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Development of new rotation concept
for rock drills
AKEPATI BHASKAR REDDY
Master of Science Thesis
Stockholm, Sweden 2015
Development of new rotation
concept for rock drills
Akepati Bhaskar Reddy
Master of Science Thesis MMK 2015:99 MKN 147
KTH Industrial Engineering and Management
Machine Design
SE-100 44 STOCKHOLM
Examensarbete MMK 2015:99 MKN 147
Utveckling av nya rotationskoncept för
bergborrar
Akepati Bhaskar Reddy
Godkänt
Examinator
Handledare
2015-09-08
Ulf Sellgren
Stefan Björklund
Uppdragsgivare
Kontaktperson
Atlas Copco Rock Drills AB
Anders Olson
Sammanfattning
Vid spränghålsborrning är det önskvärt att borra så nära tunnels periferi som möjligt. Det minsta
avståndet från kanten definieras av borrmaskinens radie. Majoriteten av dagens borrmaskiner har
en rotationsmotor som roterar en axel parallell till nackadaptern, vilket i sin tur roterar
nackadaptern genom en drevuppsättning. Detta gör borrmaskinen otymplig.
Detta examensarbete, som utfördes på Atlas Copco Rock Drills AB, Örebro, undersöker
rotationsmekanismen hos en borrmaskin. Projektets huvuduppgift var att utveckla alternativa
koncept till rotationsmekanismen som använder nackadaptern som en del av rotationsmekanismen
och reducerar borrmaskinens otymplighet.
För att hitta ett lämpligt alternativ genererades åtta olika koncept för en hydraulisk motor (med
och utan transmission) som sedan utvärderades. Två koncept, 1) ”multi-kam vingmotor” och 2)
”hydrauliskt driven töjningsvåg-växel”, valdes för ytterligare funktionell konstruktion. Båda
koncepten konstruerades under ideala förhållanden (100% effektivitet) för att uppnå det
deplacement som krävs.
För ”multi-kam vingmotor” utfördes ett flertal konstruktionsförbättringar och effekten av olika
konstruktionsparametrar analyserades. Olika varianter av motorn togs fram samt analyserades. En
grov design genomfördes för ”hydrauliskt driven töjningsvåg-växel”, vilken valdes på grund av
att designen inte förekommit I någon litteratur. CAD-modeller för båda koncepten samt relaterade
varianter togs fram för att föreslå monteringslayouter och ventilmekanismer.
De två koncepten skulle reducera otympligheten hos borrmaskinen. För –och nackdelarna hos de
olika varianterna har diskuterats. Koncepten måste utvecklas ytterligare för att kunna
implementeras i en borrmaskin.
Nyckelord: hydraulisk motor, borrmaskin, vridning
i
Master of Science Thesis MMK 2015:99 MKN 147
Development of new rotation concept for rock
drills
Akepati Bhaskar Reddy
Approved
Examiner
Supervisor
2015-09-08
Ulf Sellgren
Stefan Björklund
Commissioner
Contact person
Atlas Copco Rock Drills AB
Anders Olson
Abstract
In blast hole drilling, it is desirable to be able to drill as close as possible to the edge of the tunnel.
The minimum distance from the edge is defined by the radial size of the rockdrill. Most of the rock
drills used today have a rotation motor that rotates an axel parallel to the shank, which further
rotates the shank through a gear set. Thus making the rock drill bulky.
This thesis project carried out at Atlas Copco Rock Drills AB, Örebro, deals with the rotation
mechanism of a rock drill. The main task of the project was to develop alternate concepts for
rotation mechanism that would use the shank as a part of rotation mechanism and reduce the
bulkiness of the rock drill.
In order to find a suitable alternative, eight different concepts for hydraulic motor (with or without
transmission) were generated and evaluated against each other. Two concepts, 1) multi-cam vane
motor concept and 2) strain wave hydraulic gear motor concept, were selected for further
functional design. Both concepts were designed at ideal conditions (100% efficiency) to achieve
the required displacement.
For the multi-cam vane motor, various design improvements were performed and the effect of
different design parameters were also analyzed. Different variants of the motor were developed
and analyzed. A rough design was performed for the strain wave hydraulic gear motor concept
which was chosen for its novelty. CAD models for both the concepts and the related variants were
developed for suggesting assembly layouts and valve mechanisms.
The two concept designs would reduce the bulkiness of the rock drill. The benefits and drawbacks
of the different variants have been discussed. The concepts must be further developed for
implementation into a rockdrill.
Keywords: hydraulic motor, rock drill, rotation
iii
FOREWORD
In this section, the help, assistance, cooperation and inspiration from others has been
acknowledged.
I would like to express my gratitude to my supervisors at Atlas Copco Rock Drills AB, Maria
Pettersson and Erik Jakobsson, for the constant engagement, useful comments and remarks
throughout this master thesis. Furthermore I would like to thank Anders Olson, R&D manager,
Rock drills & rotation units, Rocktec division, for providing me an opportunity to work on this
project.
This thesis would not have been possible without the help from my university supervisor, Stefan
Björklund, and MSc project coordinator, Ulf Sellgren.
I would like to thank everyone at Atlas Copco Rock Drills AB and KTH Royal Institute of
Technology, who have, directly or indirectly, helped me achieve my goals.
Finally, all thanks to almighty and my parents for their support and blessings, without which
nothing would have been possible.
Akepati Bhaskar Reddy
Stockholm, September 2015
v
NOMENCLATURE
In this section the symbols and abbreviations used in the report are listed. The symbols are listed
in the order of their first appearance in the report.
Notations
Symbol
Description
Unit
nD
Required rotation speed
rpm
DB
Drill bit diameter
mm
f
Impact frequency
Hz
zB
Bit movement per blow
mm
Pin
Input fluid pressure
Pa
Pout
Return fluid pressure
Pa
Q
Flow rate
m3/min
 r , 
Polar coordinate system
RO
Radius of mean circle of cam profile (sine curve)
m
a
Amplitude of sine curve
m
n
Twice the number of cams
k
Number of vanes
B
Width of motor unit
m
Wp
Width of vane
m
rp
Head radius of vane
m

Angle of rotation
rad
 X C , YC 
Coordinates of point of contact
 X R , YR 
Coordinates of centre of circular part of vane

Angle of rotation of contact point
mn
Slope of normal to the sine profile

Angle formed by the line joining the centre of circular part of the rad
vane to the point of contact on the sine profile, with the horizontal
axis
RC
Distance of point of contact from the origin
m
RR
Distance of centre of circular part of the vane from the origin
m
rad
vii
viii

Angle formed by line joining centre of circular part of vane and the rad
point of contact, with the axis of vane
 ,
Angles in the depiction of contact geometry (Figure 26 and Figure rad
63)
RI
Radius of circular profile (rotor or stator)
m
1,  2
Angular position of vane 1, 2
rad
Achamber
Area of chamber
m2
Aa , Ab , Ac
Parts of chamber area
m2
Ab1 , Ab 2 , Ab3
Parts of area component Ab
m2
RR1 , RR 2
RR values for vane 1,2 corresponding to angular position  1 ,  2
respectively
m
Ac1 , Ac 2
Parts of area component Ac
m2
Ac1_1 , Ac1_ 2
Parts of area component Ac1
m2
Ac 2 _1 , Ac 2 _ 2
Parts of area component Ac 2
m2
1 , 2
Angle of contact for vane 1,2 corresponding to angular position rad
1,  2
Achamber
Change in area of chamber
m2
A   d
Total change in area for small rotation d 
m2
dA
Change in area for one sub-cycle of rotation
m2
dV
Volume of fluid added
m3
i
Instantaneous speed
rpm
Ta , Tb , Tc , Td
Torque constituents per unit width
Nm/m
FCX , FCY
Contact force (per unit width) in direction normal to vane axis and N/m
parallel to vane axis respectively

Half angle subtended by width of the vane on the circular profile
rad
z
Distance of contact point from origin along the vane axis
m
l
Distance of end of guide from origin along the vane axis
m
P1 , P2
Fluid pressure
Pa
Tvane
Torque on one vane per unit width
Nm/m
Tchamber
Torque corresponding to one chamber (per unit width)
Nm/m
Ttotal
Total torque (instantaneous)
Nm
C1
Contact point between rotor and stator profiles
C2
Contact point between vane and the cam profile

Angular position of C1
rad
AVi
Chamber area associated with vane i
m2
dAVi
Change in area of chamber associated with vane I
m2
 X ,Y 
Vane coordinate system
FX , FY
Force components vane coordinate system
N
RX , RY
Reaction force components in vane coordinate system
N
Rx , Ry
Reaction force components in motor coordinate system
N
m
Gear module
m
D
Pitch circle diameter of gear
m
Z
Number of teeth
Z
Difference in number of teeth on circular gear and flex gear

Deflection of flex gear
ZC , Z f
Number of teeth on circular gear and flex gear respectively
DC , D f
Pitch circle diameter of circular gear and flex gear respectively
m
ri , ro
Inner radius and outer radius of approximated ring
m
t
Thickness of approximated ring
m
F
Deflecting force acting on the ring
N
ds
Small section of quarter ring
m
d
Angle subtended by ds at the centre
rad
Fr , F
Force component on the small section, ds , in radial and tangential N
direction respectively
M
Bending moment at the small section ds
Nm
U
Total strain energy
J
dU
Strain energy on small section ds
J
dU1 , dU 2 , dU 3
Component of dU due to normal load, shear load and bending J
moment respectively
C
Correction factor for a rectangular cross-section in shear
MO
Bending moment at free end
Nm
A
Cross-section area
m2
E
Elastic modulus
Pa
G
Shear modulus
Pa
I
Area moment of inertia
m4
 i , o
Stress at inner fibre and outer fibre respectively
Pa
m
ix
x
ci , co
Distance from neutral axis to inner fibre and outer fibre m
respectively (for curved beams with rectangular cross-section)
e
Distance from centroidal axis to neutral axis for curved beams with m
rectangular cross-section
r
Radius of neutral axis for curved beam with rectangular cross- m
section
rc
Radius of centroidal axis for curved beam with rectangular cross- m
section
Ka
Stress concentration factor at gear root
 Ji ,  Jo ;  Ki ,  Ko
Stress at inner and outer fibre at section J and K respectively
Pa
 i _ mean ,  o _ mean
Mean stress at inner and outer fibre respectively
Pa
 i _ amp ,  o _ amp
Amplitude of alternating stress at inner and outer fibre respectively Pa
S yt
Material yield strength
Pa
Se
Material endurance limit
Pa
fs
Safety factor
rt
Any radius on an involute gear profile
m
tt
Thickness of gear tooth at rt
m
do
Addendum diameter for external tooth OR dedendum diameter for m
internal tooth
db , rb
Base circle diameter and radius respectively
m
p
Pressure angle
rad

cos 1  rb / rt 
rad
t pt , rpt
tt and rt at pitch circle
m
dof
Addendum circle diameter of flex gear
m
dbf
Base circle diameter of flex gear
m
d rf
Root diameter of flex gear
m
d rC
Root diameter of circular gear
m
dbC
Base circle diameter of circular gear
m
doC
Addendum circle diameter of circular gear
m
dC
Mid-section circle diameter of flex gear
m
PC
Perimeter of mid-section circle of flex gear
m
ae
Major axis of ellipse
m
Pe
Perimeter of ellipse
ee
Eccentricity
be
Minor axis of ellipse
X
j
, Yj 
m
m
Coordinates of point on circular gear profile (flexgear)
j
Angle subtended at the centre by arc between horizontal axis and rad
X j , Yj
C
 j   j 1

X
X
je
X
j
jC
rad

Point corresponding to  X
, Y jC 

Point corresponding to X j , Y j on mid-section circle
, Y je 
,Yj


jC
, Y jC  on elliptical mid-section

Point on elliptical gear profile corresponding to X j , Y j

pc
Length of arc bounded by two points on mid-section circle
m
p'
Length of elliptical arc
m
e
Angle subtended by elliptical arc at the centre
rad
j
Angle subtended at centre by elliptical arc bounded by horizontal rad
axis and X je , Y je
te
Elliptical parameter in parametric equation of ellipse
X C , YC
Vector difference of corresponding points on mid-section circle m
and elliptical mid-section
u
Number of pairs of meshing teeth supplied with fluid
A1 , A2 , A3 ,....
Area trapped between meshing pairs of teeth
rs
Speed ration
vwg
Speed of wave generator
rpm
vf
Speed for flex gear
rpm
X

jm
, Yj m


Mid-point of line joining two successive points on elliptical gear
profile
lj
Distance between two successive points on elliptical gear profile
mj
Slope of perpendicular to the line joining two successive points on
elliptical gear profile
X
jk
,Yj k

m
Point of intersection of line perpendicular to line joining two
successive on elliptical gear profile passing through the midpoint
X j m , Y j m ; and the perpendicular to it passing through the origin

