arXiv:0705.0856v1  [cs.RO]  7 May 2007
Proceedings of DETC’2007
2007 ASME Design Engineering Technical Conferences
September 4-7, 2007, Las Vegas, Nevada, USA
DETC2007/XXX
THE MULTIOBJECTIVE OPTIMIZATION OF A PRISMATIC DRIVE
´Emilie Bouyer, St´ephane Caro, Damien Chablat
Institut de Recherche en Communications
et Cybern´etique de Nantes
UMR CNRS n◦6597
1 rue de la No¨e, 44321
Nantes, France
Email: {stephane.caro, damien.chablat}@irccyn.ec-nantes.fr
Jorge Angeles
Department of Mechanical Engineering &
Centre for Intelligent Machines
McGill University
817 Sherbrooke West, H3A 2K6,
Montreal, QC, Canada
Email: angeles@cim.mcgill.ca
ABSTRACT
The multiobjective optimization of Slide-o-Cam is reported
in this paper. Slide-o-Cam is a cam mechanism with multiple
rollers mounted on a common translating follower. This trans-
mission provides pure-rolling motion, thereby reducing the fric-
tion of rack-and-pinions and linear drives. A Pareto frontier is
obtained by means of multiobjective optimization. This optimiza-
tion is based on three objective functions: (i) the pressure angle,
which is a suitable performance index for the transmission be-
cause it determines the amount of force transmitted to the load
vs. that transmitted to the machine frame; (ii) the Hertz pressure
used to evaluate the stresses produced on the contact surface be-
tween cam and roller; and (iii) the size of the mechanism, char-
acterized by the number of cams and their width.
NOMENCLATURE
p:
pitch of the transmission;
e:
distance between the axis of the cam and the line of centers
of the rollers;
r:
radius of the roller;
dcs:
diameter of the camshaft (dcs = 2(e−r));
L:
the width of the contact between the cams and the rollers;
ψ:
input of the mechanism, i.e., the angle of rotation of the
cam;
s:
output of the mechanism, i.e., the displacement of the fol-
lower;
µ:
pressure angle;
f:
force transmitted from the cam to the roller;
κc and κp:
curvature of the cam proﬁle and the pitch curve,
respectively;
ρc and ρp:
radii of curvature of the cam proﬁle and the pitch
curve, respectively;
m:
number of cams mounted on the camshaft;
n:
number of lobes per cam;
P:
Hertz pressure;
SM:
size of the mechanism.
1
INTRODUCTION
In robotic and mechatronic applications, whereby motion is
controlled using a piece of software, the conversion from ro-
tational to translational motion is usually realized by means of
ball-screws or linear actuators. While both are gaining popular-
ity, they present some drawbacks. On the one hand, ball-screws
comprise a high number of moving parts, their performance de-
pending on the number of balls rolling in the shaft groove. More-
over, they have a low load-carrying capacity due to the punctual
contact between balls and groove. On the other hand, linear bear-
ings are composed of roller-bearings to ﬁgure out the previous is-
sue, but these devices rely on a form of direct-drive motor, which
makes them expensive to produce and maintain.
A novel transmission, called Slide-o-Cam, is depicted in
Fig. 1 as introduced in [1] to transform a rotational motion into
a translational one. Slide-o-Cam is composed of four main ele-
ments: (i) the frame; (ii) the cam; (iii) the follower; and (iv) the
1
Copyright 2007 by ASME
rollers. The input axis on which the cams are mounted, named
camshaft, is driven at a constant angular velocity by means of
an actuator under computer-control.
Power is transmitted to
the output, the translating follower, which is the roller-carrying
slider, by means of pure-rolling contact between the cams and
the rollers. The roller comprises two components, the pin and
the bearing. The bearing is mounted to one end of the pin, while
the other end is press-ﬁt into the roller-carrying slider. Conse-
quently, the contact between the cams and rollers occurs at the
outer surface of the bearing.
