Lecture notes in Mechanical engineering

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Introduction to mechanical engineering lecture notes Csaba Hos Botond Erdos September 10, 2013 11 A short summary of the basics 1.1 Physical quantities, units and working with units The value of a physical quantity Q is expressed as the product of a numerical value Q and a unit of measurement Q: Q =Q Q (1) For example, if the temperature T of a body is quanti ed (measured) as 25 degrees Celsius this is written as: o o T = 25 C = 25 C; (2) where T is the symbol of the physical quantity "temperature", 25 is the o numerical factor and C is the unit. By convention, physical quantities are organized in a dimensional system built upon base quantities, each of which is regarded as having its own di- mension. The seven base quantities of the International System of Quantities (ISQ) and their corresponding SI units are listed in Table 1. Other conven- tions may have a di erent number of fundamental units (e.g. the CGS and MKS systems of units). Name Symbol for Symbol for SI base Symbol for quantity dimension unit unit Length l, x, r, etc. L meter m Time t T second s Mass m M kilogram kg Electric current I, i I ampere A Thermodynamic T  kelvin K temperature Amount of sub- n N mole mol stance Luminous inten- I J candela cd v sity Table 1: International System of Units base quantities All other quantities are derived quantities since their dimensions are derived from those of base quantities by multiplication and division. For example, the physical quantity velocity is derived from base quantities length and time and has dimension L/T. Some derived physical quantities have dimension 1 and are said to be dimensionless quantities. The International System of Units (SI) speci es a set of unit pre xes known as SI pre xes or metric pre xes. An SI pre x is a name that precedes 4a basic unit of measure to indicate a decimal multiple or fraction of the unit. Each pre x has a unique symbol that is prepended to the unit symbol, see Table 2. n Pre x Symbol 10 9 giga G 10 6 mega M 10 3 kilo k 10 2 hecto h 10 1 deca da 10 1 deci d 10 2 centi c 10 3 milli m 10 6 micro  10 9 nano n 10 Table 2: International System of Units pre xes. A quantity is called: extensive when its magnitude is additive for subsystems (volume, mass, etc.) intensive when the magnitude is independent of the extent of the system (temperature, pressure, etc.) Units can be used as numbers in the sense that you can add, subtract, multiply and divide them - with care. Much confusion can be avoided if you work with units as though they were symbols in algebra. For example:  Multiply units along with numbers: (5 m) (2 sec) = (5 2) (m sec) = 10 m sec. The units in this example are meters times seconds, pronounced as `meter seconds' and written as `m sec'.  Divide units along with numbers: (10 m) / (5 sec) = (10 / 5) (m / sec) = 2 m/sec. The units in this example are meters divided by seconds, pronounced as `meters per second' and written as `m/sec'. This is a unit of speed.  Cancel when you have the same units on top and bottom: (15 m) / (5 m) = (15 / 5) (m / m) = 3. 5In this example the units (meters) have cancelled out, and the result has no units of any kind This is what we call a `pure' number. It would be the same regardless what system of units were used.  When adding or subtracting, convert both numbers to the same units before doing the arithmetic: (5 m) + (2 cm) = (5 m) + (0.02 m) = (5 + 0.02) m = 5.02 m. Recall that a `cm', or centimeter, is one hundredth of a meter. So 2 cm = (2 / 100) m = 0.02 m.  You can't add or subtract two numbers unless you can convert them both to the same units: (5 m) + (2 sec) = ??? 1.2 Understanding the words "steady-state" and "unsteady" TODO 1.3 Linear motion Linear motion is motion along a straight line, and can therefore be de- scribed mathematically using only one spatial dimension. It can be uniform, that is, with constant velocity (zero acceleration), or non-uniform, that is, with a variable velocity (non-zero acceleration). The motion of a particle (a point-like object) along the line can be described by its position x, which varies with t (time). An example of linear motion is that of a ball thrown straight up and falling back straight down. The average velocityv during a nite time span of a particle undergoing P P P linear motion is equal tov = x= t, where x is the total displacement P and t denotes the time needed. The instantaneous velocity of a particle in linear motion may be found by di erentiating the positionx with respect to the time variablet:v =dx=dt. The acceleration may be found by di erentiating the velocity: a = dv=dt. By the fundamental theorem of calculus the converse is also true: to nd the velocity when given the acceleration, simply integrate the acceleration with respect to time; to nd displacement, simply integrate the velocity with respect to time. This can be demonstrated graphically. The gradient of a line on the displacement time graph represents the velocity. The gradient of the velocity time graph gives the acceleration while the area under the velocity time graph gives the displacement. The area under an acceleration time graph gives the velocity. 