- Email: [email protected]
EPSI validation for cylindrical gears
ARTICLE IN PRESS
Optics and Lasers in Engineering 42 (2004) 447–459
EPSI validation for cylindrical gears Jean-Pierre de Vaujany*, Miche" le Guingand Laboratoire Dynamique des Machines et des Structures, INSA de Lyon, 20, avenue Albert Einstein, Villeurbanne cedex 69621, France Received 28 July 2003; accepted 30 December 2003
Abstract Shape deviations of spur gears obtained by using Electronic Speckle Pattern Interferometry are compared with the results of numerical simulation. The dual illumination method produces a speckle pattern by simultaneous illumination of a sample by two laser waves symmetrically in the direction of observation. Two images obtained, respectively, before and after deformation are subtracted to obtain the field of displacement. Furthermore, a study of meshing stiffness was performed for different types of setting for an internal gear. r 2004 Elsevier Ltd. All rights reserved. Keywords: Speckle video; Gears; Meshing stiffness
1. Introduction Gears used in aerospace applications must have very thin-walled structures to reduce flying weight. They must also be designed within a very short time. The design of power transmission devices such as cylindrical gears is based on numerical studies for modelling mechanical behaviour. Optimisations of the tooth profile are increasingly precise and concern very small values. Furthermore, small modifications of geometry involve considerable variations of load on the teeth. Different models have been developed to calculate load sharing. The validity of current models is based on the results of stress distribution in the tooth roots. Certain models make use of measurements with strain gauges [1,2]. The value of these deformations is also estimated with laser holographic interferometry [3–5]. On the other hand, little experimental work has been performed to verify the load sharing between the *Corresponding author. Tel.: +33-472-43-8546; fax: +33-472-43-8930. E-mail address: [email protected] (J.-P. de Vaujany). 0143-8166/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2003.12.005
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different teeth in contact, since it is difficult to measure this load distribution directly. However, indirect methods can be used. The aim of this work is to validate this load sharing by measuring the full displacement field on the teeth under load, taking into account experimental and numerical models. The speckle video method or Electronic speckle pattern interferometry (E.S.P.I) has been used for several right cylindrical gears (internal and external). This study compares the non-uniform strain fields measured to those obtained by numerical calculations (finite prism method). Then, using a numerical model, a study of the meshing stiffness as a function of the fastening of a ring will be presented and discussed.
2. Principle of the speckle measuring technique ESPI is a non-contact full field measurement system that measures the deformations under mechanical loads along three perpendicular axes (for a 3D system). Deformations can be measured on samples with relatively large dimensions and aspect ratios, such as test specimens and components. A speckle effect is produced by the interference between reflected rays when a surface is illuminated with a laser beam. The image of the object is recorded by a video camera. The light waves reflected from the object’s surface interfere with each other and produce a variable pattern known as speckle. The pattern is defined by the microscopically rough topography of the sample surface. Any deformation of the surface results in a change of the speckle pattern. This speckle pattern and deformed object state is stored by the image processor. When the sample deforms, the speckle pattern changes. Correlation fringes are created by comparing the speckle pattern for the object in a reference state with that for the object in a deformed state (Fig. 1). These fringes represent the deformation of the object’s surface in the direction concerned. Evaluation software is then used to analyse and count the correlation fringes and transform them into a quantitative set of deformation and strain data. The object studied is illuminated with two lasers. The dual illumination method produces a speckle pattern by simultaneous illumination of the sample by two laser waves symmetrically about the observation direction. The resulting speckle pattern represents the phase difference f between the two light paths from the laser via the object surface to the camera. This speckle pattern is stored as a reference. Movement
Speckle pattern of undeformed surface
Speckle pattern of deformed surface
Difference of the two speckle patterns
Fig. 1. Examples of speckle patterns.
