GEAR TRANSMISSION ERROR MEASUREMENT ACCURACY USING LOW-COST DIGITAL ENCODERS

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1 11 International RASD Conference th 1- July 1 Pisa GEAR TRANSMISSION ERROR MEASUREMENT ACCURACY USING LOW-COST DIGITAL ENCODERS A. Palermo 1,*, L. Britte, K. Janssens, D. Mundo 1, W. Desmet 1 Dipartimento di Ingegneria Meccanica, Energetica e Gestionale Universita della Calabria Via P. Bucci, 6C, 876 Rende, Italy palermo.antonio@unical.it, d.mundo@unical.it Department of Mechanics, PMA Division Katholieke Universiteit Leuven Celestijnenlaan B, 1 Leuven, Belgium antonio.palermo@mech.kuleuven.be, wim.desmet@mech.kuleuven.be LMS International, A Siemens Business Interleuvenlaan, 68, 1 Leuven, Belgium laurent.britte@lmsintl.com, karl.janssens@lmsintl.com Keywords: gear, transmission error, encoder, measurement, accuracy. ABSTRACT Gear Transmission Error (TE) is widely recognized as the main internal source of vibration in power transmissions. It quantifies the deviations from a perfectly kinematic motion transmission which in a real case are introduced by deflections, misalignments and manufacturing errors. The TE can then be directly linked to durability, noise and diagnostics assessment. Given the relevance of this physical quantity and since gears usually attain very high stiffness, experimental techniques have been developed for measuring it with a tight constraint on measurement accuracy. Measurement procedures using encoders with a very high number of divisions (above 18), paired with dedicated acquisition systems is the common way of achieving the required accuracy. Few methods have been proposed which use low-cost digital encoders and multi-purpose acquisition systems. The accuracy of such techniques was discussed empirically for a few case studies. In the proposed paper, the authors determine numerically how errors in encoder spacing and in acquisition clock affect gear dynamic TE measurement for an experimental case study. It is concluded that it is unnecessary to seek for increased sampling time accuracy for the speed range of interest as encoder accuracy dominates the measuring chain. 1. INTRODUCTION 1.1 Gear Transmission Error Gear Transmission Error (TE) is defined as the deviation in the position of the driven gear (for any position of the driving gear), relative to the position that the driven gear would

2 occupy if both gears were geometrically perfect and undeformed [1]. The fundamental causes of TE can be traced to the process of meshing, whose smoothness is determined by aspects related to teeth: variable deflections (global tooth bending, local tooth contact), misalignments, manufacturing errors, microgeometry modifications. Therefore TE is typically expressed as a relative displacement between the gears, oriented in the direction of the total meshing force exchanged by the teeth (line of action). TE has a cyclic trend with tooth passing periods and its magnitude is in the order of microns (a few to a few tens), regardless of the teeth size and given the manufacturing quality class of the gears []. Although very small, this relative displacement is the root of substantial contact force fluctuation, because it acts on a mesh stiffness which is conversely very high (typically N/m). Furthermore, due to tooth contact nonlinearity, TE variability substantially depends on the transmitted load and is typically optimized to be minimal at a specific load. As such, TE is widely recognised as the main internal source of vibration in transmissions and therefore an important metric to be considered from the experimental point of view []. Measuring these very small relative displacements represents a challenging task, and the complexity of the measurement is further increased when stepping from quasi-static to dynamic conditions. Several measurement techniques have been developed for TE measurement in academia and industry: synchronised high-speed photography has been used by Harris [], strain gauges by Hayashi and Hayashi [5], laser vibrometers by Rosinski [6], encoders by Smith [], accelerometers by Kang and Kahraman [7]; measurements from accelerometers and encoders, currently the most established instruments for measuring TE, have been compared by Houser and Blankenship [8] and by Smith and Echeverria-Villagomez [9]. Successful measurements have been performed in a laboratory environment on tailored test rigs, typically using specialized measuring systems 1, while industrial practicality on real-case application environment is still far from being achieved. Therefore current research is focused on obtaining the required measurement accuracy by multi-purpose and minimally invasive measurement instrumentation [11]. The present paper focuses on TE measurement on a gear test rig using common encoders, which can be easily installed when access to shaft ends is possible. 