A novel approach to the inspection of gears with a co-ordinate measuring machine - theoretical aspects

Transcription

1 A novel approach to the inspection of gears with a co-ordinate measuring machine - theoretical aspects C.H. Gao, K. Cheng, D.K. Harrison Department of Engineering, Glasgow Caledonian University Cowcaddens Road, Glasgow G4 DBA, UK k.cheng@gcal.ac.uk Abstract The accuracy of gears is a key to the performance of a mechanical transmission system. In this paper a novel approach is proposed for inspecting gear profiles with a Co-ordinate Measuring Machine (CMM). The gear profile inspection models are described with a set of equations. The theoretical aspects of the models are explored with respect to the minimum number of inspection points, profile evaluation method and the factors influencing the inspection accuracy. The co-ordinates of a certain number of points on the gear tooth surface are measured and a theoretical gear tooth surface expressed as a function of the CMM co-ordinate system is estimated by the method of least squares, so that the surface fits the measured data. The gear profile models and their simulations are implemented via MATLAB. 1. Introduction Gear inspection is often the bottleneck of the gear production process. The introduction of special gear measuring equipment provides efficiency for gear inspection. However, the equipment available normally has difficulty in inspecting high precision gears because of the difficulty in obtaining higher precision master gears. The larger the gear size, the more difficult it maybe. Although there are available commercially specific CNC gear measurement machines, these are expensive for general gear manufacturing SMEs.

2 228 Laser Metrology and Machine Performance Direct coordinate measurement of points along the gear teeth profiles is the most accurate means of inspecting the profiles. However, it is a very tedious, exacting, and slow process requiring delicate mechanical measurement of angle and distance, thus limiting the technique to special inspections rather than routine production use. Therefore, a feasible production inspection approach is essential and much needed for gear manufacturing SMEs. In this paper a novel approach to spur/helical gear inspection is proposed. The approach aims to be production feasible and easy to operate. The work presented is focused on the theoretical aspects of the approach, and is part of a research project being undertaken at Glasgow Caledonian University. 2. The equations of the helical gear teeth surfaces 2.1 The equations of the involute planar curve a) The current point M of an involute curve is determined by the following vector equation as shown in Figure 1:*'^ where, (1) OP = r (2) (3) b) Due to rolling without sliding, there is PM = arc(pn) = r^o Here 0 is the angle of rotation in the rolling motion. c) Equations (1) to (4) yield: (4) Involute I Involute II Figure 1: Planar involute curves

3 Laser Metrology and Machine Performance 229 OM = [x y sin(0 0 (5) = r. [cos( sin( 0 sin( 0 + q>)-0 cos( 0 + Using this approach, the following equations for a planar involute can be obtained: x = >; = ±r^ [sin(6> + p)- The upper and lower signs correspond to involute I and involute II, respectively. 2.2 The helical gear tooth surface equations A helical gear tooth surface is generated by an involute curve that performs a screw motion. The position vector OM for a current point of the surface side I as shown in Figure 2 (a) for a right-hand helical gear is expressed as: OM = (7) II (a) ^ (b) Figure 2: A tooth surface of a helical gear The approach used to obtain the solution is similar to that described in section 2.1, and shown in Figure 2 (b)/ (pg is the base helix angle), so there are: 00, =[0 0 *I ]' L tan /?,, O.M = r, [cos( 0 + cp + <f>) + 0 sin( 0 + (p + ( sin( 0 + cp Based on equations (7) to (9), there are: (8) \-0 cos(0 + cp + <j)) 0]^ (9)

4 230 Laser Metrology and Machine Performance OM = /v [cos( 0 + <p + <f>) + 6 sin( 0 4- ^ 4- $)] sin( #4-^4-^)- (10) 4* w ) f* i (p I tan $ 1 o i? do Where ZQ, nin and a«correspond to the number of the teeth, module and the pressure angle. Similarly to the above, the equations of the helical gear tooth surface are expressed as below: x = r, [cos(0 4- cp + (/>) + 6 sin(0 + (p + </>)] y = ±rfr[sm(0 + (p + (l>)-ocqs(9 + (p + <l>)] (12) The upper and lower signs correspond to the tooth side surface I and side surface II on the right-hand helical gear as shown in Figure 2 (a). The vector equation for a helical gear tooth surface is: = n [cos(# + y> + </>) + & sm(0 4- (p 4- </>)] / ± r, k The surface unit normal is represented as:*'^ where: dr dr - {r^[cos(0 + <p + </>) + & s'm(0 + <p + </)}] i± r,[sm(0 + (p + </>)- 0 cos( 9 4- (p + 0)] j + (r(j> I tanfi^ ) k} x - (r [cos( 6> + <p -f <zj) + 6> sin( 0 ^ + </>)] i ± r^ [sin( 0 + ^? + 0) - 0 cos( 0 4- ^ + ^)] j + (r^<j) I tan /3^)^k] = ±A ^ 0sin(0 + ^4-^)/tan /?^ / -r^b cos(0 + <p 4- </>) I tanfi^ j±r^0»k \N \= sqrt {[±r^0sm( 0 + <p + f ) / tan ^ ]2 + (16) [-r^5icos( 0 + ^ + ^ ) / tan /? ^ + (±^0)2 ^ Equations (14) to (16) yield the following surface unit normal of the helical gear tooth:

5 Laser Metrology and Machine Performance 231 *k (17) The upper and lower sign (Equations (13) - (17)) correspond to side surface I of the right-hand helical gear (its angles 9, cp and < > are measured counterclockwise) and side surface II (its angles 6, cp and are measured clockwise) as illustrated Figure 2 (a). % is the angle of rotation about the z axis when the j-th tooth is, located at the position of the first tooth. %=2%(j-l)/zo+(po (18) Where ZQ is the gear teeth number. So using cpj and (12) (17), the equations of j-th tooth surface will be obtained. The equations of the left-hand gear teeth surface can be derived in the same way. 3. The principle of helical gear measurement Using a CMM, a set of measuring data of a point M (x^yi,z,) on the helical gear tooth surface can be easily sampled/ Equations (12) provide a foundation for obtaining the angles 81, cpi, and \. x,=tb[cos y,=ib[sin ( z, =r So they can be derived as: =arcsin( On the other hand, supposing the start angle is: cp=(po, solving equations (12) and (20) yields the new set of values of the same point: Xn yn, z\\ The error of this point is: Vyi=yn_yi (21) Vz,=z,i-z,

6 232 Laser Metrology and Machine Performance Using the method of least squares (Newton method in MATLAB)/' * ^ the optimum value of the supposed start angle <po of this point, 901 will be obtained. So the error of the start angle on this point will be: (Dm - (Di \ *-) The errors of other points on the gear tooth surface can be determined by using the same method. 4. The coordinate transformation and its errors In the above sections, only the gear itself is considered. It will be set arbitrarily when the gear is measured on a CMM. The co-ordinate system of the CMM is: Oc-XcVcZc, and the 2* axis is vertical (see Figure 3). The coordinate system of the gear is: Og-XgVgZg, and the Zg axis is vertical too. Its origin is Og(x^, Vgc, ZgJ. The point A (Xac, yac, Zac) is the centre of the probe, the point B is be measured one which is at gear tooth surface. The vector O%B is represented thus: (23) where: (24) Equations (23) to (24) yield: Xbg ~ Xac " Xgc - fyngx (25) Figure 3: The tooth surface measurement

7 Laser Metrology and Machine Performance 233 Where TQ is the radius of the probe. In the period of coordinate transformation, the errors will be produced/' * Using the method of least squares, the errors (including the effect of the coordinate axis rotating and parallel moving) of the coordinate transformation (Vxgc, Vyg<,, VZgJ can be determined. Under this transformation, Equations (25) are expressed in the following form: = Yac - Ygc - rongy - Vy^ (26) 5. Experiment method 5.1 pitch error Using the way described in section 3 ~ 4, and the measured data, the pitch error can be determined. When the radius of the measurement point is equal to the pitch circle radius r (using linear interpolation and least squares to obtain the Vcpj on the pitch circle), the error is pitch error. So the single pitch error is: The cumulative pitch error is: 5.2 helix angle error (27) x - jmin (28) One of the gear tooth surfaces is selected, and the co-ordinates of many points on the tooth surface are measured at random. Use equations (1 1), (12), and the way above the helix angle error may be obtained. 5.3 profile error Profile error of helical gear is usually inspected on the plane perpendicular to the axis. In measuring, the movement of probe is fixed in the direction of the z^ axis. Then the data of many points are measured from gear root to tip along the direction of profile. The profile error is the difference between the real profile and theoretical involute helicoid, which is related with the determined (% and pg The tooth profile diagram may be plotted, the profile error may be obtained using the way above described.

8 234 Laser Metrology and Machine Performance 6. Gear inspection and its simulation 6.1 The gear inspection model The gear is fixed arbitrarily on a CMM table. Using the CMM, the data (x,, y;, zj of any points on the gear tooth surface may be obtained. Considering the effecting factors, such as the CMM accuracy, the probe position and its radius, and the transformation error between the CMM and the gear measurement coordinate system, the gear errors will be determined, and this is illustrated in the flow chart in Figure 4/' * * Figure 4 shows a gear inspection and measurement model which is still under implementation. Figure 4: A flow chart of the gear inspection model 6.2 The gear inspection simulation model Using the equations (12) and (18) and the measurement data obtained on the CMM, the equations of the helical gear teeth surface can be determined. These exercises have been simulated with MATLAB which is a powerful engineering simulation tool and the approach is shown in Figure 5.** ** *' * The measurement of pitch error, helix angle error, and the profile errors for a helical gear were primarily simulated. The results are very promising. The authors will present this work in a future paper.

9 Laser Metrology and Machine Performance 235 Estimate Equations V Plot the gear model (in MATLAB) Gear Error Results Pitch errors Helix angle errors Profile errors Figure 5: A flow chart of the gear inspection simulation model 7. Concluding remarks In this paper, the equations of the helical gear inspection and some associated issues are discussed. Using the principle and inspection models described, the gear errors can be obtained on a CMM. The simulation and preliminary trials show the approach is feasible and are very promising for industrial usage. 8. Acknowledgements The authors particularly thank Mr Ian Hamilton at Glasgow Caledonian University for his kind assistance in this research work. 9. References 1. Litvin, F.L., Gear Geometry and Applied Theory, Prentice-Hall Inc, Englewood cliffs, New Jersey, Litvin, F.L., Theory of Gearing, NASA Reference Publication 1212, 1989.

10 236 Laser Metrology and Machine Performance 3. Max, F., Handbook of Mathematical, Scientific, and Engineering Formulas, Tables, Functions, Graphs, Transforms, 2nd edition, Research and Education Association, New York, Dudley, T.D.P., Gear Handbook Second Edition, McGraw-Hill, London, Graham C.G., Dynamic System Identification: Experiment Design and Data Analysis, Academic Press, New York, Biran A. and Breiner ML, MATLAEfor Engineers, Addidon Wesley, Harlow, England, Gear Design - Manufacturing and Inspection Manual, AE-15, Society of Automotive Engineers, Inc, Authony, D. M., Engineering Metrology, Pergamon Press, Oxford, International Standard (ISO1328 &1122) and British Standard (BS ISO 10064).