 Theory for the design and calculation of:

(1)

Belt and Chain drives

BMMAE-3- Machine Elements

(2)

Objectives

 Descriptions & analysis of several belt types

 Theory for the design and calculation of:

– Flat belt, v-belt, poly-belt transmissions – Statics, dynamics, vibrations, life…

– Chain and timing belt drives

(3)

Outline

1. Introduction

2. Flat, V, poly- V belts

3. Timing belts

4. Chains

(4)

History

1. Introduction

(5)

History

1. Introduction

(6)

Several types of flexible links

Chains Belts

V-belt Poly-V

Timing

1. Introduction

(7)

Several types of flexible links

1. Introduction

(8)

Several types of flexible links

1. Introduction

(9)

Several types of flexible links

1. Introduction

(10)

Several types of flexible links

(11)

Context of use for ‘flexible links’

 Driving in rotation or in translation

 Moving, handling object, power transmission

 Easy to design, flexible, adaptable; flexibility for positionning the driver and driving axes

 Large range of power

 Cheap (compared to gears/chain)

 Reduction and dampening of vibrations

 Easy to maintain

1. Introduction

(12)

 Obstacles

– Chains

– Synchronous belt (toothed, timing belt)

Driving principles

 Adhesion

– Flat belts – V-belt

– Multi-ribbed belt(Poly-V)

1. Introduction

(13)

2. Flat & trapezoïdal (V) & multi ribbed belts

 Aim

– Transmit power from one rotating shaft to another

 Principle

– contact surface, belt-pulley which permits transferring the torque from the driver pulley , transmitting it to the cord member, then to restore it to the driven pulleys

– Cord member, capable of transferring the friction force at the belt pulley interface to a tensile load in the belt strand between the pulleys.

– The initial tension of blets is a key parameter to insure adhesion (no slip) & power transmission

2.1 Introduction

2. Flat, V, Poly-V Belts

(14)

2.1 Introduction



Possibilities

– Multiplication or reduction given the ratio of diameters

– Continuous Variable Transmission (CVT) – Inversion of rotation

– Transmission with non // axes



Advantages

– Dampening of vibrations, – “Silent”

– High speeds are possible – Large center distances

Inverter with additional pulleys Inverter with crossed spans

(15)

2.2 Composition and definition

2.2.1 Flat Belt

belt width 16 20 25 32 40 50 63 71 80 90

40 50 63 80

100 125 140 160

500 560 630 710 800 900 180 200 250 315

1000 1120 1250 1400 1600 1800 2000 2240 2500 400 500 630 800

2800 3150 3550 4000 4500 5000 1000 1250 1600 2000

belt material

0.8 1.3 1.8 2.8 3.3 5 6.3

15 25 60 60 110 240 340

4.5 5.2 7.2 7.9

76 89 115 >=150

6.4 9.5 12.7

38 à 50 57 à 76 76 à 100

flat belts : some dimensions (NF ISO 22)

pulley diameter (ISO)

standard length corresponding

pulley width 20 25 32 40 50 63

2.3

10 à 13 13 à 19 13 à 19

80 90 100

71

belt thickness (mm) minimum pulley diameter (mm)

belt section diameter (mm) minimum pulley diameter (mm)

belt characteristics polyamid (f=0,5 à

0,8)

elastomer (f=0.7)

leather (f=0,4)

1.6 2

9.2

>=230 19 127 à 180 round section belt

(elastomer f=0,7)

belt thickness (mm) minimum pulley diameter (mm)

ω

width b

Thicknesse

pulley

(16)

2.2 Composition and definition

2.2.2 V and poly-V belts

fabric

cable made of polyester (kevlar etc.)

Rubber (polychloropren)

V-belt

poly-v

parallel

classical toothed

Elastomer/rubber Reinforcement

fibers

textile

β

β/2

(17)

