The detection of gear noise computed by integrating the Fourier and Wavelet methods

Transcription

1 The detetion of gear noise omputed by integrating the Fourier and Wavelet methods NIOLA VINCENZO, QUAREMBA GIUSEPPE, FORCELLI ANIELLO Department of Mehanial Engineering for Energetis University of Naples Federio II Via Claudio, 85, Napoli ITALY DVH Transmission Department ELASIS S.C.p.A. Via ex Aeroporto, s.n., 88, Pomigliano d Aro (NA) ITALY Abstrat: - This paper presents a new gearbox noise detetion algorithm based on analyzing speifi points of vibration signals using the Wavelet Transform. The proposed algorithm is ompared with a previouslydeveloped algorithm assoiated with the Fourier deomposition using Hanning windowing. Simulation arried on real data demonstrate that the WT algorithm ahieves a omparable auray while having a lower omputational ost. This maes the WT algorithm an appropriate andidate for fast proessing of noise gear box. Key-Words: - Signal proessing, gear noise, Wavelet Transform, multiresolution analysis. Introdution Gears are one of the most ommon and important mahine omponents in many advaned mahines. The most ommon method for gear noise assessment is subetive evaluations. Evaluators either drive the vehile or listen to the tape on whih the noise data was reorded to ran the noise. The subetive evaluations have some drawbas. First of all, the subetive ratings are not onsistent and may hange from time to time. Seondly, the resolution of the ratings is limited by the auditor's ability to distinguish small differene in the gear noises. Additionally, the ost for onduting a subetive evaluation is muh higher than for an obetive one. Subetive evaluation often requires more people to be involved. Noise and vibration properties of ars have beome important riteria in the ompetition between automotive manufaturers and together with new legislation are an important reason to redue noise and vibrations. Moreover, noise and vibrations are seen as indiators of the quality of the produt: noisy produts reate a heap impression while silent produts gives the impression of quality. Moreover, ar manufaturers demands, on one hand, to redue the total vehile weight and on the other hand, to inrease power at lower speed. As a onsequene noise due to the gearbox is expeted to inrease and to be less mased by the engine noise. For this reason, this wor deals with the obetive measurement of the gear noise, in partiular with the analysis of the gear whine whih represents transmission noises resulting from gears engaged in the torque flow, i.e. teeth engagement noise. In this paper we present results obtained by our basi noise researh on the vibration of gears using a traditional Fourier approah ompared to an innovative Wavelet approah. The Problem Many vehile manufaturers, researh and aademi institutions have foused their attention on vehile aousti design. Noise emissions produed by the transmission play a signifiant role. In fat, transmission aousti rating is one of the ustomer s aeptane riterion. The most maddening vibrations are indued by the engagement impulse (f = nz / 6, n is the speed and z is the number of teeth). The greater the load, the more important the entity of this impulse is. The angular speed of the mating gears has a relevant influene too. Aousti optimization has to find solutions that minimize suh impulses. ISSN: Issue, Volume 4, Marh 8

2 Our researh ativities therefore have been planned aording to the following development roadmap: st phase: noise generation i.e. teeth engagement nd phase: noise transfer i.e. gearbox housing and bearings rd phase: noise emission i.e. the transmission housing to the human pereption. Typially, to redue the gear whine noise, two options are possible: optimization of the gear s maro-geometry e.g. using high ontat ratio gears that leads to minor noise emissions in onuntion with higher transmitted power levels optimization of the gear s miro-geometry e.g. by trying to balane load-indued teeth defletions with profile orretions whih generally leads to less noisy transmission effets. This is not a suitable solution for an overall woring range; therefore profile orretions must be determined statistially to tae into aount manufaturing deviations whih will overlap their effet. This paper foused on the st phase. The goal is to find a method to proess the data in order to find the teeth engagement disontinuity during the torque transmission whih aused noise. All the experimental ativities are performed in Elasis using the Virtual Engine Simulator desribed in the next paragraph. The automotive transmission tool The Virtual Engine Simulator (VES) is the latest tool for automotive transmissions Noise Vibration and Harshness (NVH) testing. A view of a VES setup for testing and evaluating the gear noise is shown in Fig.. extremely important in reduing the produt time to maret. Sine VES an aurately reprodue mean value of both speed and ombustion and inertia fores, ar manufatures are able, in this way, to test prototype gearbox earlier in the proess, before prototype engines are available. Fig. shows a shemati diagram of the virtual engine simulator. As shown in this figure, unlie onventional test rigs, this partiular virtual engine simulator has a mehanial oupling between the input dynamometer and a high performane dynami pulse generator. Input dyno Left dyno Hydrauli pulsator Axle-shaft Gear box Cluth Flywheel Fig.. Virtual engine simulator: shematis Right dyno In order to load the gearbox output shafts, eah half-shaft is equipped with additional dynamometers to simulate road loads. The output dynamometers are sized to over a wide range of vehile speeds and torques. In addition to the simulated road loads, flywheels are installed on the output dynamometer to represent vehile inertias. The virtual engine simulator an be ontrolled by providing exitation time histories generated either by omputer software (synthesized) or measured on a real vehile. 4 The test-rig As shown in Figure, the test-rig is installed in a semi anehoi hamber where dynamometers are soundproofed inside enlosures. In the same way, in order to ahieve appropriate baground onditions for gear noise measurements, the floor is overed with soundproofing material. Fig.. Virtual engine simulator The maor benefit of using VES in gear noise testing is its ability to reprodue engine output irregularities, therefore, gearboxes an be obetively evaluated and tested without the need to build their assoiated engines. This apability is ISSN: Issue, Volume 4, Marh 8

