Simulating gear and bearing interactions in the presence of faults: Part I. The combined gear bearing dynamic model and the simulation of localised bearing faults

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Abstract

This paper presents a simulation model for a gearbox test rig, in which a range of bearing faults can be implemented. Bearing faults sometimes manifest themselves by their interaction with meshing gears, and to simulate this it is necessary to model a whole system of gears and shafts supported in bearings. This has now been done for an experimental test rig through the incorporation of a time-varying, non-linear stiffness bearing model into a previously developed gear model. The incorporated bearing model is based on Hertzian contact theory, which relates the raceway displacement to the bearing load, and also accounts for the slippage between the elements. It has the capacity to model localised spalls (inner race, outer race and rolling elements), which are discussed in this part of the paper and extended inner and outer race faults (rough surfaces), which are discussed in part II. Even though the whole gearbox has not been modelled in detail, the non-linear time-varying gear-meshing operation is modelled in some detail. Both simulated and experimental localised fault signals (acceleration signals) were subjected to the same diagnostic techniques; namely spectrum comparisons, Spectral Kurtosis (SK) analysis and envelope analysis. The processed simulated signals showed a similar pattern to that observed in their measured counterparts and were found to have a characteristic, referred to in the literature as double pulses, corresponding to entry into and exit from the localised fault. The simulation model will be useful for producing typical fault signals from gearboxes to test new diagnostic algorithms, and possibly prognostic algorithms.

Introduction

In order to facilitate the development of diagnostic and prognostic techniques for rolling element bearings in real systems, a necessity existed for producing simulation models, where faults can be implemented under different operating conditions rather than waiting for these to occur naturally, or alternatively having them seeded in the laboratory.

Such fault simulation can be very valuable in machine diagnostics and prognostics in order to produce signals with well-defined characteristics. For example, the signals could be used to train neural networks to perform diagnostics and prognostics of a range of different fault types and locations in machines. These usually require so much data to train them that it would not be economical to actually experience the number of faults of each type required to accomplish the training.

This paper presents a simulation model for a gearbox test rig, in which a range of bearing faults can be implemented. The gearbox test rig model described in Ref. [1] is presented in Section 2. This was updated by using a bearing model (two-dimensional model) similar to that in Ref. [2], with the capability of introducing geometrical faults (localised spalls). Section 3 discusses the bearing model, which includes the slippage, the inner/outer race localised fault modelling and the modelling of localised faults in rolling elements. Note that in the bearing model, the gearbox casing was included (not discussed in previous models) so that the vibration response, even at high frequencies, could be extracted for comparison with that done experimentally. Noise was added to the obtained response signals so that a reasonable signal-to-noise ratio (SNR) is achieved (15–25 dB). This noise represents the random part of the excitation, which includes all the disturbances (load variations, bearing friction, etc.). Section 4 discusses the dynamics of the gears, while Section 5 presents the combined gear-bearing model building on the presentation of 2 The spur gear test rig and its initial simulation models, 3 Rolling element bearing modelling, 4 Gear dynamics (based on Ref.. The comparison between the simulated results and the experimental results is presented in Section 6 for the three types of localised faults in the bearing, viz. inner/outer races and rolling elements.

While this part of the paper (part I) is concerned with the modelling of localised faults, the second part (part II) discusses the modelling of extended faults and compares the simulated results with those experimentally obtained for extended inner and outer race faults.

Some of the theoretical development of this paper was presented in conference papers [3], [4].

Section snippets

The spur gear test rig and its initial simulation models

The gearbox test rig (Fig. 1) under investigation was built by Sweeney [5] to investigate the effect of gear profile errors on transmission error (TE). In this test rig, the single stage gearbox (in this case a spur gear set with 1:1 ratio and 32 teeth on each gear) is driven primarily by a three-phase electric motor, but with circulating power via a hydraulic pump/motor set. The input and output shafts of the gearbox are arranged in parallel and each shaft is supported by two double row ball

A review of the dynamic modelling of rolling element bearings

The first few major works on rolling element bearing dynamic modelling were performed by Lundberg and Palmgren [9] and Harris [10], who described both radial and axial load deflections using non-linear stiffness coefficients, but did not conventionally address the total non-linearity and time-varying characteristics of the rolling element bearings. The first attempt to complete a dynamic model of rolling element bearings was reported by Gupta [11] through solving the generalised differential

Gear dynamics (based on Ref. [1])

Using a lumped parameter model, a pair of meshing gears can be modelled as a pair of cylindrical masses (Fig. 7) connected by a position-dependant stiffness variable km(θ), a damping coefficient cm and a combined effect of tooth topography deviations and misalignment of the gear pair et(θ) [1].

In this model [1], the et(θ), i.e. the effect of Geometric Transmission Error (GTE) was estimated based on combined waveform expressions of the run-out and the toothprofile errors as follows:et(θ)=erunout+

The new gear-bearing simulation model

There are 34 DOF in the new model as opposed to the 16 DOF in the previous model [1]. This is illustrated in the schematic diagram of Fig. 8. The extra 18 DOF are due to the inclusion of the five-DOF bearing model (Fig. 3), and to the fact that translational DOF are now considered both along the Line of Action (LOA) and perpendicular to it (Fig. 3, Fig. 7).

The main assumptions, on which the new model is based, are as follows.

  • (1)

    Shaft mass and inertia are lumped at the bearings or at the gears.

  • (2)

Experimental and simulated results

Different kinds of faults were introduced to a number of bearings, in a previous study by Ho [25], to investigate the use of Self Adaptive Noise Cancellation (SANC) for bearings in a gearbox environment. These have been used here to investigate the effectiveness of the simulation model.

In order to compare the simulation results to those obtained experimentally, the different types of faults were tested under a 50-N m load, while setting the output shaft speed to 10 Hz (600 rpm). Vibration signals

Conclusion

This paper presented a combined gear/bearing dynamic model for a gearbox test rig to study the interaction between gears and bearings in the presence of faults. The 34-DOF model of the test rig is a lumped mass parameter model which now has the capacity of modelling different fault types in the different parts of the bearing (inner race, outer race and the rolling elements). This is in addition to its original capacity for modelling spalls and cracks in the gears.

The new model takes into

Acknowledgement

This work is supported by the Australian Defence Science and Technology Organization (DSTO) as a part of their Centre of Expertise scheme.

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