dj
m2



Distance of X j k , Y j k from origin
m
xi
Tj
Torque contribution from a small region on elliptical gear profile Nm
on all four symmetric positions
Twg
Torque on wave generator
Nm
Tf
Torque on flex gear
Nm
L
Length of flex gear cup
m
tc
Thickness of flex gear cup
m
 mean ,  alt
Mean and alternating stress respectively
Pa
Abbreviations
xii
CAD
Computer Aided Design
1-D
One dimensional
TABLE OF CONTENTS
SAMMANFATTNING ..............................................................................................I
ABSTRACT ............................................................................................................ III
FOREWORD .......................................................................................................... V
NOMENCLATURE ............................................................................................ VII
TABLE OF CONTENTS ...................................................................................XIII
INTRODUCTION ............................................................................................ 1
1.1
1.2
1.3
1.4
1.5
BACKGROUND .................................................................................................. 1
ABOUT ATLAS COPCO ...................................................................................... 2
PROBLEM DESCRIPTION.................................................................................... 2
PURPOSE........................................................................................................... 3
DELIMITATIONS ................................................................................................ 3
FRAME OF REFERENCE ............................................................................. 5
2.1
2.2
2.3
2.4
ROTARY PERCUSSIVE ROCK DRILLING .............................................................. 5
ROCK DRILL (DRIFTER) .................................................................................... 5
ROTATION MECHANISM .................................................................................... 6
HYDRAULIC MOTORS ...................................................................................... 10
CONCEPTS .................................................................................................... 11
3.1 REQUIREMENT SPECIFICATION ....................................................................... 11
3.2 CONCEPTS GENERATED .................................................................................. 11
3.3 CONCEPT EVALUATION................................................................................... 16
MULTI-CAM HYDRAULIC VANE MOTOR CONCEPT ...................... 19
4.1
4.2
4.3
4.4
4.5
4.6
WORKING PRINCIPLE...................................................................................... 19
DESIGN CALCULATIONS .................................................................................. 21
CONCEPT IMPROVEMENT ................................................................................ 30
CAM VANE MOTOR: VANES ON ROTOR ........................................................... 32
CAM VANE MOTOR: VANES ON STATOR .......................................................... 47
DIFFERENT NUMBER OF VANES ....................................................................... 63
STRAIN WAVE GEAR HYDRAULIC MOTOR CONCEPT .................. 69
5.1
5.2
5.3
5.4
WORKING PRINCIPLE...................................................................................... 69
MINIMUM NUMBER OF TEETH ON FLEXGEAR .................................................. 70
MATHEMATICAL MODEL ................................................................................ 76
MOTOR ASSEMBLY DESIGN ............................................................................. 85
DISCUSSION AND CONCLUSIONS.......................................................... 91
xiii
6.1 DISCUSSION .................................................................................................... 91
6.2 CONCLUSIONS ................................................................................................ 92
FUTURE WORK............................................................................................ 93
REFERENCES ............................................................................................... 95
APPENDIX A: INTEGRATION FOR AREA ENCLOSED ............................ 97
APPENDIX B: CALCULATION OF AREA COMPONENTS AB AND AC 101
xiv
INTRODUCTION
This chapter provides a brief introduction, the background, the problem description, purpose and
the delimitations of the thesis work.
1.1 Background
Blasthole drilling has been in practice since centuries for mining. To make better use of explosive
force, miners started to place the explosives (gunpowder) in holes drilled into rock. One man
drilling with the help of a drill steel and sledgehammer was an establish technology used in the
eighteenth century (Figure 1). This physically demanding technology evolved slowly to give rise
to powered drills (Atlas Copco Drilling Solutions LLC 2012). The need for drilling bigger and
deeper holes into rocks at a faster rate led to the development of heavy and high power rock drills
(e.g. Figure 2) mounted on movable drilling rigs, for e.g. the underground drilling rig XE4 from
Atlas Copco (Figure 3).
Figure 1. Drilling using a drill steel and sledgehammer
Figure 2. A rock drill
1
Figure 3. Underground Drilling rig type XE4 for tunnelling.
1.2 About Atlas Copco
Atlas Copco is a world leading provider of sustainable productivity solutions. The group serves
customers with innovative compressors, vacuum solutions and air treatment systems, construction
and mining equipment, power tools and assembly systems. (Atlas Copco : Facts in brief n.d.)
The Mining and Rock Excavation Technique business area provides equipment for drilling and
rock excavation, a complete range of related consumables and service through a global network.
The business area innovates for sustainable productivity in surface and underground mining,
infrastructure, civil works, well drilling and geotechnical applications. Principal product
development and manufacturing units are located in Sweden, the United States, Canada, China
and India. (Atlas Copco : Organisation- Mining Technique n.d.)
Atlas Copco Rock Drills AB in Örebro is a part of Mining and Rock Excavation Technique. The
division is involved in development and manufacture of rock drilling machines, automation
systems, tunnelling and mining equipment for various underground applications, procurement,
warehousing and distribution of spare parts and shipping of finished drilling rigs, trucks and
loaders to the sales companies and customers around the world. (Atlas Copco Rock Drills n.d.)
1.3 Problem Description
Before a part of tunnel can be blasted, a number of holes have to be drilled in a specific pattern for
charging with explosives. Most of the drilling patterns have contour holes (Figure 4), which are
fairly closely spaced in the roof and walls of the tunnel and charged with less powerful explosive.
By this means, crack formations in the side of the rock nearest to the tunnel surface are reduced
and overbreak is avoided (Martin Lindfors 1985). It is advisable to choose a smaller diameter for
the contour holes and drill them as close as possible to the edge of the tunnel.
2
Figure 4. Different holes drilled during tunnelling (drifting)
The current layout of the rotation mechanism of the rock drills involves a rotation motor which
rotates an axle parallel to the shank. This rotation is transferred to the shank using a gear set (Figure
6). This layout makes the rock drill bulky. This limits the minimum distance at which the rock drill
can be operated from the edge of the tunnel, therefore, limiting the minimum distance of the
contour holes from the edge.
Therefore, a high emphasis is being laid at Atlas Copco for developing a rotation mechanism which
would reduce the bulkiness of the rock drill.
1.4 Purpose
The goal of this thesis project was to evaluate the possibility to use the shank as a part of the
rotation mechanism. The project involves development of new concepts that would address the
problem and could replace the existing rotation mechanism.
The detailed goals of the project are as follows:

Enlist requirement specifications for the rotation mechanism

Develop concepts that would use the shank as a part of the rotation mechanism

Shortlist concepts on the basis of requirement specifications

Perform functional design calculations for the concepts

Develop CAD models for the shortlisted concepts
The project was carried out for a particular model of rock drill. The concept can be later extended
for use on different models of rock drills.
1.5 Delimitations
The design limitations state that no other sub-system of the rock drill (percussion, damping, feed
and flushing) must be affected by the introduction of new concepts. It is advised to not increase
the number of hoses and cables to the rock drill.
3
The mathematical models for the concepts have been developed only for ideal case, i.e. considering
100% efficiency. The torque and speed calculations have been performed for constant pressure
difference and constant flow rate.
CAD models depicting the assembly layout of the motors were developed. The CAD model of the
rock drill was not changed. Therefore, physical interfaces of the developed concept with the rock
drill were not designed.
4
FRAME OF REFERENCE
This chapter provides information about percussive rock drilling, rock drills, rotation mechanism
and hydraulic motors.
2.1 Rotary percussive rock drilling
The drilling principle is based upon the impact of a steel piece (piston) that hits a utensil (shank)
which at the same time transmits the impact energy to the bottom of the blast hole by means of a
final element called the bit.
The rotary percussive rigs are classified in two large groups, (Jimeno, Jimeno and Carcedo 1995)

Top hammer: Rotation and percussion are produced outside the blast hole, and are
transmitted by the shank adaptor and the drill steel to the drill bit.

Down the hole hammer: Percussion is delivered directly to the drill bit, whereas rotation is
performed outside the hole.
Percussive rock drilling involves four major processes; percussion, rotation, feed (thrust load) and
flushing. (Figure 5)
Figure 5. Processes of percussive rock drilling (Jimeno, Jimeno and Carcedo 1995)
The four processes are described below, (Jimeno, Jimeno and Carcedo 1995)
1.
Percussion: The impacts produced by repeated blows of the piston generate shock waves that
are transmitted to the bit through the drill steel (in top hammer) or directly upon it (down the
hole)
2.
Rotation: Rotation of the drill string is required so that the inserts on the drill bit crush the
rock in a new position for each percussion impact. This helps in uniform drilling and having
good penetration rates.
3.
Feed (thrust force): In order to maintain the contact of the drill bit with the rock, a thrust load
or feed force is applied to the drill string.
4.
Flushing: Flushing is the removal of broken rock (drill cuttings) from the blast hole by using
pressurized air or water.
2.2 Rock drill (Drifter)
In modern drilling rigs, the four processes of percussion drilling are performed by drifters. A drifter
is a hydraulic or pneumatic powered rock or ground drill placed on top of a feed which is like a
rail on which the drill travels (drifts) on. Drilling using a drifter is known as drifting. The feed is
attached with a flexible boom to a stationary or a mobile unit that contains the powerpack. (Drifter
2014)
5
The various sub-systems on a modern drifter rock drill from Atlas Copco (top hammer) are shown
in Figure 6. The percussion system consists of a hydraulically actuated percussion piston which
strikes against the shank adapter. The kinetic energy of the piston is transmitted, in the form of
stress wave, via the shank adapter, drill steel and the drill bit to the rock, where it is used for
crushing. The rotation is provided by a hydraulic motor, usually an orbital gerotor motor, which
rotates the shank adapter through an axel and a gear train. The rotation torque needed to overcome
the forces at the bit and the drill rod is also used to keep the threads on the drill string tightened.
The flushing air/water, introduced into the shank, is carried through a channel in the drill steel till
the drill bit. Feed force is provided by moving the rock drill on the feed using a hydraulic feed
cylinder (not in the picture).
Figure 6. Sub-systems of a rock drill
2.3 Rotation mechanism
As mentioned in the previous section, the shank rotation is provided by a hydraulic motor, usually
an orbital gerotor motor (explained in 2.3.2), which rotates the shank adapter through an axel and
a gear train.
The default rotation speed is such that the peripheral buttons on the drill bit are moved by a button
diameter between each stroke. A high speed will lead to increased wear on the buttons, whereas a
low speed would lead to abrasion as the buttons would lock into the already crushed rock. The
rotation also provides sufficient torque to keep the threaded joints between drill rods tight.
(Grundström and Nordin 2007)
The required rotation speed ( nD ) can be calculated for using the following formula, (Wijk 1995)
nD  60
zB
f
 DB
(1)
Where DB is the bit diameter, f is the impact frequency in Hz and z B is the bit movement per blow.
Value of z B depends on the type of button bit and the rock hardness. The recommended values are
as follows,

Button bits, normal rock : z B  10 mm

Button bits, very hard rock: zB  8 mm

Blade bits, normal rock: z B  7.5 mm
For a 42 mm bit diameter ( DB ), 65 Hz impact frequency ( f ) and button bits on normal rock (
z B  10 mm ), the required rotation speed would be 295.5 rpm.
6
2.3.1 Rotation hydraulics
The rotation system provides a constant flow when the directional valve is actuated. The pump has
a constant speed. The simplified rotation hydraulic system is shown in Figure 7.
The pump is connected to the valve block which contains a pressure controlled valve and a
directional valve that creates a constant flow to the motor. The directional valve is controlled by
pilot pressures (XA and XB) set by the operator. The valve block also contains two pressure
limiting valves that protect the system against extremely high pressure.
Figure 7. Rotation hydraulic system (Grundström and Nordin 2007)
The rock drill considered for this project comes with different sizes of rotation motor depending
upon the application. The oil flow for rotation is,

Maximum continuous: 75 litre/minute

Maximum intermittent: 90 litre/minute
The 80 cc motor (smallest) provides a shank rotation speed of 300 rpm with approximately 52
litre/minute of oil flow.
7
2.3.2 Gerotor motor
A gerotor motor is a compact hydraulic motor that delivers high torque at low speeds. It consists
of two units, an inner and outer rotor. The inner rotor is located off-axis and has one less tooth
than the outer rotor. Each tooth of the inner rotor remains continuously in contact with the outer
rotor. A gerotor can either have two fixed axes at an offset for each rotor (both the rotors spin) or
a fixed outer rotor with the inner rotor revolving around the centre of the outer rotor while rotating
about its own axis. The motor used in the rock drills are of the later type. The inner rotor has a
trochoidal profile (formed by the locus of a point on a circle rolling on a circle). The outer rotor is
formed by a circle and intersecting circular arcs.
The inner and outer rotors form sealed pockets of fluid. The pressurized fluid fed to the motor acts
directly on the exposed inner rotor tooth via appropriate porting or a distributor valve. The inner
rotor is thus caused to rotate relative to the stationary outer rotor.
The classification of gerotor motors based on the type of valve mechanism is as follows,

Spool valve motors

Disc valve motors

VIS (valve-in-star) motors
The motor is shown in Figure 8. The outer rotor has rollers forming the teeth. Such a motor is also
called a geroller.
Figure 8. Gerotor motor
The ports and distributor valve (for disc valve motor) are shown in Figure 9.
8
Figure 9. Ports and distributor valve
The off-axis rotation of the rotor is transmitted to the output shaft through a cardan shaft that has
splined ends. The cardan shaft and the output shaft are shown in Figure 10.
Figure 10. Cardan shaft and output shaft
Gerotor motor produces high torque at low speed. The function diagram for MLHS 200 (disc valve
motor with 200 cc displacement) is shown in Figure 11. The function diagram data is for average
performance of randomly selected motors at back pressure 5+10 bar and oil with viscosity of 32
mm2/s at 50oC. (M+S Hydraulic n.d.)
9
Figure 11. Function diagram for MLHS 200 motor from M+S Hydraulic (M+S Hydraulic n.d.)
2.4 Hydraulic motors
Hydraulic power transmissions are known to be compact for high power transmission. Figure 12
shows the most commonly acceptable regions of applicability for hydraulic, electrical and
pneumatic power transmission media for rotation applications. It shows that hydraulic
transmission media is better suited for high torque applications. (Hunt and Vaughan 1996)
Figure 12. Comparative operational regions (Hunt and Vaughan 1996)
Hydraulic motors commonly used are of the following types:
10

