- Email: [email protected]
The rounding off theory of centerless grinding
Int. J Mach Tool Des Res Voi 21. No I. pp 4~) 55. t%1 Punted m Great Britain
0020-'35 ~ 81 '01 0049-0? $02 0(I 0 Pergamon Press Lid
THE R O U N D I N G OFF THEORY OF CENTERLESS GRINDING A. Y. CHIEf* (Originally receired 24 April 1980: in final form
22
September 1980)
A l ~ t r a e t - - O p t i m u m c o n f i g u r a t i o n in centerles~ g r i n d i n g is i n v e s t i g a t e d a n d stability c h a r t s are p r o d u c e d by a simple method. These c h a r t s serve as a guide for a v o i d i n g g e o m e t r i c instabilit? and for o b t a i n i n g high accurac~ in grinding.
NOMENCLATURE
0 r, r(O) S So
S~ Ai
7 h D, D~ d fl
q; ~p 01
02 Ci Ar Ag~ Ag2 A
Central angle which determines workpiece position in polar coordinates Radius of the cross-section circle of the workpiece Vector Component of the vector S, parallel to the error lever or to the vector of the initial roundness error of the workpiece Component of the vector S, perpendicular to the error lever or to the vector of the initial roundness error of the workpiece Rounding coefficient Included angle between the grinding wheel contact normal and the control wheel contact normal. This will be referred to as the 'included tangents angle' or the 'center height angle' Height of workpiece center Grinding wheel diameter Control wheel diameter Workpiece diameter Angle between the grinding wheel contact normal and the workblade contact normal Angle between the workblade contact normal and the control wheel contact normal Angle between the line joining the wheel centers and the inclined face of the workblade Angle between the line joining the wheel centers and the grinding wheel contact Angle between the line joining the wheel centers and the control wheel contact line The amplitude of the ith waviness on the workpiece A positive irregularity on the workpiece (increment in workpiece radius) Movement of workpiece center normal to grinding wheel surface when Ar contacts the control wheel (position 1) Movement ofworkpiece center normal to grinding wheel surface when Ar contacts the workblade (position 2) Roundness error
INTRODUCTION
NUMEROUS studies of the basic parameters of centerless grinding, that aim at obtaining highest grinding accuracy, are reported in the literature. In particular, Rowe and Barash [1] simulated the centerless grinding process on a digital computer, having first assumed a machining elasticity parameter. They obtained good agreement between the predicted and
* Q i n g h a i A g r i c u l t u r e a n d A n i m a l H u s b a n d r 3 Machiner.~ Factory. C h i n a 49
50
A. Y.
CHIFN
F)(J. I.
the experimental results. However, their method requires a considerable amount of calculations and is not convenient for use under production conditions. This paper was developed from their work, and offers a simplified theory of rounding offin centerless grinding. Based on the 'lever postulate of the roundness error' and using simple mathematics, it enables one to consistently obtain optimum grinding configurations for different conditions. THE LEVER POSTULATE OF THE ROUNDNESS ERROR The radius r of the workpiece is a function of the angle 0 (Fig. 1), which is continuous, single-valued and with the period of 2~. Let the workpiece cross-section form or 'profile' be described by a Fourier series: r(O} = ro + ~ Cicos(iO + ~i) i=2
in which the eccentricity error (the first harmonic) should be disregarded. Clearly A is given by the equation: A = r(O)m,, - r(O)r,i,.
Now that the roundness error has been expressed in terms of Fourier series, an example of generating the roundness error in rotation will be presented. In Fig. 2, 001 represents a bar of length r 0 and OE which is termed 'the error lever' is equivalent to Ci. The bar 00] rotates about the fixed point 01 with angular velocity co = dO/dt, while OE rotates about the point O with the velocity ico. It means that point E of the error lever rotates about point O1 in addition to its rotation about point O. The curve which is traced by point E on the plane is described by the equation r(O) = r o + Ci cos iO. It can be easily shown that the roundness error of the curve is only determined by the length of the OE vector and is independent of its direction. Ifa vector S which can be resolved into two components So and S. is added to OE, only So would affect the magnitude of the roundness error of the curve. This we call the above-mentioned lever postulate of the roundness error, which is put forward in the paper.