1. Measurement techniques using high-resolution encoders An overview of the main types of test rigs and instrumentation used to measure TE by encoders (and also accelerometers) can be found in the work from Houser and Blankenship [8]. Pierz [1] describes a test machine with adjustable gear centre distance and torque loading, where the TE is measured using a commercial dedicated measuring system. Two incremental encoders having 18 divisions are used to generate square wave signals (digital TTL) which are acquired at 1 MHz sampling frequency, synchronised by a 1 MHz clock. Sasaoka [1] describes the measurement method adopted by another commercial dedicated measuring system, using two optical encoders with 6 or 8 divisions. The squarewave trains are conditioned with a phase difference processor, able to resolve 1/ of a division interval, before being sent to the acquisition system. Munro et al [1] use encoders having 6 divisions and an interpolation unit to increase their resolution. Interpolation is mentioned also by Kurokawa et al [15], who explain that interpolation conditioning is performed to generate square-wave signals from analogue encoders providing a sinusoidal output. The encoders have 5 divisions and are based on optical interference from laser diffraction on the gratings. In all the above-mentioned measurement methods, when aiming at dynamic TE measurement, the high numbers of divisions impose severe rotational speed limitations for signal acquisition. It is exemplary that an encoder having 18 divisions connected to a shaft rotating at just 6 rpm generates a square-wave train with a frequency of 18 KHz. 1 Terms in italic are used according to the definition provided by the International Vocabulary of Metrology (VIM) [1].

3 1. Measurement techniques using low-cost encoders In the direction of overcoming speed limitations, reduce the cost of the required encoders and avoiding dedicated acquisition systems, several measurement techniques have been devised using common encoders having at most 5 divisions. Digital TTL encoders having 18, 8 and 96 lines were used by Remond [16] and processed using the pulse timing method. The author identifies the timer measurement precision as the main contribution to measurement error on TE and concludes that calibration for the encoder gratings produces insignificant improvement on measurement accuracy. Encoder measurement accuracy is also addressed by Du and Randall [18] using two encoders with 6 divisions and one highresolution encoder with related calibration diagram. The authors measured single and combined encoder spacing errors and conclude that both of them are non-negligible only at low shaft orders, in line with a similar statement by Smith [19], but for a different type of encoders. The present paper aims towards a better understanding of two main points left open by the conclusions referenced above. A measurement model is used to verify the measuring chain selected for a gear test rig which will later be used for experimental TE measurement and gear contact modelling validation. The first point arises because for the latter validation it is important to compare TE in angle domain instead of frequency domain. Therefore, given the specifications from the manufacturers, is the combination of selected encoders and acquisition system suitable for performing accurate TE measurement? Second, how do errors affect TE measurement in the speed range to be tested? Third, since high precision gears with low TE variability will be tested, can measurement accuracy be improved by calibrating the encoders?. TRANSMISSION ERROR MEASURING CHAIN AND GEAR TEST RIG The application case for TE measurement is a high-precision power-circulation gear test rig [][1] which is instrumented to provide redundant measurement of static and dynamic TE, in loaded and unloaded conditions. A view of the test gears side is shown in Figure 1. Encoders Figure 1. Test side of the gear test rig, showing only low-cost encoders on free shaft ends. Shaft ends can be instrumented at the same time with two high-resolution and two low-cost incremental encoders (one order of magnitude difference in cost); both are manufactured by Heidenhain and are respectively of types RON 85C [] and ERN 1 []. Encoder specifications for the two types are reported in Table 1. Each high-resolution encoder is The pulse timing method is outlined in Section of the present paper; see also [17].

