1771
A novel method for the dynamic synthesis of cam
mechanisms with an imposed driving force profile
H Erdelyi∗ and D Talaba
Transilvania University of Brasov, bdul. Eroilor, Brasov, Romania
The manuscript was received on 30 March 2009 and was accepted after revision for publication on 11 November 2009.
DOI: 10.1243/09544062JMES1649
Abstract: In the car industry, the ergonomic and haptic properties of controls are gaining high
priority. The cam mechanisms of switches and knobs, such as those of car turn switches, are usually required to comply with a certain user force profile, according to human physical limitations
and ergonomic requirements. In order to fine-tune the force input and the function performed by
the mechanism, the cam profile is normally subjected to a detailed synthesis having as input the
desired user force profile. Nevertheless, conventional cam synthesis and analysis methods do not
provide an adequate tool for designing the cam profile based on the required driving effort. In this
article, an energy-based method is proposed for designing a cam mechanism able to accurately
replicate the desired profile of the driving effort. According to the proposed method, the cam profile is determined as a finite series of points and takes into consideration all contact forces between
the cam and the follower, including friction. The method is non-linear, considering a combined
Stribeck–Therfall friction model as well as the damping of the system. Numerical validation is
provided considering the case of the cam mechanism of a conventional car turn switch.
Keywords: cam design, cam mechanism, cam profile, driving force profile, energy-based
calculation, finite series
1
INTRODUCTION
Cam mechanisms provide a wide variety of motion
generation methods, due to their different geometrical configurations and different combinations of cams
and followers. Because of their versatility and ease of
adaptability, cam mechanisms are widely spread in
all types of mechanical devices and machines, like
agricultural machines, the textile industry, packaging
machines, machine tools, printing presses, the automotive industry, switches, or control systems [1, 2].
Cam mechanisms, being a kind of mechanical program carrier, are also used in robotic design when it is
not possible to satisfy a desired position function by
means of other systems [3].
Usually, the industrial design of cam mechanisms
considers primary guidelines such as [1, 4, 5]
(a) transfer function;
(b) design speed;
∗ Corresponding author: Product Design and Robotics, Transilvania
University of Brasov, bdul. Eroilor nr.29, Brasov 500036, Romania.
email: erdelyi@unitbv.ro
JMES1649
(c)
(d)
(e)
(f )
(g)
(h)
(i)
(j)
kinematic constraints;
cam–follower acceleration curve;
maximum follower acceleration;
maximum pressure angle;
manufacturing considerations;
surface wear;
torque on the camshaft;
cost and space requirements.
However, there is a special area of application
where the above-mentioned guidelines are only of secondary importance, namely those cam mechanisms
that are actuated manually, such as switches for example. These mechanisms operate usually at low speeds
and involve relatively low forces. In these mechanisms, also the user force feedback properties (haptic
properties) are considered, and sometimes they constitute the primary design parameter. A good example
of such human-operated mechanisms is switches in
general and, in particular, the switches used in the
automotive industry. In the last few years, the major
car manufacturers have been conducting research on
the issue of deliberately shaping the haptic sensation of knobs and switches in terms of ergonomics,
comfort, and pleasure in manipulating them [6, 7].
Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science
Downloaded from pic.sagepub.com at WEST VIRGINA UNIV on June 20, 2015
1772
H Erdelyi and D Talaba
By manufacturing a switch that looks pleasant, a
positive impression is created in the customer and
this provides a competitive advantage in the market.
Therefore, in addition to the purely functional requirements, designers now face the challenge of having to
design the ‘user feel’ of the controls as well. To solve
this problem in the best possible way, the designer
would need straightforward methods for determining the mechanism’s characteristics based on a desired
force feedback (haptic) property.
Previous work on the haptic design of such switches
proposed mainly tools for aiding the analysis of the
user feel; investigating, for example, the characterization of the haptic parameters [8, 9] or focusing on
developing haptic systems [10–12], but not considering the design of the user feel itself through a specific
cam mechanism design.
The usual kinematic approach of cam mechanism
synthesis deals with the establishment of geometrical parameters and transmission functions, velocities,
accelerations, and pulses [2, 5, 13]. In the dynamic
analysis of cam mechanisms, the system is generally treated from the point of view of vibration [2, 13]
considering rigid and elastic camshafts, and one or
several degrees of freedom spring and damper models. The motion of the equivalent mass of the follower
is described by differential equations. Lederer [14] proposes a two-degree-of-freedom spring and damper
model to analyse the influence of the adopted transfer
function on the dynamics of the cam mechanism. He
first solves the differential equations in order to determine the transfer function and then an optimization
procedure is applied, in case of inadequate stiffness
and damping parameters.
