Engineering mechanics and hydraulics

Transcription

1 Engineering mechanics and hydraulics In this section we will examine the nature of friction, as well as looking at stress and strain. Friction When you walk on a rough surface, such as a footpath, then try to walk on an ice rink there are two very different results. To be able to walk on a rough surface such as the footpath, friction is used to allow you to make progress. To try to walk on an ice rink, where there is little or no friction, is almost impossible. Friction allows you to walk, the lack of friction allows you to iceskate, but not to walk on ice. Friction is the resistance to motion that occurs when two bearing surfaces slide, or tend to slide, over each other. 1 lace your right index finger, against the palm of your left hand. 2 Very lightly push your finger across your palm. 3 epeat, but slightly increase the downward force. 4 ow really increase the downward force and try moving your finger. Did you notice that as you increased the downward force, that the friction force increased? Did you also notice that your hand got hotter as the force increased? Were you able to apply a big enough downward force that prevented you moving your finger across your palm? Your palm has just applied a braking force to your finger. epeat the activity, pushing your finger against various surfaces such as your desk top, a book cover, your computer keyboard, a TD set square and an eraser. art 3: Engineering mechanics, hydraulics and communication 1 3

2 Did you notice that the same results occurred as before? Did you also notice that the frictional resistance was different for the different materials used in the activity? Were you again able to apply a braking force? Did the braking force vary with the different material? ow you will see how these activities apply to friction forces. A friction force is the reactive force opposing the movement of two touching surfaces. The friction force varies directly with the applied force that pushes the bearing surfaces together, and therefore with the normal reaction. F Applied force ushing surfaces together Frictional force (reactive) Opposing motion ormal reaction Applied force tending to move the body Figure 3.1 Analysis of forces Friction forces in braking systems Friction forces occur in braking systems due to the reaction between the specially compounded materials of the brake linings/pads, and the cast iron or aluminium alloy metals of the brake drums/discs. How brakes work Brakes are able to slow or stop a moving vehicle by retarding the rotation of the wheels through developing a controlled friction that converts kinetic energy of the vehicle into heat (energy). The heat energy is then dissipated into the surrounding air through the brake drums or brake discs. KE = 1 2 mv2 From this formula, when brakes are applied a combination of the velocity of the vehicle and mass of the vehicle determine the amount of kinetic energy that must be converted to heat energy. However, it is actually the 4 Braking systems

3 rolling friction between the tyres and the road surface that eventually brings the vehicle to a stop. Coefficient of friction Coefficient of friction µ (Mu) is the ratio of the limiting frictional resistance to the normal reaction. The coefficient of friction is a constant for any two materials in contact. m = F Coefficient of friction in braking materials If the coefficient of friction between the materials of the brake liners/pads and the materials of the brake drums/discs is excessive, the abrasion would quickly wear down both the liners/pads and the drums/discs. A high coefficient would also cause the brakes to lock. Brake materials are therefore manufactured with a range of coefficients from low friction, through to high friction, art 3: Engineering mechanics, hydraulics and communication 1 5

4 Analysis of friction problems Friction force Friction force is a reaction or a reactive force that opposes motion or pending motion that occurs due to an applied force. F = m Force analysis Analysis should commence with the determination of the direction of the friction force. The friction force should then be drawn on the force analysis diagram. All other forces acting on the body should then be drawn on the analysis diagram. These forces should then be drawn as a freebody diagram that shows only the previously analysed forces. ote the friction force always opposes motion or pending motion. = Applied force m = mass of body g = 9.8 m/s 2 = ormal reaction = Frictional resistance Force analysis Free body diagram Figure 3.2 Force analysis, free body diagram Solve, using the linear equilibrium equations; SH = 0 SV = 0 Two important formulae which must be known. Limiting friction Limiting friction is the frictional resistance acting when a body is on the point of moving. 6 Braking systems

