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Design synthesis of a gyrogrinder using direct search optimization
Mechanismand Machine Theory Vol. 17, No. 1, pp. 33-45, 1982 Printed in Great Britain.
0094-114XI82/010033.-13503.0010 Pergamon Press Ltd.
Design Synthesis of a Gyrogrinder Using Direct Search Optimization M. O. M. Osmant S. Sankar, and R. V. Dukkipatl§ Received for pul:llication 27 April 1981 Abstract A novel grinding mechanism operating on the principle of gyroscopic tracking of arbitrary contours is presented. The mathematical model describing the mechanism of gyrogrinding and the conditions under which the grinding wheel remains in contact with the periphery of template are established. The design parameters that greatly influence the performance of gyrogrinding are identified as the radius of the grinding wheel, the length of the spin axis, rotational speed of grinding wheel, and the mass polar moment of inertia of rotor and inner gimbal. In this paper, the design synthesis of a gyrogrinder is formulated as an optimization problem so that the optimal values of the design variables are calculated using a direct search multiparameter optimization method for continuous grinding with no dynamic shattering of grinding wheel. Three typical templates: elliptical, Limacon of Pascal and Four Leaved Rose, which represent possible shapes in cam production were considered and the optimal values of design parameters for each cam configurations were found using the optimization strategy. Results show that optimal design parameters lie within a small range for the three cam configurations. The mean optimal values of the design parameters were calculated and used in the dimensioning of the rotor and inner gimbal. A prototype was then built based on this investigation and when tested functioned effectively grinding the template. This paper illustrates the use of computer-aided optimization in the design synthesis of a gyrogrinder. Nomenclature A, B, C triode of fixed axes at 0 mass polar moment of inertia of rotor about x, y, z mass polar moment of inertia of inner gimbal about x, y, z I Ix,+ Ixe I , + I , , = tz,+ ~,, L distance from O to P, the point of contact (see Fig. 2) Lo distance from 0 to O' O' origin of W, V R, radius of grinding wheel R, normal reaction at P, the point of contact of the grinding wheel and the template
tProfessor of Engineering, Department of Mechanical Engineering, Concordia University, Montreal, Quebec, Canada. Member, ASME. SAssociate Professor of Engineering, Department of Mechanical Engineering, Concordia University, Montreal, Quebec, Canada. Associate Member, ASME. §ASsociateResearch Officer,Division of Mechanical Engineering, National Research Council of Canada, Ottawa, Ontario, Canada. Member, ASME. 33
34 gt tangential reaction at P, the point of contact of the ~inding wheel and the template $ dp/dy
T~, T~, T~ components of external torque along A, B, C
components of external torque along x, y, z t time v , w vertical and horizontal axes 0' v, tangential velocity of P Vo, V, velocity components of P along V and W Ot angle between outward normal to the contour and the W-axis ./ angle of P0' with W-axis coefficient defined by eqns (13a) 0 180 deg-angle between OP and the tangent to the contour at P 6 angular displacement of x-axis about the Z-axis, measured from Q-axis 4, angular displacement of Z-axis about the B-axis, measured from C-axis % angular velocity of the spin axis a angle between O'P and straight line segment of template
1. introduction
GRINDINGis a finishing operation used for removal of small amounts of material (0.4-2200/~m) from a workpiece that has been previously produced to the required shape by numerically controlled machining or by hydraulic copying mechanisms. The kind of surface machined largely determines the type of grinding machine; thus, a machine grinding cylindrical surfaces is called a cylindrical grinder. This paper presents the technique of gyrogrinding and provides a computer-aided design synthesis of a gyrogrinder for grinding any arbitrary contour of workpiece using multiparameter optimization. A recent study by Mansour and Pavlov[1, 2] on the mechanism of Gyroscopic tracking has led the authors to examine the potential application of Gyroscope for the grinding of arbitrary contour of a template. A grinding wheel, when secured to the spin axis of the gyroscope, exerts a force on the template and tracks the contour of the template. However if a workpiece is used as a template in a gryogrinder, then the force exerted on the workpiece due to the gyroscopic effect will provide the necessary cutting force for the grinding operation. Hence in this paper, the term template refers to the workpiece that needs grinding. Grinding of an arbitrary contour of a template is achieved by allowing the axis of the grinding wheel to come in contact with the periphery of the template. The grinding wheel moves around the periphery of the template with continuous contact under certain conditions. The conditions for the grinding wheel to start and remain in continuous contact with the template are influenced by many factors including the geometry of the template and the set up, the coefficient of fraction of the two materials at the point of contact, the radius of the grinding wheel, the length of spin axis and the normal and tangential reactions at the point of contact. The present investigation examines the mathematical model governing the mechanism of gyrogrinding and the details describing the strategy of analysis and synthesis. This strategy of analysis and synthesis was used to design a gyrogrinder for the grinding of a given geometrical configuration of the template. Figure 1 shows the first generation of a gyrogrinder built in 1973 to demonstrate the feasibility of metal removing by gyroscopic effect. The model proved to be successful in this respect, yet the grinding wheel of the spin axis tends to depart the cam surface especially at small template curvature. This inspired the requirement of complete dynamic analysis of gyrogrinding system to eliminate overshooting. In this paper, the design synthesis of a gyrogrinder is formulated as an optimization problem so that the optimal values of the design variables are calculated using a multiparameter optimization technique for continuous grinding with no overshooting. This resulted in designing and building a second generation of the gyrogrinder. Figure 2 shows this model which was completed in 1974. The tie-rods allow flotation of template in all directions in space and maintain parallelism between the spin axis and the normal to the template during tracking. Upon testing the model, the system was found to be satisfactory in grinding the cam surface. The design and operation of the gyrogfinder will provide a low cost, mass production gyrogrinding technique with accuracy comparable to more advanced and expensive numerically controlled grinding operation.
35
Figure 1. First generation gyrogrinder.
Figure 2. Second generation gyrogrinder. 2. Mathematical Modelling of the Gyrogrlnder System The mathematical model of a gyrogrinding system can be established by first considering a schematic diagram of a tracking gyro as shown in Fig. 3(a). In this configuration, the spinning rotor is suspended in such a manner that the rotation of the inner and outer gimbals about the spin axis are prohibited. The principal axes attached to the inner gimbal are identified by a triad x-y-z. The x-axis is chosen to coincide with the spin axis and z-axis with the axis of rotation of inner gimbal. Figure 3(b) shows the coordinate system in space drawn with respect to the axis of rotation of inner gimbal. A triad A-B-C fixed in inertial space with origin at 0 and 0' is defined. The point 0 is the center of mass of the spinning rotor and point 0' is any convenient point in the plane of the template. The template is considered to be in W-V plane. Using the sets of axes defined above, the governing equations of the gyrogrinding system can be derived [I].
36 Y "/ ~
SRN AXIS
SPINNINORoTOR
X
~
~.
N IN E R GIMBAL
TANGENTIAL V Y
GIMBAL
ill ~
'
R~
V,v
. ~B
BEARING
Figure 3(e). Schematic diagram of a tracking gyro [1].
