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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;
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.
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).
3. Plot the output shaft velocity (experimental and theoretical) versus input shaft
angle for α1 = 15º, 30º and 45o.
From the result in part 2:
1. Plot the output shaft angle versus the input shaft angle for both readings, parallel
and same side readings in the same graph.
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DISCUSSION
1. Comment on the output shaft angle versus input shaft angle plot that you have
obtained.
2. What is the significance of the tan (input shaft angle) versus tan (output shaft
angle) plot? 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?
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
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RESULT SHEET
Hooke’s Coupling Experiment Part 1
For alpha1, α1 = degrees
Input, θ Output, φ Tan θ Tan φ, Output Velocity
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
310
320
330
340
350
360
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Mechanical Design And CAD Laboratory
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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
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
310
320
330
340
350
360
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