Note: Descriptions are shown in the official language in which they were submitted.
CA 02287676 1999-10-27
AN IMPROVED GROOVE DESIGN FOR WIRE ROPE DRUMS AND SHEAVES
BACKGROUND OF THE INVENTION
This invention relates to drums and sheaves used to support wire rope in wire
rope
systems for lifting heavy loads, such as used in electric mining shovels and
walking
draglines.
It is important that wire rope be properly supported when it is wound up on a
drum if it is not properly supported the wire rope will become mashed. This
will result in
the breaking of the wires in the central area of the wire rope. Breaking of
the wires in the
central area of the wire rope can lead to premature failure of the rope
because it is not
possible to visually inspect rope internal damage. This is an especially
dangerous form of
wire rope failure. When the wire rope is supported on a drum a spiral groove
is placed in
the outside periphery of the drum. In the prior art, the shape of the groove
was in the
form of a half radius of a circle. It was important for the height of the side
of the groove
to be sufficiently great in order to properly support the wire rope.
SUMMARY OF THE INVENTION
This invention provides a device adapted to support a wire rope of known
nominal
diameter in a system where the fleet angle of rope departure from the device
is other than
perpendicular to the device, the device comprising a cylindrical body, and a
groove
around the cylindrical body, the groove having a contour adapted to support
the wire
rope, the groove contour being such that the wire rope is adapted to be
supported through
a rope support angle somewhere between 100 degrees and 160 degrees, the rope
support
angle being the included angle from the nominal rope's center to each end of
the
CA 02287676 1999-10-27
supporting groove contour on each side of the groove, the improvement
comprising the
groove contour being parabolic in shape over at least about the upper half of
the groove
contour.
This invention also provides such a device such that the improvement comprises
the groove contour having a radial length from the nominal rope center to the
contour
greater than at the bottom of the groove over at least about the upper half of
the groove
contour.
This invention also provides a method of making such a device, the method
comprising designing the groove contour such that as the wire rope leaves the
groove
there is a single line of contact between the rope and the respective side of
the groove.
BRIEF DESCRIPTION OF THE DRAWINGS
Figure 1 is an illustration of a drum with generic drum nomenclature.
Figure 2 is an illustration of a drum with generic sheave nomenclature.
Figure 3 is an illustration of a rope wound on a drum showing rope to drum and
rope to rope scrubbing.
Figure 4 is an illustration showing the relationship between groove pitch,
groove
angle and drum tread diameter.
Figure 5 is an illustration of a drum and head sheave arrangement showing the
relationship between the fleet angle and the groove angle.
Figure 6 illustrates a groove and the rope helix geometry.
Figure 7 is a graphical view of the rope groove contour in the "XZ" plane.
Figure 8 is a graphical view of the rope geometry as it leaves the drum.
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Figure 9 is a graphical view of the ellipse shown in Figure 8.
Figure 10 is a graphical view of the rope in the "YZ" plane
Figure 11 is a plot of the solution to equation 3.30
Figure 12 is a plot of the solution to equation 3.31.
Figure 13 is an illustration of a drum design using existing methods.
Figure 14 is an illustration of a sheave design using existing methods.
Figure 15 is an illustration of a groove contour of this invention created
using an
engineering graphical solution program.
Figure 16 is an illustration of a final drum design using the method of this
invention.
Figure 17 is an illustration of a sheave contour of this invention
superimposed on
the standard sheave contour.
Figure 18 is an illustration of a groove contour of this invention compared to
a
standard groove designs.
Before one embodiment of the invention is explained in detail, it is to be
understood that the invention is not limited in its application to the details
of the
construction and the arrangements of components set forth in the following
description or
illustrated in the drawings. The invention is capable of other embodiments and
of being
practiced or being carried out in various ways. Also, it is to be understood
that the
phraseology and terminology used herein is for the purpose of description and
should not
be regarded as limiting. Use of "including" and "comprising" and variations
thereof as
used herein is meant to encompass the items listed thereafter and equivalents
thereof as
CA 02287676 1999-10-27
well as additional items. Use of "consisting of and variations thereof as used
herein is
meant to encompass only the items listed thereafter and equivalents thereof.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
The surface mining industry utilizes large equipment to extract the Earth's
many
raw materials: e.g. gold, copper, coal and phosphate. Electric mining shovels
and
walking draglines are the two main types of machines that are used to move or
uncover
these raw materials. Both of these types of excavating equipment utilize large
quantities
of large diameter wire rope.
CA 02287676 1999-10-27
The most common means of driving a rope to do work is through the use of a
drum. One end of the wire rope is attached to the drum and then the drum is
rotated
which allows the rope to wind onto it. Drums on walking draglines are
cylindrical in
shape, have flanges on both ends and have a series of grooves for the rope to
wind onto.
The grooves provide an excellent rope support which reduces the radial
pressure and
helps resist crushing, which improves rope life, when compared to an ungrooved
wire
rope drum. The grooves also provide a path for the rope to follow as it winds
onto the
drum. This keeps adjacent wraps on the drum from touching one another. The
flanges on
grooved drums are to provide structural support and may be used to attach the
dead end
of the wire rope or a bull gear which would provide the torque needed to
rotate the drum.
The nomenclature of a generic drum assembly is shown in Figure 1. This
illustration greatly exaggerates the rope pitch diameter, drum tread diameter
and the rope
diameter to better illustrate the groove angle and the rope pitch.
The hoisting drums on walking draglines are grooved and have only a single
layer
of rope wound onto it. Most of the drums have two ropes which are attached at
the
drum's center. Each rope would then wind towards the flanges of the drum.
Drums on walking draglines are typically designed to the minimum recommended
tread diameter to minimize the manufacturing and operating costs. Due to the
enormous
lengths of wire rope required to operate a Walking Dragline the length of the
drum (i.e.
number of active grooves) becomes very long. This is extremely dangerous from
a
scrubbing point of view.
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T'he sheaves on a Walking Dragline are very similar to a drum. They both have
a
rope that rides on them and have a groove contour to support the rope. The
only real
difference is that the sheave has no groove pitch. Figure 2 shows the
nomenclature for a
generic sheave. This nomenclature will be used throughout this thesis.
Wire rope is one of the most uniform and reliable mechanical products ever
invented. If a wire rope is properly used and maintained it can provide an
excellent
service life. But, if a wire rope were to be abused in shipping, installation
or during
operation the service life of the rope can and probably will be much less than
satisfactory.
The causes surrounding the issue of rope life can vary from rope loading to
rope
construction and application.
Drum diameters are generally referenced as a ratio of the diameter of the
sheave
or drum (D) with respect to the diameter of the rope (d). This is the D/d
ratio. Due to the
limited deck space and the cost of manufacturing and operating a large drum on
walking
draglines and Electric mining shovels the drum D/d ratios are nearly always
made to the
recommended minimum value of 24.
When drums and sheaves are used in a system, the sheave diameters should be
generally larger in diameter than the drum. This is due to the rope only being
bent once
at the drum and twice at the sheave for each direction of travel. When the
rope travels
over each sheave the rope will be bent to conform the radius of the sheave as
it enters and
then bent straight again as it exits the sheave. This is very important on
walking
draglines since the portion of wire rope operating at the drum may never
operate at the
boom point sheaves. If the fatigue accumulation is twice as great at the boom
point as at
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the drum, the rope will fail prematurely at the boom point. This phenomenon is
simply
based on the design.