,,} y
NORMAL '
Figure 3(b). Selection of coordinates [1].
B
. L.f(cosC~c,os~.}Z+(cos~s~n~hanr_s:n~)2"
\°'1
sP,N . J
iB
:
,:
V
~o~ ~' T ;
"--
', Lo, p. C" ~ / 4 ~
AXIS OF ROT--°ONATI
! ~ /Z
OF INNER CYMBAL
-
- TEMPLATE DIRECTION OF TRACKING
Figure 3(¢). Forces and velocity components [1].
C -~ ~- ~ i
,
~
_~:~--~I-
L~.~
%
TANGENTIt~L
-
w
.44~'I.;-: Lsin~
- - - *0 / * (=s~n/3 ,
- L cos¢ s ~ A TEMPLATE
! SPIN AXIS Figure 3(d). The coefficients [1].
The mathematical model governing the system comprises four basic sets of relations. The first describes the dynamics of the gyro; the second relates the external torques to the reaction forces at the point of contact; the third describes the tangential velocity of tracking; and the fourth provides the condition for the grinding wheel to remain in the track-roll mode. (1) Dynamics or the gyro The dynamic equations of motion of the gyro can be written from Fig. 3(b) as [1]: Tx = I(6 sin $ + 66 cos ~) + / , / ~ Ty = I,(6 cos $ - 26~ sin $) + (I¢) sin 4' + I~#~)6 T~ = I~(~ + $2 sin $ cos ~) - (I$ sin $ + / , ~ ) 6
(1)
cos ~b
where ~ and ~ are the two Eulerian angles as shown in Fig. 3(c) and a dot represents a differentiation with respect to time. (2) Relationship between the external torque and reaction [orces Let the template be given by the general expression: p = F(y), or F(O, y) = O. From Fig. 3(c), the component external torques along axes A, B, C can be expressed in terms of the normal and tangential reactions (R., R,) at the point of contact (P) of the grinding wheel and the template. Therefore TA
=
- -
(R. cos a - R, sin a)p sin y
+ (R. sin a + Rt cos a)p cos y
37
T8 = Lo(R. cos
a
-
Rt sin a)
(2)
Tc = Lo(R. sina + Rt cos a). Using the angular relationship between the xyz- and ABC-coordinate systems, eqn (2) can be written as: T~ = (Ta cos 4' - T~ sin 4')cos 4, + TB sin 4' Ty = Ts cos 4' - (TA cos 4' - Tc sin 4')sin 4'
(3)
Tz = TA sin 4' + Tc cos 4'. It is possible to eliminate TA, Te, Tc, 4', and 4' from the previous relations by using the following expressions relating (4', 4') to (p, y):
sin y COS 0 = p cos y tan 4' = P -, tan 4' Lo Lo
(4)
Combining eqns (2), (3) and (4) gives: R. = %/(Lo2 + p2 COS y)" X [(Lo 2 +
Tz sin a + Lo~(Lo: + p:)
p2 cos 2 ~t) cos o[ -I- p2 sin 3' cos y sin a]}
Ty . / ( L o 2 + p2 COS2 ~) Rtsina=R. cosa-Lo¥~ Lo2+p2 • (3) Tangential velocity of tracking The kinematic relationship of the velocity components, Vw and Vv, of the point P in the W-V plane are given by:
Vw = ~bcos y - p~ sin y Vv = ti sin 3' + P'/' cos y.
(6)
Differentiating (4) with respect to time and substituting in (6) we obtain:
Vw= cLo°~s~ Vw = cos 2 4'L0cos z 4' (• cos 4' + ~ sin 4' cos 4' sin 4').
(7)
(4) Track-roll mode For an effective grinding operation using a gyrogrinder, the first condition is that the grinding wheel should remain in contact with the template contour at all times from start to finish. The second condition relates to the actual grinding process which requires a continuous rolling of the grinding wheel over the template. If these two conditions are met, then the gyrogrinder is said to be in the track-roll mode. The first condition will be satisfied, if the normal force between the grinding wheel and the template is greater than zero. It should be noted that in the track-roll mode, the grinding wheel does not have a pure rolling, however, it rolls with slipping, i.e.
R.>0.
(8)
To satisfy the second condition, the ratio IRt/R. [ which represents the coefficient of friction
38
between the grinding wheel and the template must satisfy the constraint:
Rr]
P.s.
(9)
In addition, given the geometry of the template contour and tracking kinematics (Fig. 3b and d), the following set of equations can be developed: sin a = p sin y - s cos y X,/(p2 + S2)
(10)
s sin y + p cos y v~(p 2+ s:)
(11)
cos a =
where s = (do/d3,), depends on the template contour. $ / / = ~ R ~ g X/(p2 + S2)
(12)
1 "~ = ~R~wg X/(p2 + s:)
(13)
where s¢ = sin 0; and from Fig. 3(d) if r = ¢d2, then ~ = cos $ (13a) _ Lo(/J cos y - p~, sin y) Lo 2 + p2 cos2 y
(14)
Lo = (Lo 2 + p5 sin s y cos 2 y)
(15)
[//sin y cos 0 + OJ' cos y cos ¢, - p sin Yd sin 0] = - 4~ sin 4'
(16)
P"= X/(P: + ~:Rx
+ X/(P 2 + s:) ~ f.
ozi + s~ "i
= v ( o : + s:) L " -
(17)
R~o~
+ v ( o ' + ,')
Lo = (Lo 2 + p2 COS2 y)2
(18) (19)
[(Lo 2 + 02 cos 2 3')(//cos 3' - 2//~, sin 3' - Oq 2 cos 3' - Oq sin 3') - (//cos 3' - p? sin 3')(20//cos ~ 3' - 202 cos 3' sin 3'~,)] )~ = C . F where Lo C = (Lo2 + p2 sin s y cos 2 q~), and
F = (L0 2 + p2 sin 2 y cos 2 qJ)[cos qt(~6 sin y + 2//j, cos y + p~ cos y - p~)2 sin y - pal2 sin y) - sin ~0(2//d sin y
- p,)q) cos y) + p~ sin y + pd~, cos y] - [(//sin y + p',) cos y) cos 0 - P4~ sin y sin q0][2 cos 2 ~0(p//sin 2 y + p:~ sin y) - 202d sin s y cos 0 sin 0]
(20)
