Chapter 10 Machine elements Bachelor Program in AUTOMATION ENGINEERING Prof. Rong-yong Zhao (zhaorongyong@tongji.edu.cn) Second Semester,2013-2014
Content 10.1 Cams 10.1.1- Synthesis of the mechanism 10.1.2- Analysis of the mechanism 10.1.3- Follower dynamics and related problems 10.2- Spur gears 10.2.1 Minimum number of teeth 10.2.2 Instantaneous efficiency 10.3- Gear trains 10.4 Clutches 10.5 Brakes 10.6 Belts 2
10.1 Cams A cam is a rotating or sliding piece in a mechanical linkage used especially in transforming rotary motion into linear motion or vice-versa. The cam can be seen as a device that rotates from circular to reciprocating (or sometimes oscillating) motion; Displacement diagram Certain cams can be characterized by their displacement diagrams, which reflect the changing position a roller follower (a shaft with a rotating wheel at the end) would make as the cam rotates about an axis. Animation showing rotating cams and cam followers producing reciprocating motion. Basic Displacement Diagram 3
Cam Classification Cam mechanism can be classified by the type of cam or by the shape, motion or location of the follower. 1) Plate cam : The most commonly used cam is the plate cam (also disc cam or radial cam) which is cut out of a piece of flat metal or plate. Here, the follower moves in a plane perpendicular to the axis of rotation of the camshaft. 4
a disk cam with an in-line knife edge follower a disk (Plate cam )cam with six different follower arrangements a disk-cam with an inline roller follower disk-cam with an offset roller follower a disk cam with an oscillating roller follower a disk cam with a reciprocating flat-faced follower a disk cam with an oscillating 5 flat face follower
analysis of a disk cam with an in-line knife edge follower(example) The follower is in-line (or radial) when its centerline passes through the center of cam rotation. This follower is theoretically interest; High contact stresses leads to not great practical 6
Cylindrical Cam 2) Cylindrical Cam A cylindrical cam or barrel cam is a cam in which the follower rides on the surface of a cylinder. In the most common type, the follower rides in a groove cut into the surface of a cylinder. These cams are principally used to convert rotational motion to linear motion parallel to the rotational axis of the cylinder. Motorcycle transmission showing cylindrical cam with three followers. Each follower controls the position of a shift fork 7
Face cam 3) Face Cam A face cam produces motion by using a follow riding on the face of a disk. The most common type has the follower ride in a slot so that the captive follower produces radial motion with positive positioning without the need for a spring or other mechanism to keep the follower in contact with the control surface. Sash window lock, traditional cam style, for double hung sash window 8
Linear Cam 4) Linear Cam A linear cam is one in which the cam element moves in a straight line rather than rotates. The cam element is often a plate or block, but may be any cross section. The key feature is that the input is a linear motion rather than rotational. 9 Key duplicating machine. The original key (mounted in the left hand holder) acts as a linear cam to control the cut depth for the duplicate
Typical application Cams are extensively used in internal combustion engines, machine tools, instruments and many other applications. A work piece Plate cam in an engine Cylinder cam in a machine tool to fix the work piece 10
Cam design A cam may be designed in two ways: a) to assume the required motion of the follower and to design the cam to give this motion (synthesis of the mechanism); b) to assume the shape of the cam and to determine what characteristics of displacement, velocity and acceleration this profile will give (analysis of the mechanism). 11
Nomenclature of cam mechanism 1 The trace point is a point on the follower that corresponds to the contact point of a fictitious knife edge follower. The trace point of a roller is the center of the roller. 2 The pitch curve is the path of the trace point relative to the cam. 3 The base circle is the smallest circle tangent to the cam surface about the center of the cam rotation. 4 The pressure angle is the angle between the direction of motion of the trace point and the common normal (the line of action) to the contacting surfaces. The pressure angle is a measure of the instantaneous force transmission properties of the mechanism. The bigger the pressure angle, the bigger instantaneous force. 5 The throw, or stroke, is the distance between the two extreme positions of the follower. 12
