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Papers by William Mark
The Journal of the Acoustical Society of America, 1984
Journal of Mechanisms, Transmissions, and Automation in Design
The vibratory excitation arising from a gear pair is widely recognized to be a consequence of the... more The vibratory excitation arising from a gear pair is widely recognized to be a consequence of the nonuniform transmission of motion by the gear pair. In the preceding paper in this issue, it is shown that a three-component transmission error is required to describe the nonuniform transmission of motion by bevel gears for vibration excitation characterization purposes. The expression for the three-component transmission error derived in that paper is combined in the present paper with an analysis of the mesh forces and mesh elasticity to yield an equation of constraint involving the six degree-of-freedom unknown vibratory displacements of the gear shaft centerliners, the three unknown components of the generalized force transmitted by the mesh, and the geometric deviations of the tooth running surfaces from perfect involute surfaces which are assumed known. This matrix equation can be combined with the equations of motion of a gear system to predict the vibratory response of the syst...
Journal of Mechanisms, Transmissions, and Automation in Design
For straight or spiral bevel gears of nominal spherical involute design, the resultant total forc... more For straight or spiral bevel gears of nominal spherical involute design, the resultant total force vector transmitted by the gear mesh, in the absence of friction, lies in the plane of tooth contact. This force vector can be characterized by three scalar components, two orthogonal force components lying in the plane of contact and the resultant moment taken about the nominal center of the zone of contact. Equations for these three generalized force components are derived. The equations are expressed in terms of tooth pair/gear body stiffnesses, bearing/bearing support flexibility influence coefficients, the shaft input torque, deviations of the tooth running surfaces from perfect spherical involute surfaces, and bearing centerline offsets from the positions occupied by the base cone axes of the perfect involute bevel gear counterparts to the actual gears under consideration. Inertial forces arising from transverse and axial vibrations of the gear bodies are assumed to be negligible ...
Journal of Mechanisms, Transmissions, and Automation in Design
For a given set of forces transmitted by the gears, each of the three components of the generaliz... more For a given set of forces transmitted by the gears, each of the three components of the generalized transmission error of spiral bevel gears is shown to be stationary with respect to small independent variations in the positions of the endpoints of the lines of tooth contact about their true values. The tangential generalized transmission error component is shown to take on a minimum value at the true endpoint positions. A computational procedure based on the method of steepest descent is described for computing the true line of contact endpoint positions and the three components of the generalized transmission error. A method for computing the Fourier series coefficients of the tooth meshing harmonics of the three generalized transmission error components also is provided.
Journal of Mechanisms, Transmissions, and Automation in Design
Journal of Mechanisms Transmissions and Automation in Design
The traditional definition of the transmission error of parallel-axis gear pairs is reviewed and ... more The traditional definition of the transmission error of parallel-axis gear pairs is reviewed and shown to be unsuitable for characterizing the deviation from conjugate action of bevel gear pairs for vibration excitation characterization purposes. This situation is rectified by generalizing the concept of the transmission error of parallel-axis gears to a three-component transmission error for spiral bevel gears of nominal spherical involute design. A general relationship is derived which expresses the contributions to the three-component transmission error from each gear of a meshing spiral bevel pair as a linear transformation of the six coordinates that describe the deviation of the shaft centerline position of each gear of the pair from the position of its rigid perfect involute counterpart.
Journal of Mechanisms Transmissions and Automation in Design
The traditional one-component transmission error of parallel-axis helical gears is generalized to... more The traditional one-component transmission error of parallel-axis helical gears is generalized to a three-component transmission error which characterizes the composite displacement in the plane-of-contact resulting from arbitrary small deviations in the positions of both gears of a meshing pair from the positions of their rigid perfect involute counterparts. A set of linear algebraic equations is derived for the contribution to the three generalized transmission error components arising from elastic deformations of the teeth and gear bodies and deviations of the tooth running surfaces from equispaced perfect involute surfaces. It is shown how to combine this set of equations with the generalized transmission error definition and the equations of motion of a gear system to predict the dynamic response of gear elements in the system. For the case of negligible gearbody and bearing/bearing support inertial forces, an additional set of algebraic equations that includes the effects of b...
Journal of Mechanisms Transmissions and Automation in Design
A method is developed for analytically reconstructing the geometric deviations of the running sur... more A method is developed for analytically reconstructing the geometric deviations of the running surface of a gear tooth from a perfect involute surface. The method uses standard profile and lead deviation measurements and is applicable to both helical and spur gears. The reconstruction is carried out by using normalized Legendre polynomials. For this class of functions, it is shown that the optimum locations of the profile and lead deviation measurements are the locations of the zeros of the Legendre polynomial of degree equal to the number of profile or lead deviation measurements taken–after appropriate normalization of the tooth width or depth, as appropriate. A least squares fit procedure for establishing a common origin of ordinates for sets of profile and lead deviation measurements is formulated, and its solution is carried out in closed form. Account is taken of the noninsignificant errors that typically arise in profile and lead deviation measurements so that the final analyt...
Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science
An expression is derived for the Fourier series spectrum of the transmission error arising from t... more An expression is derived for the Fourier series spectrum of the transmission error arising from tooth-spacing errors on a single spur gear meshing with a perfect involute mating gear for the case of a contact ratio of unity and no elastic deformations present. This expression is found to be in exact agreement with previously derived results. The expression illustrates the role of the discrete Fourier transform in transmission error analysis and interpretation.