61.4 Circular motion Circular motion is rotation along a circle: a circular path or a circular orbit. It can be uniform, that is, with constant angular rate of rotation, or non-uniform, that is, with a changing rate of rotation. Examples of circular motion are: an arti cial satellite orbiting the Earth in geosynchronous orbit, a stone which is tied to a rope and is being swung in circles (cf. hammer throw), a racecar turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic eld, a gear turn- ing inside a mechanism. Circular motion is accelerated even if the angular rate of rotation is constant, because the object's velocity vector is constantly changing direction. Such change in direction of velocity involves acceleration of the moving object by a centripetal force, which pulls the moving object towards the center of the circular orbit. Without this acceleration, the object would move in a straight line, according to Newton's laws of motion. For motion in a circle of radius R, the circumference of the circle is C = 2R. If the period for one rotation is T , the angular rate of rotation, also known as angular velocity, rad=s is: 2 = : (3) T In mechanical engineering, the revolution number is often used:   number of rotations n =  60 rpm = (4) 2 minute The speed of the object travelling the circle is 2R v = =R: (5) T The angle  swept out in a time t is t  = 2 =t: (6) T The acceleration due to change in the direction of the velocity is found by analysing the change of the velocity vector in (small) time interval t. As = const:, we havejvj =jvj :=v. From the triangle we see that 1 2 v ' ' v o 2 = sin  for ' 5 = ' =t: (7) v 2 2 v Thus, we have 2 v v 2 a = =v =R = (8) t R and is directed radially inward. 2 The angular acceleration " rad=s is  " = : (9) t 71.5 Newton's rst law Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed. This law states that if the resultant force (the vector sum of all forces acting on an object) is zero, then the velocity of the object is constant. Consequently:  An object that is at rest will stay at rest unless an unbalanced force acts upon it.  An object that is in motion will not change its velocity unless an unbalanced force acts upon it. Newton placed the rst law of motion to establish frames of reference for which the other laws are applicable. The rst law of motion postulates the existence of at least one frame of reference called a Newtonian or inertial reference frame, relative to which the motion of a particle not subject to forces is a straight line at a constant speed. 1.6 Newton's second law The second law states that the net force on a particle is equal to the time rate of change of its linear momentum p in an inertial reference frame: dp d dv F = = (mv) =m =ma; (10) dt dt dt where we assumed constant mass. Thus, the net force applied to a body produces a proportional acceleration. For circular motion, we have M ="; (11) withM Nm being the torqueM =Fr," denotes angular acceleration and 2  kgm is the moment of inertia. 1.7 The moment of inertia The moment of inertia of an object about a given axis describes how dicult it is to change its angular motion about that axis. The standard notation is  with the actual axis in the subscript, e.g.  meaning moment x of inertia with respect to axis x. 8Consider a point of massm rotating around an axis with circumferential velocity v. The angular velocity is and the radius of the circle is r. The kinetic energy is 1 1 1 2 2 2 2 E = mv = m (r) = mr : (12) z 2 2 2  Some equations for the moment of inertia: 2  a mass point rotating on a circle of radius r:  =mr 2  a thin ring of radius r rotating around its own axis:  =mr 1 2  a thin disc of radius r rotating around its own axis:  = mr 2  a thin rod of mass m and length l, rotating around the axis which 1 2 passes through its center and is perpendicular to the rod:  = ml 12  a thin rod of mass m and length l, rotating around the axis which 1 2 passes through its end and is perpendicular to the rod:  = ml 3  a solid ball of mass m and radius r, rotating around an axis which 2 2 passes through the center:  = mr 5 Let us calculate the moment of inertia of a disc of heightb, radiusR and uniform density  at its own axis. We divide the radius into N + 1 rings: th r =iR=N =ir. The moment of inertia of the i ring is i   2 R R R 2 2  =mr = 2r r br = 2i  b i (13) i i i i i z N N N circumference z area By summing up these rings we obtain   N N 4 X X R 3  = lim  = lim 2 b i disc i N1 N1 N i=1 i=1   4 R 1 1 1 2 2 4 2 = lim 2 b N (1 +N ) = 2R b = mR (14) N1 N 4 4 2 The parallel axis theorem (or Huygens-Steiner theorem) can be used to determine the moment of inertia of a rigid body about any axis, given the moment of inertia of the object about the parallel axis through the object's centre of mass and the perpendicular distance (r) between the axes. The moment of inertia about the new axis z is given by: 2  = +mr (15) z cm 9where  is the moment of inertia of the object about an axis passing cm through its centre of mass,m is the object's mass andr is the perpendicular distance between the two axes. For example, let us compute the moment of inertia of a thin rod rotating around the axis which passes through its end:   2 l 1 1 1 2 2 2  = +m = ml + ml = ml : (16) end cm 2 12 4 3 Finally, the moment of inertia of an object can be computed simply as the sum of moments of inertia of its "building" objects. 