39 The equations (1)-(20) completely describe the kinematic and dynamic equations governing the gyrogrinder system.
3. Optimization strategy for Design Synthesis of a Gyrogrlnder In the design synthesis of a gyrogrinder, it is required to identify six major design parameters: length of spin axis, mass polar moment of inertias L Ii and Ix,, radius of grinding wheel, and angular velocity of the grinding wheel. One way to select these parameters is to initially assume a value for each one of them and to calculate the reaction forces Rn and RI from eqn (5) after solving for the rest of the governing equations (1)-(4), (6)-(7) and (10)-(20). Since the values of R, and Rt must satisfy eqns (8) and (9) for continuous rolling contact of the grinding wheel without separation, a check is made for any violation of eqns (8) and (9). If it is not violated, then the starting values of the parameter are feasible design values. Similarly, by trial and error, it is possible to obtain several set of such feasible parameters. Then it is the choice of the designer to select the most suitable (optimal) set of parameter that meets the design requirements. It is seen from the above description that the selection of design parameters directly influence the magnitude of R, and Rt. This paper utilizes this fact advantageously, in order to formulate an optimization strategy for the design synthesis of gryogrinder. Resorting to this type of strategy eliminates any trial and error procedure in selecting feasible design parameters. From the point of view of the grinding process, R~ and Rt represent the thrust force on the grinding wheel and the cutting force respectively. From eqn (5), it can be seen that both R, and R~ are generated due to the gyroscopic action and their magnitudes depend on the gyroscopic torques, radius of curvature and geometry of the template, and the geometry of the mechanism. Hence for a gyrogrinder, the values of R, and Rt while grinding an arbitrary template contour vary continuously around the contour. Especially at small radii of curvature, the value of R, becomes very small or even negative and thus causing the grinding wheel to depart from the template. Based on the above fact, the thrust force R~ must be a maximum positive quantity for a satisfactory grinding operation. In other words I1/RnI must be a minimum. In a grinding operation, the cutting force depends on the depth of cut, feed rate, and material properties of the workpiece and the grinder. Based on a particular grinding operation, the amount of cutting force required can be estimated and hence in a gyrogrinder, this cutting force must be generated. As stated earlier, Rt represents the cutting force in a gyrogrinder and is generated due to gyroscopic torques. Since the magnitude of Rt continuously varies around the template contour due to its dependence on the radius of curvature and geometry of the template, and the geometry of the mechanism, it is required that the minimum value of RI is at least equal to the required cutting force Ft. Thus, by preselecting the minimum value of Rt, i.e. (R/)mi.as equal to the required cutting force for grinding and by combining it in the minimization scheme, the new objective criterion becomes: Minimize I ~
I
where (Rt)min = Fo the cutting force. This criterion together with other conditions in the gyrogrinder operation can be mathematically expressed as follows: Objective Function:
J~l
minimize L, £ I , Ix,, ~os, Rg
(21)
max"
Subject to Constraints tt < tts; Rn > O; Rt >- (Rt)=i,. LI<_L<_L "
It~r<_Ix,.<_I~r
IJ<_I<_I "
w~t <- w~ <- tog~ I~~ <- 11 <- I~"
Rg~~ Rg ~ Rgu
(22)
40
where the subscripts l and u correspond to lower and upper bounds of each variable and = IR,/R.I.
It should be noted that for higher values of Fc, the optimization procedure may not yield a feasible solution due to the violation of the constraint Rt->(RDmi,. This indicates that in gyrogrinder, the cutting force generated purely due to the gyroscopic effect will not be sufficient and may require additional mechanism to control the cutting force. 4. Case Studies In order to illustrate the optimization strategy, three case studies of different template profiles are considered. They are: elliptical template, Limacon of Pascal, and Four-leaved rose. For each template, the equation of the radius vector p, the rate of change of p with respect to angle (s), and s are defined. Using these three equations and starting with the initial values for the six design parameters, the objective function evaluation and constraints checking can be carried out by solving the governing equations. A non-linear programming technique based on the sequential interior penalty function method[3-5] is used to convert the constrained optimization problem of this paper into a sequence of unconstrained problems. The Hooke and Jeeves[3] pattern search optimization technique is then used to carry out the unconstrained minimization of the objective function and to determine the optimal design parameters. It should be noted that the design parameter, "the length of spin axis L" is considered to be equal to Lo, the distance 00' (Fig. 3(b)) in the design. Case 1: The elliptical template
The elliptical template as shown in Fig. 4 is symmetrical about the V and W axes and has the following equations: p = ala2/x/(al 2 sin2 y + a22 cos 2 y)
(23)
where a], a2 are the semi major and minor axes of the ellipse taken along W and V axes respectively. s
_ dp
dy
SlS2S3
-
(24)
s4
where Sl-
ala2
2
S2 = a l 2 - 022
s3 = sin 2y s, = (al 2 sinE1, - aE: cos = y)3/=
=d/dp~ = dt \ d y ]
~i
//
ss(s6- sTs2s3)5/
\",
"~"~ - -"- ~ \ -
_ ._ DIRECTION OF TRACKING
Figure 4. Elliptical template (case 1).
(25)
41
where s 5 = ~ 4s~ S6 = 2S4 COS
2y
1 S7 = 2 S3~v/(S4) S2 = ( a l 2 -
a2 2)
s3 = sin 23,.
Case 2: The Limacon of Pascal The Limacon of Pascal is shown in Fig. 5 and its equation in polar coordinates are given by: p = a, cos y + a2
(26)
s = - a, sin y
(27)
= - alq cos 3,.
(28)
It can be seen that the Limacon displays both negative and positive curvatures and the zone around 270° simulates the inside tracking of noncircular holes.
Case 3: The four-leaved rose The equations for the four-leaved rose shown in Fig. 6 are given as: p = a l COS
2y
(29)
(30)
s = - 2al sin 2y = - 4a1~, cos 2y.
(31)
5. Optimization Algorithm The design synthesis of a gyrogrinder has been formulated as a nonlinear programming problem in previous sections. There are many algorithms available to seek the solution of a nonlinear programming problem. In this paper an interior penalty function method in conjunction with the Hooke Jeeves method is used. Some of the special forms of the interior penalty
V
I /
Oi W
w
L ~. '/ • c,~//
~
DIRECTION OF TRACKING
H ~ DRECTIONOF TRACKING
Figure 5. Limacon of Pascal template (case 2).
Figure 6. Four lezved rose template (case 3).
42 function methods, which are used extensively, are as follows: Consider a nonlinear optimization problem: Minimize f(2); 2 = xl, x2... xn Subject to gj($) ~<0 j = 1, 2 . . . . p. For this task, McCormick and Fiacco's version of interior penalty function is given below [3]: ¢(2, r) = f(2) - r
gS-O"
The modified objective function ¢($, r) can be minimized using any unconstrained optimization routine. The penalty parameter r is sequentially decreased between the unconstrained minimization and each unconstrained optimization begins where the preceeding one terminates. For this method, the initial starting point must satisfy the inequality constraints. Another form proposed by Siddall[6] is as follows: $(-f) = f(-f) + 102° ~ i=1
Ig ( )l
for gj(2) > 0.