,,,.
F)G. 2.
The Rounding Off Theory of Centerless Grinding
51
r~_ 0 " C'
++/I~
Ag:
0~8Ag+
0.
~ro~ position I
A r at posi~on 2
Fit +. 3.
MATHEMATICAL
RELATIONSHIPS
Figure 3 shows the geometrical relationships that connect the workpiece, grinding wheel. control wheel and workblade in centerless grinding. We assume these to be constant because of the negligible change in workpiece diameter during grinding. The following relationships are obtained from Fig. 3: sin O~ -
2h D~+ d
2h sin 02 = - D~ + d
7r
n=~-~o-O~ 7I ~=~+~o-02.
Because 7 is small and d is negligible compared to D 8 or Dc we have
01
sin 0x
Dc + d
D~
- - ~ - - = - ~--=constant. 02 sin 0: D+ + d D+ It means that the geometrical configuration is determined by three values: y, tp and D+/D s. If an irregularity on the workpiece occurs at either the workblade contact or the control wheel contact as shown in Fig. 3+ the center of the workpiece will be displaced and this movement will change the depth of cut at the grinding wheel contact. We shall analyze these
52
A.Y. CHIF~
relationships. An irregularity Ar on the workpiece at the control wheel contact position I will cause the center of the workpiece to move parallel to the inclined face of the workblade. In the vector diagram in Fig. 3, 00' represents this displacement. The component of this displacement normal to the grinding wheel surface at the contact point is given by Ag~ = 00'cos (¢p + 01) = 00'sin ~7 =
Ar sin r/ sin~, '
Similarly Ar at the workblade contact position 2 will cause the center to move in a direction tangential to the control wheel. This is represented by 00" in Fig. 3 and has a component normal to the grinding wheel surface given by Ag2 = - 0 0 " s i n T =
-00"sin(q+0)=
-Ar
sin (~/+ 0) sin ~O
where the " - " sign means that the direction of Ag: is opposed to Agl. If Ar is positive, Agl increases the depth of cut while Ag2 reduces it. As mentioned before, the profile of the workpiece being ground can be described as follows : r(O) = ro + ~,, Cicos(iO + =~). i=2
If the center of the workpiece was to be held stationary during grinding, the'roundness error of the workpiece would be completely removed by the grinding process. The equation of the depth of cut would then be Ar(0) = Ar0 + ~ Ci cos (iO + ~ti). In reality this is not possible. The depth of cut is modified by increments imposed by the control wheel and the workblade. The phase angle between the grinding wheel contact and the control wheel contact is equal to q + 0. It can be shown that the profile at contact point 1 is given by
r0 + Y Cicos [i(0-,;-¢) + ,,~]. The depth of cut imposed by the control wheel is Agl = sin r/ {Ar0 + ~ Ci cos [i(O - rl - ¢,) + ai]}. The phase angle between the grinding wheel contact and the workblade contact is ~/. The profile at contact point 2 can be shown to be ro + ~Cicos[i(O-rl)
+ ~i].
The depth of cut imposed by the workblade is Ag2 =
sin (r/+ 0) {Aro + ~ C, cos [ i ( O - rl) + 0q]}. sin ¢
Therefore the depth of cut in actual grinding is the sum of the three components mentioned above. After rearranging we obtain [ __sinr/ A r ( 0 ) = A r o 1 + sin0
sin (r/+ ¢,)]_: sm
+ ~ C , cos(iO + ~t,)
sin r/
+ Zs~n~C, cos[i(o-,;-¢) + ~,J _
~ sin (q + ¢,) C~ si--n~. cos [i(O -- rl) .+ ;ti]
The Rounding Off Theory of CenterlessGrinding
.
Gnndingwheel
~ /
53
WorKoiec e
S
//
'// /Workblode FiG. 4.
in which the first term that contains Ar0 has no effect on the roundness error of the workpiece. The roundness error is only determined by the remaining three terms, especially the last two. STABILITY CHARTS Consider one harmonic of a certain order. Let Sl be the depth of cut of the ith harmonic. S'~ = C~ cos (iO + ~ } sin rl + ~ c, cos [i(o-,1- ~,) + ~] sin (~/ + ~,) C~ cos [i(0-~) + ~]. sin qs
(1)
Through transformation of trigonometric functions, equation (1) can be changed into S'i = S icOS (iO -F
~i).
