BELT DRIVES

An investigation into the relationship between tensions in a slipping pulley system

JONATHAN USBORNE 13-MAR-2012 SUMMARY

Power transmission and gearing are essential components of many mechanical systems. Belt drives provide both at low cost and generally high efficiency. Their versatility has lead to extensive application and a general understanding of the engineering principles associated with such systems is paramount. This report outlines an investigation to compare the mechanical theory of belt-driven transmission to practical observations by measuring tension in a slipping belt. Measuring the tension before and after contact with a driving pulley at several different angles of contact allows for the comparison of torque, power and efficiency under varying loads and angles. A peak in efficiency of 61.48% was observed and the relationships between belt tension ratios examined.

ME10010: Belt Drive Jonathan Usborne

2

TABLE OF CONTENTS

1. Introduction .................................................................................................... 2 2. Theory ........................................................................................................... 3 3. Method .......................................................................................................... 4 4. Results .......................................................................................................... 5 5. Discussion ..................................................................................................... 7 6. Conclusion ..................................................................................................... 8 7. Bibliography ................................................................................................... 8 Appendix A: Raw Data .................................................................................... 9 Appendix B: Derived Data ............................................................................. 10

1. INTRODUCTION

Belt drives are one of the oldest and still most common methods of power transmission. Such low cost, low maintenance systems provide numerous design advantages over alternative approaches with efficiency levels averaging 95%[1]. The introduction of Vee belts allowed for higher levels of transmission over larger areas, and further provided the opportunity for more dynamic gearing via variable diameter pulleys. Slippage of belts over pulleys acts as both an advantage in terms of machinery protection / redundancy, but also a shortcoming when it comes to efficiency. Unintended slip can occur due to incorrect pulley profiles, insufficient friction or excessive torque. This report outlines an investigation into the relationship between the tensions in a slipping pulley (and consequently the torque) comparing real power efficiency to theoretical estimations.

ME10010: Belt Drive Jonathan Usborne

3

2. THEORY

Pulley systems rely on friction to allow the grip of the belt on the pulley to transmit power. To enable this however the belt must be in tension, even whilst stationary. At rest the tension in the belt will, by definition, be equal around the entire system. In order to transmit power however, the driving pulley will be required to exert a torque to move the belt, resulting in imbalanced forces either side. The increased tension on the side of the belt leading onto the pulley will be offset by the deceased tension on the other side. Knowing Coulomb's law of friction and given a coefficient of friction of , an angle of contact between belt and pulley of , and a Vee belt angle of 2 (illustrated in fig 1), the ratio of lower to higher tension (T1 and T2 respectively) can be derived by integrating the tension difference (required to produce a torque) with respect to between limits = 0, [2], giving !! = !" !"#! The string used in this experiment simulates a flat belt as opposed to a Vee. In other words 2 = 180. Given sin 180 2 = 1, eqn [1] can be simplified to !! = !" The torque () exerted is defined as [2]. Given the two opposing forces either side of the pulley, the net torque is calculated through eqn [3] where the pulley radius is . = ! ! The mechanical power of a wheel under torque is the product of the torque and angular velocity[3], and thus given a rotational speed !"#, = !"# = 2 !"#60 The power of the electric engine driving the pulley is known to be = ; as such the efficiency of the system can be given by

= !"#!" = ! ! 2 !"# 60

(1)

(2)

(3)

(4)

(5)

ME10010: Belt Drive Jonathan Usborne

4

3. METHOD

The apparatus consists of a variable DC motor directly driving a fixed radius pulley of 50mm. A flat belt pinned to a load cell supports a variable mass at one end. Several idler pulleys around the rig allow for the angle of contact between the belt and driven pulley () to be set at 2, , 3 2 and 2 rad as illustrated in figure 2. The DC motor was delivered a constant 10V supply during the experiment and connected to ensure a count-clockwise rotational velocity on the pulley such that the tension was decreased on the load cell and conversely increased on the mass. (! ! > 1) T2 was altered by applying weight in 1N stages to the suspended end of the belt. At each stage T1 (as measured by the load cell) was recorded. This process was repeated for all four measures of . For = 2 the current over the motor and rotational speed of the driven pulley !"# , as measured by the DC power supply and a tachometer respectively, were also recorded for use in efficiency calculations. Figure 1: Vee belt angle Figure 2: Pulley configuration

= 2! =

= 3 2! = 2

weight

load cell

T1

T2

50mm 2

ME10010: Belt Drive Jonathan Usborne

5

4. RESULTS

Tension in the belt after [T1] and before [T2] the pulley (as measured in N by the load cell and weight respectively) was tabulated and is reproduced in Appendix A. T1 was plotted as a function of T2 for all four values of . Figure 3: Lower to higher belt tension at varied angles of contact

