Congress on Technical Diagnostics 1996

Transcription

1 Congress on Technical Diagnostics 1996 G. Dalpiaz, A. Rivola and R. Rubini University of Bologna, DIEM, Viale Risorgimento, 2. I-4136 Bologna - Italy DYNAMIC MODELLING OF GEAR SYSTEMS FOR CONDITION MONITORING AND DIAGNOSTICS Summary. A non-linear lumped parameter model of a gear system is proposed in order to study the modifications in torsional vibration due to faults. The model takes into account the mass distribution, torsional shaft stiffness and variable tooth-meshing stiffness and is validated by comparison with experimental results. Faults localized in one or a few teeth produce dynamic effects that are typically transient and timelocalized. Then, the wavelet transform - that produces a time-frequency representations suited for resolving very short-lived high frequency phenomena in the time domain - is applied to both numerical and experimental results for the detection of damaged teeth. The effectiveness of model both in the case of sound and damaged gears is discussed. 1. INTRODUCTION An effective and reliable monitoring system makes it possible to avoid unexpected failures and to perform predictive maintenance of machinery. Vibration analysis is a widely used condition monitoring method for gears as well as for other mechanical components. In gears, vibrations are mainly due to the parametric excitations produced by time-varying mesh stiffness, by tooth spacing and profile errors, and by manufacturing or mounting misalignments; all of these causes may produce variability of the transmitted force and, consequently, variable bearing forces and vibrations of the gear unit casing. Such vibrations can give information about gear conditions; for example in the case of a fatigue crack occurring in one of the gear teeth, the lower stiffness of the cracked tooth would cause a changing in the casing vibration pattern in correspondence to the meshing of that tooth. The most common vibration analysis techniques, like spectral analysis, are generally unable to detect failures developing in gears at an early stage. As a matter of fact, classical methodologies are based on the assumption of stationarity and globally characterize the signals; consequently, they are not fully suitable for detecting faults at an early stage of development, as faults are generally localized in one or a few teeth; thus low sensitivity can be the result and time-localization and identification of faults are impossible. Other techniques proposed in literature in the field of gear vibration analysis are cepstrum analysis, time synchronous averaging, amplitude and phase demodulation of narrow band signals on forced or natural frequencies of mechanical systems, [1-6], time-frequency representations [7-8]. In particular, in 1986, McFadden [3] proposed a technique based on the evaluation of the amplitude and phase modulation of one of the tooth meshing harmonics of the vibration signal. In 199 Dalpiaz [6] developed the above technique in order to increase its sensitivity and compare it

2 G. Dalpiaz, A. Rivola and R. Rubini to another technique based on the analysis of the demodulated acoustic emission signal. It is noteworthy that the sensitivity of the phase modulation method to faults can be very different when different frequency bands are processed. In 1994 Staszewski and Tomlinson [7] and McFadden [8] applied the wavelet transform (WT) to the early detection of gear failure. The WT algorithm is one of the more recent mathematical tools adopted for processing transient signals, and it is well known for its ability to treat non-stationary signals [9, 1]. Moreover, the normal procedure for monitoring and diagnostic purposes is to pick up the vibration of the gear casing. In fact, it is difficult to directly achieve the torsional vibration signal in operating condition; on the other hand, information about faults may be hidden in the casing vibration, due to the effect of the transfer function between gears and casing [11]. However, the relationship between gear torsional vibrations and casing vibrations can be expressed by an appropriate experimental filter function. Thus, an inverse-filtering procedure can be applied so as to recover the gear torsional vibration from the vibration signal picked up on the casing [12, 13]. In addition, mathematical models of gear systems can be used in order to study the effects of faults on gear torsional vibrations to the end of developing more suitable techniques for monitoring and diagnostics. Many models have been proposed and used for gear vibration analysis [14]. In 199 Dalpiaz and Meneghetti [12] presented a mathematical model, as a tool useful to predict the effect of faults on gear vibration. In fact, using the model it might be possible to foresee the effects of faults of different types and at different development stages on the vibration of the specific gear system under consideration. This information could be used for the selection of the most suitable signal processing techniques and for diagnostic purposes. The specific gear system considered in [12] and in the present paper is a gear test ring both in sound conditions and with a fatigue crack in one tooth. The non-linear lumped-parameter model takes into account the mass distribution, the torsional shaft stiffnesses and the time-varying mesh stiffness. The equations of motion are numerically integrated. The model is validated by comparison with experimental results. In this paper the value of some model parameters is modified in order to obtain more agreement between numerical and experimental results in sound conditions and the effectiveness of the model in the case of damaged gears is discussed. In particular, the effect of a crack on gear vibrations is evaluated using the WT both for the numerical and experimental signal and the results are compared. 2. MATHEMATICAL MODEL The system under consideration is a power circulating gear testing machine composed of two identical single-stage gear units mounted back to back, with a locked-in torque [12]. In order to study its torsional vibration, a model with lumped parameters has been used. The model is the same as that reported in Dalpiaz and Meneghetti [12], see Fig. 1. The torsional stiffnesses are reduced to base circles and are represented by linear springs. Analogously, the rotational inertias are represented by translating masses. Time-varying mesh stiffness and mesh damping are included in the model. A spring in parallel with a damper having respectively constant stiffness and constant damping is put between every two masses to model shafts and couplings. Damping coefficients are taken as being proportional to the corresponding spring stiffness. Bending shaft compliance, bearing dynamic and the effect of geometric errors are not included. The model can take into account tooth separation, but this actually does not take place in the considered operating conditions, as confirmed by experimental results.

