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International Journal of Mechanical Engineering Education Vol 29 No 2

Determination of coefficientof friction from oscillationsunder Coulomb frictiondamping

R. VENKATACHALAM*, Professor of Mechanical Engineering,Regional Engineering College, Warangal - 506 004, [email protected]

* Currently, on a temporary assignment, Associate Professor of Mechanical Engineering, Faculty ofEngineering, Garyounis University, P.O. Box 1308, Benghazi, Libya.

1. NOMENCLATURE

L length of the pendulum

T, T1, T2 tensions

W12 work done between positions 1 and 2

d diameter of the rod, which is acting as a hinge for the pendulum

h1, h2 heights of the bob above the datum

m mass of the bob

KEand PE change in kinetic and potential energies, respectively

semi groove angle of a grooved pulley

angle of wrap on a pulley

coefficient of friction

angular displacement of the pendulum measured from the vertical passing

through the hinge

Received 6th January 2000

The oscillations of a vibrating system in the presence of Coulomb friction damping is well

known. In this paper, an attempt is made to use this concept, to measure the coefficient of friction. For this purpose, a simple pendulum, whose hinge is not free from Coulomb

friction, is considered. An experimental set-up is designed and fabricated. Experiments areperformed with different materials and the coefficients of friction are determined. Satisfac-tory results are obtained. The work described here may be incorporated as a laboratory

experiment in applied mechanics.

Key words: friction, damping, pendulum, free oscillations

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148 R. Venkatachalam

2. INTRODUCTION

In many engineering applications, friction is one of the important design criteria. There are

many situations in which the designer intends to reduce the friction; for instance, hinges,

sliding contact bearings, gears in mesh, etc. There are also occasions where the designer

intends to introduce friction, as in the case of rolling of objects, clutches, and brakes, etc.

The coefficient of friction is an important parameter in the mechanical engineering de-

signs whenever the application involves rubbing of surfaces. The coefficient of friction may

be determined directly by observing the amount of force required to move an object of

known weight, on a horizontal surface. It may also be determined, by measuring the mini-

mum angle of inclination at which the object slides on an inclined plane. Practical difficul-

ties are involved in these methods, owing to the fact that the kinetic coefficient of friction is

less than the static coefficient of friction.

The coefficient of friction may be determined by rotating one friction surface over

another under a known load and measuring the amount of torque required to rotate the

surface. Pal and Basu [1] used a method in which a normal force was applied through a small

friction surface resting on a horizontal rotating friction surface, by means of a cantilever

beam which had stiffness only in the horizontal plane. The bending of the beam in the

horizontal plane was caused by the friction force. The bending deflection of the beam was

measured, from which the friction force was estimated and thereby the coefficient of friction

was calculated.

Venkatachalam and Sitharamarao [2] proposed a method for determining the kinetic

coefficient of friction by generating self excited oscillations. The method involved

measuring the kinetic coefficient of friction from the period of oscillations of a beam kept on

two rotating discs. This method was further extended to measure the coefficient of friction

between two test surfaces, with respect to a standard pair of friction surfaces whose coeffi-

cient of friction was known. The pair of test surfaces was kept at one disc and the standard

pair of friction surfaces was kept at the other disc. With this arrangement, it was found

possible to measure the coefficient of friction from the amplitude of oscillations, instead of

the time period of oscillations. Venkatachalam and Sitharamarao [3] further suggested

another simple technique through which both the static and kinetic coefficients of friction

could be measured. This method involved observing the stick slip motion of a body kept on a

moving surface. The experimental measurements were only displacements and did not in-

volve any time measurements. This made the method more attractive.

In this paper an attempt is made to measure the kinetic coefficient of friction by observ-

ing the oscillations of a simple pendulum to which the friction is introduced at the hinge.

3. DESCRIPTION OF THE MODEL

Fig. 1 shows a schematic arrangement of the model considered. It consists of a fixed round

circular rod to act as a hinge. A wire bent in the for of a U is placed over the rod. The free

ends of the bent wire are fastened to a plate too which the pendulum wire is attached. Thebent wire, the plate, the pendulum wire and the bob together form the simple pendulum.