ARTICLE IN PRESS J.-P. de Vaujany, M. Guingand / Optics and Lasers in Engineering 42 (2004) 447–459
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of the object produces a change in the two light paths relative to one another, creating a new phase relation f þ D: This creates a new speckle pattern. The difference D between both patterns is represented by correlation fringes. The deformation of the object’s surface is obtained by counting the number of fringes at every object point. The direction of measurement is orthogonal to the viewing direction and is produced in the plane of the two illumination directions. To obtain the field of displacement, two images obtained, respectively, before and after deformation are subtracted from each other. After this operation, the resulting image shows variations of luminous intensity as a function of phase differences between the two signals subtracted. This difference of phase is given by: 2p Dj ¼ 2u cosðyÞ; ð1Þ l where l is wavelength of the laser light, u is displacement following a direction X on the surface of the object, y is angle of impact between the axis X and the direction of the laser. When this difference of phase is equal to 2np; the two waves are in phase and the luminous intensity is maximal. For a difference of phase of ð2n þ 1Þp; the waves are out of phase and the intensity is minimal. The final image is characterized by a representative fringe system of curves of displacement following the direction chosen for the study. The clear fringe equation is nl l u¼ and the interfringe is : d ¼ : ð2Þ 2 cosðyÞ 2 cosðyÞ In particular, the fringe of order n ¼ 0; associated with a null displacement, is theoretically a white fringe. Due to the subtraction introduced during numerical image processing, the fringe from order n ¼ 0 (noted Ro) is associated with a black fringe for the experimental results presented here. On the other hand, the intermediate displacements are represented by clear furrows. The images resulting from these operations contain intrinsic noise due to the E.S.P.I. technique. Several numerical contrast and filtering improvement operations are necessary to obtain the best possible quality images of the fringe system [6].
3. Experimental test 3.1. Geometrical characteristics of the gears The study was carried out with right teeth for external and internal gears and concerns normalized teeth. Their geometrical characteristics are given in Tables 1 and 2. Different criteria were chosen to design the gears: *
contact ratio from 2 to 3 so that the load is always shared between at least two couples of teeth,
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Table 1 Geometrical characteristics of the external gear
Module (mm) Pressure angle ( ) Face width (mm) Number of teeth
Pinion
Gear
1.5 15.0 10.0 58
1.5 15.0 10.0 62
Pinion
Gear
1.5 15.0 10.0 60
1.5 15.0 10.0 160
Table 2 Geometrical characteristics of the internal gear
Module (mm) Pressure angle ( ) Face width (mm) Number of teeth
*
*
*
a small module to ensure that at least two teeth are systematically visible in the field of the camera, good quality gear (Agma 12) so as to minimize the effects of errors (pitch error, etc.), a small pressure angle to obtain mainly flexion deformations. The latter condition allows obtaining displacement in the circumferential directions (X ) greater than displacement in the perpendicular directions (Y ). These Y displacements are thereafter considered as negligible.
3.2. Experimentation The architecture of the optical setting is represented in Fig. 2. Two configurations of this setting were used. In the first, an optical device performs the measurements in two orthogonal directions (X and Y ; Fig. 5). This allows successive recording of systems of fringes associated with these two directions without modifying the positioning of the teeth or the loading. This choice required creating four distinct optical paths with coherent beams of symmetrical illumination on the surfaces studied. The resolution of displacement X and Y increased when angle y (Fig. 2) decreased. The average values of y are equal to 20 . A binocular was introduced in the acquisition system because the zone studied was very small. Therefore it was necessary to increase the resulting image resolution. The setting to position and load the gears is presented in Fig. 3. It intended to have the following characteristics: *
great rigidity of the system as a function of the teeth,
ARTICLE IN PRESS J.-P. de Vaujany, M. Guingand / Optics and Lasers in Engineering 42 (2004) 447–459
θ
451
7
θ
5
4 6
Binocular Camera CCD Video monitor
3
2
1
1 : Laser 2 : Obturator 3 : Space filter 4 : Lens 5 , 6 : Mirors 7 : Gears
Computer
Fig. 2. Optical acquisition diagram of images in speckle video [6].
Fig. 3. Diagram of gear setting. *
*
the possibility to modify the relative positions of the gear (angular position and centre-to-centre distance), the possibility of studying external and internal gears.
4. Numerical model The calculations were performed as a function of the geometrical characteristics and loading conditions. For 5 teeth of the pinion and the gear, the simulation determines the potential contact points on the flanks for several kinematical positions (Fig. 4). The finite prism method (FPM) provides information on elastic
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Fig. 4. Kinematical positions.