The mechanism uses two con-
jugate cam-follower pairs, which alternately take over the mo-
tion transmission to ensure a positive action; the rollers are thus
driven by the cams throughout a complete cycle. Therefore, the
main advantages of cam-follower mechanisms with respect to
the other transmissions, which transform rotation into translation
are: (i) lower friction; (ii) higher stiffness; (iii) low backlash;
and (iv) reduction of wear. The multiobjective optimization of
PSfrag replacements
Roller
Follower
Conjugate cams
Figure 1.
Layout of Slide-o-Cam
Slide-o-Cam is reported in this paper. This optimization is based
on three criteria: (i) the pressure angle, a suitable performance
index for the transmission because it determines the amount of
force transmitted to the load vs. that transmitted to the machine
frame; (ii) the Hertz pressure, a measure of the stresses produced
in the contact surface between the cams and the rollers; and (iii)
the size of the mechanism, characterized by the number of cams
and their width.
2
SYNTHESIS OF PLANAR CAM MECHANISMS
Let the x-y frame be ﬁxed to the machine frame and the u-v
frame be attached to the cam, as depicted in Fig. 2. O1 is the ori-
gin of both frames, O2 is the center of the roller, and C is the con-
tact point between cam and roller. The geometric parameters are
illustrated in the same ﬁgure. The notation used in this ﬁgure is
based on the general notation introduced in [2-4], namely, (i) the
pitch p, i.e., the distance between the center of two rollers on the
same side of the follower; (ii) the distance e between the axis of
the cam and the line of centers of the rollers; (iii) the radius r of
f
e
p
s
d
x
y
P
C
µ
u
v
δ
b1
b2
r
ψ
θ
O1
O2
Figure 2.
Parameterization of Slide-o-Cam
PSfrag replacements
O1
p
x
y
u
v
x
s(0)
Figure 3.
Home conﬁguration of the mechanism
the roller-bearing, i.e., the radius of the roller; (iv) the angle of ro-
tation ψ of the cam, the input of the mechanism; (v) the position
s of the center of the roller, i.e, the displacement of the follower,
which is the output of the mechanism; (vi) the pressure angle µ;
and (vii) the force f transmitted from the cam to the roller.
The above parameters as well as the surface of contact on
the cam are determined by the geometric relations derived from
the Aronhold-Kennedy Theorem [2]. As a matter of fact, when
the cam makes a complete turn, i.e., ∆ψ = 2π, the displacement
of the roller is equal to the pitch, i.e., ∆s = p. Furthermore,
if we consider that Fig. 3 illustrates the home conﬁguration of
the roller, the latter is below the x-axis when ψ = 0. Therefore,
s(0) = −p/2 and the input-output function s is deﬁned as:
s(ψ) = p
2πψ−p
2
(1)
The cam proﬁle is determined by the displacement of the contact
point C around the cam. The Cartesian coordinates of C in the
u-v frame take the form [5]
uc(ψ) = b1 cosψ+ (b2 −r)cos(δ−ψ)
(2a)
2
Copyright 2007 by ASME
vc(ψ) = −b1 sinψ+ (b2 −r)sin(δ−ψ)
(2b)
The expressions of coefﬁcients b2, b3 and δ, as obtained in [6-
10], are:
b1 = p
2π
(3a)
b2 = p
2π
q
(2πη−1)2 + (ψ−π)2
(3b)
δ = arctan
 ψ−π
2πη−1

(3c)
where η = e/p, a nondimensional design parameter.
From Eq.(3c), we can notice that η cannot be equal to
1/(2π).
Moreover, an extended angle ∆was introduced in [6]
replacements
O1
x
y
u
v
C
ψ
(a)
PSfrag replacements
O1
x
y
u
v
C
ψ
x,u
y,v
∆
-u
-v
(b)
PSfrag replacements
O1
x
y
u
v
C
ψ
x,u
y,v
∆
-u
-v
u
v
(c)
Figure 4.