(2)
The vector diagram Fig. 4 illustrates the addition of trigonometric functions with different phase angles. Si = Ci + Cias~ + Cia~ where Ci stands for the initial error of the workpiece. Assuming the direction o f C i tO be on the line 001 and resolving Si into two components So and S~ we have
Si = So + S~ tan 6i =
S~ S0
~l = 2nn -
i(r/+~k)
~82 = 2 n n -
irl
in which n is the number of workpiece revolutions made up to the present moment. Therefore sin r/ So = c i [ 1 + _.._~, cos i{,r+~,) sln~
sin (~ + ~) cos it/] sin
(3)
54
A.Y. Cmrx
Fsin (r/ + qj) sin q Sj_ = CiL ~ n ~ siniq - s i - ~ s i n i(~/+ ~,)].
(4)
The lever postulate of the roundness error mentioned above has shown us that only So determines the roundness error of the workpiece. When both sides of equation (3) are divided by C~ one obtains So
sin q = 1 +~cosi(q+~)
sin (rt + ¢) sin~, c o s i r / = 1 +A~
(5)
where sin i/ Ai = ~ cos i (~ + q')
sin (7 + ¢) cos irt. sin" q,
/6)
Ai is defined as the rounding coefficient of the ith harmonic and is only determined by the magnitudes of angles q and ¢. Let us discuss Ai in three different cases: (1) Ai -- - 1, i.e. So = Ci(A~ + 1) = 0. The initial error Ai cos (iO + ~) persists without increasing or decreasing during the grinding period. (2) A~ < - 1, which means that the angle 6~is larger than 90 °. When the highest point of the initial error is at the grinding wheel contact, the center of the workpiece moves away from the grinding wheel, an d when the lowest point of the error is at the grinding contact the center moves toward the grinding wheel. Obviously the roundness error increases with grinding. This is referred to as grinding instability. (3) A~ > - 1, 1 + A~ > 0. It means that the direction of So coincides with C~. The roundness error consistently decreases in grinding. If the magnitude of Ai is close to zero, i.e. So is close to C~, the initial error will be quickly diminished. Optimum grinding results will be obtained. In order to simplify the work of arranging the geometrical configuration in centerless grinding, stability charts of geometrical configurations together with all harmonics lower than the 20th (Fig. 5) or the 14th (Fig. 6) have been plotted. The closed curves in these figures represent the equation A i = - 1. The regions where Ai < - 1 are shaded and the numbers indicate the order of the harmonic. These are the regions of geometrical instability in terms of the indicated harmonics. When adjusting the centerless grinding set-up, one should select the correct blade angle ~0 and the included tangents angle y. The stability charts have ~0 and 7 as the coordinate axes.
50*
3
40
>,
i -4*
c"
~.
~'. ¥
Fl(~. 5.
,~.
,~.
The Rounding Off Theory of Centerless Grinding bC~
/
z
/
55
/n
>')'.7) ;:.,/.//
~o
"///"
12 - 4"
0"
4o
~.
,'2.
,'~.
FI6.6.
The ratio Dc./Dg also changes the shape of the curves in the stability charts. However. calculation results obtained with equation (6) show that the changes in shape are very small if the ratio is in the range of 0.5-1. Figures 5 and 6 are drawn with the ratio being equal to 0.7. Figure 5 considers only the harmonics of an order lower than 20, ignoring the higher ones, because the inertia of the workpiece and the presence of frictional damping will always tend to oppose the vibration of workpiece center at higher frequencies. If workpiece diameter is small, one considers only harmonics of the 3rd, 5th and 7th order. The configuration of the centerless grinding set-up is to be arranged according to stability charts that are based on the effect of harmonics that occur in the grinding process. The charts show the effect of 7 and tp on the roundness error of the workpiece. Many suggestions derived from the charts have been successfully applied in mass production. The lever postulate is the core of the stability charts. It exposes the essential property of the rounding off theory in centerless grinding using Simpler mathematics than those employed in other methods.