behaviour. It can be considered as a special form of finite elements. The finite element method (FEM) uses polynomial functions of displacement in all directions, whereas the FPM uses polynomial functions solely in two directions and series functions in the third direction. This method, developed by Cheung [7], was adapted to the analysis of the elastic behaviour of cylindrical gears by Olakorede [8] and Kim [9]. The advantage of this method in comparison to the FEM method is that it decreases calculation times and storage space. Furthermore, in the model the surface effects (contact deformations) are computed using Boussinesq’s method [10]. By knowing the coefficients of structure, surface effects and initial gaps between each couple of teeth, it is possible to determine the contact pressure distribution on each tooth by resolving the deformation compatibility relationship [11,12]. The loading corresponding to each meshing position is applied on the flank of the teeth and the displacements are calculated in the three directions. For our study, the comparison between the displacements calculated numerically and those of the experiment are performed mainly for displacements in direction X :
5. Experimental results 5.1. External gear For the case of the external gear, Fig. 5 presents the experimental displacement field obtained in the first measurement context (normal angle of impact y) and the numerical results obtained with the model. The test and calculation parameters are the following: * * * *
interference fringe in direction X: 0.674 mm interference fringe in direction Y: 0.778 mm Torque: 57.8 N m Young’s module: 2.1E+11 N/m2 and Poisson’s ratio: 0.3
The results obtained results were in good agreement, especially following X : These results show that the deformation occurs mainly close to X ; which is logical, considering the small pressure angle (15 ). Extremely low values of displacement
ARTICLE IN PRESS J.-P. de Vaujany, M. Guingand / Optics and Lasers in Engineering 42 (2004) 447–459
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Fig. 5. Interference fringe patterns (computed and measured), First measurement context.
in Y ; as well as the utilization of a substantial interference fringe can explain in part the small differences in this direction. Nevertheless, the appearance of the field of displacement (completed by lines of a half interfringe and -dotted curves) correlates with the experimental results and the number of fringes obtained on the most heavily loaded tooth is correct. In order to evaluate these initial results, we analysed the progression of the field of displacement (direction X) for different meshing positions. These positions were chosen to visualise the entry of a new tooth in the contact. This study was carried out in a second measurement context with a narrower angle of impact (y) and thus with higher resolution.
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The test and calculation parameters in this measurement configuration are: * * * *
interference fringe in direction X: 0.358 mm the displacement field in direction Y is not studied because the values are low, torque: 57.8 N m Young’s module: 2.1E+11 N/m2 and Poisson’s ratio: 0.3
Fig. 6(b) shows the progression of the field of displacement (direction X) calculated for 4 successive meshing positions. This simulation is associated with the entry of tooth no. 4 in the contact. Fig. 6(a) shows experimental displacement fields obtained in the same context, in particular, the progression of the field of
Fig. 6. Interference fringe patterns in direction X (interference fringe=0.358 mm). External gear (4 meshing positions).
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displacement during the entry of a tooth in the contact is visualized in the three first photographs. Very good agreement is obtained between the experimental and numerical results. The choice of analysis positions and the imperfections of the real teeth are sufficient to explain the very small differences observed in these different figures. 5.2. Internal gear The parameters for measuring interference fringe patterns were identical to those used for the external gear. As in the case of the external gear, the values of displacement in direction Y were low and the interference fringe used is large. These two points can justify in part the differences obtained between the experimental and numerical results in direction Y : The deformation of the structure is mainly in direction X : Fig. 7 shows the progression of the interference fringe patterns in direction X measured and calculated for four meshing positions. There is a very good agreement between the experimental results in Fig. 7(a) and numerical results in Fig. 7(b).
6. Variation of meshing stiffness depending on the setting Changing the number of teeth in contact modifies the progression of the meshing stiffness. This variation is a source of internal excitation that, due to its fluctuations, produces vibrations and noise. For example, in the case of internal gears load sharing is shown in Fig. 8. The geometrical characteristics of the gear studied are presented in Table 3. The study was carried out with several torques of 300 with 2400 N m. For one meshing period (total rotation of 0.12 rad with 0.28 rad), the division of the load is represented in Fig. 8 for the 3 torques: 300, 1200 and 2400 N m. We note that the pace of progression of tooth loading differs according to loading case. For the lowest torque, the progression is broken down into crenels, contrary to the loaded case where the variation is more gradual. This phenomenon occurs with the deflection of the teeth and the rotation of the base of the teeth (rim). Naturally, these displacements are more considerable for a high loading. They make it possible for the teeth to return into contact earlier and ensure more continuous contact through time. Thus the period during which only two teeth are in contact is shorter (Fig. 8: total rotation between 0.19 and 0.2 rad is approximately 6% of the meshing period). An internal gear can be fixed to the crankcase with the help of bolts or pins around the contour of the rim. Following the gap between each fixing point, the planet moves successively opposite and between these points. In the case of a thin rim thickness, the deformation of the structure is influenced by the bolt or pin arrangement. To quantify the phenomenon, the meshing stiffness was calculated for different types of setting (Figs. 9 and 10). The torque used for the calculations was 1200 N m.