Performance-Based Gear Metrology
Random Vibration in Mechanical Systems
The Journal of the Acoustical Society of America, 1984
Journal of Mechanisms, Transmissions, and Automation in Design
The vibratory excitation arising from a gear pair is widely recognized to be a consequence of the... more The vibratory excitation arising from a gear pair is widely recognized to be a consequence of the nonuniform transmission of motion by the gear pair. In the preceding paper in this issue, it is shown that a three-component transmission error is required to describe the nonuniform transmission of motion by bevel gears for vibration excitation characterization purposes. The expression for the three-component transmission error derived in that paper is combined in the present paper with an analysis of the mesh forces and mesh elasticity to yield an equation of constraint involving the six degree-of-freedom unknown vibratory displacements of the gear shaft centerliners, the three unknown components of the generalized force transmitted by the mesh, and the geometric deviations of the tooth running surfaces from perfect involute surfaces which are assumed known. This matrix equation can be combined with the equations of motion of a gear system to predict the vibratory response of the syst...
Journal of Mechanisms, Transmissions, and Automation in Design
For straight or spiral bevel gears of nominal spherical involute design, the resultant total forc... more For straight or spiral bevel gears of nominal spherical involute design, the resultant total force vector transmitted by the gear mesh, in the absence of friction, lies in the plane of tooth contact. This force vector can be characterized by three scalar components, two orthogonal force components lying in the plane of contact and the resultant moment taken about the nominal center of the zone of contact. Equations for these three generalized force components are derived. The equations are expressed in terms of tooth pair/gear body stiffnesses, bearing/bearing support flexibility influence coefficients, the shaft input torque, deviations of the tooth running surfaces from perfect spherical involute surfaces, and bearing centerline offsets from the positions occupied by the base cone axes of the perfect involute bevel gear counterparts to the actual gears under consideration. Inertial forces arising from transverse and axial vibrations of the gear bodies are assumed to be negligible ...
Journal of Mechanisms, Transmissions, and Automation in Design
For a given set of forces transmitted by the gears, each of the three components of the generaliz... more For a given set of forces transmitted by the gears, each of the three components of the generalized transmission error of spiral bevel gears is shown to be stationary with respect to small independent variations in the positions of the endpoints of the lines of tooth contact about their true values. The tangential generalized transmission error component is shown to take on a minimum value at the true endpoint positions. A computational procedure based on the method of steepest descent is described for computing the true line of contact endpoint positions and the three components of the generalized transmission error. A method for computing the Fourier series coefficients of the tooth meshing harmonics of the three generalized transmission error components also is provided.
Journal of Mechanisms, Transmissions, and Automation in Design
Journal of Mechanisms Transmissions and Automation in Design
The traditional definition of the transmission error of parallel-axis gear pairs is reviewed and ... more The traditional definition of the transmission error of parallel-axis gear pairs is reviewed and shown to be unsuitable for characterizing the deviation from conjugate action of bevel gear pairs for vibration excitation characterization purposes. This situation is rectified by generalizing the concept of the transmission error of parallel-axis gears to a three-component transmission error for spiral bevel gears of nominal spherical involute design. A general relationship is derived which expresses the contributions to the three-component transmission error from each gear of a meshing spiral bevel pair as a linear transformation of the six coordinates that describe the deviation of the shaft centerline position of each gear of the pair from the position of its rigid perfect involute counterpart.
Journal of Mechanisms Transmissions and Automation in Design
The traditional one-component transmission error of parallel-axis helical gears is generalized to... more The traditional one-component transmission error of parallel-axis helical gears is generalized to a three-component transmission error which characterizes the composite displacement in the plane-of-contact resulting from arbitrary small deviations in the positions of both gears of a meshing pair from the positions of their rigid perfect involute counterparts. A set of linear algebraic equations is derived for the contribution to the three generalized transmission error components arising from elastic deformations of the teeth and gear bodies and deviations of the tooth running surfaces from equispaced perfect involute surfaces. It is shown how to combine this set of equations with the generalized transmission error definition and the equations of motion of a gear system to predict the dynamic response of gear elements in the system. For the case of negligible gearbody and bearing/bearing support inertial forces, an additional set of algebraic equations that includes the effects of b...
Journal of Mechanisms Transmissions and Automation in Design
A method is developed for analytically reconstructing the geometric deviations of the running sur... more A method is developed for analytically reconstructing the geometric deviations of the running surface of a gear tooth from a perfect involute surface. The method uses standard profile and lead deviation measurements and is applicable to both helical and spur gears. The reconstruction is carried out by using normalized Legendre polynomials. For this class of functions, it is shown that the optimum locations of the profile and lead deviation measurements are the locations of the zeros of the Legendre polynomial of degree equal to the number of profile or lead deviation measurements taken–after appropriate normalization of the tooth width or depth, as appropriate. A least squares fit procedure for establishing a common origin of ordinates for sets of profile and lead deviation measurements is formulated, and its solution is carried out in closed form. Account is taken of the noninsignificant errors that typically arise in profile and lead deviation measurements so that the final analyt...
Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science
An expression is derived for the Fourier series spectrum of the transmission error arising from t... more An expression is derived for the Fourier series spectrum of the transmission error arising from tooth-spacing errors on a single spur gear meshing with a perfect involute mating gear for the case of a contact ratio of unity and no elastic deformations present. This expression is found to be in exact agreement with previously derived results. The expression illustrates the role of the discrete Fourier transform in transmission error analysis and interpretation.
Performance-Based Gear Metrology
Random Vibration in Mechanical Systems