3 Fig.. Virtual engine simulator settled in semi anehoi hamber Suh a set-up, unlie what happens on the vehile or in a power train test-ell, unouples the noise generated by the gearbox from the other noises present on the vehile or generated by the engine. For this reason, the noise problem is pronouned and onsequently, noise investigation beomes easier and more obetive. The test rig is equipped with the following signals: engine shaft veloity flutuation measured by magneti sensors at the flywheel and at the primary shaft sound pressure level (SPL) measured by using apaitive mirophones (sensitivity.5 V/Pa), loated at one meter far from transmission housing piezoeletri aelerometers (sensitivity pc/ms-) plaed on the gearbox housings. The gearbox oil temperature is also monitored beause of the strong influene of the gear whine with the oil visosity. All the tests are performed at 7 C. A five-speed manual shifted transmission is used for the analysis presented in this paper. The time history aquired on the ar is first reorded, during a wide-open throttle ondition with the seond gear engaged, and then it is reprodued on the test rig. The seond gear is seleted beause it leads to very audible gear whine noise inside the vehile abin, espeially up to 55 round per minute (). 5 Data proessing and results 5. Fourier s method The aelerometer used in this test has been plaed on the gearbox housing (Fig. 4), in orrespondene of the ontat point of gears. Fig. 5 shows the time history of the seond gear for the engine speed (green, red) and aelerometer vibrations (blue). Real F F B Time Taho Time Taho 47:A_s_mb s Fig. 5. Time history and vibrations Duration of signals is roughly 8 seonds and it onsists of one slow aeleration in seonds, a stationary phase ( seonds) and than a deeleration in seonds. The onversion from time-domain data to frequeny domain is performed using the Fast Fourier Transform (FFT) [] tehnique. In this appliation, where the engine speed is steadily inreasing and dereasing (engine run up & run down ase respetively), a family of FFT operations is obtained to san the range of the engine speed of interest. Thus, two Campbell diagrams [] have been reated, one for the aeleration phase and the other for the deeleration phase. Aquired signal has been proessed with a sampling frequeny of 5 hz, using Hanning windowing [] and a resolution of 5. Figg. 6-7 show the ontribution in frequeny aording to engine speed. In order to indiate the signal amplitude a hromati sale is used Real ( m/s ) Taho (T) Amplitude (m/s ) Hz 4 Fig. 6. Campbell diagram during aeleration phase Fig. 4. Aeleration measure point ISSN: Issue, Volume 4, Marh 8

