This work aims at developing linear mathematical models of single-mesh spur and helical gears mounted on compliant parallel shafts, and single span of a serpentine belt with pulleys also mounted on parallel compliant shafts. Both the geared shaft models are hybrid discrete-continuous ones where the gears modeled as rigid disks along with the mesh spring form the discrete elements while the elastic shafts having transverse as well as torsional flexibility constitute the continuous elements. The non-dimensional governing equations along with the natural boundary conditions are developed using the Hamilton's principle. The governing equations of the flexural and torsional shafts vibrations and the equations of motion of the disks are written in an extended operator form to prove the self-adjointness of the system. The assumed modes method is used to discretize the system equations where the matching conditions are incorporated with the use of Lagrange multipliers. Orthonormal global basis functions for flexure and torsion are chosen from separate families forming complete sets. The sensitivities of the natural frequencies of different modes to mesh stiffness, torsional and flexural rigidities of the shafts, and lengths of the shafts are examined and the results are correlated with the modal energy distributions. Excitation in the form of the loaded static transmission error at the gear mesh is identified and converted to the discretized form and the response for the same is calculated.
Torsional spring at the gear mesh in the helical gear-shaft model accounts for the energy stored due to the relative tilting of the gears. The rotation speed is high and therefore, the gyroscopic effect is non-negligible. Hamilton's principle is used to obtain the non-dimensional governing equations and the equations of motion of the disks. Excitation in the form of the loaded static transmission error at the gear mesh is incorporated in the equations of motion. The extended operator formulation is employed to simplify the system equations to a compact analytical form, which is prototypical of a gyroscopic system. Galerkin's method is used to discretize the continuous system equations. Natural frequency sensitivity to various system parameters such as the rotation speed and mesh stiffness is determined and explained using modal strain energies. Response due to the loaded static transmission error is obtained by performing modal analysis after reducing the discretized system to a first order form. The study shows that the gyroscopic effect is present even for short lengths of the shafts. For coupled frequencies, response of the system is found to be approximated by the modal superposition of a smaller number of modes.
The mathematical model and analysis of the belt-pulley-shaft system are similar to the helical gear-shaft model, where the belt is modeled as a combination of an extensional and torsional spring. The axis of the torsional spring is different from the previous model. Periodic load fluctuation from the engine in the form of a force on the shaft attached to one of the pulleys is considered and the response in the form of tension fluctuation of the extensional spring is determined.