Coupled Flexural-Torsional Vibration Analysis of a Double-Cantilever Structure for Nanomaching Application
This dissertation aims to investigate the coupled flexural-torsional vibrations of a piezoelectrically-actuated double-cantilever structure for nanomachining applications. The structure of interest consists of two identical Euler-Bernoulli cantilever beams connected by a rigid tip connection at their free ends. The double-cantilever structure in this study vibrates in two distinct modes: flexural mode or coupled flexural-torsional mode. The flexural mode refers to the in-phase flexural vibrations of the two cantilever beams resulting in transverse motion of the tip connection, while the coupled flexural-torsional mode refers to the coupled flexural-torsional vibrations of the cantilever beams resulting in the rotational motion of the tip connection. The latter is the main interest of this research. The governing equations of motion and boundary conditions are developed using Hamilton’s principle. Two uncoupled equations are found for each beam: one corresponding to the flexural vibrations and the other one corresponding to the torsional vibrations of the cantilever beam. The characteristic equations for both the flexural and the coupled flexural-torsional vibration modes are derived and solved to find the corresponding natural frequencies. The orthogonality condition among the mode shapes is derived and utilized to determine the modal coefficients corresponding to each mode of vibration. The time response to the forced vibrations of the structure is found using the Galerkin approximation method. The effects of the dimensional parameters, including the length of the cantilever beams and the length of the tip connection, and the piezoelectric input voltage on the natural frequencies and the amplitude of vibrations of the structure are analyzed. An experimental setup consisting of a piezoelectric double-cantilever structure is designed and utilized to verify the analytical results. First, the coupled flexural-torsional fundamental frequencies of the structure with various configurations are obtained experimentally, which are in good agreement with the analytically-determined values. Moreover, the experimental results verify the analytical results stating that the natural frequencies of the structure decrease as either the length of the cantilever beams or the length of the tip connection is increased. Next, the amplitudes of the coupled flexural-torsional vibrations of different configurations of the structure excited at their natural frequencies with a range of input voltages are obtained. The results of the effect of the dimensional parameters and the piezoelectric input voltage on the angle of rotation of the tip connection are presented.