χF~mχdχSubstituting (48a) and (48b) back into (22) results in the displacement of the cantilever. It should be noted that the tip displacement must satisfy condition (15).


## 2.3. Numerical Results of Transient Response and Discussion

The influences of the excitation frequencyfe1, the excitation amplitude We1, and tip-sample distance D0 on the tip transient response are investigated and shown in Figures 2(a) and 2(b). Figure 2(a) demonstrates that the interacting distance is D0=4 nm. The initial displacement and velocity are zero and are listed in (13). One has the natural frequency of the cantilever without the interacting force,f1 = 53525.9 Hz. When the frequency of root excitation is far from the natural frequency such as 48000 and 60000 Hz the small vibration of the cantilever is harmonically excited. When the frequency of excitation such as 53000 Hz approaches the natural frequencyf1, the vibration response is significantly increased. It is obvious that the frequency shift occurs due to the interacting force. Figure 2(b) demonstrates that the interacting distance is D0=2 nm. When the frequency of excitation is 52000 Hz, the vibration response is significantly increased. It is obvious from Figures 2(a) and 2(b) that the smaller the interacting distance D0 is, the greater the frequency shift is.Figure 2
The influences of the excitation frequencyfe1, the excitation amplitude We1, and tip-sample distance D0 on the tip transient response [A0=10nm,B = 45 μm, H = 2.50 μm,E = 70.3×109Pa,ρ = 2.5×103 kg/m3, mt=0.06723×10-13kg,L = 200 μm,k = 1.5447 N/m, andf1 = 53525.9 Hz,].
(a)
W e 1 = 0.05  n m , W e 2 = 0 , D 0 = 4  n m (b)
W e 1 = 0.01  n m , W e 2 = 0 , D 0 = 2  n mFigure3 demonstrates the influence of the distance between the tip and a sample surface on the tip response spectrum. It is shown that the smaller the interacting distance is, the larger the resonance frequency shift is.Figure 3
The influence of the distance between the tip and a sample surface on the tip response spectrum [A0=10nm,B = 45 μm, H = 3.50 μm,E = 70.3×109Pa,ρ = 2.5×103 kg/m3, mt=0.06723×10-13kg,L = 200 μm,k = 4.2386 N/m,f1 = 74941.4 Hz, We1=0.01nm, We2=0, and d0=3, 5, 10 nm].Figures4(a) and 4(b) demonstrate the influence of the size parameter δ on the tip response spectrum forD0= 3, 5 nm. It is observed that the larger the size parameter δ is, the greater the frequency shift is. Moreover, the smaller the interacting distance D0 is, the greater the frequency shift is.Figure 4
The influence of the size parameterδ on the tip response spectrum [A0=10nm,B = 45 μm, H = 2.50 μm,E = 70.3×109Pa,ρ = 2.5×103 kg/m3, mt=0.06723×10-13kg,L = 200 μm,k = 1.5447 N/m,f1 = 53525.9 Hz, W1e=0.01 nm, and W2b=0.0 nm].
(a) (b)Further, the influence of two excitation modes on the tip response spectrum is investigated; Figure5 demonstrates that if the second frequency of excitation fe2 = 76500 Hz is far from the natural frequencyf1 = 53525.9 Hz, the response spectrum is as shown Figures 3 and 4 with one-mode excitation. Moreover, the influence of the interacting distanceD0 on the response spectrum is slight. However, if the second frequency of excitation fe2 = 56500 Hz is close to the natural frequencyf1 = 53525.9 Hz, several resonant responses occur. Moreover, the influence of the interacting distanceD0 on the response spectrum is significant. It is concluded that the response spectrum of two-mode excitations is significantly different to that of one-mode excitation.Figure 5
The influence of the two-mode excitation on the tip response [δ=0.1, A0=10nm,B = 45 μm, H = 2.50 μm,E = 70.3×109Pa,ρ = 2.5×103 kg/m3, mt=0.06723×10-13kg,L = 200 μm,k = 1.5447 N/m,f1 = 53525.9 Hz, and W1e= W2e=0.01 nm].


