1. Automation of Cam Design for Radial Piston Hydraulic Motor
Madhu Soodhanan Govindarajan Bharadwaj Ramakrishnan
Department of Mechanical Engineering Department of Mechanical Engineering
Pennsylvania State University The Ohio State University
State College, PA 16803 Columbus, OH 43210
mxg975@psu.edu ramakrishnan.17@osu.edu
Dominic Savio Fernando Dr. K. Srinivasan
Indian Institute of Management Department of Mechanical Engineering
Shillong, India 793014 Anna University, Chennai-25, India
a.dominic.savio.fernando@gmail.com drksrin@annauniv.edu
1 ABSTRACT
The Radial Piston Hydraulic Motor is a
hydraulic energy converting device. These
motors are widely used in low speed, high torque
applications [1]. They are expected to deliver
optimum torque with minimum variation. As the
torque output is of prime concern, various
parameters affecting it are studied and the
influence of their variation on the torque output
is analyzed in this work. As these motors are
cam operated, variation of its profile is the most
important parameter which will affect the output.
Other parameters such as roller radius, number
of rollers, pressure of the fluid used are also
analyzed in this work. Since manually drawing
the profile and computing the torque output from
it, is an exerting as well as time consuming task,
a code which could automatically perform both
these tasks is developed [2].The main emphasis
of this paper is to find out the value of various
parameters that deliver optimum torque.
2 OME CLATURE
C Angle subtended by the preceding region
r Roller radius (mm)
S Follower displacement (mm)
t Time interval (seconds)
v Velocity of the follower (mm/s)
vmax Maximum velocity of the follower (mm/s)
x,y Horizontal & vertical Co-ordinates of a
a point on the cam profile (mm)
α Acceleration of the follower (mm/s2
)
θ Angle subtended by a point on cam profile
to the cam centre (degrees)
3 I TRODUCTIO
Radial piston hydraulic motors find application
in manufacturing and mechanical power
production. The radial piston hydraulic motor
consists of a cam on which multiple rollers
rotate. The center of these rollers is coupled to
the output shaft. Each of the roller acts as a
hydraulic actuator, with the compressed fluid
being supplied via the cylinder. As the roller
makes contact with the cam it intends a specific
pressure angle. The force supplied by means of
the compressed fluid, is resolved into
perpendicular components based on the pressure
angle. One component helps in gripping with the
surface, while the other assists in producing
torque. A higher pressure angle causes more
torque but lesser force that assists in gripping; a
lower pressure angle produces the vice-versa. So
an optimum pressure angle must be chosen for
the trade-off. Also the variation in pressure angle
must be such that the motor produces minimum
torque fluctuation and highest mean torque. [3].
So in other words, a suitable cam profile must be
chosen. The general cam profiles that ease their
way into manufacturing as well (at least in India)
consists of a uniform acceleration region,
followed by uniform velocity region and by a
uniform deceleration region. Dwell regions are
given in between each of those regions in order
to assist manufacturing feasibilities.
The development of cam profile implies
situational application of laws of motion [4]. The
general equation, which holds well, is:
S=u*t+.5*α*t² (1)

2. In the region of constant acceleration, the initial
velocity is zero. In the region of constant
velocity, the same equation can be used with
α=0[5]. In the uniform deceleration period,
however, it should be noted that roller
decelerates but displacement keeps increasing.
Hence it would be logical to depend on the area
under the velocity-time curve and acceleration-
time for displacement value[5]. Thus a formula
can be arrived at by integrating the area under
acceleration-time curve twice and reducing it to
simplest form. The equation is:
S=.5*t*(vmax+v) (2)
4 DESIG A D A ALYSIS
The equations mentioned above are used to find
out the displacement values at each and every
point. Using these values, profile of a cam can be
drawn. As the objective of this work is to select
suitable values of input parameters which would
deliver the optimum torque, there is a need for
altering the input values frequently. This
necessitated development of a code. The
developed code requires input values such as
maximum lift, roller radius, and base circle
radius, fluid pressure, piston diameter and angles
of dwell, uniform velocity, uniform acceleration,
and uniform deceleration part. Uniform
acceleration value can be found out using the
displacement equations and by equating the
maximum lift with the algebraic sum of
displacements at each region. By superimposing
the displacement values on the base circle,
profile can be generated.
