a study of analysis and simulation of the valve ball’s lapping instrument with a cup shaped cutter
TRANSCRIPT
A study of analysis and simulation of the valve ball’s lapping instrument
with a cup shaped cutter
Pixiang Lan1, a, Guoqing Hu1, b, Ziwen Zheng2,c , Xiaofan Deng1, d,
Fengyuan Hou1, e 1Dept. of Mechanical & Electrical Engineering, Xiamen University, Xiamen 361005, China
2Fujian Wide Plus Precision Instruments Co., Ltd, Fuzhou 350015, China.
Keywords: valve ball, lapping, cup shaped cutter, simulation
Abstract. The valve ball is one of the most important parts in the ball valve. Lapping is a very
important pressing method in valve ball’s production. A valve ball's lapping instrument with a cup
shaped cutter, and build the lapping mathematics model is studied in this paper. To build the
mathematical of valve ball, theoretical analysis and the simulation, the results proven that some key
parameters could be properly controlled to improve the lapping process.
Introduction
Ball valve is widely used in modern industry. Sealing effect of ball valve is one of the most important
indicators. In order to achieve a good performance of sealing and satisfactory service life, steradian
and surface roughness of valve ball are very important in production. Lapping process is the main
method in valve ball's finishing machining. Lapping methods include hand-lapping, double-plate
lapping, special lapping machine, cup-shaped cutter lapping, and so on [1,2,3]. Cup-shaped lapping
instrument is very simple, so it is easy to design and produce. It is installed in an existing machine and
very economical. It meets the needs of both small size and large size balls; it is a very convenient tool.
Cup-shaped Lapping Instrument
The principle of operation is shown in Fig.1. The motion of lapping consists of two parts, the first part
is the rotation of ball valve driven by the spindle; the second one is the rotation of lapping instrument
driven by the servo motor[4]. The cup-shaped lapping cutter presses the valve ball with a certain
pressure. Abrasive situated in the ball and the lapping cutter will remove the metal from the surface of
the ball gradually as a result of the force and relative motion.
Fig.1 Cup-shaped lapping instrument
Valve ball
Cutter:connected
with a servo
Fixture：Connected
with the spindle
Advanced Materials Research Vols. 468-471 (2012) pp 569-573Online available since 2012/Feb/27 at www.scientific.net© (2012) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.468-471.569
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The Mathematical Model of Lapping Instrument
Solution of the lapping instument’s trajectory. As is shown in Fig.2, the radium of valve ball is R,
the pitch radium of lapping cutter is r, the angular velocity of spindle is w1, and the angular velocity of
lapping cutter is w2.
Fig.2 Mathematical model
When w1 = 0, which means the ball doesn't move, the trajectory of point P situated in the pitch
diameter is:
••
−
••
=
)2sin(
)2cos(
22
twr
rR
twr
z
y
x
(1)
Thus 2222 Rzyx =++ , the trajectory of point P is on a sphere.
When w1≠0, b1, b2 are the contact lines between the cutter and the sphere at the moment of t1
and t2, which can be considered as a rotating lapping cutter revolving around the sphere. Using the
relationship of coordinate transformation[5], we can describe the coordinate transformation matrix
as： T=
••
•−•
)1cos()1sin(0
)1sin()1cos(0
001
twtw
twtw , the trajectory of point P is:
••
−
••
••
•−•=
=
)2sin(
)2cos(
)1cos()1sin(0
)1sin()1cos(0
00122
1
1
1
twr
rR
twr
twtw
twtw
z
y
x
T
z
y
x
(2)
Derived from the nature of the matrix operations:
2222
1
1
1
)(1 Rzyx
z
y
x
T
z
y
x
T
z
y
x
=++•=
=
=
(3)
Thus 2212121 )()()( Rzyx =++ , the trajectory of point P is also on a sphere. Relative motion occurs
between the cutter and the sphere. This is the forming theory that the lapping instrument is capable of
sphere's lapping[6].