4 54 Taho (T) 65. ( m/s ) Amplitude. Hz 4 Fig. 7. Campbell diagram during deeleration phase Spetral ontent is alulated to extrat the overall levels and engagement orders for both run up and run down maneuvers. The engagement orders are referred to: the gear order (order 9) and its first harmoni (order 8) the final redution order (order 7.4)and its first harmoni (order 4.8). Fig. 8 reports the order analysis for both aeleration and deeleration phase. F Overall level A_s_mb F Order 7.4A_s_mb F Order 4.8A_s_mb F Order 9.A_s_mb F Order 8.A_s_mb Fig. 8. overall level and engagement orders 5. The method of Wavelet Transform In the following subsetions we desribe the main onepts regarding the Disrete Wavelet Transform. The word wavelet is used in mathematis to denote a ind of orthonormal bases in L with remarable approximation properties. Wavelets allow to simplify the desription of a ompliated funtion in terms of small number of oeffiients. Often there are less oeffiients neessary than in the lassial Fourier analysis. Wavelets are adapted to loal properties of funtions to a larger extent than the Fourier basis. The adaptation is done automatially in view of the 65. Amplitude ( m/s). F Overal level A_s_mb F Order 7.4A_s_mb F Order 4.8A_s_mb F Order 9.A_s_mb F Order 8.A_s_mb Amplitude (m/s ) existene of a seond degree of freedom: the loalization in time (or spae, if multivariate funtions are onsidered). The vertial axis in the next graphs denotes always the level, i.e., the partition of the time axis into finer and finer resolutions. The advantage of this multiresolution analysis is that we an see immediately loal properties of data and thereby influene our further analysis. There were attempts in the past to modify the Fourier analysis by partitioning the time domain into piees and applying different Fourier expansions on different piees. But the partitioning is always subetive. Wavelets provide an elegant and mathematially onsistent realization of this intuitive idea [4]. In summary, wavelets offer a frequeny/time representation of data that allows us time (respetively, spae) adaptive filtering, reonstrution and smoothing. Reall that a mother wavelet ψ is a funtion of zero h-th moment (e.g., see [5], [6], [7], [8]) + x h ψ ( x) dx =, h N. From this definition, it follows that, if ψ is a wavelet whose all moments are zero, also the funtion ψ i (x): = / ψ( x ) is a wavelet. Now onsider a wavelet ψ and a funtion φ suh that {{ ϕ }, {ψ }, Z, =,,, } is a omplete orthonormal system. In this ase, a given signal s(t), deomposed by wavelet (i.e., CWT) is represented in the following detail funtion oeffiients + τ d = s( τ ) ψ dτ and in the approximating saling oeffiients as follows + s( τ) a ( τ ) = φ dτ Note that, for any, d an be regarded, as a funtion of. Consequently, if the signal s(t) is a smooth funtion, then the relative details are zero, sine, as said before, a wavelet has zero moments (for a detailed argumentation see [5]). The sequene of spaes { V, Z}, generated by φ is alled a multiresolution analysis (MRA) of L (R) if it satisfies the following properties V V+, Z and V is dense in L (R). ISSN: Issue, Volume 4, Marh 8

5 It follows that if { V, Z}, is a MRA of L (R), we say that the funtion φ generates a MRA of L (R), and we all φ the father wavelet. Besides, based on Parseval theorem, for any s L (R), it follows that s( t) = a φ ( t) + d ψ ( t) (4) = The relation (4) is alled a multiresolution expansion of s. This means that any s L (R) an be represented as a series (onvergent in L (R)), where a and d are some oeffiients, and { ψ }, Z, is a basis for W, where we define W = V V Z. In { () t } +, ψ is a general basis for W. The spae W is alled resolution level of multiresolution analysis. In the following, by abuse of notation, we frequently write resolution level or simply level. We employ these words mostly to designate not the spae W itself, but rather the oeffiients d and the funtion ψ on the level. As the Fast Fourier Transform, the Disrete Wavelet Transform (DWT) is a fast and linear operation operating on a data array of length equal to a power of and that transforms it in an array of equal length but numerially different. Both FFT and DWT ould be onsidered as a transformation from the original dominion (i.e., time) to a different dominion. In both the ases the funtions used to operate the transformation form a Complete Orthonormal System (CONS). Unlie trigonometrial basis, whih defines one only Fourier transform, infinite wavelet bases exist that differ for their loalization in the dominion of the time and for their regularity. A partiular wavelet basis is haraterized by numerial filters. In the present wor it has been applied the filter proposed by Daubehies, whih inludes both wavelets strongly loalized and wavelets strongly regular. A filter is haraterized by L oeffiients denoted as: h,, hl. We onsidered the Daubehies family of length L= 4, h,, h. The first step of wavelet transform was represented by the alulation of the following produt J J w = W x where { x, x,, xn } x is the vetor of N = J data of whih the wavelet transform have J to be alulated. While w is the wavelet vetor transform (of length L) after the first step of J alulation; W is the N order wavelet transformation matrix W J h h h h g g g g h h h h g g gg = h h h h g gg g h h h h g g g g The white elements are zero. It is important to observe the matrix struture. The first row generates the first element of onvolution between x and the h filter. Liewise the third, fifth..., and generally the odd rows of matrix generate the third, fifth..., element of onvolution respetively. The even rows generate the same type of onvolution but with the filter g rather than h. Filter g is alled also the onugated one of h and represents a pass-high filter. It is uniquely determined by means of h as the following relation g = ( ) hl, =,, L. The h and g filters are also named quadrature mirror filters (QMF). Note that g is suh to return null values if the vetor of whih we want to alulate the transform is suffiiently regular: in pratial the oeffiients g have p = L / null moments (in the following it will be espliitate suh a ondition named "p-order approximation ). Therefore the output of the filter h is the vetor x represented in a oarse shape, while the output of the filter g represents the detail that added to the oarse information allows to reonstrut the original vetor. We still notie that in the last two rows the oeffiient h and the orrespondent high-pass filter g are present due to the regularity onditions stated for the vetor x. By means of the inverse transform it is possible to reonstrut the original vetor x of length N by means of vetors of length N/ omposed of ISSN: Issue, Volume 4, Marh 8