4.1 Calculation of pressure angle
Pressure angle can be found out using the
methodology mentioned below:
The slope of the tangent say ‘m’ is found out at
the point A of the cam profile.
The slope of its normal say AB is found out
using the formula m1 *m = -1
“C(0,0)” is assumed to be origin to find “l”
using distance formula
Then the slope of line AC is found out to be
m2=y1/x1. (3)
This formula is used to find out “a”
tan a = | (m1-m2)/ (1+ (m1*m2))| (4)
Cosine formula is used to find out “n”
n² = l² + r² -2* l * r * cos a (5)
Then using sine formula given below “b” is
found out.
(n /sin a) = (r /sin b) (6)
Then Pressure angle (p) = a + b (7)
As x,y &s are functions of ‘θ’ implies,
s = f(θ) ,θ=ω*t, where ‘t’ is time (in seconds).
Then substituting :√(x ² + y ²) = s + rb
& tan θ = (y / x) in f (θ). (8)
Then if ‘rb’ is the base circle radius, (s + rb) is the
radial distance
Therefore (s + rb) = f (θ) + rb (9)
It is known that
x = (s + rb) cos θ
y = (s + rb) sin θ
dy/dθ = (∂y / ∂θ) + (∂y / ∂s) * (ds / dθ)
= (s+ rb) cos θ + sin θ (ds/ dθ) (10)
The pressure angle at all the points is required to
find out the components of force applied at the
piston.
Notation: r-AB
l- AC
n- BC
FIGURE 1: Logic explanation
dx/dθ = (∂x / ∂θ) + (∂x / ∂s) * (ds / dθ)
= - (s+ rb) sin θ + cos θ (ds/ dθ) (11)
(dy / dx) = (dy/dθ) ÷ (dx/dθ) [Obtained from
equations 10 &11]
This slope does not account for the fact, that the
temporary x-axis is rotated by an angle c. Hence
m = {(dy/dx) + tan c}/ (1 – (dy/dx)*tan c)
(12)

3. 4.2 Calculation of torque
The pressure angle value obtained at all the
points and fluid pressure are used to compute the
force acting on the roller centre which could be
resolved into 2 perpendicular components. The
horizontal component when multiplied with the
perpendicular distance between the cam centre
and the point of contact will give the driving
torque produced by a roller at a given point. The
perpendicular distance of the roller centre from
the cam centre, can be found out using the
following formulation.
Consider a point on the profile where the roller
makes contact as (x1,y1) then the general form of
the tangent equation passing through it is
y = mx + c, (13)
Then the constant is given by c = y1– mx1
The perpendicular distance from the cam centre
to the point of contact can be found out using
pd = | y1– mx1| / ((1 + m2
) ^.5)
and the perpendicular distance from the roller
centre(pdr) can be obtained by using
pdr = pd – r (14)
4.3 Calculation of cumulative torque
The individual torque available at each of the
rollers is algebraically added. This gives the
value for cumulative torque available at the
motor shaft.
4.4 Variation of cumulative torque
The variation of cumulative torque available at
the shaft is the ratio of the difference between
maximum and mean torque to the mean torque.
Variation of cumulative torque = (max. torque –
mean torque) / mean torque (15)
5 A ALYSIS
5.1 ANALYSIS OF CUMULATIVE TORQUE
WHEN THERE IS A VARIATION IN
i) NUMBER OF ROLLERS
Using the code the optimum no. of rollers can be
found out. Whenever the variation in cumulative
torque is minimal then the number of rollers is
optimum. For our case the optimum no. of rollers
were found out to be 8. The table below shows
various values of individual torque at each point
and the cumulative torque with respect to the
first roller position. The variation in cumulative
torque was found out to be 3.89 % in our case.
A graph showing the variation in cumulative
torque can also be generated automatically using
the code. A graph that was developed using our
code is shown next
.FIGURE 2: Cumulative torque Vs Angular
position of roller 1
x – Angular position of 1st
roller , in degrees
y - Cumulative torque produced by the motor,
in N mm
ii) PRESSURE OF THE FLUID
The mean and highest torque varies
proportionally with fluid pressure. So it would be
better to have the force value as constant and
study the other vital parameters for the
development of cam profile.