Although the cam design and manufacturing
methods presented in the available literature are well
established, they do not provide an adequate tool for
designing a cam profile based on a required driving
effort. Therefore, this article proposes a new method
devoted specifically to designing a cam mechanism
according to a desired profile of the driving force or
driving torque. Using energy-based calculations, the
cam profile is determined as a finite series of points in
polar coordinates and takes into consideration all contact forces between the cam and the follower, including friction. The method is non-linear, considering a
combined Stribeck–Therfall friction model as well as
the damping of the system. Numerical validation is
provided considering the case of a cam mechanism
from a conventional car turn indicator switch.
2
or torque’s profile contains all the necessary information about the cam geometry. Considering the driving
torque’s profile as a function of the cam rotation angle
(as a design input), with the method presented in this
article it is possible to accurately design the cam geometry corresponding to the required force profile. The
method has two main stages: first, an idealized model
is considered, neglecting any type of losses, for which
a so-called ‘preliminary cam profile’ is determined. At
the second stage, an iterative method is implemented,
considering losses due to friction and damping, by
which one can correct the preliminary cam profile
determined previously.
For the method presented below, the time evolution
of the driving force’s profile, the cam rotation angle,
and the angular velocity are considered as known and
the goal is to determine the cam profile.
2.1
Determining the preliminary cam profile
For the preliminary design stage, a simplified model
of a typical cam mechanism is presented in Fig. 1,
where K indicates the spring of the follower, C is the
damper, and Tr stands for the translational joint that
guides the follower. As the driving torque T acts on the
cam, the follower–cam point of contact moves along
the cam profile (Po Pa Po ) displacing the follower and
simultaneously compressing the spring of the follower.
Since the follower displacement and the spring compression are equal in this case, one can determine the
cam profile using the spring compression data as a
function of the cam rotation angle. The cam displacement can be represented using polar coordinates as
a function of the rotation angle of the cam α and the
polar radius ρ: the distance between the follower and
cam pivot axes (Fig. 1).
A CAM DESIGN METHOD BASED ON THE
DRIVING TORQUE’S PROFILE
The method presented in this article relies on the
idea that for a cam mechanism, the driving force’s
Fig. 1
Schematic model of the cam mechanism
Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science
Downloaded from pic.sagepub.com at WEST VIRGINA UNIV on June 20, 2015
JMES1649
A novel method for the dynamic synthesis of cam mechanisms
According to the proposed method, for the preliminary stage, the joint compliances and losses due to
friction and damping are neglected. In this case the
energy balance of the system is
dW T = dE P + dE K
(1)
where dW T is the work done by the driving torque T ,
dE P is the potential energy of the system, and dE K is
the kinetic energy.
At this point, considering low speeds and low
masses, the kinetic energy term dE K ≈ 0 will also be
preliminarily neglected, leaving a simple form of the
energy balance
dW T = dE P
(2)
The potential energy term dE P in the above equation
(equation (2)) is the potential energy stored in
the spring of the follower and therefore it can be
expressed as
dE P = F el dx = kx dx
(3)
where F el is the elastic force of the spring, k is
the stiffness coefficient, and x is the compression of
the spring.
Using the trapezoidal formula, one can numerically
approximate the potential energy between two consecutive values of spring compression (x0 and x1 ) as
follows
x1
k
F el dx ≈ (x12 − x02 )
(4)
2
x0
From the above equation (equation (4)), the spring
compression x1 can be determined as
x1 ≈
2
k
x1
x0
F el dx + x02
(5)
Following the above algorithm, the potential energy
stored in the spring for the next compression cycle (x1
to x2 ) can be written as
x1
x2
k
k 2
2
x2 −
F el dx ≈ (x22 − x12 ) =
F el dx + x02
2
2
k x0
x1
(6)
From equation (6), x2 can be determined as shown
below
x2
x1
2
el
el
x2 ≈
F dx +
F dx + x02
(7)
k x0
x1
1773
of spring compression with the formula
x2
xn
2 x1 el
el
el
F dx + F dx + · · · +
F dx + x02
xn ≈
k x0
x1
xn−1
n xi
2
=
F el dx + x02
(8)
k i=1 xi−1
On the other hand, according to equation (2), the
work done by the driving torque is equal to the potential energy of the spring. Furthermore, by considering
that any α interval of rotation has a corresponding
x compression of the spring, the energy balance is
xn
αn
F el dx =
dW T
(9)
xn−1
αn−1
In the above equation (equation (9)), the work of the
driving torque T can be determined as
dW T = T dα
(10)
where T is the driving torque and α is the rotation angle
of the cam.
