Module STAY SAX Laboratory Teaching Assistants UNIVERSITY OF LIVERPOOL Year 1 Laboratories 9: TBC BF Belt Friction ENGG110 Solids and Structures 1 School of Engineering 2 Introduction Belts provide a convenient means of transmitting mechanical power between shafts, and are widely used in vehicles and machinery. On automobiles, ancillary components such as the electrical generator (alternator), the water pump, the power steering pump, the air conditioning pump are usually driven from the engine by belts. On some commercial vehicles, additional ancillaries such as air compressors may also be belt-driven from the engine. Friction provides the mechanism for transmission of power between a belt and pulley. As well as providing drive and power transmission, belts in the form of the band-brake can be used to prevent or to arrest motion: for example, passenger lift and crane winch winding mechanisms. The maximum power transmission on a belt-drive system is limited by the amount of friction available between the belt of the pulley, by the angle that the belt wraps around the pulley (the lap angle), and ultimately, by the strength of the belt itself. The cross-section of the belt may be rectangular (flat), wedge (V) shaped, or circular. V-section belts are often employed with V-grooved pulley rims to prevent belts slipping off and to increase the performance. Toothed belts - employed where no appreciable slip can be tolerated, such as in the case of timing-belts driving camshafts on internal combustion engines - require a different analysis to that used here. 2.1 Theory and Equipment The theory is explained in the context of the experiment you are about to do. The apparatus is depicted in Fig. 1. The belt is assumed inextensible, completely flexible in bending, and of negligible mass. It passes around the circular pulley, the contact length subtending an angle 8 (called the lap angle) at the pulley's centre. The pulley can rotate about its centre. The theory assumes that when slippage occurs between belt and pulley, it occurs simultaneously at all points of contact over the lap angle 8 (see Fig. 1). The right hand end of the belt is anchored on a peg, so the belt will essentially not move during the procedures. The portion of the belt in contact with the pulley is held stationary, in equilibrium, under the action of the tension forces T₁ and T₂ and the friction and normal reaction forces exerted on it by the pulley. In the experiment, the pulley will be driven by a clockwise torque (moment) G. Let us assume that T₁ is constant and that the torque G is increased steadily from zero; a point will eventually be reached at which the pulley will start to slip past the stationary belt and hence to rotate. When the belt is on the point of slipping, simple theory for a flat belt predicts that T₁ T₂ =e48 where is expressed in radians. μ is the dimensionless coefficient of static friction. For a well-fitted V- belt with the cross-section shown in Fig. 2, Eqn (1) has to be slightly modified to μe ve sing 17=e T₂ (1) 3/11 (1a) Safety Students are reminded that they are required by Law to comply with the Department's basic rules of laboratory safety, given to them at the start of the semester. The principle hazard in this experiment relates to potential injury - particularly to toes and feet - caused by metal weights of up to 5kg falling. Care should therefore be exercised at all times and particularly when attaching and removing the weights. All weights and hangers should be removed from the apparatus before attempting to alter the lap angle. You are advised not to stand with feet where weights could fall. Notation Symbol g G m, M R T В μ 0 ● Details Acceleration due to gravity (9.81) External drive torque applied to pulley Mass Pulley Radius Belt Tension V-Belt Angle Upon successful completion of this lab, you should be better able to: Coefficient of Static Friction between Belt and Pulley Lap Angle 1 Aim & Objectives The aim of this lab is to help illustrate and reinforce concepts taught in Solids and Structures relating to static friction, forces and equilibrium. The friction in a belt and pulley system will be investigated and quantified. ● ● Collect and record data; ● Understand the mechanics of frictional forces between belts and pulleys; Appreciate the applicability and the limitations of the theory; Carry out laboratory experiments; Estimate accuracy and assess errors; Understand Least-Squares data fit. The technical objectives of this lab are to: Investigate how well the standard mathematical model describes belt friction; Determine the coefficient of static friction between a flat belt and a steel pulley; Compare the effectiveness of rectangular and V- section belts; Simulate the effect of wear or bad fit on a V-section belt. Units m.s2 N•m kg m N deg, rad 2/11 deg, rad 4 Experimental Procedure and Data Gathering This section describes what you actually need to do during the lab session. Read it especially carefully! 