REFERENCE
[1] W. B. RowF and M. M. B.aR-xSH,Ira. J. Math. Tool Des. Res 4, 91 I1964~
0020-'35 ~ 81 '01 0049-0? $02 0(I 0 Pergamon Press Lid
THE R O U N D I N G OFF THEORY OF CENTERLESS GRINDING A. Y. CHIEf* (Originally receired 24 April 1980: in final form
22
September 1980)
A l ~ t r a e t - - O p t i m u m c o n f i g u r a t i o n in centerles~ g r i n d i n g is i n v e s t i g a t e d a n d stability c h a r t s are p r o d u c e d by a simple method. These c h a r t s serve as a guide for a v o i d i n g g e o m e t r i c instabilit? and for o b t a i n i n g high accurac~ in grinding.
NOMENCLATURE
0 r, r(O) S So
S~ Ai
7 h D, D~ d fl
q; ~p 01
02 Ci Ar Ag~ Ag2 A
Central angle which determines workpiece position in polar coordinates Radius of the cross-section circle of the workpiece Vector Component of the vector S, parallel to the error lever or to the vector of the initial roundness error of the workpiece Component of the vector S, perpendicular to the error lever or to the vector of the initial roundness error of the workpiece Rounding coefficient Included angle between the grinding wheel contact normal and the control wheel contact normal. This will be referred to as the 'included tangents angle' or the 'center height angle' Height of workpiece center Grinding wheel diameter Control wheel diameter Workpiece diameter Angle between the grinding wheel contact normal and the workblade contact normal Angle between the workblade contact normal and the control wheel contact normal Angle between the line joining the wheel centers and the inclined face of the workblade Angle between the line joining the wheel centers and the grinding wheel contact Angle between the line joining the wheel centers and the control wheel contact line The amplitude of the ith waviness on the workpiece A positive irregularity on the workpiece (increment in workpiece radius) Movement of workpiece center normal to grinding wheel surface when Ar contacts the control wheel (position 1) Movement ofworkpiece center normal to grinding wheel surface when Ar contacts the workblade (position 2) Roundness error
INTRODUCTION
NUMEROUS studies of the basic parameters of centerless grinding, that aim at obtaining highest grinding accuracy, are reported in the literature. In particular, Rowe and Barash [1] simulated the centerless grinding process on a digital computer, having first assumed a machining elasticity parameter. They obtained good agreement between the predicted and
* Q i n g h a i A g r i c u l t u r e a n d A n i m a l H u s b a n d r 3 Machiner.~ Factory. C h i n a 49
50
A. Y.
CHIFN
F)(J. I.
the experimental results. However, their method requires a considerable amount of calculations and is not convenient for use under production conditions. This paper was developed from their work, and offers a simplified theory of rounding offin centerless grinding. Based on the 'lever postulate of the roundness error' and using simple mathematics, it enables one to consistently obtain optimum grinding configurations for different conditions. THE LEVER POSTULATE OF THE ROUNDNESS ERROR The radius r of the workpiece is a function of the angle 0 (Fig. 1), which is continuous, single-valued and with the period of 2~. Let the workpiece cross-section form or 'profile' be described by a Fourier series: r(O} = ro + ~ Cicos(iO + ~i) i=2
in which the eccentricity error (the first harmonic) should be disregarded. Clearly A is given by the equation: A = r(O)m,, - r(O)r,i,.
Now that the roundness error has been expressed in terms of Fourier series, an example of generating the roundness error in rotation will be presented. In Fig. 2, 001 represents a bar of length r 0 and OE which is termed 'the error lever' is equivalent to Ci. The bar 00] rotates about the fixed point 01 with angular velocity co = dO/dt, while OE rotates about the point O with the velocity ico. It means that point E of the error lever rotates about point O1 in addition to its rotation about point O. The curve which is traced by point E on the plane is described by the equation r(O) = r o + Ci cos iO. It can be easily shown that the roundness error of the curve is only determined by the length of the OE vector and is independent of its direction. Ifa vector S which can be resolved into two components So and S. is added to OE, only So would affect the magnitude of the roundness error of the curve. This we call the above-mentioned lever postulate of the roundness error, which is put forward in the paper.
,,,.
F)G. 2.