Over a small interval of rotation the work done by
the driving torque will be
α0 +α
α0 +α
T
dW =
T dα
(11)
α0
α0
For a finite sequence of rotation intervals of the cam,
the above equation can be approximated with the
trapezoidal formula determining the generic term as
follows
αn
α
WnT ≈
(12)
(Tn−1 + Tn )
2
αn−1
Considering equation (2), the potential energy term
from equation (8) can be replaced with the driving
torque work determined in equation (12). Considering
the above-mentioned algorithm, one can determine a
finite series of spring compression data as a function
of the cam rotation and the driving torque
2 n α
(13)
xn ≈ (Ti−1 + Ti ) + x02
k i=1 2
Finally, one can compute a series of follower displacement ρn based on the compression data of the
spring – provided by equation (13)
ρn = r0 + rr + xn ≈ r + R
2 n α
+
(Ti−1 + Ti ) + x02
k i=1 2
(14)
Using the same algorithm, one can determine for a
finite series of compression intervals the generic term
where r0 is the minimum radius and rr is the roller
radius as presented in Fig. 1.
JMES1649
Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science
Downloaded from pic.sagepub.com at WEST VIRGINA UNIV on June 20, 2015
1774
H Erdelyi and D Talaba
Equation (14) determines the cam profile as a series
of points in polar coordinates, having the generic term
ρn , being a function of the driving torque and cam
rotation angle α. Nevertheless, this cam profile is just
a preliminary approximation needed in the next stage
of the method, since the friction, damping, and kinetic
energy of the system have not yet been considered.
2.2
Actual cam profile calculus
After determining the preliminary cam profile for the
simplified model, losses due to friction and damping
as well as kinetic energy of the system can be introduced. This is done in an iterative manner based on the
preliminary results provided by the simplified model.
First, the mechanical work of the friction and damping
forces will be determined as well as the kinetic energy
of the cam mechanism. The next step is to implement
them into an iterative method to determine the actual
cam profile.
2.2.1
Damping force
F d = c ẋ
(15)
where F d is the damping force, c the coefficient of
damping, and ẋ the displacement velocity of the
follower.
Considering a series of displacements xn , the displacement velocity ẋ can be approximated as
xn
xn
xn+1 − xn
=
n =
n
tn
αn
αn+1 − αn
(16)
where n is the mean angular velocity of the cam,
between αn and αn+1 .
Equations (15) and (16) give the generic term of the
damping force
Fnd ≈ c
xn
xn
xn+1 − xn
=c
n = c
n
tn
αn
αn+1 − αn
The work of the damping force is given by
x
x
dW D =
−F d dx
x0
Friction model
Friction forces should be considered for all kinematic
joints as well as the cam–tappet contact. Figure 2
presents friction types which are typical for kinematic
joints and which are discussed in this article. In the
method presented in this article, the usual formulae
known from the literature were adopted for the calculus of the friction forces in the kinematic joints. Their
effect will be taken into consideration by a separate
energy term cumulating the friction energy in all the
joints of the actual mechanism.
In the case of the cam mechanisms used in switches
and knobs, the sliding friction at the contact between
the cam and the follower has a greater overall effect
than journal or circular crown type friction encountered at the revolute joints of the rotating parts.
However, for the sake of completeness, also the rotary
friction term will be discussed.
In the case of a journal type contact [15, 16], the
friction torque will be determined with the formula
TnfrJ = µQn rj
The damping force in the follower spring unit can be
determined as
ẋ ≈
2.2.2
(17)
(20)
where TnfrJ is the friction torque, µ is the friction coefficient, Qn is the radial load, and rj is the journal radius.
The work of the journal type friction torque is given
as follows (based on equation (20))
αn
αn−1
dWnfrJ = −µQn rj αn
(21)
For a circular crown contact, the friction torque will
be determined as [15, 16]
TnfrC =
r 3 − ri3
2
µPn e2
3
re − ri2
(22)
where TnfrC is the friction torque in the case of the circular crown contact, Pn is the axial load, re is the outer
radius, and ri is the inner radius.
The generic term for the work WnfrC corresponding to
the friction torque in the circular crown is given below
αn
(18)
r 3 − ri3
2
dWnfrC = − µPn e2
αn
3
re − ri2
αn−1
(23)
x0
D
where W represents the work of the damping force
over a portion of the displacement (x0 . . . x) of the
follower.
