The optimization of kinematical response of gear transmission

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1 Proeeings of the 7 WSEAS Int. Conferene on Ciruits, Systems, Signal an Teleommuniations, Gol Coast, Australia, January 7-9, 7 The optimization of inematial response of gear transmission VINCENZO NIOLA Department of Mehanial Engineering for Energetis University of Naples Feerio II Via Clauio, 8, Napoli ITALY GIUSEPPE QUAREMBA Department of Mehanial Engineering for Energetis University of Naples Feerio II Via Clauio, 8, Napoli ITALY Abstrat: - A brief theory of wavelet transforms an their effetive omputation metho with an emphasis on the onsierations of the hoie of pressure angle are presente in this paper. This is followe by the numerial results with relate graphs. The proeure use in this paper inlue the numerial simulation of the ynamis of a gear transmission system. Important avanements in preventive maintenane of gear transmission systems are urrently being sought for the evelopment of an aurate mahine health iagnosti system. Suh a iagnosti system shoul use vibration or aousti signals erive iretly from the gear transmission. Finally, the improvement of the inematial quality of gear transmission was teste by means of the Wavelet Transform. Key-Wors: - Spur gear, Disrete Wavelet Transform, numerial simulation, gear transmission, inematial quality, mehanial iagnostis. Introution Gears are one of the most ommon an important mahine omponents in many avane mahines. The aim of the wor is to show the importane of etermining the shape of the line of ation in orer to obtain the best inematial harateristis in terms of quality performane. Starting from a proeure for esigning spur gear sets using the quarati parametri tooth profiles a numerial simulation was evelope []. Finally, the improvement of the inematial quality of gear transmission was teste by means of the Wavelet Transform (WT). In this ase the appliation of Wavelet Transform is also able for the ientifiation an quantifiation of amage tooth base on the numerially generate vibration signal. In fat the obetive of this wor represents only the first step in orer to perform the vibration signature analysis proeures for health monitoring an iagnostis of a gear transmission system. The proeure use in this paper inlue the numerial simulation of the ynamis of a gear transmission system. The avantage of using both a mathematial moel of gear profile erive from the line of ation an the Wavelet Transform for proessing the regularity of suh a line is that the mathematial moel omprises only one single variable of the line of ation an the multiresolution analysis, performe by means of wavelet transform, provies more information regaring the inematial quality of the esigne gear pair. Gear Transmission Gears are mahine elements use to transmit rotary motion between two shafts, normally with a onstant ratio. The pinion is the smallest gear an the larger