Orientations of the cam found when vc = 0: (a) ψ = ∆; (b)
ψ = π; and (c) ψ = 2π−∆
to obtain a closed cam proﬁle. ∆is deﬁned as a root of Eq.(2b).
As far as Slide-o-Cam is concerned, ∆is negative, as shown in
Fig. 4. Consequently, to close the cam proﬁle, ψ must vary within
∆≤ψ ≤2π−∆.
2.1
Pitch-Curve Determination
The pitch curve is the trajectory of O2, the center of the
roller, distinct from the trajectory of the contact point C, which
produces the cam proﬁle. The Cartesian coordinates (e,s) of
point O2 in the x-y frame are depicted in Fig. 2. Hence, the Carte-
sian coordinates of the pitch-curve in the u-v frame are
up(ψ) = ecosψ+ s(ψ)sinψ
(4a)
vp(ψ) = −esinψ+ s(ψ)cosψ
(4b)
2.2
Curvature of the Cam Proﬁle
The curvature κp of the pitch curve is given in [10] as
κp = 2π
p
[(ψ−π)2 + 2(2πη−1)(πη−1)]
[(ψ−π)2 + (2πη−1)2]3/2
(5)
provided that the denominator does not vanish at any value of ψ
within ∆≤ψ ≤2π−∆, i.e., η ̸= 1/(2π).
Let ρc and ρp be the radii of curvature of the cam proﬁle
and the pitch curve, respectively, and κc the curvature of the cam
proﬁle. Since the curvature is the reciprocal of the radius of cur-
vature, we have ρc = 1/κc and ρp = 1/κp. Furthermore, due to
the deﬁnition of the pitch curve, it is apparent that
ρp = ρc + r
(6)
From Eq. (6), the curvature of the cam proﬁle can be written as
κc =
κp
1 −rκp
(7)
In [9], the authors claimed that the cam proﬁle has to be fully
convex for machining accuracy. Such a proﬁle can be obtained
if and only if η > 1/π. In order to increase the range of design
parameters, we include non-convex cams within the scope of this
paper. Nevertheless, the sign of the local radius ρc has to remain
positive as long as the cam pushes the roller. In this vein, the cam
is convex when η ∈]1/(2π), 1/π] and ψ ∈]∆, π] [11]. Moreover,
–
–
–
–
–40
–30
–20
–
20
30
–
PSfrag replacements
ψ
1/κc
x
y
15
15
15
10
10
10
10
10
5
5
5
5
0
0
-5
-1
1
1
2
3
Figure 5.
Cam proﬁle and local curvature of the cam
according to [9], ρc is a minimum when
ψ = ψmin = π−
√
4n2πh −n2−4n2π2h2
n
(8)
where n is the number of lobes per cam. Therefore, the cam pro-
ﬁle is not feasible when ρc(ψmin) < 0. If this inequality becomes
an equality, the roller will block the cam, as depicted in Fig. 5.
3
MULTIOBJECTIVE OPTIMIZATION PROBLEM
We introduce in this section the multiobjective optimization
of Slide-o-Cam. Indeed, such an optimization is needed to prop-
3
Copyright 2007 by ASME
erly dimension the mechanism. First, the objective functions are
deﬁned. Then, a sensitivity analysis of the mechanism is reported
in order to choose shrewdly the design variables of the optimiza-
tion problem. Finally, the results of the latter are illustrated by
means of a Pareto frontier as the objective functions are antago-
nistic.
3.1
The Objective Functions
The optimization of the mechanism is based on three ob-
jective functions: (i) the maximum pressure angle µmax; (ii) the
maximum Hertz pressure Pmax related to the contact between the
cams and the rollers; and (iii) the size of the mechanism SM. As
a matter of fact, we want to simultaneously minimize these three
functions.
