The Limiting Value of the Fleet Angle of a Rope Running off a Sheave

Moses F. Oduori, E–mail: foduori@uonbi.ac.ke and mfwedida@yahoo.com. (Corresponding author).Thomas O. Mbuya, E–mail: tmbuya@uonbi.ac.keDepartment of Mechanical Engineering, University of Nairobi, P. O. Box 30197, NAIROBI, KENYA.

Abstract

The problem of determination of the upper limiting value of the fleet angle of a rope

running off a sheave is encountered in the design of hoisting mechanisms such as that of an

electric overhead travelling (EOT) crane, or an elevator, among other applications. In this

paper, a mathematical expression for the determination of this limiting value of the fleet

angle is derived from first principles. The expression obtained here is then compared to,

and contrasted with another that is given in the literature. The earlier expression in the

literature is found to generally overestimate the upper limiting value of the fleet angle.

Introduction

The problem of determination of the upper limiting value of the fleet angle (Fig. 1) is

encountered in the design of hoisting mechanisms such as that of an electric overhead

travelling (EOT) crane or an elevator [1]1 If, in a given application, the actual maximum

fleet angle is allowed to be greater than the limiting value, intense abrasion will occur

between the rope and the sides of the sheave groove. Moreover, the rope will be pinched

by the upper edge of the sheave groove, leading to high contact stress that may result in

1 Numbers in square brackets refer to cited literature as listed in the references section.

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intense abrasive wear of both the rope and the sheave, possible crushing of the rope, and

strand nicking – the result of adjacent strands pressing and rubbing against one another.

The end result is shortened rope and sheave service life [2,3].

Iwai and Ishikawa [4] present a graphical method for determining this limiting value.

However, graphical methods have the disadvantage of having to be laid out to scale, in

entirety, in every case, and can be time consuming. Where an analytical formula is

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available, it is generally to be preferred to graphical methods. According to Rudenko [5],

the upper limiting value of the fleet angle may be determined by use of an equation that

can be written in the following form ( see Fig. 3 ):

(1)

where;

is one half of the sheave's groove angle,

max is the upper limiting value of the fleet angle,

D is the nominal diameter (pitch circle diameter) of the sheave,

h1 is the depth of the sheave's groove measured from the top of the groove to the

bottom of groove,

1 is the ratio of the depth of the sheave's groove to the nominal diameter of the

sheave; hence .

Unfortunately, Rudenko [5] presents neither the theoretical basis nor the procedure of

derivation of equation (1). In this paper, an equation for the determination of max is

derived from first principles and then its application is discussed, compared to, and

contrasted with equation (1).

Analysis

The co-ordinate system to be used in this analysis is illustrated in Fig. 2. The sheave may

rotate about the y–axis of the fixed co–ordinate system. In practice, sheave form and

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dimensions would be according to standards commonly used in the crane and elevator

industry.

It should be evident in Fig. 3 that the following relationship holds true:

(2)

where;

D is the nominal diameter (pitch circle diameter) of the sheave,

h is the nominal depth of the sheave's groove measured from the top of the groove

to the centreline of the rope (pitch line).

It shall be assumed that the surfaces of the sides of the sheave's groove are conical. Thus, a

diametrical section of the sheave's groove would be a straight–sided V–shape, except for the

bottom of the groove, which is rounded.

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The design objective is to limit the fleet angle so that the rope shall not be forced to

sharply bend over the edge of the sheave's groove, at the point S (Fig. 3) where it runs off

the sheave. As mentioned earlier, such an occurrence would shorten the service life of

the rope. Thus, the limiting value of the fleet angle is the largest value that may be used

without the rope being so bent over the edge of the sheave’s groove. This sharp bending

of the rope over the edge of the sheave’s groove would be avoided if the rope should run

off the sheave at a tangent to the surface of the sheave's groove.