6 output of the onvolution with the low-pass filter h and the high-pass filter g. The value of the elements of the vetor filter h an be obtained by imposing the orthonormality ondition for the matrix W J as follows h + h + h + h = hh + hh = and the approximation ondition of p= L/= order g + g + g + g = g + g + g + g = In the present wor (i.e., L = 4 ) the solution of ondition is [5] + h = 4 + h = 4 h = 4 h = 4 The DWT onsists in applying the W matrix in J x W of length hierarhial way to the vetor ( ) N =, then to the oarse vetor obtained by the onvolution of x with the low-pass filter h (of N/= J- length, with the W J- matrix), therefore still to the vetor of N/4 length obtained from the next onvolution with the filter h, and so on until to a prefixed level J or when the onvolution with the low-pass filter supplies a single element. The last proedure taes the name of pyramidal algorithm. In order to explain the proedure let us onsider the ase N=6= 4. Therefore the proedure is sensitized as follows J x x d () d x d () d x d () d d x 4 d d 4 d x 5 d 5 d d x6 6 d x7 d 4 W 7 x P W 8 4 d d P d W P W x 9 d 4 d x 5 d x d5 d x 6 d 4 x d 6 d 5 x 4 7 d 6 x 5 d 7 d 7 where P is a permutation matrix of elements of vetor x whih orders all the oeffiients of type "" (i.e., oarse oeffiients) and type d (i.e., detail oeffiients). Pratially the W matrix of order ats on oarse oeffiients of level, while the detail oeffiients of the same level are unhanged. Therefore at the end the wavelet transform vetor will be formed as following () () ( d d d d d d d d d d d4 d5 d6 d 7 ) () where means the oarse oeffiient obtained at the fourth step of wavelet transform, d () indiates the detail oeffiient obtained on the same step, d, d are the detail oeffiients obtained at the third step of transform, the d, =,, represent the detail oeffiients obtained at the seond step and finally d, =,,7 the detail oeffiients obtained at the first step of transform. Sine the proedure is based on orthogonal linear operations equally the WT will show the same feature. For the alulation of the inverse transform, it will be suffiient to repeat the steps of the transform in the inverse order. The performane of the WT algorithm is evaluated for real data. In this first experiment, we onsidered the real data shown in Fig. 5 as the input signal and applied the DWT to eah signal deomposition level. The Fig. 9 below shows two singularities at 5 th and 6 th deomposition level. T ISSN: Issue, Volume 4, Marh 8

