Worm gear meshing stiffness calculations based on experiments

Worm gear meshing stiffness calculations based on experiments X. Wang & L. Morrish School of Computing and Engineering, Hudders-eld University, UK. Abstract Meshing stiffness is an important factor used in modelling of gear transmission errors (TE) and wear, as load sharing has to be taken into account. However, as far as the authors are aware, there is no published information that can be used to calculate tooth stiffness for a particular worm gear pair. This paper presents a method for calculating worm gear meshmg stiffness based on experimental measurements. The resulting stiffness value compare reasonably well with the estimations htherto made by analogy with spurlhelical gears. 1 Introduction Worm gears are widely used in various industrial and every day life applications because of their compactness, high precision, smoothness in operation. However, worm gears efficiency and wear resistance are normally lower than these of other types of gears of similar size, due to high sliding inherently present in worm gear contact. Research into worm gear wear is going on worldwide, which is mainly experimental (Hoehn [l], Octrue [2], Octrue [3], Houser [4], Wang [5]). Based on the experimental results, a method for overall worm gear wear estimation has been developed within the DIN 3996 and IS0 [6] standards. However, there is a need for modelling of progressive wear. This will enable predictions of progressive changes of gear geometry of given design, as well as

372 Laser Metrology und Muchiize Performance VI predictions of gears life length. The model will take into account different meshing conditions (e.g. loads, materials, oils etc.), Egorov [7], Wang [S]. Load sharing is an important factor whch influences wear distribution on tooth surface of worm gears. It has to be taken into account in wear calculations, and the value of tooth stiffness is required to do so. As far as the authors are aware, there is no published information on how tooth stiffness for worm gears can be calculated, so an estimate has to be made. It seems reasonable to base this estimate on similar stiffness values for spurhelical gears, using one of the available methods (e.g. IS0 [9], Munro [10], Weber [l l], Steward [12]) and this is the approach used by Fish [13]. This paper presents a method for calculating worm gear meshng stiffness, which is believed to be the first publication on ths subject. It is based on experimental measurements made on the worm gear test rig at Huddersfield. The resulting stiffness value is compared with the estimation made by Fish [13]. 2 Test rig and test gears The test rig used for this research is described in detail in Fish [13]-[15]. It provides means for running worm gears under loads ranging from zero up to 2000 Nm, applied via hydraulic brake, and at speeds ranging from 100 up to 200 rpm. Two Heidenhain encoders are fitted on the worm and wheel shafts respectively. Their readings are processed by a GFM GP36 computer system to continuously monitor the gear transmission error (TE). Details of the worm and wheel used in the tests are as follows: Module Number of worm starts Number of teeth Pressure angle Worm reference diameter Reference circle diameter of wheel Centre distance Contact ratio Face width of wheel 3 Measurements description Table 1 lists parameters of the test gearbox and transmission parameters measured during the tests. Figure 1 shows positions of encoders and probes used for these measurements.

Laser Metrology and Machine Performance VI 373 Table 1 : List of measured parameters. I I I Measured bv I Notation 0 lwm 9:' yl X2 X3 X4 X5 Description input (worm) shaft angular position Subscript i in Table 1 denotes the measurement number made at different worm positions. The procedure of measurements is as follows. Initially, worm is positioned as convenient, and its shaft angular position is taken as a datum (erm= 0). Then the worm shaft is rotated manually and its position is registered. Corresponding angular positions of the wheel and probes readings are also recorded. encoder 1 encoder 2 - output (wheel) shaft angular position worm deflection axial worm shaft movement worm structure movement relative to ground wheel box top part movement relative to ground wheel deflection Probes (see Figure i) encoder 1 encoder 2 probe 1 probe 2 probe 3 probe 4 probe 5 O - O Figure 1 : Encoders and probes positions on the test gear box.