1.8 Work In physics, mechanical work is the amount of energy transferred by a force acting through a distance. In the simplest case, if the force and the displacement are parallel and constant, we have W =Fs: (17) It is a scalar quantity, with SI units of joules. If the direction of the force and the displacement do not coincide (e.g. when pulling a bob up to a hill) - but they are still constant - one has to take the parallel components: W = F v =jFjjvj cos =Fvcos; (18) where  is the angle between the force and the displacement vector and stands for the dot product of vectors. In situations where the force changes over time, or the path deviates from a straight line, equation (17) is not generally applicable although it is possible to divide the motion into small steps, such that the force and motion are well approximated as being constant for each step, and then to express the overall work as the sum over these steps. Mathematically, the calculation of the work needs the evaluation of the following line integral: Z W = Fds; (19) C C where C is the path or curve traversed by the object; F is the force vector; and s is the position vector. Note that the result of the above integral de- pends on the path and only from the endpoints. This is typical for systems in which losses (e.g. friction) are present (similarly as the actual fare of a taxi from point A to B depends heavily on the route the driver chooses). 101.9 Energy Energy is a quantity that is often understood as the ability to perform work. This quantity can be assigned to any particle, object, or system of objects as a consequence of its physical state. Energy is a scalar physical quantity. In the International System of Units (SI), energy is measured in joules, but in some elds other units such as kilowatt-hours and kilocalories are also used. Di erent forms of energy in- clude kinetic, potential, thermal, gravitational, sound, elastic and electro- magnetic energy. Any form of energy can be transformed into another form. When energy is in a form other than thermal energy, it may be transformed with good or even perfect eciency, to any other type of energy, however, during this con- version a portion of energy is usually lost because of losses such as friction, imperfect heat isolation, etc. In mechanical engineering, we are mostly concerned with the following types of energy:  potential energy: E =mgh p 1 2  kinetic energy: E = mv k 2  internal energy: E =c mT (with a huge number of simpli cations...) t p Although the total energy of an isolated system does not change with 1 time , its value may depend on the frame of reference. For example, a seated passenger in a moving airplane has zero kinetic energy relative to the air- plane, but non-zero kinetic energy (and higher total energy) relative to the Earth. A closed system interacts with its surrounding with mechanical work (W ) and heat transfer (Q). Due to this interaction, the energy of the system changes: E =W +Q; (20) where work is positive if the system's energy increases (e.g. by lifting ob- jects their potential energy increases) and heat transfer is positive if the temperature of the system increases. 1 There is a fact, or if you wish, a law, governing all natural phenomena that are known to date. There is no known exception to this lawit is exact so far as we know. The law is called the conservation of energy. It states that there is a certain quantity, which we call energy, that does not change in manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number and when we nish watching nature go through her tricks and calculate the number again, it is the same. The Feynman Lectures on Physics 111.10 Power Power is the rate at which work is performed or energy is converted. If W is the amount of work performed during a period of time of duration t, the average power P over that period is given by W P = : (21) t The average power is often simply called "power" when the context makes it clear. The instantaneous power is then the limiting value of the average power as the time interval t approaches zero. In the case of constant power P , the amount of work performed during a period of duration T is W = PT . Depending on the actual machine, we have   m mechanical (linear motion) power: P =Fv N s   rad mechanical (circular motion) power: P =M Nm s electrical power: P =UI VA h i 3 m hydraulic power: P =Qp Pa s The dimension of power is energy divided by time J=s. The SI unit of power is the watt (W), which is equal to one joule per second. A common non-SI unit of power is horsepower (hp), 1hp = 0:73549875kW . 