## 3. Prediction of Natural Frequencies

### 3.1. The System with the Size Effect

The dimensionless governing equation and boundary conditions are(49)1+δ∂4wδ∂ξ4+∂2wδ∂τ2=0The clamped boundary conditions are as follows.Atξ= 0:(50)wδ=0,(51)and∂wδ∂ξ=0Atξ= 1:(52)∂2wδ∂ξ2=0(53)1+δ∂3wδ∂ξ3-μtip∂2wδ∂τ2=fvδwhere fvδ=-cvδ/6dδ2 and cvδ=AHRδL3/EIL03, and the tip-surface distance is dδ=1-wδ(1,τ).(A) Solution Method. Assume(54)wδξ,τ=w-δξcos⁡ωδτ,Substituting (54) into (49), one obtains(55)d4w-δdξ4-ω-δ2w-δ=0where ω-δ=ωδ/1+δ. Substituting (54) into the boundary conditions (50)-(52), one obtains the following.Atξ= 0:(56)w-δ=0,(57)anddw-δdξ=0Atξ= 1:(58)d2w-δdξ2=0Substituting (54) into the boundary conditions (53) and multiplying it by cos⁡ωτ and integrating it from 0 to the period T, 2π/ω, (53) becomes(59)d3w-δdξ3+μtipω-δ2w-δ=-c-vδw-δ131-w-δ213/2where c-vδ=cvδ/1+δ. Further, the semianalytical solution can be determined by using the method by Lin [2].


### 3.2. The Classical System without the Size Effect

The dimensionless governing equation and boundary conditions are(60)∂4w0∂ξ4+∂2w0∂τ2=0The clamped boundary conditions are as follows.Atξ= 0:(61)w0=0,(62)and∂w0∂ξ=0Atξ= 1:(63)∂2w0∂ξ2=0(64)∂3w0∂ξ3-μtip∂2w0∂τ2=fv0where fv0=-cv0/6d2 and cv0=AHR0L3/EIL03, and the tip-surface distance is d0=1-w0(1,τ).(A) Solution Method. Assume(65)w0ξ,τ=w-0ξcos⁡ω0τ,Substituting (65) into (60), one obtains(66)d4w-0dξ4-ω02w-0=0Substituting (65) into the boundary conditions (61)-(63), one obtains the following.Atξ= 0:(67)w-0=0,(68)anddw-0dξ=0Atξ= 1:(69)d2w-0dξ2=0Substituting (65) into the boundary conditions (64) and multiplying it by cos⁡ωτ and integrating it from 0 to the period T, 2π/ω, (64) becomes(70)d3w-0dξ3+μtipω02w-0=-cv0w-0131-w-0213/2Further, the semianalytical solution can be determined by using the method by Lin [2].


### 3.3. Relation between Natural Frequencies and Size Parameter

It is discovered that if the following relations are satisfied(71)ω-δ=ω0orωδ1+δ=ω0(72)cv0=c-vδorAHR0L3EIL03=AHRδ/1+δL3EIL03the system composed of (55)-(59) is similar to that composed of (66)-(70) and w-0=w-δ.In other words, if the parametersE,I,L,L0 of the two systems are the same, the tip radius relation R0=Rδ/1+δ and w-0=w-δ, the relation of natural frequencies (71) must be satisfied.Tables1 and 2 demonstrate the comparison of the first two natural frequencies determined by using the analytical method [2, 4] and formula (71). It is found that these results are significantly consistent. When the interacting distance is large, the results are the same. When the interacting distance D0=5nm, the errors of the first two frequencies are 0.08% and 0.002%, respectively.Table 1
Effects of the interacting distanceD0, the axial deformation, and the size dependency parameter δ on the first frequency [SiO2: E = 70.3×109Pa, ρ = 2.5×103 kg/m3, B =45 μm, H = 3.5 μm, L=200 μm, R = 1.5×10-7m, AH = 10-19Joule, and mtip=3.18×10-13kg; the first natural frequency of beam with δ=0, f1 = 74355.803 Hz, and W-L=3nm].
D 0 (nm) f 1(Hz) Error(%) δ = 0.0 δ = 0.1 Semianalytical method Based on (71) 5 73686.61 77347.30 77283.17 0.0829  6 74051.34 77694.80 77665.68 0.0375  7 74187.16 77824.20 77808.15 0.0206  8 74251.28 77885.32 77875.40 0.0127  9 74286.01 77918.47 77911.82 0.0085  10 74306.66 77938.20 77933.48 0.0061  11 74319.82 77950.69 77947.28 0.0044  12 74328.64 77959.10 77956.54 0.0033  13 74334.81 77964.93 77963.01 0.0025  14 74339.14 77969.18 77967.55 0.0021  15 74342.38 77972.18 77970.95 0.0016  20 74350.29 77979.84 77979.24 0.0008  40 74354.29 77983.50 77983.43 0.0001  80 74355.73 77984.95 77984.95 0.0000Table 2
Effects of the interacting distanceD0, the axial deformation, and the size dependency parameter δ on the second frequency f2 [SiO2: E = 70.3×109Pa, ρ = 2.5×103 kg/m3, B =45 μm, H = 3.5 μm, L=200 μm, R = 1.5×10-7m, AH = 10-19Joule, and mtip=3.18×10-13kg; the first natural frequency of beam with δ=0, f1 = 74355.803 Hz, and W-L=3nm].
D 0 (nm) f 2(Hz) Error(%) δ = 0.0 δ = 0.1 Semianalytical method Based on (71) 5 465927.01 488678.25 488668.37 0.0020  6 465983.22 488731.79 488727.32 0.0009  7 466004.20 488751.78 488749.33 0.0005  8 466014.11 488761.27 488759.72 0.0003  9 466019.52 488766.43 488765.40 0.0002  10 466022.69 488769.43 488768.72 0.0001  11 466024.77 488771.43 488770.90 0.0001  12 466026.19 488772.76 488772.39 0.0001  13 466027.02 488773.59 488773.26 0.0001  14 466027.77 488774.26 488774.05 0.0000  15 466028.27 488774.76 488774.57 0.0000  20 466029.52 488775.93 488775.88 0.0000  40 466030.18 488776.44 488776.58 0.0000  80 466030.34 488766.74 488766.74 0.0000