(23)
The modified objective function as given by Siddall is used in this paper and the unconstrained minimization is carried out by Hooke and Jeeves pattern search technique. For the gyrogrinder design, the inequality constraints/z - (Rt)=in and R~ > 0 can be directly implemented into the equation (23) by considering the following: g , ( ~ ) = ( ~ - ~,,); f ( ~ ) = - Rn
and g3(2) = (Rt)min- Rt. Once the constrained optimization problem is reduced to the form in eqn (23), the Hooke and Jeeves unconstrained optimization method can be directly used. The Hooke and Jeeves method is a direct search routine for minimizing a function f($) of several variables $ = (Xl, x2, ... xn). The augment $ is varied until the minimum of f(2) is obtained. The pattern search routine determines the sequence of value of $, an independent routine computes the function values of f(J). A detailed description of the algorithm is given in Ref. 3. 6. Numerical Results By employing an interior penalty function of the form proposed by Sidall and using the Hooke and Jeeves direct search optimization technique, the optimal values of the gyrogrinder design parameters are obtained for each case studies considered. Optimization is performed with different set of initial values of the design variables. For each trial, the results indicate that the optimal values and the minimum value of the objective function are close to one another with a r.m.s, deviation less than 0.015. Table 1 shows the optimal results for one set of such starting values. It can be seen that the optimal values for L, L toe and R e are very close to each other for all three case studies. However, there is a slight discrepancy in the calculated optimal values of I and Ixr between the three cases. Since the three cam configurations represent the majority of all possible shapes in cam production, an optimal design of a gyrogrinder can be achieved by calculating the mean of the optimal values from each case study. They are:
L = 0.70753 m I1 = 0.3386 kg.m 2
450
350
~9 ( r e d / s )
Case 3:
Case 2:
Case 1 : ; a2 - 0.0508 m;
0.0127
eI ,, 0.5080m
; ;45" <_~r<_45° ;
a 1 - 0,254 re ; a2 - 0.0762 m;
aI - 0.254m
0.0064
] . 695
.566
Ixr(kg.m2)
Rg (m)
1.808
.904
(kg.m 2)
[
0.339
.113
Iz (kg°m2)
0.7112
•
0.0076
400.0
.904
1.017
.1695
0.6604
STARTING VALUE
Fc - .35.60 N
Fc • 35.60 N
CASE 2
CASE 3
0.0124
400.3
.7300
1.1910
.338
0.7000
0.0076
400.0
.904
1.017
.1695
0.6604
0.0124
400.42
.8170
1.2972
0.339
0.71;12
0.0075
400.0
.904
1.017
.1695
0.6604
0.0124
401.36
.6746
1.2464
0.3395
0.7112
OPTINUH STARTING OPTIHUH STARTING OPTIHUf4 VALUE VALUE VALUE VALUE VALUE
CASE 1
Numerical results
F¢ - 35.60 N
UPPER BOUND VALUE
0.635
LOMER BOUND VALUE
L(~)
DESIGN PARAHETER
Table
44 l : 1.2448 kg.m 2 [xr : 0.7405 kg.m 2 % = 400.693 rad/s Rg = 0.01245 m. Once the mean optimal values of the design parameters are calculated, a detailed dimensioning of the rotor and inner gimbal can be carried out for the final design.
7. Summary and Conclusion The instabilities that occurred during operation of the first generation gyrogrinder are investigated and the gyrogrinder redesigned by the use of an optimization scheme. This eliminated the tedious algebraic analysis. The following three shapes were considered as typical cam profiles: 1. ELLIPTICAL 2. LIMACON OF PASCAL 3. FOUR-LEAVED ROSE The dimensions obtained from the three cases were used to calculate the detailed dimensions for the rotor and the inner gimbal. Analysis shows that the optimal design parameters of the gyrogrinder lie within a small range for the three cam configurations. Since the three cam configurations represent all possible shapes in cam production, it is thus possible to utilize this gyrogrinder for machining complex curved surfaces. A gyrogrinder was designed and operated based on the optimal values obtained from this investigation. The grinder functioned effectively removing material from the workpiece (template) without any separation, thus providing supporting evidence for the technique of gyrogrinding. In this paper, no attempt is made to include the mechanism of controlling the depth of cut or to provide actual measurements on the accuracy of gryogrinding. However this will be the subject of our future investigation. Acknowledgements--The support of the National Research Council of Canada, Grant Nos. A5181and A3685, and the Formation de Chercheurs et d'Action Concertee, of the Governmentof Quebec, Grant No. 042-110 is gratefully acknowledged.
References 1. W. M. Mansourand D. Pavlov,The mechanismof GyroscopicTracking--Part 1. Trans ASME, J. Engng Industry, 95, Series B, No. 2, 430-436(1973). 2. W.M.Mansourand D. Pavlov,TheMechanismof GyroscopicTracking--Part2. TransASMEEngnglndustry. 95, Series B, No. 2, 437--444(1973). 3. D. M. Himmelblau,Applied Nonlinear Programming. McGraw-Hill,New York (1972). 4. R. L. Fox, Optimization Methods/or EngineeringDesign. Addison-Wesley,Reading,Mass. (1977). 5. A. V. Fiacco,and G. P. McCormick,NonlinearProgramming, Wiley,New York (1968). 6. J. N. Sidall,Analytical Decision Making in EngineeringDesign. Prentice-Hall,EnglewoodCliffs,New Jersey(1972).
CONCEPTION
SYNTHETIQUE
M.O.M.
Osman,
R~sum~
- On p r ~ s e n t e
sent~e
un n o u v e a u m ~ c a n i s m e
de c o n t o u r s
gyroscopique
la p ~ r i p h ~ r i e
GYROSCOPIQUE
et
sont pr~sent~s.
com~ae un p r o b l ~ m e
d'optimisation;
cherche
tudi6 trois p~tales,
sont c a l c u l ~ e s
directe
pour
A RECHERCHE
gabarits
typiques, des
continu,
fonctionnant
Le m o d u l e m a t h ~ m a t i q u e dans l e s q u e l l e s La c o n c e p t i o n
ainsi,
DIRECTE
soit elliptique,
sur le p r i n c i p e d~crivant
optimales
le m ~ c a n i s m e avec
est pr~-
des v a r i a b l e s
multiparam~trique
de
~ re-
de la meule.
On a ~-
en limaqon de Pascal et en rose
~ quatre
de cames,
dynamique
du pis-
reste en c o n t a c t
d'une meule g y r o s c o p i q u e
d'optimisation
sans ~ c l a t e m e n t
formes p o s s i b l e s
la meule
les valeurs
~ l'aide d'une m ~ t h o d e
un m e u l a g e
qui r e p r ~ s e n t e n t
de m e u l a g e
arbitraires.
les c o n d i t i o n s
du g a b a r i t
conception
PAR O P T I M I S A T I O N
S. S a n k a r et R.V. D u k k i p a t i
rage g y r o s c o p i q u e de m e u l a g e
D'UNE ~ U L E
et les v a l e u r s
optimales
des pa-
45 ram~tres de conception pour chaque forme de came ont ~t~ trouv~es par la m~thode d'optimisation.
Les r~sultats montrent que les valeurs optimales des param~tres de conception sont
comprises dans une plage limit~e pour chacune des trois formes de came.
Les moyennes opti-
males des param~tres de conception ont ~t~ calcul~es et utilis~es pour le choix des dimensions du rotor et du joint de cardan
int~rieur.
n~es de cette ~tude et, soumis ~ l'essai,
Un prototype a ~t~ construit avec les don-
il s'est av~r~ ~tre tr~s efficace.