Bent-axis axial piston motor

Swash plate axial piston motor

Vane motors

Gear motors

Lobe rotor motors (gerotor)
CONCEPTS
In this chapter a brief description of the concepts generated and their evaluation based on the
requirement specification has been presented.
3.1 Requirement Specification
Requirement specification for the concepts have been listed in Table 1. The demanded
specifications are necessary to be fulfilled by the generated concepts.
Table 1. Requirement specification
1
2
3
4
5
Requirement specification
Space
High torque / low speed
High Efficiency
Proper shock wave propagation
Serviceability
6
Independent system
7
Rotation direction
8
Number of hoses
9
Integration
10
11
Simple design
Variable displacement
Requirement
Radial diameter max. 150 mm
Displacement approx. 200 cc
No discontinuities in the shank
Shank easily removable /
replaceable
Independent rotation and
percussion
Both clockwise and anticlockwise
≤ 2 (less than existing number of
hoses)
Easy integration with the rock
drill
No complicated parts or systems
Possible switch between high
torque low speed to low torque
high speed
Demand/Wish
Demand
Demand
Demand
Demand
Demand
Demand
Demand
Wish
Wish
Wish
The requirement for efficiency has not been specified because the efficiency calculations have not
been performed in the project.
The motor (without any transmission) must be able to rotate the shank with a speed of 300 rpm
with 60 litre/minute oil flow. Therefore, the displacement requirement was set to 200 cc.
The existing gear coaxial with the shank has an outer diameter of 147 mm. Therefore, the
maximum allowable radial size was set to 150 mm so that the new concept could replace the gear
in position without much changes required in the size of the casing.
3.2 Concepts Generated
Concepts for only hydraulic motors were generated because a pneumatic or electrical motor would
require additional pneumatic or electrical lines running to the rock drill. A total of eight motor
concepts were generated. The brief description of the concepts is given below.
3.2.1 Gerotor motor – With geared output shaft
The working of the motor would be same as a gerotor motor being used on the rock drill. The
motor would surround the shank. The cardan shaft would be replaced by a geared output shaft
11
which would be supported by bearings on either side of the motor. The output shaft would be
placed coaxially with the motor stator. The rotor would rotate the output shaft through gearing.
The hollow output shaft further rotates the shank which passes through it. (Figure 13)
Figure 13. Concept: Gerotor motor – With geared output shaft
The radial size of the motor would be high as there is unused radial space between the output shaft
and the off-axis rotor. The displacement of the motor would be high and comparable to the existing
motor design.
3.2.2 Gerotor motor – With shank passing through hollow cardan shaft
In this concept, the gerotor surrounds the shank. The shank passes through a hollow cardan shaft.
The cardan shaft transmits rotation from rotor to an output hub. The output hub further transmits
rotation to the shank through the driver. (Figure 14)
Figure 14. Concept: Gerotor motor – With shank passing through hollow cardan shaft
The axial size of the motor would be high because of the long cardan shaft and the output hub. The
displacement of the motor would be high and comparable to the existing motor design.
3.2.3 Axial piston motor – with swash plate
In this concept, the shank and the driver pass through a motor body. The working of the motor is
same as a traditional axial piston motor. The high pressure hydraulic oil pushes the pistons out
against a swash plate causing the swash plate (in case of a fixed cylinder drum) or the cylinder
12
drum (in case of fixed swash plate) to rotate. The shank would be rotated by the rotating part of
the motor (swash plate or cylinder drum). (Figure 15)
Figure 15. Concept: Axial piston motor with swash plate
One complete stroke of piston would result in one revolution. Therefore, the displacement of the
motor would be low and an additional transmission would be required. The motor would have low
radial size and high axial size.
A mechanism to shut off fluid supply to select cylinders could be implemented to create a variable
displacement motor.
3.2.4 Axial piston motor – with circular cam plate
The working of the concept is similar to the axial piston motor described in 3.2.3. The swash plate
is replaced by a multi lobe circular cam plate. (Figure 16)
Figure 16.Concept: Axial piston motor with circular cam plate
This motor would have higher displacement than the motor with swash plate, because for one
revolution of the motor the number of strokes of a piston required would depend upon the number
13
of lobes on the cam plate. Therefore, no additional transmission would be required. The motor
would also have high axial size like the previous concept.
A mechanism to shut off fluid supply to select cylinders could be implemented to create a variable
displacement motor.
3.2.5 Radial piston motor multi-lobe cam
In a radial piston motor, hydraulic pressure would move the pistons radially in the rotating unit.
This movement of the pistons against the cam surface would cause the rotation of the rotating unit.
The rotating unit would further rotate the shank through a driver. (Figure 17)
Figure 17. Concept: Radial piston motor multi-lobe cam
The motor would have a high displacement as the number of strokes of piston required for one
revolution are dependent upon the number of lobes of the cam.
3.2.6 Multi-cam vane motor
In a multi-cam vane motor, the vanes are free to slide in guides on the rotation unit. The vanes are
pressed against the profile of external cam. The volume trapped between two vanes is pressurized
with hydraulic oil to cause the rotation of the rotating unit. (Figure 18)
Figure 18. Concept: Multi-cam vane motor
14
3.2.7 Strain wave hydraulic gear motor
A strain wave hydraulic gear motor concept is derived from the harmonic drive (Slatter and Slatter
2005) which allows high reduction ratios in mechanical drives. In the concept, the high pressure
hydraulic oil is applied between the meshing teeth on one side of the wave generator major axis
and on the diametrically opposite side. This would deform the flex gear resulting in rotation of the
wave generator. A reduced rotation would be obtained on the flexgear through the strain wave
gearing. The flex gear would rotate an output shaft which would further rotate the shank. (Figure
19)
Figure 19. Concept: Strain wave hydraulic gear motor
The motor is expected to exhibit very high displacement because of the high reduction ratio offered
by the strain wave gearing. The efficiency of the motor might be poor because of difficult sealing
of fluid chambers with high pressure fluid.
3.2.8 Screw motor
In a screw motor, the pressure of the hydraulic oil moving along the threads of the screws would
cause the rotation of the screws. In the concept, the central screw would be hollow and the shank
would pass through it. The central screw would rotate the shank through a driver. (Figure 20)
Figure 20. Concept: Screw Motor
15
The motor displacement would be high and an additional transmission would be required.
3.3 Concept evaluation
The concepts were evaluated for their compliance with the requirement specifications. Concepts
were first evaluated with respect to the polar properties (yes/no). The concepts were supposed to
satisfy all the mandatory properties. For the optional properties, an additional transmission was
not desired, as it would complicate the design, and a variable displacement mechanism without
complicating the design was desired (not mandatory). The evaluation for polar properties is shown
in Table 2.
Table 2. Evaluation for polar properties
Concepts
Axial
piston
Radial
motor piston
with
motor
circular
multi
cam
cam
plate
Gerotor
motor with
geared
output
shaft
Gerotor
motor hollow
cardan
shaft
Axial
piston
motor with
swash
plate
no shank
discontinuity
Y
Y
Y
Y
shank easily
removable
Y
Y
Y
independent
rotation and
percussion
Y
Y
rotation both
directions
Y
free moving
shank
less or same
number of
hoses
Properties
Mandatory
Optional
Transmission
required
Variable
displacement
Multicam
vane
motor
Strain
wave
hydraulic
gear
motor
Screw
motor
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
N
N
N
Y
N
N
N
Y
N
N
Y
Y
N
N
N
N
The generated concepts were then evaluated using a weighted PUGH matrix. Each concept is
compared to each other with respect to different factors as listed below:
16

Radial size: A smaller radial size of the motor is desired. This property has high weightage
because reducing the overall radial size of the rock drill is the main purpose of the project.

Axial Size: A smaller axial size of the motor is desired. This property has a low weightage
because a longer motor would only lead to slightly longer rock-drill which is not a major
concern.

Displacement: As the rotation mechanism is to be mainly used for high torque low speed
application, a high displacement of the motor is desired.

Efficiency: A higher efficiency of the motor is desired.

Integration: The motor assembly must be easy to assemble and must also be easily mounted
onto the rock drill.

Simplicity: A simple design of the motor is desired.

Robustness: The design must be robust.
The factors were assigned weights from 1 to 5. Each concept was scored from 1 to 5 for each
factor. No baseline score was used as there is no existing design with motor unit surrounding the
shank. The weighted PUGH matrix is shown in Table 3.
Table 3. Weighted PUGH matrix
Concepts
Properties
Weight
Radial size
5
Axial size
2
displacement
4
efficiency
5
Integration
3
simplicity
3
robustness
4
weighted score
Gerotor
motor with
geared
output
shaft
Gerotor
motor hollow
cardan
shaft
Axial
piston
motor
- with
swash
plate
Axial
piston
motor with
circular
cam
plate
Radial
piston
motor
multi
cam
Multicam
vane
motor
Strain
wave
hydraulic
gear
motor
Screw
motor
3
5
4
4
4
4
4
101
3
2
4
4
4
4
4
95
5
2
1
5
4
5
5
105
5
2
4
4
4
4
5
109
4
5
4
4
4
3
5
107
4
5
5
4
5
4
5
117
5
5
5
2
3
4
4
102
5
1
1
5
5
4
5
103
Based on the PUGH matrix, the multi-cam vane motor concept was chosen for further design.
Additionally, the strain wave hydraulic gear motor concept was also chosen for further design
because of its novelty.
Both the concepts were designed at ideal conditions (100% efficiency) with constant pressure
difference and oil flow rate. The pressure and flow parameters used are listed below:

Input fluid pressure, Pin  120 bar  1.2 107 Pa

Return fluid pressure, Pout  1 bar  1105 Pa

Fluid flow, Q  60 l/min  0.06 m3 /min
17
MULTI-CAM HYDRAULIC VANE MOTOR
CONCEPT
In this chapter the design of the multi-cam vane hydraulic motor, which is one of the selected
concepts, is presented. The various improvements on the initial concept and the reasoning behind
them are also discussed. The different versions of the concept have also been described.
4.1 Working Principle
The motor has a rotating cylindrical shaft (rotor) and a stationary housing (stator). The rotor also
forms the output shaft that rotates the driver, which in turn rotates the shank adapter. The inner
profile of the stator forms a cam ring. The volume chambers are separated by vanes which slide
inside guides in the rotor. Alternatively, the rotor can have a cam ring profile and the vanes can be
housed inside the stator. (Figure 21)
For design calculations, only the former system (Figure 21a) is considered.
a
b
Figure 21. Concept versions, a) with vanes housed in rotor; b) with vanes housed in stator
The volume enclosed by the stator, rotor and the vanes is the active volume. The minimum and
the maximum volumes are shown in Figure 22.
Figure 22. Maximum and minimum possible chamber volumes
19
High pressure hydraulic fluid through a port, whose position is fixed relative to the stator, would
rotate the rotor such that the chamber volume increases from minimum to maximum. The direction
of rotation can be reversed by swapping the pressure line with the return line. The vanes would
regulate the opening and closing of fluid ports by uncovering and covering them respectively. The
fluid port diameter must be smaller than the width of the vane. (Figure 23)
Figure 23. Operation by high pressure hydraulic fluid input and swapping of pressure ports with return ports to
change direction of rotation
High pressure hydraulic fluid from the pressure line would be supplied to the vane guides so that
the vanes are always pressed against the cam profile and the contact is never lost. (Figure 24)
Figure 24. High pressure hydraulic fluid used for maintaining contact between vane and cam profile
20
4.2 Design calculations
4.2.1 Stator cam profile
The profile of the cam ring is considered to be a sine profile because it can be easily represented
as an equation which further simplifies the calculations. The profile in polar coordinates  r ,  
can be given as follows,
n

n 
r  a sin       Ro   a sin     Ro
2

2 
(2)
Where RO is the radius of the mean circle of the sine curve, a is the amplitude of the sine curve,
n is twice the number of cams. Number of vanes (k ) was considered to be equal to n. Each vane
was equally placed along the circumference of the rotor (angular spacing of 2 / n radians). The
modification of    was made so that at an initial condition, when vane 1 is at   0 , the volume
2 

enclosed by vane 1 and vane 2    0 
 would be the minimum possible volume and the
n 

2
4 


volume enclosed by vane 2 and vane 3   
 would be the maximum possible
n
n 

volume. The cam profile and the maximum/minimum chamber volume are shown in Figure 25.
Figure 25. Stator cam profile, position of vanes at initial condition and the maximum/minimum chamber volumes
A repetitive sub-cycle for rotation would be rotation of vane 1 from   0 to  
2
and
n
2
4
. Therefore, the chamber volume
to  
n
n
2
enclosed by vane 1 and 2 would go from minimum to maximum and a rotor rotation of
would
n
n
2
be obtained. The same sub-cycle occurs at
locations for a rotation of
radians. Therefore,
2
n
corresponding rotation of vane 2 from  
21
for calculation of torque and speed, only one sub-cycle is analysed and then extended to the
complete cycle.
Width of the stator and rotor unit is denoted by B.
4.2.2 Evaluation of points of contact
The considered cross section shape of the vane was rectangular (width Wp ) with circular head
(radius rp ) as shown in Figure 26a. The contact of the vane with the cam profile is shown in Figure
26b.
a
b
Figure 26. a) Vane structure; b) contact geometry between vane and cam profile
In the figure,  is the actual angle of rotation of the vane from   0 , f  x, y  is the equation of
the cam profile in Cartesian coordinates,  X C , YC  are the coordinates of the corresponding point
of contact, i.e. C,  X R , YR  are the coordinates of the centre of the circular part of the vane, i.e. R,
and  is the angle of rotation of the contact point in the same coordinate system as used for  .
For a profile with equation r  f   in polar coordinates, the coordinates of the point on profile
at angle  ' would be,
x  r ( ') cos  ' ; y  r ( ') sin  ' 
dx dr

d d
cos( ')  r sin( ');
  '
dy dr

d d
sin( ')  r cos( ')
(3)
(4)
  '
Slope  mn  of the normal to the profile at any point is given by,
dr
dx
dx
d
mn  
  d  
dy
dr
dy
d
d
For the profile given in equation (2),
22
cos( ')  r sin( ')
  '
(5)
sin( ')  r cos( ')
  '
dr
1
n 
  an cos   
d
2
2 
(6)
At an assumed angle  , angle  can be given by,
dr
d
  tan 1 (mn )  
dr
d
cos( )  r sin( )
 
(7)
sin( )  r cos( )
 
Distance of the point of contact form the origin, RC ,
RC  r ( )
(8)
Coordinates of the point of contact  X C , YC  ,
X C  r   cos   ; YC  r   sin  
(9)
Coordinates of the centre of the circular head of the vane  X R , YR  were calculated as,
X R  X C  rp cos    ; YR  YC  rp sin   
(10)
The corresponding angle of rotation    of the vane was calculated as,
 YR 

 XR 
  tan 1 
(11)
The distance of the centre of the circular part of the vane from the origin is given by,
RR 
X R 2  YR 2
(12)
The angle   between the axis of the vane and the line joining the point of contact with the centre
of the circular head of the vane (Figure 26b) is calculated as follows,
RR sin(   )  rp cos( )
 R sin(   ) 
   cos 1  R


rp



(13)
    
(14)
       
(15)

2
The design parameters used for calculations are given in Table 4.
Table 4. Design parameters
Number of crests and troughs n
Number of vanes k
Amplitude of sine profile a
Radius of vane head rp
4 mm = 4 103 m
7 mm = 7 103 m
Width of vane WP
7 mm = 7 103 m
12
12
23
 42
Radius of rotor profile RI
Radius of mean circle of sine profile RO
For   

2


2
 3a  = 54 mm= 54 103 m


 Wp   
RI  rp 1  cos sin 1 
 a = 59 mm
 2r   

 p   

= 59 103 m
, corresponding values of  were calculated. (Figure 27)
Figure 27. Corresponding values of parameters γ and α for φ = -90  90 deg
For further calculations, values of  , RC , RR and  were needed for any rotation angle  . These
were obtained by first obtaining  value for any rotation angle  by reverse interpolation of
tabular data of  vs  using inbuilt Matlab function for 1-D interpolation by table lookup, interp1.
Spline interpolation method is used. This method uses not-a-knot end conditions. The interpolated
value at a query point is based on a cubic interpolation of the values of neighbouring grid points
in each respective dimension (Support: Documentation: interp1 n.d.). Then the RC , RR and 
values were obtained for the calculated value of  using equations (6) to (15).
4.2.3 Speed calculation
k
chambers (alternate chambers) would be pressurised simultaneously, change in volume of
2
the pressurized chambers must be calculated to calculate the speed of the motor.
As
For volume calculation, the chamber bounded by vanes 1 and 2 was considered (Figure 25). For
any rotation angle (  ), vane 1 would be at angular position,  1   , and vane 2 would be at angular
2
position  2   
. The chamber area was calculated by the following equation,
k
AChamber  Aa  Ab  Ac
(16)
Aa is the area bounded by the cam profile and the circular inner profile for an angle between
   1   2 . Ab is the portion of Aa occupied by the vanes. Ac is the area that is added or removed
from Aa because contact points between vane and the cam profile might not lie on the axis of
vanes. (Figure 28)
24
Figure 28. Parts of area bounded by the stator, rotor and two vanes
Area bounded by a curve r  f   and lines   1 ,2 (Figure 29) in polar coordinates is given
by, (Lewis 1986)
area 
2
1
  r ( ) 
 2
2
d
(17)
1
Figure 29. Area under a curve in polar coordinates
Therefore, area Aa , bounded by the cam profile equation, as given in equation (2), and the circular
rotor profile  r  RI  is calculated as follows,
2
2

2
1

1
n 
Aa    a sin     RO  d   RI 2 d
2 

1 2 
1 2
 n 
 n 
n   2   1   a 2  2 RI 2  2 RO 2   a 2  sin(n  1 )  sin(n  2 )   8aRO cos  1   8aRO cos  2 
 2 
 2 

4n
(18)
The detailed derivation for this expression is given in Appendix A.
25
a
b
Figure 30. Detail of Ab and Ac
Ab was calculated as follows,
Ab  Ab1  Ab 2  Ab3
(19)
Ab1 , Ab 2 and Ab3 are constituents of Ab as described in Figure 30a.
 rp 2 m W p rp cos( m ) 
Ab1  2 