A fleet angle is the angle between the rope, as it leaves the drum and enters
the
head sheave, and the plane perpendicular to the axis of the drum. If the angle
becomes
too great the rope will touch "scrub" onto the groove of the drum or the
adjacent wrap on
the drum, or both. If a rope is scrubbing, there will be a visible thin band
of worn
material parallel to the axis of the rope and stretching a long distance along
the rope.
This wear is often subject to high pressure, heat and abrasion. If the heat
generated by
scrubbing raises the local temperature beyond the steels critical temperature,
martensite
will form on the wires. Martensite is very brittle phase of steel. This will
form surface
cracks in the wires when the rope is bent around sheaves and drums. Eventually
these
cracks will propagate through the wire. If no heat is generated the outer
wires will simply
wear very thin and break.
Either way, the rope will fail prematurely in one localized band along the
rope.
This will severely reduce the service life of the rope. Figure 3 shows a CAD
mock up of
a drum, which demonstrates both "Rope to Drum" and "Rope to Rope" scrubbing.
Scrubbing is a purely geometrical concept. It is the wiping action of the rope
onto
another rope or the groove contour of the drum. Scrubbing is mainly caused by
operating
at too high of a fleet angle. Although, scrubbing is also dependent upon the
system
parameters such as groove contour, rope pitch, drum and rope diameters. When a
drum is
designed, manufactured and put into service, all of these parameters become
fixed. The
CA 02287676 1999-10-27
only geometric parameter that changes is the fleet angle. For every position
on a drum
there exists a unique fleet angle value.
Every rope which is wound onto a drum is subject to a fleet angle and a groove
angle. The fleet angle is the angle the rope needs to follow in order to reach
the head
sheave as it is wound off a drum. The fleet angle changes for every position
along the
drum which is defined as the angle between the rope, as it leaves the drum and
enters the
head sheave, and the plane perpendicular to the axis of the drum. In the prior
art, the
groove angle, on the other hand, always remains the same for any position on
the drum.
The groove angle, sometimes referred to as lead angle, equals the arc-tangent
of the
drums groove pitch divided by the circumference at the groove tread diameter.
Figure 4
is a CAD layout illustrating the relationship between the groove pitch, groove
angle and
the drum tread diameter.
Groove Angle = arcta~ Gr°°ve Pitch
~ Drum Tread Diameter 7z ( 1.1 )
The fleet angle plus the groove angle is called the total fleet angle. Figure
5
shows a schematic of a sheave and drum system. The drum shown has a right hand
lead
and a left hand lead grooving with the typical 2 to 2.5 dead wraps (for normal
safe
operation) and the rest of the grooves are termed active grooves. As shown in
Figure 5,
the helix angle will either add to or subtract from the groove angle to form
the total fleet
angle. This is dependent on the direction of grooving. When the fleet angle is
large and
the drum groove pitch is large the head sheave may need to be shifted to
either the right
or the left in order to equalize the total fleet angle seen by the drum. If
the fleet angles
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CA 02287676 1999-10-27
are large, the pitch of the grooves can be increased to eliminate "Rope to
Rope"
scrubbing. However, this will not eliminate the "Rope to Drum" scrubbing.
Figure 3 illustrated both types of scrubbing.
Scrubbing can be a very serious and costly mode of rope failure. When a rope
system experiences severe scrubbing, rope life on that machine will be a
fraction of what
is expected and the cost of altering the machine while its in operation can
cost literally
cost millions of dollars.
Rope to rope scrubbing is directly dependent upon the pitch of the grooving.
It
can be easily understood, that if the groove pitch were to be increased, at
some point the
"Rope to Rope" scrubbing will obviously go away.
The "Roebling Wire Rope Handbook" derives a set of close form equations
needed to predict the fleet angle at which rope scrubbing will occur based on
known
values: drum diameter, pitch of grooving and the diameter of wire rope. The
following
equation defines the total fleet angle that will just cause contact with the
preceding wrap
on the drum.
Tatal_ fret angle = arc 'Y
Y2+YD+Yd (1.2)
The values for "X", "Y", "D" and "d" are defined below:
Y:= ~d2 - (X- h)2~~5 (1.3)
X= h-
2 2 Ch2 + 8 d2l
(1.4)
Where "h" equals the pitch of the grooving , "D" equals the tread diameter of
the
drum and "d" equals the diameter of the rope.
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Using equation 1.2 we can easily find out what angle will cause scrubbing for
any
given position on the drum. What if we wanted to know what pitch diameter will
cause
scrubbing based on a known maximum fleet angle; such as 2.000°? With
the use of a
computer and some iterative math software a program can iterate through these
equations
and find the exact pitch required. To select a proper rope pitch, the program
is used to
find the pitch that will just induce scrubbing based on a certain fleet angle,
drum and rope
diameter. Then, when the rope is at its maximum material condition a small
percentage
of the ropes diameter is added to the pitch and a gap is defined.
Drum groove scrubbing is when the fleet angle of the rope leaving the drum
would be great enough to cause interference between the actual drum profile
and the
rope; See Figure 3. When scrubbing occurs the outer strands are abrasively
worn away in
a thin band on only one side of the rope. Groove and rope scrubbing will
appear to be the
same on the damaged rope, but are caused by two completely different modes of
generation.
Drum groove design has been virtually unchanged for many years. The drum
groove contour has a specifically defined radius for each given rope diameter
which is
stated in "Wire Rope Handbook". The maximum allowable fleet angle of the rope
is also
said to be no greater than 1.50° for smooth drums and 2.00° for
grooved drums. The
Handbook does state however, that "Fleet angles larger than these suggested
limits can
cause such problems as ....... the rope rubbing against the flanges of the
sheave grooves."
The rope support angle is recommended to be between 135° and
150° and the minimum
drum D/d ratio is to be 22 for drag ropes and 24 for hoist ropes .
to
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No reference has been found which gives any indication as to what values
design
parameters like drum diameter, rope diameter, fleet angle, groove lead and
groove radius
should have to reduce or eliminate drum groove scrubbing. Now that CNC
machining
and computer technologies has evolved so rapidly, the possibility of
optimizing the
groove contour and pitch to eliminate both "Rope to Rope" and "Rope to Drum"
scrubbing is possible.
The initial concept for formulating the governing equations was that, if the
rope
were to touch the groove, in three dimensional space, then a common point, or
set of
points, will exist between the rope and the groove. If there are many points,
chances are
they will form one or more three dimensional continuos curves. When the rope
and the
groove touch at one point it will be defined as having the coordinates
<"Xend","Yend","Zend">.
The next step is to define the equation of the helical rope groove contour as
a
function of its governing variables in each of the three Cartesian coordinate
directions.
Then the equations in the "X", "Y", "Z" directions are defined as being equal
to "Xend",
"Yend', "Zend' respectively. This equation definition can also be done for the
ropes
surface. If a point on the rope is to be common to a point on the groove, in
order for
scrubbing to occur, then the "Xenc~' of the rope is equal to the "Xend" of the
groove. This
coordinate equalization will be the basis for formulating all the governing
relationships.