In Fig. 3, if i, j and k are unit vectors in the x, y and z directions respectively, the quantity

denoted r can be represented by the following vector equation:

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Thus, application of the Pythagorean theorem, in Fig. 3, gives the following equation:

(3)

Hence,

(4)

With the use of trigonometry in Fig. 3, it can be shown that:

(5)

Thus, by using equations (4) and (5), one obtains the following:

(6)

The value of the fleet angle can then be found by use of the differential calculus as

follows:

(7)

Furthermore, the following too is evident in Fig. 3:

(8)

Thus from equations (4) and (8):

(9)

Hence, using equations (7) and (9), one obtains:

(10)

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Application

The First Approach

As a design approach, it may be assumed that the design of the sheave is as shown in

section A–A of Fig. 3, so that the following constraint is imposed upon the dimensions of

the sheave:

(11)

Substituting equation (11) into equation (4) leads to the results:

(12)

Using equation (12) and Fig. 3, the following equations are found to hold true:

(13)

With the constraints expressed in equations (11) and (12) in play, equation (9) becomes:

(14)

Hence, using equations (10) and (14), one obtains an expression for the limiting value of

the fleet angle as follows:

(15)

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The Second Approach

A more general, alternative design approach may be adopted in which the value of h is

not in any way constrained, in relation to the value of D. Then, with reference to Fig. 3,

it follows that;

(16)

Hence, by use of the Pythagorean theorem in Fig. 3:

(17)

which may be simplified into:

(18)

Note that:

Therefore, it follows that:

(19)

Introducing the notation , one then obtains the following:

(20)

Discussion

As was seen earlier, the equation according to Rudenko is as follows:

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(21)

Sheave groove dimensions are normally standardized and Table 1 below gives a sample

of such dimensions, derived from Japanese Industrial Standards JIS B 8807 [4].

Table 1 – Some Standard Sheave Data According to JIS B 8807

Wire Rope Diameter, d mm

Sheave Pitch Diameter, D mm

12.5 250.5 0.05489 0.07984 1.4545

14 280 0.05536 0.08036 1.4516

16 315 0.05397 0.07937 1.4706

18 355 0.05352 0.07887 1.4737

20 400 0.05375 0.07875 1.4651

Average Values 0.0543 0.07944 1.4631

Standard Deviations 0.000789 0.000673 0.009732

Coefficients of Variation 1.4535% 0.8473% 0.6651%

Using the information from Table 1, we may write the following relationship:

(22)

Now, from equations (21) and (22), for the sheave dimensions used in Table 1,

Rudenko’s equation may now be rewritten in the following form:

(23)

which facilitates the comparison of Rudenko’s equation and the authors’ equation.

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For purposes of comparison, two quantities denoted 1 and 2, which are the normalized

forms of Rudenko’s equation (21) and the authors’ equation (20), were computed using

the following mathematical equations:

(24)

(25)

Values of 1 and 2 are plotted against those of in Fig. 4. This figure reveals that

Rudenko's equation consistently overestimates the upper limiting value of the fleet angle

for the full range of plotted values.

Further comparison of fleet angles as calculated by Rudenko’s formula, and by the

authors’ formula are given in Table 2 below, for a groove angle of 38 degrees. The

consequences of allowing the maximum value of the fleet angle to be too large have

already been mentioned in the introduction to this paper. Suffice to say that

overestimating the upper limiting value of the fleet angle is unacceptable.

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Table 2 – Comparison of the Results of Rudenko’s and the Authors’ Equations

Wire Rope Diameter, d mm

Sheave Pitch Diameter, D mm

Rudenko’s max, degrees

The Authors’ max, degrees

12.5 250.5 0.05489 19.815 18.717

14 280 0.05536 19.888 18.780

16 315 0.05397 19.669 18.591

18 355 0.05352 19.597 18.529

20 400 0.05375 19.634 18.561

Average Values 19.721 18.636

For the values of r occurring in Table 2, above, the upper limiting values of the fleet

angle, as calculated by Rudenko’s equation is 5.823% higher than that obtained by use of

the Authors’ equation.

