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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.

3

Animation showing rotating cams and cam followers producing reciprocating motion.

Basic Displacement Diagram

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 (Plate cam )cam with six different follower arrangements

5

a disk cam with an in-line knife edge follower

a disk-cam with an in-line 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 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.

7

Motorcycle transmission showing cylindrical cam with three followers. Each follower controls the position of a shift fork

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.

8

Sash window lock, traditional cam style, for double hung sash window

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.

10

Plate cam in an engine Cylinder cam in a machine tool to fix the work piece

A work piece

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)

13

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, opposing the motion of the follower and even preventing its motion. r0: Base circle radius

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.

14

Conclusion : to prevent unnecessary wearing of cam edge, with a roller follower friction is strongly reduced.

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 knife edge follower

• 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;

• 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;

17

oscillation center O is at point 10, point C0 is in 10 on the base circle r0; point C must be in 10’

Oscillating angle: a10

Oscillating height:

𝑎 = 𝑎10

Parameter design

• Given the base circle diameter, stroke h;

• To design: rest angle 2𝜑1

stoke angle 2(𝜋 − 𝜑3)

Assume: stepwise acceleration as dot line

18 Due to symmetry, we can consider a rotation angle of the disk cam equal to π

Continue

• Positive acceleration : a+, negative acceleration: a-

19

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

Continue

• Considering a stepwise law of the acceleration, the angle determine:

20

accelerations are constant

accelerations have to be fixed according to the inequality

Division number in grid of Fig.10.7a

Continue

• in order to limit the value of the negative acceleration

21

the angle is determined

displacement y of the follower

Continue

22 This negative value has to be verified keeping contact between the cam profile and the follower.

Numerical example

23

Input the given conditions t into the equations above

Continue

24

double integrating the acceleration curve , displacement or the profile of can be calculated

𝑦 𝑡 = 𝑑𝑦2

𝑑𝑡2

𝜋𝜔

0

dtdt

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

27

• the cam mechanism system can be reduced to that of a slider-crank equivalent system.

Cam: rotation Follower: up-down sliding, reciprocating

slider-crank Cam-follower

Note: only instantaneous, not whole cycle duration

Instantaneous Reduction

28

• the cam mechanism system can be reduced to that of a four bar linkage equivalent system.

Cam: rotation Follower: oscillation

Four –bar linkage Cam-follower

Note: only instantaneous, not whole cycle duration

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 related problems

• the dynamic equilibrium of the follower, without consideration of friction

30

is the push force from the cam

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

31

𝑹𝒏 > 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.

10.2- Spur gears

• Transmission of motion between parallel axes

32

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:

Continue

• Based on the relative velocity, we can classify

33

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

Conjugate profiles

• With respect to fix profile 1, the frame :

34

points O2 and M2 of the profile 2 have the following velocities:

POR :the instantaneous center of the relative velocities of body 2, due to crossover point of two normal lines of

Continue

• Respect to the instantaneous center POR :

35

Conclusion : In order to have a constant gear ratio P0R must occupy a fixed position on the line O1O2

Continue

• Relative velocity in M2 is

36

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.

Input Eq.(10.4):

Two circular profiles

• The gear ratio has the desired constant value:

37

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.

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 tooth-wheel gear;

38

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

the construction of the involute of a circle

Continue

• rolling without slipping of the straight line on the circle, we have:

39

the analytical expression of the involute function:

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;

40

the construction of the involute of a circle

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.

41

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

Continue

• the profiles slide with a relative velocity:

42

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).

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;

43

procedure to machining a spur gear by means of a cutter rack tool

cutter rack tool

44

A gear hob in a hobbing machine with a finished gear.

a hob

Gear Terminology

• Common abbreviations:

45

Gear terminology (see. Figs. 10.18, 10.19)

Gear Terminology (continue) ① Gear or wheel. The larger of two interacting gears;

② Pinion. The smaller gear in a pair;

③ Path of contact. The path followed by the point of contact between two meshing gear teeth;

④ Line of action(Pressure line). The line along which the force between two meshing gear teeth is directed;

⑤ Axis. The axis of revolution of the gear(center line of the shaft);

⑥ Pitch point (P0). The point where the line of action crosses a line joining the two gear axes;

⑦ 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;

⑧ 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;

⑨ 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 all the pitch circles as one moves along the axis. Eg., for bevel gears it is a cone. 46

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 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.

② Diametral pitch (Pd). The ratio of the number of teeth to the pitch diameter.

③ 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.

④ 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'.

⑤ Interference. Contact between teeth other than at the intended parts of their surfaces.

⑥ 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 Point of contact: any point at which two tooth profiles touch each other.

② 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);

③ 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.

④ 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.

53

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

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.

58 A coupling A gear coupling

10.2.1 – Minimum number of teeth

• involute gear profile : to keep a correct rack meshing, the limit position of the rack:

59

From eq.10.15

In the modular design,

For θ = 20° (value frequently used)

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.

61

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:

Continue

• the force Rn passes through POR, these moments have also the expressions (see Fig. 10.24)

62

If consider kinetic friction of coefficient kμ a tangent force:

Continue

• driving moment Mm and resistant moment Mr

63

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,

10.3- Gear trains

• The gear rotation axes are fixed in a still supporting frame

64

Gear box

Continue

• Keeping the senses of the angular velocities

65

the two angular velocities have the same sense of rotation , same direction

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.