10.1.1- Synthesis of the mechanism design the disk-cam depends on a given law of motion y=y(x) Reference : a knife edge, in-line follower The translation profile, is the desired law of displacement of the follower, is ideally wrapped around the base circle of radius, circle radius value influences the value of the pressure angle α. The increase of pressure angle α worsens the mechanism performance; Reason: the horizontal component Rh of the contact force Rn increases, thus determining higher values of the reaction force component H, causing higher friction forces, r 0 : Base circle radius opposing the motion of the follower and even preventing its motion. 13
Analysis for friction Assume : no friction. With the same height of follower motion, Then, the larger base circle radius, the smaller pressure angle α, but the higher is the sliding speed and the larger the size of the mechanism. Conclusion : to prevent unnecessary wearing of cam edge, with a roller follower friction is strongly reduced. 14
the way to design a cam mechanism an in-line roller follower: the roller center follow the trace point curve thus including by tangency the actual cam profile. Design targets: the follower stroke h; Constraints: limited inner space 15
Design a disk-cam with an offset of The values of y(x) of the translating profile of Fig. 10.4 are measured along the tangent of a circle of radius e equal to the offset, at the prescribed angles; the knife edge follower a roller follower the design follows the same procedure as previously shown. 16
Design for a disk-cam with an oscillating (knife edge) follower relative motion cam- oscillating follower: to fix cam and rotate the oscillating follower; referring to the generic position 10 on the circle by the pin; oscillation center O is at point 10, point C 0 is in 10 on the base circle r 0 ; point C must be in 10 Oscillating angle: a 10 a = a 10 Oscillating height: 17
Parameter design Given the base circle diameter, stroke h; To design: rest angle 2φ 1 stoke angle 2(π φ 3 ) Assume: stepwise acceleration as dot line Due to symmetry, we can consider a rotation angle of the disk cam equal to π 18
Continue Positive acceleration : a +, negative acceleration: a - To determine the acceleration positive-negative switching angle(position): The condition for the follower to have null velocity at the end of the stroke is 19
Continue Considering a stepwise law of the acceleration, the angle determine: accelerations are constant accelerations have to be fixed according to the inequality Division number in grid of Fig.10.7a 20
Continue in order to limit the value of the negative acceleration the angle is determined displacement y of the follower 21
Continue This negative value has to be verified keeping contact between the cam profile and the follower. 22
Numerical example Input the given conditions t into the equations above 23
Continue double integrating the acceleration curve profile of can be calculated y t = 0 π ω, displacement or the dy 2 dt 2 dtdt 24
Note In order to prevent the abrupt force changes on the follower, the acceleration curve should be smoothed. One of feasible method: 1, to keep follower contact with cam with spring; 2, to smooth the cam edge 25
10.1.2- Analysis of the mechanism To analyze the kinematic performance of a cam of an assigned profile. Profile type: a circular profile or joining circular arcs; roll follower type : reciprocating,oscillatory type; 26
Instantaneous Reduction the cam mechanism system can be reduced to that of a slider-crank equivalent system. Cam: rotation Follower: updown sliding, reciprocating Cam-follower Note: only instantaneous, not whole cycle duration slider-crank 27
Instantaneous Reduction the cam mechanism system can be reduced to that of a four bar linkage equivalent system. Cam: rotation Follower: oscillation Cam-follower Note: only instantaneous, not whole cycle duration Four bar linkage 28
Other analysis approach Vector approach: an in-line knife edge follower: instantaneous motion can be analyzed in vector form, by means of relative reference system rotating with the cam disk (at the left) complex numbers approach :by using the complex number form, in analogy to the slider crank mechanism, as indicated in Fig. 10.9 at the right side. 29
10.1.3- Follower dynamics and the dynamic equilibrium of the follower, without consideration of friction related problems is the push force from the cam 30
Analysis of contact The follower will lose the contact with the cam profile when therefore we must keep it positive, further spring force should be large enough R n > 0 Follower-cam contact The modular of the negative acceleration a series of cams arranged on a camshaft to be installed on an internal combustion engine. 31
10.2- Spur gears Transmission of motion between parallel axes points M1 (belonging to the profile in O1) and M2 (belonging to the profile in O2) are the points currently in contact Contacting points M1 and M2, velocities relative velocity: 32