When it is oscillating, the U shaped wire rubs against the surface of the rod and experiences

the friction which tries to dampen the oscillations. The mass of the wires and the plate are

very small in comparison to the mass of the bob and hence the former is neglected in the

analysis to follow.

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Oscillations under Coulomb friction damping 149

Fig. 1. Schematic arrangement of the model.

4. MATHEMATICAL DEVELOPMENT

If the hinge is frictionless and there are no other losses, then the pendulum exhibits simpleharmonic motion, the amplitude of oscillations remaining constant forever [4], as shown in

Fig. 2(a). But if there are losses, the amplitude 1 on one side of oscillation reduces to 2 on

the other side of the oscillation as shown in Fig. 2(b). In the model considered, the friction at

the hinge is dominating all the other losses such as the one that may arise due to the air drag.

Hence, the effect of friction alone is considered, neglecting all other losses.

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In the analysis, the attention is focused on the decrease of the amplitude of oscillation

from 1 to 2. For this, the principle of work energy [5]

KE= PE+ W12 (1)

is applied between the positions 1 and 2 as shown in Fig. 3. Substituting

(b)(a)


Fig. 2. Oscillations of the pendulum. (a) When there are no losses; (b) when losses exist.

=

=

KE

PE mg h h

0

1 2( )

(2a)

(2b)

where Tis the tension in the pendulum wire.

W d T T 1 2 1 2 1 22 = +( ) ( ) ( ) (2c)

in equation (1), the work energy relation may be obtained as

0 21 2 1 2 1 2= + +mg h h d T T ( ) ( ) ( ) ( ) (3)

where T1 and T2 are the tensions in the bent wire, which are different because of the friction

between the wire and the rod.

For small oscillations, the heights from the datum

h L ii i= =( cos ), ,1 1 2 (4)

may be approximated as

h L ii i =( ) , ,2 1 22 (5)

and the tensions T1 and T2 may be approximated as

T T T mg1 2+ = (6)


and

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Fig. 3. Application of the workenergy principle.

Using equation (5) in equation (3), the decrease in the amplitude of oscillations may be

expressed as

= ( ) ( )d L T T mg1 2 (7)

where

= 1 2 (8)

The tensions T1 and T2 may be related as

Equation (10) gives the reduction in the amplitude in the half-cycle of oscillation. It also

shows that this reduction is constant. Therefore, the reduction in amplitude in one complete

oscillation will be 2, and in two oscillations will be 4, and so on, as illustrated in Fig. 4.

= + ( ) ( ) ( )d L e e1 1 (10)

where is the coefficient of friction between the rod and the wire, and is the angle of

contact of the wire on the rod, which may be taken as in the present case.Using equations (6) and (9) in equation (7), the decrease may be re-expressed as

T T e1 2 = (9)

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This idea of introducing Coulomb friction at the hinge of a simple pendulum to dampen the

oscillations is taken from reference [6].

5. EXPERIMENTATION

Equation (10) may be rewritten for convenience as

= + ln ( ) ln ( )1 1Z Z (11)

where

Z L d = ( ) (12)

In an experiment, if could be measured, then by using equation (11) the coefficient of

friction may be calculated. However, ifL d, which refers to Z 1, equation (11)

does not give any value for . This is because, the development of equation (10) is based on

the application of the work energy, equation (1), with an assumption that the pendulum

moves to the other side of the vertical static equilibrium position by an amount 2 as shown

in Fig. 3.A simple pendulum is constructed as shown in Fig. 1. A mild steel rod of diameter

d= 18 mm, is used to serve as a hinge. A groove is made on this rod in order to prevent the

slipping of the wire on the rod. The length of the pendulum L is chosen as 1 m. A mass of

1 kg is used for the bob. Provision is made to change the U shaped wire which sits on the

circular rod. With this arrangement, experiments may be performed with different materials.