374 Laser Metrology und Muchiize Performance VI The procedure is repeated twice, first under no load (when no deflections occur), and then under some known heavy (750Nm) load (load is applied by mounting a lever arm to the coupling of the output shaft, in horizontal position, with known weights attached to it). Two sets of output shaft angular positions, f3,wio and g,":,,,, are recorded. Gear box parts movements and worm and wheel shafts deflections under load have to be taken into account in calculation. These are measured using probes 1-5. The readings of probes 1, 3-5 hardly change at different worn1 positions; they are given in Table 2. Table 2: Readings of the probes under 750Nm. I Probes I 1 l 3 I 4 l 5 I symbol Readings (pm) 4.5 X3-12.5 X4-7.0 Xj -3.0 Readings of the probe 2 (axial worm movement), as well as angular positions of the worm and wheel shafts measured by encoders 1 and 2, are given in Table 3. position No Table 3 : Measured parameters. Worm angle 8Ym (Degree) 0 15 30 Wheel angle (Preload) 'i wl no load (Degree) 0.0000 0.3015 0.6015 1.201 8 1.8023 2.4026 3.0029 3.6025 4.2015 4.5008 4.8000 5.3993 5.9993 6.6000 6.8992 7.201 1 Wheel Axial worm angle movement 1 (75ONm) I e,w:,ac, X, 2 (Degree) (pm) -0.0171-5.5 0.2829-7.5 0.5830-6.5 1.1832-7.0 1.7837-7.5 Measured data from Tables 2 and 3 are used in further theoretical analysis.

Laser Metrology and Machine Performance VI 375 The accuracy of measurements recorded when the input shaft is rotated incrementally and manually has to be proved. This is done by measuring TE when the shaft is rotated uniformly by a motor, and comparing the measured TE with that calculated using the data in Table 3. TE can be calculated as follows. In the case of zero load: TE = In the case when load (750Nm) is applied: TE = Here: u - transmission ratio (50) - initial wheel shaft angular position (-0.01 7 1 O) (All angles in eqns (1) and (2) are in degrees). TE calculated using formulae (1) and (2) are plotted in Figures 2a and 3a respectively. Measured TE under no load and 750 Nm load are given in Figures 2b and 3b. A comparison of measured and calculated TE shows that manual measurements used for the principle stiffness calculations are sound. 4 Calculation of tooth deflection and meshing stiffness 4.1 Tooth deflection Wheel tooth deflection, A:, worm position i as follows: under 750 Nm load can be calculated for every Here: r, - reference radius of wheel G,- - gap increase due to deflection of worm shaft at middle section, Y,, and radial movement of wheel in vertical direction, ywh. G,"" and ywh can be calculated:

376 Laser Metrology und Muchiize Performance VI 1 TE (no load)... l 1-5 0 1... l ' Angle of worm rotatlon h Zoom of Commsite Transmission Errors I FR 03/27/82 11:28:27 1-5.0! 0.000.005.B10.@l5.B20 Revolutions (DriuL*l Gear) F'1 = 8.9 um l - 1 I - Figure 2: Transmission error under no load, (a) - worm shaft is rotated manually, (b) - worm shaft is rotated by a motor. Glwm =(l; +ywh)tana= 1; +- 3 tann Here: a - pressure angle P - radial load (P = 2 184 N) k - radial stiffness of wheel bearings (two bearings are assumed to have the same stiffness, k =l640 Nlpm). Radial load, P, is found from the known applied load, 750 Nm, and gear geometry. The value for wheel bearing stiffness has been evaluated by Fish 1131.