A single point on the groove contour is then geometrically described. This
point
is then swept out in the form of a helix in order to simulate the drum groove
pitch. This
will then define the helix in the "X", "Y" and "Z" directions. Three Cartesian
equations
a
CA 02287676 1999-10-27
for the rope surface will then be defined. Extensive variable substitution and
elimination
then yields one equation in two unknowns. Lastly, the final equation is
optimized in
order to find the absolute minimum value for the drum radius, "Helix Z", that
will cause
"Rope to Drum" scrubbing to occur.
Once the solution for one single point was found and verified, then a computer
program was written to solve for the ideal groove height for many values along
the axis
of the drum. This multitude of points will generate a curve which represents
the complete
"Ideal Groove Contour". A solution using the computer program then prints and
plots the
data that will create the full groove contour.
The governing equation derivation was broken into two main parts. The first
part
was to symbolically define the groove profile on the drum and the second part
was to
symbolically define the surface of rope as it leaves the drum. Both of these
parts were
defined in three dimensional space using the Cartesian coordinate system as a
set of
closed form equations.
After this set of equations was defined, a step by step variable substitution
and
elimination was performed. These equations were then reduced to one equation
and two
unknowns (a height "Helix Z" and an angle).
The last step was to set the equation equal to zero and solve for the variable
that is
quadratic in nature (a height "Helix Z"). This resulted in one equation with
one unknown
(an angle "A"). The minimum value of this equation defines the exact angle at
which
scrubbing will first occur.
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Once the angle for the minimum value was determined, the groove's height was
found by substituting it back into the original equation.
Figure 6 illustrates a helix wound onto a drum, which will take the form of a
groove. From this picture, the relationship between the drum axis, global axis
and groove
contour can easily be seen. Equations defining the position along the helical
spiral of the
drum were defined in the "X" and "Z" directions. One random point on the
contour was
picked to start the helix and the helix was swept out through 360° in a
left handed
positive direction.
We start by defining the equation of a helix in the "X" direction.
1 6 Pitch - h,~ j=x X = Xend
2 TI -
(3.1)
The distance "Xend" propagates along the drum axis and is directly
proportional
to the "helix angle". "Xend" is defined as the final solution point in the "X'
direction.
The final solution will be where the rope and the groove will just start to
intersect. This
is where scrubbing will just start to occur. In 360° of rotation, the
distance traveled along
the drum axis is equal to the pitch of the groove. The "Helix X" term is an
arbitrary axial
offset in the negative "X" direction. Later on this value will be a
predetermined
incremental step which will be used to generate the "Ideal Groove Contour".
In the definition of a helix, the "Yend' term can be expressed in terms of
"Zend"
by simply changing the Sine and Cosine values of the helix roll angle. From a
mathematical point of view this relationship will form redundant equations in
the "Y" and
"Z" directions. For this reason, only the equations in the "Z" and "X"
directions will be
13
CA 02287676 1999-10-27
required. The loss of one degree of freedom will be compensated for by
introducing the
general equation of an ellipse. When defining the equations for the rope, an
ellipse is
created when the cylindrical wire rope is cut with a plane at the fleet angle.
Next, define the equation of the helix in the "Z" direction as the following
relationship:
Helix Z cos(8) = Zend (3.2)
The same point used in equation 3.1 was also used as the starting point on the
groove profile for equation 3.2. The height of "Helix Z" is directly
proportional to the
cosine of the helix angle ("8"). The "Zend" term will be defined as the final
solution in
the "Z" direction. Figure 7 shows how "Helix X" relates to "Helix Z". For any
value of
"Helix X" there exists a "Helix Z" value, such that scrubbing on the drum will
initially
start. Our goal will be to derive an equation that equates the radial height,
"Helix Z",
directly to all known quantities.
The equation of the rope surface as it fleets off of the drum is then defined.
First,
assume that the rope fleets off of the drum in a straight line, it is centered
within the
drum's radius and that the rope returns to a cylindrical shape very shortly
after leaving
the drums surface. The global coordinate axis will be centered on the axis of
the drum
with the centerline of the rope passing through the "Z" axis and the axis of
the rope
running directly down the "Y" axis while at a zero degree fleet angle. This is
the same
global position as stated above.
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The fleet angle is the angle at which the rope leaves the drum. This fleet
angle is
zero only at when the rope leaves the drum exactly perpendicular to the "X Z"
plane. As
the rope winds onto the drum the fleet angle will either increase of decrease
depending on
the position of the head sheave. For a single layer hoisting drum, the maximum
fleet
angle usually occurs when the drum is completely full. This is when the rope
has wound
onto the last and final wrap available on the drum. For our analysis, a random
position on
the drum was chosen to be the global axis system. The "~' axis of this system
passes
through the center of the groove and the rope. The fleet angle will be
positive when the
rope rotates about the positive "Z" axis. Figure 8 shows the basic geometry of
a rope as
it leaves the drum when subjected to a fleet angle.
Three equations are needed to describe the final solution. There will be an
equation in the "X" and "Z" directions along with the general equation of an
ellipse.
Figure 9 is a view in the "XZ" plane located at the "PLANE OF ELLIPSE" shown
in Figure 8. On the contour of the ellipse we define the final solution to be
point "P".
This solution point will have the final solution coordinates <"Xenc~',"Zend">.
At any
distance along the "Y" axis there exists a plane that cuts through the rope
parallel the
"XZ" plane.
Figure 9 shows how the cutting plane creates an ellipse as it intersects the
rope at
a fleet angle. Remember, the axis of the rope only rotates about the "Z" axis
and remains
in a plane parallel to the "X Y" plane. Figure 10 shows a view looking in the
"YZ" plane
which shows the relationship of the helical roll angle, "Yend', and the
solution point "P"
as it relates to the plane of the ellipse .
CA 02287676 1999-10-27
The first step is to define the relationship between the center of the ellipse
and the
fleet angle as equation 3.3. To better see where this relationship comes from,
view
Figures 3.2-3, 3.2-4 and 3.2-5 all at the same time.
Xend -XEllipse = Y~nd tan(oc) (3.3)
From Figure 10 we can see that the rope angle "0" is related to "Vend" and
"Zend"
by the following equation.
Yend = Zend sin( A ) (3.4)
Note that the "Helix Z" term shown in Figure 7 is exactly the same as the
"Zend"
term stated in Figure 10. Then the "Zend" term in equation 3.4 can be replaced
with
"Helix 2" thus creating the equation 3.5 below.
Yend = Helix Z sin( 8) (3.5)
By substituting equation 3.5 into equation 3.3, the relationship 3.6 is then
created.
This relationship defines the solution on the ellipse in the "X" direction.
Xend + XEllipse = -Helix Z sin( 9) tan( a) (3.6)
In order to completely define the equation in the "X" direction, we will need
solve
the above relationship for "Xend' and create equation 3.7 below. The distance
from the
center of the ellipse to the final solution point in the "X" direction is
given by
"X Ellipse". Again, the "Xend" term is the final solution.
-Helix Z sin( 9) tan( o~) - X Ellipse = Xend (3.7)
The next step is define the final solution in the "Z" direction. The value in
the "Z"
direction is the distance to the center of the ellipse minus the distance back
to the contour
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of the ellipse at the solution point "P". This is simply one half the drum
rope pitch
diameter "DRPD" minus the "Y Ellipse". This is shown below as equation 3.8.