3.1.1
The Pressure Angle
The pressure angle µ of a
cam-roller-follower mechanism is deﬁned as the angle between
the normal to the contact point C between the cam and the roller
and the velocity of C as a point of the follower [3]. As illustrated
in Fig. 2, µ is a signiﬁcant parameter in cam design. In fact, the
smaller µ1, the better the transmission. The expression for µ is
given in [3]; in terms of η, we have
tanµ = n −2nπη
nψ−π
(9)
g replacements
active part
µ = µmax
P = Pmax
(a)
(b)
x
y
Figure 6.
Active parts of: (a) a two- and (b) three-conjugate cam mech-
anisms
Figure 6 illustrates the active parts of a two- and a three-
conjugate-cam mechanisms. It turns out that the pressure angle
is a maximum at the ends of the active parts for the two mecha-
nisms. In this paper, µmax denotes the maximum pressure angle
1µ is a real number and can be either positive or negative. However, within
the scope of this paper, µ remains positive. Therefore, µ = |µ|, | · | denoting the
absolute value.
along the active part of the cam proﬁle; it is an objective function
in this optimization problem.
3.1.2
The Hertz Pressure
When two bodies with
curved surfaces, for example, a cam and a roller, are pressed to-
gether, contact takes place not along a line but along a surface,
due to the inherent material compliance. Moreover, the stresses
developed in the two bodies are three-dimensional. Those con-
tact stresses may generate failures as cracks, pits, or ﬂaking
in the surface material.
To quantify these stresses, Heinrich
PSfrag replacements
x
y
B
O1
ρc
L
r
Figure 7.
The width B of the contact between a cam and a roller
Rudolf Hertz (1857–1894) proposed some formulas to evaluate
the width of the band of contact between two cylinders and the
maximum pressure of contact, called Hertz pressure. In Slide-o-
Cam, the rollers and the cams are the bodies in contact. Unlike
the roller, the cam is not a cylinder, but can be approximated by
a cylinder with radius identical to the radius of curvature of the
cam at the contact point. The width B of the band of contact is
illustrated in Fig.7, and given by Hertz as
B =
r
16F(K1 + K2)Requ
L
(10a)
Requ =
rρc
r + ρc
(10b)
F being the magnitude of the axial load f while Requ is the equiv-
alent radius of contact, L the width of the contact between the
cams and the rollers, and K1 and K2 the coefﬁcients that char-
acterize the materials of the cams and the rollers, respectively,
i.e.,
K1 = 1 −ν2
1
πE1
,
K2 = 1 −ν2
2
πE2
(11)
4
Copyright 2007 by ASME
where ν1 and ν2 are the Poisson ratios of the materials of the
cam and the roller, respectively, while E1, E2 their corresponding
Young moduli. Accordingly, the Hertz pressure P of the contact
between the cams and the rollers takes the form:
P = 4F
LπB
(12)
Let us notice that P depends on ψ, as F is a function of this
variable and B is a function of F.
Let us assume that F is constant. As L and r are constant and
Requ is monotonic with respect to (w.r.t) ρc as long as ρc > −r,
from Eq.(10a), the lower ρc, the lower B. From Eq.(12), the
lower B, the higher P. According to [11], ρc is a minimum when
ψ = π/n−∆for a two-conjugate cam mechanism. Therefore, P
is a maximum when ψ = π/n −∆for such a mechanism.
Figure 6 illustrates the active parts of a two- and a three-
conjugate-cam mechanisms. It turns out that the Hertz pressure is
a maximum at the ends of the active parts for the two mechanisms
as ρc is a minimum at those ends. In this paper, Pmax denotes the
maximum Hertz pressure along the active part of the cam proﬁle;
it is an objective function in this optimization problem.
The maximum Hertz pressures allowed for some materials
are obtained from [12] and recorded in Table 1. The second col-
umn gives the allowable pressure Pstat for a static load. As a
matter of fact, it is recommended not to apply more than 40% of
Pstat in order to secure an inﬁnite fatigue life. The corresponding
values Pmax are given in the third column of Table 1. Obviously,
Table 1.