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Fig. 7. Interference fringe patterns in direction X (interference fringe=0.358 mm). Internal gear (4 meshing positions).
With only one central fixing point (Fig. 9(b)), the value of the meshing stiffness was located between those calculated for the two conditions: free rim (Fig. 9(a)) and completely fixed rim (Fig. 9(f)). The more the number of fixing points the more the progression of stiffness resembles the case of a rim completely blocked at its edges. The average stiffness increases with the number of fixing points except in the case of a meshing centred between two fixing points (Fig. 9(c)). In this case, the average rigidity (1.2E+6 N/mm) is weaker than the case of only one central fixing point (Fig. 9(b) and Fig. 10: 1.35E+06 N/mm), because the rim is not blocked on the side
ARTICLE IN PRESS J.-P. de Vaujany, M. Guingand / Optics and Lasers in Engineering 42 (2004) 447–459
tooth 2
tooth 3
tooth 4
tooth 5 Torque = 2400 N.m
12000
2500
10000
2000
Load (N)
Load (N)
Torque = 300 N.m 3000
1500 1000
8000 6000 4000
500
2000
0 0.10
457
0.15
0.20
0.25
0 0.10
0.30
Pinion angular position (rd)
0.15
0.20
0.25
0.30
Pinion angular position (rd)
Torque = 1200 N.m 12000
Load (N)
10000 8000 6000 4000 2000 0 0.10
0.15
0.20
0.25
0.30
Pinion angular position (rd) Fig. 8. Load sharing.
Table 3 Gear characteristics and torque condition
Number of teeth Module (mm) Pressure angle ( ) Face width (mm) Rim thickness (mm) Torque (N m)
Pinion
Ring
51 2.53 20 58 5 module 1200
195 2.53 20 58 4module
of the meshing zone and the deformations are greater. The deformation of the rim depends on the position of the planets next to the fixing elements and can vary by approximately 50%. It is therefore necessary to model the setting very precisely to obtain a representative stiffness of the gear’s mechanical behaviour.
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Fig. 9. Definition of fixing points.
Meshing stiffness (N/mm)
1.80E+06 1.60E+06 1.40E+06 free 1 fixing 2 fixings
1.20E+06 1.00E+06 8.00E+05
3 fixings 5 fixings clamping
6.00E+05 4.00E+05 2.00E+05 0.00E+00 0.10
0.15
0.20
0.25
0.30
Pinion angular position (rad)
Fig. 10. Meshing stiffness dependent on the rim fixing.
7. Conclusion Experimental simulation of the behaviour of gears with deformation gauges in the tooth roots are very difficult to carry out. The E.S.P.I method was used because it allows visualizing the tooth deflection with precision. By using a small module, the displacement fields are obtained on several teeth, thus making it possible to verify the numerical results of the load sharing. For external and internal gears, the comparisons between the numerical and experimental results were found to be in good agreement. The numerical simulation highlighted the variation of the meshing stiffness versus the setting of the ring (50%). This study shows that it is necessary to model the structure below the teeth correctly.
Acknowledgements This research work was carried out with the financial support of MESR and Eurocopter France. We also thank Mr. M. Atouf, M. Conte and C. Bard for their assistance.
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References [1] Oda S, Nagamura K, Aoki K. Root stresses of thin rimspur gears by finite element method. Bull ASME 1981;24(193):1273–80. [2] Li S. Gear contact model and loaded tooth analysis of a three-dimensional, thin-rimmed gear. J Mech Design 2002;124:511–7. [3] Drago RJ, Lutthans RV. Combined effects of rim thickness and pitch diameter on spur gear tooth stresses. J Am Helicopter Soc 1983;28(3):13–9. [4] Chong TH, Suzuki T, Aida T, Fujio H. Bending stresses of internal spur gear. Bull JSME 1982;25(202):679–86. [5] Von Eiff H, Hirschmann KH, Lechner G. Influence of gear tooth geometry on tooth stress of external and internal gears. J Mech Design Trans ASME 1990;112:575–83. [6] Atouf M. Etude du comportement m!ecanique des dentures de courroies synchrones. Application aux courroies de type HTD. Th"ese de Doctorat, Institut des Sciences Appliqu!ees de Lyon, No 92 ISAL 0063, 1992. 208p. [7] Cheung YK. Finite strip method in structure analysis. Oxford: Pergamon Press; 1976. 233p. ISBN 0-08-018308-5. [8] Olakorede AA. R!epartition des charges et r!esistance en conception d’engrenages cylindriques. Application de la m!ethode des Prismes Finis en C.A.O. Th"ese de Doctorat, Institut des Sciences Appliqu!ees de Lyon, No 90 ISAL 0049, 1990. 157p. [9] Kim HC. Etude des charges et des contraintes dans les engrenages cylindriques avec voile. Simulation num!erique par la M!ethode des Prismes Finis. Th"ese de Doctorat: Institut National des Sciences Apliqu!ees de Lyon, No 96 ISAL 0045, 1996. 152p. [10] Boussinesq J. Application des potentiels a" l’!etude de l’!equilibre et du mouvement des solides e! lastiques, Paris: Gauthiers-Villard; 1885. [11] Bard C. Mod!elisation du comportement dynamique des transmissions par engrenages. Th"ese de Doctorat, Institut des Sciences Appliqu!ees de Lyon, No 95 ISAL 0031, 1995. 296p. [12] De Vaujany JP. Comportement d’engrenage cylindrique int!erieur. Effets de jante et de voile. INSA de Lyon, No 96 ISAL 0128, 1996. 178p.