Laser Metrology alzd Machine Performance VI 377 Angle of worm rotatlon l l a Zoom of Composite Transmission Errors FR 63/27/92 14:54:21 5.0 T Revolutions (Dr~uerf Gear) 4.4 urn l I I - Figure 3 : Transmission error under 750Nm load, (c) - worm shaft is rotated manually, (d) - worm shaft is rotated by a motor. Calculated wheel tooth deflection, A: ; under 750 Nm load is plotted in Figure 4. Slight changes occur with worm rotation due to various factors, e.g. load sharing and the specific geometry of teeth contacting area. The average tooth deflection is 23 microns. 4.2 Meshing stiffness Meshing stiffness can be calculated as:

378 Laser Metrology und Muchiize Performance VI j 26 l I Tooth deflection (750Nm) 90 180 270 360 Angle of worn rotation Figure 4: Tooth deflection under 750 Nm. Here: T, - applied load (T, =750 Nm) b - wheel facewidth (b = 35.56 mm) - A - mean tooth deflection Mean tooth deflection can be calculated using eqn (7): It is found (section 4.1) to be = 23.0 pm. From eqn (6), meshing stiffness is found to be 7.3 N/mm/p. This figure is very similar to 5.9 Nlmmlp originally used by Fish, [13], which was based on experience and analogy with spur gears Munro [10], and therefore proves that the method presented is sound. However, more investigations are desirable to provide further verification. Ths will be achieved from future research, by modelling, implementing tests and possibly by the use of FEA. 5 Conclusions Meshing stiffness of a particular worm gear pair has been calculated using test box deflection measurements and theoretical considerations. The results are qualitatively reasonable. They do not conflict with previous estimates produced (Fish 1131). However, further verification is required. Acknowledgements The authors would llke to thank for financial support and in-kind assistance:

Laser Metrology and Machine Performance VI 379 Engineering and Physical Sciences Research Council, DSTL British Gear Association, Holroyd Machine Tool, Rotors, & Precision Gears Ltd., Renold Gears Ltd., David Brown Ltd., The University of Huddersfield References Hoehn, B.-R., Neupert, K. & Steingroever, K. "Wear Load Capacity and Efficiency of Worm Gears", VD1 Berichte, No. 1230, pp.409-425, 1996. Octrue, M. "Relationshp between wear and pitting phenomena in worm gears". American Gear Manufacturers Association, paper 97FTM9, 1997. Octrue, M., "Evolution des methodes de calcul de la capacite de charge des engrenages a r'oue et vis tangentes pression et usure", 4th World Congress on Gearing and Power Transmission, Paris, France, 1, pp. 405-417, 1999. Houser, D. R., Su, X. & Vaishya, M. "Effects of wear on the meshing contact of worm gearing", AGMA 99FTM18,1999. Wang, X. & Morrish, L. "Transmission error and wear measurement of involute helicoidal worm gears", IFToMM, Int. J. of Gear and Transmissions, 3, pp. 77-82,2001. ISOITC 60lSC 11GT 7N 201, "Load capacity calculation of worm gears", 2001 (Preparatory). Egorov, I.M. & Morrish, L. "Digital Approach For The Solution Of Gearing Problems", the paper accepter for presentation at the Power Transmission and Gearing conference, USA, 2003. Wang, X. & Morrish, L. "Predictions of Wear and Transmission Errors of Cylindrical Worm Gears", the paper accepter for presentation at the Power Transmission and Gearing conference, USA, 2003. IS0 6336-1, ISOITC-60,-"~alculation of load capacity of spur and helical gears", Part 1, 14'~ April 1994. [l01 Munro, R.G., Palmer, D. & Morrish, L. "An experimental method to measure gear tooth stiffness through and beyond path of contact", proceeding of IMechE, part C, Vol. 215, pp. 793-803,2001. [l l] Weber, C. Sponsored research (Germany), DSIR London, Report 3, 1949. [l21 Steward, J.H. "The compliance of solid, wide-faced spur gears". Trans. ASME, J. Mech. Des, 112, pp.590-595, 1990. [l31 Fish, M. "Transmission errors in precision worm gear drives", PhD thesis, University of Huddersfield, 1998. [l41 Fish, M. & Munro, R. G. "Analysis of Marking Patterns and Transmission Errors in Worm Gears", Annual Congress of British Gear Association, 1995. [l51 Fish, M. & Munro, R. G. "Kinematic Errors in Precision Worm Gears", LAMDAMAP Conference, pp. 283-292, 1997.