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Experimental and theoretical investigations about the nonlinear vibrations of rectangular atomic force microscope cantilevers immersed in different liquids

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Abstract

Nonlinear flexural vibrations of rectangular atomic force microscope cantilever have been investigated by both the theoretical model and experimental works. As for the theoretical model, the Timoshenko beam theory which takes the rotatory inertia and shear deformation effects into consideration has been adopted. To increase the accuracy of the theoretical model, all necessary details for cantilever and sample surface have been taken into account. Differential quadrature method as a simple and fast numerical method has been used for solving the differential equations. During the investigation, the softening behavior was observed for all cases. It was also seen that raising the amplitude of vibrations led to a decrease in the nonlinear resonant frequency to linear resonant frequency ratio. The effects of different parameters such as normal and lateral contact stiffness, cantilever thickness, the angle between cantilever and sample surface and tip height in the presence of air as environment on the softening behavior were also examined. It was also demonstrated that increasing the lateral and normal contact stiffness, but decreasing the Timoshenko beam parameter would lead to an increase in the amplitude of vibrations for the first and second modes. The vibrational behavior of cantilever immersed in different liquids including water, methanol, acetone and carbon tetrachloride has been studied. Results show that increasing the liquid density reduces the nonlinear frequency. Furthermore, experimental works were compared with theoretical model for water and air environments. Results show good agreement.

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Abbreviations

A :

Area of the cross section

c :

Damping coefficient of the beam material

\(c_{a} \) :

Additional hydrodynamic damping

\(c_{\infty }\) :

Hydrodynamic damping when the cantilever is vibrating in free liquid

\(c_{s}\) :

Hydrodynamic damping when the cantilever is close to the surface

AFM:

Atomic force microscope

\(C_{ij}^{(n)} \) :

The weighting coefficients for the nth order derivative for the cantilever

d :

Distance between the lower edge of the cantilever and the centroid of the cross section

D:

Equilibrium tip–sample separation between the cantilever tip and the sample surface

E :

Young’s modulus

\(E_{\text {t}} \) :

Young’s modulus of the tip

\(E_{\text {s}} \) :

Young’s modulus of the sample

\(f_{\text {t}} ,f_{\text {n}} \) :

Interaction forces in tangential and normal directions of sample surface

\(f_{d} \) :

Hydrodynamic force

G :

Shear modulus

\(G_{\text {t}} \) :

Shear modulus of the tip

\(G_{\text {s}} \) :

Shear modulus of the sample

H :

Distance from the natural axis of the beam to the top of the tip

h(x, t):

The transient distance between the cantilever and the surface

I :

Moment of the cross section

k :

Shear coefficient

\(k_{\text {c}}\) :

Cantilever contact stiffness

\(k_{n} ,k_{t} \) :

Linear normal and lateral contact stiffness of the sample surface

\(k_{n1} ,k_{n2} ,k_{t1} ,k_{t2} \) :

Nonlinear normal and lateral contact stiffness of the sample surface

kAG :

shear rigidity

\(m_{\text {tip}} \) :

tip mass

N :

The number of total discrete grid points in the domain

r :

Timoshenko beam parameter to describe the rotatory inertia effect

\(R_{\text {t}} \) :

Tip radius

s :

Parameter to describe the shear deformation effect

t :

Time

w :

Transverse deflection of the cantilever

\({\overline{w}} \) :

Dimensionless transverse deflection of the cantilever

x :

Longitudinal coordinate

y :

General transverse deflection of the cantilever

t :

Time

\({\overline{t}} \) :

Dimensionless time

\(Z_{\text {0}} \) :

Surface offset

\(\xi \) :

Longitudinal coordinate parameter for the cantilever

\(\alpha \) :

Angle between the cantilever and sample surface

\(\lambda \) :

Non-dimensional resonant frequency parameter

\(\rho \) :

Mass density

\(\rho _{\text {a}} \) :

Additional mass per length

\(\rho A \) :

Mass per unit length

\(\mu \) :

Dimensionless damping parameter

\(\varPhi \) :

Bending angle of the cantilever

\(\omega \) :

Circular resonant frequency

\(\sigma _{\text {n}} \) :

Nonlinear frequency to linear frequency ratio

\(\delta _{0} \) :

Static contact deformation

\(\eta \) :

Density of the liquid

\(\varLambda \) :

Normal contact stiffness relative to the cantilever stiffness

\(\nu \) :

Poisson’s ratio

\(\nu _{\text {t}} \) :

Poisson’s ratio of the tip

\(\nu _{\text {s}} \) :

Poisson’s ratio of the sample

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  1. Renewable Energy Research Center, Damavand Branch, Islamic Azad University, Damavand, Iran

    Ali Hossein Gholizadeh Pasha & Ali Sadeghi

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  1. Ali Hossein Gholizadeh Pasha
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Pasha, A.H.G., Sadeghi, A. Experimental and theoretical investigations about the nonlinear vibrations of rectangular atomic force microscope cantilevers immersed in different liquids. Arch Appl Mech 90, 1893–1917 (2020). https://doi.org/10.1007/s00419-020-01703-5

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  • DOI: https://doi.org/10.1007/s00419-020-01703-5

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