Optics and Lasers in Engineering 42 (2004) 447–459
EPSI validation for cylindrical gears Jean-Pierre de Vaujany*, Miche" le Guingand Laboratoire Dynamique des Machines et des Structures, INSA de Lyon, 20, avenue Albert Einstein, Villeurbanne cedex 69621, France Received 28 July 2003; accepted 30 December 2003
Abstract Shape deviations of spur gears obtained by using Electronic Speckle Pattern Interferometry are compared with the results of numerical simulation. The dual illumination method produces a speckle pattern by simultaneous illumination of a sample by two laser waves symmetrically in the direction of observation. Two images obtained, respectively, before and after deformation are subtracted to obtain the field of displacement. Furthermore, a study of meshing stiffness was performed for different types of setting for an internal gear. r 2004 Elsevier Ltd. All rights reserved. Keywords: Speckle video; Gears; Meshing stiffness
1. Introduction Gears used in aerospace applications must have very thin-walled structures to reduce flying weight. They must also be designed within a very short time. The design of power transmission devices such as cylindrical gears is based on numerical studies for modelling mechanical behaviour. Optimisations of the tooth profile are increasingly precise and concern very small values. Furthermore, small modifications of geometry involve considerable variations of load on the teeth. Different models have been developed to calculate load sharing. The validity of current models is based on the results of stress distribution in the tooth roots. Certain models make use of measurements with strain gauges [1,2]. The value of these deformations is also estimated with laser holographic interferometry [3–5]. On the other hand, little experimental work has been performed to verify the load sharing between the *Corresponding author. Tel.: +33-472-43-8546; fax: +33-472-43-8930. E-mail address: [email protected] (J.-P. de Vaujany). 0143-8166/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2003.12.005
ARTICLE IN PRESS 448
J.-P. de Vaujany, M. Guingand / Optics and Lasers in Engineering 42 (2004) 447–459
different teeth in contact, since it is difficult to measure this load distribution directly. However, indirect methods can be used. The aim of this work is to validate this load sharing by measuring the full displacement field on the teeth under load, taking into account experimental and numerical models. The speckle video method or Electronic speckle pattern interferometry (E.S.P.I) has been used for several right cylindrical gears (internal and external). This study compares the non-uniform strain fields measured to those obtained by numerical calculations (finite prism method). Then, using a numerical model, a study of the meshing stiffness as a function of the fastening of a ring will be presented and discussed.
2. Principle of the speckle measuring technique ESPI is a non-contact full field measurement system that measures the deformations under mechanical loads along three perpendicular axes (for a 3D system). Deformations can be measured on samples with relatively large dimensions and aspect ratios, such as test specimens and components. A speckle effect is produced by the interference between reflected rays when a surface is illuminated with a laser beam. The image of the object is recorded by a video camera. The light waves reflected from the object’s surface interfere with each other and produce a variable pattern known as speckle. The pattern is defined by the microscopically rough topography of the sample surface. Any deformation of the surface results in a change of the speckle pattern. This speckle pattern and deformed object state is stored by the image processor. When the sample deforms, the speckle pattern changes. Correlation fringes are created by comparing the speckle pattern for the object in a reference state with that for the object in a deformed state (Fig. 1). These fringes represent the deformation of the object’s surface in the direction concerned. Evaluation software is then used to analyse and count the correlation fringes and transform them into a quantitative set of deformation and strain data. The object studied is illuminated with two lasers. The dual illumination method produces a speckle pattern by simultaneous illumination of the sample by two laser waves symmetrically about the observation direction. The resulting speckle pattern represents the phase difference f between the two light paths from the laser via the object surface to the camera. This speckle pattern is stored as a reference. Movement
Speckle pattern of undeformed surface
Speckle pattern of deformed surface
Difference of the two speckle patterns
Fig. 1. Examples of speckle patterns.
