7/29/2019 6. Univesal Joints

1/16

1

6. UNIVERSAL JOINTS A universal joint is a connection between two intersecting rotating

shafts which are coplanar and are inclined at an angle with respect

to each other. The angle b/n the shafts may vary during operation

Used to transmit rotational motion

For a constant angular velocity of the driver, the velocity of thefollower fluctuates b/n a certain maximum and a certain minimum.

7/29/2019 6. Univesal Joints

2/16

2

6.1. Velocity Ratio of Shafts

Consider two shafts A and B which are the driver and follower,respectively.

The axes of the two shafts are inclined at an angle from the planview as shown below.

7/29/2019 6. Univesal Joints

3/16

3

If observed from the direction of A, when the shafts rotate,

A-A traces a circle while B-B traces an ellipse

The ellipse is a projection of the circle traced by b-b of the figure

above If shaft A turns through an angle from AA to A1A1, then the

projection of BB will also turn through angle to B1B1.

During this time the angle turned by shaft B is as observed fromthe axis of shaft B.

The projection of B1 andB2 on AA are C1 and C2.

From the geometry of the projections

)2(tan

)1(tan

11

2

22

2

11

1

BC

OC

BC

OCand

BC

OC

7/29/2019 6. Univesal Joints

4/16

4

Combining the above two equations we get

From the plan view it can be observed that

The relationship b/n , the angular displacement of shaft A and ,the angular displacement of shaft Bis obtained to be

Differentiating equation (5) with respect to time, the output shaftvelocity can be related to the input shaft velocity.

)3(tan

tan

1

1

2

1

OB

OC

OC

OC

)4(cos1

1 OB

OC

)5(costantan

)6(cossecsec22

dt

d

dt

d

7/29/2019 6. Univesal Joints

5/16

5

Where is a constant

The velocity relationship b/n the two shafts is thus obtained to be

From trigonometric relations

Substituting for tan from equation (5)

)8(

)7(

B

A

dt

dand

dt

d

)9(cossecsec 22 BA

22 tan1sec

2

22

2

2

2

cos

tancos

cos

tan1sec

7/29/2019 6. Univesal Joints

6/16

6

from equation (9) we obtain the equation that relates the input andoutput velocities.

Upon simplification, the velocity relation is obtained to be

Hence, the ratio of the angular velocities is given by

The ratio B/ A has a maximum value when for which

= 0 or = 180o or any multiple of 180o. For this condition,

)10(sec

1

coscos

tancos22

22

BA

)11(cos

cossin122

BA

)12(

cossin1

cos22

A

B

,1cos

)13(

cos

1

sin1

cos2

max

A

B

7/29/2019 6. Univesal Joints

7/16

7

The ratio B/ A has a minimum value when cos = 0, for which = 90o or = 270o, . or any multiple of 90o. For this condition,

6.2. POLAR ANGULAR VELOCITY DIAGRAM

Polar angular velocity diagramshows the velocity of the driver and

follower for a complete revolution of the joint.

)14(cosmin

A

B

7/29/2019 6. Univesal Joints

8/16

8

Since the angular velocity of the driver is assumed constant, it isrepresented by a circle.

The angular velocity of the follower is shown as an ellipse, since its

magnitude varies b/n a maximum and a minimum.

The ellipse crosses the circle at four points, in which case, during acycle the angular velocities of the driver and driven shaft are equal.

For this condition

Equation (15) yields

)15(1cossin1

cos22

)16(cos1

1

sincos1cos 2

2

7/29/2019 6. Univesal Joints

9/16

9

Upon simplification we obtain

Solving for tan we get

Thus, the driver and follower have the same speed when

)17(tan1cos1sec22

)18(costan

costan

7/29/2019 6. Univesal Joints

10/16

10

6.3. COEFFICIENT OF SPEED FLUCTUATION

The difference b/n the maximum and minimum speeds of thefollower expressed as a ratio of the driving shaft speed for constant

angle is defined as the coefficient of speed fluctuation.

Substituting for (B )max and (B )min yields

)19(minmax

A

BBq

)21(

cos

sin

cos

cos1cos

cos

1

)20(

coscos

1

2

2

q

or

qA

AA

7/29/2019 6. Univesal Joints

11/16

11

From equation (21) the coefficient of speed fluctuation is obtainedto be

For small angle , sin = , and tan = , hence, the coefficient ofspeed fluctuation is given by

where is in radians.

Having obtained the coefficient of speed fluctuation q, the totalfluctuation of speed is then given by

)22(tansin q

)23(2q

)24(2

Aspeedofnfluctuatiototal

7/29/2019 6. Univesal Joints

12/16

12

6.4. ANGULAR ACCELERATION OF DRIVEN SHAFT

Assuming A to be constant, for a constant inclination b/n thedriver and follower, the angular velocity of the follower is

The angular acceleration of the driven shaft is then obtained from

Which yields the angular acceleration of the driven shaft to be

)25(cossin1

cos22 AB

)26(cossin1

cos22

A

B

dt

d

dt

d

)27()cossin1(

2sinsincos222

22

AB

7/29/2019 6. Univesal Joints

13/16

13

For maximum angular acceleration, the acceleration term isdifferentiated with respect to time t and set equal to zero to give theposition for which the acceleration is maximum or minimum. i.e.

Upon simplification

For small values of ,

)28(0)cossin1(

2sinsincos222

2

dt

d

dt

d B

)29(

sin2

2cos2sin2cos

2

22

)30(sin2

sin22cos

2

2

7/29/2019 6. Univesal Joints

14/16

14

6.5. DOUBLE HOOKS JOINT

In an automobile, if only a single Hooks joint were used, either the

speed of the engine or that of the car would have to vary during

each revolution of the drive shaft. However, the inertia at both ends would resist this occurrence.

High stresses would occur on the transmission shaft and slippage on thetires.

This problem is solved by employing a double Hooks joint

which provides a uniform velocity b/n the input and output ends, Limits the variation of speed to the intermediate shaft.

7/29/2019 6. Univesal Joints

15/16

15

If the driver and follower are inclined equally relative to theintermediate shaft,

The fluctuation of speed will be confined to the intermediate shaft.

The intermediate shaft can then be made short and light in order toreduce the inertia in the transmission.

the relation b/n2, speed for the driver, and

4, speed for the

follower, is obtained as follows.

For angle which the driver turns through in a given time,

where is the angle turned by the intermediate shaft during thesame time.

Also

where is the angle turned through by the follower or output shaft.

)31(costantan

)32(costantan

7/29/2019 6. Univesal Joints

16/16

16

From these relations we have

i.e. the driving and driven shaft turn through the same angle in thesame time.

If the forks on the intermediate shaft are set at right angles, thespeed of the follower 4 will fluctuate b/n;

)34(

)33(tantan

or

)35(24

22

2

2

cos

1

cos

and