Page 452 - Cam Design Handbook

THB13 9/19/03 7:56 PM Page 440

440 CAM DESIGN HANDBOOK

nomial indicates finite acceleration values y c≤ at the ends (infinite jerk); vibrations result.
This condition can be alleviated by the use of special ramps. However, if we desire that
the cam acceleration curve have finite values of jerk, a fourth-order boundary condition
[for follower y = f(q)] equal to zero at the ends is required. The 5-6-7-8-9 polynomial
family having the following basic equation fulfills this condition:
y =-126q 5 + 420540q 7 + 315q 8 - 70q . (13.88)
-
q
9
6
1
This curve gives a lift of 1.0 at the start (q = 0) and a lift and first four derivatives of
0.0 at the end (q = 1). This family is suggested for most high-speed, high-flexibility
machinery.
The best approach is to establish basic equations with certain simplifying approxima-
tions. The significance of the approximations must be compared with the accuracy of the
given data and the cutting of the cam. The procedure for design is as follows:
1. Choose a polynomial equation, y = f(q), with proper control at the end points. The
5-6-7-8-9 family is often suggested.
2. Establish the follower system flexibility relationship, using equations similar to Eq. (13. 83).
3. Combine steps 1 and 2, plot displacement, velocity, and acceleration curves of both the
cam and the follower end to check the reasonableness of assumptions. This will give
the cam shape to be cut. The reader should remember that a comparison between y and
y c must be made after the lever ratio and gear ratio are considered, if they are part of
the follower linkage system.

13.7.8 Example Using the 3-4-5 Polynomial End Mass Movement
For simplicity we shall solve a polydyne example, using the impractical 3-4-5 polynomial
(for the end mass movement) to indicate the method. We know that the cam will have
finite values at the end points, requiring a special ramp to meet these values smoothly.
Note that for reasonable accuracy, a larger number of significant figures is generally nec-
essary than that shown in the calculations of this problem.

EXAMPLE A cam for a high-speed textile machine rotates at 1000 rpm with the follower
rising 3/4in in 60° of cam rotation. The follower linkage spring rate is 25,000lb/in, and
the helical compression spring rate is 400lb/in. The effective weight of the follower
(including the lever) is 5 1/2lb on the cam and 3 1/2lb on the follower end. The lever
arms are 4in and 6in long (Fig. 13.21).
The clearance or backlash in the linkage is negligible. The external load is 100lb, and
the initial spring load is 150lb. Plot curves of the cam and follower displacement and
acceleration. Describe the ramp necessary.

Solution The equation for a 3-4-5 polynomial meeting the boundary conditions for this
problem is
Ê q ˆ 3 Ê q ˆ 4 Ê q ˆ 5
y = .75 - .7 5 +11 .25 - .45 . (13.89)
Ë 60 ¯ Ë 60 ¯ Ë 60 ¯
Differentiating
Ê q ˆ 2 Ê q ˆ 3 Ê q ˆ 4
y ¢ =-.375 + .75 -.375 (13.90)
Ë 60 ¯ Ë 60 ¯ Ë 60 ¯