Furthermore, using the trapezoidal approximation
method, one can write the work equation as
xn
x d
dWnD ≈ −
Fn−1 + Fnd
2
xn−1
(xn − xn−1 ) xn+1 − xn
c
n
2
αn+1 − αn
(19)
Fig. 2 Typical types of friction encountered in kinematic
joints
Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science
JMES1649
=−
Downloaded from pic.sagepub.com at WEST VIRGINA UNIV on June 20, 2015
A novel method for the dynamic synthesis of cam mechanisms
To model the sliding friction of the cam–follower
contact, numerous methods are available in the literature ranging from simple – like the basic Coulomb
friction model – to more complex ones like the
Rooney and Deravi friction model [17] or the Threlfall
model [18]. Considering that cam mechanisms are
usually lubricated, a viscous friction model is adopted
within the proposed method, namely the Stribeck friction model [19] that considers the variation of friction
force with the sliding speed as
i vn
Fnfr = FnC + (FnS − FnC ) e−(|vn |/vS )
+ µvisc vn
|vn |
Fnfr
FnC
=
µstat FnN
tan σ =
vn
(1 − e−(3vn /vr ) ) if
|vn |
|vn | < vr
(26)
where µ is the sliding friction coefficient, FnN is the
normal contact force, and vr is a small characteristic velocity. In the above equation (equation (26)),
1 − e−(3v/vr ) is the regulation factor that smoothes out
the discontinuity of the friction force.
Figure 3 presents the Threlfall friction, the Stribeck
friction, and the combined friction model, which
tan σn ≈
JMES1649
Different friction models
(27)
yn
yn+1 − yn
=
xn
xn+1 − xn
(28)
Turning to polar coordinates in equation (28), one
will arrive at the following
σn ≈ arctan
ρn+1 cos αn+1 − ρn cos αn
ρn+1 sin αn+1 − ρn sin αn
(29)
On the other hand, it can be proved that a relation
exists between the pressure angle, the cam rotation
angle and σn as follows
σ n = αn − β n
(30)
where βn is the pressure angle.
Fig. 4
Fig. 3
dy
dx
For discrete values of a finite series, equation (27)
can be approximated as
(25)
where µstat is the static friction coefficient.
In order to avoid discontinuity at zero relative tangential velocity and to obtain a continuous friction
force for the Coulomb friction force term FnC , the
Threlfall friction model [18] is used
FnC = µFnN
eliminates the discontinuity at zero of the standard
Stribeck model.
Before writing the equation for the normal contact
force, first the pressure angle will be determined. In
Fig. 4, the pressure angle is denoted by βn and the
tangent to the pitch curve is denoted by σn . FnN is the
contact force and its two orthogonal components are
denoted by Fnl and Fnt .
Considering a coordinate system attached to the
cam – as shown in Fig. 4 – the tangent to the pitch
curve in the contact point can be defined as
(24)
where
is the friction force,
is the Coulomb sliding friction, FnS represents the maximum static friction
force, vn is the sliding speed, vS is the sliding speed
coefficient, µvisc is the viscous friction coefficient, and
i is an exponent. The generic term of the maximum
static friction force series FnS can be written as
FnS
1775
Schematic representation of the contact force and
important angles
Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science
Downloaded from pic.sagepub.com at WEST VIRGINA UNIV on June 20, 2015
1776
H Erdelyi and D Talaba
From equations (29) and (30) the pressure angle βn
can be determined as
ρn+1 cos αn+1 − ρn cos αn
(31)
βn ≈ αn − arctan
ρn+1 sin αn+1 − ρn sin αn
FnN
can be determined as a funcThe normal force
tion of the pressure angle βn and the follower spring
and damper forces
FnN = (Fnel + Fnd ) cos βn
(32)
where Fnel is the elastic force of the follower’s spring,
having the generic term
Fnel = kxn = k(xn+1 − xn )
(33)
Finally, considering equations (17) and (33), the
normal force will be
xn+1 − xn
N
n cos βn
(34)
Fn = k(xn+1 − xn ) + c
αn+1 − αn
The sliding speed v, tangent to the cam profile,
can be determined as a function of the displacement
velocity of the follower and the pressure angle
v = ẋ sin β
The total work of friction forces, considering sliding
friction of the follower, circular crown and journal type
of contacts, is given below
xn
xn−1
dWnTFr =
xn+1 − xn
n sin βn
αn+1 − αn
(36)
The work done by the sliding friction force over a
portion of displacement (S0 . . . S) can be determined as
S
dW fr =
S0
−F fr ds
(37)
Sn−1
αn−1
dWnfrJ +
αn
αn−1
dWnfrC
where WnTFr is the total work of friction forces.
2.2.3
Kinetic energy
Cam mechanisms used in human-operated controls
are usually driven at a low speed and are lightweight.
As such, the kinetic energy term is considerably lower
than the potential energy of the spring. However,
for the completeness of the method, also the kinetic
energy term will be considered.