2 DRPD - Z Ellipse = Zend
(3.8)
Now, determine the position of a solution on the ellipse. This is done by
defining
the general equation of an ellipse which is stated as equation 3.9, below.
2 2
Z + x =1
rl 2 r2 2 (3.9)
This equation can be solved for the "x" term which will define the "x"
position on
the ellipse. This "x" position has the same meaning as "X Ellipse". Since we
know the
solution to scrubbing will always occur within the fourth quadrant; See Figure
9, we can
eliminate the negative square root term and define equation 3.10 as the
following.
.5
X Ellipse = 1 - Z r2 2
rl 2
(3.10)
By substituting equation 3.10 into 3.7 we can create equation 3.11, shown
below.
.5
-Hekx Z sin( 8 ) tan( a ) - 1- Z r2 2 = Xend
rl 2
(3.11)
Equation 3.9 can also be solved for the "z" value, just like we solved for the
"x"
value before. Again, we know where to look for our solution, quadrant IV,
therefore we
can again eliminate the negative square root term.
.5
Z Ellipse = 1 - x rl 2
r2 2
(3.12)
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By substituting equation 3.12 into equation 3.8, we create equation 3.13,
shown
below.
.5
1 DRPD - 1 - x rl 2 - Zend
2 r2 2
(3.13)
Equations 3.9, 3.11 and 3.13 are the three final equations which define the
final
solution on the surface of the rope. The next step is to define the values for
"rl" and
"r2". The quantity "rl" is the minor axis of the ellipse, which is defined as
the drum
rope radius "DRR" since the axis of the rope is parallel to the "X Y" plane.
rl = DRR (3.14)
The quantity "r2" is the major axis of the ellipse which is defined as the
drum rope radius
divided by the Cosine of the fleet angle "a". The "intersecting plane" cuts
the cylindrical
rope at precisely that angle.
r2 - DRR
cos(cc) (3.15)
Start by substituting equation 3.14 and 3.15 into equation 3.11, 3.13 and 3.10
as
shown below, creating equations 3.16, 3.17 and 3.18 respectively.
.5
1 _ z DRR2
2
-Helix Z sin( 8 ) tan( oc ) - D~ 2 = Xend
cos(oc) (3.16)
.5
2 DRPD- 1 _ x2 cos(2 )2 D~2 =end
DRR
(3.17)
is
CA 02287676 1999-10-27
z22+x2cos(2)2=1
DRR DRR (3.18)
Equations 3.1 and 3.2 above are restated here for the definition of the groove
helix
geometry so that the final five equations are all on one page.
1 8 Pitch - H~lzx X = Xend
2 II -
(3.1)
Helix Z cos(8) = Zend (3.2)
Eliminate the variable "Xend" from equations 3.1 and 3.16. The following
equation, 3.19, is the result from subtracting equation 3.11 from equation
3.16.
.5
1- z DRR2
2
0 = -Helix Z sin( 6) tan( cc) - D~ 2 - 2 a ~Lch + Helix X
cos(a) (3.19)
Then eliminate the variable "Zend" from equation 3.2 and 3.17. The following
equation, 3.20, is the result from subtracting equation 3.2 from equation
3.17.
2 2 'S
0 =1 DRPD- 1- x cos oc D~2 - H~lix Z cos(8)
D~2 (3.20)
Notice that equation 3.19 and 3.20 have a "~" and a "x2" term respectively. We
can use equation 3.18 to bridge the relationship between "x" and "y" in these
equations.
Solve equation 3.18 for "z2" and create equation 3.21 below.
z2- x2 cos(a)2 _ 1 DRR2
(3.21 )
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CA 02287676 1999-10-27
Then take equation 3.21 and substitute it back into equation 3.19 and create
equation 3.22, shown below.
.5
0 = -Helix_Z sin( 8) tan( c~c) - ~ x 2~ - 2 g ~~ch + Helix_X
(3.22)
Note that equation 3.20 and 3.22 both contain "x2" terms. Solve both of them
for
"x2" and create equations 3.23 and 3.24 respectively, as shown below.
x 2 = Helix Z2 sin( 6) 2 tan( a) 2 + Helix Z sin( 9~an( a) 6 Pitch _ 2. Helix
Z sin( A) tan( oc) Helix X
+ . 2500000000 e2 ~ h 2 _ 1. 8 Prtch~Helix ~' + Helix X 2
(3.23)
x2 __ DRR2 - .2500000000 DRPD2 +DRPD Helix_Z cos(9) - 1. Helix Z2 cos(9)2
cos(a)2
(3.24)
Notice that in equation 3.23 and 3.24 both have unknown quantities only in
"Helix Z" and "A". By subtracting equation 3.23 and 3.24, there will be one
equation and
two unknowns thus creating Equation 3.25. It can plainly be seen that this
equation is
quadratic in "Helix Z" and highly nonlinear in "0". This is the final
equation.
0 = ~II2 - cos(cc)2 II2 + cos(6)2 cos(oc) II2~ Helix Z2 +
-2 Helix XII2 + 8 Pitch IZ~ cos(o~) sin(A) sin(tx) - DRPD cos(8) IZ2~ Helix Z
+ ~.25 82 Fitch2 - 8 Filch Helix h'II+Helix X2 II2~ cos(ec)2
+ ~ -DRR2 + . 25 DRPD2~ II2
(3.25)
CA 02287676 1999-10-27
When the value of this equation is equal to or less than zero scrubbing is
taking
place. Our intent will be evaluate where this equation equals zero. This is
where the
groove contour and the rope contour will be touching.
Optimizing the solution is defined as finding the precise point at which
scrubbing
will occur for any given value of "Helix X" and the system constants. This
will be the
minimum value of "Helix Z" such that the value of the equation 3.25 is equal
to zero.
Notice that equation 3.25 is quadratic in "Helix ~'. Therefore, by using the
quadratic
equation, both roots in "0" can be easily defined. Equation 3.26, below, is
the general
form of a quadratic equation.
.5
_1 -bb - { bb 2 - 4 as cc~
2 as (3.26)
The constants "aa", "bb" and "cc" are derived from equation 3.25.
as II2- cos(o~)2IZ2+cos(6)2 cos(c~c) II2 (3.27)
bh ~ -2 Helix X II2 + 8 Pitch IZ~ cos( cx) sin( 8) sin( c~c) - DRPD cos( 8)
II2 (3.28)
cc = ~.25 62 Pyitch2 - APitch Helix XII+Helix X2 II2~ cos(a)2
+ { -DRR2 + .25 DRPD2~ II2 (3.29)
If we substitute "aa", "bb" and "cc" into equation 3.26 we will get one
equation in
one unknown "8". There are two roots, the negative root represents the
intersection at
the bottom of the rope and the positive root represents the intersection at
the top of the
rope. Due to geometry, only the negative solution is of interest to us.