3 Dynamic Modelling of Gear Systems for Condition Monitoring and Diagnostics m 7 COUPLING PULLEY PINION SHAFT k1 k2 k3 k4 k5 m m m m d d d d d k7 d 7 GEAR BOX "L" m 6 k6 d6 WHEEL SHAFT Figure 1. Lumped parameter model. m 5 GEAR BOX "R" The variation of mesh stiffness is the main source of gear vibration. Many expressions have been proposed for calculating stiffness [15]. A simple stiffness time variation was chosen; it consists of two half-sinusoids of different amplitude and duration, superimposed on a constant term (for more details see [12]). When a cracked tooth meshes, the mesh stiffness suffers an appropriate reduction. This changing is the primary cause of gear vibration modification. The equations of motion are numerically integrated. The numerical results can be obtained under different conditions by varying the angular speed and the locked-in torque. 3. TESTS AND MODEL VALIDATION Experimental tests have been carried out using the above-mentioned power circulating gear testing machine. A spur gear pair of module 3 mm is contained in each gear unit. The pinion and the wheel have respectively 28 and 55 teeth. Further data about the gears are as in [6]. The results presented in this paper are relative to a wheel torque of 1139 Nm and wheel speed of 764 rpm. Tests were performed both in sound conditions and after introduction of a crack in one of the teeth of the wheel mounted in gear box L (Fig. 1). The tooth was precracked before mounting the wheel on the testing machine, using an appropriate device in order to apply a fatigue load to the tooth in a similar way to when the tooth is loaded during meshing. The crack affected the whole tooth flank, extending between the two wheel faces. The crack length was about 1.2 mm, corresponding to about 2% of the tooth thickness at the radius affected by the crack, i.e. near the root. Gear torsional vibration was measured using a pair of accelerometers fastened on a gear in the tangential direction and in diametrically opposite positions. The signal was collected by means of a slip ring set. In addition, a one-per-wheel revolution tachometer signal was taken using an inductive proximity probe. These signals were recorded on magnetic tape and were then acquired through a DIFA Scadas data acquisition unit and analysed using the LMS CADA-X software working on a HP 9/715 workstation.

4 G. Dalpiaz, A. Rivola and R. Rubini 6 (a) 15 (b) Amplitude -6 6 (c) Amplitude 15 (d) -6.1 Time / Rev. Period Frequency [Hz] 5 Figure 2. Gear torsional acceleration reduced to base circles in the case of undamaged teeth: time synchronous average(left); frequency spectrum (right). (a), (b) experimental results; (c), (d) numerical results. Since the model takes into account only excitations of frequency corresponding to the wheel rotation frequency and its harmonics, the model validation is performed by comparing numerical results with experimental results, after a processing of time synchronous averaging over 21 wheel revolutions, performed in order to reduce the effects of noise and vibration sources other than gear pairs. A particular average methodology was used, so as to compensate slight - but quite important - variations of the machine angular speed occurring during wheel revolution. The antialiasing cut-off frequency was 5 Hz. The time is described with 248 samples per wheel revolution. Results are presented with reference to the nominal wheel angular speed of 764 rpm, that is to say, the wheel revolution frequency (order 1) and the meshing frequency (order 55) correspond respectively to Hz and 7 Hz. The results shown in Fig. 2 are relative to the tangential acceleration of the wheel of gearbox L (corresponding to mass m 7 of the model, see Fig. 1), reduced to the base circles. In order to better match the experimental results [Figs. 2(a), (b)], the values of the stiffnesses have been modified with respect to [12], while the mass values are unchanged. Data are reported in Table 1 (a mean value is assumed for variable mesh stiffness and natural frequencies). The coefficient of proportionality between damping coefficients and stiffnesses was assumed to be s. The comparison between numerical and experimental results can be considered very satisfactory both in time and in frequency domain (Fig. 2).