Solution of the lapping speed. Lapping speed is one of the most important parameters in lapping
process. There is a great difference in speed between rough lapping process and fine lapping
process[7]. Here we use previous geometry model to solve the synthesis speeds. As shown in Fig.3,
the relative speed of point P on pitch diameter of the lapping cutter and the valve ball is the synthesis
of sphere's speed v1 and lapping cutter's speed v2.
570 Automation Equipment and Systems
Fig.3 Speed's solution
Supposing the angle between the connecting line from point P to centre of the lapping cutter and x
axis as c（c=w2*t）, the coordinates of P is ( )2cos(2 twr •• ,22 )2(rR −, )2sin(2 twr •• ).
Curve a is the intersection line between sphere∑and plane∑1. Curve b is the intersection line
between sphere∑and plane∑2. Thus we get the equation of curve a and b:
a：2222),,( RzyxzyxG −++=，
)2cos(2),,( twrxzyxF ••−= (4)
b：2222),,( RzyxzyxH −++=，
22 )2(),,( rRyzyxK −−= (5)
From equation (4), we get the tangent vector of curve a:
( )yzyxxzzy
FF
GG
FF
GG
FF
GGT
yx
yx
xz
xz
zy
zy2,2,0
01
22
,10
22
,00
22
,,1 −=
=
=
(6)
Substitute T1 in the coordinates of point P, and we get the tangent vector of curve a at point P:
))2(),2sin(2,0(1 22 rRtwrT P −−••=. Then we get the direction cosine of T1P with further
calculation.
0cos =α ,
)2(cos)2(
)2sin(2cos
222 twrR
twr
••−
••=β
,
)2(cos)2(
)2(cos
222
22
twrR
rR
••−
−−=γ
.
With the equation of1)2(cos)2(111 222 wtwrRwrv •••−=•=,
the vector expression of speed v1:))2(1),2sin(21,0(1 22 rRwtwrwv −•−•••=，
similarly, the vector expression of speed v2: ))2cos(22,0),2sin(22(2 twrwtwrwv ••••••−= .
Thus, vector expression of the synthetic speed v is:
))2(1)2cos(22),2sin(21),2sin(22(21 22 rRwtwrwtwrwtwrwvvv −•−•••••••••−=+= (7)
And numeric value of the synthetic speed is:
)2cos()2(2212)2(cos)1()2()1()2()2( 222222222 twrRrwwtwwrwRwrv ••−••−••−•+•= (8)
Advanced Materials Research Vols. 468-471 571
Parametric analysis
The lapping trajectory has important effect on the homogenization of the ball’s surface, especially in
the solid-abrasive case, the more dispersed the trajectory is, the influence got from the defect of the
lapping tool is smaller; the lapping speed also affects the homogenization of the ball’s surface: if the
pressure is uniform, the elimination speed of the metal will be bigger when the lapping speed is bigger,
that is because when the speed is bigger, the amounts of abrasive get through the surface will be more,
then the abrasive will cut more metal[7].
The parameters’ effects on the trajectory. It can be found from equation(2), trajectory of point
P during lapping process is determined by the parameters of R, r, w1, w2 and t. R and r are determined
by the size of valve ball. So they couldn't be controlled during the lapping process; time is a variable
that couldn't be controlled either. So we will analyze the affect of w1, w2 over lapping trajectory
below.
Supposing at time t0, the sphere turns n1 laps while the lapping cutter turns n2 laps, when the
lapping motion gets back to the initial state, lapping trajectory begin to duplicate, so 12/10 nwt =• π ;
22/20 nwt =• π .
From the formula 00 tt = , we can get 2/121 wwnn •= . When w1>w2, if w1 and w2 are
irreducible, and the difference between them are bigger, the value of n1 will be larger. This means that
the lapping trajectory can be distributed widely on the ball’s surface, and vice versa. Here, we use
MATLAB to simulate w1 and w2's effects on lapping trajectories. Supposing R=7, r=5, running time
is 20s, and sampling time interval is 1/1000 s.
