§9 － 2 Follower Motion Curves §9 － 3 Graphical Cam Profile Synthesis §9 － 4 Cam Size and Physical Characteristics §9 － 1 Applications and Classification of Cam Mechanisms Chapter 9 Cam Mechanisms

一、 Applications internal combustion engine coiler Cylindrical cam conveyer 二、 Characteristics Cam mechanisms are the simplest mechanisms to transfer a simple motion into any desired complicated motion. Cam mechanism can not transmit heavy loads because of the point or line contact between the cam and the follower. § 9 － 1 § 9 － 1 Applications and Classification of Cam Mechanisms

Plate cam (or disc cam) 三、 Classifications of Cam Mechanisms 1. By the cam shape Translating camCylindrical cam 2. By the shape of the follower end

Characteristics ： Knife-edge follower——Although it is simple, the stresses at the line of contact are excessive and cause rapid wear on the follower and the cam surface. Roller follower——The roller follower greatly reduces wear because the contact is almost entirely rolling rather than sliding. Flat-faced follower——This follower is used often in high-speed cam mechanisms since a dynamic pressure oil film can be formed easily.

3. By the manner of keeping the cam and the follower in contact ： Form-closed cam mechanism Force-closed cam mechanism （ by a preloaded spring, or by gravity ） B’ B

D O O s r0r0 ω h B/B/ A B C 一、 Motion curve of the follower 1. Technology terms Prime circle, Prime circle radius: r 0 Cam angle for rise: δ 0 Cam angle for inner dwell: δ 02 Total follower travel ： h Cam angle for outer dwell: δ 01 Cam angle for return: Follower Motion Curves: s = s ( t ) v = v ( t ) a = a ( t ) = s ( δ ) = v ( δ ) = a ( δ ) § 9 － 2 § 9 － 2 Follower Motion Curves

s ｏ δ v ｏ δ a ｏ δ 2. Polynomial Motion Curve The standard polynomial equation s=C 0 + C 1 δ+ C 2 δ 2 +…+C n δ n where s —— displacement of the follower, δ ——rotation angle ， C i —— the constants. （ 1 ） Constant Velocity Motion Curve s = C 0 + C 1  v = ds / dt = C 1 ω a = dv / dt = 0 h δ0δ0 Depend on the boundary conditions: δ=0 ， s=0 δ=δ 0 ， s=h Therefore C 0 = 0 ， C 1 = h/δ 0 Equations for rise s ＝ hδ/δ 0 v ＝ hω /δ0 a ＝ 0 Rigid impulse——The consequent large inertia force will result in a very large amount of shock.

（ 2 ） Constant Acceleration and Deceleration Motion Curve The follower is given a constant acceleration during the first half of the rise and a constant deceleration during the second half of the rise. Constant acceleration motion equations for rise: s = 2hδ 2 /δ 0 2 v = 4hωδ /δ 0 2 a = 4hω 2 /δ ４ 1 0 s ｏ δ v ｏ δ a ｏ δ h/2 A 2hω/δ 0 4hω 2 /δ 0 2 Constant deceleration motion equations for rise: s = h-2h(δ 0 –δ) 2 /δ 0 2 v = -4hω(δ 0 -δ)/δ 0 2 a = -4hω 2 /δ 0 2 δ0δ0 B C Soft impulse —— Since every link in a earn mechanism has elasticity, an abrupt change in the magnitude and/or direction of inertia force would initiate undesired vibration. There are three more abrupt changes in the acceleration of the follower(points A 、 B 、 C)