ARTICLE IN PRESS J.-P. de Vaujany, M. Guingand / Optics and Lasers in Engineering 42 (2004) 447–459
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of the object produces a change in the two light paths relative to one another, creating a new phase relation f þ D: This creates a new speckle pattern. The difference D between both patterns is represented by correlation fringes. The deformation of the object’s surface is obtained by counting the number of fringes at every object point. The direction of measurement is orthogonal to the viewing direction and is produced in the plane of the two illumination directions. To obtain the field of displacement, two images obtained, respectively, before and after deformation are subtracted from each other. After this operation, the resulting image shows variations of luminous intensity as a function of phase differences between the two signals subtracted. This difference of phase is given by: 2p Dj ¼ 2u cosðyÞ; ð1Þ l where l is wavelength of the laser light, u is displacement following a direction X on the surface of the object, y is angle of impact between the axis X and the direction of the laser. When this difference of phase is equal to 2np; the two waves are in phase and the luminous intensity is maximal. For a difference of phase of ð2n þ 1Þp; the waves are out of phase and the intensity is minimal. The final image is characterized by a representative fringe system of curves of displacement following the direction chosen for the study. The clear fringe equation is nl l u¼ and the interfringe is : d ¼ : ð2Þ 2 cosðyÞ 2 cosðyÞ In particular, the fringe of order n ¼ 0; associated with a null displacement, is theoretically a white fringe. Due to the subtraction introduced during numerical image processing, the fringe from order n ¼ 0 (noted Ro) is associated with a black fringe for the experimental results presented here. On the other hand, the intermediate displacements are represented by clear furrows. The images resulting from these operations contain intrinsic noise due to the E.S.P.I. technique. Several numerical contrast and filtering improvement operations are necessary to obtain the best possible quality images of the fringe system [6].
3. Experimental test 3.1. Geometrical characteristics of the gears The study was carried out with right teeth for external and internal gears and concerns normalized teeth. Their geometrical characteristics are given in Tables 1 and 2. Different criteria were chosen to design the gears: *
contact ratio from 2 to 3 so that the load is always shared between at least two couples of teeth,
ARTICLE IN PRESS 450
J.-P. de Vaujany, M. Guingand / Optics and Lasers in Engineering 42 (2004) 447–459
Table 1 Geometrical characteristics of the external gear
Module (mm) Pressure angle ( ) Face width (mm) Number of teeth
Pinion
Gear
1.5 15.0 10.0 58
1.5 15.0 10.0 62
Pinion
Gear
1.5 15.0 10.0 60
1.5 15.0 10.0 160
Table 2 Geometrical characteristics of the internal gear
Module (mm) Pressure angle ( ) Face width (mm) Number of teeth
*
*
*
a small module to ensure that at least two teeth are systematically visible in the field of the camera, good quality gear (Agma 12) so as to minimize the effects of errors (pitch error, etc.), a small pressure angle to obtain mainly flexion deformations. The latter condition allows obtaining displacement in the circumferential directions (X ) greater than displacement in the perpendicular directions (Y ). These Y displacements are thereafter considered as negligible.
3.2. Experimentation The architecture of the optical setting is represented in Fig. 2. Two configurations of this setting were used. In the first, an optical device performs the measurements in two orthogonal directions (X and Y ; Fig. 5). This allows successive recording of systems of fringes associated with these two directions without modifying the positioning of the teeth or the loading. This choice required creating four distinct optical paths with coherent beams of symmetrical illumination on the surfaces studied. The resolution of displacement X and Y increased when angle y (Fig. 2) decreased. The average values of y are equal to 20 . A binocular was introduced in the acquisition system because the zone studied was very small. Therefore it was necessary to increase the resulting image resolution. The setting to position and load the gears is presented in Fig. 3. It intended to have the following characteristics: *
great rigidity of the system as a function of the teeth,
ARTICLE IN PRESS J.-P. de Vaujany, M. Guingand / Optics and Lasers in Engineering 42 (2004) 447–459
θ
451
7
θ
5
4 6
Binocular Camera CCD Video monitor
3
2
1
1 : Laser 2 : Obturator 3 : Space filter 4 : Lens 5 , 6 : Mirors 7 : Gears
Computer
Fig. 2. Optical acquisition diagram of images in speckle video [6].
