Hooke's Experiment

Universiti Tenaga Nasional , 2010

Machine Design And CAD Laboratory

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HOOKE’S COUPLING EXPERIMENT

OBJECTIVES

The objective of this experiment is to investigate the variation in displacement for a

single joint at various angles and to show that when two joints are used together with the

same intermediate angle, the variation in displacement is cancelled out.

THEORY

A flexible coupling or universal joint is frequently used to link two shafts and transmit

circular motion from the other. Indeed continuous circular motion is perhaps the single

largest thing that mankind produces in the world with the available energy. A universal

joint is simply a combination of machine elements which transmit rotation from one axis

to another. A universal join can accommodate larger angles between the shafts. An

arbitrary and accepted lower limit is 3o. In general if the angularity between two shafts is

less than 3o a flexible coupling is used. A universal joint is used where the angularity

between the shafts is intentional. Kinematically universal joints may be divided into two

types the Hooke’s Cardan coupling ant the constant velocity joint. These name although

frequently used do not clarify the difference between them. A Hooke’s coupling is a fixed

arm coupling and a constant velocity joint is a variable arm coupling. We are concerned

here with a fixed arm coupling only.

The following analysis will show that the angle as the angle between the shafts increases

there is a periodic speed and hence torque fluctuation. Such fluctuation cannot be

tolerated in machinery so it is usual to have two coupling with small intermediate shaft.

The second coupling introduces equal and opposite fluctuations, thus the overall effect is

of smooth and uniform transmission. However both the input and the output shaft must

make the same angle with the intermediate shaft for this to work.

A Hooke’s coupling consists of a cruciform spider which pivots in two fork ends formed

in the end of the shafts. For practical manufacturing reasons the fork ends are made as



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separate pieces to which the shafts are attached. Thus standard coupling can be simply

fixed to any length of shaft.

A Hooke’s coupling is shown schematically below

The driven shaft OB is inclined at an angle to the driving shaft in plan view. The axes of

the shafts are on the same horizontal plane. In the figure below the spider arm CD moves

through an angle θ to C1D1.



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P P’ is the plan view of the plane of rotation of CD and Q Q’ is the plan view of the plan

view of the plane rotation of EF. If we now draw RO at right- angles to C1D1 and project

R to R1 in the plan view we can take radius O R1 and draw an arc to cut Q Q’ at R2. We

may now project this point to meet RS at T1. Then angle TOS equals, the angle moved

through by the arm EF in the plane of rotation where P P’ is the plan view

Consider;

SO

RSTan =θ

SO

TSTan =φ

so;

θφ TanSO

TSTan = ……………………….. (1)

θφ TanOR

OTTan

1

1=

θφ TanOR

OTTan

2

1=

αφθ CosTanTan .=∴ …………………… (2)

The above equation gives the displacement. The velocity equation may be obtained by

differentiating equation (2)

Velocity Ratio;

θα

α

ω

φ22 cos.sin1

cos

−=

dt

d ………………….. (3)

The graph below shows the relationship between input and output angles from 0 to 90o

for displacement where the joint angle is 10o to 50

o



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The graph below gives the velocity variation for a single coupling for shaft angle from

10o to 50

o.



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APPARATUS

Hooke’s Coupling Apparatus



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PROCEDURE

Part 1

1. Set the left hand Bracket over so that the angle α1 between the shaft is 45º

2. Set the other joint to be straight, α2 = 0º

3. Start the scale on the input shaft at zero. Rotate the shaft by 10º at each turn. Note

the reading on the input and output scales. (you have to minus the zero error to get

the output angle)

4. Repeat for 360º that is one complete rotation for the shaft input.

5. Tabulate results in the data sheet

6. Repeat procedures 1 through 5 for α1 = 15º and 30º

Part 2

1. Set both α1 and α2 equal at 30º, both bend at the same side.

2. For one revolution of the input shaft (at10º intervals), take readings of the scale of

the output shaft

3. Tabulate results on the data sheet.

4. Repeat the experiment with the input and output shaft parallel but still keep at the

same value of 30º.

5. Tabulate results in the table given.

GRAPH AND RESULT

From the result in part 1:

1. Plot the output shaft angle versus the input shaft angle for α1 = 15º, 30º and 45o

(all in one plot).

2. Plot tan (input shaft angle) versus tan (output shaft angle)1.

3. Plot the output shaft velocity versus input shaft angle for α1 = 15º, 30º and 45o.

From the result in part 2:

1 Only plot from 100

o to 260

o for both x and y-axis as Tan 90,270 = infinity.



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1. Plot the output shaft angle versus the input shaft angle for both readings, parallel

and same side readings in the same graph.

DISCUSSION

1. Comment on the output shaft angle versus input shaft angle plot that you have

obtained2.

2. What is the significance of the tan (input shaft angle) versus tan (output shaft

angle) plot?3 In other words, what information can you obtain from the plot? Does

the plot show the validity of the analysis that was shown in the laboratory?4

3. Comment on the output shaft velocity versus input shaft angle plot that you have

obtained. How well do the velocity variations you found compare with those

predicted in theory.

4. What is the purpose of the second part of the experiment? Does the result justify

the purpose?

5. What is the effect if α1 is not the same as α2

2 Minimum 3 comments, hint(1): what happen at 0

o& 90

o& 180

o& 270

o& 360

o

3 Refer equation 2, recap order of the equation

4 Please recap the boundary of cos(θ).



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RESULT SHEET

Hooke’s Coupling Experiment Part 1

For alpha1, α1 = degrees

Input, θ Output, φ Tan θ Tan φ, Velocity Ratio

0

10

20

30

40

50

60

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90

100

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360



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RESULT SHEET

Hooke’s Coupling Experiment Part 2

For alpha1, α1 = alpha2, α2 = 30 degrees (same / parallel)

Input, θ Output, φ

0

10

20

30

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Hooke's Experiment

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