Assuming that the cam mechanism is composed of
k elements with translation movements (with masses
mi and velocities ẋi ) and p elements with fixed axis
rotation movements (with moments of inertia Jl and
angular velocities α̇l ), the total kinetic energy can be
determined as [15]
EK =
i=1
mi (ẋi )2
+
2
p
l=1
Jl (α̇l )2
2
(42)
For the particular case presented in Fig. 4, considering a constant angular velocity of the cam, the series
of the kinetic energy will have the generic form
EnK =
mf [(xn+1 − xn /αn+1 − αn )n ]2
Jc (n )2
+
2
2
(43)
S0
Using the trapezoidal formula, one can integrate the
above equation in an approximated form as
Sn
αn
(41)
k
S
xn−1
dWnfr +
(35)
Given equations (16) and (35), the generic term of
the sliding speed is
vn =
xn
dWnfr ≈ −
s fr
(F
+ Fnfr )
2 n−1
(38)
The displacement S = S − S0 can be determined
as a function of spring compression x
s = x sin β
(39)
Taking into account the above, one will get
xn
xn sin βn fr
Fn−1 + Fnfr
dWnfr ≈ −
2
xn−1
=−
(xn − xn−1 ) sin βn fr
Fn−1 + Fnfr
2
(40)
where the friction force Fnfr is given in equation (24).
where mf is the mass of the follower, Jc is the moment
of inertia of the cam, and n is the angular velocity of
the cam.
2.2.4
Numerical integration and cam profile
iterative calculation
The iterative method presented below implies that the
preliminary cam profile is known.
In the first iteration, with equations (43), (19), and
(41), the kinetic energy, the work of the damping force
and that of the friction force, respectively, will be determined for the preliminary cam profile, determined by
equation (14) for the idealized model.
A new energy balance equation will subsequently
be written, considering all the losses and the kinetic
energy determined for the preliminary cam profile
dWnT = dEnP + dEnK − dWnTFr − dWnD
Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science
Downloaded from pic.sagepub.com at WEST VIRGINA UNIV on June 20, 2015
(44)
JMES1649
A novel method for the dynamic synthesis of cam mechanisms
From the above equation, the potential energy term
will be determined
xn
xn
xn
xn
dEnP =
dWnT −
dEnK +
dWnTFr
xn−1
xn−1
xn
+
xn−1
xn−1
xn−1
dWnD
(45)
Using equations (45) and (14), the cam profile
equation resulting within the first iteration is
xi
n 2
α
+
T
)
−
dEiK0
(T
i−1
i
k
2
xi−1
ρn1 = r0 + rr + xi
xi
i=0
TFr0
D0
+ x02
+
dWi
+
dWi
xi−1
xi−1
(46)
where the terms of the kinetic energy E K0 of the work
of the friction force W TFr0 and of the damping force
W D0 are determined for the preliminary cam profile
with equations (43), (41), and (19), respectively. Note
that since this article deals with a synthesis problem,
all parameters from equation (43) (mf , Jc , and ωn ) are
known as input data.
After having obtained the new cam profile ρn1 for the
first iteration, the method goes on with the next iterations by determining the kinetic energy, the work of
the friction force and that of the damping forces for the
newly determined cam profile. The terms E K1 , W TFr1 ,
and W D1 will thus be obtained using equations (43),
(41), and (19) for ρn1 . E K1 , W TFr1 , and W D1 will be substituted into equation (14), while the cam profile ρn2
will be obtained as
xi
n 2
α
(T
+
T
)
−
dEiK1
i−1
i
k
2
x
i−1
ρn2 = r0 + rr + xi
xi
i=0
TFr1
D1
+ x02
+
dWi
+
dWi
xi−1
xi−1
are the kinetic energy, the work of the friction force,
and the work of the damping force, determined for the
cam profile ρnm−1 , obtained in the preceding (m − 1)th
step of the iterative process, respectively.
After some steps the process stabilizes, and the
cam profile, determined with the iterative process, falls
within a margin of error that suits the application, and
the calculation can therefore be stopped. The number
of steps required for stabilization is application dependent, but for a large set of torque data the process
usually stabilizes after just a few steps.
3
NUMERICAL EXAMPLE AND EXPERIMENTAL
VALIDATION
For the experimental validation, the turn switch of a
car was considered. The switch has a cam mechanism with a fixed cam and a moving follower, as shown
in Fig. 5.
The contact between the follower and the cam
is ensured by a spring. The follower is connected to the
lever of the switch via a translational joint. Between the
lever and the cam, there is a revolute joint, ensuring
the rotation of the lever around the lever pivot axis as
one turns the blinkers on. For the experimental study,
the electrical contacts have been removed, leaving just
the cam mechanism to be considered.
The objective of the experimental research was to
determine the driving force profile of the switch and
the real cam profile, using a three-dimensional (3D)
digitizer. Finally, the dynamic synthesis method presented before was applied to compute the cam profile
based on the measured driving force. The results
were compared to the original cam profile to see if
the proposed method delivered back the cam mechanism that experimentally provided the input user force
profile.