2.2 Composition and definition

2.2.2 V and poly-V belts

Narrow series

E

SPZ

CD AB

Z

SPA SPB

SPC Classic series

a

h

40°

f g

lp

β k

m

pulley dp

Z A B C D E SPZ SPA SPB SPC

a 10 13 17 22 32 38 10 13 16 22

h 6 8 11 4 9 25 8 10 13 18

lp 8.5 11 14 19 27 32 8.5 11 14 19

f 7 9 11.5 16 23 28 7 9 11.5 16

g 12 15 19 25.5 37 44.5 12 15 19 25.5

k(mini) 2 2.75 3.5 4.8 8.1 9.6 2 2.75 3.5 4.8

m(mini) 7 8.7 10.8 14.3 19.9 23.4 8.5 11 14 19

dp (usuel) 50 to 630

75 to 800

125 to 1120

200 to 2000

355 to 2000

500 to 2500

63 to 630

90 to 800

140 to 1120

224 to 2000

parameters

(mm) classical series Narrow series

Trapezoidal belt section ISO 4183

(18)

2.4 Transmission geometry Case with 2 pulleys

( ) ( )

2

a 2

d 1 D

a cos 2

d sin D

cos . a 2 d D d 2 D L



 



 −

−

=

− α

= α

α +

α

− + π +

=

For small α

( ) ( )

d D

2

2 2 2

2 2

a 4

d d D

2 D a 2 L

a 2 d D 21 1 . a a 2 2 d d D D d 2 D L

1 2 sin 1 a cos

2 d sin D

θ

− π

= α + π

= θ

+ − π +

+

≅



 



 − −

− +

− + π +

=

−α

≅ α

−

=

− α

≅ α

θ ^D θ ^d

α Center distance a

∅^D

∅ ^d α

ω ^D

ω ^d ^α

Non crossed belt spans

Crossed belt spans

θ ^D

α Center distance a

∅^D

∅ ^d α

ω ^D ω ^d

θ ^d

( ) ( )

2 2

2

2 arcsin 2 4 1

2

2 2 4

d D

D d a

L a D d D d if small

D d

L a D d

a

θ θ π θ

θ α

π

+

 

= = +  =

 

= − + + +

= + + + + 2

*D 2

*d D v

d

D d

d

D = =ω =ω

ω ω 2. Flat, V, Poly-V

Belts

(19)

( +¹ ) (² +¹ )²

= − + −

i i i i i

e x x y y

1 1

1

i i

a ,i

i

i i

b ,i

i

r r

a cos ( brin ext.) e

r r

a cos ( brin croisé) e

γ γ

−

− +

−

 =



 = +



( )

2 2

1 1 1

2 2

1

i i i i

l e r r brin ext

l e r r brin croisée

( .)

( )

− − −

+

 = − −



 = − +



( ) (² )²

2 2

1 1 1 1 1

2 1

i i i i i i

i

i i

e e x x y y

a e e

β ⁻ ⁺ ⁻ ⁺ ⁻

−

+ − − − −

= cos ^αⁱ ⁼²^π ⁻

(

^{β γ}ⁱ ⁺ ^{a i}^, ⁺^γ^{b i}^,

)

O_i(x_i,y_i)

O_i-1(x_i-1, y_i-1) O_i+1(x_i+1, y_i+1)

r_i r_i+1

r_i-1 li

li-1

ei

e_i-1 αi

βi

γa,i

γb,i

2.4 Transmission geometry Generic case with n pulleys

Ext. span

Crossed span

Ext. span Crossed span

(20)

7 1

6

5

3

4

2

x y

N°

pulley Type Xi(mm) Yi(mm) Di (mm) Ratio w_i (tr/min)

Pi_abs (W)

Cr_i (N.m)

1 Crankshaft 0.00 0.00 150.00 2000

2 water pump 73.31 147.88 115.00 1.30 2608.7 70 0.3

3 idler 1 -74.00 109.00 70.00 2.14 4285.7

4 compressor -337.50 -72.00 110.00 1.36 2727.3 3500 12.3 5 alternator -190.00 -62.60 52.00 2.88 5769.2 4000 6.6

6 idler 2 -251.70 -20.70 65.00 2.31 4615.4

7 tensioner -131.03 30.58 76.20 1.97 3937.0

2.4 Transmission geometry

example

(21)

i E_i(mm) Croisé γ^a_i γ^b_i l_i β_ⁱ α^_i ^l_α^_i

1 165.1 1.0 32.8 83.9 164.1 103.2 140.1 183.3

2 152.4 1.0 96.1 81.5 150.7 48.8 133.6 134.0

3 319.7 1.0 98.5 93.6 319.1 160.3 7.6 4.7

4 147.8 1.0 86.4 78.7 144.9 30.8 164.1 157.5

5 74.6 -1.0 101.3 38.3 46.3 37.8 182.5 82.8

6 131.1 1.0 38.3 92.4 131.0 57.2 172.0 97.6

7 134.5 -1.0 87.55 32.8 72.9 143.8 95.8 63.7

1028.93 723.62

L (mm) 1752.55

7 1 6

5

3

4

2

yx

2.4 Transmission geometry example

(22)