The ratio ! ! is defined as the belt tension ratio. Using a linear line of best fit to estimate the gradient of the four plots the ratios were established. Table 1: Belt Tension Ratios

Angle of Contact () (rad) 1.5510 2.2508 3.1446 4.2673 Knowing the BTR and for each dataset, the coefficient of friction () can be derived from the gradient of the relationship rearranged from eqn 2 giving the linear function: !! = (6)

ME10010: Belt Drive Jonathan Usborne

6

Figure 4: Best fit line through eqn [6]

The gradient of the plot in fig 4, and hence the coefficient of friction (), was calculated as 0.2146. Using this figure, and respective values of , a theoretical plot of BTR was produced and compared to the experimental observations at those four points. Figure 5: Theoretical and experimental belt tension ratios

ME10010: Belt Drive Jonathan Usborne

7

At position = 2 data regarding the current across the motor and the rotational velocity of the pulley were recorded for each change in load. Input and output power was then derived and an overall efficiency established using eqn [5]. The torque at each data point (given by eqn [3]) was plotted as a function of efficiency. The raw values are reproduced in Appendix B. Figure 6: Torque (Nm) efficiency curve at = 2

5. DISCUSSION

Figure 5 illustrates a general conformity between experimental and hypothetical values regarding belt tension ratios, thus positively affirming the overall validity of the theory outlined in section 2. The slight divergence of practical data points as increases can be attributed to marginal systematic error. The system efficiency behaves as expected, parabolically with respect to the torque, and a maximum efficiency (found by differentiating the quadratic line of best fit) was 61.48% at a torque of 0.222Nm. Whilst this is significantly lower than predicted, it can be largely credited to estimations and assumptions made during calculations. Most importantly this system is of a constantly slipping belt as opposed to a traditional transmission system. Another large factor is the assumed efficiency of the electric motor driving the main pulley. In reality there would be power losses, due predominantly to internal friction, however for this report power loss through the engine was not considered.

ME10010: Belt Drive Jonathan Usborne

8

6. CONCLUSION

The maximum observed efficiency of the system was 61.48% and the coefficient of friction estimated at 0.2146. Whilst discrepancies existed between these figures and theoretical estimations, they were explained by the constantly slipping nature of the apparatus. Most significantly the general theory was confirmed, suggesting the efficiency and belt tension ratio are related to the angle of contact between the belt and pulley such that an increased angle (suggesting larger contact area and higher nominal friction) will increase the ratio. The efficiency profile highlighted the need to properly gear and load the system, optimising efficiency by maintaining the optimal level of torque through control of the tension differential and pulley geometry. It further identified the potential advantages of slip in situations where torque is excessive. Characteristics and trends were observed which revealed the main advantage in belt-drive transmission systems. On top of factors related to cost and efficiency, the level of control available through several variables, not to mention belt type, means the continued widespread application of pulley systems is unsurprising.

7. BIBLIOGRAPHY

[1] Carlisle Power Transmission Products LTD. Energy Loss and Efficiency of Power Transmission Belts. Belt Technical Center. Springfield, Missouri Available online at http://bit.ly/zfOfLX (accessed 12-MAR-12)

[2] Dunn, D. J. Solid Mechanics Dynamics Pulley Drive Systems. Freestudy. Available online at http://bit.ly/zXJTT8 (accessed 12-MAR-12)

[3] Darling, J. Belt Drive Laboratory Exercise (ME10010 lab sheet). University of Bath. 10-FEB-12

ME10010: Belt Drive Jonathan Usborne

9

APPENDIX A: RAW DATA

T1 (N) T2 (N) (A)

0.98 0.75 0.35 0.1 0.1 1.5 1382 1.96 1.15 0.7 0.4 0.3 2.1 1333 2.94 1.8 1.15 0.75 0.5 2.8 1282 3.92 2.35 1.65 0.95 0.8 3.4 1234 4.9 3.05 2.15 1.3 0.9 4 1183 5.88 3.7 2.55 1.45 1.2 4.6 1133 6.86 4.3 2.9 1.95 1.4 5.3 1082 7.84 4.95 3.4 2.3 1.6 6 1027 8.82 5.7 3.8 2.55 1.9 6.5 978.7 9.8 6.3 4.2 2.9 2.2 7.2 913.2

ME10010: Belt Drive Jonathan Usborne

10

APPENDIX B: DERIVED DATA

T1 (N)

0.98 0.75 0.35 1.96 1.15 0.7 2.94 1.8 1.15 3.92 2.35 1.65 4.9 3.05 2.15 5.88 3.7 2.55 6.86 4.3 2.9 7.84 4.95 3.4 8.82 5.7 3.8 9.8 6.3 4.2