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Abstract The paper presents some considerations about the optimum design of
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cam mechanisms with reciprocal and oscillating roller followers. The objective
function express the dimension of the cam, while the constraints limit the size of
the pressure angle and / or the magnitude of the unit stress to the higher con-
tact between cam and follower or cam and roller. Finally, the result of solving a
numerical example is presented.7197
Keywords Cam mechanisms · Optimization · Contact stress · Hertz1 Introduction
The relative motion of the follower with respect to the cam is a combination of a
relative turning motion and a relative oscillating motion. In this relative motion, the
knife edge follower plots the pitch curve of the cam [8, 9, 10]. The cam profile is
obtained as the envelope of the circles family which represents the relative positions
of the roller with respect to the cam. The centers of the circles are situated on the
pitch curve. The optimum design of the cam implies the solving of a nonlinear pro-
gramming problem [2, 4, 5]. The objective function expresses one of the following
features: the maximum radius of the pitch curve of the cam, the overall size of the
cam mechanism, the efficiency of the mechanism, the actuating moment etc., or a
combination of these.
The constraints which must to be considered are: the maximum value of the
pressure angle, the maximum value of the unit stress at the cam – follower contact
point, the maximum value of the negative curvature radius of the cam pitch curve,
the maximum magnitude of follower acceleration, the minimum value of the posi-
tive curvature radius of the cam pitch curve, to avoid the cam undercutting, and the
load capacity of the higher pair between cam and follower.A very important constraint limits the maximum value of the stress to the high
contact between cam and roller or cam and follower. This value is influenced by: the
magnitude of the pressure angle, the value of positive curvature radius of the cam
profile at the contact point, and the size of the resistance force which acts on the
follower, which include the inertial force and elastic force of back spring.
But, the last constraint does not eliminate the restriction which limits the
magnitude of the pressure angle, which is a working kinematics condition.
2 Optimum Synthesis of Cam Mechanisms
The objective function, which expresses the maximum radius of the pitch curve of
a cam with reciprocating roller follower (Fig. 1), is:For the cam with oscillating roller follower (Fig. 2), the objective function is:
where: l is the length of follower, d is the distance between axes of the revolute
pairs between cam and frame and follower and frame, ψ0 is the initial angle of the
follower, θ is the follower angular stroke.
The pressure angle, of movement transmission from cam to reciprocating
follower, is determined by using very well known formula [10]:
and to cam mechanism with oscillating roller follower:
The extreme of the function δ = δ(ϕ) - for reciprocating follower - is in a point
defined by ϕ∗ as solution of the equation:The magnitude of the admissible pressure angle is calculated in terms of dimen-
sions of the follower and guiding [10], and the minim size of the curvature radius
is calculated in terms of maximum load capacity imposed to the higher contact
between cam and roller. For oscillating follower cam mechanism, the ratio
l
For solving may be used one of specific methods, so as Lagrange method, penalty
functions method etc.
The other constraints which must to be considered are:
a) the maximum value of the negative curvature radius of the cam profile do not
exceed the roller radius.
b) the minimum value of the positive curvature radius of the cam profile not be less