## 3.1. The System with the Size Effect

The dimensionless governing equation and boundary conditions are(49)1+δ∂4wδ∂ξ4+∂2wδ∂τ2=0The clamped boundary conditions are as follows.Atξ= 0:(50)wδ=0,(51)and∂wδ∂ξ=0Atξ= 1:(52)∂2wδ∂ξ2=0(53)1+δ∂3wδ∂ξ3-μtip∂2wδ∂τ2=fvδwhere fvδ=-cvδ/6dδ2 and cvδ=AHRδL3/EIL03, and the tip-surface distance is dδ=1-wδ(1,τ).(A) Solution Method. Assume(54)wδξ,τ=w-δξcos⁡ωδτ,Substituting (54) into (49), one obtains(55)d4w-δdξ4-ω-δ2w-δ=0where ω-δ=ωδ/1+δ. Substituting (54) into the boundary conditions (50)-(52), one obtains the following.Atξ= 0:(56)w-δ=0,(57)anddw-δdξ=0Atξ= 1:(58)d2w-δdξ2=0Substituting (54) into the boundary conditions (53) and multiplying it by cos⁡ωτ and integrating it from 0 to the period T, 2π/ω, (53) becomes(59)d3w-δdξ3+μtipω-δ2w-δ=-c-vδw-δ131-w-δ213/2where c-vδ=cvδ/1+δ. Further, the semianalytical solution can be determined by using the method by Lin [2].


## 3.2. The Classical System without the Size Effect

The dimensionless governing equation and boundary conditions are(60)∂4w0∂ξ4+∂2w0∂τ2=0The clamped boundary conditions are as follows.Atξ= 0:(61)w0=0,(62)and∂w0∂ξ=0Atξ= 1:(63)∂2w0∂ξ2=0(64)∂3w0∂ξ3-μtip∂2w0∂τ2=fv0where fv0=-cv0/6d2 and cv0=AHR0L3/EIL03, and the tip-surface distance is d0=1-w0(1,τ).(A) Solution Method. Assume(65)w0ξ,τ=w-0ξcos⁡ω0τ,Substituting (65) into (60), one obtains(66)d4w-0dξ4-ω02w-0=0Substituting (65) into the boundary conditions (61)-(63), one obtains the following.Atξ= 0:(67)w-0=0,(68)anddw-0dξ=0Atξ= 1:(69)d2w-0dξ2=0Substituting (65) into the boundary conditions (64) and multiplying it by cos⁡ωτ and integrating it from 0 to the period T, 2π/ω, (64) becomes(70)d3w-0dξ3+μtipω02w-0=-cv0w-0131-w-0213/2Further, the semianalytical solution can be determined by using the method by Lin [2].