5 Dynamic Modelling of Gear Systems for Condition Monitoring and Diagnostics Table 1. Masses, stiffnesses and natural frequencies of the numerical model. m 1 = 5.19 kg k 1 = N/m f 1 =. Hz m 2 = 3.39 kg k 2 = N/m f 2 = 382. Hz m 3 = kg k 3 = N/m f 3 = Hz m 4 = 4.15 kg k 4 = N/m f 4 = 69.7 Hz m 5 = 2.91 kg k 5 = N/m f 5 = Hz m 6 = 2.1 kg k 6 = N/m f 6 = Hz m 7 = 2.81 kg k 7 = N/m f 7 = 294. Hz 4. APPLICATION OF THE WAVELET TRANSFORM 4.1. Theoretical background Because wavelets are a local function of the time, each with a predetermined frequency content, wavelet analysis provides a good means studying how the signal frequency content changes with time. In particular the WT is a linear transformation decomposing an arbitrary time signal x(t) into elementary functions ha, b( t ) = 1 a h t b a given by the translation and dilation of an analysing wavelet h(t): the parameter b is the translation and describes the time localization of the wavelet; the parameter a (a > ) is the dilation and determines the width or scale of the wavelet. Thus the WT is defined as 1 W( a, b ) = x( t ) h a * t b dt a where h * (t) is the complex conjugate of the analysing wavelet h(t) [9]. One of the most interesting complex functions used as an analysing wavelet is the Gaussian wavelet, defined as (1) (2) j π f t k t h( t ) = e 2 e 2 2 (3) where f and k are constants. The WT has relevant advantages in resolution with respect to other time-frequency methods, such as the Short Time Fourier Transform, whose time window remains constant. In fact, for the WT, the width of the window in the time domain is proportional to a, while the bandwidth in the frequency domain is proportional to 1/a [7-1]. Consequently, both time and frequency resolutions depend on parameter a; the latter property makes the WT suitable for the detection of transient signals. In this paper the distribution will be displayed in the time-frequency domain, using the relationship f=f /a between the frequency and scale. The WT was performed by the algorithm included in Time Data Processing Monitor module of the LMS CADA-X software, using a Gaussian analysing wavelet. Although the WT is a complex-valued function, only the amplitude will be considered and represented in linear scale in the following.

6 G. Dalpiaz, A. Rivola and R. Rubini 6 (a) 6 (b) (c) (d) 17 Frequency [Hz] Amplitude Amplitude 6 5 (e) 7 Hz 3 (f) 7 Hz 1 (g) 832 Hz (h) 832 Hz (i) 14 Hz (j) 14 Hz 1 Time / Rev. Period Time / Rev. Period Figure 3. Gear torsional vibration in the case of one cracked tooth: experimental results (left); numerical results (right). (a), (b) time domain; (c), (d) wavelet transform; wavelet cross-section corresponding to (e), (f) 7 Hz; (g), (h) 832 Hz; (i), (j) 14 Hz.

