ANALYSIS OF FRICTION BRAKES

Transcription

1 CHAPTER III ANALYSIS OF FRICTION BRAKES This chapter gives a brief account of all the governing equations and brake design parameters. These governing equations and design parameters are discussed with appropriate details. 3.1 BRAKE TORQUE It is the moment of braking force about the centre of rotation. A moment exists when two parallel, equal and opposite forces acts on a body separated by a distance. These two parallel forces are the drag forces between brake pads and brake rotor, and the bearing force. The distance is measured from rotor drag force to the centre of the bearing spindle. The brake torque is the product of total drag force acting on the rotor and the effective rotor radius. The braking force between tire and road responds to brake torque [12]. T b = F b x R e (3.1) Here R e is the effective radius of disc brake rotor 3.2 BRAKE FACTOR It is defined as the ratio of total drag force F d to the application force F a. Brake Factor = F d / F a (3.2) For a disc brake mechanism with brake pads producing drag forces on the inboard and outboard sides of the rotor, the brake factor is BF = 2F d / F a = 2 (µ F a ) / F a BF= 2 µ (3.3) Where µ is the coefficient of friction. 34

2 3.3 BRAKE FACTOR SENSITIVITY The brake factor sensitivity of a brake is measured as the change of brake factor with respect to change in coefficient of friction between brake rotor and brake pad. A slope of curve obtained for brake factor and coefficient of friction represents brake sensitivity. It is the amount by which brake factor varies for a change in coefficient of friction or how steep the brake factor curve is. Sensitivity of a disc brake: S = d/dµ (BF) = d/dµ (2µ) S = 2 A comparison of the brake factors for different brakes has been shown in the figure 3.1 [12]. Figure 3.1 Brake factor comparison 35

3 3.4 BRAKE FACTOR AND BRAKE SYSTEM DESIGN The brake factor is important for the proper design and effectiveness of a braking system. The brake design engineer must select a brake factor as well as a friction pad/rotor material combination that maintains the designed brake factor for all operating conditions and ensures proper braking effectiveness [12]. The pad and rotor coefficient of friction is a function of many variables. It is always preferable to choose a quality brake pad material. 3.5 BRAKE FACTOR ANALYSIS OF DRUM BRAKES A distinguishing characteristic of a drum brake is the higher brake factor compared with disc brake. The higher brake factor is because of self energizing within the brake. A brake design cannot be rated superior to others in all respects Brake lining pressure distribution and wear The brake drum and brake shoes are assumed to be rigid and therefore entire deformation occurs within the lining material. Compression in the lining is measured by the rotation angle of the shoe about its pivot, is related to the strain and the original lining thickness d Lo by ε = d L /d Lo (3.4) Where d L - Lining compression d Lo - Original lining thickness ε - Strain of lining material If the shoe is long then the pressure will not be uniform. It is necessary to determine the distribution of pressure along the lining; the pressure distribution should be favourable for maintaining uniform wear. As the brake drum is made of a hard material like cast iron or steel, wear occurs on friction lining, which is 36

4 attached to the shoe. The lining is necessary to retain the cylindrical shape of the brake drum when wear occurs Self energizing and self locking of drum brake A drum brake with leading shoe is illustrated in figure 3.2. The principle of self energizing for leading shoe with the shown direction of drum rotation, the moment of the frictional force adds to the moment of the application force, F a. As a consequence, the required actuation force needed to create a known contact pressure is much smaller than that if this effect is not present. This phenomenon of frictional force aiding the brake actuation is referred to as selfenergizing. The application force F a applied at the tip of the shoe pushes the brake block against the drum. The counter clockwise rotation of the drum produces a drag force F d. The moment balance around the shoe pivot point yields F a h + F d c - F n b = 0 F a h + F d c = (F d / μ) b F d = μ F a F n = F a Where b - Brake dimension c -Brake dimension h - Brake dimension µ - Coefficient of friction Solving for the ratio of drum drag F d to application force F a yields the brake factor BF of the leading shoe as BF = F d / F a 37

5 BF = μ h / (b - μ c) (3.5) The drum drag rotates the brake shoe such that it will increase the normal force of the block pushing against the drum. This increased normal force causing an additional increase in drum drag is the self-energizing effect of the brake. The self-energizing shoe is called the leading shoe. The ratio of drum drag to application force as expressed by the above equation, will increase for smaller denominators, and will be infinite when the denominator is zero. The lining friction coefficient at which the denominator will be zero for the brake geometry given is b/c, designated as μ L. Figure 3.2 Drum brake If the actual lining friction coefficient were equal to μ L, then a brake application would cause ever increasing self-energizing until the brake locked. Even releasing the application force would not disengage the brake block from the drum. Although self locking generally is not a problem because b > c, brake 38