Allowable pressures
Material
Pstat [MPa]
Pmax [MPa]
Stainless steel
650
260
Improved steel
1600 to 2000
640 to 800
Grey cast iron
400 to 700
160 to 280
Aluminum
62.5
25 to 150
Polyamide
25
10
the maximum allowable pressure depends also on the shape of
the different parts in contact. A thick part will be stiffer than a
thin one. Nevertheless, we only take into account the material
of the cams and rollers for the determination of the allowable
pressures within the scope of this research work. Finally, let us
notice that only improved steel is appropriate for a Slide-o-Cam
transmission in case of high Hertz-pressure values.
3.1.3
Size
The size of the mechanisms SM is deﬁned as
SM = mL
(13)
where m is the number of cams. From [9], a Slide-o-Cam with
only one cam, i.e., m = 1, is not feasible. Besides, the smaller
SM, the less bulky the mechanism.
3.2
The Design Variables
The design variables of the optimization problem are: (i) the
diameter dcs of the camshaft (dcs = e−r); (ii) the radius r of the
rollers; (iii) the width L of the contact between cam and roller;
and (iv) the number of cams m.
3.3
Sensitivity Analysis
We conduct here the analysis of the sensitivity of the perfor-
mance of Slide-o-Cam to the variations in its design parameters.
Such an analysis is needed to both determine the tolerance of the
design variables and obtain a robust design.
PSfrag replacements
µ (degree)
ψ(rad)
12
10
8
7
6
5
4
3
2
0
-2
-4
-6
-20
-40
-60
-80
20
40
60
80
the came pushes
to the left
to the right
(a)
(b)
Figure 8.
Pressure-angle distribution for (a) two conjugate-cam and (b)
three conjugate-cam mechanisms with one lobe, p = 50, r = 10 and
e = 9
3.3.1
Sensitivity of the Pressure Angle
Figure 8
illustrates the pressure-angle distribution for two conjugate- and
three conjugate-cams with one lobe. We can notice that the pres-
sure angle decreases with the number of cams. Consequently,
we can use conjugate cams, namely, several cams mounted on
the camshaft, to reduce the pressure angle. Below is a list of the
effects of some design parameters on the pressure angle:
1. The lower η, the lower the pressure angle, with η ≥1/π;
2. the lower r, the lower the pressure angle;
3. the lower n, the lower the pressure angle,[9];
4. the higher m, the lower the pressure angle.
where m is the number of cam(s) mounted on the camshaft.
As the pressure angle increases with the number of lobes,
we consider only mono-lobe cams, i.e. n = 1.
5
Copyright 2007 by ASME
3.3.2
Sensitivity of the Hertz Pressure
Pmax de-
pends on the geometry of the cam, the number of conjugate
cams, the material of the parts in contact and the load applied.
Therefore, we have different ways to minimize the Hertz pres-
sure, namely,
1. The higher m, the lower Pmax;
2. the lower the axial load, the lower Pmax;
3. the more compliant the material, the lower Pmax;
4. the higher L, the lower Pmax.
In order to analyze the sensitivity of P to r, η, p and L, we use
a ﬁrst derivative model of P w.r.t. the corresponding parameters,
i.e.,
δP ≈cδq
(14)
with
c =


∂P/∂r
∂P/∂η
∂P/∂p
∂P/∂L

, δq =


δr
δη
δp
δL


(15)
If the values of the parameters are known, we will be able to
evaluate c. Let us assume that r = 4 mm, η = 0.18, p = 50 mm
and L = 10 mm. The partial derivatives have to be normalized to
replacements
∂P
∂qi
× qi0
ψ
w.r.t r
w.r.t η
w.r.t p
w.r.t L
w.r.t Cm
0
5.0
6.0
7.0
200
100
−100
−200
−300
ψ = π/n −∆
ψ = 2π/n −∆
Figure 9.
Inﬂuence of the variations in r, η, p and L on P
be compared. In this vein, we divide each of them by its nominal
value. Now, we can plot each partial derivative with respect to
the angle of rotation of the cam ψ, as illustrated in Fig. 9.
