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Effective dampings and frequency shifts of several modes of an inclined cantilever vibrating in viscous fluid
Precision Engineering 34 (2010) 320–326
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Precision Engineering journal homepage: www.elsevier.com/locate/precision
Effective dampings and frequency shifts of several modes of an inclined cantilever vibrating in viscous fluid Shueei-Muh Lin Mechanical Engineering Department, Kun Shan University, 949, DaWan Road, Tainan, 710-03, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 28 December 2008 Received in revised form 25 June 2009 Accepted 5 August 2009 Available online 21 August 2009 Keywords: Liquid Inclined cantilever Q-factor Resonant frequency
a b s t r a c t In this study, one investigates the dynamic behavior of an inclined non-uniform cantilever vibrating in fluid and close to a sample’s surface. The closed-form solution of this beam model is presented. In fact, there must exist the hydrodynamic loading to the cantilever. Its effects include the viscous shear damping, the squeeze film damping and the added liquid mass attached to the cantilever. These depend on the material and geometrical properties and the operational conditions, e.g. the inclined angle of a cantilever to a sample’s surface. For simplicity, the effective damping and the added mass are usually expressed as some formula. It is found here that these conventional formula are inaccurate for the case of the cantilever close to a sample’s surface. For understanding the detailed mechanism of motion, Basak and Raman (2006) [1] analyzed the 3D fluid-structure interaction of a cantilever vibrating in liquid and close to a solid surface. The Q-factors and the resonant frequencies of different modes were presented. But the effective damping and the added mass attached to the cantilever were not presented. Via the present solution method the effective damping and the added mass are easily determined. It is very helpful for constructing the mathematical model and understanding the AFM behavior clearly. © 2009 Published by Elsevier Inc.
1. Introduction Atomic force microscopy (AFM) subjected to a damping force has been widely developed as a powerful technique for obtaining atomic-scale images and the surface properties of the soft materials, such as polymers and biological samples [2,3]. In general, a biological sample is measured in liquids. So far, some studies have been made on the influence of the resonance frequency and quality factor of the AFM cantilever in a viscous fluid [4–10]. A better understanding of the dynamic behavior of AFM is generally required to be further improved its performance. Owing to the significant effect of the hydrodynamic force on the performance of an AFM cantilever, some literature were devoted to find out its effective damping and added mass coefficients. Sader [5] and Green and Sader [6] derived the hydrodynamic force distribution on a cantilever torsionally oscillating in viscous fluid. The effect of gap between the cantilever and surface of sample was neglected. The effective damping and added mass coefficients was not investigated. Maali et al. [7] produced the experimental work to show that the errors in estimation of the damping and added mass coefficients in the mass-spring-damper model by using the hydrodynamic force model [5] were great especially for the higher modes. Green and Sader [8] derived hydrodynamic
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force distribution on a cantilever flexurally oscillating in viscous fluid and close to a surface. However, this method is based on a similar formulation as that [5] and thus suffers from the same limitations. Moreover, the quality factor is not computed explicitly and the model is not applicable for micro-cantilevers inclined to a surface—a situation commonly found in AFM applications. Naik et al. [9] investigated experimentally the dynamic response of a cantilever (10 mm × 1 mm × 0.5 mm) vibrating in liquid near a solid wall. The uniform gap between the cantilever and a solid surface was considered and varied from millimeter to micrometer range. The effective damping and added mass coefficients in a beam model were proposed. It should be noted that the dimension of the cantilever is greatly larger than common AFM cantilever. The effect of dimension on the AFM behavior should be investigated. Using the commercial ADINA software, Basak and Raman [1] developed a fully three-dimensional finite element flow-structure interaction model to predict the hydrodynamic loading of micro-cantilevers in fluids and close to a solid surface. They discovered that the added mass coefficient for the first bending mode is obviously larger than those for the second bending and torsional modes. In addition, the added mass coefficients increase as the gap is decreased for all the three modes. Based on the effective damping and added mass coefficients, some literature [11–15] investigated the AFM behavior in the effective spring-mass-damper model. However, the accuracy of this discrete model is relatively low. Because the mathematical problem of the beam model is difficult to be solved, few studies are devoted
S.-M. Lin / Precision Engineering 34 (2010) 320–326
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the numerical transaction error. mt is the tip mass, W the flexural displacement, x the coordinate along the beam, the mass density per unit volume, and ˝ the oscillating frequency. Based on Eq. (1), the dimensionless governing differential equation of the beam is expressed as ∂2 ∂ 2
∂2 w b() 2 ∂
+ ctotal
∂w ∂2 w + mtotal () 2 = 0, ∂ ∂
(2)
where the total damping coefficient ctotal is ctotal = c0 + cv
Fig. 1. Geometry and coordinate system of an inclined cantilever scanning a sample.
to investigate the distributed beam system. Rankl et al. [10] used the ‘squeeze-film theory’ to examine the viscous damping when an AFM cantilever oscillated. The cantilever was considered to be inclined at an angle to a sample’s surface. Because the hydrodynamic force depends on the position, its mathematical problem requires lots of effort to solve. Therefore, this force was approximated to be constant thoroughly. In addition, a uniform beam was considered so that all the mathematical coefficients were constant and the equations of motion were easy to solve. However, due to the limitation of these approximations, a few complicate phenomena are hardly discovered and understood. So far, little researches have devoted to the dynamic response of an inclined beam operating in liquid and close to a solid surface. In this paper, the closed-form solution of the dynamic response of an inclined beam operating in liquid and close to a solid surface is derived. The effective damping and added fluid mass of the system is examined. Because the resonance response, the quality factor and the response ratio of the AFM cantilever oscillating in fluid are the crucial parameters affecting the scan speed and the sensitivity, the effects of several parameters on these properties are canvassed here. 2. Governing equation and boundary conditions In this study, one consider an AFM cantilever vibrating in gas or liquid. In the typical AFM experiments, the cantilever is inclined at an angle to the substrate surface. When the beam is excitated harmonically by a shaker at the root end, the beam is vibrating harmonically, as shown in Fig. 1. Basak and Raman [1] found that far from the sample’s surface, micro-cantilever damping arisen from localized fluid shear near the edges of the micro-cantilever. Closer to the surface, however, the damping arisen due to a combination of squeeze film effects and viscous shear near the edge. In addition to viscous dissipation, the surrounding fluid also creates an added mass effect on the micro-cantilever. This results in wet natural frequencies that are much lesser than the dry natural frequencies. These effects are investigated here. The cross-section of beam is considered to be non-uniform. The beam material is homogenous. The following dimensionless quantities are defined b() =
I(x) , I(0)
m0 =
W (x, t) w(, ) = , L0 t = 2 L
EI(0) , A(0)
(x)A(x) , (0)A(0) x = , L
ω = ˝L
2
¯t= m
in which the cantilever material damping c0 and the squeezing damping coefficient cv . The squeezing damping is due to the hydrodynamic force. If the cantilever approaches the surface the water must be squeezed out, and the viscous resistance causes a hydrodynamic force against the movement of the cantilever. Therefore, there exists the corresponding viscous damping with the cantilever. In addition, the total mass coefficient mtotal is mtotal () = m0 () + ma ()
(4)
where m0 is the mass of cantilever per unit length and ma is the added liquid mass to the canilever. Because the mechanism of fluidstructure interaction of a cantilever in liquid is very sophisticate [1], any accurate semi-analytical model is quite helpful. Two recent semi-analytical models are discussed as follows: Firstly, Sader [5] investigated the motion of a cantilever far from the surface in liquids by solving the Navier–Stokes equation of the hydrodynamical problem numerically. Rankl et al. [10] used this model to investigate the hydrodynamic force acting on a magnetically excited cantilever over a wide range of tip–surface separations. The viscous damping coefficient is expressed as cv = cs + c∞
(5)
where cs is squeezing damping coefficient, cc wb3 /h3 in which wb is the width of the cantilever beam, h is the gap between the cantilever and the sample’s surface and cc = L2 / E(0)I(0)(0)A(0). is the viscous friction damping of the absolute viscosity of liquid. c∞ is the cantilever in free fluid, 3cc (1 + wb 2liquid ˝/4). In addition, the liquid mass attached to the cantilever is expressed as 3/2
ma =
0.6liquid wb L1/2 (0)A(0)
.
