Distortion of a Harmonic Elastic Wave Reflected From a Dry Friction Support

This study is motivated by the need to understand the elastodynamic response of belts in frictional contact with pulleys. To this end, a simplified model for belt~pulley contact is used to investigate the dynamic response of a belt subject to a train of harmonic tension waves. Through a nondimensionalization, a single dimensionless parameter f~ is identified which governs the dynamic response. A numerical solution is developed and exercised over a wide range of values of fL An approximate closedform solution is derived assuming the belt stretches quasi-statically, and is shown to yield accurate results for small values of ~. Reported results include the distortion of an initially harmonic tension wave, the energy reflected from the frictional support, and the distance harmonic waves penetrate into the support. The results suggest that the quasi-static stretching assumption may be further utilized as a modeling 1 simplification for belt drives characterized by values of f~ < -j. This study is principally motivated by belt drive mechanics which have been studied by many researchers, starting with Leonard Euler (1762). Euler proposed the well-known capstan formula for the tension distribution in a belt wrapped around a fixed pulley or a capstan (a fixed post used in docking a ship). A comprehensive survey of belt drive mechanics after Euler and up to 1981 is provided by Fawcett (1981). Much of the research cited in Fawcett (1981) and work since its appearance has been concerned with either (1) frictional contact (slip) between a belt and a steadily rotating pulley (neglecting vibratory effects) or (2) vibration of belts neglecting frictional contact. Early research on frictional contact was carried out by Grashof (1883), who studied frictional mechanics of belt drives under steady operating speeds and applied torques. The belt was treated as a string and the mechanism of elastic creep of the belt along the pulley was shown to yield a single slip arc on the exit region of the pulley. In this classical creep theory, the transition from low to high tension (or vice-versa) occurs in this slip arc; see also (Johnson, 1985). Later investigators, including Firbank (1970) and Gerbert ( 1991, 1996), considered the influence of belt thickness as well as its length, and proposed revised estimates of the extent of the slip zone based on belt creep, shear, radial deformation, and seating/unseating behavior. These new analyses predicted tension transitions in the adhesion arc as well as in the slip arc. Finally, Townsend and Salisbury (1988) used a control volume approach to calculate the energy efficiency of a two-pulley belt drive based solely on the stiffness per unit length of the belt and the transmitted torque. Again, the operating conditions of the drive were assumed to be steady. More recently, substantial research has concentrated on serpentine belt drive systems which include an automatic tensioning element, as commonly employed in vehicle front end