Synthesis of Cam mechanism based on given program

Synthesis of Cam mechanism based on given program Ahmet Shala1*, Ramë Likaj2*, Mirlind Bruqi3* 1

*University of Prishtina, Faculty of Mechanical Engineering, Prishtina, Kosovo e-mail: [email protected] 2 *University of Prishtina, Faculty of Mechanical Engineering, Prishtina, Kosovo e-mail: [email protected] 3 *University of Prishtina, Faculty of Mechanical Engineering, Prishtina, Kosovo e-mail: [email protected] Abstract: Synthesis of cam mechanisms used for the exact design of cam profile for desired movement program of follower is presented. This method uses continues trajectory conditions in the points where path, velocity or acceleration starts-ends the changes. Usually the diagram of acceleration and maximal displacement of the follower are known. Based on desired conditions in this paper is described design of exact profile of the Cam mechanism. The method of continues trajectory, usually converts the problem of cam design in problem of solving the system of equations of unknown coefficients. For the simulations results we have use MathCad and Working Model software’s. Keywords: Cam; mechanism; follower; trajectory; design; velocity; acceleration; program. 1. INTRODUCTIONS Cam mechanism usually consists by cam, follower and roller. Follower refers only to that part of the follower link which contacts the cam. There are three types of followers: flatfaced, mushroom (curved), and roller. The roller follower has the advantage of lower (rolling) friction than the sliding contact of the other two, but can be more expensive. Flatfaced followers can package smaller than roller followers for some cam designs; they are often favored for that reason as well as cost for some robotic valve trains. Many robotic systems today use roller followers for their lower friction. Roller followers (Figure 1) are commonly used in robotics and automotive where their ease of replacement and availability from bearing manufacturers’ stock in any quantities are advantages. Grooved or track cams require roller followers. Roller followers are essentially ball or roller bearings with customized mounting details. One basic rule in cam design is that motion curve must be continuous and the first and second derivatives (corresponding to the velocity and acceleration of the follower) must be finite even at the transition points.

The desired movement program of the follower can be accurately described if the cam profile is calculated using a numerical analytical method [1]. The present paper refers to the exact synthesis of the curve defined as the profile of the rotary cam and roll or flat follower without a reversing movement. 2. SYNTHESIS OF CAM PROFILE BASED ON DESIRED MOVEMENT PROGRAM Desired movement program usually is the diagram of acceleration (Figure 2) and maximal displacement of the follower (L = smax).

Figure 2. Desired movement program – Diagram of follower acceleration in function of cam angle θ.

Figure 1. Cam mechanism with roller slider-follower *corresponding author: [email protected]

Diagram of acceleration in Fig. 2 is linear form; different phases; (A) maximal and minimal values are unknown. Now for the given-desired maximal displacement of the slider-follower (L = smax) it is important to design exact form of acceleration, velocity and displacement (a, v, s) of sliderfollower analytically. Parameters (a, v, s) at the same time have relations with kinematic parameters of peripheral point of the cam. s – Represents translation of follower and cam profile

ds – Represents velocity of the follower and velocity d component of the peripheral point of the cam in follower movement direction dv d 2 s a ( )   – Represents acceleration of the follower d d 2 and acceleration component of the peripheral point of the Cam in follower movement direction Cam mechanism will be synthesis by acceleration given phases. Unknown coefficients will be named Ci, where i =0,1…n Phase for θ = 0 ÷ θ1 Acceleration in Fig. 2 is linear form (first order) so velocity will be a second order equation and displacement will be a third order equation. s1 ( )  C 0  C1    C 2   2  C3   3    v1 ( )  C1  2  C 2    3  C3   2  ............................(1)  a1 ( )  2  C 2  6  C3    For angle θ = 0 all kinematic parameters are zero s1 (0)  v1 (0)  a1 (0)  0 From Fig. 2, diagram of acceleration has its maximum for θ = θ1 so: a1 (1 )  A . Using previous conditions from equations (1) are derived equations (1.1): 0  C0   0  C1   .....................................................(1.1) 0  2  C2  2  C 2  6  C3  1  A v( ) 

Phase for θ = θ1 ÷ θ2 Acceleration in Fig. 2 is zero a 2 ( )  0 (constant) so velocity will be constant and displacement will be a first order equation.