4 provided with a calibration diagram and will be used both for measuring TE at low rotational speed and for calibrating the low-cost encoders. Encoder Model Number of Divisions Signal Type One-revolution Accuracy Spacing Accuracy Recommended Measuring Step RON 85C 18 Analog Sine ±5 arcsec ±1%.6 arcsec ERN 1 5 Digital TTL ± arcsec ±5% - Table 1: Encoder Specifications. Signals are acquired using an LMS SCADAS acquisition system [] equipped with voltage input channels for the analogue encoders ( bit ADC sampling at.8 KHz) and an RV module for rotational vibration providing two channels for digital encoders (input pulse rate up to.8 KHz synchronised with an 8 MHz clock). The 8 MHz clock yields a time resolution of 1. ns. The measuring chain is therefore very simple and independent for each encoder: angular positions are acquired separately connecting each encoder to the SCADAS and the TE is calculated off-line by postprocessing the acquisition. TE measurement is expected to be alias-free since both gears have 57 teeth and in the worst case (encoders with the least line count) more than gear mesh orders fit in half of 5 divisions. This statement is true provided that the Nyquist limitations on sampling frequency are respected also for analogue signal acquisition and that the maximum input pulse rate is not saturated for digital signals. For high-resolution encoders, taking 7 points to reconstruct a sinusoid, the maximum allowed shaft rotational speed is 1 rpm. Quasi-static static TE measurement will therefore be performed using an angular measuring step of 1 arcsec, circa three times higher than the minimum measuring step recommended by the manufacturer. This angular resolution yields on the gear base radii of mm a displacement resolution along the line of action equal to ±. μm for a single encoder. The combined resolution of ±.68 μm is expected to be sufficient to resolve the TE oscillations, expected in the order of 5 μm peak to peak. For low-cost encoders, the maximum input pulse rate is reached for a shaft rotational speed of 5 rpm. By skipping one pulse, using the related functionality available for the SCADAS RV module, the shaft rotational speed can be extended to reach the maximum speed of the test rig, equal to 5 rpm. The speed range covered is sufficiently wide to observe dynamic phenomena happening during meshing of the test gears. Measurement overlap in quasi-static conditions will be used for cross-validation of the TE measurements using high-resolution and low-cost encoders. From a design point of view, a set of precautions has been taken to avoid influence of assembly errors on encoder accuracy and on gear meshing. Test gears have been first case hardened and then precision ground to quality ISO. Manufacturing tolerances after execution are in the order of ±1 μm for any of the characteristic tooth errors defined in the norm ISO 18 [5]. These tight tolerances are aimed at achieving a very repeatable tooth contact pattern and a very repeatable gear excitation on a tooth-passing period [6]. Another important precaution, as highlighted by Sweeney and Randall [17], can be found in the selection of a 1:1 gear ratio. This ratio ensures that each tooth on one gear will always mesh with the same tooth on the mating gear, yielding a very repeatable gear excitation on a revolution period. Gear shafts have been manufactured from through-hardened and preground cylindrical steel bars; shaft radial oscillation tolerance with respect to bearing seats has been measured in μm for both shafts. High precision bearings (quality P5) have been selected to restrict radial oscillation for the inner race to 5 μm maximum (% w.r.t. standard). Self-centring expansion elements have been used to fasten the gears to the shafts. Combined radial oscillation measured in the assembled test rig using a dial indicator did not.8 KHz / (7 samples/div * 18 div) =.165 Hz shaft rotational frequency..8 KHz / 5 div/revolution =.96 Hz shaft rotational frequency.