6 engineers must guard against it by ensuring that neither friction levels nor brake geometries are such that self locking may occur [12]. When the direction of rotation is changed, the moment of frictional force will be opposing the application force and hence greater magnitude of force is needed to create the same contact pressure. The shoe on which this is prevailing is known as a trailing shoe. For reversed rotation of the drum, the leading shoe turns into a trailing shoe. The drum drag force would be directed upward attempting to lift the brake block off the drum, thus reducing the effect of the application force. The brake factor of the trailing shoe is given as F a h - F d c - F n b = 0 F a h = F d c + F n b F a h = F d c + (F d / µ) b F a h = F d [c + (b / µ)] F d / F a = h / [(b / µ) + c] BF = µ h/ [b + µc] (3.6) The plus sign in the denominator indicates the decrease in brake factor with increasing lining friction coefficients, i.e, non self energizing of the trailing shoe. The trailing shoe does not contribute to self-energizing. 3.6 DISC BRAKES All modern cars have disc brakes on the front wheels and some have disc brakes on all four wheels. Disc brake is a part of the braking system which actually works for deceleration or completely stopping the car. A disc brake with single-piston floating calliper is preferred on modern cars. The single piston floating calliper is of self centering and self-adjusting type. It is able to slide from side to side so that each time it will move to the centre when brakes are applied. As there is no spring to pull the pads away from the disc; the pads are always 39

7 in slight contact with the rotor. Usually the rubber piston seal and any tremble in the rotor may pull the pads a small distance away from the rotor. This is important because the pistons in the wheel cylinders are much larger in diameter than the master cylinder. If the pistons in the wheel cylinders are retracted then it may take several applications of brake pedal to pump the fluid into the wheel cylinders while applying the brakes. If the vehicle is provided with disc brakes on all four wheels then an emergency brake has to be actuated by a separate mechanism than the primary brakes in case of a total primary brake failure. A cable is preferred to actuate the emergency brake. Some of the vehicles are with separate drum brake integrated into the hub of the rear wheels. This drum brake is only for the emergency brake system and it is actuated by the cable. Figure 3.3 Disc brake assembly Disc brake shown in figure 3.3 is considered the most satisfactory brake design. It can give stable braking performance in spite of the brake temperature. The small gap between brake disc and brake pad in an un-actuated state minimizes the excessive pedal travel during braking. Also it gives more uniform wear of the friction material for the same distribution of pressure on the friction surface. 40

8 The major types of high performance brake rotors are drilled and slotted rotors. Figure 3.4 illustrates a drilled brake rotor with holes drilled in them. A rotor full of holes means there is less surface area for the brake pads to grab and stop the vehicle, but there are few reasons; for which it makes sense to use drilled rotors. The first reason is heat. When brake pad grabs the rotor it creates friction which generates heat. If this heat doesn t escape, it leads to brake fade. The second reason is gas build up. The gas that builds up between the rotor surface and friction pad during braking limits the stopping power of a brake. The last reason is water. The brake rotors can get wet; when a vehicle drives through a pond, a car wash or even a rain storm. A wet brake rotor is slippery and difficult for the brake pads to grab. Brake rotor having drilled holes; makes it easy for heat, gas and water to be quickly moved away from the rotor surface and maintains effective braking performance. Figure 3.4 Disc brake rotor with cross drills The downside of using drilled brake rotors is that all of those holes tend to weaken the rotor. Even rotors can crack after repeated stressful driving. Figure 3.5 shows the slotted brake rotor having slots engraved into the flat metal surface to move the gases, heat and water away from the surface of the rotor. They are found to be little more durable than drilled brake rotors but tend to wear down brake pads very quickly. 41

9 Figure 3.5 Disc brake rotor with slots 3.7 MATHEMATICAL MODELLING OF HEAT DISSIPATION IN DISC BRAKE ROTOR Newton s law of cooling: Q = h A (T s -T ) (3.7) Where: Q - Rate of heat transfer (W) h- Convective heat transfer coefficient A - Surface area of the rotor in m 2 T s - Surface temperature of the brake rotor ( C) T - Ambient air temperature ( C) It can be seen from this expression that in order to maximize the heat transfer from the rotor and keep the rotor temperature (T s ) to a minimum value, the value of heat transfer coefficient (h), or the surface area (A) is required to be increased. As it is required to keep T s to a minimum value, improvements must be made by 42