s 2 ( )  C 4  C5     v 2 ( )  C5  ...........................................................(2)  a 2 ( )  0  For angle θ = θ1 is known that velocity and displacement diagram should be continues so: v1 (1 )  v 2 (1 ) s1 (1 )  s 2 (1 ) Using previous conditions from equations (2) are derived equations (2.1):  C1  2  C 2  1  3  C3  12  C5  ............(2.1) C0  C1  1  C 2  12  C3  13  C 4  C5  1  Phase for θ = θ2 ÷ θ3 Acceleration in Fig. 2 is linear form (first order) so velocity will be a second order equation and displacement will be a third order equation.

s3 ( )  C 6  C7    C8   2  C9   3    v3 ( )  C7  2  C8    3  C9   2  ............................ (3)  a3 ( )  2  C8  6  C9    For angle θ = θ2 is known that velocity and displacement diagram should be continues so: v 2 ( 2 )  v3 ( 2 ) and s 2 ( 2 )  s3 ( 2 ) From Fig. 2, diagram of acceleration is symmetric so: a3 ( 2 )   A . For θ = θ3 from Fig. 2 is a3 ( 3 )  0 and this position is the end of rise period so s3 ( 3 )  L  s max . Using previous conditions from equations (3) are derived equations (3.1):  C 5  C 7  2  C8   2  3  C 9   2 2  C 4  C 5   2  C 6  C 7   2  C8   2 2  C 9   2 3   2  C8  6  C 9   2   A  ..... (3.1)  2  C8  6  C 9   3  0   C 6  C 7   3  C8   3 2  C9   33  L  s max  Phase for θ = θ3 ÷ θ4 Acceleration in Fig. 2 is linear form (first order) so velocity will be a second order equation and displacement will be a third order equation. s 4 ( )  C10  C11    C12   2  C13   3    v 4 ( )  C11  2  C12    3  C13   2  ....................... (4)  a 4 ( )  2  C12  6  C13    For cam angle θ = θ3 is known that acceleration, velocity and displacement diagram should be continues so: a3 ( 3 )  a 4 ( 3 ) ; v3 ( 3 )  v4 ( 3 ) ; s3 ( 3 )  s 4 ( 4 ) For cam angle θ = θ3 acceleration is: a 4 ( 3 )  0 For cam angle θ = θ4 acceleration is: a 4 ( 4 )   A Using previous conditions from equations (4) are derived equations (4.1): 2C8  6C9   3  2  C12  6  C13   3

  C7  2C8 3  3C9 3  C11  2C12 3  3C13 3   ....... (4.1) C6  C7 3  C8 32  C9 33  C10  C11 3  C12 32  C13 33   2C12  6C13 3  0   2C12  6C13 4   A  2

Phase for θ = θ4 ÷ θ5 Acceleration in Fig. 2 is zero

2

a5 ( )  0 so velocity will be

constant and displacement will be a first order equation. s 5 ( )  C14  C15     v5 ( )  C15  ........................................................ (5)  a 5 ( )  0  For cam angle θ = θ4 is known that velocity and displacement diagram should be continues so: v 4 ( 4 )  v5 ( 4 ) s 4 ( 4 )  s5 ( 4 )

Using previous conditions from equations (5) are derived equations (5.1):  C11  2  C12   4  3  C13   4 2  C15  (5.1) C10  C11   4  C12   4 2  C13   4 3  C14  C15   4  Phase for θ = θ5 ÷ θ6 Acceleration in Fig. 2 is linear form (first order) so velocity will be a second order equation and displacement will be a third order equation.

s 6 ( )  C16  C17    C18   2  C19   3    v 6 ( )  C17  2  C18    3  C19   2  .....................(6)  a 6 ( )  2  C18  6  C19    For cam angle θ = θ5 is known that velocity and displacement diagram should be continues so: v5 ( 5 )  v 6 ( 5 ) s 5 ( 5 )  s 6 ( 5 ) For cam angle θ = θ5 acceleration is: a6 ( 5 )  A For cam angle θ = θ6 acceleration is: a 6 ( 6 )  0 Using previous conditions from equations (6) are derived equations (6.1): C15  C17  2  C18  5  3  C19   5 2 2

 

 C 0   0.0000       C1   0.0000  C     2   0.0000   C3   0.3112       C 4    0.2118   C   0.4550   5     C 6    1.7520  C     7   3.4409   C8    1.7923       C9   0.3112   C    2.6520   10     C11    3.4409   C   1.7923   12     C13    0.3112       C14   1.5353   C    0.4550   15     C16   17.6162     13.7636    C17     C18   3.5845       C19    0.3112    A   1.3035    In Figures in view are represented exact diagrams for acceleration, velocity and displacement of the follower per degree of cam rotation.

