CA2287676A1 - An improved groove design for wire rope drums and sheaves - Google Patents
An improved groove design for wire rope drums and sheaves Download PDFInfo
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Abstract
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 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. In other words, the groove contour has 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.
Description
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 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.
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 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.
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 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.
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.
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 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 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 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 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.
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 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 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.
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 (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 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.
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 .
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 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.
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) 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 -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 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 (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 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 ) 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) 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 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) -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 value resides. Equation 3.31 is the partial derivative of equation 3.30 with respect to "0", shown below.
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) - 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~
(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) 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 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°.
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 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 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.
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 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.
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 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 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.
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.
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.
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 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.
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 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.
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.
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 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 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 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 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.
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 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 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.
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 (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 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.
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 .
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 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.
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) 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 -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 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 (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 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 ) 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) 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 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) -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 value resides. Equation 3.31 is the partial derivative of equation 3.30 with respect to "0", shown below.
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) - 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~
(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) 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 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°.
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 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 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.
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 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.
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 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 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.
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.
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.
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 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.
Claims (3)
1. 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 said cylindrical body, said groove having a contour adapted to support the wire rope, said 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, said rope support angle being the included angle from the nominal rope's center to each end of the supporting groove contour on each side of said groove, the improvement comprising said groove contour being parabolic in shape over at least about the upper half of said groove contour.
2. 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 said cylindrical body, said groove having a contour adapted to support the wire rope, said 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, said rope support angle being the included angle from the nominal rope's center to each end of the supporting groove contour on each side of said groove, the improvement comprising said groove contour having a radial length from the nominal rope center to said contour greater than at the bottom of the groove over at least about the upper half of said groove contour.
3. A method of making 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 spirally around said cylindrical body and having a groove pitch, said groove having a contour adapted to support the wire rope, said 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, said rope support angle being the included angle from the nominal rope's center to each end of the supporting groove contour on each side of said groove, the method comprising designing said groove contour such that as the wire rope leaves said groove there is a single line of contact between the rope and the respective side of the groove.
Applications Claiming Priority (2)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| US20699998A | 1998-12-07 | 1998-12-07 | |
| US09/206,999 | 1998-12-07 |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CA2287676A1 true CA2287676A1 (en) | 2000-06-07 |
Family
ID=22768802
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA 2287676 Abandoned CA2287676A1 (en) | 1998-12-07 | 1999-10-27 | An improved groove design for wire rope drums and sheaves |
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| Country | Link |
|---|---|
| AU (1) | AU5933999A (en) |
| CA (1) | CA2287676A1 (en) |
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| CN103693046A (en) * | 2013-12-24 | 2014-04-02 | 兖州煤业股份有限公司 | Rail type separating steel wire rope supporting device |
| US10047848B2 (en) | 2014-07-10 | 2018-08-14 | National Oilwell Varco, L.P. | Sheave with structured web |
| US10557540B2 (en) | 2014-04-14 | 2020-02-11 | National Oilwell Varco, L.P. | Fleet angle tolerant sheave |
| CN113483848A (en) * | 2021-06-28 | 2021-10-08 | 阳光学院 | Water level measuring device for geotechnical engineering investigation with good stability |
| US11279601B2 (en) | 2017-04-03 | 2022-03-22 | National Oilwell Varco, L.P. | Hoisting and tensioning bearing saver |
-
1999
- 1999-10-27 CA CA 2287676 patent/CA2287676A1/en not_active Abandoned
- 1999-11-11 AU AU59339/99A patent/AU5933999A/en not_active Abandoned
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN103693046B (en) * | 2013-12-24 | 2016-01-20 | 兖州煤业股份有限公司 | Rail mounted separating and supporting rope device |
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| US10047848B2 (en) | 2014-07-10 | 2018-08-14 | National Oilwell Varco, L.P. | Sheave with structured web |
| US11279601B2 (en) | 2017-04-03 | 2022-03-22 | National Oilwell Varco, L.P. | Hoisting and tensioning bearing saver |
| CN113483848A (en) * | 2021-06-28 | 2021-10-08 | 阳光学院 | Water level measuring device for geotechnical engineering investigation with good stability |
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