(6)
Secondly, Naik et al. [9] investigated experimentally the dynamic response of a cantilever in air and fluorinent liquid. The uniform gap between the cantilever and a solid surface is considered. Naik et al. [9] presented the squeezing damping coefficient cv
⎡
cv = cc ⎣B0 +
⎤ 3 w j b ⎦ Bj
j=1
h
,
h ≥ 0.01 wb
(7)
where B0 = 1.16Rk0.73 , B1 = 97.5 + 0.0260Rk , B2 = 0.207 − 2.27 ×
mt , (0)A(0)L
10−4 Rk − 2.78 × 10−9 Rk2 , B3 = −2.73 × 10−3 + 7.10 × 10−7 Rk +
2.67 × 10−11 Rk2 , Rk =
A(0) EI(0)
(3)
(1)
where A denotes the cross-sectional area, E the Young’s modulus of beam, I the area moment of inertia, L the length of beam, and L0 the characteristic length. A small value of L0 is introduced to avoid
liquid ˝b2 4
,
(8)
in which Rk is the Reynold number. Also, the liquid mass attached to the cantilever is ma () = cm
liquid beam
m0 (),
(9)
322
S.-M. Lin / Precision Engineering 34 (2010) 320–326
where cm =
4
j
j=0
A0 = 1.48,
3.2. General solution
(wb /h) , h/wb ≥ 0.01
A1 = 0.183,
A3 = 2.29 × 10−5 ,
A2 = −2.91 × 10−3 ,
A4 = −6.47 × 10−8 .
The fundamental solution of the characteristic differential Eq. (16) is assumed to be (10)
The cantilever is harmonically excited at the root. All the associated boundary conditions are written as At = 0: w = A0 cos ˚ cos ω,
(11)
∂w = 0. ∂
(12)
At = 1: ∂2 w = 0, ∂ 2 ∂ ∂
b()
(13)
∂2 w ∂ 2
¯t −m
∂2 w = 0, ∂ 2
(14)
where A0 and ω are the dimensionless amplitude and frequency of root excitation, respectively. ˚ is the inclined angle of the cantilever to a sample’s surface, as shown in Fig. 1. The gap h = [htip /(1 − sin ˚)][1 − (sin ˚)x/L].
⎡ ¯ Wc,1 ¯ ⎢W ⎢ ¯ c,1 ⎢ Wc,1 ⎢W ⎢ ¯ c,1 ⎢W ⎢ ¯ s,1 ⎢W ⎢ ¯ s,1 ⎣W ¯ s,1
¯ W s,1
⎡
(15)
¯ () = ¯ s ()/W ¯ c2 + W ¯ s2 , tan = W ¯ c (), is the phase where W W angle. Substituting the solution into the governing Eq. (2) and the boundary conditions (11–14) and taking ‘cos ω’ and ‘sin ω’ apart, the coupled differential equations can be obtained d2 d 2 d2 d 2
¯c d2 W b() d 2
b()
¯s d2 W d 2
2
¯ s − ω mtotal ()W ¯ c = 0, + ωctotal ()W
(16a)
¯ c − ω2 mtotal ()W ¯ s = 0, − ωctotal ()W
(16b)
At = 0: ¯ c (0) = A0 cos ˚ W
(17a)
¯ s (0) = 0, W
(17b)
¯ c (0) dW = 0, d
(18a)
¯ s (0) dW = 0. d
(18b)
At = 1: ¯ c (1) d2 W d 2
¯ c,2 W ¯ W c,2 ¯ W c,2 ¯ W c,2 ¯ s,2 W ¯ W s,2 ¯ W s,2 ¯ W
¯ s,i () W
¯ c,3 W ¯ W c,3 ¯ W c,3 ¯ W c,3 ¯ s,3 W ¯ W s,3 ¯ W s,3 ¯ W
s,2
1
0
¯ c () cos ω + W ¯ s () sin ω] = W ¯ () cos(ω − ), w(, ) = [W
Ci
i=1
⎢0 ⎢0 ⎢ ⎢0 =⎢ ⎢0 ⎢ ⎢0 ⎣0
The solution of the system is assumed
=
8 ¯ c,i () W
,
(21)
satisfy the following normalization conditions at the origin of the coordinated system:
3.1. Characteristic governing equations and boundary conditions
¯ s () = W ¯ () sin W
where the eight linearly independent fundamental solutions ¯ s,i ()]T , i = 1, 2, . . ., 8, of Eq. (16) are chosen such that they ¯ c,i () W [W
3. Solution method
¯ c () = W ¯ () cos , W
¯ c () W ¯ s () W
¯ c,4 W ¯ W c,4 ¯ W c,4 ¯ W c,4 ¯ s,4 W ¯ W s,4 ¯ W s,4 ¯ W
s,3
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
s,4
¯ c,5 W ¯ W c,5 ¯ W c,5 ¯ W c,5 ¯ s,5 W ¯ W s,5 ¯ W s,5 ¯ W
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0
s,5
⎤
¯ c,6 W ¯ W c,6 ¯ W c,6 ¯ W c,6 ¯ s,6 W ¯ W s,6 ¯ W s,6 ¯ W s,6
¯ c,7 W ¯ W c,7 ¯ W c,7 ¯ W c,7 ¯ s,7 W ¯ W s,7 ¯ W s,7 ¯ W s,7
⎤
¯ c,8 W ¯ ⎥ W c,8 ¯ ⎥ W c,8 ⎥ ¯ ⎥ W c,8 ⎥ ¯ s,8 ⎥ W ⎥ ¯ ⎥ W s,8 ⎥ ¯ ⎦ W s,8 ¯ W s,8
=0
0 0⎥ ⎥ 0⎥ 0⎥ ⎥, 0⎥ ⎥ 0⎥ 0⎦ 1
(22)
where primes indicate differentiation with respect to the dimensionless spatial variable . Substituting Eq. (21) into the boundary conditions (17–20), one obtains the coefficients C1 = A0 cos ˚, C2 = C5 = C6 = 0 and
⎡
⎤
⎡
⎤
¯ (1) W ¯ (1) W ¯ (1) W ¯ (1) −1 W C3 c,7 c,3 c,4 c,8 ¯ (1) W ¯ (1) W ¯ (1) W ¯ (1) ⎥ ⎢W ⎢ C4 ⎥ s,7 s,4 s,8 ⎦ ⎣ C ⎦ = −A0 cos ˚⎣ s,3 H4 H7 H8 H3 7 C8 G3 G4 G7 G8
⎡ ¯ ⎤ Wc,1 (1) ¯ (1) ⎥ ⎢W × ⎣ s,1 ⎦. H1 G1
(23)
Substituting the fundamental solutions and coefficients (23) back into Eqs. (21) and (15), the general solution w() is obtained. 4. Relation between energy dissipation and Q-factor
= 0,
(19a)
The energy dissipation is usually described by the Q-factor. Its definition is
¯ s (1) d2 W = 0, d 2
(19b)
Q -factor = 2
¯ c = 0, ¯ t ω2 W +m
(20a)
¯ s = 0. ¯ t ω2 W +m
(20b)
where Etotal is the total energy and Eloss the energy lost per cycle [16,17]. It is obvious that the phase angle is a function of the position variable x. It means that when the tip is at the top dead position, i.e., the velocity of the tip is zero; the velocity at the other position of beam is not zero. According to the fact, the total energy
d d d d
b
¯c d2 W d 2
¯s d2 W b d 2
Etotal ,
Eloss
(24)
S.-M. Lin / Precision Engineering 34 (2010) 320–326
323
Fig. 2. Influence of the distance d between the beam and a sample’s surface on the induced damping and added mass coefficients of a uniform cross-sectional cantilever in water [L = 200 m, temperature of water T = 27 ◦ C, water = 1 × 103 kg/m3 , the inclinded ratio 2 = 0, the width of cross-section of beam b¯ = 45 m, the thickness of cross-section of beam h¯ = 3.5 m, b = m = 1, mt = 3.18 × 10−13 kg, E = 70.3 × 109 Pa, = 2.5 × 103 kg/m3 ].