3

C14  C15   5  C16  C17  5  C18   5  C19   5  .......... 6.1) 2  C18  6  C19   5  A   2  C18  6  C19   6  0 

For cam angle θ = θ6 ÷ 360 deg all kinematic parameters are zero. Equations (1.1),(2.1),(3.1),(4.1),(5.1) and (6.1) represents 21 equations with 21 unknowns (20 coefficients C0÷19 and maximal value of acceleration A).

3. CAM SYNTHESIS DESIGN RESULTS

1.303 0.652 a(  )

0  0.652  1.303

0

40

80

120

160

200

240

280

320

360

 deg

Figure 3. Desired movement program – follower acceleration in function of cam angle θ, maximal acceleration A=1.303 m/s2.

For Cam synthesis design results are used cam phase angles: 0.455

θ1 = 40 deg; θ2 = 70 deg; θ3 = 110 deg; θ4 = 150 deg; θ5 = 180 deg; θ6 = 220 deg;

0.227 v( )

Maximal displacement of the slider-follower: L = smax = 0.45m To avoid lose of contact or contact in two points between roller and cam is preferred cam base circle radius to be Rb = 0.5m > L Results for 20 coefficients C0÷19 and maximal value of acceleration A are shown in view:

0

 0.227  0.455

0

40

80

120

160

200

240

280

320

360

 deg

Figure 4. Velocity of the follower per degree of cam rotation

0.45 0.337 s(  ) 0.225 0.112 0

0

40

80

120

160

200

240

280

320

360

 deg

Figure 5. Displacement of the follower per degree of cam rotation

  120

150

Rb s( ) Rb

180

90 0.95 0.855 0.76 0.665 0.57 0.475 0.38 0.285 0.19 0.095 0

60

30

0

Figure 7. Design and simulation of Cam mechanism 4. CONCLUSIONS

210

330

240

300 270 

Figure 6. Designed Cam profile - red color, base circle – blue color. Velocity and acceleration of the follower in function of time, if cam has angular velocity  can be calculated using equations in view: v(t )  a (t ) 

ds ( ) d ds ( )    d dt d d 2 s ( ) d

If angular velocity

2

2 

ds ( ) d   d d

 is constant, than:

v(t )  v( )  

a(t )  a( )   2 Simulation of designed cam is realized using Working Model Software.

In this paper is represented successfully analytical synthesis for exact design of cam profile. For the desired inputs (in our example is known form of the diagram of the follower acceleration) is described analytical way for exact design of cam profile. Based on simulations results in Figure 7, it is clear that manufacturing profile error indicates too much on kinematic and dynamic characteristics of Cam mechanisms. For the future work will be focused on force analysis of the cam mechanisms during different applications in robotics. 5. REFERENCES [1] Rehwald, W., Luck, K.,(2007) Research of Coupler Curves of Geometrical Poles of first Order, Journal of Mechanisms and Manipulators, vol. 6, nr. 2, pp. 5-16;. [2] Umesh Chavan., at all, (2010). Synthesis of coupler curves with combined planar cam follower mechanisms by genetic algorithm. Proveeding of 2nd International Conference on Computer Engineering and Technology, Chengdy, China. [3] U.S. Chavan, S.V. Joshi. (2010). Synthesis and analysis of coupler curves with combined planar cam follower mechanisms, International Journal of Engineering, Science and technology, Vol. 2, No. 6, pp. 231-243 [4] Eres Soylemez, Lecture notes: Mechanisms, http://ocw.metu.edu.tr/course/view.php?id=132 [5] A. Shala, Lecture notes and examples: Analysis and Synthesis of Mechanisms, www.ahmetshala.tk