3.1
(47)
1777
Driving force profile measurement and
reference cam profile determination
where ρnm is the cam profile determined for the mth
step of the iterative process, E Km−1 , W Frm−1 , and W Dm−1
The driving force profile needed for the cam profile
calculus was measured with a Series 244 hydraulic
actuator. The switch was rigidly fixed on a workbench,
and it was driven by the piston of the actuator at
a constant speed of 0.044 rad/s, while the force and
position were recorded and processed to obtain the
driving force profile. The layout of the measurement is
presented in Fig. 6.
The recorded dataset contained 3797 data pairs of
force value as a function of lever rotation angle, for a
period of 3.5 s. The data were smoothened to obtain
a continuous evolution of the force profile (as presented in Fig. 7), which constituted the input for the
calculation.
The cam profile was carefully digitized with a 3D
digitizer. The resulting dataset was interpolated to
JMES1649
Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science
This algorithm is carried out for a number of steps,
always determining the terms of the friction and
damping force’s work W TFr and W D and the kinetic
energy E K for the cam profile obtained in the previous
step of the iterative process
ρnm = r0 + rr
xi
n 2
α
(T
+
T
)
−
dEiKm−1
i−1
i
k
2
x
i−1
+
i=0
xi
xi
TFrm−1
Dm−1
+ x02
+
dWi
+
dWi
xi−1
xi−1
(48)
Downloaded from pic.sagepub.com at WEST VIRGINA UNIV on June 20, 2015
1778
H Erdelyi and D Talaba
Fig. 5
Car turn switch and its cam mechanism
obtain the real cam profile that was used as a reference
profile for the validation of the method.
3.2
Cam profile calculus
The schematic representation of the cam system of
the turn switch is given in Fig. 8. For the numerical
calculus, the following parameters were used.
Fig. 6
Fig. 7
Driving force profile measurement
Measured driving force profile
k = 1.857 (N/mm)
c = 0.1 (Ns/mm)
x0 = 5 mm
ri = 2.5 mm
re = 3.5 mm
rj = 2.5 mm
µstat = 0.2
µ = 0.13
n = 0.044 (1/s)
mf = 2.137 × 10−3 kg
Jlever = 39.374 kg mm2
Spring stiffness coefficient
Damping coefficient
Spring initial compression
Inner radius of the revolute joint
Outer radius of the revolute joint
Radius of the revolute joint
Static friction coefficient
Sliding friction coefficient
Mean angular velocity
Follower mass
Lever moment of inertia
The mass of the follower and the moment of inertia
of the lever were used to compute the kinetic energy
with equation (42).
With the computation method presented in section
2 using the measured driving force profile (Fig. 7), the
preliminary cam profile was determined in polar coordinates. In Fig. 9, the preliminary cam profile is shown
with a continuous line and the reference cam profile is
shown with a dashed line.
The preliminary cam profile differs considerably
from the reference profile, since in its calculation
the mechanical losses and the kinetic energy are not
included. At the next step, the mechanical losses and
the kinetic energy of the system were computed for the
preliminary cam profile.
Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science
Downloaded from pic.sagepub.com at WEST VIRGINA UNIV on June 20, 2015
JMES1649
A novel method for the dynamic synthesis of cam mechanisms
Fig. 8
Schematic representation of the turn switch
Fig. 10
Fig. 9
Preliminary cam profile versus reference cam
profile
The friction force in the revolute joint was computed
using the method presented in section 2. The revolute
joint had two pairs of surfaces in contact, as presented
in Fig. 10.
The contact between the lever and the cam is realized by a compression spring, while the contact inside
the revolute joint was maintained by the resultant of
the external forces acting in the plane perpendicular
to the lever pivot axis.
For the circular crown contact, the friction torque
TnfrC was determined using equation (22) in which, for
the present case, the axial load Pn is the preload of the
spring as seen in Fig. 10.
The generic term for the journal type friction
torque TnfrJ in the revolute joint was determined based
on equation (20). Furthermore, Qn (as presented in
Fig. 10) is the resultant of the external forces, which
act in the plane perpendicular to the lever pivot
JMES1649
1779
Revolute joint of the turn switch
axis. These forces are as follows: the driving force of
the lever Fndrv and the friction and normal forces at
the cam–follower contact Fnfr and FnN , respectively. The
friction and normal forces are orthogonal; thus their
resultant is
Rn =
(Fnfr )2 + (FnN )2
(49)
The generic term of the angle θn between the
first resultant Rn and the driving force Fndrv can be
determined as
θn =
π
− (αn + βn + σn )
2
(50)
where σn is the angle between the normal force FnN and
the first resultant Rn , determined with equation (49)
F fr
σn = arctan nN
Fn
(51)
Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science
Downloaded from pic.sagepub.com at WEST VIRGINA UNIV on June 20, 2015
1780
H Erdelyi and D Talaba
Finally, the resultant of the driving force and the first
resultant can be determined with the relation
2
2
(52)
Qn =
Fndrv + Rn1 + 2Fndrv Rn1 cos(σn )
The generic term of the cam profile ρnm at the step
m of the iterative process will be determined using
equation (48).