Continue Based on the relative velocity, we can classify profiles are called primitive profiles, their contact is of pure rolling; Case 1:if the two profiles have a common normal n and the does not have a component along n, then the profiles are called conjugate profiles; Case 2: if the two profiles are colliding, and are not of any practical use for the transmission of motion 33
With respect to fix profile 1, the frame : Conjugate profiles points O2 and M2 of the profile 2 have the following velocities: P OR :the instantaneous center of the relative velocities of body 2, due to crossover point of two normal lines of 34
Continue Respect to the instantaneous center P OR : Conclusion : In order to have a constant gear ratio P 0R must occupy a fixed position on the line O 1 O 2 35
Relative velocity in M 2 is Continue Input Eq.(10.4): M1 and M2 occupy the same geometric point M of contact The sliding velocity depends also on the distance of the point of contact from the relative velocity center. 36
The gear ratio has the desired constant value: Two circular profiles from the equilibrium of disc 1 rolling without sliding condition: determines a strong limitation to the maximum value of the torque (power) that can be transmitted without slipping. 37
Continue To avoid this limitation it is necessary to find other kind of profiles, capable to transmit forces along the normal to the profile; Leonhard Euler proposed to use the involute of the circle for the shape of the teeth of toothwheel gear; the construction of the involute of a circle The involute of a curve is the locus of points traced out by a chosen point of a straight line (a taut string) rolling without sliding on the curve itself 38
Continue rolling without slipping of the straight line on the circle, we have: the analytical expression of the involute function: 39
Fundamental property The line tracing the curve results to be the normal to the curve itself; This is a fundamental property to satisfy conditions: 1) to keep a constant gear ratio; 2) to be able to transmit a driving force not relaying on friction, allowing the transmission of significant power; the construction of the involute of a circle 40
Other property The involute of a circle has some properties that makes it extremely important to the gear industry. If two intermeshed gears have teeth with the profile-shape of involutes, then their relative rates of rotation are constant; the gears always make contact along a single steady line of force; Other teeth shape: the rotation speeds and transmitted forces rise and fall as successive teeth engage, resulting in vibration, noise, and excessive wear. For this reason, nearly all modern gear teeth bear the involute shape. Line remains tangent to them and normal to the involute profiles. Every contact between the involute two profiles the normal n intersects the line joining the centers of the two base circles 41
Continue the profiles slide with a relative velocity: Relative motion becomes of pure rolling only when the contact point M is in P0R In order to obtain a continuous rotation it is necessary that, before two profiles loose contact, another pair of profiles are in contact. A series of profiles (teeth of the spur gear) must be provided around the base circle, at a constant distance form one another (circular pitch). 42
Rack tool One way is to use a rack tool, as the European system, is designed on the basis of a parameter m called module(unit of millimeter); The flanks of rack tool are rectilinear, inclined of 2θ to each other, (θ being the pressure angle) and capable of machining a spur gear starting from a disk of suitable diameter; procedure to machining a spur gear by means of a cutter rack tool 43
cutter rack tool a hob A gear hob in a hobbing machine with a finished gear. 44
Common abbreviations: Gear Terminology Gear terminology (see. Figs. 10.18, 10.19) 45
Gear Terminology (continue) 1 2 3 4 5 6 7 8 9 Gear or wheel. The larger of two interacting gears; Pinion. The smaller gear in a pair; Path of contact. The path followed by the point of contact between two meshing gear teeth; Line of action(pressure line). The line along which the force between two meshing gear teeth is directed; Axis. The axis of revolution of the gear(center line of the shaft); Pitch point (P 0 ). The point where the line of action crosses a line joining the two gear axes; Pitch circle. A circle, centered on and perpendicular to the axis, and passing through the pitch point. also called the 'pitch line', although it is a circle; Pitch diameter (Dp). Diameter of a pitch circle. Equal to twice the perpendicular distance from the axis to the pitch point. The nominal gear size is usually the pitch diameter; Pitch surface. For cylindrical gears, this is the cylinder formed by projecting a pitch circle in the axial direction. More generally, it is the surface formed by the sum of 46 all the pitch circles as one moves along the axis. Eg., for bevel gears it is a cone.