5 overcome 15- μm, both on gear collars and on encoder cylindrical surfaces. These tolerances are well within the prescribed values by the encoder manufacturer. Assembly errors and manufacturing tolerances are not expected to degrade measurement accuracy of the encoders. Also the use of hollow-shaft encoders substantially yields improved performances in this direction. Solid-shaft encoders were excluded because they require a coupling to be connected to the rotating shaft. This coupling introduces a kinematic error in motion transfer and a limitation in speed due to a lowered torsional natural frequency. Also grating drums with separate scanning heads were excluded, since their alignment is difficult.. MEASUREMENT MODEL AND MAIN SOURCES OF ERROR As stated in the previous section, TE is calculated off-line after having performed the angle acquisition. A measurement model has been used to evaluate the effects of the main sources of measurement errors on the low-resolution encoders. Two main sections can be identified in the measurement model and will be discussed hereafter: the first section reproduces the measurement chain as signals are generated by the encoders and acquired by the measurement system; the second section performs the required post processing to calculate the TE. The measurement chain adopted for measuring encoder angles is illustrated in Figure. The pulse timing method is used to calculate angles as it allows exploiting the accuracy of the acquisition system clock and avoiding increasing the number of encoder divisions. Considering an encoder with 5 divisions, the encoder angular resolution is equal to 6 /5 divisions = 59. arcsec, meaning that angle readings will be available every 59. arcsec travelled in one direction by the shaft. However angular resolution does not affect the measurement error, since at each reading the total error on the new angular increment,, will be (Figure ): ( ) ( ) (1) Here ( ) represents the spacing error for the encoder gratings, represents the error due to the time required for the square wave rising edge to reach the acquisition trigger level, represents the cable travelling time for the signal to reach the acquisition system and represents the error due to the timer resolution. represents the instantaneous angular velocity of the measured shaft, which converts the error on time in angle units. Electrical and temperature disturbances are assumed to be negligible since tests are performed in laboratory environment. Figure. Measuring chain for one encoder. Values for the above-mentioned sources of error can be found on the specifications provided by the encoder manufacturer [][] and the acquisition system manufacturer [].

6 These values are used to assess measurement accuracy. ( ) represents a source of systematic measurement error as it derives from manufacturing errors which make the gratings unequally spaced. The systematic nature of ( ) was confirmed in experiments by Smith [7]. This error can be therefore compensated by encoder calibration. Moreover, although the number of divisions does not affect directly accuracy, for the type of encoders used, the spacing error on the gratings is within 1/ th of the grating period and therefore inversely proportional to the number of divisions; therefore the accuracy on the angle measurement improves with the number of divisions. All the other errors on time are delays in acquiring samples and become more important with linear trend as angular velocity increases. is typically constant and depends on the performance of TTL signal electronics. is similarly constant and depends on the cable length to be travelled by the signal. arises from the time quantisation performed by the acquisition system clock and generates random error. It is not possible, in fact, to predict the time period before a rising edge is detected by the following clock tick. While and affect angle measurement trueness, affects angle measurement precision. Error values for highresolution and low-cost encoders can be compared in Table for high and low speed values. Encoder type Divisions Shaft speed [rpm] [arcsec] [arcsec] [arcsec/1m cable] [arcsec] RON 85C 18 5 ± to +.6 ERN ± to to +.1 Table : Error values for high-resolution and low-cost encoders at high and low speed values. Since pairs of encoders of the same models and having the same cables length are used, the related and are very similar, and since TE is calculated by the angles difference these two errors cancel out. Effects due to ( ) and will be analysed in the next section assuming they arise from uniform error distributions. The second section of the model performs the signals post-processing required to obtain the TE. An example of the two square-wave trains generated by the encoders can be found in Figure for a sinusoidal TE trend having exaggerated magnitude for the sake of clear visualisation. The square wave train appears to be expanded when the torsional vibration is out of phase with the direction of rotation (lower speed), while it appears to be tighter when the torsional vibration in phase with the direction of rotation (higher speed). Encoder 1 - TTL Signal [V] Encoder - TTL Signal [V] Tooth number Figure. Square-wave trains generated by low-resolution encoders showing torsional vibration at tooth-passing frequency.

