機械設計外文翻譯-凸輪速度對凸輪系統影響的實驗研究【中文4500字】【PDF+中文WORD】
機械設計外文翻譯-凸輪速度對凸輪系統影響的實驗研究【中文4500字】【PDF+中文WORD】,中文4500字,PDF+中文WORD,機械設計,外文,翻譯,凸輪,速度,系統,影響,實驗,研究,中文,4500,PDF,WORD
⑥陽酬。因
侃”，4-114x份制刷7-9
Mech. Mach. Theory Vol 31, No. 4, pp. 397-412, I”6 Copyright ⑥ 1996 El回vier Science Ltd
Printed in Great Britain. All rights re甜”“
0094-l 14X/96 $15.00 + 0.00
AN EXPERIMENTAL STUDY OF THE EFFECTS OF CAM SPEEDS ON CAM-FOLLOWER SYSTEMS
H. S. YAN and M. C. TSAI
Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, Republic of China
M. H. HSU
Department of Mechanical Engineering, Kung Shan Institute of Technology and Commerce, Yungkang,
Tainan 71016, Taiwan, Republic of China
(Received 9 September 1994; received for publication 26 October 1995)
Abstract-Traditionally, in a cam-follower system, the cam is often operated at a constant sp臼d and the motion characteristics of the follower are determined once the cam displacement curve is designed. From the kinematic point of view, the approach by varying cam input driving speed is an alternative way for
improving the follower motion characteristics. Here we show how to find a polynomial speed tr句“tory
for reducing the peak values of the motion characteristics. Furthermore, constraints and systematic design procedures for generating an appropriate tr句“tory of the cam angular velocities are developed. Design examples are given to illustrate the procedure for getting an appropriate speed trajectoηas variable speed
cam-follower systems. Furthermore, an experimental setup 叨th a servo controller is developed to study the feasibility of this approach. Experimental data show that the results are very close to those of theory.
NOME NCLATUR E
a-acceleration of the follower
A, A0-normalized acceleration of the follower
c,d,e爪，孔，孔，x, y--constant parameters
h-maximum displacement of the follower
J一jerk of the follower
J, J0-normalized jerk of the follower
s--displacement of the follower
S-normalized displacement of the follower t一time for the cam to rotate through angle (}
T，凡，T 凡，normalized time
v--velocity of the follower
→
V, V0-normalized velocity of the follower
→：
F am angle rotation for total rise h
A ，品，島，A am rotation angle
1扣一normalized cam angle of rotation 0一切m an阱 of rotation
τ time of cam rotation for total rise h
h，勺，勺，馬一time of cam rotation α護-cam angluar velocity
w.,.-average cam angular velocity of a complete cycle
ω時 w,2, w,3 , w54-average cam angular velocity in a follower motion period φ the 1st derivative of w
d頭 the 2nd derivative of w
。一
'1-normalized cam angular velocity 0-the 1st derivative of n
the 2nd derivative of n
INTR ODUCTION
In a cam-follower system, the load produced by inertia forces is prone to deflection and creates vibrations; and the load introdu臼d by jerks may cause vibrations as well. These will d創 the operating life of the cam. Therefore, the design of motion cu凹臼 to minimize dynamic loading is of importan四 for high sp臼d cam mechanisms. It is well known that the velocity and acceleration cu凹es are required to be continuous and to have smaller peak values. In addition, the jerk curve should be finite.
397
398 H. S. Yan et al.
A cam is often assumed to be operated at a constant speed in designing a cam-follower system. However, the motion characteristics of the follower are changed as the cam speed varies. Traditionally, to achieve the desired motion is an application of synthesis for obtaining new displacement curves which have better dynamic characteristics. In this paper, we propose an alternative method by varying the cam speeds. The concept of using variable speeds in a cam-follower system design was seldom studied in the literature. Rothbart [1] designed a variable speed cam mechanism in which the input to the cam is the output of a Withworth quick-return mechanism. Tesar and Matthew [2] derived the motion equations of the follower by considering the case of variable speed cams. The criteria for selecting proper angular velocities which will eliminate the discontinuity in motion characteristics of the follower are investigated by Yan et al. [3]. From the kinematic point of view, the objective of this work is to find cam speed tr哉jectoη for reducing the peak values of the follower-output motion. Furthermore, constraints and system design procedures for generating a proper trajectoη of the cam angular velocities are deveoped. Design examples are given to illustrate the procedure for getting a proper angular speed for a given follower system. An experimental cam-follower system is set up in which a servo motor is controlled to generate the desired speed trajectory for performance evaluations.