Gear terminology(continue) 1 Angle of action. Angle with vertex at the gear center, one leg on the point where mating teeth first make contact, the other leg on the point where they disengage. 2 Arc of action. The segment of a pitch circle subtended by the angle of action. 3 Pressure angle (θ ). The complement of the angle between the direction that the teeth exert force on each other, and the line joining the centers of the two gears. For involute gears, the pressure angle is constant. 4 Outside radius, diameter (Re, De). Diameter of the gear, measured from the tops of the teeth. 5 Root radius, diameter (Ri, Di). Diameter of the gear, measured from the base of the tooth space. 6 Addendum (a). The radial distance from the pitch surface to the outermost point of the tooth. 47
Gear terminology(continue) 1 Dedendum (d). The radial distance from the depth of the tooth trough to the pitch surface; 2 Whole depth (ht). Whole depth (tooth depth) is the total depth of a tooth space, equal to addendum plus dedendum, also equal to working depth plus clearance. 3 Clearance. Clearance is the distance between the root circle of a gear and the addendum circle of its mate. 4 Working depth. Working depth is the depth of engagement of two gears, that is, the sum of their operating addendums. 48
Gear terminology(continue) 1 2 3 4 5 6 Circular pitch (p). The distance from one face of a tooth to the corresponding face of an adjacent tooth on the same gear, measured along the pitch circle. Diametral pitch (Pd). The ratio of the number of teeth to the pitch diameter. Base circle (radius ρ ). Applies only to involute gears, where the tooth profile is the involute of the base circle. The radius of the base circle is somewhat smaller than that of the pitch circle. Base pitch (pb). Applies only to involute gears. It is the distance from one face of a tooth to the corresponding face of an adjacent tooth on the same gear, measured along the base circle. Sometimes called the 'normal pitch'. Interference. Contact between teeth other than at the intended parts of their surfaces. Interchangeable set. A set of gears, any of which will mate properly with any other. 49
Gear geometry A spur gear with z number of teeth, designed on the basis of the module m (metric system) 50
Contact nomenclature 1 2 3 4 Point of contact: any point at which two tooth profiles touch each other. Line of Contact: a line or curve along which two tooth surfaces are tangent to each other, line laying on the flanks of two mating teeth (normal to the plane of Fig.10.20); Path of action: is the locus of successive contact points between a pair of gear teeth, during the phase of engagement. For conjugate gear teeth, the path of action passes through the pitch point. It is the trace of the surface of action in the plane of rotation. Line of action: The line of action is the path of action for involute gears. It is the straight line passing through the pitch point and tangent to both base circles, line T1-T2 in Fig.10.20. 51
Contact nomenclature Length of action: the distance on the line of action through which the point of contact moves during the action of the tooth profile, length of line M-N in Fig.10.20. Arc of action: the arc of the pitch circle through which a tooth profile moves from the beginning to the end of contact with a mating profile(arcs in Fig.10.20)) Arc of approach: the arc of the pitch circle through which a tooth profile moves from its beginning of contact until the point of contact arrives at the pitch point (arc in Fig.10.20). Arc of recess: the arc of the pitch circle through which a tooth profile moves from contact at the pitch point until contact ends ( in Fig.18.20). 52
Contact nomenclature Contact ratio: the ratio f between the arc of action and the angular pitch p. Also,It can be calculated measuring the length of action M-N due to the property of the involute curve Doing the same for the recess arc, the contact ratio is 53
Continue Target: a new pair of profiles has to come in contact before the preceding pair looses contact, therefore : Practical minimum values: Limit diameter: the diameter on a gear at which the line of action intersects the maximum (or minimum for internal pinion) addendum circle of the mating gear. This is also referred to as the start of active profile, the start of contact, the end of contact, or the end of active profile. 54
Backlash Backlash is the error in motion that occurs when gears change direction; Backlash, sometimes called lash or play, is clearance or lost motion in a mechanism caused by gaps between the parts. One source defines it as the maximum distance through which one part of something can be moved without moving a connected part. An example, in the context of gears and gear trains, is the amount of clearance between mated gear teeth. It exists because there is always some gap between the tailing face of the driving tooth and the leading face of the tooth behind it on the driven gear, and that gap must be closed before force can be transferred in the new direction; 55