The most inﬂuential variables are those with the highest ab-
solute value of their corresponding partial derivative for a given
value of ψ. As the maximum value of the Hertz pressure is ob-
tained for ψ = π/n −∆and ∆= −1.2943 rad, the partial deriva-
tives can be evaluated for ψ = π/n−∆. The sensitivity of Pmax to
δq is recorded in Table 2. The plots in Fig. 9 show the sensitivity
Table 2.
Inﬂuence of the variations in r, η, p and L on Pmax
qi
r
η
p
L
qinit
4 mm
0.18
50 mm
10 mm

∂Pmax
∂qi
(qinit)

103.32
83.25
362.03
232.67
Order of importance
3
4
1
2
of the Hertz pressure w.r.t the different parameters for different
values of ψ and for the active part of the cam proﬁle. However, it
is more relevant to calculate the rms value of each partial deriva-
tive, as recorded in Table 3.
As a matter of fact, Table 2 and
Table 3.
Global inﬂuence of the variations in r, η, p and L on P
qi
r
η
p
L
qinit
4
0.18
50
10
s
n
π
Z
ψ( ∂P
∂qi
)2dψ
156.59
20.21
261.85
207.79
Order of importance
3
4
1
2
Table 3 provide the same results in terms of order of importance
of the variations in r, η, p and L. Finally, in order to minimize
the variations in the Hertz pressure, we had better minimize the
variations in p, L r and η in descending order.
3.3.3
Sensitivity of the Size of the Mechanism
The sensitivity analysis of SM is trivial. Indeed, from eq. (13),
the higher m, the higher SM. Likewise, the higher L, the higher
SM.
3.4
Problem Formulation
A motivation of this research work is to implement a Slide-
o-Cam transmission in the Orthoglide, a low-power machine tool
6
Copyright 2007 by ASME
introduced in [8]. To that end, the transmission has to transmit a
torqueCt of 1.2 Nm with a pitch of 20 mm. In case of high-speed
operations, i.e., when the velocity of the cams is higher than
50 rpm, the pressure-angle is recommended to be smaller than
30◦. Table 1 shows that the maximum value of the Hertz pres-
sure has to be smaller than 800 MPa as the cams and the rollers
are made up of steel. Moreover SM is supposed to be smaller than
90 mm with a view to limiting the size of the mechanism. Be-
sides, the Slide-o-Cam transmissions under study are composed
of two- or three- conjugate cams as a Slide-o-Cam with only one
cam is not feasible and such a mechanism with more than three
conjugate cams would be too bulky, i.e., m = {2,3}. Conse-
quently, the optimization problem can be formulated
min
x (µmax, Pmax, SM)
s.t.
µmax ≤30◦
Pmax ≤800 MPa
SM ≤90 mm
xl ≤x ≤xu
where x = [dcs, r, L, m]T, while xl and xu denote the lower and
upper bounds of the design variables, respectively. Here, xl =
[0 mm, 4 mm, 0 mm, 2] and xu = [0 mm, 10.5 mm, Lmax, 3],
Lmax being equal to SMax/m knowing that SMax = 90 mm .
3.5
Results
replacements
µmax [deg]
SM [m]
Pmax [MPa]
njugate cams
njugate cams
5
10
15
20
25
30
5
10
15
20
25
30
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0
0.02
0.04
0.06
0.08
0.1
400
600
800
400
500
600
700
800
Figure 10.
Pareto frontier of a two- and a three- conjugate cam mecha-
nisms
The optimization problem deﬁned in Section 3.4 is multi-
objective with objective functions of a different nature. For this
reason, the optimum solutions of the problem can be illustrated
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0
0.02
0.04
0.06
0.08
0.1
400
600
800
400
500
600
700
800
Figure 11.