7 Dynamic Modelling of Gear Systems for Condition Monitoring and Diagnostics 4.2. Cracked tooth: experimental gear vibration Fig. 3(a) shows the gear tangential acceleration for cracked tooth, after time synchronous averaging. In the time domain it is not easy to detect the presence of the crack. By means of the wavelet analysis it is possible to find out the time-localization of the crack, because it causes a transient phenomena that the WT is able to track. In this case, the crack effect is detected only in the frequency range lower than the second meshing harmonics; thus the WT representation of Fig. 3(c) is limited to the frequency range covering the first to the third meshing harmonics. Crosssections of the WT - related to three different constant values of the frequency - were considered, so as to give a clearer representation. Figs. 3(e), (g), (i) illustrate the amplitude of the crosssections respectively obtained for the frequency value of 7 Hz, 832 Hz and 14 Hz. By visual inspection of this feature, the time-localization of the transient phenomena is easier for the crosssection performed at 832 Hz Cracked tooth: numerical gear vibration In order to study the modifications in torsional vibration due to faults, the presence of the cracked tooth was introduced in the numerical model by appropriately reducing the meshing stiffness. The result of this analysis in the time domain is reported in Fig. 3(b). The numerical response of the model was also analysed using the WT. The result of the wavelet analysis is illustrated in Fig. 3(d). Finally, Figs. 3(f), (h), (j) show the amplitude of the cross-sections of the WT, obtained at the same frequencies of the experimental cross-sections. In the WT of the model results, the presence of the cracked tooth appears less evident than in the experimental case. Moreover, the cross-sections show different behaviour with respect to the experimental ones. However, the cross-sections of the WT performed on the model are able to detect and well localize the presence of the crack. The cause of the disagreements might be the incorrect modelling of the crack. 5. CONCLUSION A non-linear lumped parameter model of a gear system is proposed in order to study the modifications in torsional vibration due to faults. The model was satisfactorily validated by comparison with experimental results in the case of undamaged tooth. As regards the modelling of the effects of a cracked tooth, the study presented is of an exploratory character. The WT has been applied both to numerical and experimental results, as a useful tool for the detection of a cracked tooth. When a fatigue crack is present at the root of one of the teeth, the agreement between the numerical and experimental results is not very good: the crack presence is detected also in numerical results, but its effects are somewhat different. This is possibly due to an unsatisfactory modelling of the meshing stiffness during the engagement of the cracked tooth. Thus, further investigations are required to improve the crack modelling. ACKNOWLEDGMENT This work was partially supported by a grant from the Italian Ministry of University and Research (MURSeT).

8 REFERENCES G. Dalpiaz, A. Rivola and R. Rubini [1] Randall R.B.: Gearbox Fault Diagnosis Using Cepstrum Analysis. Proceedings of Fourth World Congress on T. of M. and M., Newcastle u. Tyne, Vol. 1, 1975, pp [2] McFadden, P.D., and Smith, J.D.: A signal processing technique for detecting local defects in a gear from the signal average of the vibration. Proceedings of the Institution of Mechanical Engineers, Part C, Vol. 199, No. 4, 1985, pp [3] McFadden, P.D.: Detecting fatigue cracks in gears by amplitude and phase demodulation of the meshing vibration. ASME J. of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 18, No. 2, 1986, pp [4] McFadden, P.D.: Determining the location of a fatigue crack in a gear from the phase of the change in the meshing vibration. Mechanical Systems and Signal Processing, Vol. 2, No. 4, 1988, pp [5] Sokolova, A., "Gear vibration diagnostics - State of art", Proceedings 9th School on Diagnostics, Rydzyna, Poland, 1989, pp [6] Dalpiaz G.: Early Detection of Fatigue Cracks in Gears by Vibration Analysis Techniques, Österreichische Ingenieur- und Architekten- Zeitschrift (ÖIAZ), Vol. 135, No. 7/8, 199, pp [7] Staszewski W. J., Tomlinson G. R.: Application of the Wavelet Transform to Fault Detection in a Spur Gear. Mechanical System and Signal Processing, Vol. 8, No. 3, 1994, pp [8] McFadden P.D.: Application of the Wavelet Transform to Early Detection of Gear Failure by Vibration Analysis. Proceedings of an International Conference on Condition Monitoring, Swansea, UK, 1994, pp [9] Rioul O., Vetterli M.: Wavelets and Signal Processing, IEEE Signal Processing Magazine, Vol. 8, No. 4, 1991, pp [1] Dalpiaz G., Rivola A.: Fault Detection and Diagnostics in Cam Mechanisms. Proceedings of the 2nd International Symposium on Acoustical and Vibratory Surveillance Methods and Diagnostic Techniques, Vol. 1, Senlis France, Courbevoie: Société Française des Mécaniciens, 1995, pp [11] McFadden P.D., Smith J.D.: Effect of Transmission Path on Measured Gear Vibration. Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 18, 1986, pp [12] Dalpiaz G., Meneghetti U.: Detection and Modelling of Fatigue Cracks in Gears. Condition Monitoring (Proceedings of 3rd Int. Conf. on Condition Monitoring, Windsor, UK, 199). Ed. McEwan J.R., London & N.Y.: Elsevier Applied Science, 1991, pp [13] Dalpiaz G., Meneghetti U.: Monitoring Fatigue Cracks in Gears. NDT&E International, Vol. 24, No. 6, 1991, pp [14] Nevzat Ozguven H. and Houser D.R.: Mathematical Models Used in Gear Dynamics - A Review. J. of Sound and Vibration, Vol. 121, No. 3, 1988, pp [15] Umezawa K., Sato T., Ishikawa J.: Simulation on rotational vibration of spur gears. Bullettin of the Japan Society of Mechanichal Engineers, Vol. 27, No. 223, 1984, pp