4
 2

Wp
 RR1  RI    RR 2  RI  
2 
W p RI
R 2 
Ab 3  2 
 2  cos( )   I 
2 
 4
Ab 2 
(20)
Here, RR1 and RR 2 are RR values for  1 and  2 respectively.
Ac was calculated as follows,
Ac  Ac1  Ac 2
(21)
Ac1  Ac1_1  Ac1_ 2 ; Ac 2  Ac 2 _1  Ac 2 _ 2
Aci _1 and Aci _ 2 ( i  1, 2 ) are described in Figure 30b. Aci _1 is the area bounded by cam profile
between angles i and  i .
i
2
1

n 
Aci _1     a sin     RO  d
2 

i 2 
 n
n   i  i   a  2 RO   a  sin(n i )  sin(n  i )   8aRO cos  i
 2

4n
2
2
2
Aci _ 2 
26
rp 2
2
i 
RCi RRi sin( i  i )
2

 n i 
  8aRO cos 


 2 
(22)
(23)
RCi , RRi and i are RC , RR and  values for  i i  1, 2 .
Derivation of expressions from calculation of Ab and Ac is given in Appendix B.
The area added to the chamber  AChamber  for rotation from  to   d  would be,
Achamber  Achamber   d  Achamber 
As simultaneously, the same change is volume occurs in
(24)
k
chambers, the total added area,  dA 
2
would be,
A   d  Achamber
The increase in area of the chamber for rotation   0 
k
2
(25)
2
,
n
2
nd 
dA
0
2
n
  Aid i 1d
(26)
i 0
For obtaining a displacement of 200 cc/rev, i.e. 200 106 m3/rev, the width ( B ) was calculated
as follows,
B
200  10 6
n.dA 2
0
(27)
n
The required motor width ( B ) was calculated to be 0.0174 m (17.4 mm).
The total volume added ( dV ) for rotation from  to   d  would be,
dV   d   A   d  B
(28)
The instantaneous speed for rotation i  in RPM, from  to   d  would be,
i 
Qd 
2 dV   d
(29)
2
is shown in Figure
n
31. The instantaneous speed at the beginning and at the end of the sub-cycle tends to infinity
because of very small change in volume for that rotation. The average of the instantaneous speeds
is 300 RPM.
The calc1ulated instantaneous speed at different rotation angles   0 
27
Figure 31. Instantaneous speeds for rotation of 2π/n rad
The displacement and the average speed of the motor were calculated to be 345 cc and 174 RPM
respectively.
4.2.4 Torque calculation
The torque on the sub-system is due to the hydraulic pressure acting on the vanes and the contact
forces on the vanes with the cam profile. The torque (per unit width) on a vane can be divided into
four constituents Ta , Tb , Tc , Td  . The forces and pressures causing the four constituents are
shown in Figure 32. FCY and FCX are components of the contact force (per unit width) in the
direction parallel to the axis of the vane and normal to the axis respectively. Anti-clockwise
rotation is considered for calculations and the torque in anti-clockwise direction is considered to
be positive.
Figure 32. Torque components on a vane
The four torque (per unit width) components Ta , Tb , Tc , Td  , were calculated as follows,
28
 Wp 

 2 RI 
z  Rr  rp cos( ); l  RI cos( )
(30)
P P
P
Ta   Pxdx  x 2  1 2  z 2  l 2 
2 l
2
l
(31)
  sin 1 
z
z
Wp
 rp sin( )

2
P P


Tb    Pxdx     P2 xdx   P1 xdx    1 2
2
Wp
rp sin( )
  Wp


2
 2

Wp
2
 W p  2
2

r
sin(

)


  p
 2 

(32)
Contact forces are caused due to high pressure fluid acting on the back of the vanes. Therefore,
FCY is equal to the resultant of force (per unit width) due to pressure beneath the vane ( PinWp ) and
the force (per unit width) due to pressure on the top of the vane in a direction parallel to the axis
of the vane.
Wp
 rp sin( )

2


FCY  PinW p   Pdx  PinW p    P2 dx   Pdx
1

Wp
Wp
rp sin( )



2
 2

Wp 

 Wp

 FCY  PinW p  P2  rp sin( ) 
 rp sin( ) 
  P1 
2 

 2

Tc   FCY rp sin( )
Wp
2
(33)
As the contact force must be perpendicular to the surface at the contact point, FCY and FCX would
be the orthogonal components of the resultant contact force which is at an angle  with the axis
of vane. Therefore, FCX can be calculated as,
FCX  FCY tan( )
Td  FCX z
(34)
The torque components (per unit width) for one activated chamber at different rotation angles
2
rad is shown in Figure 33. The component Ta has the maximum contribution towards
 0
n
the total torque.
Total torque (per unit width) on each vane,
Tvane  Ta  Tb  Tc  Td 
(35)
For vane 2, P1  Pin and P2  Pout
For vane 1, P1  Pout and P2  Pin
Torque from one pressurized chamber is the sum of the torque from the two bounding vanes.
Tchamber  Tvane1  Tvane 2  Ta  Tb  Tc  Td vane1  Ta  Tb  Tc  Td vane 2
 Ta  Tb  Tc  Td chamber
(36)
29
a
b
Figure 33. a) Components of torque (per unit width) for one pressurized chamber; b) total torque from one
pressurized chamber (per unit width)
Total torque in the system at a certain angular position,
Ttotal  Tchamber
k
B
2
The calculated torque in the system at different rotation angles   0 
(37)
2
is shown in Figure 34.
n
Figure 34. Total torque during rotation of 2π/n rad
The average torque for the motor was calculated to be 379 Nm.
4.3 Concept improvement
As shown in Figure 34 and Figure 31, the motor would result in a very jerky motion, due to high
range of fluctuation in torque at constant hydraulic pressure. The motor would also stall completely
since torque is zero at some positions.
The following design changes can be used to reduce the range of fluctuation in torque.
30
4.3.1 Double motor
The motor could be composed of two motor units combined with a phase difference of

as shown
n
in Figure 35.
Figure 35. Double motor
This design would reduce the range of torque fluctuation by superposition of two high fluctuating
torque profiles as shown in Figure 36. The width (B) of individual units was reduced to half to
maintain the same displacement of 200 cc.
Figure 36. Torque profile for double motor
The average torque for the motor was calculated to be 379 Nm.
4.3.2 Less number of vanes; sealing by rotor and stator
If the stator and rotor profiles are allowed to have contact  RO  RI  a  , the number of vanes can
be reduced by having rotor and stator to seal the chambers at contact points. (Figure 37)
31
n
 1 , would create a phase
2
difference between the characteristics (torque, chamber volume change) associated with each vane.
Unlike the previous case, the characteristics for each vane would lead the characteristics of the
1 2
trailing vane by a phase angle 2    (Figure 37). Because of this phase angle, all the vanes
k n
would deliver different amount of torque at a particular moment. When one vane would be
delivering minimum torque, the remaining vanes would be delivering higher torque so that the
superimposition would result in a torque in the proximity of the average value, therefore reducing
the fluctuation range.
Having the number of vanes one less than the number of cams, i.e. k 
Figure 37. Motor with sealing between stator and rotor; less number of vanes
This motor can be designed in two configurations as below:

With vanes on rotor and cam profile on the stator

With vanes on stator and cam profile on the rotor
The valves must be capable of pressurizing the chamber on one side of the vane with high pressure
and the chamber on other side of the vane with return pressure.
The detailed design of both design configurations are described in the sections 4.4 and 4.5
respectively.
In further text, such a motor is referred to as “cam vane motor”.
4.4 Cam vane motor: Vanes on rotor
For simplifying calculations, the equation of the sine curve was established so that vane 1 would
be at the completely retracted position at   0 . Therefore, the equation established was,
n 
r   a cos     Ro
2 
32
(38)
As an initial condition, if vane 1 is considered to be at   0 , a sub-cycle would be rotation of vane
4
1 till  
. (Figure 38)
n
Figure 38. Sub-cycle of rotation
4.4.1 Contact Points


, corresponding values of parameters  , RC , RR and  (Figure 26) were
2
2
calculated using the method shown in 4.2.2. As the considered equation for the profile is as shown
in equation (38), the derivate in equation (6) was modified to,
For   

dr 1
n 
 an sin   
d 2
2 
(39)
The value of  , RC , RR and  for any rotation angle    were obtained by interpolation.
 
The width of the vane Wp was calculated so that the contact point never reaches the edge of the
vane.
Wp  2rp sin( max )
(40)
4.4.2 Speed Calculation
The change in volume of pressurized chambers was derived to calculate the speed of the motor.
The sub-cycle volume change (on one side of vane 1) was considered. The pressurized chamber is
bounded by the contact between the vane and the cam profile at one end ( C2 ) and the contact
between the cam profile and the rotor profile at the other end ( C1 ) as shown in Figure 39.
W 
4
For vane 1, for     sin 1  p  , the contact C1 is at   
and for    , the contact C1
n
 2 RI 
is at   0 .
For the equations the angular position of C1 would be denoted by .
33
Figure 39. Pressurized volume for
   and   
For a rotation angle    , the chamber volume was calculated as follows,
AV 1  Aa  Ab  Ac
(41)
Figure 40. Area components
Aa is the area bounded by the cam profile, the circular inner profile for an angle between   
(Figure 40) , and is calculated using the equation (17) as follows,

2

1

1
n 
Aa    a cos     RO  d   RI 2 d
2 

 2
 2
 n
n      a 2  2 RI 2  2 RO 2   a 2  sin(n  )  sin(n )   8aRO sin 
 2

4n

 n 
  8aRO sin 


 2 
The derivation for the above expression is given in Appendix B.
Ab is the part of Aa occupied by a vane (Figure 40) and it is calculated as follows,
34
(42)
Ab  Ab1  Ab 2  Ab 3
Ab1 
Ab 3 
rp 2 m
2
W p rp cos( m )

4
Ab 2 
Wp
W p RI
 2  cos( )  
4
2
(43)
 RR  RI 
RI 2
2
Ab1 , Ab 2 and Ab3 are constituents of Ab as described in Figure 40.
Ac is the area that is added or removed from Aa because the contact point between vane and cam
profile may not lie on the axis of vane (Figure 40). It is calculated as follows,
Ac  Ac _1  Ac _ 2

2
1

n 
Ac _1    a cos     RO  d
2 

 2
 n 
 n 
n       a 2  2 RO 2   a 2  sin(n  )  sin(n  )   8aRO sin    8aRO sin  
 2 
 2 

4n
2
rp
R R sin(   )
Ac _ 2 
 C R
2
2
(44)
Ac _1 and Ac _ 2 are described in Figure 40.
Derivation of expressions for calculating Ab and Ac are given in Appendix B.
The change in area related to vane 1  AV 1  for rotation from  to   d  is given by,
AV 1  AV 1   d  AV 1 
(45)
4
were calculated and concatenated together.
n
The change in area corresponding to the remaining vanes were calculated by having a successive
1 2
phase difference of 2    with vane 1.
k n
The change in area for   0   and    
The total change in area  A   d  for rotation from  to   d  ,
k
A   d    AV i
(46)
1
The increase in area of the chamber for rotation   0 
4
,
n
4
nd 
dA
0
4
n
  Aid i 1d
(47)
i 0
For obtaining a displacement of 200 cc/rev, i.e. 200 106 m3/rev, the width ( B ) was calculated
as follows,
35
B
2  200 106
n.dA 4
0
(48)
n
The total volume added ( dV ) for rotation from  to   d  would be,
dV   d   A   d  B
(49)
The instantaneous speed for rotation i  in RPM, from  to   d  would be,
i 
Qd 
2 dV   d
(50)
4.4.3 Torque Calculation
The torque acting on one vane is calculated using the method described in 4.2.4. The pressures on
the vane are P1  Pin and P2  Pout . The torque profile is calculated for vane 1 for one sub-cycle,
4
i.e. rotation from 0 to
. The torque for the remaining vanes were calculated by having a
n
1 2
successive phase difference of 2    with vane 1.
k n
4.4.4 Parameter dependencies
Three design parameters which were considered to be important were:

Amplitude of sine wave (a)

Number of vanes (k)

Radius of the vane head ( rp )
The different characteristics, eg. Average torque, torque fluctuation, speed etc. were calculated by
varying one design parameter at a time keeping the other two constant.
Effect of amplitude (a)
The width of the vane is kept constant. The torque and displacement of the motor increases with
increase in amplitude of the sine curve as shown in Figure 41.
Figure 41. Average torque, displacement vs. amplitude of sine curve
The minimum possible outer radius of the motor for different amplitude of sine curve is shown in
Figure 42.
36
Figure 42. Minimum possible outer diameter of the motor vs. amplitude of sine curve
For having a constant displacement of 200 cc, the width (B) of the motor is varied with increase
in amplitude. (Figure 43)
Figure 43. Width of the motor vs. amplitude of sine curve for maintaining constant displacement
At constant displacement, the range of fluctuation of output torque and output speed initially
increase with increase in amplitude of sine curve but remains almost constant on further increase
in amplitude. (Figure 44)
Figure 44. Range of torque fluctuation, range of speed fluctuation vs. amplitude of sine curve at constant
displacement
37
The maximum contact force (per unit width) between the vane and the cam profile increases with
increase in amplitude of sine curve as shown in Figure 45.
Figure 45. Maximum contact force between vane and cam profile (per unit width) vs. Amplitude of sine curve at
constant displacement
Effect of number of vanes (k)
The number of crests and troughs (n) also changed with number of vanes (k) as per equation
n
k   1. The width of the vane was kept constant.
2
The output torque and the displacement of the motor increase with increase in number of vanes as
shown in Figure 46.
Figure 46. Average torque, displacement vs. number of vanes
For having a constant displacement of 200 cc, the width (B) of the motor was varied with the
increase in the number of vanes. (Figure 47)
38
Figure 47. Width of the motor vs. number of vanes for maintaining constant displacement
The range of fluctuation in torque and speed remains almost constant and low for more than three
vanes as shown in Figure 48.
Figure 48. Range of torque fluctuation, range of speed fluctuation vs. number of vanes (at constant displacement)
For 2 vanes the range of fluctuation is very high as there would be only two characteristic (torque
or speed) profiles, one from each vane, for superposition. As the number of vanes increases the
number of characteristic profiles to be superpositioned increases. This reduces the range of
fluctuations. The increase in number of crests and troughs also increases the contact force (FCX) as
the vanes have a steeper path along the cam profile. Therefore, the contribution of contact force
towards torque, Td, also increases, thus increasing the range of fluctuations. The increase and
decrease in range of fluctuations nearly cancel each other, hence keeping the range of fluctuations
almost constant with increase in number of vanes beyond three.
The maximum contact force (per unit width) between the vane and the cam profile increases with
increase in number of vanes as shown in Figure 49.
39
Figure 49. Maximum contact force between vane and cam profile (per unit width) vs. number of vanes at constant
displacement
 