MOTIO N EQU ATIO NS
For a cam-follower system, the follower displacement, s( t ), is a function of cam rotation angle
O( t ). Mathematically, it can be expressed as:
．
．
、
且
，，
·
a，·飛
s(t ) =f (O( t ))
where O ( t ) is the cam rotation angle at time t. The follower velocity, v( t ), of the follower is then given by:
v( t ) =f ’（O)ro(t) (2)
where f' (O ) = df 刷刷，and ro(t) = dO( t )/d t is the cam angular velocity. Furthermore, the corresponding follower acceleration, a( t ), and jerk, j( t ), are:
a( t ) =f "
j( t ) = f "'(O)ro3(t) ＋ ν，＇（O)ro(t )<.iJ(t ) +f' (O ）φ（t ）
(3)
(4)
where f "(O) = df 2(() )/d0 2, f' ”（0) = df 3(0)/d03，φ（t ) = dw( t )/d t, and co( t ) = dw 2( t )/d t 2. Equations (1)-(4) present the relationship between cam input angular velocity w(t) and follower-output motions s( t), v( t ), a( t ), and j(t ). Obviously if w( t ) is a constant, they can be greatly simplified.
Let h be the total displacement of the follower as the cam rotates an angle P in time period τ．
Furthermore, denote T = t /t, y 咐，and S = s/h. Now we have T e[O, 1], 'v't e [O，τ ］ ，y e [O, 1], 'v'O e[O, P] and Se[O, l], 'v's e[O, h]. Then, equations (1)-(4) can be rewritten in terms of their normalized forms as follows:
= h
s( t )
S(T) = g( y )
V(T) = g ’ ）Q(T)
A(T ) = g 氣y )Oi(T ) +g ’ ）Q(T)
J(T) = g ”’（y )03(T) + 3g 氣y )0(T)Q( T) +g ’ ）凸（T)
(5)
(6)
(7)
(8)
where O(T) = dy (T)/dT is the normalized cam angular velocity and V(T), A(T ), and J(T) are the normalized velocity, acceleration, and jerk of the follower, respectively. The relationship between equations (1) to (8) can be found as:
s = hS (9)
V＝雪 V
t
(10)
‘
、
l
－－且
，SE飛
Effects of cam speeds on cam-follower systems 399
α ＝A
r
i
與
τ
EF
(12)
When the cam operates at a constant speed, i.e. Q(T) = 1, the normalized velocity, Vc (T),
acceleration, Ac(T ), and jerk, Jc (T), of the follower can be expressed as:
where y (T) = T.
F二 （ T) = g'(y )
Ac (T) = g"( y )
Jc (T) = g ’ （y )
(13)
(14)
(15)
CR ITER IA FOR DESIG N ll( T)
For a given cam-follower system, the peak values of the normalized velocity, acceleration, and jerk resulting from a constant driving speed may possibly be reduced if we properly control its input speed trajectory !l(T). For example, to redu臼the peak values of normalized velocity, !l(T) can
be chosen so that IV(Tpv )I 副Vc (Tpv )I where Ve has peak values at normalized time Tpv· Then, from equations (6) and (13), we select that !l(T) must satisfy the condition:
一 1 ζ Q(Tpv) ζ l. (16)
“
For the case that we want to reduce the peak values of the normalized acceleration, i.e.
IA (Tpa )I 運 ／Ac(T pa)I were A0 has peak values at normalized me Tpa . Based on equations (7) and
(14），（T) should be chosen such that
一［ l + 02(Tp，）］ 《 g'(y (1盧））!l(T歸. ） 《 ［ I -
2
Q (T )]. (lη
g"(y (T陽. ）） 陽
Note that g氣y (Tpa )) must be nonzero. Similarly, if it requires /J （馬）｜ 運 ｜孔（馬）I where J0 has peak values at normalized time TPi ' then from equations (8) and (15), we need some Q(T) which satisfies
一［ l + Q3(Tpi )] as g"'( y (Tpi )) # 0.
3g 氣y （馬））Q(Tpi)!l(Tpi ) +g'(y (Tpi ))fi(T.陽）
ζ ［ 1 -!l3(T凹）］ (18)
g’叨叨（y (Tp))
For avoiding excessive vibration in the follower, the harmonics of !l(T) should be chosen as low as possible. Here, we choose an appropriate speed trajectory.