2 Proeeings of the 7 WSEAS Int. Conferene on Ciruits, Systems, Signal an Teleommuniations, Gol Coast, Australia, January 7-9, 7 gear is alle the gear wheel. A ra is a retangular prism with gear teeth mahine along one sie; it is in effet a gear wheel with an infinite pith irle iameter. In pratie the ation of gears in transmitting motion is a am ation eah pair of mating teeth ating as ams. Gear esign has evolve to suh a level that throughout the motion of eah ontating pair of teeth the veloity ratio of the gears is maintaine fixe an the veloity ratio is still fixe as eah subsequent pair of teeth ome into ontat. When the teeth ation is suh that the riving tooth, moving at onstant angular veloity, proues a proportional onstant veloity of the riven tooth the ation is sai to onstitute a onugate ation. The teeth shape universally selete for the gear teeth is the involute profile. The vast maority of gear appliations use the stanar involute system beause of its goo ombination of bening an surfae pressure strength, involute insensitivity to errors in enter istane an relative ease of manufaturing. Most gears are proue by hobbing or other generationtype proesses, where a straight-tooth ra or equivalent tool proues the involute woring gear tooth surfae as well as a trohoial root fillet. Despite the benefits of this system, it is generally felt that a higher bening strength an hene loa arrying apaity shoul be obtaine. This is espeially true with small numbers of teeth (less than or 7 epening on the tip raius of the hob), where the stanar involute teeth are suseptible to unerutting. This is a situation where the tip of the utter removes material from the involute profile in a seonary utting ation. The resulting teeth have smaller thinesses near their roots, where the ritial setion is usually loate, an this severely hampers loa arrying apaity. From an other point of view, the harateristis of motion are stuie by the tehniques of tooth ontat analysis in the fixe oorinate system. Many authors propose moel an solution in orer to approah the problem of esigning an optimum tooth profile. For example [,] propose a mathematial moel of parametri tooth profiles for spur gears using pressure angle as a parametri variable. In [] was propose a metho of esigning high-ontat-ratio spur gears using quarati parametri tooth profiles for the shorter aenum without unerut an [-7] stuie the effets of the linear profile moifiation on the ynami tooth loa an stress for high-ontat ratio gearing. In [8] was propose an optimum tooth profile of spur gear rotary pumps metho to reue the elivery flutuation. The avantage of using both a mathematial moel of gear profile erive from the line of ation an the Wavelet Transform for proessing the regularity of suh a line is that the mathematial moel omprises only one single variable of the line of ation an the multiresolution analysis, performe by means of wavelet transform, provies more information regaring the inematial quality of the esigne gear pair [9]. Mathematial bagroun The wor wavelet is use in mathematis to enote a in of orthonormal bases in L with remarable approximation properties. Wavelets allow to simplify the esription of a ompliate funtion in terms of small number of oeffiients. Often there are less oeffiients neessary than in the lassial Fourier analysis. Wavelets are aapte to loal properties of funtions to a larger extent than the Fourier basis. The aaptation is one automatially in view of the existene of a seon egree of freeom: the loalization in time (or spae, if multivariate funtions are onsiere). The vertial axis in the next graphs enotes always the level, i.e., the partition of the time axis into finer an finer resolutions. The avantage of this multiresolution analysis is that we an see immeiately loal properties of ata an thereby influene our further analysis. There were attempts in the past to moify the Fourier analysis by partitioning the time omain into piees an applying ifferent Fourier expansions on ifferent piees (e.g., Fourier Fast Transform). But the partitioning is always subetive. Wavelets provie an elegant an mathematially onsistent realization of this intuitive iea []. In summary, wavelets offer a frequeny/time representation of ata that allows us time (respetively, spae) aaptive filtering, reonstrution an smoothing. Reall that a mother wavelet ψ is a funtion of zero h-th moment (e.g., see [], [], []) + x h ψ ( x) x =, h N. () From this efinition, 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 onsier a wavelet ψ an a funtion φ suh that {{ ϕ }, {ψ }, Z, =,,, } is a omplete orthonormal system. In this ase, a given signal s(t),

3 Proeeings of the 7 WSEAS Int. Conferene on Ciruits, Systems, Signal an Teleommuniations, Gol Coast, Australia, January 7-9, 7 eompose by wavelet (i.e., Continuous Wavelet Transform) is represente in the following etail funtion oeffiients + τ = s( τ ) ψ τ () an in the approximating saling oeffiients as follows + a s( τ) ( τ ) = φ τ. Note that, for any, an be regare, as a funtion of. Consequently, if the signal s(t) is a smooth funtion, then the relative etails are zero, sine, as sai before, a wavelet has zero moments (for a etaile argumentation see []). The sequene of spaes { V, Z}, generate by φ is alle a multiresolution analysis (MRA) of L (R) if it satisfies the following properties V V Z an +, V is ense in L (R). It follows that if { V, Z}, is a MRA of L (R), we say that the funtion φ generates a MRA of L (R), an we all φ the father wavelet. Besies, base on Parseval theorem, for any s L (R), it follows that s( t) = a φ ( t) + ψ ( t). () = The relation () is alle a multiresolution expansion of s. This means that any s L (R) an be represente as a series (onvergent in L (R)), where a an are some oeffiients, an { ψ }, Z, is a basis for W, where we efine In () { () t } W = V V Z. +, ψ is a general basis for W. The spae W is alle resolution level of multiresolution analysis. In the following, by abuse of notation, we frequently write resolution level or simply level. We employ these wors mostly to esignate not the spae W itself, but rather the oeffiients an the funtion ψ on the level. As the Fourier Fast Transform (FFT), the Disrete Wavelet Transform (DWT) is a fast an linear operation operating on a ata array of length equal to a power of an that transforms it in an array of equal length but numerially ifferent. Both FFT an DWT oul be onsiere as a transformation from the original ominion (i.e., time) to a ifferent ominion. In both the ases the funtions use to operate the transformation form a Complete Orthonormal System (CONS). Unlie trigonometrial basis, whih efines one only Fourier transform, infinite wavelet bases exist that iffer for their loalization in the ominion of the time an for their regularity. A partiular wavelet basis is haraterize by numerial filters. In the present wor it has been applie the filter propose by Daubehies, whih inlues both wavelets strongly loalize an wavelets strongly regular. A filter is haraterize by L oeffiients enote as: h,, hl. We onsiere the Daubehies family of length L=, h,, h. The first step of wavelet transform was represente by the alulation of the following J J prout w = W x where x { x, x,, xn } is the vetor of N = J ata of whih the wavelet J transform have to be alulate. While w is the wavelet vetor transform (of length L) after the first step of alulation; W J is the N orer 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 where the white elements are zero. It is important to observe the matrix struture. The first raw generates the first element of onvolution between x an the h filter. Liewise the thir, fifth..., an generally the o raws of matrix generate the thir, fifth..., element of