Effect of radius of the vane head rp
The width of the vane also varies with the radius of the vane head according to the equation (40).
The change in average torque and displacement with increase in vane head radius is negligible.
(Figure 50).
Figure 50. Average torque, displacement vs. vane head radius
At constant displacement of 200cc, the range of fluctuation of torque and speed increased with
increase in vane head radius as shown in Figure 51.
40
Figure 51. Range of torque fluctuation, range of speed fluctuation vs. vane head radius (at constant displacement)
From equations (31) and (34), it is evident that the two major contributors to torque (i.e. Ta and
Td )
are
dependent
upon
rp sin( ) and rp cos( ) . As the range
of
fluctuation
in
rp sin( ) and rp cos( ) increases with increase in vane head radius, the fluctuation in the torque
components and the total torque also increases. The comparison between torque components for
head radius 3 mm and 9 mm is shown in Figure 52.
rp  3 mm
rp  9 mm
 max  0.41 rad ;  min  0.41 rad
 max  0.45 rad ;  min  0.45 rad
 r sin( ) 
 r cos( ) 
p
max
p
 1.19 ;  rp sin( ) 
max
 3 ;  rp cos( ) 
min
min
 1.19
 2.75
 r sin( ) 
 r cos( ) 
p
max
p
 3.95 ;  rp sin( ) 
max
 9 ;  rp cos( ) 
min
min
 3.95
 8.09
Figure 52. Comparison between torque components for vane head radius 3 mm and 9 mm
The contact forces, FCX and FCY , are also depended upon rp sin( ) and rp cos( ) , as evident from
equations (33) and (34). Therefore, the maximum contact force (per unit width) between the vane and
the cam profile increases with increase in vane head radius as shown in Figure 53.
41
Figure 53. Maximum contact force between vane and cam profile (per unit width) vs. vane head radius at constant
displacement
4.4.5 Ideal torque and speed
The geometric design parameters used are shown in Table 5. The vane is considered to have a
height of  3a  , where a is the amplitude of the sine curve, so that at the most extended position of
the vane, a height equal to one third of the height of the vane  a  is still inside the guide. The
dimension RI is such that the rotor has enough space to house the vane in the most retracted
position with some clearance.
Table 5. Geometric design parameters
2  k  1
Number of crests and troughs n
 0.043
Radius of rotor profile RI
 3a  m
RI  a m
Mean circle radius of sine curve RO
The required ideal displacement of the motor can be obtained by changing the number of vanes,
amplitude of sine curve and/or width of the motor.
It is desirable to have the lowest possible amplitude of sine curve so as to have lowest possible
size of the motor. The amplitude of the sine curve was chosen such that with further decrease in
amplitude, the required width of the motor does not increase by a large amount. (Figure 43)
The number of vanes is chosen such that the torque and the speed fluctuations are low. (Figure 48)
A high vane head radius is desirable for having low contact stresses (Fischer-Cripps 2007). The
radius of vane head has negligible effect on average torque and displacement from the motor. The
radius is selected such that the range of fluctuation of torque and speed are less while keeping the
radius high.
The selected parameters are shown in Table 6.
Table 6. Values of design parameters.
Number of vanes k
Amplitude of sine curve
a
Radius of piston head rp
42
5
4 mm = 4 103 m
7 mm = 7 103 m
The width of the motor ( B ), required to have a displacement of 200 cc, was calculated to be 0.0274
m (27.4 mm).
The individual torque profiles of each vane and the superimposed total torque is shown in Figure
54, a and b respectively.
a
b
Figure 54. a) Individual torque profiles for each vane; b) total torque
As evident from the torque profiles, each vane has high range of torque fluctuation but the total
superimposed torque has very small range of torque variation. The average torque is 379 Nm.
The individual change in volume profiles related to each vane (for every small rotation of 0.03
degrees) and the instantaneous speed of the motor is shown in Figure 54, a and b respectively.
a
b
Figure 55. a) Volume of fluid added to pressurised chamber related to each vane; b) instantaneous speed of the
motor
The average speed is 300 RPM.
4.4.6 Bearing load
The net force in the vane coordinate system ( X , Y ) was calculated by equating the pressures and
contact forces shown in Figure 32.
43
FX   FCX   P1  P2  z  l   B

 Wp

 Wp

FY   FCY  P1 
 rp sin( )   P2 
 rp sin( )   B
 2

 2


(51)
The reactions acting at bearing would be RX   FX and RY   FY .
The reaction forces were converted into the motor coordinate system ( x, y ) using the following
expressions, (Figure 56)
Rx  RY cos   RX sin 
Ry  RY sin   RX cos 
(52)
Figure 56. Reaction forces due to loads on one vane
The reaction forces due to the remaining vanes (placed at an angular interval of
2
and having a
k
1 2
phase difference of 2    in force characteristics) were calculated and added to obtain the
k n
4
net radial load on the bearing. The net resultant radial load for a rotation of 0 
with 120 bar
n
fluid pressure is shown in Figure 57.
Figure 57. Radial load on the bearings
44
Along with the alternating amplitude, the direction of radial load also oscillates about the centre.
The radial loads are taken up by two angular contact roller bearings placed on both sides of the
rotor. For CAD purposes, SKF 75x115x25 angular contact roller bearings (tapered roller bearings)
were used which have a static load capacity of 163 kN.
4.4.7 Valve mechanism and assembly
As the vanes rotate with the rotor, the valve ports supplying hydraulic fluid to the chambers (high
pressure on one side of the vane and return pressure on other side) must also rotate with the rotor.
Therefore, the distributor was integrated into the rotor design. The two main ports (inlet and return)
would be on the back face of the motor which would allow easy connection to the rock drill fluid
lines. The internal channels in the motor back housing and valve plate supply hydraulic fluid to
two circular cavities between the distributor part of the rotor and the valve plate. The cavities lead
to individual chamber ports that supply hydraulic fluid on either side of the vane. The valve
mechanism is shown in Figure 58. The high pressure is represented by red colour and the return
pressure is represented by blue colour. The ports can be swapped, i.e. high pressure in blue and
return pressure in red for a reversed rotation direction.
Figure 58. Valve mechanism
For supplying high pressure fluid to the back side of the vanes (to maintain continuous contact), a
simple selector valve mechanism was integrated into the valve plate. The valve receives hydraulic
fluid from both channels and delivers the high pressure fluid to a groove on the rotor that connects
all the guides of the vanes. The valve uses a ball which is acted upon by fluid from both channels
and slides to block the flow of return fluid and allow flow of high pressure fluid. (Figure 59)
45
Figure 59. Selector valve mechanism
The exploded view of the assembly and the part list are given in Figure 60. The section views are
shown in Figure 61.
Figure 60. Exploded vied of the assembly and part list
46
Figure 61. Section views of the assembly
The assembly has an axial length of 135 mm and a side-to-side distance of 150 mm.
4.5 Cam vane motor: Vanes on stator
For simplifying calculations, the equation of the sine curve was established so that vane 1 would
be at the completely retracted position at   0 . Therefore, the equation established was,
n 
r  a cos     Ro
2 
(53)
To make the calculations simpler, rotor frame of reference was used, i.e. rotor was assumed to
stand still while the stator rotates. As an initial condition, if vane 1 is considered to be at   0 , a
4
sub-cycle would be rotation of vane 1 till  
. (Figure 62)
n
47
Figure 62. Sub-cycle of rotation
4.5.1 Contact Points


, corresponding values of parameters  , RC , RR and  (Figure 63) were
2
2
calculated using the method shown in 4.2.2. Because the considered equation for the profile is as
shown in equation(53), the derivate in equation (6) was modified to,
For   

dr
1
n 
  an sin   
d
2
2 
(54)
Figure 63. Contact geometry
As the vane is on the outside of the cam profile (Figure 63), equation (10) was replaced by the
following equation,
48
X R  X C  rp cos    ; YR  YC  rp sin   
(55)
For calculation of α, equation (15) was replaced by the following equation,
   
(56)
The value of  , RC , RR and  for any rotation angle    can be obtained by interpolation.
 
The width of the vane Wp was calculated as follows,
Wp  2rp sin( max )
(57)
4.5.2 Speed Calculation
The speed of the motor was calculated using the same method as described in section 4.4.2.
W 
Pressurized volume for     sin 1  p  and    is shown in Figure 64.
 2 RI 
Figure 64. Pressurized volume for
Figure 65. Description of
   and   
Aa , Ab and Ac
49
The equations for calculating Aa , Ab and Ac (Figure 65) were slightly modified because of the
position of the vane outside the cam profile.
The equations are given below.
AVi  Aa  Ab  Ac

(58)

2
1
1

n 
Aa   RI 2 d    a cos     RO  d
2 

 2
 2
 n
n      a 2  2 RI 2  2 RO 2   a 2  sin(n )  sin(n  )   8aRO sin 
 2

4n

 n 
  8aRO sin 


 2 
(59)
Derivation of the above expression is given in Appendix A.
Ab  Ab1  Ab 2  Ab 3
Ab1 
Ab 3 
rp 2 m
2

W p rp cos( m )
4
Ab 2 
Wp
W p RI
 2  cos( )  
4
2
(60)
 RI  RR 
RI 2
2
Ac  Ac _ 2  Ac _1

2
1

n 
Ac _1    a cos     RO  d
2 

 2
 n
n       a 2  2 RO 2   a 2  sin(n  )  sin(n  )   8aRO sin 
 2

4n
2
RC RR sin(   ) rp 
Ac _ 2 

2
2

 n 
  8aRO sin  

 2 
(61)
Derivation of expressions for calculating Ab and Ac are given in Appendix B.
4.5.3 Torque Calculation
The torque acting on one vane was calculated using the method described in 4.4.3 and 4.2.4. The
equations for calculating z , Ta , Tb and Tc were slightly modified because of the vane being on the
outside of the cam profile according to Figure 66.
50
Figure 66. Torque components on the vane
The modified equations are given below,
z  Rr  rp cos( )
(62)
P P
P
Ta   Pxdx  x 2  1 2  l 2  z 2 
2 l
2
z
(63)
z
l
Wp
2

Tb 

rp sin( )

Pxdx 
Wp
2

Wp
Wp
P P
P2 xdx   P1 xdx  1 2
2
rp sin( )
2
 W p  2
2

r
sin(

)





p
 2 

(64)
2
Tc  FCY rp sin( )
(65)
4.5.4 Parameter dependencies
The design parameters considered and the method of their evaluation were same as explained in
4.4.4.
Effect of amplitude (a)
The width of the vane is kept constant. The torque and displacement of the motor increases with
increase in amplitude of the sine curve as shown in Figure 67.
51
Figure 67. Average torque, displacement vs. amplitude of sine curve
The minimum possible outer radius of the motor for different amplitude of sine curve is shown in
Figure 68
Figure 68. Minimum possible outer diameter of the motor vs. amplitude of sine curve
For having a constant displacement of 200 cc, the width (B) of the motor is varied with increase
in amplitude. (Figure 69)
52
Figure 69. Width of the motor vs. amplitude of sine curve for maintaining constant displacement
At constant displacement, the range of fluctuation of output torque and output speed increase with
increase in amplitude of sine curve (Figure 70)
Figure 70. Range of torque fluctuation, range of speed fluctuation vs. amplitude of sine curve at constant
displacement
The maximum contact force (per unit width) between the vane and the cam profile increase with
increases in amplitude of sine curve as shown in Figure 71.
Figure 71. Maximum contact force between vane and cam profile vs. Amplitude of sine curve at constant
displacement
53
Effect of number of vanes (k)
The number of crests and troughs (n) also changed with number of vanes (k) as per equation
n
k   1. The width of the vane was kept constant.
2
The output torque and the displacement of the motor increase with increase in number of vanes as
shown in Figure 72.
Figure 72. Average torque, displacement vs. number of vanes
For having a constant displacement of 200 cc, the width (B) of the motor was varied with the
increase in the number of vanes. (Figure 73)
Figure 73. Width of the motor vs. number of vanes for maintaining constant displacement
The range of fluctuation in torque and speed remains varies very little for more than three vanes
as shown in Figure 74.
54
Figure 74. Range of torque fluctuation, range of speed fluctuation vs. number of vanes (at constant displacement)
For 2 vanes the range of fluctuation is very high as there would be only two characteristic (torque
or speed) profiles, one from each vane, for superposition. As the number of vanes increases the
number of characteristic profiles to be superpositioned. This reduces the range of fluctuations. The
increase in number of crests and troughs also increases the contact force (FCX) as the vanes have a
steeper path along the cam profile. Therefore, the contribution of contact force towards torque, Td,
also increases, thus increasing the range of fluctuations. The increase and decrease in fluctuations
add up, hence keeping the change in range of fluctuations very low with increase in number of
vanes beyond three.
The maximum contact force (per unit width) between the vane and the cam profile increases with
increase in number of vanes as shown in Figure 75.
Figure 75. Maximum contact force between vane and cam profile vs. number of vanes at constant displacement
 
Effect of radius of the vane head rp
The width of the vane also varies with the radius of the vane head according to the equation(40).
The change in average torque and displacement with increase in vane head radius is negligible.
(Figure 76).
55
Figure 76. Average torque, displacement vs. vane head radius
At constant displacement of 200 cc, the range of fluctuation of torque and speed increased with
increase in vane head radius as shown in Figure 77.
Figure 77. Range of torque fluctuation, range of speed fluctuation vs. vane head radius (at constant displacement)
The maximum contact force (per unit width) between the vane and the cam profile increase with
increases in vane head radius as shown in Figure 78.
Figure 78. Maximum contact force between vane and cam profile vs. vane head radius at constant displacement
56
The range on fluctuation of torque and speed is higher in this cases as compared to the motor with
vanes on the rotor (4.4.4). For comparing the two motors, motor designs with identical cam profile
(equal value of RO for both cases) were studied. An amplitude of 6 mm was used as the range of
fluctuation is high at higher amplitude. The vane head radius and the vane width were maintained
same for the two cases. The torque per unit width (Nm/m) for one vane and for the complete motor
are shown in Figure 79.
The motor with vanes on stator exhibits higher torque per unit width. For one vane, the torque is
slightly higher (at all angular positions) for the motor with vanes on stator, as in this case the vanes
are on the outside of the cam profile. Therefore, the applied force (due to fluid pressure) is at a
farther distance from the centre.
For one vane
For complete motor
Figure 79. Comparison of torque per unit width for cam vane motor with vanes on rotor and vanes on stator
4.5.5
Ideal torque and speed
The geometric design parameters used are shown in Table 7. The vane is considered to have a
height of  3a  , where a is the amplitude of the sine curve, so that at the most extended position of
the vane, a height equal to one third of the height of the vane  a  is still inside the guide (in the
stator).
Table 7. Geometric design parameters
Number of crests and troughs n
Mean circle radius of sine profile RO
Stator profile radius RI
2  k  1
 0.045  a  m
 RO  a  m
The required ideal displacement of the motor can be obtained by changing the number of vanes,
amplitude of sine curve and/or width of the motor.
It is desirable to have the lowest possible amplitude of sine curve so as to have lowest possible
size of the motor. The amplitude of the sine curve was chosen such that with further decrease in
amplitude, the required width of the motor does not increase by a large amount. (Figure 69)
The number of vanes is chosen such that the torque and the speed fluctuations are low. (Figure 48)
A high vane head radius is desirable for having low contact stresses (Fischer-Cripps 2007).The
radius of vane head has negligible effect on average torque and displacement from the motor. The
radius is selected such that the range of fluctuation of torque and speed are less while keeping the
radius high.
57
The selected parameters are shown in Table 6.
Table 8. Values of design parameters.
Number of vanes k
Amplitude of sine curve a
Radius of piston head rp
5
4 mm = 4 103 m
7 mm = 7 103 m
The width of the motor ( B ), required to have a displacement of 200 cc, was calculated to be 0.0318
m (31.8 mm).
The individual torque profiles of each vane and the superimposed total torque is shown in Figure
80, a and b respectively.
a
b
Figure 80. a) Individual torque profiles for each vane; b) total torque
The average torque is 379 Nm.
The individual change in volume profiles related to each vane (for every small rotation of 0.03
degrees) and the instantaneous speed of the motor is shown in Figure 81, a and b respectively.
a
b
Figure 81. a) Volume of fluid added to pressurised chamber related to each vane; b) instantaneous speed of the
motor
The average speed is 300 RPM.
58
4.5.6 Bearing load
The bearing loads are calculated in the same way as shown in 4.4.6.
The equations used for the calculation are as follows, (refer to Figure 66 and Figure 82)
FX   FCX   P1  P2  l  z   B

