A note on the contact problem in an ultrasonic travelling wave motor

Characterization of the Damping of a Free Vibrating Piezoelectric Motor Stator by Displacement Measurements

“…Research toward producing analytical models of the motor have focused on neglecting the rotor when determining the vibration frequency for driving the motor, and then developing an analytical contact model to describe the stator-to-rotor interaction [1][2][3][7][8][9]. Schmidt et al [10] developed an analytical model which includes the rotor-to-stator contact in the vibration frequency calculation by modeling the stator geometry as an equivalent beam. Models of piezoelectric motors including rotor-to-stator contact have also been created using the explicit dynamic finite element method [4,11].…”

“…This can be demonstrated by substituting equation (2-1) into equation (2)(3) and noting that (2)(3)(4)(5) therefore, (2)(3)(4)(5)(6) or (2)(3)(4)(5)(6)(7) requiring that (2)(3)(4)(5)(6)(7)(8) making use of the general identity for matrices, (2)(3)(4)(5)(6)(7)(8)(9) Taking advantage of the symmetry of such that, results in, (2)(3)(4)(5)(6)(7)(8)(9)(10) As with the coefficients, the manufacturer's literature seldom contains data for the permittivity values evaluated at constant strain . Instead, the permittivity values at constant stress are listed.…”

“…Setting equation (2)(3)(4) equal to equation , (2)(3)(4)(5)(6)(7)(8)(9)(10)(11) rearranging, (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) substituting in equation (2)(3) (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) Modeling Piezoelectric Ceramics Using ABAQUS and ANSYS…”

“…This results in differences in the ordering of the stress and strain vectors, the and coefficient matrices, and the and elastic property matrices. ANSYS also transposes the and coefficient matrices, such that equation (2)(3) and equation (2)(3)(4) are written in the following form with respect to the ANSYS program: (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) (2-15) Table 1 makes a comparison of these three formats. The transverse vibration of both the stator of the piezoelectric motor and the disk samples used in the damping experiment are very similar to the vibration of thin circular plates.…”

“…The loss of energy in the system is modeled as a viscous damper with the assumption that for low velocities, the force in the damper is proportional to the velocity of the mass. For the steady state response, it is convenient to choose the initial time reference such that the phase angle is zero and equation (3)(4)(5)(6)(7)(8)(9)(10)(11) becomes (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) The solution to this differential equation is of the form (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) where: The value of the motion amplitude is dependent on the magnitude of the forcing function , the stiffness of the spring , the amount of damping in the system, and how close the forcing frequency is to the system natural resonant frequency .…”