Equation 3.30
21
CA 02287676 1999-10-27
shown below is the negative root. This equation describes the exact
intersection between
the rope and the groove.
p = 2 -~ -2 Helix XII2 + A Pitch II~ cos( cx) sin(8) sin(oc) + DRPD cos( 8)
II2 -
-2 Helix X II2 + 8 Pitch II~ cos(oc) sin( 9) sin(oc) - DRPD cos(8) II2~2 - 4
~II2 - cos(cc)2II2 + cos(A)2 cos(oc) II2~ ~ ~.25 82 Pitch2 -
8 Pitch Helix_X IZ + Helix_X2 IZ2~ cos( oc) 2
.5
+ ~-DRR2 +.25 DRPD2~ 1Z2~~ l ~IZ2 - cos(oc)2 IZ2 + cos(8)2 cos(ot) IZ2
(3.30)
Both, the positive and the negative, roots are plotted in Figure 11. Notice
how the
bottom curve is concave upward. This has the same type of characteristics that
were
experienced above where the optimum contour was derived using CAD. The curve
in
CAD was also concave upward and had a relative minimum value occurnng at a
value of
"8" just greater than zero. The curve becomes very flat near the bottom. This
can be a
tangency problem within CAD but, by using symbolic notation all we need is for
the
curve to be smooth and continuous.
In looking at Figure 11 we can see the vertical axis is "Helix Z" which is the
drum radius. Therefore by inspection, the lower curves represents the
underside of the
rope, which is where scrubbing is likely to occur. The upper curve represents
the top side
of the rope, which is meaningless for groove scrubbing.
The minimum point on this lower curve is the helix angle where scrubbing will
start. In other words, where the derivative of this function is equal to zero,
the minimum
22
CA 02287676 1999-10-27
value resides. Equation 3.31 is the partial derivative of equation 3.30 with
respect to "0",
shown below.
23
CA 02287676 1999-10-27
0 = -2. Helix_Xcos(8)3 sin(a) II cos(a)3 %1
+2. Helix X cos(A) sin(a) II cos(a)3 %1 + 2. Helix X cos(6) sin(a) IZ cos(a)
%1
+4. II2 APitch sin(8) cos(a)2 Helix X cos(8) - 2. Helix X sin(a) TI3 cos(a)3
DRPD
- 1. A Pitch sin(a) II2 cos(a) DRPD+ 8 Pitch sin(a) II2 cos(a)3 DRPD
+ 2. Helix X sin(a) IZ3 cos(a) DRPD- 4. II3 Helix X2 sin(8) cos(a)2 cos(8)
- 2. II2 Pitch Helix X cos(a)2 cos(A)2 - 2. I-I2 cos(6)4 cos(a)4 Pitch Helix X
+ A Pitch cos(A)3 sin(a) cos(a)3 %1 - 1. 8 Pitch cos(8) sin(a) cos(a)3 %1
+II cos(A) cos(a)4 sin(8) 82 Pitch2 + sin(8) DRPDII cos(6)2 cos(a)2 %1
- 1. I1 cos(8)2 cos(a)4 APitch2 + cos(A)3 cos(a)4 sin(9) II A2 Pitch2
- 1. Pitch sin(A) sin(a) cos(a) %1 +Pitch sin(8) sin(a) cos(a)3 %1
- 1. Pitch sin(8) sin(a) cos(a)3 cos(6)2 %1 - 1. sin(A) DRPDII %1
+ sin(8) DRPD II cos(a)2 %1 - 1. APitch cos(6) sin(a) cos(a) %1
+4. IZ3 cos(A) cos(a)4 siri(A) Helix X2 + 2. >-I2 cos(8)2 cos(a)4 Pitch Helix
X
+4. cos(6)3 cos(a)~ sin(6) tI3 Heliz X2
-4. cos(A)3 cos(a)4 sin(9) >-I2 APitch Helix X
- 2. cos(8)2 DRPDII3 Helix Xsin(a) cos(a)3
+4. cos(6)4 DRPDII3 Helix Xsin(a) cos(a)3- 4. IZ3 cos(8)3 cos(a)4 siri(6) DRR2
+cos(6)3 DRPDIZ2 Pitch sin(9) sia(a) cos(a)3+II3 cos(8)3 cos(a)4 sin(A) DRPD2
- 1. II3 cos(8) cos(a)4 sin(6) DRPD2 + cos(8)2 DRPD IZ2 8 Pitch sin(a) cos(a)3
+IIcos(9)'~ cos(a)'~ BPitch2+IZBPitch2 cos(a)2 cos(8)2
- 1. II 92 Pitch2 sin(9) cos(a)2 cos(6)
- 1. cos(8) DRPD II2 Pitch sin(8) sin(a) cos(a)3
- 2. cos(9)~ DRPD II2 8 Pitch siu(a) cos(a)3 +4. II3 cos(8) cos(a)'~ sin(8)
DRR2
- 1. cos(A)3 DRPD2 II3 sia(9) cos(a)2 - 4. II3 cos(6) cos(a)2 siri(8) DRR2
-4. II2 cos(6) cos(a)4 sin(8) BPitch Xe&x X
- 4. cos(A)2 DRPD II3 Helix X si~(a) cos(a) + cos(8) DRPD2 IZ3 sia(9)
+2. cos(A)2DRPDII2 BPrlch sin(a) cos(a)
+cos(6) DRPDII2 Pitch siri(8) sin(a) cos(a)
%1 =-1. II2 ~-1. cos(6)2 DRPD2II2 + 62 Pitch2 cos(a)2 cos(6)2
- 4. II2 cos(6)2 cos(a)2 DRR2 +II2 cos(A)2 cos(a)2 DRPD2
+ 2. 8 Pitch sin(6) sin(a) II cos(a) cos(9) DRPD
- 4. Helix Xsin(6) sin(a) II2 cos(a) cos(9) DRPD+4. II2 cos(a)2 DRR2
- 1. IZ2 cos( a) 2 DRPD2 - 4. DRR2 II2 + DRPD2 II2
+4. 1-I2 Helix X2 cos(a)2 cos(6)2 - 4. IZ BPitch Helix Xcos(a)2 cos(6)2~
24
CA 02287676 1999-10-27
(3.31)
Where "Pitch" = Pitch of the drum, "DRR" = Drum rope radius, "DRPD" = Drum
rope pitch diameter, "a" = Fleet angle and "8" = Roll angle of the helix.
Figure 12 shows a plot of this equation 3.31. Note that the equation passes
through zero at one single value of "8". This is the value of "A" where
scrubbing will
first start.
Once the angle has been defined using equation 3.31, it can be substituted
back
into equation 3.25 and then solved for the "Helix Z" value. As a reminder,
"Helix Z"
was the radius of the drum at which scrubbing will occur.
The next step is to write a computer program that will iterate through
multiple
values of "Helix X" and find all of the corresponding values of "Helix Z" by
utilizing
equations 3.25 and 3.31.
In many ways drums and sheaves are the same. They both have fleet angles,
groove radii and D/d diameter ratios. The only big difference is that a sheave
has only
one straight groove and therefore has no "Pitch". When the "Pitch" in
equations 3.25 and
3.31 is set equal to zero they become simplified to equations 3.32 and 3.33
respectively.
These equations are shown below.