4 Proeeings of the 7 WSEAS Int. Conferene on Ciruits, Systems, Signal an Teleommuniations, Gol Coast, Australia, January 7-9, 7 6 onvolution respetively. The even raws generate the same type of onvolution but with the filter g rather than h. The filter g is also alle the onugate one of h an it represents a pass-high filter. It is uniquely etermine by means of h as the following relation g = ( ) h, =,, L. (6) L + h = + h = h = h =. (9) The h an g filters are also name as quarature 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 onition name as "p-orer approximation ). Therefore the output of the filter h is the vetor x represente in a oarse shape, while the output of the filter g represents the etail that ae to the oarse information allows to reonstrut the original vetor. We still notie that in the last two raws the oeffiient h an the orresponent high-pass filter g are present ue to the regularity onitions state for the vetor x. By means of the inverse transform it is possible to reonstrut the original vetor x of N length by means of vetors of N/ length ompose of output of the onvolution with the low-pass filter h an the high-pass filter g. The value of the elements of the vetor filter h an be obtaine by imposing the orthonormality onition for the matrix W J as follows h + h + h + h = hh + hh = (7) an the approximation onition of p= L/= orer g + g + g + g = (8) g + g + g + g =. In the present wor (i.e., L = ) the solution of onition is [] The DWT onsists in applying the W matrix in a J hierarhial way to the vetor ( W ) J x of length N =, then to the oarse vetor obtaine 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/ length obtaine from the next onvolution with the filter h, an so on until to a prefixe level J or when the onvolution with the low-pass filter supplies a single element. The last proeure taes the name of pyramial algorithm. In orer to explain the proeure let us onsier the ase N=6=. Therefore the proeure is synthesize as follows x () x () x () x () x () x () x 6 6 () x 7 W 7 x P () 8 W P x 9 x x x 6 x 6 x 7 6 x 7 7 () ( ) () () () ( ) () () ( ) () () ( ) () () () () () W P W. () where P is a permutation matrix of elements of vetor x whih orers all the oeffiients of type ""