Le present ex-
pos~ illustre l'utilisation de la conception assist~e par ordinateur pour la conception d'une meule gyroscopique.
0094-114XI82/010033.-13503.0010 Pergamon Press Ltd.
Design Synthesis of a Gyrogrinder Using Direct Search Optimization M. O. M. Osmant S. Sankar, and R. V. Dukkipatl§ Received for pul:llication 27 April 1981 Abstract A novel grinding mechanism operating on the principle of gyroscopic tracking of arbitrary contours is presented. The mathematical model describing the mechanism of gyrogrinding and the conditions under which the grinding wheel remains in contact with the periphery of template are established. The design parameters that greatly influence the performance of gyrogrinding are identified as the radius of the grinding wheel, the length of the spin axis, rotational speed of grinding wheel, and the mass polar moment of inertia of rotor and inner gimbal. In this paper, the design synthesis of a gyrogrinder is formulated as an optimization problem so that the optimal values of the design variables are calculated using a direct search multiparameter optimization method for continuous grinding with no dynamic shattering of grinding wheel. Three typical templates: elliptical, Limacon of Pascal and Four Leaved Rose, which represent possible shapes in cam production were considered and the optimal values of design parameters for each cam configurations were found using the optimization strategy. Results show that optimal design parameters lie within a small range for the three cam configurations. The mean optimal values of the design parameters were calculated and used in the dimensioning of the rotor and inner gimbal. A prototype was then built based on this investigation and when tested functioned effectively grinding the template. This paper illustrates the use of computer-aided optimization in the design synthesis of a gyrogrinder. Nomenclature A, B, C triode of fixed axes at 0 mass polar moment of inertia of rotor about x, y, z mass polar moment of inertia of inner gimbal about x, y, z I Ix,+ Ixe I , + I , , = tz,+ ~,, L distance from O to P, the point of contact (see Fig. 2) Lo distance from 0 to O' O' origin of W, V R, radius of grinding wheel R, normal reaction at P, the point of contact of the grinding wheel and the template
tProfessor of Engineering, Department of Mechanical Engineering, Concordia University, Montreal, Quebec, Canada. Member, ASME. SAssociate Professor of Engineering, Department of Mechanical Engineering, Concordia University, Montreal, Quebec, Canada. Associate Member, ASME. §ASsociateResearch Officer,Division of Mechanical Engineering, National Research Council of Canada, Ottawa, Ontario, Canada. Member, ASME. 33
34 gt tangential reaction at P, the point of contact of the ~inding wheel and the template $ dp/dy
T~, T~, T~ components of external torque along A, B, C
components of external torque along x, y, z t time v , w vertical and horizontal axes 0' v, tangential velocity of P Vo, V, velocity components of P along V and W Ot angle between outward normal to the contour and the W-axis ./ angle of P0' with W-axis coefficient defined by eqns (13a) 0 180 deg-angle between OP and the tangent to the contour at P 6 angular displacement of x-axis about the Z-axis, measured from Q-axis 4, angular displacement of Z-axis about the B-axis, measured from C-axis % angular velocity of the spin axis a angle between O'P and straight line segment of template
1. introduction
GRINDINGis a finishing operation used for removal of small amounts of material (0.4-2200/~m) from a workpiece that has been previously produced to the required shape by numerically controlled machining or by hydraulic copying mechanisms. The kind of surface machined largely determines the type of grinding machine; thus, a machine grinding cylindrical surfaces is called a cylindrical grinder. This paper presents the technique of gyrogrinding and provides a computer-aided design synthesis of a gyrogrinder for grinding any arbitrary contour of workpiece using multiparameter optimization. A recent study by Mansour and Pavlov[1, 2] on the mechanism of Gyroscopic tracking has led the authors to examine the potential application of Gyroscope for the grinding of arbitrary contour of a template. A grinding wheel, when secured to the spin axis of the gyroscope, exerts a force on the template and tracks the contour of the template. However if a workpiece is used as a template in a gryogrinder, then the force exerted on the workpiece due to the gyroscopic effect will provide the necessary cutting force for the grinding operation. Hence in this paper, the term template refers to the workpiece that needs grinding. Grinding of an arbitrary contour of a template is achieved by allowing the axis of the grinding wheel to come in contact with the periphery of the template. The grinding wheel moves around the periphery of the template with continuous contact under certain conditions. The conditions for the grinding wheel to start and remain in continuous contact with the template are influenced by many factors including the geometry of the template and the set up, the coefficient of fraction of the two materials at the point of contact, the radius of the grinding wheel, the length of spin axis and the normal and tangential reactions at the point of contact. The present investigation examines the mathematical model governing the mechanism of gyrogrinding and the details describing the strategy of analysis and synthesis. This strategy of analysis and synthesis was used to design a gyrogrinder for the grinding of a given geometrical configuration of the template. Figure 1 shows the first generation of a gyrogrinder built in 1973 to demonstrate the feasibility of metal removing by gyroscopic effect. The model proved to be successful in this respect, yet the grinding wheel of the spin axis tends to depart the cam surface especially at small template curvature. This inspired the requirement of complete dynamic analysis of gyrogrinding system to eliminate overshooting. In this paper, the design synthesis of a gyrogrinder is formulated as an optimization problem so that the optimal values of the design variables are calculated using a multiparameter optimization technique for continuous grinding with no overshooting. This resulted in designing and building a second generation of the gyrogrinder. Figure 2 shows this model which was completed in 1974. The tie-rods allow flotation of template in all directions in space and maintain parallelism between the spin axis and the normal to the template during tracking. Upon testing the model, the system was found to be satisfactory in grinding the cam surface. The design and operation of the gyrogfinder will provide a low cost, mass production gyrogrinding technique with accuracy comparable to more advanced and expensive numerically controlled grinding operation.
35
Figure 1. First generation gyrogrinder.
Figure 2. Second generation gyrogrinder. 2. Mathematical Modelling of the Gyrogrlnder System The mathematical model of a gyrogrinding system can be established by first considering a schematic diagram of a tracking gyro as shown in Fig. 3(a). In this configuration, the spinning rotor is suspended in such a manner that the rotation of the inner and outer gimbals about the spin axis are prohibited. The principal axes attached to the inner gimbal are identified by a triad x-y-z. The x-axis is chosen to coincide with the spin axis and z-axis with the axis of rotation of inner gimbal. Figure 3(b) shows the coordinate system in space drawn with respect to the axis of rotation of inner gimbal. A triad A-B-C fixed in inertial space with origin at 0 and 0' is defined. The point 0 is the center of mass of the spinning rotor and point 0' is any convenient point in the plane of the template. The template is considered to be in W-V plane. Using the sets of axes defined above, the governing equations of the gyrogrinding system can be derived [I].
36 Y "/ ~
SRN AXIS
SPINNINORoTOR
X
~
~.
N IN E R GIMBAL
TANGENTIAL V Y
GIMBAL
ill ~
'
R~
V,v
. ~B
BEARING
Figure 3(e). Schematic diagram of a tracking gyro [1].
,,} y
NORMAL '
Figure 3(b). Selection of coordinates [1].