0 = ~ 1 - cos(a)2 + cos(A)2 cos(a)2~ Helix Z2
- (2 sin(A) sin(o~) Helix X cos(cx) + DRPD cos(8)) Helix Z
+ H~lix X2 cos( oc) 2 - DRR2 + .25 DRPD2 (3.32)
CA 02287676 1999-10-27
0 = cos(8) DRPD2 IZ2 sin(6) - 2. IZ2 Helix X sin(a) cos(a)3 DRPD
- 4. II2 Helix X2 sin(8) cos(a)2 cos(0) + 2. Helix X sin(a) II2 cos(a) DRPD
- 4. cos(A)2 DRPDII2 Helix Xsin(cx) cos(oc)
- 2. Helix X cos(6)3 sin(a) cos(a)3 %1 + sin(8) DRPD cos(8)2 cos(oc)2 %1
+2. Helix Xcos(9) sin(a) cos(a)3 %1 +2. Helix Xcos(9) sin(a) cos(a) %1
+ sin(A) DRPD cos(a)2 %1 +4. II2 cos(6) cos(cc)4 sin(8) Helix_X2
+4. cos(6)3 cos(ct)4 sin(8) IZ2 Helix_X2 - 1. IZ2 cos(0) cos(cc)~ sin(8) DRPD2
+4. >-I2 cos(6) cos(a)4 sin(8) DRR2 +4. II2 cos(8)'l DRPD Helix_X sin(a)
cos(cc)3
- 1. II2 cos(0)3 DRPD2 sin(0) cos(c~c)2 - 2. II2 cos(8)2 DRPD Helix X sin(ct)
cos(a)3
- 4. II2 cos(6) cos(a)2 sin(8) DRR2 - 1. sin(9) DRPD %1
-4. II2 cos(8)3 cos(a)4 sin(A) DRR2+IZ2 cos(8)3 cos(a)4 sin(8) DRPD2
%1 :=IZ4 ~-4. Helix X2 cos(a)2 cos(A)2+4. Helix Xsin(8) sin(a) cos(a) cos(8)
DRPD
+ cos(A)2 DRPD2 + 4. DRR2 - 1. DRPD2 - 4. cos(a)2 DRR2 + cos(a)2 DRPD2
+4. cos(A)2 cos(a)2 DRR2 - 1. cos(A)2 cos(o~)2 DRPD2~
(3.33)
Where "DRR" = Drum rope radius, "DRPD" = Drum rope pitch diameter,
"a" = Fleet angle and "0" = Roll angle of the helix.
The solution for the ideal drum groove contour was presented above and it
turns
out that final solution, equation 3.25 and equation 3.31, is extremely
nonlinear in "8".
Therefore, numerical methods are the most practical mechanisms to solve for
its roots.
An engineering equation solving package called TkSolver was used to model the
final
equations. The numerical method used by the TkSolver software is a Modified
Newton-
Raphson procedure which is widely applicable in situations like this and is
well known
for its robustness.
Drum groove scrubbing will be evaluated at every value of "Helix X" along the
axis of the drum up to one half of the rope pitch. Beyond that point, the
solution would
26
CA 02287676 1999-10-27
be meaningless; the start of the neighboring groove will obviously lie beyond
that point.
The value of "Helix X" can be placed into a list and be incremented by very
small
amounts up to a value of half the rope pitch. Then the value of "Helix Z" can
be solved
for each and every value of "Helix X" by utilizing equations 3.25 and 3.31.
These
consecutive solutions will simulate a curve which represents the "Ideal Groove
Contour"
of the drum that will eliminate "Rope to Drum" groove scrubbing.
Since the only unknown variable in equation 3.31 is "A", TkSolver can then
iterate
until a solution is found. TkSolver will then utilize that value of "0" in
equation 3.25 and
solve for "Helix Z". The drum radii that will just allow scrubbing to occur is
the value
"Helix Z".
Tk-Solver provides the solution that at exactly 2.500" along the axis of the
drum,
for these system constants, the roll angle of the helix is "0" = 2.972°
and the ideal drum
The rope support angle is defined as the angle the rope is supported within
the
drums groove if the rope were under high tension. The rope will flatten
(elliptical) into
the groove contour. If a line from the end of support is drawn through the
theoretical
center of the rope and a line is drawn from the other end of support through
the
theoretical center of the rope, then the angle included between these two
lines is defined
as the rope support angle. Figure 1 illustrates exactly how to measure the
rope support
angle.
Every standard rope groove will have some rope support angle associated with
it.
Typically, the rope support angle should be between 120° to
150°.
27
CA 02287676 1999-10-27
If the rope support angle is too small then the rope will be lacking support
and
will then be flattened and elliptical shaped. This will cause the rope to fail
by interstitial
strand penning and nicking which will eventually cause the internal wires fail
prematurely. This is one of the most dangerous forms of wire failure; since it
cannot be
found by a simple visual inspection. The internal broken wires are only
noticed after they
have worked their way to the outside of the rope.
If the rope support angle is too great, then the rope will scrub against the
groove
and cause "Rope to Drum " scrubbing. The rope will actually scrub against the
groove
contour, even when the fleet angle may be very low. A wire rope has an
increased risk of
"Drum to Groove" scrubbing when the drum diameter, pitch or the fleet angle
are too
large. Figure 3 shows "Rope to Drum" and "Rope to Rope" scrubbing. The
question to
ask is; What groove contour will provide adequate rope support and at the same
time
prevent or minimize groove scrubbing?
The ideal groove contour for a dragline drum is determined as shown by the
following example.
There are three specific parameters required to design a new groove. They are
groove pitch, groove radius and rope support angle. There are several
parameters that are
set by the surrounding envirorunent and machinery. These parameters are
usually defined
early in the design process which usually occur long before a designer
actually
designs the drum groove contour. Some of these parameters are rope diameter,
drum
capacity (which affects the drum tread diameter and drum length) and the
sheave and
drum placement (which affects the maximum operating fleet angle). When these
28
CA 02287676 1999-10-27
parameters are being designed, the designer needs to foresee the possible
future
consequences of fixing their values.
The first to be defined is the nominal rope diameter. The rope diameter is
usually
chosen very early in the design. It is based on the rated suspended load the
Walking
Dragline is expected to hoist or drag. The rope diameter typically ranges from
2.00" to
6.00" in diameter. The ropes are manufactured to a tolerance to +0% to +$% of
nominal
rope diameter. For this example, a nominal rope diameter of 4.00" will be
used.
The second issue is drum capacity and D/d Ratio. Drum capacity is a function
of
the number of wraps wrapped on the drum and the drums tread diameter. The drum
diameter is usually set to the minimum recommended diameter for the wire rope
being
used. The drum ratio typically used is a D/d equal to 24.
The maximum operating fleet angle is the last of the three predefined
variables the
designer needs to contend with. The maximum fleet angle is directly affected
by where
the sheaves and the drum are placed on a machine. The maximum operating fleet
angle
for a grooved drum is typically 2.00°. The designer needs to consider
the placement of
the drum and sheaves such that the fleet angle will never exceed this value.
For this
example, the rope diameter will be 4.00", the drum tread diameter will be
24*4.00" _
96.00" and the maximum allowable fleet angle will be set at 2.00°.
As stated above, there are three specific parameters a designer has control
over
when designing a new groove contour. They are groove pitch, groove radius and
rope
support angle. The groove pitch or lead of the rope is chosen in order to
prevent rope to
rope scrubbing, as described earlier. The equations initially stated above
predict the exact
29
CA 02287676 1999-10-27
total fleet angle at which "Rope to Rope" scrubbing will occur. Using this
relationship,
the equations 1.2, 1.3 and 1.4 are programmed into Tk-Solver and its iterative
equation
solver is used to find the value of any one variable in terms of all the
others. More
specifically, Tk-Solver is used to find the groove ip'tch required in order to
just cause
"Rope to Rope" scrubbing based on a known drum tread diameter, true rope
diameter and
a maximum fleet angle.