5 Coefficient of friction µ (Mu) and friction formula Coefficient of friction µ (Mu) is the ratio of the limiting frictional resistance to the normal reaction. m = F Alternative method of writing formula F = m ote: this formula only applies when limiting friction is involved. Worked example 1 In the following five examples different forces are being applied to a body on a horizontal plane. The problem is usually described in words accompanied by a diagram. The diagram is called a space diagram. You will use this diagram for the analysis of the problem and will refer to the diagram as an analysis diagram. To solve the problem: 1 Analysis diagram The analysis diagram is used to determine and show all of the forces acting on the body. As the friction force opposes motion or pending motion the first step must be to determine and show the friction force on the analysis diagram. Having shown the friction force, show all of the other forces acting on the body. emember that if the body is to be in equilibrium the two linear equilibrium equations, S H = 0, S V = 0, must be satisfied. 2 Free body diagram The free body diagram is used to show only the forces acting on the system. It is easier for analysis if the forces are drawn with the arrows pointing away from the point of intersection as shown in the solutions. This method uses the principle of transmissibility, and thus does not alter the question. 3 Equation for Write an equilibrium equation for using the two linear equilibrium equations, S H = 0, S V = 0 and F =µ. ote: the friction force always opposes motion or pending motion. art 3: Engineering mechanics, hydraulics and communication 1 7

6 Body at rest Analysis diagram Free body diagram Equation for = 0 o horizontal force is acting Body at rest force horizontal (given) = (given) (ÂH = 0) Body at point of moving force horizontal = m = m (given) Body at point of moving force downward, cos 30 sin 30 = m = m( + sin 30 ) Body at point of moving force upward, sin 30 cos 30 = m = m( - sin 30 ) Figure 3.3 Analysis diagrams 8 Braking systems

7 An alternative method, the angle of friction Angle of friction f, (phi) also only applies to limiting friction. If the friction force and normal reaction are replaced by a resultant force,, the angle that makes with the normal is f, the angle of friction, and tan f = µ. To solve the problem: 1 Analysis diagram The analysis diagram is again used to determine and show all of the forces acting on the body. The first step is to determine and show the friction force on the analysis diagram and then show all of the other forces acting on the body. 2 Angle of friction method eplace the friction force and normal reaction with a resultant force,. Indicate the angle that makes with the normal as f, the angle of friction 3 Free body diagram On the free body diagram show the three forces acting on the system,, and the resultant force. The solution is now found using a force diagram. You solve the problem using a graphical method or using trigonometry. Once you learn this method of analysis you will find it much easier and quicker to use than the previous method when solving limiting friction problems. Sample solution ending motion f Force analysis Free body diagram Figure 3.4 Angle of friction The solution using a force triangle as shown is a much quicker method. art 3: Engineering mechanics, hydraulics and communication 1 9

8 Worked example 2 epeat the force analysis for parts iii, iv and v, using the angle of friction method. Body at point of moving force horizontal f f (given) Body at point of moving force downward, 30 f 30 f F Body at point of moving force upward, f f Figure 3.5 Inclined lane 1 Inclined lane 2 10 Braking systems

9 Friction on an inclined plane Basic introduction. In Landscape products you were introduced to the analysis of forces on an inclined plane. You should revise this work before commencing this basic introduction to friction on an inclined plane. Limiting friction on an inclined plane When a body is at rest on an inclined plane, and is on the point of moving, (ie pending motion), the angle of inclination, q, is equal to the angle of friction, f, and tan f = µ. tan f = µ This angle, q, is sometimes called the angle of repose. q = f, and tan f = µ Figure 3.6 Inclined plane 1 Inclined plane 2 Inclined planes A body at rest on lane 1 is in equilibrium. The friction force is equal in magnitude, but opposite in sense to the weight component down the plane. A body on lane 2 is on the point of slipping (that is, pending motion). Again the friction force is equal in magnitude, but opposite in sense to the weight component down the plane. As it is on the point of moving, (that is, pending motion), the angle of inclination, q, is equal to the angle of friction, f, and tan f = µ. Determine the coefficient of friction between two selected materials. Using a ream of paper, or a brick, glue a material, such as a fabric, or sheet of garnet paper, to one side and another material to the other. Method 1 Determine the horizontal force required to move the body across a surface. You may be able to secure a spring balance to the body. art 3: Engineering mechanics, hydraulics and communication 1 11