The Rounding Off Theory of Centerless Grinding
51
r~_ 0 " C'
++/I~
Ag:
0~8Ag+
0.
~ro~ position I
A r at posi~on 2
Fit +. 3.
MATHEMATICAL
RELATIONSHIPS
Figure 3 shows the geometrical relationships that connect the workpiece, grinding wheel. control wheel and workblade in centerless grinding. We assume these to be constant because of the negligible change in workpiece diameter during grinding. The following relationships are obtained from Fig. 3: sin O~ -
2h D~+ d
2h sin 02 = - D~ + d
7r
n=~-~o-O~ 7I ~=~+~o-02.
Because 7 is small and d is negligible compared to D 8 or Dc we have
01
sin 0x
Dc + d
D~
- - ~ - - = - ~--=constant. 02 sin 0: D+ + d D+ It means that the geometrical configuration is determined by three values: y, tp and D+/D s. If an irregularity on the workpiece occurs at either the workblade contact or the control wheel contact as shown in Fig. 3+ the center of the workpiece will be displaced and this movement will change the depth of cut at the grinding wheel contact. We shall analyze these
52
A.Y. CHIF~
relationships. An irregularity Ar on the workpiece at the control wheel contact position I will cause the center of the workpiece to move parallel to the inclined face of the workblade. In the vector diagram in Fig. 3, 00' represents this displacement. The component of this displacement normal to the grinding wheel surface at the contact point is given by Ag~ = 00'cos (¢p + 01) = 00'sin ~7 =
Ar sin r/ sin~, '
Similarly Ar at the workblade contact position 2 will cause the center to move in a direction tangential to the control wheel. This is represented by 00" in Fig. 3 and has a component normal to the grinding wheel surface given by Ag2 = - 0 0 " s i n T =
-00"sin(q+0)=
-Ar
sin (~/+ 0) sin ~O
where the " - " sign means that the direction of Ag: is opposed to Agl. If Ar is positive, Agl increases the depth of cut while Ag2 reduces it. As mentioned before, the profile of the workpiece being ground can be described as follows : r(O) = ro + ~,, Cicos(iO + =~). i=2
If the center of the workpiece was to be held stationary during grinding, the'roundness error of the workpiece would be completely removed by the grinding process. The equation of the depth of cut would then be Ar(0) = Ar0 + ~ Ci cos (iO + ~ti). In reality this is not possible. The depth of cut is modified by increments imposed by the control wheel and the workblade. The phase angle between the grinding wheel contact and the control wheel contact is equal to q + 0. It can be shown that the profile at contact point 1 is given by
r0 + Y Cicos [i(0-,;-¢) + ,,~]. The depth of cut imposed by the control wheel is Agl = sin r/ {Ar0 + ~ Ci cos [i(O - rl - ¢,) + ai]}. The phase angle between the grinding wheel contact and the workblade contact is ~/. The profile at contact point 2 can be shown to be ro + ~Cicos[i(O-rl)
+ ~i].
The depth of cut imposed by the workblade is Ag2 =
sin (r/+ 0) {Aro + ~ C, cos [ i ( O - rl) + 0q]}. sin ¢
Therefore the depth of cut in actual grinding is the sum of the three components mentioned above. After rearranging we obtain [ __sinr/ A r ( 0 ) = A r o 1 + sin0
sin (r/+ ¢,)]_: sm
+ ~ C , cos(iO + ~t,)
sin r/
+ Zs~n~C, cos[i(o-,;-¢) + ~,J _
~ sin (q + ¢,) C~ si--n~. cos [i(O -- rl) .+ ;ti]
The Rounding Off Theory of CenterlessGrinding
.
Gnndingwheel
~ /
53
WorKoiec e
S
//
'// /Workblode FiG. 4.
in which the first term that contains Ar0 has no effect on the roundness error of the workpiece. The roundness error is only determined by the remaining three terms, especially the last two. STABILITY CHARTS Consider one harmonic of a certain order. Let Sl be the depth of cut of the ith harmonic. S'~ = C~ cos (iO + ~ } sin rl + ~ c, cos [i(o-,1- ~,) + ~] sin (~/ + ~,) C~ cos [i(0-~) + ~]. sin qs
(1)
Through transformation of trigonometric functions, equation (1) can be changed into S'i = S icOS (iO -F
~i).
