Non-probabilistic interval process method for analyzing two-stage straight bevel gear system with uncertain time-varying parameters

Abstract

A two-stage straight bevel gear system is a gear system that can be used in various applications. The straight bevel gear is known for its complex tooth geometry. Due to the variation of the number of pairs of teeth in contact, the mesh stiffness function can be considered as a time-varying function. However, the mesh stiffness for the straight bevel gear is sensitive to measurement and modeling errors. Thus, at each time step, its value can not assigned to deterministic one. Generally, the uncertain parameters are assumed to be time-independent. In this paper, the interval process method has been used to represent the time-varying uncertain parameters, whose bounds are determined through the potential energy method. The lumped parameter model of two-stage straight bevel gear has been proposed. We have considered that the masses of the straight bevel gear system components and bearing stiffnesses along with time-varying mesh stiffnesses are uncertain parameters which can be represented by the interval process model. The Chebyshev polynomial expansion has been used to approximate the response of the two-stage straight bevel gear system with respect to the interval variables. The lower and higher bounds of the eigenvalues of the system have been determined. The bounds of dynamic displacements of the straight bevel gear system have been computed and compared with those computed by the Monte Carlo method.

Effects of the interval geometric deviation and crowning parameters on the automotive differential dynamics

“…Modelling mesh stiffness for the straight bevel gear Time-varying mesh stiffness reflects the flexibility of gear teeth in mesh and the variation of the number of teeth pairs in contact. In various studies, the mesh stiffness of a gear system is assumed to be the sole source of excitation 50,51,28 . Thus, the mesh stiffness should be modelled accurately so that the proposed dynamic model may reflect the actual dynamic characteristics of a gear system.…”

“…In contrast to the parallel-shaft gear systems, the tooth geometry of the straight bevel gear precludes the modelling of the mesh stiffness due to its complex geometry. To this end, Tregold approximation and the slice theory 21,28 have been used in order to approximate the straight bevel gear as a set of virtual spur gears as shown in Figure 5. The tooth of the straight bevel gear is sliced into a set of slicing planes as represented in Figures 5 and 6.…”

“…To sum things up, the ranges of the displacements of the differential system have been determined by using the algorithm illustrated in Figure 14. The reason for using the algorithm mentioned in the work of Lafi et al 28 in favour of the algorithm presented in this paper is that the presented algorithm can be used in a system with multiple interval variables, and the simulation time of the algorithm is much shorter. The scanning method 60,61 is used in this paper as a reference method to validate the computational algorithm.…”

“…Wang and Zhang 26 have scrutinized the dynamic behaviour of a spur gear when mesh stiffness, an external load, backlash and a transmission error have been regarded as interval parameters. Two studies 27,28 developed a time-variant interval process model to examine the dynamic response of the uncertain system when the uncertain parameters are time-dependent. Despite the importance of the interval process model to deal with the dynamic model with uncertain time-varying parameters, the interval process model can be computationally expensive when the number of uncertain parameters exceeds 4.…”

Effects of interval friction coefficients on the differential mechanism dynamics

“…Some works [17][18][19] developed a time-variant interval process model to examine the dynamic response of the uncertain system when the uncertain parameters are time-dependent. The authors implemented their methods into three mechanical systems.…”