7 TE [mm] TE [mm] Each rising edge triggers the angle reading to be incremented of one encoder division. Detecting only the rising edge of the square wave, as opposed to both rising and falling, is sufficient since the number of divisions of the encoder is times higher than the value to verify the Nyquist theorem in angular domain. Rising edges however do not happen at the same time instants for the two encoders, therefore a direct comparison of angular position between the two encoders is not possible. Under the assumption of constant angular velocity between subsequent angle readings [16], the TE can be calculated by merging the both the sets of angle readings generated by the encoders (Figure ). A subtraction between the two angles can be directly made since gears have 1:1 gear ratio and the result can be multiplied by the gears base radius to obtain the TE as a relative displacement along the line of action.. EVALUATIONS To perform evaluations on measurement accuracy, a sinusoidal TE trend having tooth-passing frequency (1 st meshing order) has been assumed with a peak to peak value of 5 μm, in line with the corresponding value obtained by simulations on gears to be tested. A reliable measurement is expected to have a measurement error at least below ±.5 μm, one order of magnitude of the oscillation to be measured. TE values are shown in the next figures for an acquisition time covering five mesh cycles. Measurement errors are discussed according to the values reported in Table. First of all, the TE measurement has been verified for the high-resolution encoder. Figure shows very good accuracy in capturing the TE sinusoid. The measurement error on each encoder is equal to ±.7 arcsec, which multiplied by the gear base circle radius yields ±. μm error on the TE. The combined error is therefore equal to ±.8 μm. 5 x Tooth number Time [s] Figure. TE measurement using high-resolution encoders spinning at 1 rpm. Low-cost encoders before calibration have then been verified only at low speed, since Figure 5 already shows poor accuracy in capturing the TE sinusoid. The spacing error dominates the overall measurement accuracy, since the combined error of ±5.9 arcsec yields ±8.8 μm on the TE, while the error due to time quantisation is at most +.6 arcsec yielding a combined value of at most +.17 μm Tooth number Time [s] Figure 5. TE measurement using low-resolution encoders spinning at 1 rpm. Calibration using the high-resolution encoders brings the accuracy on grating spacing for the low-cost encoders to ±1/6 of a division. The spacing error after calibration lies in the same

8 TE [mm] TE [mm] TE [mm] range as the one for the reference instrument (±. μm). The combined error on the TE becomes therefore ±.8 μm. Marked improvement can be observed in Figure 6. 5 x Tooth number Time [s] Figure 6. TE measurement using calibrated low-resolution encoders spinning at 1 rpm. When repeating the measurement at 5 rpm, no degradation is observed comparing Figure 7 to Figure 6. In fact, the error due to time quantisation increases but it still limited to +. μm, which is one order of magnitude below the spacing error. Similarly, no degradation is observed when skipping one pulse and performing the TE measurement at 5 rpm (Figure 8). The error due to time quantisation in this case becomes at most +.81 μm. 5 x Tooth number 5. e- 8.6 e- Time [s] 1. e- 1.7 e-.1 e- Figure 7. TE measurement using calibrated low-resolution encoders spinning at 5 rpm. - x Tooth number 5. e-.7 e- Time [s] 7. e- 9. e- 1. e- Figure 8. TE measurement using calibrated low-resolution encoders spinning at 5 rpm. 5. CONCLUSIONS A measurement model has been used to evaluate the impact of the main sources of measurement error for the TE measurement on a gear pair test rig. The selected combination of encoders and acquisition system appears suitable to perform TE measurement both in quasi-static condition and at high angular speed. Based on specifications provided by the manufacturers, encoder grid spacing error appears to be the main contributor to measurement accuracy. The time resolution for the acquisition system appears to be abundantly sufficient as it impacts the measurement error one order of magnitude less than the spacing error. Measurement accuracy can be improved for low-cost encoders by performing a calibration against the high-resolution encoders. However, still the residual spacing error appears to be the limiting factor for measurement accuracy.