 Wp

 Wp

FY   FCY  P1 
 rp sin( )   P2 
 rp sin( )   B
 2

 2


Rx  RX sin   RY cos 
Ry  ( RY sin   RX cos  )
(66)
(67)
Figure 82. Reaction forces due to loads on one vane
The net resultant radial load for a rotation of 0 
4
with 120 bar fluid pressure is shown in
n
Figure 83.
Figure 83. Radial load on the bearings
Along with the alternating amplitude, the direction of radial load also oscillates about the centre.
The radial loads are taken up by two angular contact roller bearings placed on both sides of the
59
rotor. For CAD purposes, SKF 75x115x25 angular contact roller bearings (tapered roller bearings)
were used which have a static load capacity of 163 kN.
4.5.7 Valve mechanism and assembly
As the vanes do not rotate with the rotor, the valve ports supplying hydraulic fluid to the chambers
(high pressure on one side of the vane and return pressure on other side) could be fixed relative to
the stator.
Two different valve mechanisms were developed for this design,

Single side valve

Double side valve
Single side valve
As the name suggests, the valve mechanism lies on one side of the chambers. The working is
similar to the valve mechanism explained in 4.4.7, except for the distributor which is separate and
not integrated with the rotor. The distributor is held fixed relative to the valve plate by the use of
locking wings. The internal channels in the motor back housing and valve plate supply hydraulic
fluid to two circular cavities between the distributor and the valve plate. The cavities lead to
individual chamber ports that supply hydraulic fluid on either side of the vane. The valve
mechanism is shown in Figure 84. The high pressure is represented by red colour and the return
pressure is represented by blue colour. The ports can be swapped, i.e. high pressure in blue and
return pressure in red for a reversed rotation direction.
Figure 84. Single side valve mechanism
For supplying high pressure fluid to the back side of the vanes, selector valve mechanism similar
to the one explained in 4.4.7 was integrated into the valve plate.
The exploded view of the assembly and the part list are given in Figure 85. The section views are
shown in Figure 86.
60
Figure 85. Exploded vied and part list for assembly with single side valve mechanism
Figure 86. Section views of the assembly
The assembly has an axial length of 135 mm and a side-to-side distance of 150 mm.
Double side valve
As the name suggests, the valve mechanism constitutes of valve plates on both sides of the
chambers.
61
One of the main port on the back housing of the motor supplies fluid to the cavity enclosed by the
surface of the back housing and the back valve plate, through channel in the back housing. The
back valve plate has chamber ports which supplies fluid from the cavity to the chambers on one
side of the vanes.
Channel from the other main port runs through the back housing, back valve plate, stator body and
the front valve plate to supply fluid to the cavity enclosed by the front valve plate and the front
housing. The front valve plate also has chamber ports which supply fluid to the other side of the
vanes. The valve mechanism is shown in Figure 87. The high pressure is represented by red colour
and the return pressure is represented by blue colour. The ports can be swapped, i.e. high pressure
in blue and return pressure in red for a reversed rotation direction.
Figure 87. Double side valve mechanism
For supplying high pressure fluid to the back side of the vanes, selector valve mechanism similar
to the one explained in 4.4.7 was integrated into the back valve plate.
The exploded view of the assembly and the part list are given in Figure 88. The section views are
shown in Figure 89.
Figure 88. Exploded vied and part list for assembly with double side valve mechanism
62
Figure 89. Section views of the assembly
The assembly has an axial length of 145 mm and a side-to-side distance of 150 mm.
4.6 Different number of vanes
In the above explained concepts, the number of vanes ( k ) is one less vane than the number of
cams ( n / 2 ). The possibility of having different number of vanes has been studied below.
4.6.1 More number of vanes than the number of cams
Figure 90 shows an illustration of cam vane motor (vanes on stator) having more number of vanes
( k  7 ) than the number of cams ( n / 2  6 ). Such a design would not be possible as high pressure
fluid and return pressure fluid would be supplied to the same chamber at certain positions as shown
in the figure.
63
Figure 90. More number of vanes than the number of cams
The same problem would occur if the number of vanes is equal to the number of cams as shown
in Figure 91.
Figure 91. Equal number of cams and vanes
Therefore the number of vanes must be less than the number of cams.
4.6.2 Symmetric motor design
A symmetric motor design can be obtained by having an even number of cams and vanes while
having the number of vanes less than the number of cams. In Figure 92, an illustration of a cam
vane motor (vanes on stator) having a symmetric design ( k  4 and n / 2  6 ) is shown. This is
viable as the high pressure and low pressure chambers are properly separated.
64
Figure 92. Symmetric motor design
Figure 93 shown the comparison of torque for motor with five vanes (concept designed) and four
vanes keeping all other geometric parameters same while only varying the width to make the
displacement equal.
B = 32 mm
B = 40 mm
Torque with developed concept having 5 vanes
Torque with motor having 4 vanes
Figure 93. Comparison of torque for motor with 5 vanes and 4 vanes
65
The motor with four vanes has very high amplitude of torque fluctuation as compared to the motor
with five vanes. This is because in case of a symmetric motor, the opposite vanes would have same
torque characteristics and the number of torque profiles to be superimposed would be only half the
number of vanes.
The benefit of having a symmetric motor would be that the bearing loads are theoretically zero.
Therefore, bearings would not be required for a symmetric motor.
4.6.3 Doubling the vanes and cams
If the number of vanes and the cams of the designed motor (5 vanes and 6 cams) is doubled, a
symmetric motor is obtained. (Figure 94)
Figure 94. Motor with double number of vanes and cams
This symmetric motor was compared to the original motor design (vanes on stator). The vane head
radius ( rp ) for both cases was changed to 2 mm ( 2 103 m ) for accommodating 10 vanes.
The sub-cycle for the motor with 10 vanes will be 30o whereas it is 60o for the motor with 5 vanes.
Therefore, for comparison only the first half of the sub-cycle was used for motor with 5 vanes.
In the motor with 10 vanes, the vanes symmetrically opposite to each other would have the same
torque characteristics. Therefore, the torque profiles of the opposite vanes were added up for
comparison with the motor having 5 vanes. The width of the motors was adjusted for having the
same displacement of 200cc. The comparison of torque characteristics is shown in Figure 95.
The motor with 10 vanes, shows high range of torque fluctuation. It also shows higher individual
torque characteristics (sum of torque characteristics of opposite vanes). The bearing load was
calculated to be zero for this motor.
66
B = 16 mm
B = 32 mm
Double vanes  10 vanes
Developed concept  5 vanes
Figure 95. Comparison of motor having 5 vanes and 6 cams with motor having 10 vanes and 12 cams (double)
The high range of fluctuation is expected to be due to steeper accent in case of the motor with 10
vanes. Another comparison was performed by halving the amplitude of sine curve for the motor
with 10 vanes while keeping the radius of mean circle equal to that of motor with 4 vanes. The
width of the motor remained almost equal for having a displacement of 200 cc. The comparison is
shown in Figure 96.
The range of fluctuation reduces but still remains higher than the fluctuation range for motor with
5 vanes.
67
a = 2 mm
B = 32 mm
a = 4 mm
B = 32 mm
Double vanes  10 vanes
Developed concept  5 vanes
Figure 96. Comparison of motor having 5 vanes and 6 cams with motor having 10 vanes and 12 cams (double) and
halved amplitude of sine curve
68
STRAIN WAVE GEAR HYDRAULIC MOTOR
CONCEPT
In this chapter the design of the strain wave gear hydraulic motor, which is one of the selected
concepts, is presented.
5.1 Working Principle
The energy transfer in this concept of motor is similar to that of a harmonic drive, which allows
high reduction ratio ranging from 50:1 to 320:1. Power transmission through multiple teeth
engagement results in high output torque capacity. (Slatter and Slatter 2005)
Similar to the harmonic gear, the motor comprises of three main parts,
1.
Circular gear: a rigid steel gear with internal teeth (fixed)
2.
Flexgear: a flexible gear with external teeth.
3.
Wave generator: an elliptical plug serving as a high efficiency torque converter
In a harmonic drive the input rotation is applied to the wave generator. The zone of tooth
engagement travels with the major elliptical axis. The flexgear, has lesser number of teeth than the
circular gear. Therefore, every time the wave generator rotates by one revolution, the flexgear and
the circular gear shift by the difference in the number of teeth. The rotation of the flexgear is in an
opposite direction to that of the wave generator.
In the hydraulic motor concept, high pressure hydraulic fluid acts between the meshing teeth on
one side of the major axis of the ellipse. This creates a rotation torque on the wave generator. A
reduced rotation is obtained on the flexgear by the same mechanism as for harmonic drive. (Figure
97)
Figure 97. Strain wave gear hydraulic motor mechanism
High pressure fluid is supplied to few meshing teeth on one side of the major axis. Therefore, the
valve ports rotate with the major axis.
69
5.2 Minimum number of teeth on flexgear
As the flexgear bends periodically under the action of the wave generator, and is in a working
condition of alternating stress, failure is thus mainly caused by fatigue fracture of the flexgear.
Zhu, et al. (2013) present a method to calculate the minimum number of teeth on flexspline for a
harmonic drive with external wave generator by analyzing the radial deformation of the mid-layer
and section stress of the flexspline under the action of wave generator based on the research model
of a smooth cylindrical shell. The same method has been modified to obtain the minimum number
of teeth on flexspline with internal wave generator based on fatigue failure. The method is
described below.
The module of a gear is given as,
m
D
Z
(68)
Where D is the pitch circle diameter and Z is the number of teeth on the gear.
For gears with module, m, and the difference in the number of teeth between circular gear and flex
gear, Z , the deflection  is given by,

DC  D f
2

m  ZC  Z f 
2

mZ
2
(69)
Where DC and D f are pitch circle diameters of circular gear and flexgear respectively and Z C
and Z f are the number of teeth on circular gear and flexgear respectively.
This is the deflection required to make the length of the major axis of the flex ellipse equal to the
pitch circle diameter of the circular gear.
The flex gear is approximated to be a ring with outer radius ( ro ) equal to the root circle radius
without considering the gears. The thickness ( t ) of the ring would be equal to the thickness of the
rim of the gear.
The flexible ring being acted upon by force F can be represented by a quarter section as shown
in Figure 98. The figure also shows the force components and the moment acting on a small section
of the ring ( ds ). Fr is the force component in radial direction, F is the force component in
tangential direction and M is the bending moment.
Figure 98. Representation of loaded circular ring by a quarter of the ring
70
By solving the free body diagram for small ring section ( ds ),
F 
F
F
Fr
cos( ); Fr  sin( ); M  M O 
1  cos( ) 
2
2
2
(70)
The strain energy in the small section is composed of strain energy due to axial load, shear load
and bending load. The strain energy due to different loading is given below, (Beer, et al. 2012)
P2 L
2 AE
T 2L
Strain energy in torsion: U 
2GJ
L
M2
Strain energy in bending: U  
dx
2 EI
0
Strain energy in tension: U 
(71)
The strain energy for a small section of curved beams is given as follows (Alexion 2011),
F 2 Rd
 due to normal load
2 AE
CF 2 Rd
dU 2  r
 due to shear load
2 AG
M 2 Rd
dU 3 
 due to bending moment
2 EI
F 2 Rd CFr 2 Rd M 2 Rd
dU  dU1  dU 2  dU 3 


2 AE
2 AG
2 EI
dU1 
(72)
Where C  1.5 is the correction factor for a rectangular cross-section in shear.
The total strain energy for the quarter ring would be,

2
U 
0


F Rd
CF Rd 2 M 2 Rd
 r

2 AE
2 AG
2 EI
0
0
2
2
2
(73)
According to Castigliano’s theorm (Patnaik and Hopkins 2003), the displacement corresponding
to any force ( P ) applied to an elastic structure and collinear with that force is equal to the partial
U
derivative of the total energy with respect to that force, i.e.
.
P
As the section at K does not rotate, the rotation corresponding to bending moment M O must be
zero, i.e.,
71
U
0
M O


 2 2

2
2
CFr Rd 2 M 2 Rd 
  F Rd



0
M O  0 2 AE
2 AG
2 EI 
0
0


(74)


  2 M 2 Rd 

0
M O  0 2 EI 




R 2
M
M
d  0

EI 0 M O

M
R
1,
As
is a non-zero constant and M could be non-zero, and
M O
EI
zero.


2
2

 Md    M
0
O

0
 M O

2
0

 Md must be equal to
0
FR
1  cos( )  d  0
2

FR 

2 

2
0


 sin( ) 02   0

FR  

  1  0
2 2 2 
FR  
2
 MO 
  1
2  2 
 MO

2

(75)
1 1 
 M O  FR   
2  
M  MO 
FR
1 1
FR
1  cos( )   FR     1  cos( ) 
2
2   2
(76)
FR 
2
M 
 cos( )  
2 

The bending moment at J and K are,
M
 
J   
 2

M K  0
FR  2 
FR
   
2  

(77)
FR  2 

1  
2  
A positive bending moment acts to straighten the beam, i.e. increasing the radius of curvature,
whereas a negative bending moment would act to decrease the radius of curvature.
According to Castigliano’s theorm, deflection due to force F would be equal to
72
U
.
F


 2 2

2
2
CFr Rd 2 M 2 Rd 
U
  F Rd




F F  0 2 AE
2 AG
2 EI 
0
0



2
  
0

F R  F

AE  F
(78)

CFr R  Fr


 d  
AG  F

0
2
MR  M


d  
EI  F

0
2

 d

The partial derivative of F , Fr and M with respect to F are as follows,
F cos  Fr sin  M R 
2

;

;
  cos   
F
2 F
2 F 2 

(79)
Substituting the partial derivatives and the expressions for F , Fr and M from equations (70), (76)
and (79) into equation (78),

2


2
2
FR cos 2 
CFR sin 2 
FR 3 
2
 
d  
d  
 cos    d
4 AE
4 AG
4 EI 

0
0
0
2

2

2

3 2
2
FR
CFR
FR 
2
 
cos 2  d 
sin 2  d 
 cos    d



4 AE 0
4 AG 0
4 EI 0 






 
FR  sin 2  2 CFR  sin 2  2 FR 3   sin 2  2 4
4

 




 2  02  sin  02  (80)
 





4 AE  2
4  0 4 AG  2
4  0 4 EI   2
4 0 



FR    CFR    FR 3   2 4 
 


 
4 AE  4  4 AG  4  4 EI  4   

FR  
C R 2   6  

   
4  4 AE 4 AG EI  4   
Solving equations (80) and (69) to obtain an expression for F, which is the point force required to
mate the flexgear to the circular gear for a given module and difference of number of teeth.
F
mZ
R 
C
R 2   6 


  
2  4 AE 4 AG EI  4   
(81)
Stress equations for stress at inner fibre and outer fibre of a curved beam according to WinklerBach formula (Srinath 2010) are as follows,
i 
Mci
Mco
; o 
Aeri
Aero
(82)
ri and ro are radius of inner and outer fibre respectively. ci and co are distance from neutral axis
to inner fibre and outer fibre respectively. e is the distance from centroidal axis to neutral axis
(Figure 99). The radius of neutral axis ( r ) and the radius of centroidal axis ( rc ) for a curved beam
with a rectangular cross-section is given as follows,
73
r
t
r
log  o
 ri
; rc  ri 
t
2



e  rc  r ; co  rc  ro ; ci  rc  ri
(83)
Figure 99. Section properties of rectangular curved beam
The stresses at inner and outer fibres of section J are only due to bending. At section K the axial
stress adds on to the bending stresses. A multiplication factor for stress concentration in the gear’s
root ( K a ) is used with the stress on outer fibre. The stresses on inner and outer fibres at sections J
and K are given as follows,
 Ji 
 Ki
M Aci
M c
;  Jo  K a A o
Aeri
Aero
 F M B co 
F M B ci