Pareto frontier of Slide-o-Cam mechanisms
by means of a Pareto frontier [13]. As the problem involves three
objective functions, i.e., µmax, Pmax and SM, the corresponding
Pareto frontier is depicted in 3D space as shown in Figs. 10 and
11. Figure 10 illustrates the Pareto frontiers of a two- and a three-
conjugate cam mechanisms. As we want to minimize the three
objective functions concurrently, the closer the Pareto frontier to
the origin, the better the design. In Fig. 10, we notice that the op-
timum solutions obtained with a three-conjugate cam mechanism
are slightly better when µmax is smaller than 24◦. Otherwise, a
two-conjugate cam mechanism turns out to be more interesting.
Nevertheless, the difference between the optimum solutions ob-
tained with a two- and a three-conjugate cam mechanisms re-
mains low. Figure 11 depicts the region closest to the origin
of the two frontiers shown in Fig. 10. It also shows the Pareto
frontier of Slide-o-Cam mechanisms, regardless of the number
of conjugate-cams.
PSfrag replacements
µmax [deg]
SM [m]
Pmax [MPa]
5
10
15
20
25
30
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Figure 12.
Pareto frontier w.r.t. µmax and SM
7
Copyright 2007 by ASME
replacements
µmax [deg]
SM [m]
Pmax [MPa]
5
10
15
20
25
30
450
500
550
600
650
700
750
800
Figure 13.
Pareto frontier w.r.t. Pmax and µmax
replacements
µmax [deg]
SM [m]
Pmax [MPa]
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
450
500
550
600
650
700
750
800
Figure 14.
Pareto frontier w.r.t. Pmax and SM
For better clarity of the results, Figs. 12, 13 and 14 illustrate
the projections of the Pareto frontier shown in Fig. 11 w.r.t µmax
and SM; Pmax and µmax; and Pmax and SM, respectively. These
ﬁgures allow us to see clearly the location the optimum and the
feasible solutions of the problem at hand.
Figures 15 and 16 illustrate the contours of µmax and Pmax
w.r.t dcs and r for a two- and a three-conjugate cam mechanisms
with SM = 0.06 m. On the one hand, the continuous lines depict
the iso-contours of µmax. On the other hand, the broken lines de-
pict the iso-contours of Pmax. Besides, Figs. 15 and 16 highlight
the location of the optimum solutions for a two- and a three-
conjugate cam mechanisms with SM = 0.06 m. We can notice
PSfrag replacements
dcs [mm]
r [mm]
M1
M2
Optimal solutions
Figure 15.
Contours of µ and P w.r.t dcs and r and the location of the
optimal solutions for a two conjugate-cam mechanism with SM = 0.06 m
PSfrag replacements
dcs [mm]
r [mm]
Optimal solutions
M3
M4
Figure 16.
Contours of µ and P w.r.t. dcs and r and the location of
the optimal solutions for a three conjugate-cam mechanism with SM =
0.06 m
that the line of optimum solutions in the space of design vari-
ables dcs and r is longer in Fig. 16 than in Fig. 15. This means
that a three-conjugate cam mechanism allows more optimal so-
lutions than its two-conjugate cam counterpart. In this vein, it is
more interesting to design a three-conjugate cam mechanism.
Figure 17 depicts the mechanisms corresponding to points
M1 and M2 that are plotted in Fig.15. For M1, dcs = 2.6 mm,
r = 4.24 mm, µmax = 3◦and Pmax = 653.83 MPa. For M2, dcs =
4.16 mm, r = 6.4 mm, µmax = 30◦and Pmax = 562.12 MPa.
Figure 18 depicts the mechanisms corresponding to points
M3 and M4 that are plotted in Fig.16. For M3, dcs = 2.2 mm,
8
Copyright 2007 by ASME
replacements
M1
M2
653.83 MPa
562.12 MPa
µmax = 3◦
µmax = 30◦
Figure 17.
Optimal two conjugate-cam mechanisms
replacements
M3
M4
654.57 MPa
579.45 MPa
µmax = 2◦
µmax = 30◦
Figure 18.