B
. L.f(cosC~c,os~.}Z+(cos~s~n~hanr_s:n~)2"
\°'1
sP,N . J
iB
:
,:
V
~o~ ~' T ;
"--
', Lo, p. C" ~ / 4 ~
AXIS OF ROT--°ONATI
! ~ /Z
OF INNER CYMBAL
-
- TEMPLATE DIRECTION OF TRACKING
Figure 3(¢). Forces and velocity components [1].
C -~ ~- ~ i
,
~
_~:~--~I-
L~.~
%
TANGENTIt~L
-
w
.44~'I.;-: Lsin~
- - - *0 / * (=s~n/3 ,
- L cos¢ s ~ A TEMPLATE
! SPIN AXIS Figure 3(d). The coefficients [1].
The mathematical model governing the system comprises four basic sets of relations. The first describes the dynamics of the gyro; the second relates the external torques to the reaction forces at the point of contact; the third describes the tangential velocity of tracking; and the fourth provides the condition for the grinding wheel to remain in the track-roll mode. (1) Dynamics or the gyro The dynamic equations of motion of the gyro can be written from Fig. 3(b) as [1]: Tx = I(6 sin $ + 66 cos ~) + / , / ~ Ty = I,(6 cos $ - 26~ sin $) + (I¢) sin 4' + I~#~)6 T~ = I~(~ + $2 sin $ cos ~) - (I$ sin $ + / , ~ ) 6
(1)
cos ~b
where ~ and ~ are the two Eulerian angles as shown in Fig. 3(c) and a dot represents a differentiation with respect to time. (2) Relationship between the external torque and reaction [orces Let the template be given by the general expression: p = F(y), or F(O, y) = O. From Fig. 3(c), the component external torques along axes A, B, C can be expressed in terms of the normal and tangential reactions (R., R,) at the point of contact (P) of the grinding wheel and the template. Therefore TA
=
- -
(R. cos a - R, sin a)p sin y
+ (R. sin a + Rt cos a)p cos y
37
T8 = Lo(R. cos
a
-
Rt sin a)
(2)
Tc = Lo(R. sina + Rt cos a). Using the angular relationship between the xyz- and ABC-coordinate systems, eqn (2) can be written as: T~ = (Ta cos 4' - T~ sin 4')cos 4, + TB sin 4' Ty = Ts cos 4' - (TA cos 4' - Tc sin 4')sin 4'
(3)
Tz = TA sin 4' + Tc cos 4'. It is possible to eliminate TA, Te, Tc, 4', and 4' from the previous relations by using the following expressions relating (4', 4') to (p, y):
sin y COS 0 = p cos y tan 4' = P -, tan 4' Lo Lo
(4)
Combining eqns (2), (3) and (4) gives: R. = %/(Lo2 + p2 COS y)" X [(Lo 2 +
Tz sin a + Lo~(Lo: + p:)
p2 cos 2 ~t) cos o[ -I- p2 sin 3' cos y sin a]}
Ty . / ( L o 2 + p2 COS2 ~) Rtsina=R. cosa-Lo¥~ Lo2+p2 • (3) Tangential velocity of tracking The kinematic relationship of the velocity components, Vw and Vv, of the point P in the W-V plane are given by:
Vw = ~bcos y - p~ sin y Vv = ti sin 3' + P'/' cos y.
(6)
Differentiating (4) with respect to time and substituting in (6) we obtain:
Vw= cLo°~s~ Vw = cos 2 4'L0cos z 4' (• cos 4' + ~ sin 4' cos 4' sin 4').
(7)
(4) Track-roll mode For an effective grinding operation using a gyrogrinder, the first condition is that the grinding wheel should remain in contact with the template contour at all times from start to finish. The second condition relates to the actual grinding process which requires a continuous rolling of the grinding wheel over the template. If these two conditions are met, then the gyrogrinder is said to be in the track-roll mode. The first condition will be satisfied, if the normal force between the grinding wheel and the template is greater than zero. It should be noted that in the track-roll mode, the grinding wheel does not have a pure rolling, however, it rolls with slipping, i.e.
R.>0.
(8)
To satisfy the second condition, the ratio IRt/R. [ which represents the coefficient of friction
38
between the grinding wheel and the template must satisfy the constraint:
Rr]
P.s.
(9)
In addition, given the geometry of the template contour and tracking kinematics (Fig. 3b and d), the following set of equations can be developed: sin a = p sin y - s cos y X,/(p2 + S2)
(10)
s sin y + p cos y v~(p 2+ s:)
(11)
cos a =
where s = (do/d3,), depends on the template contour. $ / / = ~ R ~ g X/(p2 + S2)
(12)
1 "~ = ~R~wg X/(p2 + s:)
(13)
where s¢ = sin 0; and from Fig. 3(d) if r = ¢d2, then ~ = cos $ (13a) _ Lo(/J cos y - p~, sin y) Lo 2 + p2 cos2 y
(14)
Lo = (Lo 2 + p5 sin s y cos 2 y)
(15)
[//sin y cos 0 + OJ' cos y cos ¢, - p sin Yd sin 0] = - 4~ sin 4'
(16)
P"= X/(P: + ~:Rx
+ X/(P 2 + s:) ~ f.
ozi + s~ "i
= v ( o : + s:) L " -
(17)
R~o~
+ v ( o ' + ,')
Lo = (Lo 2 + p2 COS2 y)2
(18) (19)
[(Lo 2 + 02 cos 2 3')(//cos 3' - 2//~, sin 3' - Oq 2 cos 3' - Oq sin 3') - (//cos 3' - p? sin 3')(20//cos ~ 3' - 202 cos 3' sin 3'~,)] )~ = C . F where Lo C = (Lo2 + p2 sin s y cos 2 q~), and
F = (L0 2 + p2 sin 2 y cos 2 qJ)[cos qt(~6 sin y + 2//j, cos y + p~ cos y - p~)2 sin y - pal2 sin y) - sin ~0(2//d sin y
- p,)q) cos y) + p~ sin y + pd~, cos y] - [(//sin y + p',) cos y) cos 0 - P4~ sin y sin q0][2 cos 2 ~0(p//sin 2 y + p:~ sin y) - 202d sin s y cos 0 sin 0]
(20)