3.3
Results and discussion
The cam profile was determined based on the iterative method presented in section 2 using the measured
driving force data. The first cam profile in the iterative process (presented in Fig. 11), determined with
equation (48), shows considerable deviation from the
reference cam profile. As the iterative process is carried out, the computed cam geometry approaches the
reference cam profile. In Fig. 11 the preliminary cam
profile is shown with a continuous line and the reference cam profile is shown with dashed line. Furthermore, two cam profiles are also shown, determined at
the first and at the third iteration step. Figure 11 reveals
that the cam profile determined at the third iteration
step falls close to the reference cam geometry.
The relative error of the cam geometry to the
reference cam geometry was determined as
Error =
ρcomp − ρref
ρref
(53)
Computed cam geometry (iteration 01 and iteration 03) versus preliminary and reference cam
geometries
Relative error of the first four cam profiles
with the iterative method, one can monitor and stop
the evolution of the process when the relative error
between two consecutive profiles is small enough. The
relative error between the cam profiles can be given as
EnIT =
where ρcomp is the computed cam profile and ρref is
the reference cam profile.
Expressed in percentage, the relative error for the
first four computed cam profiles is presented in Fig. 12.
As can be seen in Fig. 12, an error margin of ±2.5 per
cent is obtained after only four iterations. Defining the
relative error between the cam profiles EnIT , obtained
Fig. 11
Fig. 12
ρn − ρn−1
ρn−1
(54)
In Fig. 13, the relative error between the consecutive
cam profiles is presented for the first four profiles.
As can be seen in Fig. 13, the relative error between
the third and fourth cam profiles, determined with
the iterative method, falls into the error margin of
Fig. 13
Relative error between the consecutive cam
profiles
Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science
Downloaded from pic.sagepub.com at WEST VIRGINA UNIV on June 20, 2015
JMES1649
A novel method for the dynamic synthesis of cam mechanisms
±1 per cent, and thus the iterative process could be
stopped. The results prove that by using the computation method presented in section 2, it is possible
to determine the cam geometry corresponding to a
desired driving force profile with a very good error
margin.
4
CONCLUSIONS
In this article, a new approach to cam mechanism
dynamic synthesis is presented, based on the driving
effort imposed on the cam mechanism. The problem is
treated using an energy-based algorithm, in which the
calculations involve the driving work of the cam mechanism, the work of friction and damping forces, as well
as the potential and kinetic energy of the system. The
computation is carried out iteratively, starting with a
preliminary synthesis considering an idealized model,
and continuing with a number of iterations that take
into consideration the mechanical losses and kinetic
energy. To model the friction between the cam and
the follower of the mechanism, an alternative friction
model is proposed that combines the Threllfall and the
Stribeck friction models.
The method is numerically and experimentally validated for the cam mechanism of a conventional car
turn indicator switch. In the validation process, the
driving force profile of the switch is recorded with
experimental measurements, using a hydraulic actuator. The cam profile of the turn switch is digitized,
and is used as the reference profile in the validation. Based on the measured force profile, the cam
geometry is determined and compared to the reference cam profile. Results show that with the method
presented in this article, after only a few iterative
steps, the computed cam profile approaches the cam
profile of the real switch with a relative error of
<2.5 per cent.
ACKNOWLEDGEMENTS
This work was supported by the research grant titled
IREAL – Haptic Interface for Virtual Prototyping in
an Immersive Environment – 132/2007, funded by the
Romanian National Council for Higher Education and
Research.
© Authors 2010
REFERENCES
1 Rothbart, H. A. Cam design handbook, 2004 (McGrawHill, New York).
2 Shigley, J. E., Mischke, C. R., and Brown, T. H. Standard
handbook of machine design, 2004 (McGraw-Hill, New
York).
JMES1649
1781
3 Sandler, B. Z. Robotics – designing the mechanisms
for automated machinery, 2nd edition, 1999 (Academic
Press, San Diego, California).
4 Alexandru, P., Talaba, D., and Alexandru, C. Functional
design of mechanisms (in Romanian), vol. 2, 2000 (Lux
Libris Press, Brasov).
5 Norton, R. Design of machinery. Introduction to the synthesis and analysis of mechanisms and machines, 1992
(McGraw-Hill, New York).
6 Burnett, G. E. and Porter, J. M. Ubiquitous computing
within cars: designing controls for non-visual use. Int. J.
Hum. Comput. Stud., 2001, 55(4), 521–531.