10 increasing the heat transfer coefficient (h) and/or the surface area (A) of the rotor. Figure 3.6 Mathematical model of disc brake The amount by which surface area can increase is confined to the diameter of the wheel and the requirements of minimizing un-sprung mass, so improvements in cooling can be made through increased value of the heat transfer coefficient. The use of an internally ventilated rotor will increase both surface area and the heat transfer coefficient due to forced convection created by the internal airflow with negligible influence on un-sprung mass. The material selection and the physical dimensions of the rotor will also have a direct bearing 43

11 on cooling ability. Figure 3.6 shows the contact surface element of the pad and the disc. P - Pressure applied on disc brake by the pad F N - Normal force applied on disc brake F T - Tangential force acting on disc brake ω - Angular velocity of disc brake rotor µ - Coefficient of friction T - Torque r 1 -Radius at inlet of rotor r 2 - Radius at outlet of rotor df N - Normal force applied on elemental ring df T - Tangential force acting on elemental ring dt - Torque exerted on elemental ring An analytical temperature field for the disc is obtained by assuming a differential element of the disc and solving an ordinary differential equation. Assumptions: 1. Pressure acting on disc is uniform 2. Angular velocity is constant df N = P da (3.8) = P 2πr dr This gives the normal force applied on the disc. df T = µ df N (3.9) = µ P 2πr dr 44

12 Torque acting on elemental ring of disc brake rotor is given as: dt = df T r (3.10) = µ P 2πr dr r Torque acting on the entire surface of disc brake rotor can be obtained by integrating above equation between r 1 and r 2 : 3 T = (2/3) µ P π [r 2 r 3 1 ] T ω = work done per sec 3 = (2/3) µ P π ω [r 2 r 13 ] (3.11) But brake pad is not occupying the entire rotor surface. It is occupying for angle ϕ so then the equation becomes: 3 = (1/3) µ P ϕ ω [r 2 r 3 1 ] (3.12) Equation 3.12 gives the work done per second by the pad on the rotor surface and which is equal to the heat generated. 3.8 THERMAL ANALYSIS OF THE ROTOR Steady state heat conduction in three dimensional heat transfer problem is governed by the following differential equation. Consider the differential cube of solid material illustrated in figure 3.7. Heat flow in a solid is governed by Fourier s law of conduction, which states that q = -k A (dt/dx) (3.13) Where q is the energy transfer per unit time (J/s), T is the temperature (K), A is the area of a cross section through which the energy flows (m 2 ), T/ x is the temperature gradient normal to the area A (K/m), and k is the thermal conductivity of the solid (W/mK), which is a physical property of the solid material. 45

13 Figure 3.7 Model of infinitesimally small element fixed in space The net rate of flow of energy into the solid in the x direction is = q(x) [q(x) + ( q(x)/ x) dx] = - ( q(x)/ x) dx = - / x (- ka ( T/ x))dx = / x (k ( T/ x)) dv Similarly net rate of flow of energy into the solid in y and z direction is = / y (k ( T/ y)) dv = / z (k ( T/ z)) dv For steady heat flow, there is no net change in the amount of energy stored in the solid, so the sum of the net rate of flow of energy in the three directions is zero. Thus, / x (k ( T/ x)) + / y (k ( T/ y)) + / z (k ( T/ z)) = 0 (3.14) Equation 3.14 governs the steady diffusion of heat in a solid. When the thermal conductivity k is constant then above equation becomes, T 2 / x 2 + T 2 / y 2 + T 2 / z 2 = 0 (3.15) Equation 3.15 is Laplace equation. 46