The first group : w1 ==4π, w2 = 8π, the result is shown in Fig.3(a);
The second group: w1 = 4π, w2 = 9π, the result is shown in Fig.3 (b);
The third group: w1 = 4π, w2 = 31π, the result is shown in Fig.3(c).
Fig.4 The lapping trajectory in different speeds
Comparison between the first group and the second shows if w1 and w2 are irreducible, there will
be an improvement in the lapping trajectory. Comparison between the second group and the third
shows the greater difference between w1 and w2, the more dispersed the trajectory is.
Simulation analysis confirms the aforementioned analysis of w1 and w2's influence on the lapping
trajectory is correct. As the lapping instrument described in this article is drove by servo motor, so we
can control the servo motor's rotating speed to improve the lapping trajectory, and reduce the
repeatability of lapping trajectory.
The parameters’ effects on the Lapping speed. According to the simulation parameters above,
w1 =4π, w2 = 31π, R = 7, r2 = 5, the running time is just for lapping cutter rotating a cycle (2π/32
π). With the simulation of MATLAB, we get the curve a of speed (equation 8) shown in Fig.5. From
the figure, it is known that the lapping speed along the axis is changing. Lapping speed of the positive
axis is higher than the negative, this will cause asymmetry between the positive and negative half
shaft. There are two ways to solve this problem: One way is to increase the difference between the
speeds of w1 and w2, the simulation results of speed curve b (w1 = 2 π, w2 = 31π) is shown in Fig.5.
This can improve the uniformity of lapping speed. The second way is to apply positive &negative
a b c
572 Automation Equipment and Systems
rotation to the cutter (sphere) or symmetrically arrange cutters on each side of the ball. As the
simulation curves shown in Fig.6, the lapping speed along the axis x will be more convergent when
these two curves combined together.
Fig.5 Speed curves at different value of w1/w2 Fig.6 Speed curves at positive &negative rotation
Conclusions
This mathematical modeling of the cup-shaped lapping instrument, and find the function expression
of both lapping trajectory and lapping speed is accomplished in the paper. With the theoretical
analysis and simulation of mathematical models, we find the key parameters (w1, w2) need to be
controlled to optimize the lapping process, and present two reasonable optimization methods: The
first is to enlarge the difference between them in the precondition of w1 and w2 are not reducible; The
second is to arrange w1 and w2 in opposite directions. Theoretical analysis serves as a very important
guide for the practice, making the practice to be clear and regular.
References
[1] Jianyu Hua, Lin Li: Valve Vol.2 (2008), p. 27-28.
[2] Xianglin Wang: Mechanical craft Vol.6 (1998),p.37
[3] Mingming Wu, Jianping Wang, Zhaozhong Zhou: Development & Innovation of Machinery &
Electrical Products Vol.1(2004), p. 94-96.
[4] Haiyu Lai, Zhizhong Yang: Valve Vol.2 (2006), p. 34-35.
[5] Hongyi Liu: Fundamentals of Robot Techniques in Chinese(Metallurgical Industry Press, China
2002)
[6] Luiyuan Liu:Research on Key Technology for Metal-seated High-temperature Wear-resistant
Ball Valve in Chinese (Zhejiang university Ph.D. Thesis 2008)
[7] Hongru Wang: The Valve’s Latest Standards and its Engineering Application Technology
Encyclopedi in Chinese (Yinxiang Av Publishing House 2004).
Advanced Materials Research Vols. 468-471 573
Automation Equipment and Systems 10.4028/www.scientific.net/AMR.468-471 A Study of Analysis and Simulation of the Valve Ball’s Lapping Instrument with a Cup Shaped Cutter 10.4028/www.scientific.net/AMR.468-471.569