;  Ko  K a 


2 A Aeri
 2 A Aero 
(84)
The ring experiences fluctuating stress where inner fibre has extreme stresses of  Ji and  Ki , and
outer fibre has extreme stresses of  Jo and  Ko . The mean stress (  mean ) and the stress amplitude
(  amp ) for the inner and outer fibres are as follows,
 Ji   Ki
   Ki
;  i _ amp  Ji
2
2
   Ko
   Ko
 Jo
;  o _ amp  Jo
2
2
 i _ mean 
 o _ mean
(85)
The Soderberg line criteria (Bhandari 2010) was used as the fatigue failure criteria as it is the most
conservative of all the failure criterion as shown in Figure 100.
74
Figure 100. Fatigue failure diagram (Bhandari 2010)
According to Soderberg line criteria, the design would be safe if the following condition is
satisfied,
 mean
S yt

 amp
Se
1
(86)
Implementing a safety factor, f s ,

 amp 
f s  mean 
  1
 S
S
yt
e


(87)
Where S yt is yield strength and S e is the endurance limit of the material.
Solving expressions (81),(84),(85) and (87), for module ( m ), the maximum permissible module
for the flexgear according to fatigue failure criteria is obtained, The maximum module further
gives the minimum number of teeth on the flexgear.
The material properties and the dimensions used for calculation are listed in Table 9.
Table 9. Material properties and ring dimension.
Material
Bending fatigue limit , S e
Elastic modulus , E
Difference in number of teeth,
Z
Ring outer radius, ro
Thickness of ring, t
Gear width (for cross-section
properties)
Stress concentration at gear root,
Ka
Safety factor, f s
Steel 30CrMnSiA
quenched and nitrided
625 MPa =
Example material in Zhu, et al.
(2013)
6.25 108 Pa
2.1105 MPa =
2.11011 Pa
2
58 mm = 58 103 m
5 mm = 5 103 m
10 mm = 10 103 m
2
Satisfying geometry conditions
Satisfying geometry conditions
Assumed
Zhu, et al. (2013)
1.5
75
The maximum permissible module was calculated to be 2.8 mm ( 2.8 103 m ).
5.3 Mathematical model
5.3.1 Gear profiles
A simple involute gear profile was considered. The dedendum diameter of the flexgear (external
gear teeth) and the addendum diameter of the circular gear (internal gear teeth) were considered to
be equal to the respective base circle diameters. Therefore, the gear was composed of only involute
part and there was no non-involute portion of the gear (usually between base circle and the
deddendum of external gear teeth).
The involute curve is modelled based on the tooth thickness tt at any radius rt of an involute gear
(Shigley and Uicker 1986),
tt  2rt
t pt
2rpt
  tan  p   p    tan    
(88)
db
d
 o , where d o is the addendum diameter for external tooth or dedendum diameter for
2
2
internal tooth. d b is the base circle diameter.
ri 
r 
Where   cos 1  b  ,  p is the pressure angle of the gear.
 rt 
D
D
are tt and rt at pitch circle respectively, and D and Z are pitch diameter
2Z
2
and number of teeth respectively.
t pt 
and rpt 
The half tooth profile was completed with base circle and addendum/deddendum circles. (Figure
101).
Figure 101. Parts of simplified gear tooth profile
The profile was mirrored and then patterned about the centre to obtain a quarter gear. (Figure 102).
76
Figure 102. Single tooth profile and quarter gear
The complete quarter gear profile is obtained in form of coordinate vectors.
The design parameters were used for generating the gear profiles are shown in Table 10.
Table 10. Gear profiles design parameters.
Difference in number of teeth,
Z
2
Module , m
2.5 mm = 2.5 103 m
Pressure angle ,  p
14.5̊
Less than maximum possible
module
One of the pressure angles
commercially used for gears
Flex gear
Circular pitch diameter, D f
Number of teeth, Z f
Addendum circle diameter, dof
Base circle diameter, dbf
Root diameter, d rf
Thickness of ring, t
Number of teeth, Z C
Circular pitch diameter, DC
Root diameter, d rC
130 mm = 130 103 m
Assumed: Based on driver size
and approximate size of wave
generator
Df
= 52
m
D f  1.4m = 133.5 mm
D f cos  p = 125.86
Small addendum to have some
clearance between addendum of
flexgear and root of circular gear
Gear tooth parameter
mm = 125.86 103 m
dbf = 125.86 103 m
Only involute profile
= 133.5 103 m
5 mm = 5 103 m
Circular gear
Z f  Z = 54
Assumed
ZC m = 135 mm =
135 103 m
DC  2m = 140 mm =
140 103 m
Deeper root to have some
clearance between addendum of
flexgear and root of circular gear
77
Base circle diameter, dbC
Addendum circle diameter, doC
DC cos  p = 130.7 mm
Gear tooth parameter
= 130.7 103 m
dbC = 130.7 103 m
Only involute profile
Single tooth profile for circular gear and undeformed flexgear are shown in Figure 103.
Figure 103. Single tooth for circular gear and undeformed flexgear
The circular gear profile of the flexgear was converted into an approximate elliptical gear profile.
This conversion was performed by moving each point on the circular profile to a corresponding
location on the elliptical profile. The procedure is described below. (Figure 104)
To match the pitch circles of the circular gear with the pitch ellipse of the flex gear, the pitch circle
of flexgear must be deflected by  ,

mZ
2
(89)
The mid-section circle of the flex gear must also be deflected by approximately  . The diameter
of mid-section of the circular profile,
dC  dbf  t
(90)
The perimeter of the mid-section of the flexgear is,
PC   dC
(91)
When the mid-section circle is converted to an ellipse, the major axis ( ae ) would be,
ae 
dC

2
(92)
The perimeter of an ellipse is approximated by Maclaurin series as follows (Chandrupatla and
Osler 2010),
78
3
5 6
 1

Pe  2ae 1  ee 2  ee 4 
ee ........
64
256
 4

Where ee  1 
(93)
be 2
, and be is the minor axis of the ellipse.
ae 2
Equating the circular perimeter with the elliptical perimeter, i.e. equating equations (91) and (93),
PC  Pe


1
4
 dC  2ae 1  ee 2 
3 4
5 6

ee 
ee ........
64
256

(94)
The minor axis of the ellipse ( be ) can be obtained by solving equation (94) for ee ,
be  ae 2 1  ee 2 
(95)
Figure 104. Moving of coordinate points on circular gear profile to elliptical gear profile



Let the coordinates of two adjacent points of the circular gear profile be X j , Y j and X j 1 , Y j 1

These points subtend an angle  j and  j 1 at the centre respectively.
 Yj
X
 j

 Y j 1 
1
 ;  j 1  tan 


 X j 1 
 C   j   j 1
 j  tan 1 
The corresponding points on the mid-section circle are
X
jC
(96)
, Y jC  and  X (j1)C , Y(j1)C  .
79
The length of arc bounded by these two points on the mid-section circle would be,
pc 
 dC
d 
C  C C
2
2
(97)
The length of arc of an ellipse ( p ' ) that subtends an angle  e from the minor axis is given by
(Banmote, Mahale and Gulhane 2003),
3
1
1 4
 1

1

p '  ae 1  ee 2  ee 4   e   ee 2  ee 2  sin  2  e  
ee sin  4  e 
64 
32 
256
 4
8
(98)
The length of arc bounded by two lines at angles  j and  j 1 is given as,
pe  p '     p '   
j
(99)
j 1
For a known value of  j , the value of  j 1 can be obtained by equating the circular length of arc
and the elliptical length of arc, i.e. equations (97) and (99),
pc  pe
d C C
 p '    p '  
e
j
e
j 1
2

(100)

For the angle  j 1 , the coordinates of point on ellipse X (j1) e , Y(j1) e corresponding to the point


on mid-section circle X (j1)C , Y(j1) C were calculated using the parametric expression of ellipse.
X (j1) e  be cos te ; Y(j1) e  ae sin te
b

te  tan 1  e tan  j 1 
 ae


(101)

The distance between the point on mid-section ellipse X (j1) e , Y(j1) e the corresponding point on


the mid-section circle X (j1)C , Y(j1) C , i.e. X C  X (j1) e  X (j1)C and YC  Y(j1) e  Y(j1)C , is added


to the point on the circular gear profile X j1 , Yj1 to obtain the approximate coordinates of the

corresponding point on the elliptical gear profile X j1 , Yj1

X j 1  X j 1  X C , Y j 1  Y j 1  YC
(102)
X j 1  X j 1  X (j1) e  X (j1)C , Y j 1  Y j 1  Y(j1) e  Y(j1)C
The shifting of points from circular gear profile to elliptical gear profile is started from the point
on the vertical axis, i.e. on the major axis after deformation. Therefore, the coordinates of the first
point would be X 1  0, Y1  Y1   as the point moves only vertically by a distance  . For this

point, 1  . The  2 calculated for the first pair of points (first and second points) would be  j
2
for the next pair of points (second and third points). Therefore, for successive pairs of points  j
would be a known parameter.
The coordinates of the first teeth were mirrored about vertical axis to obtain the complete tooth.
The gear profiles with undefomed flexgear and deformed flex gear are shown in Figure 105.
80
A
b
Figure 105. Quarter gear profiles with a) undeformed flexgear and b) deformed flex gear
5.3.2 Speed calculation
The high pressure fluid is supplied to the gaps between u meshing teeth as shown in Figure 106.
The area trapped between each meshing pair of teeth ( Ai ) was calculated using the inbuilt Matlab
function polyarea, (Support: Documentation: Polyarea n.d.) which returns the area of the polygon
specified by the vertices in the input vectors. The input vectors for each trapped volume were the
x and y coordinates of the points that enclose the volume.
Figure 106. Supply of high pressure fluid to pockets between meshing teeth
The trapped area for each chamber when volume trapped between five pairs of teeth is pressurized
(i.e. u  5 ) is shown in Figure 107.
81
Figure 107. Trapped area for each chamber
2
,
ZC
the major axis rotates, therefore, the volume trapped by one pair of mashing gears is increased to
the volume of subsequent pair of meshing gears.
When the wave generator rotates by an angle equivalent to one tooth on the rigid gear, i.e.
Before rotation
After rotation
Figure 108. Change in chamber volume for a small rotation
As shown in figure, the area trapped between gear tooth 2 and circular gear after rotation is equal
to the area trapped between gear 3 and circular gear before rotation. Therefore, the change in
2
trapped area for a rotation by an angle equivalent to one tooth on the rigid gear, i.e.
, would be
ZC
the difference in the trapped area of the adjacent trapped volumes.
 Ai 1k 1  Ai 1  Ai
(103)
As the same change in area takes place on diagonally opposite part of the gear as well, the total
change in area would be,
82
k 1
dA
0
2
ZC
 2  Ai
(104)
1
For obtaining a displacement of 200 cc/rev, i.e. 200 106 m3/rev, the width ( B ) was calculated
as follows,
B
200 106.rs
ZC .dA 2
0
(105)
ZC
Where rs is the speed ratio of a strain wave gear given as follows,
rs 
Z
Zf
(106)
Volume of fluid required for one revolution of wave generator,
dV  dA
0
2
ZC
.ZC .B
(107)
The speed of wave generator ( vwg ),
vwg 
Q
dV
(108)
The speed of flexgear ( v f ), which is the output speed of the motor, is calculated as follows,
v f  vwg rs
(109)
With the fluid supplied to 5 chambers, i.e. u  5 , the width of the motor ( B ), required to have a
displacement of 200 cc, was calculated to be 0.0095 m (9.5 mm).
5.3.3 Torque calculation
The fluid pressure acting on the teeth of flexgear applies a torque on the wave generator.
The torque from each pair of points on the elliptical flexgear profile is calculated. As the contact
between the flexgear teeth and the teeth of the circular gear is assumed to be a line contact, the
pressure is assumed to be acting on the complete length of both sides of the flexgear tooth. For the
last pressurized tooth, the contact point is assumed to be around half height of the tooth, therefore,
the pressure on top half of the contact side is considered. The pressure variation within the
chambers is also neglected, therefore, Pin  const and Pout  const .
Figure 109. Torque due to fluid pressure between two successive points on gear profile
83
Let A  X j , Y j  and B  X j 1 , Y j 1  be coordinates of two successive points on the profile as shown
in Figure 109. The length ( l j ) of the line AB is given as,
lj 
X
j 1
  Y
2
Xj
j 1
 Yj

2
(110)
The mid-point coordinates  X j m , Y j m  of the line AB would be,
Xjm 
X j  X j 1
2
Y j  Y j 1
;Yj m 
2
(111)
The slope of perpendicular ( m j ) to the line AB is given as,
mj  
X j  X j 1
Y j  Y j 1
(112)
The coordinates of the point C  X j k , Y j k  are,
X jk 
m j X j m  Yj m
1
mj 
mj
Yj k  
(113)
1
Xjk
mj
The perpendicular distance ( d j ) of the line perpendicular of AB passing through the mid-point
from the origin would be,
dj 
2
X j k  Yj k
2
(114)
The high pressure ( Pin ) acts on the similar region on two diametrically opposite locations and
return pressure ( Pout ) acts on two other diametrically opposite locations. The torque contribution
( T j ) from this small region on all four locations would be,
T j  2  Pin  Pout  l j d j B
(115)
Considering clockwise torque to be positive, the sense of torque contribution (positive or negative)
could be determined by values of X j k and X j k according to the Figure 110.
84
Xjk
Yj k
Tj
Positive
Positive
Positive
Negative
Positive
Negative
Positive
Negative
Positive
Negative Negative Negative
Figure 110. Sense of torque
The total torque on the wave generator would be,
Twg   T j
(116)
The torque on the flex gear, which is also the output torque from the motor, was calculated using
the speed ratio ( rs ) calculated in equation (106).
Tf 
Twg
rs
(117)
With the fluid supplied to 5 chambers, i.e. u  5 , the ideal torque from the motor was calculated to
be 375.5 Nm. The slight error in the torque valve (as compared to 379 Nm) is expected to be due
to the approximations considered.
5.4 Motor assembly design
5.4.1 Torque Transfer Mechanism
Two different torque transfer mechanisms for transfer of torque from flexgear to the output shaft
were considered (Figure 111):

Castellated coupling

Bolted flange coupling
A castellated coupling offers maximum compactness but have low positioning accuracy and
efficiency (Musser 1960). It would also have lot of friction as the lugs slide continuously inside
the grooves. Therefore, a bolted flange coupling was considered for further design.
85
Figure 111. Torque transfer mechanisms
Flexgear cup design
The minimum length of the cup of the flexgear was determined by approximating it to be a
mZ
cantilever beam with unit width undergoing an alternating deflection (  
) at the free end.
2
(Figure 112)
Figure 112. Cantilever beam approximation of the flexgear cup
The displacement and stress equations for a cantilever beam with rectangular cross-section,
FL3
3 EI
F 3
3EI
L
(118)
FL
(stress on outermost fibre)
W
(119)


Where W 
wt 2
wt 3
t2
t3
I

W

I

w

1
and
. As
(unit width),
and
.
12
6
6
12
Substituting W and I into equations (119) and (118)