1.11 Problems Problem 1.1 A spring with sti nesss = 100N=mm is compressed from its initial length of L = 20cm to L = 10cm. 0 1  Calculate the force. (F = 10kN)  Calculate the work. (W = 0:5kJ) Problem 1.2 We drive by car for 4 hours, after which we refuel 32l of gasoline. The car has a 55kW motor (75hp) and it can be assumed that during the journey this was the useful power. The heating value of gasoline is 35MJ=l.  Calculate the useful work (W = 220kWh = 792MJ), input energy u (E = 1120MJ) and eciency ( = 70:7%). i Problem 1.3 A 210MW coal plant consumes 4100t of coal per day. The heating value of lignite is 17MJ=kg. 12 Calculate the eciency of the plant ( = 26%). Worked problem 1.4 A rotating wheel of a vehicle is stopped by two brakes as seen in Figure 1. The friction coecient is  = 0:13, the diameter of the wheel is 910mm, the initial velocity of the vehicle was 65km=h. The pushing force is F = 6000N. Fs o F F m Fs Figure 1: Braking a rotating wheel.  Calculate the friction force. F =F = 0:78kN. f  Calculate the (overall) braking torque acting on the wheel. D M = 2F = 0:7098kNm f f 2  Calculate the power of braking at the start of the breaking. 2v 0 P =M =M = 28:17kW 0 f f D  Assuming constant torque and linearly decreasing velocity (i.e. con- stant deceleration), compute the time needed to stop a 20t vehicle with six braked weels. 1 2 The initial kinetic energy of the vehicle is E = mv = 3:26MJ. k 0 2 The overall work done by the six brakes W = 6M T (T is yet f 2 unknown). The initial kinetic energy is fully dissipated by the braking work: E =WT = 38:6s k 13 D2 Steady-state operation of machines 2.1 The sliding friction force due to dry friction Dry friction resists relative lateral motion of two solid surfaces in con- tact. The two regimes of dry friction are static friction between non-moving surfaces, and kinetic friction (sometimes called sliding friction or dynamic friction) between moving surfaces. Coulomb friction is an approximate model used to calculate the force of dry friction: jFjN: (22) f where  F is the force exerted by friction (in the case of equality, the maximum f possible magnitude of this force).   is the coecient of friction, which is an empirical property of the contacting materials,  N is the normal force exerted between the surfaces. The Coulomb friction may take any value from zero up to N, and the direction of the frictional force against a surface is opposite to the motion that surface would experience in the absence of friction. Thus, in the static case, the frictional force is exactly what it must be in order to prevent motion between the surfaces; it balances the net force tending to cause such motion. In this case, rather than providing an estimate of the actual frictional force, the Coulomb approximation provides a threshold value for this force, above which motion would commence. This maximum force is known as traction. The force of friction is always exerted in a direction that opposes move- ment (for kinetic friction) or potential movement (for static friction) between the two surfaces. For example, a curling stone sliding along the ice experi- ences a kinetic force slowing it down. For an example of potential movement, the drive wheels of an accelerating car experience a frictional force pointing forward; if they did not, the wheels would spin, and the rubber would slide backwards along the pavement. Note that it is not the direction of move- ment of the vehicle they oppose, it is the direction of (potential) sliding between tire and road. In the case of kinetic friction, the direction of the friction force may or may not match the direction of motion: a block sliding atop a table with rectilinear motion is subject to friction directed along the line of motion; an automobile making a turn is subject to friction acting perpendicular to the line of motion (in which case it is said to be 'normal' to it). The direction of the static friction force can be visualized as directly opposed to the force that would otherwise cause motion, were it not for the 14Materials Dry and clean Lubricated Aluminum Steel 0.61 Copper Steel 0.53 Brass Steel 0.51 Cast iron Copper 1.05 Cast iron Zinc 0.85 Concrete (wet) Rubber 0.30 Concrete (dry) Rubber 1.0 Concrete Wood 0.62 Copper Glass 0.68 Glass Glass 0.94 Metal Wood 0.2-0.6 0.2 Polythene Steel 0.2 0.2 Steel Steel 0.80 0.16 Steel Te on 0.04 0.04 Te on Te on 0.04 0.04 Wood Wood 0.25-0.5 0.2 Table 3: Approximate coecients of friction static friction preventing motion. In this case, the friction force exactly can- cels the applied force, so the net force given by the vector sum, equals zero. It is important to note that in all cases, Newton's rst law of motion holds. 