chapter 1 evolution of a successful design

4.2.2 Polynomial Family
The basic polynomial equation is
θ
s = C0 + C1
+ C2 2
+ C3 3
+ ⋅⋅⋅ (4.18)
with constants C i depending on assumed initial and final conditions.
This family is especially useful in the design of flexible cam systems, where values
of the dynamic factor are µ d ≥ 10 −2 . Dudley (1947) first used polynomials for the synthesis
of flexible systems, and his ideal later was improved by Stoddart [4.9] in application
to automotive cam gears.
The shape factor s of the cam profile can be found by this method after a priori
decisions are made about the motion y of the last link in the kinematic chain. Cams
of that kind are called polydyne cams.
When flexibility of the system can be neglected, the initial and final conditions
([4.3], [4.4], and [4.8]) might be as follows (positive drive):
1. Initial conditions for full-rise motion are
θ
β0
= 0 s = 0 s′=0 s″=0
2. Final (end) conditions are
θ
= 1
β0
s = s0 s′=0 s″=0
The first and second derivatives of Eq. (4.18) are
s′= C1 + 2C2
s″= 2C 2 + 6C3
+ 3C3 2
+ 4C4 3
β0 β0
+ 12C4 2
β0 β0
+ ⋅⋅⋅
+ ⋅⋅⋅
(4.19)
Substituting six initial and final conditions into Eqs. (4.18) and (4.19) and solving
them simultaneously for unknowns C0, C1, C2, C3, C4, and C5, we have
s = 10s0 3
− 1.5 4
+ 0.6 5
θ θ θ
and for a jerk s″′ = ds″/dθ,or
s′= 30 2
s0
β0
− 2 3
β0
s″= 60 − 3 2
s0
β0 2
CAM MECHANISMS
4.14 MACHINE ELEMENTS IN MOTION
β0
θ
β0
θ
β0
θ
β0
θ
β0
θ
β0
β0
+ 4 θ
+ 2 3
β0
s″′ = 60 1 − 6 + 6 2
s0 θ θ
β0
3
θ
β0
θ
θ
θ
β0
θ
β0
β0
θ
β0
θ
β0
(4.20)
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