7 Moreau, G., Fuchs, P., and Stergiopoulos, P. Applications of virtual reality in the manufacturing industry:
from design review to ergonomic studies. Mecanique
Ind., 2004, 5(2), 171–179.
8 Weir, D. W., Peshkin, M., and Colgate, J. E. The haptic
profile: capturing the feel of switches. In Proceedings
of the IEEE 12th International Symposium on Haptic
interfaces for virtual environment and teleoperator systems (HAPTICS 2004), Chicago, IL, 27–28 March 2004,
pp. 186–193.
9 MacLean, K. E. Haptic camera: a technique for characterizing and playing back haptic properties of real
environments. In Proceedings of the ASME Dynamic
Systems and Control Devision, Atlanta, GA, 1996,
pp. 245–252.
10 Colton, M. B. and Hollerbach, J. M. Reality-based haptic force models of buttons and switches. In Proceedings
of the IEEE International Conference on Robotics and
automation (ICRA ’07 ), Rome, Italy, 10–14 April 2007,
pp. 497–502.
11 Colton, M. B. and Hollerbach, J. M. Haptic models of an
automotive turn-signal switch: identification and playback results. In Proceedings of the WorldHaptics 2007:
Second Joint EuroHaptics Conference and IEEE Symposium on Haptic interfaces for virtual environments and
teleoperator systems, Tsukuba, Japan, 22–24 March 2007,
pp. 243–248.
12 Erdelyi, H., Antonya, Cs., and Talaba, D. Haptic feedback generation for a car direction indicator switch. In
ASI on: product engineering: tools and methods based
on virtual reality (series: intelligent systems, control and
automation: science and engineering), vol. 35, 2007,
pp. 257–273 (Springer, Berlin).
13 Mathew, G. K. and Tesar, D. Cam system design: the
dynamic synthesis and analysis of the one degree
of freedom model. Mech. Mach. Theory, 1976, 11,
247–257.
14 Lederer, P. Dynamische synthese der übertagungsfunktion eines kurvengetriebes. Mech. Mach. Theory, 1993,
28, 23–29.
15 Kreith, F. Mechanical engineering handbook, 1999 (CRC
Press, Boca Raton, Florida).
16 Talaba, D. Articulated mechanisms. Computer aided
design (in Romanian), 2001 (Transilvania University
Publisher, Brasov).
17 Rooney, G. T. and Deravi, P. Coulomb friction in mechanism sliding joints. Mech. Mach. Theory, 1982, 17,
207–211.
18 Threlfall, D. C. The inclusion of coulomb friction
in mechanisms programs with particular reference to
DRAM. Mech. Mach. Theory, 1978, 13, 475–483.
Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science
Downloaded from pic.sagepub.com at WEST VIRGINA UNIV on June 20, 2015
1782
H Erdelyi and D Talaba
19 Andersson, S., Söderberg, A., and Björklund, S. Friction
models for sliding dry, boundary and mixed lubricated
contacts. Tribol. Int., 2006, 40(4), 580–587.
APPENDIX
Notation
In this article, subscripts are used to represent a
generic term of a series. For example, xn represents
the generic term of a finite series of spring compression data.
c
E IT
EK
EP
FC
Fd
F drv
F el
F fr
F l and F t
FN
FS
Jc
Jl
Jlever
k
mf
mi
P
damping coefficient
relative error between the cam profiles
kinetic energy
potential energy
Coulomb sliding friction
damping force
driving force
elastic force
sliding friction force of a cam–follower
contact
orthogonal components of F N
normal contact force
maximum static friction force
moment of inertia of the cam
moment of inertia of an element with
rotational motion
moment of inertia of the lever
stiffness coefficient
mass of the follower
mass of an element with translational
motion
axial load
Q
re
ri
rj
rr
r0
R
s
W frJ
WT
W TFr
radial load
outer radius
inner radius
journal radius
roller radius
minimum radius
resultant of F fr and F N
displacement of the follower along the
pitch curve
driving torque
friction torque of a circular crown type
contact
friction torque of a journal type contact
sliding speed
compression of the spring
initial compression of the spring
work of the damping force
work done by the sliding friction force
work of the circular crown type friction
torque
work of the journal type friction torque
work done by the driving torque
total work of the friction forces
α
β
θ
µ
µstat
µvisc
ρ
ρcomp
ρref
σ
cam or lever rotation angle
pressure angle
angle between R and F drv
sliding friction coefficient
static friction coefficient
viscous friction coefficient
cam profile
computed cam profile
reference cam profile
tangent to the pitch curve
angular velocity of the cam or lever
T
T frC
T frJ
v
x
x0
WD
W fr
W frC
Proc. IMechE Vol. 224 Part C: J. Mechanical Engineering Science
Downloaded from pic.sagepub.com at WEST VIRGINA UNIV on June 20, 2015
JMES1649