is considered to be an average value of a cycle as follows: Etotal =
1 T
The energy lost per cycle due to the damping is
T
(Es (t) + Ek (t))dt,
(25)
= ˛ E˜ s +
0
1 2 ω E˜ k 2
where E(0)I(0) 2 ˛= Lc , L3
E˜ k =
1 2
1 E˜ s = 4
(26a)
1
1
¯ s ())2 ]d, + (W
1 ¯ c2 (1) + W ¯ s2 (1)]. ¯ t [W m 2
¯ b3 () w d¯ 3 ()
0
(28a)
¯ s2 ())d, ¯ c2 () + W (W
1
¯ c2 () + W ¯ s2 ())d, (W
E˜ c∞ = ¯ c2 () + W ¯ s2 ()]d + m[W
(27)
where
0
1 0
3 ˜ ¯ 2 (1)}
Eloss = ˛ω{[cc rbd Ecs + (c0 + c∞ )E˜ c∞ ] + ct W
E˜ cs = ¯ c ())2 b[(W
∂w(x, t) dw(x, t)dx . ∂t
Considering the semi-analytical model given by Rankl et al. [10], the corresponding energy loss is derived as
L
ctotal (x)
0
where Es and Ek are the strain and kinetic energies, respectively. Substituting the solution (15) into Eq. (25), the total energy is expressed as Etotal
Eloss =
0
(26b)
¯ c2 (1) + W ¯ s2 (1)). ¯ 2 (1) = (W W
(28b)
324
S.-M. Lin / Precision Engineering 34 (2010) 320–326
Similarly, considering the semi analytical model given by Naik et al. [9], the corresponding energy loss is derived as 3 ˜ 2 ˜
Eloss = ˛ω{[cc B3 rbd Ecs3 + cc B2 rbd Ecs2 + cc B1 rbd E˜ cs1
¯ 2 (1)} + (c0 + cc B0 )E˜ c0 ] + ct W where
1 0
¯ b () w ¯ d()
0
¯ b () w ¯ d()
0
¯ b () w ¯ d()
E˜ cs3 =
1 E˜ cs2 =
1 E˜ cs1 =
3 ¯ c2 () + W ¯ s2 ())d, (W
2 ¯ c2 () + W ¯ s2 ())d, (W
¯ c2 () + W ¯ s2 ())d, (W
1
¯ c2 () + W ¯ s2 ())d. (W
E˜ c0 =
(29a)
(29b)
0
5. Numerical results and discussion First, the numerical results by using semi-analytical models given by Rankl et al. [10] and Naik et al. [9] and the 3D fluidstructure interaction analysis by Basak and Raman [1] are compared as follows: Fig. 2a shows the influence of the constant gap between the cantilever and a sample’s surface on the resonant frequency of modes 1 and 2. It is found that for a small gap the resonant frequencies increase greatly with the gap. Moreover, the first resonant frequency in the three models given by Naik et al., [9], Rankl et al. [10] and Basak and Raman [1] are almost consistent. But the second resonant frequencies are significantly different. Meanwhile, Fig. 2b shows the influence of the constant gap on the Q-factor. In particular, it is found that the Q-factors of modes 1 and 2 in the three models are significantly different. Despite of that, the Q-factor in the model given by Naik et al. [9] are almost independent to the gap. This result is against the experiment. On the other hand, the Q-factor in the other two models given by Rankl et al. [10] and Basak and Raman [1] increases with the gap. This trend is reasonable. Further, the effect of the inclined angle on the Q-factor and the resonant frequency is considered. Fig. 2c shows the relation between the first two resonant frequencies and the tip gap htip at the inclined angle of 11◦ . It is observed from Fig. 2a with c that the relations between the resonant frequencies and the tip gap in the model given by Basak et al. [1] depends significantly on the inclined angle . Noteworthily, based on the models given by Rankl et al. [10] and Naik et al. [9], the effect of the inclined angle on the relations is negligible. These results in the last two models are unreasonable. Similar phenomenon is found in Fig. 2b and d. It is concluded that the models given by Rankl et al. [10] and Naik et al. [9] is not suitable to study the effect of the inclined angle on the resonant frequency and the Q-factor. Although the resonant frequency and the Q-factor of a cantilever vibrating in liquid can be determined by Basak at al. [1] by using the commercial ADINA software, the effective damping and added mass coefficients are interesting and accommodating for understanding the mechanism of motion. Based on the present solution method and the resonant frequency and the Q-factor determined by Basak at al. [1], one can determine the effective damping coefficient cv and the added liquid mass to the cantilever madd without effort. Fig. 3a shows that the damping coefficient cv of mode 2 is larger than that of mode 1. The damping coefficient cv decreases with the gap. Comparing the damping coefficient cv in different models, it is found that the damping coefficient cv given by Naik et al. [9] is
Fig. 3. Comparison of the resonant frequencies and the quality factors of a uniform cantilever made of single crystal silicon, determined by different methods. [L = 197 m, temperature of water T = 27 ◦ C, 1 = 2 = 0, the base amplitude A0 cos ˚ = 1 nm, the inclinded ratio 2 = 0, the width of cross-section of beam b¯ = 29 m, the thickness of cross-section of beam h¯ = 2 m, Lc = 10 nm, b = m = 1, mt = 0, E = 169 × 109 Pa, = 2.32 × 103 kg/m3 , the first two natural frequencies in vacuum, f1 = 71018.72 Hz, f2 = 445066.86 Hz].
the maximum, then that given by Rankl et al. [10] and that given by Basak et al. [1] is the minimum. Because the Q-factor decreases with the damping coefficients cv , Fig. 2b and d shows that the quality factor by Naik et al. [9] is the minimum, then that by Rankl et al. [10] and that by Basak et al. [1] is the maximum. Further, Fig. 3b shows that the added mass in the models [9,10] are independent to the tip gap. In addition, the added mass of mode is the same as that of mode 2. Obviously, this phenomenon is against the experiment [1]. In the model [1] the added masses of modes 1 and 2 are different and decrease with the gap. 6. Conclusions The mathematical model of an inclined cantilever vibrating in fluid is established. Meanwhile, the closed-form solution of this system is obtained. Via this solution method the effective damping
S.-M. Lin / Precision Engineering 34 (2010) 320–326
and the added mass of an inclined cantilever vibration in liquid can be easily calculated. It is found here that the influence of the gap on the Q-factor and the resonant frequencies is significant. Moreover, the semi-analytical models given by Rankl et al. [10] and Naik et al. [9] are inaccurate for the cases of an AFM cantilever close to a sample’s surface.
