An Approach to Fault Diagnosis of Gearbox Based on an Instantaneous Angular Acceleration. Experimental Study

2020. Т. 20, No 1. С. 89–99 89 Introduction Gears are widely used in operating mechanisms in many industries. Variable loads and speeds lead to gear wear. Moreover, the growth of the defect rapidly evaluates and can be a factor of irreparable harm. In addition, the diagnosis of the gears is highly important but it is a difficult problem because of non-steady-state operations, as previously mentioned. The most common method of gears diagnosis is vibration diagnostics [1, 2]. Nevertheless, even a defect-free gear pair has some work vibration, which has a wide frequency band. The vibration is the result of key factors: cyclic variation of gear teeth stiffness in a meshing phase and manufacturing and assembly errors [3]. The cyclic variation and constant meshing tooth error excite oscillations on a gear-meshing frequency fz 1 1 2 2 , z f z f z f   where z1, z2 are teeth number of pinion and gear, respectively. The f1, f2 are rotation frequencies of the pinion and gear, respectively [4]. The variable meshing tooth appears on the rotation frequencies kf1, kf2 (k = 1, 2, 3, ...), and the modulated frequencies mfz ± nf1, mfz ± nf2 (m, n = 1, 2, 3, ...) [5]. At the same time, the frequency spectrum has components that are associated with cutting error of pinion and gear fd , 1, 2, , d d f kz f k   where zd is the number of teeth of the gear wheel of the gear cutting machine. Operating defects of gears Приборостроение, метрология и информационно-измерительные приборы и системы


Introduction
Gears are widely used in operating mechanisms in many industries. Variable loads and speeds lead to gear wear. Moreover, the growth of the defect rapidly evaluates and can be a factor of irreparable harm. In addition, the diagnosis of the gears is highly important but it is a difficult problem because of non-steady-state operations, as previously mentioned.
The most common method of gears diagnosis is vibration diagnostics [1,2]. Nevertheless, even a defect-free gear pair has some work vibration, which has a wide frequency band. The vibration is the result of key factors: cyclic variation of gear teeth stiffness in a meshing phase and manufacturing and assembly errors [3]. The cyclic variation and constant meshing tooth error excite oscillations on a gear-meshing frequency f z 1 1 2 2 , z f z f z f  where z 1 , z 2 are teeth number of pinion and gear, respectively. The f 1 , f 2 are rotation frequencies of the pinion and gear, respectively [4]. The variable meshing tooth appears on the rotation frequencies kf 1 , kf 2 (k = 1, 2, 3, …), and the modulated frequencies mf z ± nf 1 , mf z ± nf 2 (m, n = 1, 2, 3, …) [5]. At the same time, the frequency spectrum has components that are associated with cutting error of pinion and gear f d mating surfaces (such as wear, microchipping, sticking, crack, chipping, broke and related) are an additional consideration of frequency spectrum change. Moreover, a broken tooth is the most dangerous type of defect in view of rapid growth from crack to broke. Operational defects of gear at an early stage have a low energy contribution in the frequency spectrum that makes more complicated detecting them. Furthermore, commonly gearboxes are operated with other mechanical nodes, which also make highlevel noises [6,7]. That fact is challenging to methods of gearbox diagnosis and successfully fault detection [8,9].
Traditional source of information about condition gearbox is accelerometers which are fixed on the gearbox body. Moreover, various methods, criteria, and indicators are used for gearbox diagnosis in the time-domain, frequency-domain, and time-frequency domain. The time-domain diagnosis is the poorly-successful and problematic approach. That fact was shown in studies [10][11][12]. On the other hand, diagnosis in frequency-domain and time-frequency domain are more successful approaches. The commonly conditional indicators in the frequency-domain are the kurtogram [13,14], kurtogram with TSA [15], adaptive SK filtering method on the grounds of Morlet wavelet [16] et alii. The commonly conditional indicators in the time-frequency domain are Hilbert-transform based methods [17,18].
The author in study [19] describes the 8-DOF dynamic model of a gearbox in detail with a local defect of the pinion. The function of gear tooth meshing stiffness varies according to the modelled defect type (chipped or broken tooth). The pinion has z 1 = 19 teeth; the gear has z 2 = 48 teeth. The frequency of pinion rotation is f 1 = 30 Hz. The result shows a spectrum changing of gear linear accelerations with various local defects, which corresponds to the "canonical" non-monotonic behaviour described in the literature. However, focusing on the traditional measurement methods, the author considered the behaviour of only the linear acceleration spectrum. On the other hand, analysis of the spectrum of an angular acceleration gear allows for increasing possibilities of a gearbox diagnosis.
The approach of the shaft acceleration measuring from the rotating shaft, including angular acceleration, studied in [20]. Moreover, the mathematical model one of the design versions of the WAS-Technology sensor described in [21]. That design allows measuring angular acceleration, two linear accelerations in the diametrical plane and the turning angle of the shaft.

Method for processing of angular acceleration
An analysis of the gear angular accelerations changing (dynamic model from Ref. [19]) under the influence of local defects shows that the changes of the spectrum discrete components are associated with harmonics of the gear frequency is close to monotonic (Fig. 1). At the same time, a modulation index of the angular acceleration spectrum around harmonics of the gear frequency increases in the same way as the linear acceleration spectrum. Moreover, defects detection based on a comprehensive analysis of discrete components, which are associated with harmonics of the gear and rotation frequencies, by the Prism-method was studied in [22,23]. On the other hand, the first pinion harmonic f 1 of the gear angular spectrum increases by increasing local defect, in other words, the level of harmonics of the frequency meshing defected tooth increases (Fig. 2).

