The quartz double-end tuning fork is composed of two parallel slender beams with their ends fixed to the proof masses, both ends of which are clamped. The structure is made of a quartz wafer which is anisotropic in stiffness. In anti-phase mode the two slender parallel beams vibrate in opposite directions and can be modelled as an Euler beam. The twist moments caused by the slender beams on the proof mass make the cross-section of the proof mass deform into a warped surface. The objective of this research is to establish the warping deformation model so that we can build up the equation of motion for anisotropic stiffness by using Hamilton’s principle and then perform theoretical analysis. The more realistic warping displacement leads the natural frequency closer to the true one. The purpose of the proof mass is to modulate the frequencies and mode shape of tuning fork beams. The advantage of anti-phase mode is that the centre of mass in unchanged during motion so that the system has a higher signal-to-noise ratio. The theoretically obtained frequency is compared with the experimental one and that obtained by the finite element method.

A self-sensing and self-actuating quartz tuning fork (QTF) can be used to obtain its frequency shift as function of the tip-sample distance. Once the function of the frequency shift versus force gradient is acquired, the combination of these two functions results in the relationship between the force gradient and the tip-sample distance. Integrating the force gradient once and twice elucidates the values of the interaction force and the interatomic potential, respectively. However, getting the frequency shift as a function of the force gradient requires a physical model which can describe the equations of motion properly. Most papers have adopted the single harmonic oscillator model, but encountered the problem of determining the spring constant. Their methods of finding the spring constant are very controversial in the research community and full of discrepancies. By circumventing the determination of the spring constant, we propose a method which models the prongs and proof mass as elastic bodies. Through the use of Hamilton’s principle, we can obtain the equations of motion of the QTF, which is subject to Lennard-Jones potential force. Solving these equations of motion analytically, we get the relationship between the frequency shift and force gradient.