Optimal three conjugate-cam mechanisms
r = 4.68 mm, µmax = 2◦and Pmax = 654.57 MPa. For M4, dcs =
4.56 mm, r = 9.28 mm, µmax = 30◦and Pmax = 579.45 MPa.
According to Figs. 15 and 16, we can notice that the higher
r, the smaller Pmax. Indeed, the maximum Hertz pressure values
corresponding to M2 and M4 are smaller than the ones corre-
sponding to M1 and M3. However, the size of the mechanism
along the x-axis is higher for M2 and M4. Moreover, this induces
a better transmission of the torque as dcs is higher. Finally, we
can notice that the proﬁles of M2 and M4 are easier to machine
as they are fully convex.
4
CONCLUSIONS
The multiobjective optimization of Slide-o-Cam was re-
ported in this paper. Slide-o-Cam is a cam mechanism with mul-
tiple rollers mounted on a common translating follower. This
transmission provides pure-rolling motion, thereby reducing the
friction of rack-and-pinions and linear drives. A Pareto frontier
was obtained by means of a multiobjective optimization. This
optimization is based on three objective functions: (i) the pres-
sure angle, which is a suitable performance index for the trans-
mission because it determines the amount of force transmitted
to the load vs. that transmitted to the machine frame; (ii) the
Hertz pressure used to evaluate the stresses produced in the con-
tact surface between the cams and the rollers; and (iii) the size
of the mechanism characterized by the number of cams and their
width. It turns out that three-conjugate cam mechanisms have
globally better performance that their two-conjugate cam coun-
terparts. However, the difference is small.
REFERENCES
Gonz´alez-Palacios, M.A. and Angeles, J., “The design of a
novel pure-rolling transmission to convert rotational into trans-
lational motion”, ASME Journal of Mechanical Design, 2003,
Vol. 125, pp. 205–207
Waldron, K. J. and Kinzel, G. L., Kinematics, Dynamics, and
Design of Machinery, John Wiley & Sons, Inc., New York,
1999.
Angeles, J. and L´opez-Caj´un,C., Optimization of Cam Mecha-
nisms, Kluwer Academic Publishers B.V., Dordrecht, 1991.
Carra, S., Garziera, R. and Pellegrini, M., “Synthesis of cams
with negative radius follower and evaluation of the pressure
angles,” Mechanism and Machine Theory, 2004, Vol. 34,
pp. 1017–1032.
Gonz´alez-Palacios, M. A. and Angeles, J., Cam Synthesis,
Kluwer Academic Publishers B.V., Dordrecht, 1993.
Lee, M.K., Design for Manufacturability of Speed-Reduction
Cam Mechanisms, M.Eng. Thesis, Dept. of Mechanical Engi-
neering, McGill University, Montreal, 2001.
Golovin A., Borisov A., Drozdova I., and Shuman B., “The sim-
ulating model of a gearing wear”, Moscow State Technical Uni-
versity named after M.E. Bauman, Moscow, Russia, 2005.
Chablat, D. and Wenger P. (2003). Architecture Optimization
of a 3-DOF Parallel Mechanism for Machining Applications,
the Orthoglide. IEEE Transactions on Robotics and Automa-
tion Vol. 19/3, 403–410, June.
Chablat D. and Angeles J. “Design Strategies of Slide-o-Cam
Transmission”, Proceedings of CK2005, International Work-
shop on Computational Kinematics, Cassino, 2005, May 4–6.
Chablat, D. and Angeles J., “The Design of a Novel Prismatic
Drive for a Three-DOF Parallel-Kinematics Machine”, ASME
Journal of Mechanical Design, 2006, Volume 128, Issue 4,
pp. 710–718, July.
Chablat, D., Caro, S. and Bouyer E. (2007) “The Optimization
of a Novel Prismatic Drive taking into account the Transmitted
Force”, IFToMM, Besanon, France, June.
http://www.matweb.com/
Collette, Y. and Siarry, P. (2006) Multiobjective Optimization:
Principles and Case Studies. Springer-Verlag Berlin Heidelberg
New York.
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