Collision detection between toolholder and workpiece on ball nut grinding
György Hegedűs University of Miskolc, Department of Machine Tools Hungary hegedus.gyorgy@uni-miskolc.hu
Introduction
This application presents numerical methods for the determination of collision detection of toolholder (quill) and workpiece on ball nut grinding. Beside the collision detection the method is capable of the determination of proper grinding angle with the prescribed safety gap between the toolholder and workpiece. The applied Newton-Raphson and Broyden numerical algorithms were executed and compared to each other. Note: For viewing the inputs go to View-->Show/Hide Contents...-->Select Inputs. For viewing the procedure for examples go to Edit-->Startup Code.
Overview of the problem
Gothic-arc profile ballscrew motion transforming mechanisms are widely used in machine tools and the demand for high-lead ballscrews is increasing due to the high-speed manufacturing. The gothic arc is a symmetrical combined curve of two arcs with equal radius and distance between their centres. These types of ballscrews are manufactured by form grinding, where the grinding tool has corresponding profile [6], ultra-precision ballscrews sometimes lapped after grinding process [4], [5]. In case of long and high lead threaded ball nut the grinding wheel is not tilted at the lead angle of the thread to avoid the collision between the quill and workpiece. Fig. 1 shows the real manufacturing process on conventional thread grinding machine.
Fig. 1. Ball nut thread grinding on conventional machine
Due to these conditions the profile obtained is not gothic-arc, because the grinding wheel tends to overcut the thread surface. The problem is well known in gear and worm manufacturing, different methods had been worked out to solve this task [3], [8], [2]. In case of long threaded ball nut the setting of optimum tilt angle is not possible due to the collision of quill and workpiece. This angle parameter has to be determined for the real manufacturing process. In this paper a numerical method will be presented for the determination of grinding wheel tilt angle on cylindrical and conical toolholders.
Collision detection between quill and ball nut
Collision detection between cylindrical bodies is widely used in three dimensional mechanical systems, for example machine tools, robots, different mechanisms. There are different software package for collision detection, for example I-Collide, V-Collide, Rapid, Solid. In this work collision detection is only determined between cylindrical-cylindrical and conical-cylindrical bodies without using of third party developed software. In case of the ball nut and the quill the determination of collision is equivalent with a minimum distance computation between cylinders or conical and cylindrical surfaces. Detecting of collision between cylindrical rigid bodies were developed using line geometry by Ketchel and Larochelle [7]. Distance computation between cylinders has four different types according to their three dimensional positions in space. Fast and accurate computation method was developed by Vranek [11]. To determine the maximum tilt angle for the grinding the minimum distance determination is required between the tilted quill axis and the edge of the ball nut represented as a circle (see Fig. 2., where is tilt angle, ϕ lead angle – optimal tilt angle – of the ball nut, length of the ball nut, pitch of the ball nut).
Fig. 2. Schematic figure of collision between cylindrical quill and ball nut
Collision detection between cylindrical quill and ball nut
Determination of minimum distance between the quill and the ball nut is equivalent with the computation of the distance between the quill axis and the circular edge of the ball nut.
Fig. 3. Spatial position of tool-workpiece
Applying notations of Fig. 3. the circle equation described by
where 2[0, 2π], is point of the circle, C is centre of the circle, is diameter of the quill and u and v are unit vectors in the plane containing the circle. The minimum distance between the quill axis and the circular edge is
where is a point on the quill axis and Q is the projection of on the circle plane. Applying the expressions from [10] a nonlinear equation system can be formulated for the unknown parameters. The equation for the minimum distance between the quill and the edge of the ball nut using the notations of Fig. 3. is written by
where is a safety gap between the quill and the ball nut, and is the tilt angle from the vk direction vector of the quill axis. Minimizing (2.2) a quartic equation can be formulated [10]
where
The roots of the nonlinear equation system from (2.3) and (2.4) are found by root finder algorithm.
Numerical algorithms for nonlinear equation systems
In this work Newton-Raphson and Broyden numerical algorithms [12] are used for the solving of the nonlinear equation system and the results are compared. Initial values are required for the two unknown parameters on both methods for correct solutions.
Newton-Raphson method for nonlinear systems
Let N functional relations to be zeroed, involving variables , i = 1, 2, . . .,n, thus
=1,2,...,n.
x the vector of values and F denotes the vector of functions . The expanded functions in Taylor series in the neighbourhood of x
The matrix of partial derivatives appearing in the above equation is the Jacobian matrix J, where
.
In matrix notation
By neglecting terms of order and higher and by setting F(x + δx) = 0, we obtain a set of linear equations for the corrections δx that move each function closer to zero simultaneously, namely
Jδx = -F.
The above matrix equation can be solved by LU decomposition. The corrections are then added to the solution vector,
and the process is iterated to convergence. In general it is a good idea to check the degree to which both functions and variables have converged. Once either reaches machine accuracy, the other won’t change.