（ 3 ） Polynomial Motion Curve s=10h(δ/δ 0 ) 3 － 15h (δ/δ 0 ) 4 +6h (δ/δ 0 ) 5 There is neither rigid impulse nor soft impulse. v =ds /dt = C 1 ω+ 2C 2 ωδ+ 3C 3 ωδ 2 + 4C 4 ωδ 3 +5C 5 ωδ 4 a =dv/dt = 2C 2 ω 2 + 6C 3 ω 2 δ+12C 4 ω 2 δ 2 +20C 5 ω 2 δ 3 Expressions : Depend on the boundary conditions: s = 0 ， v = 0 ， a = 0 Whenδ= 0; s = h, v = 0 ， a = 0 Whenδ =δ 0. Results in C 0 = C 1 = C 2 = 0, C 3 = 10h/δ 0 3 C 4 =15h/δ 0 4, C 5 = 6h/δ 0 5 s =C 0 + C 1 δ + C 2 δ 2 + C 3 δ 3 + C 4 δ 4 +C 5 δ 5 Displacement equation v a h s svasva O δ(t)δ(t)

3. Triangle Function Motion Curve (1) Cosine Acceleration Motion Curve (or Simple Harmonic Motion Curve) motion equations for rise: s=h[1-cos(πδ/δ 0 )]/2 v =πhωsin(πδ/δ 0 )δ/2δ 0 a =π 2 hω 2 cos(πδ/δ 0 )/2δ 0 2 Although the motion curve avoids the sudden reverse of acceleration at the middle-point of the travel, the acceleration will still change abruptly at the beginning and end points. That would produce soft impulses. v O δ s O δ a O δ h δ0δ0 a max = 4.93hω 2 /δ 0 2 v max =1.57hω/2δ 0

v O δ s O δ a O δ h δ0δ0 a max =6.28hω 2 /δ 0 2 v max = 2hω/δ 0 （ 2 ） Sine Acceleration Motion Curve (or Cycloid Motion Curve) motion equations for rise: s = h[δ/δ 0 - sin(2πδ/δ 0 )/2π] v = hω[1-cos(2πδ/δ 0 )]/δ 0 a = 2πhω 2 sin(2πδ/δ 0 )/δ 0 2 In this motion curve, the velocity and the acceleration always begin from zero, and change smoothly. There is no sudden change in velocity and acceleration at either end of the stroke, even if there are dwell periods at the beginning or the end of the travel. Therefore there is neither rigid impulse nor soft impulse.

一、 The Fundamental Principle of Cam Profiles Design -ω ω O e r0r0 Cam: rotation Follower: reciprocation Cam mechanism: in the direction opposite to ω (-ω ) Cam ： stationary Follower : reciprocation + rotation The relationship between the follower travelling distance along its guide-way and the inverse angle is exactly the same as the desired motion curve. This is called the principle of inversion. Principle of inversion § 9 － 3 § 9 － 3 Graphical Cam Profile Synthesis

r0r0 -ω-ω ω 80 ° 二、 Disk Cam with reciprocating Follower 1. Plate cam with translating offset knife-edge follower Given: prime circle radius r 0, angular velocityω, offset e and motion curve of follower. Design the cam contour. 120 ° 60 ° 100 ° ’1’ 6’6’ 2’2’ 3’3’ 4’4’ 5’5’ ’7’ 11 ’ 8’8’ 9’9’ 10 ’ e Procedure 1) Select scaleμ l ， draw the prime circle and the offset circle. 2 ） Divide the offset circle into a number of equal segments in the direction opposite to ω, and assign station numbers to the boundaries of these segments. 4 ） The corresponding position of the knife edge can be located. 3 ） Draw line from these points, making them all tangent to the offset circle. 5)The pitch curve is obtained by drawing a smooth curve.

r0r0 -ω-ω ω 80 ° 120 ° 60 ° 100 ° ’1’ 6’6’ 2’2’ 3’3’ 4’4’ 5’5’ ’7’ 11 ’ 8’8’ 9’9’ 10 ’ e Given: prime circle radius r 0, angular velocityω, offset e, roller radius r r and motion curve of follower. Design the cam contour. 2. Plate Cam with Translating Roller Follower 1) The motion curve of the knife-edge is the same as that of the roller center, or that of the translating roller follower. The locus of the roller center, relative to the cam can be designed according to the methods mentioned in the last section. Procedure 2 ） The locus is called the pitch curve of the cam. 3 ） Many roller circles are drawn with roller radius r r and center at the points on the pitch curve. Since the roller is tangent to the cam contour at all times, the cam contour(cam surface) is the envelope of the family of the roller circles.