Acknowledgement The support of the National Science Council of Taiwan, ROC, is gratefully acknowledged (Grant number: Nsc96-2221-E168-023).
Appendix A. Appendix A.1. Exact fundamental solutions In general, the closed-form fundamental solutions of two coupled differential equations with variable coefficients are not available. However, if the coefficients of the equations, which involve the material properties and/or geometric parameters, can be expressed in matrix polynomial form, then a power series representation of the fundamental solutions can be constructed by the modified method of Frobenius. Upon expanding the governing characteristic differential Eq. (16a), one can obtain the following equation with coefficients expressed in the polynomial form ∈ (0, 1)
(A1a)
¯ c,1 : For W ¯ c,2 : For W ¯ c,3 : For W ¯ c,4 : For W ¯ c,5 : For W ¯ c,6 : For W ¯ c,7 : For W ¯ c,8 : For W ¯ s,1 : For W ¯ s,2 : For W ¯ s,3 : For W ¯ s,4 : For W ¯ s,5 : For W ¯ s,6 : For W ¯ s,7 : For W ¯ s,8 : For W
ak k ,
B˜ =
k=0
3
bk k ,
C˜ =
k=0
4
ck k ,
˜ = D
k=0
3 k=0
˛j,m+4 = +
4
˜ a¯ k k = A,
B¯ =
k=0
3
˜ b¯ k k = B,
k=0
C¯ =
= ˛1,3 = ˛2,3 = ˛3,3 = ˛1,3
ˇ5,1 ˇ6,0 ˇ7,0 ˇ8,0
= ˇ5,2 = ˇ6,2 = ˇ7,2 = ˇ8,1
= ˇ5,3 = 0, = ˇ6,3 = 0, = ˇ7,3 = 0, = ˇ8,2 = 0, (A3b)
= 0, = 0, = 0, = 0, = 0, = 0, = 0, = 0,
m
m
dk ˇj,m−k
k=0
bk (m − k + 3)(m − k + 2)(m − k + 1)˛j,m−k+3
(A4a)
dk k .
(A1b)
(A2a)
= 0, = 0, = 0, = 0,
ak (m − k + 4)(m − k + 3)(m − k + 2)(m − k + 1)˛j,m−k+4 ,
m
+
−1 a¯ 0 (m + 4)(m + 3)(m + 2)(m + 1)
c¯ k ˇj,m−k +
k=0 m
+
m k=0
m
d¯ k ˛j,m−k
k=0
b¯ k (m − k + 3)(m − k + 2)(m − k + 1)ˇj,m−k+3
a¯ k (m − k + 4)(m − k + 3)(m − k + 2)(m − k + 1)ˇj,m−k+4 .
k=1
4
= ˛1,2 = ˛1,2 = ˛3,1 = ˛1,2
k=0
where A¯ =
˛1,1 ˛2,0 ˛3,0 ˛1,1
k=1
Similarly, upon expanding the governing characteristic differential Eq. (16b), one can obtain the following equation with coefficients expressed in the polynomial form ∈ (0, 1)
= ˛5,3 = ˛6,3 = ˛7,3 = ˛8,3 = ˇ1,3 = ˇ2,3 = ˇ3,3 = ˇ4,3
−1 a0 (m + 4)(m + 3)(m + 2)(m + 1)
ck ˛j,m−k +
ˇj,m+4 =
¯s ¯s d3 W d4 W ¯ ¯ ¯ ¯ s + D() ¯ ¯ c = 0, + B() + C() W W A() d 4 d 3
= 1, = 1, = 1/2, = 1/6, = ˛5,1 = ˛5,2 = ˛6,1 = ˛6,2 = ˛7,1 = ˛7,2 = ˛8,1 = ˛8,2 = ˇ1,1 = ˇ1,2 = ˇ2,1 = ˇ2,2 = ˇ3,1 = ˇ3,2 = ˇ4,1 = ˇ4,2 = 1, = 1, = 1/2, = 1/6,
These eight fundamental solutions satisfy the normalization condition (20). Upon substituting Eq. (A3) into Eqs. (A1) and (A2) and collecting the coefficients of like powers of , the following recurrence formula can be obtained:
+
4
˛1,0 ˛2,0 ˛3,2 ˛4,3 ˛5,0 ˛6,0 ˛7,0 ˛8,0 ˇ1,0 ˇ2,0 ˇ3,0 ˇ4,0 ˇ5,0 ˇ6,1 ˇ7,2 ˇ8,3
k=0 m
where ˜= A
and
m
¯c ¯c d4 W d3 W ˜ ˜ W ˜ ¯ c + D() ˜ W ¯ s = 0, A() + B() + C() 4 d d 3
325
(A4b)
With these recurrence formulas, one can generate the eight exact normalized fundamental solutions of the differential Eqs. (A2) and (A3).
˜ c¯ k k = C,
k=0
3
¯ = D
˜ d¯ k k = −D
(A2b)
k=0
Because the coefficients of the coupled differential Eqs. (A1) and (A2) can be expressed in matrix polynomial form, then a power series representation of the fundamental solutions can be constructed by the modified method of Frobenius. One can assume that the eight fundamental solutions of Eqs. (A1) and (A2) are in the form of
¯ c,j W ¯ s,j W
=
∞ ˛j,k k k=0
ˇj,k k
,
j = 1, 2, . . . , 8
(A3a)
References [1] Basak S, Raman A. Hydrodynamic loading of microcantilevers vibrating in viscous fluid. Journal of Applied Physics 2006;99:114906. [2] Giessibl FJ. Advances in atomic force microscopy. Reviews of Modern Physics 2003;75:949–83. [3] Garcia R, Perez R. Dynamic atomic force microscopy methods. Surface Science Report 2002;47:197–301. [4] Chen GY, Warmack RJ, Thundat T, Allison DP, Huang A. Resonance response of scanning force microscopy cantilevers. Reviews of Science Instrument 1994;65:2532–7. [5] Sader JE. Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. Journal of Applied Physics 1998;84:64–76. [6] Green CP, Sader JE. Torsional frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. Journal of Applied Physics 2002;92:6262.
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[7] Maali A, Hurth C, Boisgard R, Jai C, Cohen-Bouhacina T, Aime JP. Hydrodynamics of oscillating atomic force microscopy cantilevers in viscous fluids. Journal of Applied Physics 2005;97:074907. [8] Green CP, Sader JE. Small amplitude oscillations of a thin beam immersed in a viscous fluid near a solid surface. Physics of Fluids 2005;17:073102. [9] Naik T, Longmire EK, Mantell SC. Dynamic response of a cantilever in liquid near a solid wall. Sensors and Actuators A 2003;102:240–54. [10] Rankl C, Pastushenko V, Kienberger F, Stroh CM, Hinterdorfer P. Hydrodynamic damping of a magnetically oscillated cantilever close to a surface. Ultramicroscopy 2004;100:301–8. [11] Chen GY, Warmack RJ. Harmonic response of near-contact scanning force microscopy. Journal of Applied Physics 1995;78(3):1465–9. [12] Delineau L, Brandsch R, Bar G, amd Whagbo MH. Harmonic responses of a cantilever interacting with elastomers in tapping mode atomic force microscopy. Surface Science 2000;448:L179–87.