## 3.3. Relation between Natural Frequencies and Size Parameter

It is discovered that if the following relations are satisfied(71)ω-δ=ω0orωδ1+δ=ω0(72)cv0=c-vδorAHR0L3EIL03=AHRδ/1+δL3EIL03the system composed of (55)-(59) is similar to that composed of (66)-(70) and w-0=w-δ.In other words, if the parametersE,I,L,L0 of the two systems are the same, the tip radius relation R0=Rδ/1+δ and w-0=w-δ, the relation of natural frequencies (71) must be satisfied.Tables1 and 2 demonstrate the comparison of the first two natural frequencies determined by using the analytical method [2, 4] and formula (71). It is found that these results are significantly consistent. When the interacting distance is large, the results are the same. When the interacting distance D0=5nm, the errors of the first two frequencies are 0.08% and 0.002%, respectively.Table 1
Effects of the interacting distanceD0, the axial deformation, and the size dependency parameter δ on the first frequency [SiO2: E = 70.3×109Pa, ρ = 2.5×103 kg/m3, B =45 μm, H = 3.5 μm, L=200 μm, R = 1.5×10-7m, AH = 10-19Joule, and mtip=3.18×10-13kg; the first natural frequency of beam with δ=0, f1 = 74355.803 Hz, and W-L=3nm].
D 0 (nm) f 1(Hz) Error(%) δ = 0.0 δ = 0.1 Semianalytical method Based on (71) 5 73686.61 77347.30 77283.17 0.0829  6 74051.34 77694.80 77665.68 0.0375  7 74187.16 77824.20 77808.15 0.0206  8 74251.28 77885.32 77875.40 0.0127  9 74286.01 77918.47 77911.82 0.0085  10 74306.66 77938.20 77933.48 0.0061  11 74319.82 77950.69 77947.28 0.0044  12 74328.64 77959.10 77956.54 0.0033  13 74334.81 77964.93 77963.01 0.0025  14 74339.14 77969.18 77967.55 0.0021  15 74342.38 77972.18 77970.95 0.0016  20 74350.29 77979.84 77979.24 0.0008  40 74354.29 77983.50 77983.43 0.0001  80 74355.73 77984.95 77984.95 0.0000Table 2
Effects of the interacting distanceD0, the axial deformation, and the size dependency parameter δ on the second frequency f2 [SiO2: E = 70.3×109Pa, ρ = 2.5×103 kg/m3, B =45 μm, H = 3.5 μm, L=200 μm, R = 1.5×10-7m, AH = 10-19Joule, and mtip=3.18×10-13kg; the first natural frequency of beam with δ=0, f1 = 74355.803 Hz, and W-L=3nm].
D 0 (nm) f 2(Hz) Error(%) δ = 0.0 δ = 0.1 Semianalytical method Based on (71) 5 465927.01 488678.25 488668.37 0.0020  6 465983.22 488731.79 488727.32 0.0009  7 466004.20 488751.78 488749.33 0.0005  8 466014.11 488761.27 488759.72 0.0003  9 466019.52 488766.43 488765.40 0.0002  10 466022.69 488769.43 488768.72 0.0001  11 466024.77 488771.43 488770.90 0.0001  12 466026.19 488772.76 488772.39 0.0001  13 466027.02 488773.59 488773.26 0.0001  14 466027.77 488774.26 488774.05 0.0000  15 466028.27 488774.76 488774.57 0.0000  20 466029.52 488775.93 488775.88 0.0000  40 466030.18 488776.44 488776.58 0.0000  80 466030.34 488766.74 488766.74 0.0000


## 4. Conclusion

The transient response of AFM probe subjected to two-mode excitations based on the modified couple stress theory is investigated. The semianalytical solution is presented. The similarity of the classical Bernoulli-Euler beam system and the system based on the modified couple stress theory is discovered. The relation between the natural frequencies of the two systems isωδ=ω01+δ. Several trends about the response spectrum due to two-mode excitations are obtained as follows.(1)
The response spectrum of two-mode excitations is significantly different to that of one-mode excitation. If the two frequencies of excitation are close to the natural frequency, several resonant responses occur.(2)
The larger the size parameterδ is, the greater the frequency shift is.(3)
The smaller the interacting distanceD0 is, the greater the frequency shift is.



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