Tk-Solver will iterate through these equations by simply giving an initial
guess
for the rope pitch. This example has a maximum fleet angle of 2.000°, a
drum diameter
of 96.000" and a nominal rope diameter of 4.000" which may be manufactured at
+S%
larger than standard, LE. 4.200". Tk-Solver iterates through the set of
equations until a
convergence is found. The ideal groove pitch is 4.5838". This is based on a
rope that is
manufactured at a S% oversize condition. If the designer wanted added security
that
scrubbing will not occur, they can simply add a small amount to the pitch. In
this
example, the groove pitch may be rounded up to 4.600". This will leave a small
gap of
.0162" even when the largest possible rope size is placed on the grooves and
the rope is
operating at the maximum fleet angle. In this particular example, the 4.600"
pitch is 13%
larger than nominal rope diameter. This is a very typical pitch for this size
of rope on
mining equipment.
In the prior art, the groove radius for a new groove is typically between 6%
and
10% larger than the nominal rope diameter. For this example, 2.139" is used as
the
groove radius. This value corresponds to 6.5% larger than nominal rope
diameter.
CA 02287676 1999-10-27
When the wire rope deforms into the groove radius, it will flatten and become
oval shaped. Because of this, the rope needs to be supported on its
circumference by
some specified angle. Many sources specify a rope support angle which can vary
from
120° to 1 SO°. Most sources recommend 135° for adequate
rope support. For this
particular example, a rope support angle of 135° will be used.
The last and final step is to define the drum groove depth. The depth of the
groove is the distance from the bottom of the groove (D/2) to the outside
diameter of the
drum. At this point the designer has already defined the pitch, groove radius
and the rope
support angle. This is all that is required in order to define the groove
depth is to create
the circle that is tangent to both of the groove radii and the two 135°
rope support angle
lines. In CATIA, a circle can be created which is multi-tangent to two circles
and a point.
This will define the "Cap" radius between two adjacent grooves. In Figure 13
there is a
layout of the standard drum groove and the cap radius we have just defined.
Note how
the "Cap" radius turned out to be .351" and the drum outside diameter is
99.074".
In conclusion, the standard drum groove design procedure for a 4.000" nominal
rope diameter with a maximum operating fleet angle of 2.000° and a
tread diameter of
96.000", will have a groove radius of 2.139", cap radius of .351", drum
outside diameter
of 99.074" and a rope support angle of 135°.
The sheave design example is very similar to the drum. The only difference is
that groove has no pitch and the depth of the groove is increased
significantly in order to
prevent the rope from bouncing out of the sheave. The only additional
parameter is the
"Throat Angle"; see Figure 2. The throat angle is nothing more than
180° minus the
31
CA 02287676 1999-10-27
rope support angle. For our 4.000" nominal rope diameter example, the throat
angle will
be 45°. The groove radius and the tread diameter will remain the same,
96.000" and
2.139" respectively.
Depending on the application, the groove depth should be 1.5 to 2.5 rope
diameters. Past experience of specific applications will be the best judge. If
the wire
rope experiences sudden changes in tension, shock load, it may jump out of a
sheave.
Some applications will never experience shock loading and therefore have a
groove depth
as small as a standard drum. Figure 14, illustrates the standard sheave groove
design with
a groove depth of 1.50 times the nominal rope diameter.
The fleet angle for a sheave, just like a drum, should not exceed
2.000°. Since
there is no pitch, there is no need to consider anything else. For a sheave
with a nominal
4.000" rope diameter and a fleet angle of 2.000° and a tread diameter
of 96.000", the
sheaves groove radius will be 2.139", throat angle will be 45°, groove
depth will be
6.000", rope support angle will be 135° and the outside diameter will
be 108.000".
The existing groove design procedure has several similarities to the newly
proposed groove design procedure. Both of these procedures have the same
parameters
which are defined by the surrounding environment and machinery; such as a
maximum
operating fleet angle, minimum drum tread diameter and nominal rope diameter.
The
pitch design procedure explained above will be exactly the same in the ideal
contour
design procedure. The only radical alteration to the existing design procedure
will be to
redefine the groove radius.
32
CA 02287676 1999-10-27
When the rope groove radius is in the form of a true radius, it does not take
into
account the effects of "Rope to Drum" scrubbing. The equations derived above
define
the "Ideal Groove Contour" that will just eliminate scrubbing for any drum or
sheave.
The ideal drum groove design example will incorporate many similar parameters
that were previously defined above. Here, the drum tread diameter was 96.000",
maximum fleet angle was 2.000° and the nominal rope diameter was
4.000". The ideal
pitch that will eliminate "Rope to Rope" scrubbing was defined as 4.600". For
consistence, the 135° rope support angle will also be kept the same.
The only change will
be in the way the rope's supporting radius is defined. The rope groove radius
is defined
as a parabolic shape ( Equation 3.25 and Equation 3.31 ). By altering the
shape of the
groove and keeping all of the other parameters the same, the "Cap" radius and
Drum
outside diameter will both be forced to change.
In order to solve for the ideal groove contour, only four parameters are
required,
drum tread diameter, actual rope diameter, maximum operating fleet angle and
rope
groove pitch. All of these parameters are defined in the existing groove
design
procedure stated above. A TkSolver program was written to solve for the ideal
drum
radius required to eliminate "Rope to Drum" groove scrubbing. The program will
utilize
"Newton's Method" for finding roots and solve for "A" and "Helix Z" by
starting with
only an initial guess. The program is set up to solve for one single solution.
The rope
diameter is chosen to be the nominal rope diameter at 0% oversize (4.000").
This
diameter represents the operating rope size during most of its service life.
This solution
will be located along the drum axis at a "Helix X" value of 1.000". The
solution
33
CA 02287676 1999-10-27
converges to a final value of "Helix 2" equal to 48.249" and a helix roll
angle "0"of
1.518°.
This TkSolver program can solve for an ideal contour at any point along the
drums axis (.01, .02, .03... 2.01, 2.02......) up to half of the groove pitch.
At any point
beyond half of the groove pitch, there will be the adjacent wrap on the drum.
These
multiple points are found by creating lists of data on
Point Axial Ideal
k
l
i
h
O
h
li
f i
T [umber DistanceHeight
vers L
st S
eet.
nce t
e
So
st o
nput variables
Helix Helix
a X Z
e all def - -
ned
then TkSolver will easil
l
f
ll
r [inches][inches]
i
,
y so
ve
or a
the positions specified. 1 0.000 48.000
2 0.100 48.002
This data was then drawn in CATIA along 3 0.200 48.009
with
4 0.300 48.021
all of the other pertinent groove and 5 0.400 48.038
drum parameters.
6 0.500 48.060
In order to draw the contour all of the 7 0.600 48.086
<Helix X,
8 0.700 48.118
di
H
li
~
t
l
tt
d
Thi
f
coor 9 O,g00 48.156
na
es were p
e
x
o
e
.
s sequence o
points was then connected by utilizing 10 0.900 48.200
the "Spline"
11 1.000 48.249
function within CATIA. The resulting 12 1.100 48.305
spline represents
13 1.200 48.369
is a parabola that defines the "Ideal 14 1.300 48.440
Drum Groove
15 1.400 48.521
Contour". 16 1.500 48.611
17 1.600 48.712
Once the contour has been created, then 1 g 1.700 48.827
the
entire la 19 1.800 48.958
out of the drum
rofile can be com
leted
y 20 1.900 49.109
p
p
.