[13] Salapaka MV, Chen DJ, Cleveland JP. Linearity of amplitude and phase in tapping-mode atomic force microscopy. Physical Reviews B 2000;61(2): 1106–15. [14] Ke SH, Uda T, Terakura K. Frequency shift and energy dissipation in on-contact atomic-force microscopy. Applied Surface science 2000;157: 361–6. [15] Lin SM, Lin CC. Phase shifts and energy dissipations of several modes of AFM: Minimizing topography and dissipation measurement errors. Precision Engineering 2009;33:371–7. [16] Lin SM. Energy dissipation and frequency shift of a damped dynamic force microscopy. Ultramicroscopy 2006;106:516–24. [17] Lin SM. Energy dissipation and dynamic response of an AM-AFM subjected to a tip-sample viscous force. Ultramicroscopy 2007;107:245–53.
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Precision Engineering journal homepage: www.elsevier.com/locate/precision
Effective dampings and frequency shifts of several modes of an inclined cantilever vibrating in viscous fluid Shueei-Muh Lin Mechanical Engineering Department, Kun Shan University, 949, DaWan Road, Tainan, 710-03, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 28 December 2008 Received in revised form 25 June 2009 Accepted 5 August 2009 Available online 21 August 2009 Keywords: Liquid Inclined cantilever Q-factor Resonant frequency
a b s t r a c t In this study, one investigates the dynamic behavior of an inclined non-uniform cantilever vibrating in fluid and close to a sample’s surface. The closed-form solution of this beam model is presented. In fact, there must exist the hydrodynamic loading to the cantilever. Its effects include the viscous shear damping, the squeeze film damping and the added liquid mass attached to the cantilever. These depend on the material and geometrical properties and the operational conditions, e.g. the inclined angle of a cantilever to a sample’s surface. For simplicity, the effective damping and the added mass are usually expressed as some formula. It is found here that these conventional formula are inaccurate for the case of the cantilever close to a sample’s surface. For understanding the detailed mechanism of motion, Basak and Raman (2006) [1] analyzed the 3D fluid-structure interaction of a cantilever vibrating in liquid and close to a solid surface. The Q-factors and the resonant frequencies of different modes were presented. But the effective damping and the added mass attached to the cantilever were not presented. Via the present solution method the effective damping and the added mass are easily determined. It is very helpful for constructing the mathematical model and understanding the AFM behavior clearly. © 2009 Published by Elsevier Inc.
1. Introduction Atomic force microscopy (AFM) subjected to a damping force has been widely developed as a powerful technique for obtaining atomic-scale images and the surface properties of the soft materials, such as polymers and biological samples [2,3]. In general, a biological sample is measured in liquids. So far, some studies have been made on the influence of the resonance frequency and quality factor of the AFM cantilever in a viscous fluid [4–10]. A better understanding of the dynamic behavior of AFM is generally required to be further improved its performance. Owing to the significant effect of the hydrodynamic force on the performance of an AFM cantilever, some literature were devoted to find out its effective damping and added mass coefficients. Sader [5] and Green and Sader [6] derived the hydrodynamic force distribution on a cantilever torsionally oscillating in viscous fluid. The effect of gap between the cantilever and surface of sample was neglected. The effective damping and added mass coefficients was not investigated. Maali et al. [7] produced the experimental work to show that the errors in estimation of the damping and added mass coefficients in the mass-spring-damper model by using the hydrodynamic force model [5] were great especially for the higher modes. Green and Sader [8] derived hydrodynamic
E-mail address: [email protected]. 0141-6359/$ – see front matter © 2009 Published by Elsevier Inc. doi:10.1016/j.precisioneng.2009.08.004
force distribution on a cantilever flexurally oscillating in viscous fluid and close to a surface. However, this method is based on a similar formulation as that [5] and thus suffers from the same limitations. Moreover, the quality factor is not computed explicitly and the model is not applicable for micro-cantilevers inclined to a surface—a situation commonly found in AFM applications. Naik et al. [9] investigated experimentally the dynamic response of a cantilever (10 mm × 1 mm × 0.5 mm) vibrating in liquid near a solid wall. The uniform gap between the cantilever and a solid surface was considered and varied from millimeter to micrometer range. The effective damping and added mass coefficients in a beam model were proposed. It should be noted that the dimension of the cantilever is greatly larger than common AFM cantilever. The effect of dimension on the AFM behavior should be investigated. Using the commercial ADINA software, Basak and Raman [1] developed a fully three-dimensional finite element flow-structure interaction model to predict the hydrodynamic loading of micro-cantilevers in fluids and close to a solid surface. They discovered that the added mass coefficient for the first bending mode is obviously larger than those for the second bending and torsional modes. In addition, the added mass coefficients increase as the gap is decreased for all the three modes. Based on the effective damping and added mass coefficients, some literature [11–15] investigated the AFM behavior in the effective spring-mass-damper model. However, the accuracy of this discrete model is relatively low. Because the mathematical problem of the beam model is difficult to be solved, few studies are devoted
S.-M. Lin / Precision Engineering 34 (2010) 320–326
321
the numerical transaction error. mt is the tip mass, W the flexural displacement, x the coordinate along the beam, the mass density per unit volume, and ˝ the oscillating frequency. Based on Eq. (1), the dimensionless governing differential equation of the beam is expressed as ∂2 ∂ 2
∂2 w b() 2 ∂
+ ctotal
∂w ∂2 w + mtotal () 2 = 0, ∂ ∂
(2)
where the total damping coefficient ctotal is ctotal = c0 + cv
Fig. 1. Geometry and coordinate system of an inclined cantilever scanning a sample.