Broyden method for nonlinear systems
Newton’s method as showed perviously above is quite powerful, but it still has several disadvantages. One drawback is that the Jacobian matrix is needed. In many problems analytic derivatives are unavailable. If function evaluation is expensive, then the cost of finite-difference determination of the Jacobian can be prohibitive. Just as the quasi-Newton methods provide cheap approximations for the Hessian matrix in minimization algorithms, there are quasi-Newton methods that provide cheap approximations to the Jacobian for zero finding. These methods are often called secant methods, since they reduce to the secant method in one dimension. The best of these methods still seems to be the first one introduced, Broyden’s method [11]. Let us denote the approximate Jacobian by B. Then the i-th quasi-Newton step is the solution of
where . The quasi-Newton or secant condition is that statisfy
where . This is the generalization of the one-dimensional secant approximation to the derivative, δF/δx. However, does not determine uniquely in more than one dimension. The best-performing algorithm to pin down in practice results from Broyden’s formula. This formula is based on the idea of getting by making the least change to consistent with the secant . Broyden showed that the resulting formula is
Early implementations of Broyden’s method used the Sherman-Morrison formula to invert the above equation analytically. Then instead of solving equation by e.g., LU decomposition, one determined
by matrix multiplication in operations. The disadvantage of this method is that it cannot easily be embedded in a globally convergent strategy, for which the gradient of requires B, not ,
Determining of the initial values
Previous analyses pointed out that correct initial guesses are needed for real solutions. In this section the determination of initial guesses described in detail.
Initial values on cylindrical quill and ball nut
Fig. 4. Quill and workpiece position for initial value determination on cylindrical quill
Assuming that the quill and the ball nut axes are in the same plane. The initial tilt angle and the initial quill axis parameter are determined by
the rotation matrix for x axis is
and the rotated points are
the direction vector
the equation for the unknown and parameters is
applying the Fig. 4. notations. Solving equation (4.5) and simplifying the results the required initial values for the iterative algorithms are
and
Initial values on conical quill and ball nut
In the previous section the equation system was formulated for the determination of the tilt angle of the quill, where the quill was cylindrical. In certain cases conical quills are used in grinding process. The determination of the conical quill tilt angle is similar to the cylindrical case, but equation (3) has to be modified according to the taper angle of the quill (see Fig. 5.).
Fig. 5. Quill and workpiece position for initial value determination on conical quill
The modified expression is
where is the smaller diameter of the quill, is the grinding wheel width and is the width of additional parts. Solving equation (4.5) similarly the initial values are
the parameter is
and the larger diameter of the conical quill is
without additional parts
and the run-out of the tool is
(as seen on Fig.5.).
Parameters and result on different ball nuts and grinding tools
In this section the parameters of the different ball nuts and tools are collected and showed which have been analysed in this work. Table 1. shows the dimensions of the ball nuts, Table 2. shows the dimensions of grinding tools and Table 3. shows the results of different applied procedures.
32x25
34
60
32.71
2.68
0.25
5
40x20
43.5
90
41.69
3.77
0.28
7.144
40x30
42
100
39.47
3.38
0.255
6.35
50x30
54
133
51.69
4.22
0.264
8
7
26.5
12
-
10
37
14
16.5
31.5
120
20
Newton method I.
Newton method II.
Broyden method
Maple's fsolve()
Number of iterations
25
78
23
71
33
101
22
Procedures for numerical algorithm
Worked out examples
Acknowledgement
This research was supported by the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP 4.2.4. A/2-11-1-2012-0001 ‘National Excellence Program’.
References
[1] Broyden, C.G. (1965), Mathematics of Computation, Vol. 19, pp. 577–593.
[2] Dudás, I. (2004), The Theory and Practice of Worm Gear Drives, pp. 320, ISBN 978-1-903996-61-4, Butterworth-Heinemann
[3] Dudás, L. (2010), New way for the innovation of gear types, Engineering the Future, Rijeka, Dudás, L. (Ed.),pp.111-140, ISBN 978-953-307-210-4
[4] Guevarra, D. S.; Kyusojin, A.&Isobe, H.& Kaneko, Y. (2001), Development of a new lapping method for high precision ball screw (1st report) – feasibility study of a prototyped lapping tool for automatic lapping process, Precision Engineering (25), pp.63-69, ISSN 0141-6359
[5] Guevarra, D. S.; Kyusojin, A. &Isobe, H. & Kaneko, Y. (2002), Development of a new lapping method for high precision ball screw (2nd report) Design and experimental study of an automatic lapping machine with in–process torque monitoring system, Precision Engineering (26), pp. 389–395,ISSN 0141-6359
[6] Harada, H.; Kagiwada, T. (2004), Grinding of high-lead and gothic-arc profile ball-nuts with free quill-inclination, Precision Engineering (28), pp. 143-151, ISSN 0141-6359
[7] Ketchel, J.;Larochelle, P. (2005), Collision Detection of Cylindrical Rigid Bodies Using Line Geometry, Proceedings of the 2005 ASME International Design Engineering Technical Conferences, pp. 3-13,ISBN: 0-7918-4744-6
[8] Litvin, F. L.;Fuentes, A. (2004), Gear Geometry and Applied Theory – Second Edition, pp. 801, ISBN 978-052-181-517-8, Cambridge University Press
[9] Mihálykó Cs., Virágh J., (2011), Közelítő és szimbolikus számítások feladatgyűjtemény, Typotex, 2011.
[10] Schneider, P. J.;Eberly, D. H. (2003).Geometric Tools for Computer Graphics,pp. 1056, ISBN 1-55860-594-0, Morgan Kaufmann Publishers, San Fransisco
[11] Vranek, D (2002). Fast and accurate circle–circle and circle–line 3D distance computation, Journal of Graphics Tools, Vol. 7(1), pp. 23–32, ISSN 1086-7651
[12] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, (1992), Numerical Recipes in C, The Art of Scientific Computing, Second Edition, Cambridge University Press
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