10 Alternately you could use a fixed pulley and attached masses. ow, using the friction formula F = µ determine the coefficient of friction. Spring balance or Sand could be added to the bucket until movement occurs Figure 3.7 Coefficient of friction on a horizontal plane Method 2 lace the body on an inclined plane. Determine the angle of repose of the inclined plane, that is, the angle of inclination of the plane when the body just commences to move. ow, determine the coefficient of friction knowing that the angle of inclination, q, is equal to the angle of friction, f, and tan f = µ. Figure 3.8 Determining the angle of limiting friction Turn to the exercise sheets and complete exercise Braking systems

11 Exercise 3.2 a A disc brake system has a force of 8 k applied to each of the pads. If the coefficient of friction between the materials of the pad and of the disc is 0.35, determine the total braking force. b A drum braking system has a force of 12 k applied by the brake shoe to the drum surface. If the coefficient of friction between the materials of the shoe liner and of the drum is 0.3, determine the braking force. 40 Braking systems

12 c A family sedan is parked with its hand brake on. The hand brake operates only on the rear wheels. A truck, attempting to park, bumps the sedan with a horizontal force of 2 k. If each of the rear wheels of the sedan supports a mass of 300 kg, and the coefficient of friction between the tyres and the surface of the parking area is 0.2, determine if the sedan will move forward as a result of the collision. art 3: Engineering mechanics and hydraulics communications 41

13 Exercise 3.3 Select the alternative, A, B, C. or D that best answers the question. 1 Braking systems are effective as a result of : a an extremely high coefficient of friction between the braking materials b an extremely low coefficient of friction between the braking materials c a range of coefficients of friction between the braking materials from 0.25 through to 0.45 d having no coefficient of friction between the braking materials. 2 Effective dissipation of heat energy is important in braking systems to: a keep the driver warm in winter b allow the conversion of kinetic energy to heat energy to continue during heavy braking operations. c allow the drums/discs to stay hot during braking. d allow fade when braking. 3 The angle of friction is: a equal to the coefficient of friction b used during calculations only when limiting friction applies c equal to the normal reaction d equal to the friction force. Exercise 3.4 a A family sedan, moving at a velocity of 100 km/h, brakes suddenly to avoid a collision. If the front wheels are fitted with disc brakes: i draw a force analysis diagram showing all of the forces acting between one of the front discs and the brake pads ii draw a free body diagram of the braking area of the disc showing all of the forces iii write an equilibrium equation that would be used to determine the magnitude of the braking force 42 Braking systems

14 ads Disc Analysis diagram Free body diagram Equation for Figure 3.36 b The front wheels equally support 2/3 of the total mass of the sedan. i Draw on the following diagram a force analysis diagram showing all of the forces acting between one of the front wheels and the road surface. ii Draw a free body diagram of the braking area of the front wheel and the road surface showing all of the forces. iii eplace the friction force and normal reaction with a single force. Clearly label the angle of friction. iv Draw a force triangle that would be used to determine the magnitude of the braking force. Wheel rotates clockwise Analysis diagram Free body diagram Force triangle Figure 3.37 c A truck is parked on the side of a gravel road. The angle of inclination of the road is 6º. Due to the loose gravel surface, the truck is on the point of sliding down the hill. Determine the coefficient of friction between the truck tyres and the gravel surface. 6 Figure 3.38 art 3: Engineering mechanics and hydraulics communications 43

15 Exercise 3.5 Complete the analysis of forces being applied to a body on a horizontal plane by showing all of the forces acting on the body, then completing the free body diagram, (showing only the forces acting), then writing an equilibrium equation for F. The first example, a body at rest, is completed for you. ote: the friction force opposes motion or pending motion. Analysis diagram Free body diagram Equation for Body at rest Body at rest force horizontal = 0 o horizontal force is acting (given) (given) Body at point of moving force horizontal = (given) = Body at point of moving force downward, = Body at point of moving force upward, = Figure 3.39 Friction analysis 44 Braking systems