4.1 Part 1 - Flat belt a. Place the flat belt in the groove on the pulley. b. Adjust the anchor point of the belt to give a lap angle of 60°. c. Add mass to the hanger on the left hand side to achieve M=1kg d. Increase the mass m on the string hanger (right hand side) until the pulley begins to rotate. 4.2 e. f. g. a. Record clearly in Table A.1 (see Annex 2, page 11) the value of m at the onset of rotation. DO NOT PERFORM ANY CALCULATIONS AT THIS STAGE! Concentrate on taking careful readings and accurately recording them! Complete the Tables in black ink! Repeat steps d-e for values of M of 2kg; 3kg; 4kg; 5kg. Repeat steps c-f for lap angles of 90°, 120°, 150° and 180°. Part 2 - V-belt (new and worn) Choose just one lap angle for these tests. Make a note of the lap angle in the captions under Tables A.2 and A.3 in Annex 2. b. Replace the flat belt with a V-belt, putting the belt in its CORRECT slot, i.e. so that the belt is NOT TOUCHING the bottom of the channel. C. For each value of M from 1kg - 5kg, determine the maximum m which can be supported, entering the results into Annex 2 Table A.2. d. Repeat (c) with the V-belt placed in the groove that makes contact with the bottom of the belt, rather than with its two sloping sides. This simulates a worn or badly-fitted V-belt. Enter the results into Annex 2 Table A.3. 5 Data Analysis The analysis described in this section is all to be conducted after the experiment is finished. It will involve calculations and graph plotting. You should use the template "Word" document provided. You may either complete the calculations by hand/using a calculator. A partially completed Microsoft Excel spreadsheet called "BF Spreadsheet.xls" is provided which you may wish to use. Likewise, the graphs can be produced from within Excel. You may alternatively draw them carefully by hand, or produce them using any other suitable software that you are familiar with, then scan or paste them into the report template. Whatever method you use to produce them, graphs must be correctly annotated, with titles, axis labels and units where applicable. If you decide to use "BF Spreadsheet.xls" you will need to transfer your raw data into the appropriate columns, then either configure the spreadsheet to perform the necessary computations (which is the recommended way) or carry them out yourself by other means. You will need to determine "best fit" lines. You can either do this by estimating, but more marks will be awarded for accurate calculations, e.g. using the theory presented in Annex 1. 5/11 M T Mg R m mg Anchor point ۳*۶ Fig 1. Flat Belt and Pulley Therefore, if is known, if T₁ is given, and if T₂ can be measured or deduced at the point of slip, a value for the coefficient of friction μ satisfying Eqn (1) can be estimated by rearranging Eqn (1). But to determine whether the equation adequately describes the physics for (all) other values of the variables we would need to repeat the process for different combinations of values of the variables. V-Belt Our procedure will be to fix 0 and T₁ and determine the T₂ value at which slip occurs; then increase T₁ and find the new corresponding value for T2, all the time keeping 0 constant. We then plot T₁ vs T₂. If the theory were accurate, we would expect a straight line graph passing through the origin with gradient = e. G(=mgR)=TR-T₂R=(T₁-T₂) R This implies that Pulley How can we determine the tensions T₁ and T₂? Rather than directly measure them, we deduce their values indirectly. First, we hang a known mass M on the free end of the belt, as shown in Fig.1. The belt supports the mass by generating an internal tension: precisely the force T₁ in Eqn (2) and Fig. 1. T₂=(M-m)g Fig. 2: V-Belt in Cross-Section T₁ = Mg The pulley remains at rest because the right side of the belt exerts an equal and opposite moment on it. At this point, T₂ = T₁. Next we apply the clockwise drive-torque to the pulley by hanging mass m on the light string that is attached to the pulley. This will cause T₂ to decrease. The drive torque G is given by Eqn (3). 4/11 G = mgR (3) After a certain point, T₂ will decrease so much that the pulley will slip. For as long as the pulley remains in equilibrium, right up to the point of slip, the clockwise drive torque is balanced by a nett anticlockwise torque exerted on the pulley by the belt, the forces T₁ and T₂ exerting anti-clockwise and clockwise torque components respectively. The equation of rotational equilibrium at all points right up to the onset of slip can be written (2) (4) (5)