9 6. ACKNOWLEDGEMENTS The authors gratefully acknowledge the IWT (Agency for Innovation by Science and Technology in Flanders) for the financial support through the project HEV-NVH (Agr. 116). REFERENCES [1] R. W. Gregory, S. L. Harris, R. G. Munro, Dynamic Behaviour of Spur Gears, Proceedings of the IMechE, Vol. 178, pp. 7 6, 196. [] J.D. Smith, Gear noise and vibration, Marcel Dekker, Cambridge,. [] R. G. Munro, A review of the theory and measurement of gear transmission error, IMechE International Conference of Gearbox Noise and Vibration, pp. -1, 199. [] S. L. Harris, Dynamic loads on the teeth of spur gears, Proceedings of the IMechE, vol. 17, pp , [5] I. Hayashi, T. Hayashi, New measuring method of a single flank transmission error of a pair of gears, Proceedings of the Fifth World ASME Congress on Theory of Machines and Mechanisms, [6] J. Rosinski, D. A. Hofmann, J. A. Pennell, Dynamic transmission error measurements in the time domain in high speed gears, Proceedings of International Gearing Conference, University of Newcastle upon Tyne, 199, pp [7] M. R. Kang, A. Kahraman, Measurement of vibratory motions of gears supported by compliant shafts, Mechanical Systems and Signal Processing, Vol. 9, pp. 91, 1. [8] D. R. Houser, G. W. Blankenship, Methods for measuring gear transmission error under load and at operating speeds, SAE Technical Paper , [9] J. D. Smith, J. S. Echeverria-Villagomez, Comparing encoder and accelerometer measurement of transmission error or torsional vibration, Proceedings of the First IMechE International Conference on Gearbox Noise and Vibration, Paper C/7, pp. -9, 199. [1] JCGM :1, International Vocabulary of Metrology - Basic and general concepts and associated terms, rd edition, [11] R. White, V. Palan, Measurement of Transmission Error Using Rotational Laser Vibrometers, Proceedings of the ASME IDETC/CIE 7, paper DETC7-. [1] M. P. Pierz, Development of a Precision Variable Load Transmission Error Test Device, SAE Technical Paper , 1. [1] S. Sasaoka, Measurement technique for loaded gear transmission error, SAE Technical paper 9797, [1] R. G. Munro, D. Palmer, L. Morrish, An experimental method to measure gear tooth stiffness throughout and beyond the path of contact, Proceedings of the IMechE, Part C, Vol. 15, pp. 79-8, 1. [15] S. Kurokawa, Y. Ariura, Y. Matsukawa, T. Doi, Evaluation of gear engagement accuracy by Transmission Error with sub-microradian resolution, International Journal of Surface Science and Engineering, Vol., pp , 9. [16] D. Remond, Practical performances of high-speed measurement of gear transmission error or torsional vibrations with optical encoders, Measurement Science and Technology, Vol. 9, pp. 7-5, [17] P. J. Sweeney, R. B. Randall, Gear Transmission Error Measurement Using Phase Demodulation, Proceedings of the IMechE, Part C, Vol. 1, pp. 1-1, 1996.

10 [18] S. Du, R. B. Randall, Encoder error analysis in gear transmission error measurement, Proceedings of the IMechE, Part C, Vol. 1, pp , [19] J. D. Smith, Gear Transmission Error Accuracy with Small Rotary Encoders, Proceedings of the IMechE, Part C, Vol. 1, pp. 1-15, [] A. Palermo, J. Anthonis, D. Mundo, W. Desmet, A novel gear test rig with adjustable shaft compliance and misalignments, Part I: Design, Proceedings of the CMMNO Conference 1, University of Ferrara. [1] A. Palermo, J. Anthonis, D. Mundo, W. Desmet, A novel gear test rig with adjustable shaft compliance and misalignments, Part II: Instrumentation, Proceedings of the CMMNO Conference 1, University of Ferrara. [] Dr. Johannes Heidenhain GmbH, Catalogue for Angle Encoders with integral bearings, available online, November 1. [] Dr. Johannes Heidenhain GmbH, Catalogue for Rotary Encoders with integral bearings, available online, November 11. [] LMS International NV, SCADAS Data Acquisition Systems, Product Brochure, available online, January 1. [5] ISO 18-1:1995, Cylindrical gears, ISO system of accuracy, Part 1: Definitions and allowable values of deviations relevant to corresponding flanks of gear teeth. [6] G. W. Blankenship, A. Kahraman, Steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type non-linearity, Journal of Sound and Vibration, Vol. 185, pp , [7] J. D. Smith, Gear transmission error accuracy with small rotary encoders, Proceedings of the IMechE, Part C, Vol. 1, pp. 1-15, 1987.