7 Then the proposed methodology will be tested in order to investigate deeper on the regularity of the line of ation. The obetive is to alulate the shape of the required modified hobbing tools and to show that the generation proess will be no more ompliated than that used urrently for the prodution of standard gears [9][][]. The wor suggests some diretions for future investigations: Fig. 9 RunUp + RunDown Wavelet Analysis 6 Disussion and Conlusion This wor deals with the obetive measurement of the gear noise, in partiular with the analysis of the gear whine whih represents transmission noises resulting from gears engaged in the torque flow, i.e. teeth engagement noise. In this paper an innovative Wavelet s approah is presented and the results on basi noise researh of gear vibrations, obtained by using a traditional Fourier s approah, were ompared. A brief theory of wavelet transforms and their effetive omputation method was showed. This is followed by the numerial results with related graphs. From a deeper mehanial point of view, it is shown the ability of wavelet transform in order to detet and to loalize, starting from the experimental data set of signals, the area where the energy and the entropy assume onsistent value indiating the potential sliding and stress surfae of tooth profile. In fat a parallel main obetive of this wor for noise redution is the use of vibration signature analysis proedures for health monitoring and diagnostis of a gear transmission system. In order to approah suh a problem, probably, we have to modify the pressure angle and orresponding arhes of irumferene, onstituting the tooth profile, until the first derivative. The first results are interesting and show that if we design a line of ation that fulfils the inematial behavior requirements of the gear pair it will be muh easier, by applying the wavelets, to ensure the inematial quality of the designed gear pair. redution of underutting and interferene problem redution of slipping speeds [] inrease loading apaity inrease rigidity of toothing redution of noise and radial fores []. Finally, in order to improve the reliability of the proposed method, investigation will be onduted further on a wider ase-wise as well as to verify this design, a series of numerial simulations will be arried out using the boundary element method (BEM), and the results will be onfirmed with subsequent laboratory testing. In this ase the appliation of Wavelet Transform is also able for the identifiation and quantifiation of gear noise based on the numerially generated vibration signal. In fat a subsequent goal of this researh wor is the use of noise-vibration signature analysis proedures for health monitoring and diagnostis of a gear transmission system []. Important advanements in preventive maintenane of gear transmission systems are urrently being sought for the development of an aurate mahine health diagnosti system. Suh a diagnosti system would use vibration or aousti signals from the gear transmission system for rapid on-line evaluation of gear wear or damage status predition of remaining gear life. Suh health diagnosti apabilities would be essential for effetive mahine event/life management and advane warning before ritial omponent failures. Finally, in order to redue the gear noise we have to examine another requirement: the shape of teeth neessary for the speed ratio to remain onstant during an inrement of rotation; this behavior of the ontating surfaes (i.e., the teeth flans) is nown as onugate ation. ISSN: Issue, Volume 4, Marh 8

8 Referenes: [] Rader C.M., Disrete Fourier Transform when the Number of Data Samples is Prime, Pro. IEEE, vol. 56, pp. 7-8, June 968 [] Genta G., A Fast Modal Tehnique for the Computation of the Campbell Diagram of Multi- Degrees of Freedom Rotors, Journal of Sound and Vibration, 55, 99 [] Harris F.J., On the use of window for harmoni analysis with the disrete Fourier transform, Pro. IEEE, vol. 66, pp. 5-8, Jan. 978 [4] Härdle W., Keryaharian G., Piard D. and Tsybaov A., Leture Notes in Statistis - Wavelets, Approximation, and Statistial Appliations, Springer, 998 [5] Daubehies I., Ten Letures on Wavelets, SIAM, 99 [6] Antoniadis A., Oppenheim G., Leture notes in Statistis - Wavelets and Statistis, Springer, 995 [7] Kaiser G., A Friendly Guide to Wavelets, Birhäuser, 999 [8] Anthony, T., Computational Signal Proessing with Wavelets, Birhäuser, Boston, 998 [9] C. Lee, H.H. Lin, F.B. Oswald and D.P. Towsend, Influene of linear profile modifiation and loading onditions on the dynami tooth load and stress of high-ontat-ratio spur gears, ASME Journal of Mehanial Design, Vol., pp [] C. Lee, H.H. Lin, F.B. Oswald and D.P. Towsend, Computer-aided design of high-ontatratio gears for minimum dynami load and stress, ASME Journal of Mehanis Design, Vol. 5, pp. 7-78, 99. [] C. Lee, H.H. Lin, F.B. Oswald and D.P. Towsend Lee, Effet of ontat ratio on spur gear dynami load with no tooth profile modifiations, ASME Journal of Mehanis Design, Vol. 8, pp , 996. [] A.L. Kapelevih and R.E. Kleiss, Diret Gear Design for Spur and Helial Involute Gears, Gear Tehnology, Sept.-Ot., pp. 9-5,. [] F.K. Choy, D.H. Mugler and J. Zhou, Damage Identifiation of a Gear Transmission Using Vibration Signatures, ASME Trans. J. Mehanial Design, Vol. 5, pp. 94-4,. ISSN: Issue, Volume 4, Marh 8