2.2 Rolling resistance Rolling resistance, sometimes called rolling friction or rolling drag, is the resistance that occurs when a round object such as a ball or tire rolls on a at surface, in steady velocity straight line motion. It is caused mainly by the deformation of the object, the deformation of the surface, or both. (Additional contributing factors include wheel radius, forward speed, sur- face adhesion, and relative micro-sliding between the surfaces of contact.) It depends very much on the material of the wheel or tire and the sort of ground. For example, rubber will give a bigger rolling resistance than steel. Also, sand on the ground will give more rolling resistance than concrete. A moving wheeled vehicle will gradually slow down due to rolling resistance including that of the bearings, but a train car with steel wheels running on steel rails will roll farther than a bus of the same mass with rubber tires running on tarmac. The coecient of rolling resistance is generally much smaller for tires or balls than the coecient of sliding friction. 15R F aR b N Figure 2: Hard wheel rolling on and deforming a soft surface. The force of rolling resistance can also be calculated by: Nb F = =C N (23) rr R where  F is the rolling resistance force,  R is the wheel radius,  b is the rolling resistance coecient or coecient of rolling friction with dimension of length,  C =b=R is the coecient of rolling resistance (dimensionless num- rr ber), and  N is the normal force. C b Description rr 0.0002...0.0010 0.5 mm Railroad steel wheel on steel rail 0.1mm Hardened steel ball bearings on steel 0.0025 Special Michelin solar car/eco-marathon tires 0.005 Tram rails standard dirty with straights and curvescitation needed 0.0055 Typical BMX bicycle tires used for solar cars 0.0062...0.015 Car tire measurements 0.010...0.015 Ordinary car tires on concrete 0.3 Ordinary car tires on sand Table 4: Approximate coecients of rolling resistance In usual cases, the normal force on a single tire will be the mass of the object that the tires are supporting divided by the number of wheels, plus 16the mass of the wheel, times the gravitational acceleration. In other words, the normal force is equal to the weight of the object being supported, if the wheel is on a horizontal surface. 2.3 Statics of objects on inclined planes (restoring forces) To calculate the forces on an object placed on an inclined plane, consider the three forces acting on it:  The normal force (N) exerted on the body by the plane due to the force of gravity,  The force due to gravity (mg, acting vertically downwards) and  the frictional force (F ) acting parallel to the plane. f We can decompose the gravitational force into two vectors, one perpendic- ular to the plane and one parallel to the plane. Since there is no movement perpendicular to the plane, the component of the gravitational force in this direction (mg cos ) must be equal and opposite to normal force exerted by the plane, N plus the normal component of the force F : mg cos =F sin +N: (24) The remaining component of the gravitational force parallel to the surface (mg sin ) plus the friction force equals the the "pulling" force F : t mg sin +F =F cos (25) f By de nition, the friction force is F =N: (26) f F N n t b a Ff G a Figure 3: Object on an inclined plane. 17Combining these three equations, we arrive at the following equation for the traction force F : sin + cos F =mg : (27) cos + sin Let us describe the three di erent cases while varying the force:  If the actual traction force is smaller thanF (but positive), the object is at rest. The value of the friction force is such that the body stays at rest: F = mg sin and points upwards to balance the tangential f component of G.  If the traction force is exactly F , the body is either at rest or moves with constant arbitrary velocity.  Finally, for traction force values beyond F the object experiences a constant acceleration.  One might want to compute the angle for which the smallest F force is needed to move the object: dF d 1 0 = =mg (sin + cos ) (cos + sin ) d z d const.   sin + cos  = const.(1) : = arctan (28)   2 (cos + sin ) Note that our equations are valid only if N 0, i.e. G cos F sin (29) max In other words, the maximal value before the force F lifts the object max is (the condition is F =F ): max sin + cos cos  = = arc cot (tan ) = : max cos + sin sin 2 max max max (30) Finally, let us de ne the eciency of the traction. The useful work is the vertical displacement h = L sin while the input work is the work done by tangential component of the force, i.e. F : t E mgh mgL sin pot  = = = ; (31) W F L mgL sin +F L F t f t whose maximum value occurs if F = 0, i.e. when the normal component f vanishes. Hence the maximum-eciency point corresponds to the an- max gle, i.e. when the normal component of force F balances the gravity and no friction occurs. 182 1.5 o α=60 1 o α=30 o α=10 0.5 0 0 20 40 60 80 β Figure 4: Solid line: the traction force needed as a function of , see (27). Dashed line: the lift-up force given by (29). The friction coecient is = 0:1. 