Backlash 56
Discussion A pair of gears could be designed to have zero backlash, but this would presuppose perfection in manufacturing, uniform thermal expansion characteristics throughout the system, and no lubricant; Reason 1: reducing the tooth thickness of each gear by half the desired gap distance; Reason 2:moving the gears farther apart; 57
Discussion for backlash application Gear couplings use backlash to allow for angular misalignment. A coupling is a device used to connect two shafts together at their ends for the purpose of transmitting power. A coupling A gear coupling 58
10.2.1 Minimum number of teeth involute gear profile : to keep a correct rack meshing, the limit position of the rack: From eq.10.15 In the modular design, For θ = 20 (value frequently used) 59
Discussion If the value prescribed in eq. 10.17 is exceeded, there will be interference ; To obtain a lower minimum number of teeth there are two usual non standard procedures: 1) Correction (called long and short addendum system ): using the same modular cutting rack, to both on the wheel and the pinion pair; 2) Use of non modular cutting rack: while the pitch remains p m π = as previously, both addendum and dedendum of this special cutting rack are reduced of a quantity α 60
10.2.2 Instantaneous efficiency Two involute profiles are in contact in a generic point M along the line of action T1T2. The relative velocity of profile in O2 over profile in O1 is (eq. (10.10)) : To define a driving moment Mm: To define a resistant moment Mr: 61
Continue the force Rn passes through P OR, these moments have also the expressions (see Fig. 10.24) If consider kinetic friction of coefficient k μ a tangent force: 62
Continue driving moment Mm and resistant moment Mr define an instantaneous efficiency η: distance δ normally is >0; when δ=0, the instantaneous efficiency η is maximum The average efficiency: With gear pairs of normal accuracy, 63
10.3- Gear trains The gear rotation axes are fixed in a still supporting frame Gear box 64
Continue Keeping the senses of the angular velocities the two angular velocities have the same sense of rotation, same direction 65
10.3.1- Epicyclic gear trains Epicyclic gearing or planetary gearing is a gear system that consists of one or more outer gears, or planet gears, revolving about a central, or sun gear; The planet gears are mounted on a movable arm or carrier; Epicyclic gearing systems may also incorporate the use of an outer ring gear or annulus 66
Continue the epicyclic gearing system in Fig. 10.27 has 2 d.o.f. and another condition is necessary to define the rotations. Gear ratio is determined by the Willis formula: as if the carrier were still gear ratio is also 67
Continue Adding a second necessary condition, Epicyclic gearbox, with brakes and clutch 68
Continue -Working modes ) Brake C closed Carrier blocked, ω 0 =0, (F,A,B free) reverse rotation between shafts 1 and 2: 2) Clutch F active: direct connection (brakes A,B,C free): 69
Continue -Working modes 3) Brake B closed outer ring blocked, ω 2 =0, (F,A,C free) advantageous reduction of velocity between shafts 1 and 0 4) Brake A closed sun gear blocked, ω 1 =0, (F,B,C free) limited increase of velocity between shafts 0 and 2 70
Fairbanks epicyclic gearing an ordinary gear train: Based on Willis formula, the ratio must be kept equal also when the carrier is rotating 71
Automotive differential In automobiles and other wheeled vehicles, a differential is the usual way to allow the driving roadwheels to rotate at different speeds. This is necessary when the vehicle turns, making the wheel that is travelling around the outside of the turning curve roll farther and faster than the other. video Car differential of a Škoda 422 72
Differential principle Input torque is applied to the ring gear (blue), which turns the entire carrier (blue). The carrier is connected to both sun gears (red and yellow) only through the planet gear (green). Torque is transmitted to the sun gears through the planet gear. The planet gear revolves around the axis of the carrier, driving the sun gears. If the resistance at both wheels is equal, the planet gear revolves without spinning about its own axis, and both wheels turn at the same rate If the left sun gear (red) encounters resistance, the planet gear (green) spins as well as revolving, allowing the left sun gear to slow down, with an equal speeding up of the right sun gear (yellow). 73
Automotive differential Two pairs of bevel gears are inside the differential gearbox, of number of teeth z 1 and z 2 ; The angular velocity of the differential box (carrier), connected to the wheel z 0 ; the gear ratio is: 74
Continue with Willis formula, i.e. the mean value of the angular velocities of the two driven wheels; If a wheel is stopped, then 75
10.4- Clutches Working principle flat rough surface circular crown Left-projection view of the crown circular crown is rotating at an angular velocity ω ; crown is pressed against a flat rough surface ; N: the pressure force; δ is the local thickness; The pressure distribution on crown is circular symmetry, depending on the distance from the axis : p=p(r) 76