Since the velocity and acceleration curves, equations (6) and (7), are required to be continuous and the jerk curve, equation （例，should be finite as well, O(T) must be at least second order
differentiable.
Considering the the continuity of !l(T), the slope of !l(T) may be zero at T = 0 and I , i.e.
0(0) = 0 and 0(1) = 0.
Furthern ore, due to the boundary conditions of the normalized cam rotation angle, y(O ) = 0 and
y(l ) = l, the integration of !l(T) must satisfy the following condition:
f !l(T) dT + c = y (T) (19)
for some constant c.
；叫小
In a variable-speed cam-follower system, the cam operates at an angular velocity w (t ) and rotates an angle f3 in a time period τ，we have:
F ρ
Sinc晴 w(t) = p IτQ(T), equation (20) is in fact equal to:
f O(T) dT = I. (21)
400 H. S. Yan et al.
Here, we only consider the case of !l(T） 》 0, i.e. the direction of cam speed is not changed. As a result, design criteria for selecting !l(T) tο reduce the peak values of follower-outputs are:
(a) (I) for the case of reducing the peak value of the normalized velocity:
-1 ::;; !l(Tp.） 運 l
(II) for the case of reducing the peak value of the normalized acceleration:
一［ l +
' T0.)] ::;; g ’ （匯.））O(Toa ) ::;; [l -
, Toa )J
Ql(
P pa
ff(
四 g"( y( T歸. ）） 陽
(III) for the case of reducing the peak value of the normalized jerk:
一［l +Q3(T.陽）］ ::,; Pl ζ ［ l -!l3(T四）］．
3g ”（y （乓，j))!l(Tpi)O(Tpi ) +g'(y(T.回））nc r.;)
g”’ （幾））
(b) !l(T) is at least second order differentiable.
(c) 0(0) = 0(1) = 0.
(d) Constant c satisfies J !l(T) d T + c = y (T) subject to the boundary conditions y (O) = 0, y(l) = I.
(e) !l(T) has the harmonics as low as possible.
( f ) JA !l(T) d T = I.
(g) !l(T) 注 0.
Let equations (5問8) represent the normalized motion characteristics of the follower in the rising period. Then, the motion characteristics in the falling period are:
S(T) = 1 -g(y)
(22)
V(T) = -g’ ）!l(T)
(23)
A(T ) = -g飛y )!l2(T) -g ’ ）O(T)
(24)
J（ ＝-g＇”（y )!l3(T) -3g ”（y )!l(T)O(T) -g ’
） （T).
(25)
It is easy to find that the absolute values of the normalized velocity, acceleration, and jerk in the falling period are equal to those in the rising period, respectively. Hence, we have the following fact:
Ifthe same displacement cu凹e is used in the rising and falling periods of a follower, the functions of !l（η in these two periods are identical.
A NGULA R VELOCITY O( T)
Consider a cam-follower system which has a cam providing a cycloidal motion cu凹e where cam-input !l(t) is a polynomial.
In the rising (or falling) period and applying criteria (a) and (g) to reduce the peak values of motion curves, we choose the following polynomial !l(T), Fig. 1:
.Q(T)
0 Ta Tb
Fig. 1. Polynomial angular velocity in rising or falling period.
Effects of cam speeds on cam-follower systems
Qσ）
401
。
T
0.5
Fig. 2. Polynomial angular velocity in dwell period.
Q(f) 1.15
1.10
1.05
1.00 ...,_ -- - 』 ，甲，『 』 ，』 『 F 』 ，， 一『 ～- - - -
。
0.25
0.5
0.75
0.95
0.90 T
。 0.25
0.5
0.75
sσ） 1.00
0.75
0.50
0.25
0.00 T
V口J 2.50
丘
2.00
〉｜
1.50
t ”
，”
。
. 『
T
0.25 0.5 0.75
A(T) 10
。
-5
。
-10 T
0.25 0.5 0.75
5
0
5
。
25
J(T) 7
5
2
-
咽h 『 『 －”
-50 。
T
0.25 0.5 0.75
一一一一一 Vanable speed
------- Constant speed Fig. 3. Cycloidal motion.
402 H. S. Yan et al.
Peak value of V Peak value of A Peak value of J
Table l. Cycloidal motion Constant Variable
anιular 飛 elocity angular velocity Difference %
2.00 1.83 -8.5
6.28 5.97 -4.9
39.48 52.55 33.1
for O 運 T T.,
for T. T ζ Tb,
for Tb ζ T ζ 1,
Q(T) = 1 + d (26a)
Q(T) = 1 + d[l -e(T - T.Y(T - Tb川 (26b) Q(T) = 1 + d (26c)
where constant parameters d，鳥 兒 y，丸，and Tb are to be determined. Parameter d presents the fluctuation of Q(T), where 一 1 < d < 1 by criterion (g).