39 The equations (1)-(20) completely describe the kinematic and dynamic equations governing the gyrogrinder system.
3. Optimization strategy for Design Synthesis of a Gyrogrlnder In the design synthesis of a gyrogrinder, it is required to identify six major design parameters: length of spin axis, mass polar moment of inertias L Ii and Ix,, radius of grinding wheel, and angular velocity of the grinding wheel. One way to select these parameters is to initially assume a value for each one of them and to calculate the reaction forces Rn and RI from eqn (5) after solving for the rest of the governing equations (1)-(4), (6)-(7) and (10)-(20). Since the values of R, and Rt must satisfy eqns (8) and (9) for continuous rolling contact of the grinding wheel without separation, a check is made for any violation of eqns (8) and (9). If it is not violated, then the starting values of the parameter are feasible design values. Similarly, by trial and error, it is possible to obtain several set of such feasible parameters. Then it is the choice of the designer to select the most suitable (optimal) set of parameter that meets the design requirements. It is seen from the above description that the selection of design parameters directly influence the magnitude of R, and Rt. This paper utilizes this fact advantageously, in order to formulate an optimization strategy for the design synthesis of gryogrinder. Resorting to this type of strategy eliminates any trial and error procedure in selecting feasible design parameters. From the point of view of the grinding process, R~ and Rt represent the thrust force on the grinding wheel and the cutting force respectively. From eqn (5), it can be seen that both R, and R~ are generated due to the gyroscopic action and their magnitudes depend on the gyroscopic torques, radius of curvature and geometry of the template, and the geometry of the mechanism. Hence for a gyrogrinder, the values of R, and Rt while grinding an arbitrary template contour vary continuously around the contour. Especially at small radii of curvature, the value of R, becomes very small or even negative and thus causing the grinding wheel to depart from the template. Based on the above fact, the thrust force R~ must be a maximum positive quantity for a satisfactory grinding operation. In other words I1/RnI must be a minimum. In a grinding operation, the cutting force depends on the depth of cut, feed rate, and material properties of the workpiece and the grinder. Based on a particular grinding operation, the amount of cutting force required can be estimated and hence in a gyrogrinder, this cutting force must be generated. As stated earlier, Rt represents the cutting force in a gyrogrinder and is generated due to gyroscopic torques. Since the magnitude of Rt continuously varies around the template contour due to its dependence on the radius of curvature and geometry of the template, and the geometry of the mechanism, it is required that the minimum value of RI is at least equal to the required cutting force Ft. Thus, by preselecting the minimum value of Rt, i.e. (R/)mi.as equal to the required cutting force for grinding and by combining it in the minimization scheme, the new objective criterion becomes: Minimize I ~
I
where (Rt)min = Fo the cutting force. This criterion together with other conditions in the gyrogrinder operation can be mathematically expressed as follows: Objective Function:
J~l
minimize L, £ I , Ix,, ~os, Rg
(21)
max"
Subject to Constraints tt < tts; Rn > O; Rt >- (Rt)=i,. LI<_L<_L "
It~r<_Ix,.<_I~r
IJ<_I<_I "
w~t <- w~ <- tog~ I~~ <- 11 <- I~"
Rg~~ Rg ~ Rgu
(22)
40
where the subscripts l and u correspond to lower and upper bounds of each variable and = IR,/R.I.
It should be noted that for higher values of Fc, the optimization procedure may not yield a feasible solution due to the violation of the constraint Rt->(RDmi,. This indicates that in gyrogrinder, the cutting force generated purely due to the gyroscopic effect will not be sufficient and may require additional mechanism to control the cutting force. 4. Case Studies In order to illustrate the optimization strategy, three case studies of different template profiles are considered. They are: elliptical template, Limacon of Pascal, and Four-leaved rose. For each template, the equation of the radius vector p, the rate of change of p with respect to angle (s), and s are defined. Using these three equations and starting with the initial values for the six design parameters, the objective function evaluation and constraints checking can be carried out by solving the governing equations. A non-linear programming technique based on the sequential interior penalty function method[3-5] is used to convert the constrained optimization problem of this paper into a sequence of unconstrained problems. The Hooke and Jeeves[3] pattern search optimization technique is then used to carry out the unconstrained minimization of the objective function and to determine the optimal design parameters. It should be noted that the design parameter, "the length of spin axis L" is considered to be equal to Lo, the distance 00' (Fig. 3(b)) in the design. Case 1: The elliptical template
The elliptical template as shown in Fig. 4 is symmetrical about the V and W axes and has the following equations: p = ala2/x/(al 2 sin2 y + a22 cos 2 y)
(23)
where a], a2 are the semi major and minor axes of the ellipse taken along W and V axes respectively. s
_ dp
dy
SlS2S3
-
(24)
s4
where Sl-
ala2
2
S2 = a l 2 - 022
s3 = sin 2y s, = (al 2 sinE1, - aE: cos = y)3/=
=d/dp~ = dt \ d y ]
~i
//
ss(s6- sTs2s3)5/
\",
"~"~ - -"- ~ \ -
_ ._ DIRECTION OF TRACKING
Figure 4. Elliptical template (case 1).
(25)
41
where s 5 = ~ 4s~ S6 = 2S4 COS
2y
1 S7 = 2 S3~v/(S4) S2 = ( a l 2 -
a2 2)
s3 = sin 23,.
Case 2: The Limacon of Pascal The Limacon of Pascal is shown in Fig. 5 and its equation in polar coordinates are given by: p = a, cos y + a2
(26)
s = - a, sin y
(27)
= - alq cos 3,.
(28)
It can be seen that the Limacon displays both negative and positive curvatures and the zone around 270° simulates the inside tracking of noncircular holes.
Case 3: The four-leaved rose The equations for the four-leaved rose shown in Fig. 6 are given as: p = a l COS
2y
(29)
(30)
s = - 2al sin 2y = - 4a1~, cos 2y.
(31)
5. Optimization Algorithm The design synthesis of a gyrogrinder has been formulated as a nonlinear programming problem in previous sections. There are many algorithms available to seek the solution of a nonlinear programming problem. In this paper an interior penalty function method in conjunction with the Hooke Jeeves method is used. Some of the special forms of the interior penalty
V
I /
Oi W
w
L ~. '/ • c,~//
~
DIRECTION OF TRACKING
H ~ DRECTIONOF TRACKING
Figure 5. Limacon of Pascal template (case 2).
Figure 6. Four lezved rose template (case 3).
42 function methods, which are used extensively, are as follows: Consider a nonlinear optimization problem: Minimize f(2); 2 = xl, x2... xn Subject to gj($) ~<0 j = 1, 2 . . . . p. For this task, McCormick and Fiacco's version of interior penalty function is given below [3]: ¢(2, r) = f(2) - r
gS-O"
The modified objective function ¢($, r) can be minimized using any unconstrained optimization routine. The penalty parameter r is sequentially decreased between the unconstrained minimization and each unconstrained optimization begins where the preceeding one terminates. For this method, the initial starting point must satisfy the inequality constraints. Another form proposed by Siddall[6] is as follows: $(-f) = f(-f) + 102° ~ i=1
Ig ( )l
for gj(2) > 0.