2.4 Transmission geometry example

(23)

2.4 Transmission geometry Case with n pulleys: example

(24)

2.5 Dynamics

2.5.1 Global equilibrium

( ) ( )

t T T

C 2 T

F T t

R . 2 T C F T

T

R R C

C

t 2 T

t d T R C

t 2 T

t D T R C

d d d 0

0

D d d

D d

D

d d

D D

+

=

−

=

−

=

+

= +

=

ω

= ω

=

−

=

−

=

−

=

−

=

C

_d

θ ^D θ ^d

α

∅ ^D

∅^d α

ω ^D ω ^d

T T

t t

Driving pulley

Driven pulley

C

_D

In operation

T₀, setting/initial tension depends on :

• torque transmitted on the small pulleyC_d

• radius on small pulley R_d,

• contact arc θ_d

Resulting force on the shaft is 2T₀.cosα θ ^D At rest

θ ^d

α

∅ ^D

∅ ^d α

T

₀

T

₀

T

₀

T

₀

Driving pulley

Driven pulley

2T₀.cosα

(25)

It is subjected to the forces:

 tensile forces: F & F+dF

 pulley normal contact force p.ds

 pulley tangent friction force p.f.ds Assuming dγ very small & dγ≈sin(dγ):

Static equilibrium of a small belt element

.

. '

. .

( ) .

^f

Fd p ds dF

d où fd

dF f p ds F

hence F K e

^γ

γ γ

γ

=

 =

 =



=

 Transmittable torque increase with contact arc

p.ds

pf.ds F F+dF

dγ/2 ^dγ/2 γ

dC

p.ds

pulley belt

2.5 Dynamics

2.5.1 Local equilibrium

(26)

γ



 



β

= γ

β γ

 =



=

β

= γ

. sin 2

f

e . K )

( F soit

) d 2 / sin(

f F

où dF ' ds d

. p . f dF

ds ).

2 / sin(

. p Fd

Static equilibrium of a small V-belt element

p.sin(β ^/2).ds

pf.ds

F+dF F

dγ/2 ^dγ/2 γ

dC

β

pulley

p.ds/2 pf.ds

p.ds/2

pulley belt

2.5 Dynamics

It is subjected to the forces:

 tensile forces: F & F+dF

 pulley normal contact force p.sin(β^/2)ds

 pulley friction force p.f.ds

Assuming dγ very small & dγ≈sin(dγ):

 Transmittable torque increase with contact arc Higher transmission capacity than flat belt

(27)

D d d

D f

f θ

= θ

 Friction coefficients on each pulley depend on contact arcs :

This permits having a continuity for the tensile force in the belt

Continuity of belt span tensions?

( )

^d

D d

. f D

f

. f

te T

et te

F

t F 0

e . K )

( F

γ θ θ

θ

θ γ θ

= θ

=

= γ

=

γ ^{( )}

( )

^{( )} ^d

D d

2 . / sin

f D

2 f sin

te T

et te

F

t F 0

e . K )

( F

β θ β γ

θ θ

β γ θ

θ

= θ

=

= γ

Tensile force variation on the large pulley :

Flat belt V-Belt

C

_d

θ ^D θ ^d

α

∅^D

∅ ^d α

ω ^D ω ^d

T T

t t

Driveing pulley

Driven pulley

C

_D

Relative sliding : creep theory cf §2.5.5

2.5 Dynamics

(28)

2.5.3 Inertia/centrifugal forces

( )

( ) ( )

. d

f 2

2

2 .

f 2

2 2 2

2 2

2 2 2

2 2

i

V e . S . t

V . S . T

V . S . e

V . S . t

F

V . S . . f F

. d f

1 dF

R . b V

. S . F p

R . V

R . . S . F R

. p . b R

. S d .

dm

0 R . d .

R dm . p . b F

2 0

R . . dm d

. R . p . b d

. F

R . . dm f

1 0

d . R . b . f . p dF

θ

γ

ρ =

− ρ

−

ρ + ρ

−

= γ

ρ

−

= γ −

→

ρ

−

=

→ ω

=

ω ρ

−

= ρ 

γ =

= γ ω

+ +

−



= ω

+ γ +

γ

−

ω

=

= γ

−

Belt tensions are reduced by ρSV² in operation:

To transmit the same power, one must increaseT⁰ of ρSV²

For a belt velocity > ~15 m/s inertia forces cannot be neglected (f_i)

F+dF

dC p.ds

pf.ds F

dγ/2 ^dγ/2 γ

f

_i

2.5 Dynamics

(29)

2.5.3 Minimum setting tension T0_MIN

The transmittable torque depending on the

contact arc, one must pay attention to the small pulley whatever it is driving or driven:

. d

f f

. f

te T

et te

F

t F 0

e . K )

( F

θ γ

γ

=

= γ

Flat belt

( )

( )^/² ^sin( )^f ^/² ^. ^d

sin f

. 2 sin f

te T

et te

F

t F 0

e . K )

( F

β θ β γ

γ β

=

= γ

V-Belt

) mV 1 (

e

1 e

R C 2 T 1

1 e

e R

T C 1

e 1 R

t C

2 f

f

d d MINI

_ 0

f f

d d f

d d

d d d

− +

= +

= −

θ θ

θ θ θ

C

_d

θ ^D θ ^d

α

∅^D

∅^d α

ω ^D ω ^d

T T

t t

Driving pulley

Driven pulley

C

_D

T ≥ T

2.5 Dynamics

(30)

C

_d

θ ^D θ ^d

α

∅ ^D

∅^d α

ω ^D ω ^d

T T

t t

Driving pulley

Driven pulley

C

_D

2.5.4 Reactions on shaft/bearings

R

y x

( )

α +

α

=

α

− π

= θ

+ θ

= θ

− θ

=

= θ + θ

=

2 2

d 2 2 d

2 0 d

2 d 2

d 2 d 2 d

2 0

d d

Y

d 0

d X

R sin cos C

. T . 4 R

. 2

cos 2 R

C sin 2

. T . 4 R

cos 2 R

C cos 2

. t T R

sin 2 . T . 2 2

sin . t T R

θ ^d

α

ω ^d

T t

C_d

θ ^d

R t

2α

T

^θ^R

+ α

= −

θ tan

t T

R

2.5 Dynamics

(31)

β

pulley

p.a.R.dγ 2paf.R.dγ

p.a.R.dγ

p.sin(β ^/2).ds

pf.ds

F+dF F

dγ/2 ^dγ/2 γ

dC

Pulley with 2 flanges

Axial load on pulley flanges

Fa Fa

( ) _



 



 β

=

 

 



 β

= −

 

 



 β

=

 

 



 β

=

= γ

 

 



 β γ

=



cos 2 f .

. R . 2

Couple cos 2

f . 2

t F T

cos 2 f .

2 F dF

cos 2 f .

2 dF dF

dF d

. R . a . f . p . 2

cos 2 . d . R . a . p dF

a

T

t a

a a

Larger is the torque to be

transmitted, larger are the axial forces on the flanges

2.5.4 Forces on pulley flanges

2.5 Dynamics

(32)

Experimental researchs have shown that it exists:

-An adhesion arc φ_a at the belt seating where T(γ) is constant -A relative sliding arc φ_g where the belt tension

increases/decreases between t &T

Driven pulley

θ ^D θ ^d

α

∅^D

∅^d α

ω ^D ω ^d

T T

t t

Driving pulley

A

C

B

φg

D

φa/d

φg

φa/D

Sliding arc (variation of T(γ)) one can use the relations

.

0 0

( )

2

1 1

.ln .ln

2

f g

f

d d

g

D D

T te et T te

T C R

T

f t f T C R

γ φ

= =

 + 

=     =  −  2.5.5 Relative sliding (pulley/belt creep)

g d

g D

D / a

g d

d / a

2 π − θ − φ

= φ

− θ

= φ

φ

− θ

= φ

Adhesion arc (different on each pulley)

2.5 Dynamics

(33)

2.5.5 Relative sliding (pulley/belt creep)

2.5 Dynamics

(34)

C_d

θ ^D θ ^d

α

∅^D

∅^d α

ω ^D ω ^d

T T

t t

Driving pulley

Driven pulley

C_D A

C

B

D On arc AB:

-The belt streches(T(γ) ↑ tT) On arc CD :

-The belt retracts (T(γ) ↓ Tt)

Relation between streching and tension for an elemental arc length

T( γ ^)+dT T( γ ⁾

dγ

δ^l

The strain for an arc dγ is given by :

( ) ( ) ( )

( ) ( )

( ) ^γ ⁼ { ( ⁻ ^ρ ) ⁺ ^ρ } ^γ

δ

ρ + ρ

−

= γ

= γ γ γ

 δ

= σ γ

= δ ε

γ γ

d . R V . S . e

. V . S . ' t

E . S l 1

V . S . e

. V . S . t

T

' E . S T d

. R

l '

E l

l

2 .

f 2

2 .

f 2

2.5.5 Relative sliding (pulley/belt creep)

2.5 Dynamics

(35)

2 f

f

d d MINI

_

0 mV

1 e

R C 2 T 1

d

d +

−

= ^θ_θ +

Summary (case of 2 pulleys transmission)

T

₀

=T

_{0_MIN}

T

₀

>T

_{0_MIN}

T

₀

<T

_{0_MIN}

Gross slip Slip limit Normal operation

= + 2

= − 2

− = =

/ = 0

/ = −

2.5 Dynamics

(36)

2.5.6 Losses by relative sliding

Transmission ratio

Adhesion on seating arc on the driven pulley Vslack_span=Rdriven.ω^driven The belt goes faster than the pulley on the sliding arc:

Vbelt_unseating_driven=(1+ε) Vslack_span= Vtight_span= Rdriving.ω^driving

True transmission ratio: η= R^driven.ω^driven/R^driving.ω^driving=1(1+ε)

Streching on driving pulley

( )

_ .

_

. '. . 1

driving g f g

driving driven

driving g g

l t

l S E f e

ϕ ϕ

ϕ

ε δ ε

= = ϕ − = −

θ ^d

∅^d α

ω ^d

T t

Driving pulley

C

φg D

φ_a/d Streching on the driven pulley

( ) { ( ) }

( )

{ }

( )( )

( )

2 . 2

0

2

. 2 2

. _

1 . . . . . .

. '

. . . . . .

. '

. . 1 . . . . .

. ' . ' 1

f

g driven f driven

f g driven

driven g

driven g f g

driven

driven driven

l t S V e S V R d

S E

l R t S V e S V d

S E

t S V

l R e S V t S V

S E f

R t l

l e

S E f l

γ

ϕ γ

ϕ

ϕ ϕ

δ γ ρ ρ γ

δ ρ ρ γ

δ ρ ρ ϕ ρ

δ ε δ

= − +

 − 

 

= − + → 

 

 

 

=  −   =

 



(

^.

)

_

. '. . 1

f g

driven g g

t e

S E f

ϕ

ϕ ⁼ ϕ ⁻

Driven pulley

θ ^D α

∅ ^D

ω ^D

T

t A

B

φg

φa/D

2.5 Dynamics

(37)

Traction

- due to the belt tension

( ) ( )

^F _S

1

= γ γ σ

Bending on pulleys

R 2

e '.

E R

2 / e

'.

E

3 3

= σ

= ε

ε

= σ

F F+dF

dγ /2 ^d^γ/2

γ

fi=dm.v²/R

σ

¹

σ

³

2.6 Fatigue and duration (Life)

2.6.1 Stresses

!!! E’ modulus of rubber << longitudinal modulus (cords)

Belt & Chain drives

37 2. Flat, V, Poly-V

Belts

(38)

A B C D

σ

^{1= t/S}

σ

^{1= T/S}

σ

^{1= t/S}

σ

^{3= E’e/(2R}^D⁾

σ

^{3= E’e/(2R}^d⁾

Large pulley Small pulley

Stress spectrum for one belt travel

T=L/v

A belt element is stressed dynamically and cyclically, hence fatigue criteria can be applied for life estimation.

C

_d

θ ^D θ ^d

α

∅^D

∅^d α

ω ^D ω ^d

T T

t t

Driving pulley

Driven pulley

C

_D

A

C

B D

2.6 Fatigue and duration (Life)

2.6.2 Stress spectrum

(39)

2.6.3 Damage at constant speed and torque

For a n pulleys transmission.

The belt passes over pulley i, it is stressed at σ1+ σ3 and its life is reduced.