(2)
The vector diagram Fig. 4 illustrates the addition of trigonometric functions with different phase angles. Si = Ci + Cias~ + Cia~ where Ci stands for the initial error of the workpiece. Assuming the direction o f C i tO be on the line 001 and resolving Si into two components So and S~ we have
Si = So + S~ tan 6i =
S~ S0
~l = 2nn -
i(r/+~k)
~82 = 2 n n -
irl
in which n is the number of workpiece revolutions made up to the present moment. Therefore sin r/ So = c i [ 1 + _.._~, cos i{,r+~,) sln~
sin (~ + ~) cos it/] sin
(3)
54
A.Y. Cmrx
Fsin (r/ + qj) sin q Sj_ = CiL ~ n ~ siniq - s i - ~ s i n i(~/+ ~,)].
(4)
The lever postulate of the roundness error mentioned above has shown us that only So determines the roundness error of the workpiece. When both sides of equation (3) are divided by C~ one obtains So
sin q = 1 +~cosi(q+~)
sin (rt + ¢) sin~, c o s i r / = 1 +A~
(5)
where sin i/ Ai = ~ cos i (~ + q')
sin (7 + ¢) cos irt. sin" q,
/6)
Ai is defined as the rounding coefficient of the ith harmonic and is only determined by the magnitudes of angles q and ¢. Let us discuss Ai in three different cases: (1) Ai -- - 1, i.e. So = Ci(A~ + 1) = 0. The initial error Ai cos (iO + ~) persists without increasing or decreasing during the grinding period. (2) A~ < - 1, which means that the angle 6~is larger than 90 °. When the highest point of the initial error is at the grinding wheel contact, the center of the workpiece moves away from the grinding wheel, an d when the lowest point of the error is at the grinding contact the center moves toward the grinding wheel. Obviously the roundness error increases with grinding. This is referred to as grinding instability. (3) A~ > - 1, 1 + A~ > 0. It means that the direction of So coincides with C~. The roundness error consistently decreases in grinding. If the magnitude of Ai is close to zero, i.e. So is close to C~, the initial error will be quickly diminished. Optimum grinding results will be obtained. In order to simplify the work of arranging the geometrical configuration in centerless grinding, stability charts of geometrical configurations together with all harmonics lower than the 20th (Fig. 5) or the 14th (Fig. 6) have been plotted. The closed curves in these figures represent the equation A i = - 1. The regions where Ai < - 1 are shaded and the numbers indicate the order of the harmonic. These are the regions of geometrical instability in terms of the indicated harmonics. When adjusting the centerless grinding set-up, one should select the correct blade angle ~0 and the included tangents angle y. The stability charts have ~0 and 7 as the coordinate axes.
50*
3
40
>,
i -4*
c"
~.
~'. ¥
Fl(~. 5.
,~.
,~.
The Rounding Off Theory of Centerless Grinding bC~
/
z
/
55
/n
>')'.7) ;:.,/.//
~o
"///"
12 - 4"
0"
4o
~.
,'2.
,'~.
FI6.6.
The ratio Dc./Dg also changes the shape of the curves in the stability charts. However. calculation results obtained with equation (6) show that the changes in shape are very small if the ratio is in the range of 0.5-1. Figures 5 and 6 are drawn with the ratio being equal to 0.7. Figure 5 considers only the harmonics of an order lower than 20, ignoring the higher ones, because the inertia of the workpiece and the presence of frictional damping will always tend to oppose the vibration of workpiece center at higher frequencies. If workpiece diameter is small, one considers only harmonics of the 3rd, 5th and 7th order. The configuration of the centerless grinding set-up is to be arranged according to stability charts that are based on the effect of harmonics that occur in the grinding process. The charts show the effect of 7 and tp on the roundness error of the workpiece. Many suggestions derived from the charts have been successfully applied in mass production. The lever postulate is the core of the stability charts. It exposes the essential property of the rounding off theory in centerless grinding using Simpler mathematics than those employed in other methods.
REFERENCE
[1] W. B. RowF and M. M. B.aR-xSH,Ira. J. Math. Tool Des. Res 4, 91 I1964~