Figure 15 is the final drum design using21 2.000 49.284
the newly
22 2.100 49.489
proposed design methods. Due to the new 23 2.200 49.733
groove
( 24 ~ 2.300 50.024
contour, a new "Cap" radius and new drum outside
Ideal Drum Groove
Contour Data
34
CA 02287676 1999-10-27
diameter have been defined. The 135° rope support angle, 4.600" pitch
and the 96.000"
tread diameter were all kept the same. The changes are subtle but evident. The
cap
radius is .374" and the new drum outside diameter is 98.853".
The ideal sheave contour will have exactly the same characteristics as the
standard
sheave groove contour. The tread diameter, outside diameter and throat angle
will all
remain the same. The groove support radius will be similar to the ideal
contour defined
above. The main difference is that a sheave has no pitch and therefore has
less of a
possibility of having "Groove Scrubbing".
The output of the ideal groove contour program is then computed for when the
ip tch is set equal to zero. This particular solution was located along the
drum axis at a
"Helix X" value of 1.000". The radial height "Helix Z" that is associated with
that value
is 48.258". Again, Tk-Solver will solve for all of the required points and
CATIA was
used to connect and analyze the data.
Figure 17 illustrates the ideal contour superimposed onto the standard groove
contour. Note that the ideal contour never penetrates the standard contour.
Lines are then
created which are parallel to the 45° throat angle and are tangent to
the "Ideal Groove
Contour". This tangency condition occurs at 140.7°, which is slightly
above the standard
135°. This illustrates that there will be an .018" gap between the
ideal and standard
groove contours. This means that scrubbing will not be a problem for this
particular
design. In this example, we would typically adopt all of the "Standard Design"
parameters.
CA 02287676 1999-10-27
This sheave will have a nominal rope diameter of 4.00", maximum operating
fleet
angle of 2.000° and a tread diameter of 96.000". The sheaves groove
radius will be
2.139", throat angle will be 45°, groove depth will be 6.000", rope
support angle will be
135° and the outside diameter will be 108.000".
The standard and ideal groove contours are very similar in shape. The standard
groove has a true radius supporting the rope and the ideal contour has a
parabola
supporting the rope. The difference between these two geometric shapes is
closely
examined below. All of the other groove geometry (rope support angle, groove
pitch,
tread diameter) remains the same.
In order to compare the drum groove geometry, we need to define a common
reference point for all of the groove shapes. The common datum is located at
the center of
the rope when the rope is manufactured at it's nominal diameter. The groove
contour
comparison will be based on the distance from the center of this rope to all
of the
proposed contours.
Every point below the "Ideal Contour" will cause scrubbing. The 6% and 10%
over nominal rope diameter curves represent the recommended maximum and
minimum
values for the groove radius which is stated in the Wire Rope Users Manual 3rd
Edition.
The rope support angle is plotted on the abscissa axis from the centerline of
the rope and
towards the right. This is the positive "Helix X" direction which was defined
above.
The total rope support angle is twice that stated in figure 18. The two
vertical reference
lines define the minimum and maximum rope support angles recommended by
several
sources.
36
CA 02287676 1999-10-27
In Figure 18 the curve defining the nominal rope diameter is a straight line.
The
radius of the rope does not change as the rope support angle increases. The
distance from
the center of the nominal rope diameter to the 6% and 10% are parabolic in
nature. This
is because the circles are not concentric. All of the circles are tangent to
the tread
diameter and therefore the centerlines are all shifted vertically by the
change in rope
radius.
If the groove radius was to be manufactured at 10% over the nominal rope
diameter the "Drum Groove" scrubbing will only occur at a rope support angle
of about
140° and greater, see Figure 18. If the groove support angle was to be
only 135° then the
rope will not scrub at all. At first, this may appear satisfactory, but when
the rope is
allowed too much freedom to deform to the shape of the groove, then the
internal fatigue
damage will greatly reduce the life of the wire rope.
If the groove was to be manufactured to 6% over the nominal rope diameter,
then
scrubbing will exist for nearly all rope support angles. At this small 6%
groove radius,
the amount of deflection due to scrubbing interference will have a dramatic
affect on the
life of the rope. The rope will not only scrub severely, it may also not allow
enough
room for the individual wires to shift positions as the rope is wound onto the
drum. This
will pinch the rope and will cause accelerated cyclic fatigue.
The goal is to allow minimum movement of the rope which will still allow the
individual wires to relocate as the rope is bent around a drum and to prevent
"Rope to
Drum" drum groove scrubbing.
37
CA 02287676 1999-10-27
In Figure 18 the "Ideal Groove Contour" is also plotted. This groove contour
is
slightly greater than the 6% curve up to about 40°. After that point,
the amount of
scrubbing increases greatly. If the groove was manufactured at this ideal
contoured
shape, the rope will not scrub on the drum, at all.
Above, the groove radius was chosen to be 2.139". This is the recommended
radius stated in the Wire rope Handbook 2nd Edition. If this curve was plotted
on the
same graph, the relative distance between it and the ideal contour could then
be
measured. These measurements are shown in the table below. Notice that the
2.139"
inch radius tracks the ideal contour nearly exactly up to 80° of total
rope support.
Beyond that value, the deviation increases exponentially. If the rope support
angle was to
be only 135° we will need to modify the recommended radius by .028". If
we wanted a
GrooveHalf StandardTrue Deviation
SupportSupportDesign Ideal From
Angle Angle ProcedureContourStandard
[degrees][degrees](inches][inches][inches]
0.0 0.00 2.000 2.000 0.000
10.0 5.00 2.000 2.001 0.000
20.0 10.00 2.002 2.002 0.000
30.0 15.00 2.004 2.004 0.000
40.0 20.00 2.008 2.008 0.000
50.0 25.00 2.012 2.012 0.000
60.0 30.00 2.017 2.017 0.000
70.0 35.00 2.024 2.024 0.000
80.0 40.00 2.031 2.031 0.000
90.0 45.00 2.038 2.040 0.001
100.0 50.00 2.047 2.051 0.004
110.0 55.00 2.056 2.064 0.008
120.0 60.00 2.066 2.080 0.014
130.0 65.00 2.077 2.099 0.022
135.0 67.50 2.082 2.110 0.028
140.0 70.00 2.087 2.123 0.036
150.0 75.00 2.099 2.152 0.053
~ ~
Standard and Ideal Drum
Groove Contour Deviations
38
CA 02287676 1999-10-27
rope support angle of 150° we will need to modify the rope support
radius by .053".
When the rope is being wound onto a drum and if the drum has a 135° of
rope support
angle, then the rope will need to be deflected by .028" in order for the rope
to pass.
When the rope is under a high tension, the actual pressure on the individual
wires can
become extreme. This will cause severe abrasive wear and burning of the rope,
which
may cause the formation martensite on the surface of the wires, which will
also greatly
accelerate the cyclic fatigue and initiate crack propagation.
In various embodiments of the invention, the groove contour may vary from side
to side as the fleet angle changes from one side of the groove to the other,
or from groove
to groove. In most embodiments however, a single groove contour will be used
over the
entire drum.
39