FL 6 FL 3 Et
 2 
W
t
2 L2
The stress in the outer fibres would fluctuate from  (tensile) to  (compressive).
86
(120)
The mean stress and the alternating stress would be,
 mean  0; alt 
3 Et
L2
(121)
Solving the fatigue failure criteria expression(87), for length ( L ), the minimum permissible length
for the flexgear cup was obtained.
For the flexgear material properties as given in Table 9 and an assumed ring thickness ( tc ) of 2.5
mm ( 2.5 103 m ), the minimum required length of the cup was calculated to be 49 mm i.e.
49 103 m .
5.4.2 Valve Mechanism
As the chamber ports need to rotate with the major axis of the wave generator, the valve ports were
integrated with the wave generator. (Figure 113)
Figure 113. Wave generator design with integrated valve ports
The valve lies on only one side of the chambers. The two main ports (inlet and return) would be
on the back face of the motor which would allow easy connection to the rock drill fluid lines. The
internal channels in the motor back housing would supply fluid to two concentric circular cavities
between the valve part of the wave generator and the back housing. The chamber ports on the valve
part of the wave generator supply high pressure fluid and return pressure fluid to either side of the
major axis of the elliptical flexgear. The valve mechanism is shown in Figure 114.
Figure 114. Valve mechanism
87
5.4.3 Assembly
Several bearings would be required for the assembly. A smooth contact between the elliptical face
of the wave generator and the flexgear can be obtained by using thin section bearings, which are
flexible and their flexibility does not affect the performance, provided the bearings are properly
mounted and the races are uniformly supported (Burgess, Miller and VanLangevelde 2014).
Needle bearings would be required to avoid sliding contact between internal surface of the wave
generator and the external surface of the output shaft which runs throughout the length of the
motor. Angular contact roller bearings would be required on either end of the motor to support the
output shaft, which would further support the wave generator and the flex gear.
The ends of the motor would be sealed by dynamic seals allowing rotation of the output shaft,
while O-rings would be used for all the static seals.
The exploded view and the section view of the assembly are shown in Figure 115 and Figure 116.
Figure 115. Exploded view of the assembly
88
Figure 116. Section views of the assembly
The assembly has an axial length of 185 mm and a side-to-side distance of 170 mm.
89
DISCUSSION AND CONCLUSIONS
In this chapter a discussion about the concepts developed has been presented along with the
conclusions that can be drawn from the project.
6.1 Discussion
In the project two concepts have been developed for having the hydraulic motor around the shank
of a rock drill. Only functional design has been performed considering ideal conditions (100%
efficiency). The parts were considered to be massless and the contacting surfaces frictionless.
Below is the discussion about the two concepts developed.
6.1.1 Multi-cam vane motor concept
The concept has been developed using a sine curve for cam profile. The feasibility of other profiles
need to be studied.
The required displacement of the motor can be obtained by varying the following design
parameters:

Width of the motor

Number of vanes

Amplitude of the sine curve
Comparison between original concept and cam vane motor
The cam vane motor concept (4.4 and 4.5) has a very low fluctuation range whereas original multicam vane motor concept has extremely high range of fluctuations which would lead to a very jerky
motion and also stall the motor due to near zero torque at some locations.
One drawback of having the rotor and stator to seal chambers is that progressive wear would lead
to increase in leakage through the contacts between rotor and stator. In the original concept, the
vanes are the only elements that seal and are always pressed against the cam profile by high
pressure on the back of the vanes, therefore, the leakage would not increase over time irrespective
of wear of cam surface.
Cam vane motor concept
The cam vane motor with vanes on stator has less rotating parts and the valve mechanism is also
fixed (not rotating). Therefore, the design is simpler as compared to the version with vanes on the
rotor.
Both the versions of the motor (vanes on rotor and vanes on stator) show similar dependencies on
amplitude of sine curve, number of vanes and the size of vanes. The displacement increases with
increase in amplitude of sine curve or number of vanes. The range of fluctuations show an
increasing trend with increase in amplitude of sine curve or head radius of vanes, whereas it
decreases with increase in number of vanes.
For an equal width and identical cam profiles, the motor with vanes on the stator gives higher
torque with higher range of fluctuations as compared to the motor with vanes on rotor (Figure 79).
This is due to the vanes being placed outside the cam profile giving a longer torque arm for the
forces produced by hydraulic oil.
A motor with number of vanes more or equal to the number of cams is not possible because both
high pressure fluid and return fluid would be supplied to the same chamber at certain positions.
Therefore, the number of vanes must be less than the number of cams. Having one less vane than
91
the number of cams would give very small range of fluctuations while the radial loads are high.
These radial loads would be taken up by bearings on both sides of the motor unit. By having a
symmetric motor, the bearing loads can be reduced to zero. Therefore, no bearings would be
required. But this would be accompanied by high range of fluctuations.
Comparison between one sided vale and two-sided valve
If the complete valve mechanism is on one side of the chambers (Figure 84), the internal cavities
carrying high pressure oil and return pressure oil lie very close to each other. Therefore, the leakage
from the high pressure cavity to the return pressure cavity would be high, thus decreasing the
volumetric efficiency of the motor. Such a problem is not a concern when the valve plates are on
both sides of the chambers (Figure 87) as the internal cavities are located far apart (one on each
side of the chamber. A double sided valve would result in a longer motor assembly.
A double sided valve would be very complicated for the motor with vanes on the rotor, as this
motor requires the valve to rotate with the rotor.
6.1.2 Strain wave gear hydraulic motor concept
A rough design for the strain wave gear hydraulic motor concept has been presented in the previous
sections. The speed and torque calculations are not accurate due to the various approximations and
assumptions considered.
A detailed study needs to be performed to obtain the most suitable tooth profile for this application.
The wear on the teeth and the leakage of hydraulic fluid across tooth contacts must also be studied
and estimated.
The design would have various sources of fluid leakage, for e.g., from unmeshed teeth, between
the valve face and the gear. These leakages would further contribute to reduction of volumetric
efficiency of the motor. Therefore, the sources of leakages need to be identified and the amount of
leakage must also be quantified so that they can be minimized.
The forces on the bearings need to be calculated and the bearings must be selected accordingly.
6.2 Conclusions
It can be concluded that it is possible to have a rotation motor that rotates the shank directly.
Therefore, reducing the size of the rock drill. The cam vane motor concept (both versions) satisfies
the requirement specifications. The strain wave gear hydraulic motor concept is slightly larger than
the allowed radial size. This size can possibly be lowered by optimisation of gear profiles and
structural components.
Additional conclusions have been listed below,

A cam vane motor is better than the original concept (with only vanes sealing) in terms of
steady output.

The cam vane motor with vanes on the stator has a simpler design because the vanes do not
rotate with the rotor and the valve plate is also fixed.

The came vane motor with vanes on rotor would have smoother output than the motor with
vanes on the stator.

A symmetric motor would have negligible bearing loads whereas the range of fluctuations
would be higher than the non-symmetric motor of same size and displacement.

The strain wave gear hydraulic motor concept needs to be analysed and studied predicting
the actual behaviour.
Hydraulic leakages from the strain wave gear hydraulic motor might be very high.

92
FUTURE WORK
In this chapter recommendations for future work have been presented.
The hydraulic motor designs presented in this report are limited to ideal case, i.e. hydraulic losses
and mechanical losses are not considered. The design must be extended to include all the factors.
Recommendations for future work are listed below.
1.
Design and analysis:
a.
Study of different cam profiles other than sine curve for cam vane motor
b.
Study of different teeth and spline profiles for strain wave gear hydraulic motor
c.
Optimisation of design parameters
d.
Real case design considering hydraulic and mechanical losses
e.
Structural design optimization (strength and fatigue calculations), for the different
parts
f.
Proper selection of bearing and seals (dynamic seals and O-rings)
g.
Dynamic analysis using Hopsan NG
h.
For the project, the rotation mechanism has been assumed to be completely isolated
from the percussion mechanism. This might not be true in real case and some
component of the shock waves might be transmitted to the rotation motor. Therefore,
the rotation motor must be designed to resist shock waves
i.
The effect of pulsating torsional load must also be studied
2.
Prototyping and testing of the motor must be performed to validate the analytical design and
calculations
3.
Changes must be performed on the rock drill design for accommodating the rotation motor.
93
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96
APPENDIX A: INTEGRATION FOR AREA ENCLOSED
Compute the definite integral:

2

1

1
n 
A    a cos     RO  d   RI 2 d
2 

 2
 2
2

1 

n 
   a cos     RO   RI 2 d
2 


 2


Factor out constants,
2


1 

n 
   a cos     RO   RI 2 d
2  
2 


Integrate the sum term by term and factor out the constants,


2
R2
1 

n 
   a cos     RO  d  I  1d
2 
2 
2 

2


n
n
n 
For the integrand  a cos     RO  , substitute u   and du  d .
2
2
2 


n
n
This gives a new lower bound u   and upper bound u   :
2
2

n
2

2
RI 2
1
a
cos
u

R
du

1d
   O
n n
2 
2
Expanding the integrand  a cos  u   RO  ,
2
n
2

RI 2
1
2
2
2
   a cos  u   2aRO cos  u   RO  du 
1d
n n
2 
2
Integrate the sum term by term and factor out constants
n
2 2
a

n
2aR
n cos  u  du  n O
2
2
n
2
n
2 2
2
2
R
n cos  u du  nO

RI 2
 1du  2  1d
n
1
1
Write cos 2  u  as cos  2u   :
2
2
n
2 2
a

n
2aR
1
1
n  2 cos  2u   2  du  n O
2
n
2
n
2 2
2
2
R
n cos  u du  nO

RI 2
1
du

1d

2 
n
97
The antiderivative of
1
1 u 1
cos  2u   is  sin  2u   constant :
2
2 2 4
u 1

a 2   sin  2u  
2 4

 
n
n
2
2aRO

n
u
n
2
n
2
n
2 2
2
2
R
n cos  u du  nO

RI 2
1
du

1d

2 
n
Evaluate the antiderivative at the limits and subtract,
u 1

a 2   sin  2u  
2 4

n
n
2
 n 1  2n
a 2   sin 
4 4  2
 
n
  2  n 1  2n
  a   sin 
4 4  2

 
n



a 2  n(  )  sin  n   sin  n  

4n
n
u
2
a  n(  )  sin  n   sin  n   2aRO

4n
n
2
n
2
n
2 2
2
2
R
n cos  u du  nO

RI 2
1
du

1d

2 
n
Apply the fundamental theorem of calculus,
The antiderivative of cos  u  is sin  u  :
n
a 2  n(  )  sin  n   sin  n   2aRO cos  u  2
R 2


 O
4n
n
n
n
u
2
n
2
 1du 
n
2

RI 2
1d
2 
Evaluate the antiderivative at the limits and subtract
n
2
2aRO sin  u 
n
n
u
 n
2aRO sin 
 2

n

 n  2aR  sin  n   sin  n  
 


O
 2aRO sin 

 2 

 2 
  2 
n
n
2
a 2  n(  )  sin  n   sin  n  
4n
  n 
 n
2aRO  sin    sin 
 2
  2 

n



R
 O
n
n
2 2

RI 2
1
du

1d

2 
n
2
Apply the fundamental theorem of calculus,
The antiderivative of 1 is u :
  n 
 n
2aRO  sin    sin 
a  n(  )  sin  n   sin  n  
 2
  2 


4n
n
2



n
R 2u 2
R 2
 O
 I
n u  n
2
2
Evaluate the antiderivative at the limits and subtract,
RO 2u
n
98
n
2
u
n
2
R 2
 I
2

 
RO 2     RI 2    


2
2

 
  n 
 n
2aRO  sin    sin 
a  n(  )  sin  n   sin  n  
 2
  2 

4n
n
2




RO 2    
 n
n      a 2  2 RI 2  2 RO 2   a 2  sin(n  )  sin(n )   8aRO sin 
 2

4n
2

RI 2    
2

 n 
  8aRO sin 


 2 
Using similar approach,


2
1 2
1

n 
 2 RI d   2  a cos  2    RO  d
 n
n      a 2  2 RI 2  2 RO 2   a 2  sin(n )  sin(n  )   8aRO sin 
 2

4n


 n 
  8aRO sin 


 2 

2
2
1

1 2
n 

a
sin


R
d


RI d

 O
 2 

2 

1 2
 n 
 n 
n      a 2  2 RI 2  2 RO 2   a 2  sin(n )  sin(n  )   8aRO cos 
  8aRO cos  
 2 
 2 

4n

2

1

1
n 
Aa    a cos     RO  d   RI 2 d
2 

 2
 2
 n
n      a 2  2 RI 2  2 RO 2   a 2  sin(n  )  sin(n )   8aRO sin 
 2

4n

 n 
  8aRO sin 


 2 
99
APPENDIX B: CALCULATION OF AREA
COMPONENTS AB AND AC
Calculation of Ab
For vanes on rotor:
W 
 m  sin 1  p 
 2r 
 p
 Wp 

 2 RI 
  sin 1 
Ab is the area occupied by the vanes. The three constituents of Ab are Ab1 , Ab 2 and Ab3 .
Ab  Ab1  Ab 2  Ab3
Ab1 and Ab3 are constant for every angular position of the vanes whereas Ab 2 changes with angular
position of the vane.
Calculation of Ab1 :
Ab1  area of sector FDE + area FCD
 rp 2
rp 2 m
area of sector FDE 
m 
2
2
Wp rp cos( m )
1  Wp 
area FCD  
  rp cos( m )  
2 2 
4
Ab1 
rp 2 m
2

W p rp cos( m )
4
101
Calculation of Ab 2 :
Ab 2  area GBCF
Ab 2 
Wp
2
 RR  RI 
Calculation of Ab 3 :
Ab 2  area ABGH - area HAG
 Wp 
area ABGH  
  RI 1  cos( )  
 2 
area HAG  area of sector OAG - area OAH
area of sector OAG 
area OAH 
area HAG 
 RI 2
R 2
 I
2
2
W
1
 RI cos( )   P 
2
 2 
RI 2 1
W 
-  RI cos( )   P 
2
2
 2 
 W p 
  RI 2 1
 W 
Ab 2  
R
1

cos(

)
-  RI cos( )   P  




 - 
 I
2
 2 
 2 
  2
Ab 2 
RI 2
 2  cos( )  
4
2
Wp RI
For vanes on stator:
Ab  Ab1  Ab 2  Ab3
For 2 vanes bounding the chamber:
102
Ab  Ab1  Ab 2  Ab3
 rp 2 m W p rp cos( m ) 
Ab1  2 


4
 2

Wp
 RR1  RI    RR 2  RI  
2 
W p RI
R 2 
Ab 3  2 
 2  cos( )   I 
2 
 4
Ab 2 
Here, RR1 and RR 2 are RR values for  1 and  2 respectively.
Calculation of AC
AC  area ABC  area OABC  area OACO = AC _1  AC _ 2
103
AC _1 is the area swept by the profile curve r   (sine curve) between angle  and  in polar
coordinates.

1
AC _1   r ( ) 2 d
 2
The area is calculated in the same way as shown in Appendix 1, neglecting the RI component.
AC _ 2  area of sector DAC + area ODC
 rp 2
rp 2
area of sector DAC 

2
2
area FCD 
R R sin(   )
1
 RC  RR sin(   )   C R
2
2
AC _ 2 
104
rp 2
2

RC RR sin(   )
2