To satisfy design criteria (b）一（喲，imply:
X +y 注 2
X = y = 0, 2, 4, . . .
(27)
(28)
Parameters x and y are determined based on the type of cam displacement curves and the design criteria (e). Furthermore, parameter e subject to desi伊 criteria (t) is given by:
30
e 一 （衛一一一
一 一
Tb)5 '
(29)
Apparently, we can properly select d，叉，and Tb to obtain the lowest peak values of motion characteristic under the polynomial Q( T) of Fig. 1. Since the cycloidal motion curve is of symmetry, we let Tb = 1 - T. for simplicity and symmetry.
In addition, when the follower is in the dwell period, from design criteria (c) and (g) and equations (26)-(29), we obtain Q(T), Fig. 2, as follows:
Q(T) = 2n (2T3 - 3T2 ) + 1 + n. (30)
Under design criterion (d) and equation (30), we have:
y (T) = n(T4 - 2T3 ) +(1 + n)T (31)
and from design criteria (c) and （剖，we imply:
O(T) = 12n(T2 - T) (32)
(Os4
Time (t)
0 'ti τI ＋τ2 τI ＋τ2＋τ3 τI＋τ2＋τ3＋τ4
Fig. 4. Angular velocity of a variable-speed cam-follower system.
Effects of cam speeds on cam-follower systems 403
g（ gbE苦HS昆寫
110
105
100
95
90
，、
υε
dh
85 。
0.12
0.24
0.36
0.48
0.6
Time(s)
ε
υ
ε
JAU
aJAU
JA
J
句3句3肉，＆噸，－’且’且
nu hυnυ
．
321
g（ g）Hgg82房擊
。 0.12
0.24
0.36
0.48
0.6
Time(s)
nunununUAU Aυnu
hu nunυ
q’
d J
m（ BE）hM忘。古〉
。
mEE）gzEU38〈
（N
5,000
2,500。
-2,500
0.12
0.24
0.36
0.48
0.6
Time(s)
-5,000 。
0.12
0.24
0.36
0.48
0.6
Time(s)
300,000
回〉白g 捋 同
（
m
u
150,000。
） HU
-150,000
-300,000 。
0.12
0.24
0.36
0.48
0.6
Time(s)
- - - --- Constant speed
一一一 Variable speed
Fig. 5. Cam-follower system with cycloidal motion (n == 0, d == 0.1, w??? == 100 rpm).
MMT 31陣 D
404 H. S. Yan et al.
。（T) = 12n(2T 一 1) (33)
where -1 < n < 1.
Assume that the cam has cyclo1dal motion which is given by:
Therefore:
g(y ) = y 一 ι si
L冗
g'(y ) = I -cos (2πy )
g”（y ) = 2n sin (2ny ) g’” ）＝4π2 cos (2πy ).
(34)
(35)
(36)
(37)
We c。nsider to reduce the peak values of both normalized velocity and acceleration for the given
cycloidal motion curves, and then choose T.= 0, Tb = 1, d = 0.1, and x = y = 2, i.e.
。（T) = I + 0.1[1 - 30T2( T - 1)2]. (38)
Substituting equations (34）一（38) into equations (5問8), we have normalized displacement, velocity, acceleration, and jerk as shown in Fig. 3, in that we can see the normalized velocity and acceleration curves being continuous and the normalized jerk curve is being 缸1ite. The peak values of V, A, and J as demonstrated in Table 1 shows the peak values of the normalized velocity and acceleration are reduced.
DESIG N EX AMPLE
Design a variable-speed cam-follower system to satisfy the following condition:
A radial roller follower dwells while the cam rotates 60°, and rises through a total displacement of 30 mm with cycloidal motion for the next 120。.The follower dwells again in 60。of cam rotation,
and then returns 30 mm with cycloidal motion for the final 120°.