5 Proeeings of the 7 WSEAS Int. Conferene on Ciruits, Systems, Signal an Teleommuniations, Gol Coast, Australia, January 7-9, 7 7 (i.e., oarse oeffiients) an type (i.e., etail oeffiients). Note that the W matrix of orer ats on oarse oeffiients of level, while the etail oeffiients of the same level are unhange. Therefore at the en the wavelet transform vetor will be forme as following () () () () () () () () ( 6 7 ) () () where means the oarse oeffiient obtaine at () the fourth step of wavelet transform, iniates the etail oeffiient obtaine on the same step, () (), are the etail oeffiients obtaine at the () thir step of transform, the, =,, represent the etail oeffiients obtaine at the seon step an finally (), =,,7 the etail oeffiients obtaine at the first step of transform. Sine the proeure is base 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 orer. In [][9] it was stuie the problem of minimizing the speifi sliing ratio of meshing profiles funtions for both the pinion an gear as well as the sliing wor. The funtion α ( λ ) whih minimize the above funtions for all λ values, will provie the optimum shape of the line of ation, where λ is the parametri variable of the line of ation an α the angle between λ an x -axis. By eriving the quote funtions we obtaine ( ) ( ) α λ α λ λ λ tanα ( λ) + λ λ α( λ) + λ tanα( λ) = λ π. where osα( λ) α( λ) T () Finally, for evaluating the features of the signal, a parameter (entropy) was efine. Given a set S: = {x i, I {,,,n}} an a funtion : x i S (x i ) R, the entropy H( ) of is efine as follows ( ) ( ) x ( ): i m x m ln i H = ( xi) m s M m s M m () where s = i I ( ) xi m M m, M: = max { ( x i ), i {,,..., n} } an m: = min { ( x i ), i {,,..., n} } The entropy measures the best ratio between the maximum ynami showe by signal an the smallest uniformity of signal. Given S = n, the entropy, as before efine, rihes its maximum value at ln(n) iff, for any i S, (x i ) = onst. Finally H() = iff, for any i {,,,n}, (x i ) = S an, for any {,,,n}-{i}, (x ) =. In the Fig. is shown a shemati Blo Diagram employe for the simulation gear box. It was use for estimating the response of DWT applie to the moel of Gear Box. To Worspae x x7 x x7 Gearbox L To Worspae6 Gear Box Simulation x To Worspae x x x x (pinion shaft) x x7 (wheel shaft). x To Worspae x x x Gearbox R Fig. Shemati Blo Diagram x To Worspae x To Worspae x To Worspae Results In the Fig. an, reporte below, it is shown the ability of wavelet transform in orer to etet an to loalize, starting from the () the instants where the entropy, efine in (), rihes onsistent values iniating the potential sliing an stress surfae of profile.

6 Proeeings of the 7 WSEAS Int. Conferene on Ciruits, Systems, Signal an Teleommuniations, Gol Coast, Australia, January 7-9, 7 8 Fig. Entropy istribution Fig. MRA analysis of entropy istribution In partiular in the Fig. it is well represente the instants where the ontat stress rihes its maximum value (i.e., from to units of time) an onsequently the surfae pressure of tooth profiles. The results are very interesting an show that if we esign 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 esigne gear pair. Conlusions Gears are one of the most ommon an important mahine omponents in many avane mahines. Moern gear esign is generally base on stanar tools. This maes gear esign quite simple (almost lie seleting fasteners), eonomial, an available for everyone, reuing tooling expenses an inventory. At the same time, it is well nown that universal stanar tools provie gears with less than optimum performane an in some ases o not allow for fining aeptable gear solutions. Appliation speifis, inluing low noise an vibration, high ensity of power transmission (lighter weight, smaller size) an others, require gears with non-stanar parameters [,,]. On the other sie, an improve unerstaning of sliing wor an surfae ontat stress is require both for the early etetion of inipient gear failure an to ahieve high reliability. A brief theory of wavelet transforms an their effetive omputation metho with an emphasis on the onsierations of the hoie of pressure angle are presente in this paper. This is followe by the numerial results with relate graphs. It is shown the ability of wavelet transform in orer to etet an to loalize, starting from the line of ation, the area where the energy an the entropy assume onsistent value iniating the potential sliing an stress surfae of profile. In orer to avoi suh a problem, probably, we have to moify the pressure angle an orresponing arhes of irumferene, onstituting the profile, until the first erivative. The results are interesting an show that if we esign 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 esigne gear pair. Then the propose methoology will be investigate in orer to investigate eeper on the regularity of the line of ation. The obetive is to alulate the shape of the require moifie hobbing tools an to show that the generation proess will be no more ompliate than that use urrently for the proution of stanar gears. The wor suggests some iretions for future investigations: reution of unerutting an interferene problem reution of slipping spees inrease loaing apaity inrease rigiity of toothing reution of noise an raial fores. The appliation of Wavelet Transform is also able for the ientifiation an quantifiation of amage tooth base on the numerially generate vibration signal. The proeure use in this paper inlue the numerial simulation of the ynamis of a gear transmission system. Important avanements in preventive maintenane of gear transmission systems are urrently being