Fig. 3. Diagram of gear setting. *
*
the possibility to modify the relative positions of the gear (angular position and centre-to-centre distance), the possibility of studying external and internal gears.
4. Numerical model The calculations were performed as a function of the geometrical characteristics and loading conditions. For 5 teeth of the pinion and the gear, the simulation determines the potential contact points on the flanks for several kinematical positions (Fig. 4). The finite prism method (FPM) provides information on elastic
ARTICLE IN PRESS 452
J.-P. de Vaujany, M. Guingand / Optics and Lasers in Engineering 42 (2004) 447–459
Fig. 4. Kinematical positions.
behaviour. It can be considered as a special form of finite elements. The finite element method (FEM) uses polynomial functions of displacement in all directions, whereas the FPM uses polynomial functions solely in two directions and series functions in the third direction. This method, developed by Cheung [7], was adapted to the analysis of the elastic behaviour of cylindrical gears by Olakorede [8] and Kim [9]. The advantage of this method in comparison to the FEM method is that it decreases calculation times and storage space. Furthermore, in the model the surface effects (contact deformations) are computed using Boussinesq’s method [10]. By knowing the coefficients of structure, surface effects and initial gaps between each couple of teeth, it is possible to determine the contact pressure distribution on each tooth by resolving the deformation compatibility relationship [11,12]. The loading corresponding to each meshing position is applied on the flank of the teeth and the displacements are calculated in the three directions. For our study, the comparison between the displacements calculated numerically and those of the experiment are performed mainly for displacements in direction X :
5. Experimental results 5.1. External gear For the case of the external gear, Fig. 5 presents the experimental displacement field obtained in the first measurement context (normal angle of impact y) and the numerical results obtained with the model. The test and calculation parameters are the following: * * * *
interference fringe in direction X: 0.674 mm interference fringe in direction Y: 0.778 mm Torque: 57.8 N m Young’s module: 2.1E+11 N/m2 and Poisson’s ratio: 0.3
The results obtained results were in good agreement, especially following X : These results show that the deformation occurs mainly close to X ; which is logical, considering the small pressure angle (15 ). Extremely low values of displacement
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Fig. 5. Interference fringe patterns (computed and measured), First measurement context.
in Y ; as well as the utilization of a substantial interference fringe can explain in part the small differences in this direction. Nevertheless, the appearance of the field of displacement (completed by lines of a half interfringe and -dotted curves) correlates with the experimental results and the number of fringes obtained on the most heavily loaded tooth is correct. In order to evaluate these initial results, we analysed the progression of the field of displacement (direction X) for different meshing positions. These positions were chosen to visualise the entry of a new tooth in the contact. This study was carried out in a second measurement context with a narrower angle of impact (y) and thus with higher resolution.
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The test and calculation parameters in this measurement configuration are: * * * *
interference fringe in direction X: 0.358 mm the displacement field in direction Y is not studied because the values are low, torque: 57.8 N m Young’s module: 2.1E+11 N/m2 and Poisson’s ratio: 0.3
Fig. 6(b) shows the progression of the field of displacement (direction X) calculated for 4 successive meshing positions. This simulation is associated with the entry of tooth no. 4 in the contact. Fig. 6(a) shows experimental displacement fields obtained in the same context, in particular, the progression of the field of
Fig. 6. Interference fringe patterns in direction X (interference fringe=0.358 mm). External gear (4 meshing positions).
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displacement during the entry of a tooth in the contact is visualized in the three first photographs. Very good agreement is obtained between the experimental and numerical results. The choice of analysis positions and the imperfections of the real teeth are sufficient to explain the very small differences observed in these different figures. 5.2. Internal gear The parameters for measuring interference fringe patterns were identical to those used for the external gear. As in the case of the external gear, the values of displacement in direction Y were low and the interference fringe used is large. These two points can justify in part the differences obtained between the experimental and numerical results in direction Y : The deformation of the structure is mainly in direction X : Fig. 7 shows the progression of the interference fringe patterns in direction X measured and calculated for four meshing positions. There is a very good agreement between the experimental results in Fig. 7(a) and numerical results in Fig. 7(b).