Let ω時 ，ro,2 , Ws3 , and ro84 denote the average angular velocities while the follower is in the first dwell, rising, second dwell, and falling period, respectively. Similarly, let {31 , /32 ，比，/34 be the cam
rotation angles; and h，勺，勺，r4 be the times in the above period. Then, we have the following facts:
＝
－自1 τ1 (39)
W,1
一/3一2 τ2
α』s2
/33
τ3
W,3
(40)
(41)
(42)
盧4
一一 τ4
α）s4
一
一2π τ1 ＋τ2 ＋τ3 ＋τ4 (43)
w???
where w??? is the average cam angular velocity of a complete cycle. The polynomial ro(t), Fig. 4, must be continuous. Based on equations (26)-(29) and (30)-(33), we have:
(1 -n )w,1 = (1 + d)w,2
(1 + d)w,2 = (1 - n)w 。
(1 + n)w,3 = (1 + d)w 抖
。＋d)ws4 = (1 + n)ro,1
(44)
(45)
(46)
(47)
Effects of cam speeds on cam-follower systems 405
J宮幸
140
120
100
80
60 。
0.12 0.24 0.36 0.48 0.6 Time(s)
35
30
、， 、、
5 、
20 、、
15 、
自.日s‘
10。s
。
0.12
-5
、、、
、、
0.24 0.36 0.48 0.6 Time(s)
400
200。
號 -200
。
-400
0.12 0.24 0.36 0.48 0.6 Time(s)
M 氣 8,000
4,000。
-4,000
-8,000 。
，’ ／
，，
，，
，，
0.12 0.24 0.36 0.48 0.6 Time(s)
。
300,000
”·、
M
-300,000
曾
-600,000 。
、．．··』 _ ,
0.12 0.24 0.36 0.48 0.6 Time(s)
--- - -- C。nstant speed Variable speed
Fig. 6. Cam-follower system with cycloidal motion (n = 0.2, d = 0.1, w...= 1傭rpm).
406 日.S. Yan et al.
y
盧
60" [
X
、
base circle Fig. 7. Profile of a cycloidal cam.
I I I + d l + d l
I -
＝一2π II P1 ＋一
n P2 ＋品＋ I + n P4 lIroave
(48)
where p1 π／3，品 2π／3, /J3 π ／3, p4 = 2π ／3, d = 0.}, and W,1 = W,3. Let Wave be ｝｛）（） 叩m and n be equal to 0. The angular velocity of the cam and the displacement, velocity, acceleration, and the jerk of follover can be calculated. Figure 5 shows the result where the dashed lines are the corresponding results when a cam-follower system operates at a constant speed of 100 rpm. Similarly, the results of the case, Wave is 100 甲m and n is equal to 0.2, are demonstrated in Fig. 6. Furthermore, Figs 5 and 6 show that the time at the peak values occurs in a variable speed cam-follower system is not the same as the constant-speed case. This implies that a proper variable cam speed design can provide a method to change the times of various period to get desired or
better motion characteristics in a cam-follower system.
Fig. 8. Schematic of experimental apparatus and instrumentation.
Effects of cam speeds on cam-follower systems
IBM-PC BUS
Fig. 9. Experimental system for DSP-based control.
407
EX PER IMENT A L SETUP A N D R ESU LTS
The experimental variable-speed cam-follower system and apparatus are used in this study. The base circle diameter of this experimental disk cam, made of S50C, is 60 mm. Consider the condition of static balance, the centroid of the disk cam as shown in Fig. 7 is (-0.478 mm, 0 mm). The thickness, area, mass, and mass moment of inertia of the cam are 13 mm, 9777.7 mm2, 1080 g, and 0.0031 kg-m2, respectively. The maximum cam pressure angle is 18° that the guide is properly designed to withstand the side thrust. The cam-follower system is mounted on a frame which is rigidly fixed to the foundation and shown in Fig. 8. η1e roller follower moves horizontally in the guide fixed to the frame, and the follower is actuated by the rotation of the ca噸·
The follower is made of a bar (THK, LBH25) of length 495 mm and diameter 20 mm at both ends. The roller is fixed at the cam-end of the bar and is of cylindrical type, 22 mm in diameter and IO mm thick, which rotates on a roller-pin of diameter 8 mm and length 36 mm. The cam-follower system is assumed to be rigid because it processes the large cross-section dimensions and is made of carbon steel. The preset of the spring (SPEC, D13160) with 0.146 kg/mm spring stiffness is 36 mm so that the roller follower can keep contact with the cam.
The angular velocities of the cam and the driving motor are assumed to be identical because a rigid Oldham coupling (MIGHTY, MJC50) is used to connect the cam shaft to motor shaft. Hence, using a d.c. servo motor (SANYO, DENKI, CN-800-10, 850W, 1000 rpm) and a driver of motor as shown in Fig. 8, it can be
收藏