r0r0 -ω-ω ω 80 ° 120 ° 60 ° 100 ° 6’6’ 2’2’ 3’3’ 4’4’ 5’5’ 11 ’ 8’8’ 9’9’ 10 ’ 1’1’ 7’7’ 3.Disk Cam with Reciprocating Flat-faced Follower Given: prime circle radius r 0, angular velocityω, and motion curve of follower. Design the cam contour. 1 ） Similar to the roller follower, the intersection of the centerline of the follower stem and the face of the follower is treated as the knife-edge of a virtual knife-edge follower which is fixed to the flat­faced follower. Procedure 2 ） A line is drawn at each of these points perpendicular to the center line of the follower to represent the position line of the flat face of the follower during inversion. 3 ） So the cam contour may be drawn as a smooth curve tangent to the family of the position lines of the flat face.

5) Plate Cam with Oscillating Knife-edge Follower The following data should be provided: the radius of prime circle radius r 0, the distance l OA between the two pivots of the cam and the follower,the length l (l AB )of the follower arm, the initial status of the follower (left or right), the direction of ω, the motion curve φ-δ. Design the cam contour. ω -ω 90 ° 180 ° φ1φ1 φ2φ2 φ3φ3 φ4φ4 φ6φ6 φ5φ5 φ7φ7 φ8φ8 A8A8 A9A9 A6A6 A7A7 A3A3 A1A1 A4A4 A2A2 A5A5 A B r0r0 B8B8 B6B6 B5B5 B7B7 B2B2 B1B1 B3B3 B4B4 B9B9

The motion programs discussed in section 9-2 dealt only with the motion of the cam follower and not with the nature of the cam that produced that motion. 一、 Pressure Angle Pressure Angle——the angle between the force that the cam applies to the follower ( neglecting friction) and the direction in which the follower is free to translate. ω  n n t t G d b l B F φ1φ1 φ2φ2 φ2φ2 ∑ F x =0 ， ∑F y =0 ， ∑M B =0 F= cos( α+φ 1 )- (1+2b/l) sin( α+φ 1 )tan φ 2 G α↑, F ↑ ； whenα ↑, →denominator =0 ， F → ∞ ， → mechanism self-lock § 9 － 4 § 9 － 4 Cam Size and Physical Characteristics

Pressure angle a changes during motion so it is necessary to control the value of the maximum pressure angle a max. α max ≤[α] Rise stroke translating follower [α] ＝ 30° oscillating follower [α] ＝ 35° ～ 45° Return stroke [α]’ ＝ 70° ～ 80° α c = arctan [1 / (1 + 2b / l) tanφ 2 ] -φ 1 The allowable pressure angle [a] in the return for a force ­ - closed earn mechanism is quite large. The oscillating follower works more smoothly than the translating follower. The allowable pressure angle[α] for cam mechanism ：

n n  E vpvp ω O r0r0 S The smaller the value of r 0 is, the larger the value of α max will be. S0S0 P ｅ C 二、 Radius of the prime circle OP= v/ ω =[ds/dδ] In △ BCP, s 0 = r e 2 For the translating roller follower, an increase in r 0 will definitely reduce α and hence α max However, larger cams require more space and this means more mass and greater inertia forces which may lead to unwanted vibrations when speeds are high. tanα =(ds / dδ-e) / [( r e 2 ) 1/2 + s ] Depend onα≤[α] ， therefore

三、 Radius of roller ρaρa ρ Pitch curve Cam surface r ρ ρ r r r

In a words ρ min > r r, and ρ amin ≥(1 ～ 5)mm. 四、 Width Requirements for a Translating Flat-faced Cam Follower l = 2l max + ( 5 ~ 7 ) mm