Fig. 2. The first harmonic angular acceleration of defected pinion with various defects
Thus, the level of the first pinion harmonic can operate as a criterion for the detection of local defects, such as a chipped and broken tooth.

Test equipment 2.1. Experimental rig
The applicability of the proposed criterion for the diagnosis of gear transmission is considered on the example of a single-stage gearbox of the experimental rig (MFS-Magnum from SpectraQuest) with bevel gears (Fig. 3).
The motor of the experimental rig is operated by the frequency drive that allows setting any motor rotation frequency in the operating range.
The pinion of the gearbox has z p = 18 teeth; the gear has z g = 27 teeth. The design of the studied gearbox allows for the simulation of the pinion defects such as the chipped tooth (Fig. 4) and the broken tooth (Fig. 5). Simulating the pinion defects due to pinion experiences more wear than the gear because pinion has fewer teeth and makes more revolutions.

WAS-Sensor
The applicability of the proposed criterion for the diagnosis of gear transmission designed the prototype of a WAS-Sensor based on WAS-Technology. The WAS-Sensor contains three MEMS-accelerometers (ADXL001-70 from Analog Devices) (Fig. 6). Each accelerometer has a measurement rate ±70g, resonance frequency 22 kHz, sensitivity 24.2 mV/g at 100 Hz. The mathematical model of the Sensor is described in the study [21]. Also, the WAS-Sensor contains ADC (AD7609 from Analog Devices) which samples of accelerometers signals at 39.4 kHz. Moreover, the WAS-Sensor contains microprocessor (STM32L476 from STMicroelectronics) which controls accelerometers and ADC, makes pre-filtering signals and writes data to NAND. Also, the WAS-Sensor has lithium batteries for power.

Experimental results
The WAS-Sensor was fixed on output shaft of their gearbox. The gearbox was diagnosed in a quasisteady operation. The simulator motor rotation frequency had 12 Hz. Thus, the output shaft was rotated with 8 Hz in accordance with gear-ratio k z = 1.5. The measured linear accelerations of the output shaft by the WAS-Sensor contain periodic high-frequency vibrations for the chipped tooth situation (Fig. 7). The frequency spectra of the accelerations measured by the sensor contain many components, including discrete peaks (Fig. 8).
Comparison of the power spectrum of each acceleration  , x and y for each gear defect (Fig. 9) clearly shows the change of the vibration signalsthe change in the energy ratio of the spectrum components. Moreover, the power spectrum grows around multiple of the tooth ripples (214 Hz, 428 Hz, 642 Hz, 856 Hz), for the case 'the chipped tooth pinion'. Furthermore, the levels of discrete components of linear acceleration x are significantly lower in the region of multiple frequencies for broken tooth case than for the chipped tooth case. On the other hand, the depth of modulation around the multiple of the tooth ripples substantially increased (Fig. 9). Thus, the paradoxical behaviour of the spectral characteristics for the different size of the tooth defect is clearly shown.
As previously mentioned, the presence of local defects affects a change of the stiffness gearmeshing function. A change of stiffness effects on a change of the force in the tooth contact pattern and a change of the angular acceleration of the gears. Moreover, a local defect affects a change of the magnitude of the frequency components which are related to the frequency meshing defected tooth. The growth of a local defect (from chipped to breaking) leads to an increase in the amplitude of the first harmonic of angular acceleration A  due to a reduction of the stiffness gear-meshing. Thus, 2  As a result, the grown of the local defect leads to a monotonic increase magnitude of the A  . Moreover, that trend continues at other rotation frequencies of the pinion (Fig. 10). The amplitude of the first rotation frequency component in angular acceleration is increased with the increasing rotation frequency of the pinion. Moreover, the amplitude of the first rotation frequency component for the faultless case has an almost linear dependence on the rotation frequency from 8 to 18 Hz (Fig. 11). , where pinion f is frequency of the pinion rotation (Fig. 12).   The ICP-accelerometer (by PCB Piezotronics) was mounted on the gearbox body for comparing the proposed approach with the traditional approach receiving diagnosis information (from a gearbox body). The ICP-accelerometer has a 3-axis with a sensitivity of 100 mV/g. The linear accelerations were measured at the same rotation frequencies and with the same defects such as the proposed approach. However, the results show the magnitude of the first harmonic of the pinion in linear accelerations does not depend on the rotation frequency and the defect (Fig. 13). Moreover, the value of the magnitude for a 'normal' case is more than the values of a 'chipped' and a 'broken' cases at some rotation frequencies.

Discussion
The proposed approach allows clearly detecting the pinion local defect on the first pinion rotation frequency. On the other hand, the local pinion defect information is hidden in data from the traditional approach (from a gearbox body). Hence, the measurement angular acceleration from a shaft allows receiving extended information about the behaviour of the mechanical system. The application of the proposed approach diagnosis allows detecting a local defect of the pinion, for example, 'chipped tooth' type easily. The approach dispenses without special signal processing methods unlike with the traditional approach. Nevertheless, the proposed approach requires an additional extended experimental investigation for various types of gearbox with various mechanical loads. It will be the topic of the future research.