pressure force N: Working principle pressure distribution on crown opposing moment M f : 77
the wear hypothesis : the volume per unit time of the material taken away by wear is proportional to the power of friction force: Determine pressure 78
Analysis on pressure wear only intervenes on the rotating circular crown; the thickness δ of the material taken away by wear is constant along the radius; the pressure can become extremely high in proximity of the rotation axis 79
Continue Input eq.(10.34) into eq.(10.30), then the constant c Input (10.34) into (10.3), the moment due to friction Conclusion :no dependence of M f on M m acting on the circular crown 80
Continue Comparison of friction moment M f and machine moment M m : Case 1 : If M f = M m sliding will continue, at constant angular velocity of the circular crown; Case 2 :If M f >M m the circular crown decelerates, finally stop; Case 3: If M f < M m sliding will continue, at increasing angular velocity of the circular crown; 81
A clutch sketch The external box, carrying the helical springs and two circular crowns, rotates with the power source unit; the central clutch disk, rotating with the user side; Power /driver side User side 82
Continue Consider n pair of mating surfaces, eq. (10.36) changed Operation of clutch the two clutch discs are pushed against the central clutch disk: 83
Power balance From power side, power balance equation of the power source unit: angular acceleration : M f =0.75M mmax Deceleration : 84
Continue From user side, the power balance: angular acceleration: Acceleration : 85
Continue 1) power drive unit has an initial angular velocity ω m0 decreases with time; 2) user side starts from a value ω u0 =0, increases with time; 3) The two angular 4) velocities become equal at t=t a representing the time of insertion of the clutch; 4) At t=ta the clutch becomes a joint, and the moment transmitted: 86
Continue the acceleration of the system: torque transmitted by the clutch: A transient dissipation of energy : corresponds to the shaded area weighted by the constant M f in Fig.10.35 87
10.5- Brakes Drum brakes with external shoes; a drum, rotating around C at an angular velocity ω; Force F (the braking force) is pushing the so called shoe, pivoting around O; Sliding friction between the drum and the shoe surfaces causes (opposite) surface forces opposing; Tangent forces depend on the local pressure acting between the mating surfaces. 88
Analysis To determine the pressure distribution the same wear hypothesis; Assume only wear affects the shoe; Point P on the shoe, would be displaced: determines a variable value of overlapping between the two mating surfaces, i.e. a wear of the shoe 89
Continue using the wear hypothesis, δ the local thickness of the material taken away from the shoe by wear, where, dv is the volume per unit time of material taken away locally from the shoe, and r is the radius of the drum Eliminating da, the local pressure acting on point P: In Fig. 10.37: Thickness: 90
Continue When α = 0, i. e. the maximum pressure p 0 along the direction n-n normal to OC p 0 = p max = local pressure p determines elementary forces 91
Continue integrating these elementary forces: the moment Mf induced on the drum by the elementary tangent forces: 92
Continue Overall result: p 0 is unknown and should be determined experimentally Equilibrium at rotation of the drum we must have M f =Th 93
Continue a complete drum brake, with two opposite shoes acted upon by the braking force F f writing the equilibrium to rotation of both shoes and keeping account of the kinetic friction equation linking the values of the tangent force to that of the normal component: 94
Continue using also eq. (10.50), then equations (10.51) and (10.52) determines two different values of the resulting tangent action: rotation given for the drum, contribution of the left shoe to the braking moment is greater than that provided by the right shoe. 95
a) The drum brake with internal shoes (Fig.10.40a) is far more frequently used in practical applications; b) The braking action is determined by the opening of the internal shoes, due to the action of a hydraulic device placed in A Continue 96
Disk brakes a typical arrangement of a disk brake for an automobile the wear of the brake pads, it appears to be accompanied by a material consumption δ constant along the radius 97
the braking moment, Continue 1) two the mating surfaces causing sliding friction 2) Comparing with the clutch, for disk brakes the force N required to produce the same braking moment M f is almost half as much resulting moment M f opposing rotation 98
10.6- Belts belt is a looped strip of flexible material, used to mechanically link two or more rotating shafts; Usage: to transmit power or to track relative movement; Belts are looped over pulleys. In a two pulley system, the belt can either drive the pulleys in the same direction; 99
