Dynamic Transmission Error Research Articles

Spectral coupling issues in a two-degree-of-freedom system with clearance non-linearities

In an earlier study [14], the frequency response characteristics of a multi-degree-of-freedom system with clearance non-linearities were presented. The current study is an extension of this prior work and deals specifically with the issue of dynamic interactions between resonances. The harmonic balance method, digital solutions and analog computer simulation are used to investigate a two-degree-of-freedom system under a mean load, when subjected to sinusoidal excitations. The existence of harmonic, periodic and chaotic solutions is demonstrated using digital simulation. The method of harmonic balance is employed to construct approximate solutions at the excitation frequency which are then used to classify weak, moderate and strong non-linear spectral interactions. The effects of parameters such as damping ratio, mean load, alternating load and frequency spacing between the resonances have been quantified. The applicability of the methodology is demonstrated through the following practical examples: (i) neutral gear rattle in an automotive transmission system; and (ii) steady state characteristics of a spur gear pair with backlash. In the second case, measured dynamic transmission error data at the gear mesh frequency are used to investigate spectral interactions. Limitations associated with solution methods and interaction classification schemes are also discussed.

A non-linear mathematical model for dynamic analysis of spur gears including shaft and bearing dynamics

A six-degree-of-freedom non-linear semi-definite model with time varying mesh stiffness has been developed for the dynamic analysis of spur gears. The model includes a spur gear pair, two shafts, two inertias representing load and prime mover, and bearings. As the shaft and bearing dynamics have also been considered in the model, the effect of lateral-torsional vibration coupling on the dynamics of gears can be studied. In the non-linear model developed several factors such as time varying mesh stiffness and damping, separation of teeth, backlash, single- and double-sided impacts, various gear errors and profile modifications have been considered. The dynamic response to internal excitation has been calculated by using the “static transmission error method” developed. The software prepared (DYTEM) employs the digital simulation technique for the solution, and is capable of calculating dynamic tooth and mesh forces, dynamic factors for pinion and gear, dynamic transmission error, dynamic bearing forces and torsions of shafts. Numerical examples are given in order to demonstrate the effect of shaft and bearing dynamics on gear dynamics.

Method for measuring the dynamic transmission error in large gear hobbing machines

A new method for measuring the dynamic transmission error between a hob and worktable in large gear hobbing machines is described in which a pair of small, highly sensitive servo-accelerometers are mounted tangentially, 180° (arc degree) apart, on the periphery of a large worktable, in order to detect rotational fluctuations of the worktable. Two acceleration signals from the servo-accelerometers mounted on the worktable are summed in order to determine the circumferential acceleration. The resulting sum will nullify the rectilinear acceleration of the worktable. After the circumferential acceleration signal transmitted by a telemeter is processed by a pulse signal from a rotary encoder attached to the hob shaft in this apparatus, the dynamic transmission error can be obtained. In tests using a large gear hobbing machine (3.3 m worktable diameter), the dynamic transmission error has been measured and recorded. By processing the recorded analogue signal with an FFT (Fast Fourier Transformation) analyser and by consulting the schematic drawing of the rotational transmission mechanism, specific parts in the mechanism causing the transmission error can be identified. Also, by processing the circumferential acceleration signal of the worktable from a standing start with an FFT analyser, the rotational natural frequency of the worktable can be easily obtained.

Dynamic analysis of high speed gears by using loaded static transmission error

A single degree of freedom non-linear model is used for the dynamic analysis of a gear pair. Two methods are suggested and a computer program is developed for calculating the dynamic mesh and tooth forces, dynamic factors based on stresses, and dynamic transmission error from measured or calculated loaded static transmission errors. The analysis includes the effects of variable mesh stiffness and mesh damping, gear errors (pitch, profile and runout errors), profile modifications and backlash. The accuracy of the method, which includes the time variation of both mesh stiffness and damping is demonstrated with numerical examples. In the second method, which is an approximate one, the time average of the mesh stiffness is used. However, the formulation used in the approximate analysis allows for the inclusion of the excitation effect of the variable mesh stiffness. It is concluded from the comparison of the results of the two methods that the displacement excitation resulting from a variable mesh stiffness is more important than the change in system natural frequency resulting from the mesh stiffness variation. Although the theory presented is general and applicable to spur, helical and spiral bevel gears, the computer program prepared is for only spur gears.

高速域における平歯車の片歯面かみあい試験

It is tried to use the S 2 type of gear rolling tester at the high rotational speed where the meshing frequency is about 1/2-2 times as large as the natural frequency of a pair of gears. This trial is based on the simplified measuring principle. As a result, it is found that the gear rolling tester can measure the dynamic transmission error of gears with the accuracy enough for practical use. Next, the dynamic transmission error of hobbed gears which have large profile errors is measured under light load condition, and the vibration of gears is studied experimentally. As a result, the following phenomena are observed : the large and irregular vibration skipping several teeth occurs near the resonance speed of gears, and it does not decrease even at the higher rotational speed than the resonance one, the period, where the teeth of gears separate and do not mesh during rotation, reaches 70%-80% of the total rotational period and also do not decrease at the higher rotational speed than the resonance one.