7 Proeeings of the 7 WSEAS Int. Conferene on Ciruits, Systems, Signal an Teleommuniations, Gol Coast, Australia, January 7-9, 7 9 sought for the evelopment of an aurate mahine health iagnosti system. Suh a iagnosti system woul use vibration [6]or aousti signals from the gear transmission system for rapi on-line evaluation of gear wear or amage status preition of remaining gear life. Suh health iagnosti apabilities woul be essential for effetive mahine event/life management an avane warning before ritial omponent failures. Referenes: [] Niola, V. an Quaremba, G., A wavelet appliation for improving the inematial quality of gear system, in press [] Chang, H.L. an Tsai, Y.C., A mathematial moel of parametri tooth profiles for spur gears, ASME Journal of Mehanial Design,, 99, pp. 8-6 [] Chang H.L. an Tsai, Y.C., A stuy of linear parametri tooth profiles using pressure angle, Journal of the Chinese Soiety of Mehanial Engineers, (), 99, pp. 7- [] Tsai, M.H. an Tsai, Y.C., Design of highontat-ratio spur gear using quarati parametri tooth profiles, Meh. Mah. Theory, (), 998, pp. -6 [] Lee, C., Lin, H.H., Oswal, F.B. an Towsen, D.P., Influene of linear profile moifiation an loaing onitions on the ynami tooth loa an stress of high-ontat-ratio spur gears, ASME Journal of Mehanial Design,, 99, pp. 7-8 [8] Costopulos, T.H., Kanarahos, A. an Pantazis, E., Reution of elivery flutuation an optimum tooth profile of spur gear rotary pumps, Meh. Ma. Theory, (), 988, pp. -6 [9] Niola, V. an Quaremba G., The shape of the line of ation optimize by means of DWT, Pro. VI WSEAS Int. Conf. on Wavelet Analysis & Multivariate Systems, Buharest (Romania), Ot. 6-8, 6 [] Daubehies, I., Ten Letures on Wavelets, SIAM, 99 [] Antoniais A., an Oppenheim G., Leture Notes in Statistis Wavelet an Statistis, Springer, 99 [] Härle, W, Keryaharian, G., Piar, D. an Tsybaov, A., Wavelets, Approximation, an Statistial Appliations, Springer, Berlin, 998 [] Fong, Z.H., Chiang, T.W. an Tsay, C.W., Mathematial Moel for Parametri Tooth Profile of Spur Gear Using Line of Ation, Mathematial an Computer Moelling, 6,, pp. 6-6 [] Duley, D.W., Hanboo of Pratial Gear Design, MGraw-Hill, New Yor, 98 [] Kapelevih, A.L. an Kleiss, R.E., Diret Gear Design for Spur an Helial Involute Gears, Gear Tehnology, Sept.-Ot., pp. 9- [6] Choy, F.K., Mugler, D.H. an Zhou, J., Damage Ientifiation of a Gear Transmission Using Vibration Signatures, ASME Trans. J. Mehanial Design,,, pp. 9- [6] Lee, C., Lin, H.H., Oswal, F.B. an Towsen, D.P., Computer-aie esign of high-ontatratio gears for minimum ynami loa an stress, ASME Journal of Mehanis Design,, Marh 99, pp [7] Lee, C., Lin, H.H., Oswal, F.B. an Towsen, Effet of ontat ratio on spur gear ynami loa with no tooth profile moifiations, ASME Journal of Mehanis Design, 8, September 996, pp. 9-

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

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