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In the application of equation (10), two alternative approaches were presented. The first

approach constrains the value of h to be equal to 0.207D while the second approach

allows the value of h to be varied freely. However, there are other factors that come into

consideration when the values of D and h are to be determined. For example, a

consideration of the rope’s flexural fatigue imposes a lower limit on the nominal sheave

diameter, which depends on the type and diameter of the rope to be used. This

relationship is adequately dealt with in the relevant literature. It has also been reported

that the depth of sheave grooves should be at least 1.5 times the rope diameter and that

one half of the sheave’s groove angle should not exceed 26 degrees [3]. Thus, one may

not be entirely free to fix the value of h to be equal to 0.207D, for example, as the values

of both these quantities (h and D) are influenced by factors such as the type and diameter

of the rope to be used.

In a given design situation, the volume of the sheave may be estimated to be that of two

identical but longitudinally opposed conical frusta (Fig. 5), which can be calculated by

use of the following formula [6]:

(26)

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Thus, it will be found that for given values of D, deeper sheave grooves lead to larger,

heavier sheaves, which is undesirable. Grooves should not be made unnecessarily deep.

The literature is seemingly inconsistent in recommending the upper limiting values of the

fleet angle. Recommended values range from 0.250 to 4.750 [2,3,4] and one

recommendation gives 150 [1] as the upper limiting value for EOT Crane applications.

The problem with most of these recommendations is that they do not give a basis upon

which the recommendations are made. Note that this paper looks at the upper limiting

value of the fleet angle from the point of view of a rope running off a sheave. In

applications utilizing sheaves and drums/barrels, the maximum value of the fleet angle

may be farther limited by the phenomenon of the rope running onto and off a

drum/barrel.

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Conclusions

(1) The problem of the determination of the upper limiting value of the fleet angle of a

rope running off a sheave is relevant to the design of hoisting mechanisms such as

those of cranes and elevators.

(2) An equation that may be used to determine the upper limiting value of the fleet angle

of a rope running off a sheave was derived from first principles. The use of this

equation was discussed and compared to one that was given by Rudenko [2] and it

was found that Rudenko's equation overestimates this upper limiting value.

(3) Two design approaches were presented and their effect on the size of the sheave

discussed. The second approach was found to be flexible in its application and

therefore should be a preferred choice for designers.

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Nomenclature

is one half of the sheave's groove angle,

max is the upper limiting value of the fleet angle,

D is the nominal diameter (pitch circle diameter) of the sheave,

h is the nominal depth of the sheave's groove measured from the top of the groove to the

centreline of the rope (pitchline),

h1 is the depth of the sheave's groove measured from the top of the groove to the bottom

of groove, thus ,

is the ratio of the nominal depth of the sheave's groove to the nominal diameter of the

sheave; hence ,

1 is the ratio of the depth of the sheave's groove to the nominal diameter of the sheave;

hence .

References

(1) Oduori, M. F. and Nyauma, G. F. (1979). “The Design of an Electric Overhead

Travelling Crane”, Final Year Project Report Submitted in Partial Fulfilment for the

Award of the Degree of Bachelor of Science in Mechanical Engineering,

Department of Mechanical Engineering, University of Nairobi, Nairobi.

(2) Dickie, D. E. (1975). “Crane Handbook”. Construction Safety Association of

Ontario, Revised UK Edition published in 1981 by Butterworths & Co. Ltd.

London.

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(3) Dickie, D. E. (1975). “Lifting Tackle Manual”. Construction Safety Association of

Ontario, Revised UK Edition published in 1981 by Butterworths & Co. Ltd.

London.

(4) Iwai, Minoru and Yoshio Ishikawa (1989). “Modern Machine Design and Drawing

Practice, Number 5. Part 4 – Design of the Power Winch. Pp 150-151. Ohm Sha,

Tokyo, Japan (in Japanese).

(5) Rudenko, N. (1969), “Materials Handling Equipment”, 2nd Edn. Mir Publishers,

Moscow.

(6) Carmichael, R. D. and. SMITH, E. R (1962), “Mathematical Tables and Formulas”,

Dover Publications, Inc., New York.

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