FIGURE 3: Cumulative torque Vs Angular
position of roller 1
x – Angular position of 1st
roller , in degrees
y - Cumulative torque produced by motor, in N
mm.

4. 1st roller Roller 1 Roller 2 Roller 3 Roller 4 Roller 5 Roller 6 Roller 7 Roller 8 CumTorq
Positio
n 1.00E+07 1.00E+07 1.00E+07 1.00E+07 1.00E+07 1.00E+07 1.00E+07 1.00E+07 1.00E+07
8 0.6999 -0.6057 -0.2586 0.8933 0.6999 -0.6057 -0.2586 0.8933 1.4577
13 1.1269 -1.10338 0 0.6598 1.1269 -1.0338 0 0.6598 1.5058
18 1.1013 0.207 -0.1037 -0.5122 1.1013 0.207 -0.1037 -0.5122 1.3847
23 0.8933 0.6999 -0.6057 -0.2586 0.8933 0.6999 -0.6057 -0.2586 1.4577
28 0.6598 1.1269 -1.0338 0 0.6598 1.1269 -1.0338 0 1.5058
33 -0.5122 1.1013 0.207 -0.1037 -0.5122 1.1013 0.207 -0.1037 1.3847
38 -0.2586 0.8933 0.6999 -0.6057 -0.2586 0.8933 0.6999 -6057 1.4577
43 0 0.6598 1.1269 -1.0338 0 0.6598 1.1269 -1.0338 1.5058
48 -0.1037 -0.5122 1.1013 0.207 -0.1037 -0.5122 1.1013 0.207 1.3847
53 -0.6057 -0.2586 0.8933 0.6999 -0.6057 -0.2586 0.8933 0.6999 1.4577
58 -1.0338 0 0.6598 1.1269 -1.0338 0 0.6598 1.1269 1.5058
63 0.207 -0.1037 -0.5122 1.1013 0.207 -0.1037 0.5122 1.1013 1.3847
TABLE 1: Torque at each roller, cumulative torque at the angular position of roller
iii) ROLLER /BASE CIRCLE RADIUS
It is obvious that increasing the base circle radius
of cam profile with a constant roller radius is the
same as decreasing the roller radius for the same
base circle radius. Two values of the roller radius
on either sides of the optimum roller radius were
taken. The mean torque and maximum torque
generated was calculated. The mean torque and
maximum torque increases with the decrease in
roller radius. However, this does not mean the
recommendation of a knife edge follower which
is effectively a roller follower with zero radiuses.
This is owing to the fact that the increase in
vertical component of torque in such a case
causes increased wear and tear.
FIGURE 4: Cumulative torque Vs Angular
position of roller1
x – Angular position of 1st
roller , in degrees
y - Cumulative torque produced by motor,
in N mm
6 RESULTS A D DISCUSSIO
i) Radial piston hydraulic motors are primarily
cam operated; variation of its profile is the most
important parameter which will affect the output
torque [6] .Hence the code gives the user an
option to vary the most basic input parameters
and generate a profile which will deliver desired
output torque.
ii) The output torque obtained needs to be
optimized or else the motor might deliver
unpredictable output. Hence other parameters
were varied to find out the optimum torque
output.
iii) When the number of rollers is increased the
variation in cumulative torque decreases initially
but after a certain value it starts increasing and
yields unpredictable torque at the motor shaft.
This value of number of rollers at which the
torque output is optimum has to be found out.
iv) Pressure of fluid has least significance on the
variation of cumulative torque. It just gives a
proportional variation to the output. So
depending on the torque output required of the
motor shaft, the pressure of the fluid has to be
chosen.
v) When roller radius is decreased continually
the cumulative torque increases but since we
need an optimum cumulative torque this can be
decreased only to an optimum value so that wear
and tear is also minimal.

5. ACK OWLEDGEME TS
We acknowledge the inputs given by
Prof .N.S. Parthasarathy, Dept. of Mechanical
Engineering, Anna University-Chennai.
We also thank Anna university- Federal republic
of Germany Institute, Anna University.
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J. Mech. Des. -- November 2004 -- Volume
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DOI:10.1115/1.1798231
[8]http://www.allbusiness.com/hydraulic-
motors/3430145-
1.html?googlesubnew=hydraulic%20motors