Fig. 4. Oscillations of the pendulum under Coulomb friction damping.

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The pendulum bob may be pulled to one side by an angle 1 and released, allowing the

pendulum to swing to an angle 2 on the other side. With these values of 1 and 2, the

coefficient of friction may be calculated using equation (11). However, in order to reduce

the experimental errors, the amplitude may be measured after two complete oscillations; thatis, 1 and 5 shown in Fig. 4 may be measured. As shown in Fig. 1, a circular scale is

provided near the bob. From the arc lengths recorded on this scale, the angular displacements

may be calculated with less errors.

6. RESULTS AND DISCUSSION

Experiments are performed with different wires and the coefficients of friction are calcu-lated. One typical experimental observation is presented in Table 1.

Table 1. A typical experimental observation

Trials 1(degrees)

5(degrees)

(1 5)

(degrees)

= (1 5)4

(degrees)

1

2

3

30

20

10

28.0

17.9

8.0

2.0

2.1

2.0

0.50

0.52

0.50

It may be noticed that the value of turned out to be same irrespective of the value of1.

The same trend has been observed in other experiments in which the wires of different

materials are tried. Table 2 presents the results of the other experiments.

Table 2. Results of the experiments with

different wires

Sample 0

Copper wire

Aluminium wire

Steel wire

Nylon wire

0.6

0.5

0.45

0.4

0.42

0.34

0.30

0.26

The values of the coefficients of friction obtained for different materials, presented in Table

2, appear to be more than the values available in the design data books. This may perhaps be

due to the groove made on the rod in which the wire slides. The effective coefficient of

friction, when a circular wire is in a grooved member is

effective = sin (13)

where, is the semi-groove angle. The coefficients of friction calculated with the present

experimental set-up must have already taken into account the effect of the groove angle.

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7. CONCLUDING REMARKS

The significant conclusions that may be drawn on the basis of the present work may be

summarized as follows.

(i) An experimental method of determining the coefficient of friction is proposed.

(ii) The experimental set-up is very simple in construction, and the experimental observa-

tions are also very simple.

(iii) Experiments are performed with different materials and the coefficients of friction are

calculated.

(iv) The decrease in amplitude of oscillation is found to be constant, as predicted by the

mathematical analysis.

(v) The proposed method may be very useful in obtaining the coefficient of friction

between belt/rope and plane/grooved pulleys.

(vi) The experiments may be performed with different angles of wrap .(vii) The work presented in this paper involves the mathematical solutions of a physical

problem followed by experimental investigations. The work can be incorporated as a

laboratory experiment with a view to illustrating a simple principle mathematically

and experimentally.

ACKNOWLEDGEMENT

The author wishes to acknowledge the Department of Mechanical Engineering, Garyounis

University, Benghazi, Libya, for providing various facilities to carry out the work.

REFERENCES

[1] Pal, D. K. and Basu, S. K., Hydrostatic lubrication of plastic guides, Proceedings of Third AllIndia Machine Tool Design and Research Conference, The Indian Institute of Technology, Bombay,

India (1969).[2] Venkatachalam, R. and Sitharamarao, T. L., Determination of coefficient of friction by generating

self excited motion, International Journal of Mechanical Engineering Education, 14(1), 2330,1986.

[3] Venkatachalam, R. and Sitharamarao, T. L., Determination of static and kinetic coefficients of

friction by generating stick slip motion, International Journal of Applied Engineering Education,3(2), 131137, (1987).[4] Beer, F. P. and Johnson Jr., E. R., Vector Mechanics for EngineersDynamics, McGraw Hill Book

Company, New York, pp. 946948, 1990.[5] Meriam, J. L. and Kraige, L. G.,Engineering Mechanics, Volume 2Dynamics, 3rd edn. John Wiley

and Sons Inc., New York, pp. 460465.[6] Steidel Jr., R. F.,An Introduction to Mechanical Vibrations, 3rd edn., John Wiley & Sons Inc., New

York, p. 194.

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