14 In an unsteady situation, however, there can be a net change with time in the amount of energy stored in the solid. The energy stored in the solid mass is given by, E = dm CT (3.16) =ρ dv CT = (ρ C T) dv Where ρ is the density of the solid material (kg/m 3 ), dv is the differential volume (m 3 ), T is the temperature (K), and C is the specific heat (J/kg-K), which is a physical property of the solid material. The sum of the net heat flow components must equal the time rate of change of the stored energy. Thus, (ρct) / t = / x (k ( T/ x)) + / y (k ( T/ y)) + / z (k ( T/ z)) (3.17) Equation 3.17 governs the unsteady diffusion of heat in a solid. When the thermal conductivity k, density ρ, and specific heat C are constant then it get simplified as: T/ t = (k/ρc) [ T 2 / x 2 + T 2 / y 2 + T 2 / z 2 ] (3.18) Thus, we can see that excessive thermal loading can damage the disc brake rotor severely. This means that effective cooling of disc brake rotor is of vital importance with respect to the life of the disc brake rotor as well as the safety of the driver and passengers of a vehicle [9, 10]. Though the ventilated disc brake rotor provides considerable cooling there is extensive scope for improvement. Therefore, prime importance has been given for the selection of better geometrical design variables to improve the thermal performance of a disc brake rotor. The vehicle designers and researchers are presently confronted with this task. The friction heat generated between brake pad and rotor surface causes thermo-elastic deformation; which alters the contact pressure distribution. This coupled thermo-mechanical process is referred to as frictionally-excited thermo- 47

15 elastic instability or TEI and leads to the development of non-uniform contact pressure and local high temperature with important gradients called hot spots. The formation of such localized hot spots is accompanied by high local stresses that can lead to material degradation and eventual failure. Also, the hot spots can be a source of undesirable frictional vibrations, known as hot judder. In this research, a novel rotor geometry configuration is presented by employing the profiles to accelerate the heat dissipation. Being familiar with the physics of the flow over these profiles, a superior shape and arrangement for vanes/pillars is suggested. The new design is analyzed for heat dissipation capacity of a rotor and its promising performance is compared with the existing rotor. It is concluded that the heat generated due to friction between the rotor surface and the brake pad should be ideally dissipated to the environment to avoid the decreasing friction coefficient between the rotor surface and the brake pad and to avoid the temperature rise in various brake components and brake fluid vaporization due to excessive heating [3, 4, 6]. Increasing the speed of air passed through the rotor flow passages has been a general guideline considered in almost all articles in the field, and we have taken full advantage of this design guideline. Also another heat transfer improving factor, to keep the flow attached to the surface of a flow passage is explored in detail through CFD simulation. The convective heat dissipation can be written as: Heat carried away by air = heat lost by the rotor h A T = m C (dt/dt) (3.19) Where m is the mass of rotor (kg), C is the specific heat (J/kg-K), A is the surface area (m 2 ), dt/dt is the rate of change of rotor temperature (K/sec), T is the temperature difference between rotor and surrounding air (K) and h is the heat transfer coefficient (W/m 2 K). Experimentally heat transfer coefficient may be obtained by using this equation. 48

16 The major aim of designing brake discs is to improve the convective heat dissipation from the disc brake rotor. In operations of braking systems, convection is the most important mode of heat transfer, dissipating the highest proportion of heat to surrounding air. 3.9 MECHANICAL STRENGTH OF DISC BRAKE ROTOR Considering weight of a particular utility vehicle as 2000 kg and running with a speed of 120 km/hr; the torque required to stop the vehicle becomes: v = 120 km/hr v = m/s Kinetic energy possessed by the vehicle K.E. = 0.5 m v 2 (3.20) K.E. = 0.5 x 2000 x (33.33) 2 K.E. = N-m Tangential force required to stop the vehicle: Total kinetic energy possessed by the vehicle is divided onto all the four wheels but approximately 80% of the braking is with front wheels. Kinetic energy expected to be dissipated by the front wheels of the vehicle becomes: = 0.8 x = N-m K.E. per second per wheel becomes: = /2 = N-m/sec This is the kinetic energy per second dissipated on each front wheel of the vehicle. Torque required to absorb this kinetic energy becomes: 2πNT/60 =

17 v =r ω ω = 2πN/60 (3.21) r = m v = m/sec ω = rad/sec x T = T = 4000 N-m Tangential force applied on disc brake rotor at equivalent radius: T = F x R (3.22) R: equivalent radius of a brake rotor 4000 = F x F = N Normal force applied on disc brake rotor at equivalent radius: Assuming 0.5 as the coefficient of friction between bake pad and rotor surface F N = F/µ (3.23) F N = /0.5 F N = N Pressure applied on brake pad: Brake pad size: 80 mm x 40 mm Area of brake pad: 3.2 x 10-3 m 2 Total area occupied by the brake pad on the rotor surface becomes: = 2 x 3.2 x

18 = 6.4 x 10-3 m 2 P = F/A (3.24) P = /6.4 x 10-3 P = N/m 2 P = MPa The designed disc brake rotor should have enough strength to sustain the pressure applied during braking [11]. 51