to investigate the distributed beam system. Rankl et al. [10] used the ‘squeeze-film theory’ to examine the viscous damping when an AFM cantilever oscillated. The cantilever was considered to be inclined at an angle to a sample’s surface. Because the hydrodynamic force depends on the position, its mathematical problem requires lots of effort to solve. Therefore, this force was approximated to be constant thoroughly. In addition, a uniform beam was considered so that all the mathematical coefficients were constant and the equations of motion were easy to solve. However, due to the limitation of these approximations, a few complicate phenomena are hardly discovered and understood. So far, little researches have devoted to the dynamic response of an inclined beam operating in liquid and close to a solid surface. In this paper, the closed-form solution of the dynamic response of an inclined beam operating in liquid and close to a solid surface is derived. The effective damping and added fluid mass of the system is examined. Because the resonance response, the quality factor and the response ratio of the AFM cantilever oscillating in fluid are the crucial parameters affecting the scan speed and the sensitivity, the effects of several parameters on these properties are canvassed here. 2. Governing equation and boundary conditions In this study, one consider an AFM cantilever vibrating in gas or liquid. In the typical AFM experiments, the cantilever is inclined at an angle to the substrate surface. When the beam is excitated harmonically by a shaker at the root end, the beam is vibrating harmonically, as shown in Fig. 1. Basak and Raman [1] found that far from the sample’s surface, micro-cantilever damping arisen from localized fluid shear near the edges of the micro-cantilever. Closer to the surface, however, the damping arisen due to a combination of squeeze film effects and viscous shear near the edge. In addition to viscous dissipation, the surrounding fluid also creates an added mass effect on the micro-cantilever. This results in wet natural frequencies that are much lesser than the dry natural frequencies. These effects are investigated here. The cross-section of beam is considered to be non-uniform. The beam material is homogenous. The following dimensionless quantities are defined b() =
I(x) , I(0)
m0 =
W (x, t) w(, ) = , L0 t = 2 L
EI(0) , A(0)
(x)A(x) , (0)A(0) x = , L
ω = ˝L
2
¯t= m
in which the cantilever material damping c0 and the squeezing damping coefficient cv . The squeezing damping is due to the hydrodynamic force. If the cantilever approaches the surface the water must be squeezed out, and the viscous resistance causes a hydrodynamic force against the movement of the cantilever. Therefore, there exists the corresponding viscous damping with the cantilever. In addition, the total mass coefficient mtotal is mtotal () = m0 () + ma ()
(4)
where m0 is the mass of cantilever per unit length and ma is the added liquid mass to the canilever. Because the mechanism of fluidstructure interaction of a cantilever in liquid is very sophisticate [1], any accurate semi-analytical model is quite helpful. Two recent semi-analytical models are discussed as follows: Firstly, Sader [5] investigated the motion of a cantilever far from the surface in liquids by solving the Navier–Stokes equation of the hydrodynamical problem numerically. Rankl et al. [10] used this model to investigate the hydrodynamic force acting on a magnetically excited cantilever over a wide range of tip–surface separations. The viscous damping coefficient is expressed as cv = cs + c∞
(5)
where cs is squeezing damping coefficient, cc wb3 /h3 in which wb is the width of the cantilever beam, h is the gap between the cantilever and the sample’s surface and cc = L2 / E(0)I(0)(0)A(0). is the viscous friction damping of the absolute viscosity of liquid. c∞ is the cantilever in free fluid, 3cc (1 + wb 2liquid ˝/4). In addition, the liquid mass attached to the cantilever is expressed as 3/2
ma =
0.6liquid wb L1/2 (0)A(0)
.
(6)
Secondly, Naik et al. [9] investigated experimentally the dynamic response of a cantilever in air and fluorinent liquid. The uniform gap between the cantilever and a solid surface is considered. Naik et al. [9] presented the squeezing damping coefficient cv
⎡
cv = cc ⎣B0 +
⎤ 3 w j b ⎦ Bj
j=1
h
,
h ≥ 0.01 wb
(7)
where B0 = 1.16Rk0.73 , B1 = 97.5 + 0.0260Rk , B2 = 0.207 − 2.27 ×
mt , (0)A(0)L
10−4 Rk − 2.78 × 10−9 Rk2 , B3 = −2.73 × 10−3 + 7.10 × 10−7 Rk +
2.67 × 10−11 Rk2 , Rk =
A(0) EI(0)
(3)
(1)
where A denotes the cross-sectional area, E the Young’s modulus of beam, I the area moment of inertia, L the length of beam, and L0 the characteristic length. A small value of L0 is introduced to avoid
liquid ˝b2 4
,
(8)
in which Rk is the Reynold number. Also, the liquid mass attached to the cantilever is ma () = cm
liquid beam
m0 (),
(9)
322
S.-M. Lin / Precision Engineering 34 (2010) 320–326
where cm =
4
j
j=0
A0 = 1.48,
3.2. General solution
(wb /h) , h/wb ≥ 0.01
A1 = 0.183,
A3 = 2.29 × 10−5 ,
A2 = −2.91 × 10−3 ,
A4 = −6.47 × 10−8 .
The fundamental solution of the characteristic differential Eq. (16) is assumed to be (10)
The cantilever is harmonically excited at the root. All the associated boundary conditions are written as At = 0: w = A0 cos ˚ cos ω,
(11)
∂w = 0. ∂
(12)
At = 1: ∂2 w = 0, ∂ 2 ∂ ∂
b()
(13)
∂2 w ∂ 2
¯t −m
∂2 w = 0, ∂ 2
(14)
where A0 and ω are the dimensionless amplitude and frequency of root excitation, respectively. ˚ is the inclined angle of the cantilever to a sample’s surface, as shown in Fig. 1. The gap h = [htip /(1 − sin ˚)][1 − (sin ˚)x/L].
⎡ ¯ Wc,1 ¯ ⎢W ⎢ ¯ c,1 ⎢ Wc,1 ⎢W ⎢ ¯ c,1 ⎢W ⎢ ¯ s,1 ⎢W ⎢ ¯ s,1 ⎣W ¯ s,1
¯ W s,1
⎡
(15)
¯ () = ¯ s ()/W ¯ c2 + W ¯ s2 , tan = W ¯ c (), is the phase where W W angle. Substituting the solution into the governing Eq. (2) and the boundary conditions (11–14) and taking ‘cos ω’ and ‘sin ω’ apart, the coupled differential equations can be obtained d2 d 2 d2 d 2
¯c d2 W b() d 2
b()
¯s d2 W d 2
2
¯ s − ω mtotal ()W ¯ c = 0, + ωctotal ()W
(16a)
¯ c − ω2 mtotal ()W ¯ s = 0, − ωctotal ()W
(16b)
At = 0: ¯ c (0) = A0 cos ˚ W
(17a)
¯ s (0) = 0, W
(17b)
¯ c (0) dW = 0, d
(18a)
¯ s (0) dW = 0. d
(18b)
At = 1: ¯ c (1) d2 W d 2
¯ c,2 W ¯ W c,2 ¯ W c,2 ¯ W c,2 ¯ s,2 W ¯ W s,2 ¯ W s,2 ¯ W
¯ s,i () W
¯ c,3 W ¯ W c,3 ¯ W c,3 ¯ W c,3 ¯ s,3 W ¯ W s,3 ¯ W s,3 ¯ W
s,2
1
0
¯ c () cos ω + W ¯ s () sin ω] = W ¯ () cos(ω − ), w(, ) = [W
Ci
i=1
⎢0 ⎢0 ⎢ ⎢0 =⎢ ⎢0 ⎢ ⎢0 ⎣0
The solution of the system is assumed
=
8 ¯ c,i () W
,
(21)
satisfy the following normalization conditions at the origin of the coordinated system:
3.1. Characteristic governing equations and boundary conditions
¯ s () = W ¯ () sin W
where the eight linearly independent fundamental solutions ¯ s,i ()]T , i = 1, 2, . . ., 8, of Eq. (16) are chosen such that they ¯ c,i () W [W
3. Solution method
¯ c () = W ¯ () cos , W