Let ri the number of revolutions the belt can do accounting only the stresses on pulley i. At each belt revolution, the life is reduced by 1/rⁱ

 When the belt passes over n pulleys, for each revolution the life is reduced by 1/r⁰.

The life is H⁰=r⁰.L/v)



=

= ⁿ

1

i i

0 r

1 1r

2.6 Fatigue and duration (Life)

(40)

2.7 Vibrations

∆θ ²

∆θ ¹

∅²

∅¹

ω ¹

3 types of decoupled vibrations are considered

Longitudinal vibrations of belt spans (pulley rotations)

∅ ²

∅¹

ω ¹

Transverse vibrations (vibrating string)

δ l δ l

torsion of belt span

(41)

2.7 Vibrations

2.7.1 Transverse vibrations ( vibrating string)

F λ /2

. 2

l ₌ n λ ^F

λ /2 λ /2

F λ /2 λ /2 λ /2

2 .

2. f

n F

f l m

ω = π

 

 =

 

l=string length F=tension

m=mass/unit length

n=order of vibration mode

!!!, here, no bending rigidity and zero velocity

(42)

Model: Pre-loaded beam at rest

.

2

. .

n

1 n F V n E I

l m F l m

π ρ π

ω = ^  ^  − ^  + ^ 

 

 

If the belt travels with a velocity V and its tension is F:

Belt bending rigidity: E.I(N.m²)

2 4

.

_n

² n .

0

n .

m T EI

L L

π π

ω = ^  ^  + ^  ^ 

   

 Belt span length L (m)

 Belt bending rigidity EI (N.m²)

 Belt mass per unit length m (kg/m)

 Belt tension T₀(N)

T₀ T₀

L y

z

EI, m

2.7 Vibrations

2.7.1 Transverse vibrations

(43)

Laser sensors

Load sensor

Model: Pre-loaded beam at rest: determination of EI

2.7 Vibrations

(44)

y = 4,2x + 19,6 R² = 0,9991 y = 16,9x + 79,6

R² = 0,9995 y = 38,3x + 201,5

R² = 0,9993

0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000

0 200 400 600 800 1000 1200

T (N)

f²

Mode 1st 2nd 3rd

EI (N.m²) 0.26 0.07 0.03

 Vibration frequency measurement

 Very small bending rigidity

0

. . 1

n 2

T

f n n f

L m

= =

Vibrating string model is sufficient

beam string diff. beam string diff. beam string diff.

300 35.7 35.6 0.30 72.1 71.2 1.17 109.7 106.8 2.59

600 50.4 50.4 0.15 101.3 100.7 0.59 153.1 151.1 1.32

1000 65.1 65.0 0.09 130.5 130.0 0.36 196.6 195.1 0.80

Tension 1st freq. 2nd freq. 3rd freq.

Model: Pre-loaded beam at rest: determination of EI

2.7 Vibrations

(45)

Torsion vibration in response to an axial excitation

3^rd bending Mode de flexion

 Visualization

2. Flat, V, Poly-V

Belts

2.7 Vibrations

(46)

 Visualization: experimental set up

laser

Pot

électrodynamique 2. Flat, V, Poly-V

Belts

2.7 Vibrations

(47)

2.7.2 Longitudinal Vibrations : introduction

A belt transmission contains several pulleys (N) mounted on different shafts.

The system can then be considered as a N- dof* system:

- N-dof are the pulley rotations - Each pulley + shaft = an inertia

- Coupling between pulley rotations is done by the belt strands

Vibration coupling beteen pulley rotations and belt span axial elongations

Belts

2.7 Vibrations

* degree of freedom

(48)

2.7.2 Longitudinal vibrations : 1st Model, case of 2 pulleys

ω ¹

T

∅ ²

∅¹

t

T

ω ²

I1 C2 I2

C1

( ) ( )



 



ϕ + ω

δ +

=

ω δ

+

=

t cos

. C C

C

t cos

. C C

C

2 2

ini _ 2 2

1 1

ini _ 1 1

Damping is neglected

θ ² θ ¹

∅ ²

∅¹

ω ¹

I1 I2

∆

k k

2 ini _ 2 1

ini _ 1 ini

ini ini

ini

2 2 1 1

r C r

t C . T

k t

t

. k T

T

. r .

r

=

 −



∆

−

=

∆ +

=

θ

− θ

=

∆

Belts