(23)
The modified objective function as given by Siddall is used in this paper and the unconstrained minimization is carried out by Hooke and Jeeves pattern search technique. For the gyrogrinder design, the inequality constraints/z - (Rt)=in and R~ > 0 can be directly implemented into the equation (23) by considering the following: g , ( ~ ) = ( ~ - ~,,); f ( ~ ) = - Rn
and g3(2) = (Rt)min- Rt. Once the constrained optimization problem is reduced to the form in eqn (23), the Hooke and Jeeves unconstrained optimization method can be directly used. The Hooke and Jeeves method is a direct search routine for minimizing a function f($) of several variables $ = (Xl, x2, ... xn). The augment $ is varied until the minimum of f(2) is obtained. The pattern search routine determines the sequence of value of $, an independent routine computes the function values of f(J). A detailed description of the algorithm is given in Ref. 3. 6. Numerical Results By employing an interior penalty function of the form proposed by Sidall and using the Hooke and Jeeves direct search optimization technique, the optimal values of the gyrogrinder design parameters are obtained for each case studies considered. Optimization is performed with different set of initial values of the design variables. For each trial, the results indicate that the optimal values and the minimum value of the objective function are close to one another with a r.m.s, deviation less than 0.015. Table 1 shows the optimal results for one set of such starting values. It can be seen that the optimal values for L, L toe and R e are very close to each other for all three case studies. However, there is a slight discrepancy in the calculated optimal values of I and Ixr between the three cases. Since the three cam configurations represent the majority of all possible shapes in cam production, an optimal design of a gyrogrinder can be achieved by calculating the mean of the optimal values from each case study. They are:
L = 0.70753 m I1 = 0.3386 kg.m 2
450
350
~9 ( r e d / s )
Case 3:
Case 2:
Case 1 : ; a2 - 0.0508 m;
0.0127
eI ,, 0.5080m
; ;45" <_~r<_45° ;
a 1 - 0,254 re ; a2 - 0.0762 m;
aI - 0.254m
0.0064
] . 695
.566
Ixr(kg.m2)
Rg (m)
1.808
.904
(kg.m 2)
[
0.339
.113
Iz (kg°m2)
0.7112
•
0.0076
400.0
.904
1.017
.1695
0.6604
STARTING VALUE
Fc - .35.60 N
Fc • 35.60 N
CASE 2
CASE 3
0.0124
400.3
.7300
1.1910
.338
0.7000
0.0076
400.0
.904
1.017
.1695
0.6604
0.0124
400.42
.8170
1.2972
0.339
0.71;12
0.0075
400.0
.904
1.017
.1695
0.6604
0.0124
401.36
.6746
1.2464
0.3395
0.7112
OPTINUH STARTING OPTIHUH STARTING OPTIHUf4 VALUE VALUE VALUE VALUE VALUE
CASE 1
Numerical results
F¢ - 35.60 N
UPPER BOUND VALUE
0.635
LOMER BOUND VALUE
L(~)
DESIGN PARAHETER
Table
44 l : 1.2448 kg.m 2 [xr : 0.7405 kg.m 2 % = 400.693 rad/s Rg = 0.01245 m. Once the mean optimal values of the design parameters are calculated, a detailed dimensioning of the rotor and inner gimbal can be carried out for the final design.
7. Summary and Conclusion The instabilities that occurred during operation of the first generation gyrogrinder are investigated and the gyrogrinder redesigned by the use of an optimization scheme. This eliminated the tedious algebraic analysis. The following three shapes were considered as typical cam profiles: 1. ELLIPTICAL 2. LIMACON OF PASCAL 3. FOUR-LEAVED ROSE The dimensions obtained from the three cases were used to calculate the detailed dimensions for the rotor and the inner gimbal. Analysis shows that the optimal design parameters of the gyrogrinder lie within a small range for the three cam configurations. Since the three cam configurations represent all possible shapes in cam production, it is thus possible to utilize this gyrogrinder for machining complex curved surfaces. A gyrogrinder was designed and operated based on the optimal values obtained from this investigation. The grinder functioned effectively removing material from the workpiece (template) without any separation, thus providing supporting evidence for the technique of gyrogrinding. In this paper, no attempt is made to include the mechanism of controlling the depth of cut or to provide actual measurements on the accuracy of gryogrinding. However this will be the subject of our future investigation. Acknowledgements--The support of the National Research Council of Canada, Grant Nos. A5181and A3685, and the Formation de Chercheurs et d'Action Concertee, of the Governmentof Quebec, Grant No. 042-110 is gratefully acknowledged.
References 1. W. M. Mansourand D. Pavlov,The mechanismof GyroscopicTracking--Part 1. Trans ASME, J. Engng Industry, 95, Series B, No. 2, 430-436(1973). 2. W.M.Mansourand D. Pavlov,TheMechanismof GyroscopicTracking--Part2. TransASMEEngnglndustry. 95, Series B, No. 2, 437--444(1973). 3. D. M. Himmelblau,Applied Nonlinear Programming. McGraw-Hill,New York (1972). 4. R. L. Fox, Optimization Methods/or EngineeringDesign. Addison-Wesley,Reading,Mass. (1977). 5. A. V. Fiacco,and G. P. McCormick,NonlinearProgramming, Wiley,New York (1968). 6. J. N. Sidall,Analytical Decision Making in EngineeringDesign. Prentice-Hall,EnglewoodCliffs,New Jersey(1972).
CONCEPTION
SYNTHETIQUE
M.O.M.
Osman,
R~sum~
- On p r ~ s e n t e
sent~e
un n o u v e a u m ~ c a n i s m e
de c o n t o u r s
gyroscopique
la p ~ r i p h ~ r i e
GYROSCOPIQUE
et
sont pr~sent~s.
com~ae un p r o b l ~ m e
d'optimisation;
cherche
tudi6 trois p~tales,
sont c a l c u l ~ e s
directe
pour
A RECHERCHE
gabarits
typiques, des
continu,
fonctionnant
Le m o d u l e m a t h ~ m a t i q u e dans l e s q u e l l e s La c o n c e p t i o n
ainsi,
DIRECTE
soit elliptique,
sur le p r i n c i p e d~crivant
optimales
le m ~ c a n i s m e avec
est pr~-
des v a r i a b l e s
multiparam~trique
de
~ re-
de la meule.
On a ~-
en limaqon de Pascal et en rose
~ quatre
de cames,
dynamique
du pis-
reste en c o n t a c t
d'une meule g y r o s c o p i q u e
d'optimisation
sans ~ c l a t e m e n t
formes p o s s i b l e s
la meule
les valeurs
~ l'aide d'une m ~ t h o d e
un m e u l a g e
qui r e p r ~ s e n t e n t
de m e u l a g e
arbitraires.
les c o n d i t i o n s
du g a b a r i t
conception
PAR O P T I M I S A T I O N
S. S a n k a r et R.V. D u k k i p a t i
rage g y r o s c o p i q u e de m e u l a g e
D'UNE ~ U L E
et les v a l e u r s
optimales
des pa-
45 ram~tres de conception pour chaque forme de came ont ~t~ trouv~es par la m~thode d'optimisation.
Les r~sultats montrent que les valeurs optimales des param~tres de conception sont
comprises dans une plage limit~e pour chacune des trois formes de came.
Les moyennes opti-
males des param~tres de conception ont ~t~ calcul~es et utilis~es pour le choix des dimensions du rotor et du joint de cardan
int~rieur.
n~es de cette ~tude et, soumis ~ l'essai,
Un prototype a ~t~ construit avec les don-
il s'est av~r~ ~tre tr~s efficace.
Le present ex-
pos~ illustre l'utilisation de la conception assist~e par ordinateur pour la conception d'une meule gyroscopique.