¯ c () W ¯ s () W
¯ c,4 W ¯ W c,4 ¯ W c,4 ¯ W c,4 ¯ s,4 W ¯ W s,4 ¯ W s,4 ¯ W
s,3
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
s,4
¯ c,5 W ¯ W c,5 ¯ W c,5 ¯ W c,5 ¯ s,5 W ¯ W s,5 ¯ W s,5 ¯ W
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0
s,5
⎤
¯ c,6 W ¯ W c,6 ¯ W c,6 ¯ W c,6 ¯ s,6 W ¯ W s,6 ¯ W s,6 ¯ W s,6
¯ c,7 W ¯ W c,7 ¯ W c,7 ¯ W c,7 ¯ s,7 W ¯ W s,7 ¯ W s,7 ¯ W s,7
⎤
¯ c,8 W ¯ ⎥ W c,8 ¯ ⎥ W c,8 ⎥ ¯ ⎥ W c,8 ⎥ ¯ s,8 ⎥ W ⎥ ¯ ⎥ W s,8 ⎥ ¯ ⎦ W s,8 ¯ W s,8
=0
0 0⎥ ⎥ 0⎥ 0⎥ ⎥, 0⎥ ⎥ 0⎥ 0⎦ 1
(22)
where primes indicate differentiation with respect to the dimensionless spatial variable . Substituting Eq. (21) into the boundary conditions (17–20), one obtains the coefficients C1 = A0 cos ˚, C2 = C5 = C6 = 0 and
⎡
⎤
⎡
⎤
¯ (1) W ¯ (1) W ¯ (1) W ¯ (1) −1 W C3 c,7 c,3 c,4 c,8 ¯ (1) W ¯ (1) W ¯ (1) W ¯ (1) ⎥ ⎢W ⎢ C4 ⎥ s,7 s,4 s,8 ⎦ ⎣ C ⎦ = −A0 cos ˚⎣ s,3 H4 H7 H8 H3 7 C8 G3 G4 G7 G8
⎡ ¯ ⎤ Wc,1 (1) ¯ (1) ⎥ ⎢W × ⎣ s,1 ⎦. H1 G1
(23)
Substituting the fundamental solutions and coefficients (23) back into Eqs. (21) and (15), the general solution w() is obtained. 4. Relation between energy dissipation and Q-factor
= 0,
(19a)
The energy dissipation is usually described by the Q-factor. Its definition is
¯ s (1) d2 W = 0, d 2
(19b)
Q -factor = 2
¯ c = 0, ¯ t ω2 W +m
(20a)
¯ s = 0. ¯ t ω2 W +m
(20b)
where Etotal is the total energy and Eloss the energy lost per cycle [16,17]. It is obvious that the phase angle is a function of the position variable x. It means that when the tip is at the top dead position, i.e., the velocity of the tip is zero; the velocity at the other position of beam is not zero. According to the fact, the total energy
d d d d
b
¯c d2 W d 2
¯s d2 W b d 2
Etotal ,
Eloss
(24)
S.-M. Lin / Precision Engineering 34 (2010) 320–326
323
Fig. 2. Influence of the distance d between the beam and a sample’s surface on the induced damping and added mass coefficients of a uniform cross-sectional cantilever in water [L = 200 m, temperature of water T = 27 ◦ C, water = 1 × 103 kg/m3 , the inclinded ratio 2 = 0, the width of cross-section of beam b¯ = 45 m, the thickness of cross-section of beam h¯ = 3.5 m, b = m = 1, mt = 3.18 × 10−13 kg, E = 70.3 × 109 Pa, = 2.5 × 103 kg/m3 ].
is considered to be an average value of a cycle as follows: Etotal =
1 T
The energy lost per cycle due to the damping is
T
(Es (t) + Ek (t))dt,
(25)
= ˛ E˜ s +
0
1 2 ω E˜ k 2
where E(0)I(0) 2 ˛= Lc , L3
E˜ k =
1 2
1 E˜ s = 4
(26a)
1
1
¯ s ())2 ]d, + (W
1 ¯ c2 (1) + W ¯ s2 (1)]. ¯ t [W m 2
¯ b3 () w d¯ 3 ()
0
(28a)
¯ s2 ())d, ¯ c2 () + W (W
1
¯ c2 () + W ¯ s2 ())d, (W
E˜ c∞ = ¯ c2 () + W ¯ s2 ()]d + m[W
(27)
where
0
1 0
3 ˜ ¯ 2 (1)}
Eloss = ˛ω{[cc rbd Ecs + (c0 + c∞ )E˜ c∞ ] + ct W
E˜ cs = ¯ c ())2 b[(W
∂w(x, t) dw(x, t)dx . ∂t
Considering the semi-analytical model given by Rankl et al. [10], the corresponding energy loss is derived as
L
ctotal (x)
0
where Es and Ek are the strain and kinetic energies, respectively. Substituting the solution (15) into Eq. (25), the total energy is expressed as Etotal
Eloss =
0
(26b)
¯ c2 (1) + W ¯ s2 (1)). ¯ 2 (1) = (W W
(28b)
324
S.-M. Lin / Precision Engineering 34 (2010) 320–326
Similarly, considering the semi analytical model given by Naik et al. [9], the corresponding energy loss is derived as 3 ˜ 2 ˜
Eloss = ˛ω{[cc B3 rbd Ecs3 + cc B2 rbd Ecs2 + cc B1 rbd E˜ cs1
¯ 2 (1)} + (c0 + cc B0 )E˜ c0 ] + ct W where
1 0
¯ b () w ¯ d()
0
¯ b () w ¯ d()
0
¯ b () w ¯ d()
E˜ cs3 =
1 E˜ cs2 =
1 E˜ cs1 =
3 ¯ c2 () + W ¯ s2 ())d, (W
2 ¯ c2 () + W ¯ s2 ())d, (W
¯ c2 () + W ¯ s2 ())d, (W
1
¯ c2 () + W ¯ s2 ())d. (W
E˜ c0 =
(29a)
(29b)
0
5. Numerical results and discussion First, the numerical results by using semi-analytical models given by Rankl et al. [10] and Naik et al. [9] and the 3D fluidstructure interaction analysis by Basak and Raman [1] are compared as follows: Fig. 2a shows the influence of the constant gap between the cantilever and a sample’s surface on the resonant frequency of modes 1 and 2. It is found that for a small gap the resonant frequencies increase greatly with the gap. Moreover, the first resonant frequency in the three models given by Naik et al., [9], Rankl et al. [10] and Basak and Raman [1] are almost consistent. But the second resonant frequencies are significantly different. Meanwhile, Fig. 2b shows the influence of the constant gap on the Q-factor. In particular, it is found that the Q-factors of modes 1 and 2 in the three models are significantly different. Despite of that, the Q-factor in the model given by Naik et al. [9] are almost independent to the gap. This result is against the experiment. On the other hand, the Q-factor in the other two models given by Rankl et al. [10] and Basak and Raman [1] increases with the gap. This trend is reasonable. Further, the effect of the inclined angle on the Q-factor and the resonant frequency is considered. Fig. 2c shows the relation between the first two resonant frequencies and the tip gap htip at the inclined angle of 11◦ . It is observed from Fig. 2a with c that the relations between the resonant frequencies and the tip gap in the model given by Basak et al. [1] depends significantly on the inclined angle . Noteworthily, based on the models given by Rankl et al. [10] and Naik et al. [9], the effect of the inclined angle on the relations is negligible. These results in the last two models are unreasonable. Similar phenomenon is found in Fig. 2b and d. It is concluded that the models given by Rankl et al. [10] and Naik et al. [9] is not suitable to study the effect of the inclined angle on the resonant frequency and the Q-factor. Although the resonant frequency and the Q-factor of a cantilever vibrating in liquid can be determined by Basak at al. [1] by using the commercial ADINA software, the effective damping and added mass coefficients are interesting and accommodating for understanding the mechanism of motion. Based on the present solution method and the resonant frequency and the Q-factor determined by Basak at al. [1], one can determine the effective damping coefficient cv and the added liquid mass to the cantilever madd without effort. Fig. 3a shows that the damping coefficient cv of mode 2 is larger than that of mode 1. The damping coefficient cv decreases with the gap. Comparing the damping coefficient cv in different models, it is found that the damping coefficient cv given by Naik et al. [9] is
Fig. 3. Comparison of the resonant frequencies and the quality factors of a uniform cantilever made of single crystal silicon, determined by different methods. [L = 197 m, temperature of water T = 27 ◦ C, 1 = 2 = 0, the base amplitude A0 cos ˚ = 1 nm, the inclinded ratio 2 = 0, the width of cross-section of beam b¯ = 29 m, the thickness of cross-section of beam h¯ = 2 m, Lc = 10 nm, b = m = 1, mt = 0, E = 169 × 109 Pa, = 2.32 × 103 kg/m3 , the first two natural frequencies in vacuum, f1 = 71018.72 Hz, f2 = 445066.86 Hz].