2.4 Pulley 2.4.1 Pulley without friction A pulley, also called a sheave or a drum, is a mechanism composed of a wheel on an axle or shaft that may have a groove between two anges around its circumference. A rope, cable, belt, or chain usually runs over the wheel and inside the groove, if present. Pulleys are used to change the direc- tion of an applied force, transmit rotational motion, or realize a mechanical advantage in either a linear or rotational system of motion. It is one of the six simple machines. Two or more pulleys together are called a block and tackle. The di erent types of pulley systems are: Fixed A xed pulley has a xed axle. That is, the axle is " xed" or anchored in place. A xed pulley is used to change the direction of the force on a rope (called a belt). Movable A movable pulley has a free axle. That is, the axle is "free" to move in space. A movable pulley is used to multiply forces. Compound A compound pulley is a combination of a xed and a movable pulley system. The block and tackle is a type of compound pulley where several pulleys are mounted on each axle, further increasing the mechanical advantage. 19 F/GFigure 5: (Left) Pulley systems. (Right) Each student nding a mistake in the cartoon gets extra 10% in the exam. It is important to note that as long as the friction due to sliding friction in the system where cable meets pulley and in the rotational mechanism of each pulley is neglected, the change in the potential energy of the weight Gh and the lifting work F s are equal. 2.4.2 Pulley with friction Figure 6 depicts a pulley with friction between the bearing and the shaft. F represents the load force (in this case, one lifts a mass) while F and 1 2;v F shows two possible pulling directions. The friction force is 2;h ( F +F for vertical arrangement, and 1 2;v q F =N; where N = f 2 2 F +F for vertical arrangement: 1 2;h (32) The torque equilibrium is given by F R =F R +Nr (33) 2 1 In the case of vertical arrangement, this leads to r 1 + R F =F (34) 2;v 1 r 1 R TODO: horizontal 2.5 Friction drive and belt drive The friction drive (see the left-hand side of Figure 7) or friction engine is a type of transmission that, instead of a chain and sprockets, uses two 20n F22 R bearing shaft 2r Fs F1 F21 Figure 6: Pulley with friction T2 T2 n2 n2 n1 n1 Fp Fp Ff T1 T1 Figure 7: (left) friction drive (right) belt drive. wheels in the transmission to transfer power to the driving wheels. This kind of transmission is often used on scooters, mainly go-peds, in place of a chain. The friction force transmitting the power is F =F =F ; (35) f 1 cl where F is the clamping force, i.e. the force pushing the two gears against cl each other. The belt drive (see the right-hand side of Figure 7) uses a belt, i.e. a loop of exible material used to link two or more rotating shafts mechanically. Belts are looped over pulleys. In a two pulley system, the belt can either drive the pulleys in the same direction, or the belt may be crossed, so that the direction of the shafts is opposite. As a source of motion, a conveyor belt is one application where the belt is adapted to continually carry a load between two points. 21Belt friction is a physical property observed from the forces acting on a belt wrapped around a pulley, when one end is being pulled. The equation used to model belt friction is, assuming the belt has no mass and its material is a xed composition:  s T =T e ; (36) 2 1 where T is the tension of the pulling side, which is typically the greater 2 force,T is the tension of the resisting side, is the static friction coecient, 1 s which has no units, and is the angle, in radians formed by the rst and last spots the belt touches the pulley, with the vertex at the center of the pulley. The tension on the pulling side has the ability to increase exponentially if the size of the angle increases (e.g. it is wrapped around the pulley segment numerous times) and as the coecient of friction grows. The force needed to be applied to the shaft is    s F =T +T =T 1 +e ; (37) cl 1 2 1 while the friction force transferring the driving torque is   2 T 2 2 T 1 2 1 M (T T )(T +T ) T 1 1 2 1 2 1 F = =T T = = f 1 2  s R T +T T (1 +e ) 1 1 2 1   2 2 s s 1e 1e =T =F (38) 1 cl  2 s  s (1 +e ) (1 +e ) z  Thus, for both drives, the connection between the clamping and friction force is given by F =F . f cl For both drives, the gear ratio of the transmission is given de ned as n 1 1 i = = : (39) n 2 2 The input power on the driving gear is P =F v =M , the output 1 f 1 1 1 power is P = F v = M . In the practical cases there is a slip between 2 f 2 2 2 the driving and the driven machine: v v v 1 2 2 s = = 1 : (40) v v 1 1 The torque, revolution number and power of the driving machine are M , 1 n and P , respectively. On the driven side, we have 1 1 R R 2 2 M =F R =F R =M ; (41) 2 f 2 f 1 1 R R 1 1 v v R R 2 2 1 1 1 n = = = (1s) =n (1s) and (42) 2 1 2 R 2 R 2R R 2 1 2 2 P =M =P (1s); (43) 2 2 2 1 22

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