6. Variation of meshing stiffness depending on the setting Changing the number of teeth in contact modifies the progression of the meshing stiffness. This variation is a source of internal excitation that, due to its fluctuations, produces vibrations and noise. For example, in the case of internal gears load sharing is shown in Fig. 8. The geometrical characteristics of the gear studied are presented in Table 3. The study was carried out with several torques of 300 with 2400 N m. For one meshing period (total rotation of 0.12 rad with 0.28 rad), the division of the load is represented in Fig. 8 for the 3 torques: 300, 1200 and 2400 N m. We note that the pace of progression of tooth loading differs according to loading case. For the lowest torque, the progression is broken down into crenels, contrary to the loaded case where the variation is more gradual. This phenomenon occurs with the deflection of the teeth and the rotation of the base of the teeth (rim). Naturally, these displacements are more considerable for a high loading. They make it possible for the teeth to return into contact earlier and ensure more continuous contact through time. Thus the period during which only two teeth are in contact is shorter (Fig. 8: total rotation between 0.19 and 0.2 rad is approximately 6% of the meshing period). An internal gear can be fixed to the crankcase with the help of bolts or pins around the contour of the rim. Following the gap between each fixing point, the planet moves successively opposite and between these points. In the case of a thin rim thickness, the deformation of the structure is influenced by the bolt or pin arrangement. To quantify the phenomenon, the meshing stiffness was calculated for different types of setting (Figs. 9 and 10). The torque used for the calculations was 1200 N m.
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Fig. 7. Interference fringe patterns in direction X (interference fringe=0.358 mm). Internal gear (4 meshing positions).
With only one central fixing point (Fig. 9(b)), the value of the meshing stiffness was located between those calculated for the two conditions: free rim (Fig. 9(a)) and completely fixed rim (Fig. 9(f)). The more the number of fixing points the more the progression of stiffness resembles the case of a rim completely blocked at its edges. The average stiffness increases with the number of fixing points except in the case of a meshing centred between two fixing points (Fig. 9(c)). In this case, the average rigidity (1.2E+6 N/mm) is weaker than the case of only one central fixing point (Fig. 9(b) and Fig. 10: 1.35E+06 N/mm), because the rim is not blocked on the side
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tooth 2
tooth 3
tooth 4
tooth 5 Torque = 2400 N.m
12000
2500
10000
2000
Load (N)
Load (N)
Torque = 300 N.m 3000
1500 1000
8000 6000 4000
500
2000
0 0.10
457
0.15
0.20
0.25
0 0.10
0.30
Pinion angular position (rd)
0.15
0.20
0.25
0.30
Pinion angular position (rd)
Torque = 1200 N.m 12000
Load (N)
10000 8000 6000 4000 2000 0 0.10
0.15
0.20
0.25
0.30
Pinion angular position (rd) Fig. 8. Load sharing.
Table 3 Gear characteristics and torque condition
Number of teeth Module (mm) Pressure angle ( ) Face width (mm) Rim thickness (mm) Torque (N m)
Pinion
Ring
51 2.53 20 58 5 module 1200
195 2.53 20 58 4module
of the meshing zone and the deformations are greater. The deformation of the rim depends on the position of the planets next to the fixing elements and can vary by approximately 50%. It is therefore necessary to model the setting very precisely to obtain a representative stiffness of the gear’s mechanical behaviour.
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Fig. 9. Definition of fixing points.
Meshing stiffness (N/mm)
1.80E+06 1.60E+06 1.40E+06 free 1 fixing 2 fixings
1.20E+06 1.00E+06 8.00E+05
3 fixings 5 fixings clamping
6.00E+05 4.00E+05 2.00E+05 0.00E+00 0.10
0.15
0.20
0.25
0.30
Pinion angular position (rad)
Fig. 10. Meshing stiffness dependent on the rim fixing.
7. Conclusion Experimental simulation of the behaviour of gears with deformation gauges in the tooth roots are very difficult to carry out. The E.S.P.I method was used because it allows visualizing the tooth deflection with precision. By using a small module, the displacement fields are obtained on several teeth, thus making it possible to verify the numerical results of the load sharing. For external and internal gears, the comparisons between the numerical and experimental results were found to be in good agreement. The numerical simulation highlighted the variation of the meshing stiffness versus the setting of the ring (50%). This study shows that it is necessary to model the structure below the teeth correctly.
Acknowledgements This research work was carried out with the financial support of MESR and Eurocopter France. We also thank Mr. M. Atouf, M. Conte and C. Bard for their assistance.
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