67

Gear ratio is determined by the Willis formula:

as if the carrier were still

gear ratio is also

Continue

• Adding a second necessary condition,

68

Epicyclic gearbox, with brakes and clutch

Continue -Working modes

1) Brake C closed – Carrier blocked, ω0 =0, (F,A,B free)

69

reverse rotation between shafts 1 and 2:

2) Clutch F active: direct connection (brakes A,B,C free):

Continue -Working modes

3) Brake B closed – outer ring blocked, ω2 =0, (F,A,C free)

70

advantageous reduction of velocity between shafts 1 and 0

4) Brake A closed – sun gear blocked, ω1 =0, (F,B,C free)

limited increase of velocity between shafts 0 and 2

Fairbanks epicyclic gearing

71

an ordinary gear train:

Based on Willis formula, the ratio must be kept equal also when the carrier is rotating

72

• 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.

Automotive differential

Car differential of a Škoda 422

video

Differential principle

73

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).

Automotive differential

• Two pairs of bevel gears are inside the differential gearbox, of number of teeth z1 and z2;

• The angular velocity of the differential box (carrier), connected to the wheel z0;

74

the gear ratio is:

Continue

• with Willis formula ,

75

i.e. the mean value of the angular velocities of the two driven wheels; If a wheel is stopped, then

10.4- Clutches

• Working principle

76

• 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)

Left-projection view of the crown

circular crown flat rough surface

Working principle

• pressure force N:

77

pressure distribution on crown

opposing moment Mf :

Determine pressure

• the wear hypothesis :

78

the volume per unit time of the material taken away by wear is proportional to the power of friction force:

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;

79

the pressure can become extremely high in proximity of the rotation axis

Continue

• Input eq.(10.34) into eq.(10.30), then the constant c

80

Conclusion :no dependence of Mf on Mm acting on the circular crown

Input (10.34) into (10.3), the moment due to friction

Continue

Comparison of friction moment Mf and machine moment Mm:

Case 1 : If Mf = Mm sliding will continue, at constant angular velocity of the circular crown;

Case 2 :If Mf >Mm the circular crown decelerates, finally stop;

Case 3: If Mf < Mm 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;

82

Power /driver side

User side

Continue

• Consider n pair of mating surfaces, eq. (10.36) changed

83

Operation of clutch

the two clutch discs are pushed against the central clutch disk:

Power balance

• From power side, power • balance equation of the • power source unit:

84

angular acceleration :

Mf=0.75Mmmax Deceleration :

Continue

• From user side, the power • balance:

85

angular acceleration:

Acceleration :

Continue

86

1) power drive unit has an initial angular velocity ωm0

decreases with time; 2) user side starts from a

value ωu0 =0, increases with time;

3) The two angular 4) velocities become equal at

t=ta representing the time of insertion of the clutch;

4) At t=ta the clutch becomes a joint, and the moment transmitted:

Continue

• the acceleration of the system:

87

torque transmitted by the clutch:

A transient dissipation of energy :

corresponds to the shaded area weighted by the constant Mf in Fig.10.35

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:

89

determines a variable value of overlapping between the two mating surfaces, i.e. a wear of the shoe

Continue

• using the wear hypothesis, δ the local thickness of the material taken away from the shoe by wear,

90

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:

Continue

• When 𝛼 = 0°, 𝑖. 𝑒. the maximum pressure p0 along the direction n-n normal to OC

91

𝑝0 = 𝑝𝑚𝑎𝑥 =

local pressure p determines elementary forces

Continue

• integrating these elementary forces:

92

the moment Mf induced on the drum by the elementary tangent forces:

Continue

• Overall result:

93

p0 is unknown and should be determined experimentally

Equilibrium at rotation of the drum we must have Mf=Th

Continue

• a complete drum brake, with two opposite shoes acted upon by the braking force Ff

94

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:

Continue

95

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.

Continue

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

96

Disk brakes

• a typical arrangement of a disk brake for an automobile

97

the wear of the brake pads, it appears to be accompanied by a material consumption δ constant along the radius

Continue

• the braking moment,

98

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 Mf is almost half as much

resulting moment Mf opposing rotation

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,

100 Reversing Belt Transmission with Friction Wheels

The drive belt: used to transfer power from the engine's flywheel. Here shown driving a threshing machine

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

102 Polyurethane Round Belt

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 cross-section shape is trapezoidal (hence the name "V").

103 V'Pulleys V'Belts Poly V'Belts Belts on a Yanmar 2GM20 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.

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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).

105 Timing Belts

Timing Pulleys

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;

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Belt transmission principle

• Neglecting power losses, let W1=W2=W be the power transmitted;

• Neglecting also any stretch of the belt;

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• neglecting the belt elasticity and excluding belt sliding, then:

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• 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.

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• along the local normal and tangent to the pulley,

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is due to a pressure acting along the contact on the pulley

the centrifugal force of inertia is kept

Simplifying eq. (10.40)

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Assume :

• the variation of tension occurs at the limit of adherence of the belt ;

• μ is the friction coefficient in these conditions;

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Substituting in eq.(10.41)

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• 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.

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• 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-by-side in an arrangement called a multi-V, running on matching multi-groove sheaves.

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with fibers like steel

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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 a higher value of allowable variation of the belt tension