the maximum, then that given by Rankl et al. [10] and that given by Basak et al. [1] is the minimum. Because the Q-factor decreases with the damping coefficients cv , Fig. 2b and d shows that the quality factor by Naik et al. [9] is the minimum, then that by Rankl et al. [10] and that by Basak et al. [1] is the maximum. Further, Fig. 3b shows that the added mass in the models [9,10] are independent to the tip gap. In addition, the added mass of mode is the same as that of mode 2. Obviously, this phenomenon is against the experiment [1]. In the model [1] the added masses of modes 1 and 2 are different and decrease with the gap. 6. Conclusions The mathematical model of an inclined cantilever vibrating in fluid is established. Meanwhile, the closed-form solution of this system is obtained. Via this solution method the effective damping
S.-M. Lin / Precision Engineering 34 (2010) 320–326
and the added mass of an inclined cantilever vibration in liquid can be easily calculated. It is found here that the influence of the gap on the Q-factor and the resonant frequencies is significant. Moreover, the semi-analytical models given by Rankl et al. [10] and Naik et al. [9] are inaccurate for the cases of an AFM cantilever close to a sample’s surface.
Acknowledgement The support of the National Science Council of Taiwan, ROC, is gratefully acknowledged (Grant number: Nsc96-2221-E168-023).
Appendix A. Appendix A.1. Exact fundamental solutions In general, the closed-form fundamental solutions of two coupled differential equations with variable coefficients are not available. However, if the coefficients of the equations, which involve the material properties and/or geometric parameters, can be expressed in matrix polynomial form, then a power series representation of the fundamental solutions can be constructed by the modified method of Frobenius. Upon expanding the governing characteristic differential Eq. (16a), one can obtain the following equation with coefficients expressed in the polynomial form ∈ (0, 1)
(A1a)
¯ c,1 : For W ¯ c,2 : For W ¯ c,3 : For W ¯ c,4 : For W ¯ c,5 : For W ¯ c,6 : For W ¯ c,7 : For W ¯ c,8 : For W ¯ s,1 : For W ¯ s,2 : For W ¯ s,3 : For W ¯ s,4 : For W ¯ s,5 : For W ¯ s,6 : For W ¯ s,7 : For W ¯ s,8 : For W
ak k ,
B˜ =
k=0
3
bk k ,
C˜ =
k=0
4
ck k ,
˜ = D
k=0
3 k=0
˛j,m+4 = +
4
˜ a¯ k k = A,
B¯ =
k=0
3
˜ b¯ k k = B,
k=0
C¯ =
= ˛1,3 = ˛2,3 = ˛3,3 = ˛1,3
ˇ5,1 ˇ6,0 ˇ7,0 ˇ8,0
= ˇ5,2 = ˇ6,2 = ˇ7,2 = ˇ8,1
= ˇ5,3 = 0, = ˇ6,3 = 0, = ˇ7,3 = 0, = ˇ8,2 = 0, (A3b)
= 0, = 0, = 0, = 0, = 0, = 0, = 0, = 0,
m
m
dk ˇj,m−k
k=0
bk (m − k + 3)(m − k + 2)(m − k + 1)˛j,m−k+3
(A4a)
dk k .
(A1b)
(A2a)
= 0, = 0, = 0, = 0,
ak (m − k + 4)(m − k + 3)(m − k + 2)(m − k + 1)˛j,m−k+4 ,
m
+
−1 a¯ 0 (m + 4)(m + 3)(m + 2)(m + 1)
c¯ k ˇj,m−k +
k=0 m
+
m k=0
m
d¯ k ˛j,m−k
k=0
b¯ k (m − k + 3)(m − k + 2)(m − k + 1)ˇj,m−k+3
a¯ k (m − k + 4)(m − k + 3)(m − k + 2)(m − k + 1)ˇj,m−k+4 .
k=1
4
= ˛1,2 = ˛1,2 = ˛3,1 = ˛1,2
k=0
where A¯ =
˛1,1 ˛2,0 ˛3,0 ˛1,1
k=1
Similarly, upon expanding the governing characteristic differential Eq. (16b), one can obtain the following equation with coefficients expressed in the polynomial form ∈ (0, 1)
= ˛5,3 = ˛6,3 = ˛7,3 = ˛8,3 = ˇ1,3 = ˇ2,3 = ˇ3,3 = ˇ4,3
−1 a0 (m + 4)(m + 3)(m + 2)(m + 1)
ck ˛j,m−k +
ˇj,m+4 =
¯s ¯s d3 W d4 W ¯ ¯ ¯ ¯ s + D() ¯ ¯ c = 0, + B() + C() W W A() d 4 d 3
= 1, = 1, = 1/2, = 1/6, = ˛5,1 = ˛5,2 = ˛6,1 = ˛6,2 = ˛7,1 = ˛7,2 = ˛8,1 = ˛8,2 = ˇ1,1 = ˇ1,2 = ˇ2,1 = ˇ2,2 = ˇ3,1 = ˇ3,2 = ˇ4,1 = ˇ4,2 = 1, = 1, = 1/2, = 1/6,
These eight fundamental solutions satisfy the normalization condition (20). Upon substituting Eq. (A3) into Eqs. (A1) and (A2) and collecting the coefficients of like powers of , the following recurrence formula can be obtained:
+
4
˛1,0 ˛2,0 ˛3,2 ˛4,3 ˛5,0 ˛6,0 ˛7,0 ˛8,0 ˇ1,0 ˇ2,0 ˇ3,0 ˇ4,0 ˇ5,0 ˇ6,1 ˇ7,2 ˇ8,3
k=0 m
where ˜= A
and
m
¯c ¯c d4 W d3 W ˜ ˜ W ˜ ¯ c + D() ˜ W ¯ s = 0, A() + B() + C() 4 d d 3
325
(A4b)
With these recurrence formulas, one can generate the eight exact normalized fundamental solutions of the differential Eqs. (A2) and (A3).
˜ c¯ k k = C,
k=0
3
¯ = D
˜ d¯ k k = −D
(A2b)
k=0
Because the coefficients of the coupled differential Eqs. (A1) and (A2) can be expressed in matrix polynomial form, then a power series representation of the fundamental solutions can be constructed by the modified method of Frobenius. One can assume that the eight fundamental solutions of Eqs. (A1) and (A2) are in the form of
¯ c,j W ¯ s,j W
=
∞ ˛j,k k k=0
ˇj,k k
,
j = 1, 2, . . . , 8
(A3a)
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