Continue belt may be crossed, direction of the shafts is opposite; Belt types : Flat belts, Round belts, Vee belts, Multi-groove belts, Ribbed belt, Film belts, Timing belts, Reversing Belt Transmission with Friction Wheels The drive belt: used to transfer power from the engine's flywheel. Here shown driving a threshing machine 100
Flat belts Flat belts were widely used in the 19th and early 20th centuries in line shafting to transmit power in factories. Application :in countless farming, mining, and logging applications, such as bucksaws, sawmills, threshers, silo blowers, conveyors for filling corn cribs or haylofts, balers, water pumps (for wells, mines, or swampy farm fields), and electrical generators.; Because flat belts tend to climb towards the higher side of the pulley, pulleys were made with a slightly convex or "crowned" surface (rather than flat) to allow the belt to self-center as it runs. 101
Round belts Round belts are a circular cross section belt designed to run in a pulley with a 60 degree V- groove. Round grooves are only suitable for idler pulleys that guide the belt, or when (soft) O-ring type belts are used Polyurethane Round Belt 102
Vee belts Vee belts (also known as V-belt or wedge rope) solved the slippage and alignment problem. It is now the basic belt for power transmission. They provide the best combination of traction, speed of movement, load of the bearings, and long service life. They are generally endless, and their general crosssection shape is trapezoidal (hence the name "V"). V'Belts Poly V'Belts V'Pulleys Belts on a Yanmar 2GM20 103 marine diesel engine
Multi-groove belts A multi-groove or polygroove belt is made up of usually 5 or 6 "V" shapes alongside each other. This gives a thinner belt for the same drive surface, thus it is more flexible, although often wider. The added flexibility offers an improved efficiency, as less energy is wasted in the internal friction of continually bending the belt. 104
Timing belts Timing belts, (also known as toothed, notch, cog, or synchronous belts) are a positive transfer belt and can track relative movement. These belts have teeth that fit into a matching toothed pulley. When correctly tensioned, they have no slippage, run at constant speed, and are often used to transfer direct motion for indexing or timing purposes (hence their name). They are often used in lieu of chains or gears, so there is less noise and a lubrication bath is not necessary. Camshafts of automobiles, miniature timing systems, and stepper motors often utilize these belts. Timing belts need the least tension of all belts, and are among the most efficient. They can bear up to 200 hp (150 kw) at speeds of 16,000 ft/min (4,900 m/min). Timing Belts Timing Pulleys 105
General principles General principles of the transmission of power by means of a belt is presented by considering a flat belt; Two pulleys rotating around axes in O1 and O2, and with tighten devices; 106
Belt transmission principle Neglecting power losses, let W1=W2=W be the power transmitted; Neglecting also any stretch of the belt; 107
Continue neglecting the belt elasticity and excluding belt sliding, then: 108
Continue the ratio of the angles of rotation of the two pulleys is not constant, thus preventing the use of this kind of transmission for many applications; more closely the way the tensions can vary along the angles of contact of the pulleys. an element of belt (see Fig. 10.48) of length rdα in limit conditions of adherence to the pulley and neglecting the centrifugal forces of inertia. The tension T changes to T+dT over an angle dϕ at the center of the pulley. 109
Continue along the local normal and tangent to the pulley, is due to a pressure acting along the contact on the pulley the centrifugal force of inertia is kept Simplifying eq. (10.40) 110
Assume : Continue the variation of tension occurs at the limit of adherence of the belt ; μ is the friction coefficient in these conditions; Substituting in eq.(10.41) 111
Continue the tension occurs along the arcs having equal extension ϕ on the two pulleys; transmitted torque is increased, in order to satisfy eq. (10.38) the difference T1 T2 will increase also, as well as the angle ϕ necessary to vary the tension. This angle has the maximum value corresponding to the smallest angle of contact between ϕ1 and ϕ2. If this angle is exceeded, sliding conditions will occur. 112
Continue Vee belts (also known as V-belt or wedge rope) are an early solution that solved the slippage and alignment problem; The "V" shape of the belt tracks in a mating groove in the pulley (or sheave),with the result that the belt cannot slip off (see Fig. 10.50); For high-power requirements, two or more vee belts can be joined side-byside in an arrangement called a multiwith fibers like steel V, running on matching multi-groove sheaves. 113
Continue Assume 1) the variation of tension occurs at the limit of adherence of the belt 2) μ is the friction coefficient in these conditions an equivalent friction coefficient higher than the one for a flat belt 114 a higher value of allowable variation of the belt tension