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HOOKE'S JOINT ANALYSIS

OBJECTIVE

The objectives of this experiment are:

(1) To investigate the kinematic characteristics of a universal joint (a Hooke's joint).

(2) To determine the effect of using this type of joint to transmit uniform motion.

DISCUSSION

A complete theoretical development of motion transmitted through a Hooke's joint can be found on pages

62 through 64 of Kinematics and Dynamics of Machines by George Martin. A summary of this motion

analysis (using the notation shown in Figure 1) is given below:

Hooke's Joint

2 2Input Shaft

3

3

3Output Shaft

Pin

Figure 1.

Angular displacement relationship:

tan = tan cos3 2 (1)

Angular velocity relationship:

3

2

2

2

2=

cos

1 - sin sin

(2)

Angular acceleration relationship

(This assumes a constant angular speed of link 3 - i.e. 2 = constant, 2 = 0):

3 2

22

22

2 2= ( )

cos sin sin

(1 - sin sin )

2 2b g(3)

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Hooke's Joint Analysis 2

PROCEDURE

All of the experiments are to be performed on the Hooke's joint apparatus.

(1) Single Joint For this section the intermediate shaft ant the output shaft are always aligned in a

straight line. Determine the angular displacement ratio between the input and output shafts for five

different angles between the input and output shaft. Use the angles for of 10o, 20o, 30o, 40o, and

50o. For each of these configurations record the output angle (3) and the error (3 - 2) for 30

o

intervals of the input shaft (2). Use Table 2 to record your results.

(2) Two Joints - Parallel Output Shaft Adjust the Hooke's joints on either end of the center shaft

such that the axes of the pins (for the two portions on the center shaft) are parallel.

(a) Arrange the shaft as shown in Figure 2. Record 3 and 4 for 30o

intervals of the input

shaft (2). Use Table 3 to record your results.

(b) Put the mechanism in the configuration of Figure 3 and record 5 for 30o intervals of the

input shaft (2). Use Table 3 to record your results.

(3) Two Joints - Perpendicular Output Shaft Adjust the Hooke's joints on either end of the center

shafts such that the axes of the pins are perpendicular.

(a) Arrange the shaft as shown in Figure 2. Record 3 and 4 for 30o intervals of the input

shaft (2). Use Table 3 to record your results.

(b) Put the mechanism in the configuration of Figure 3 and record 5 for 30o

intervals of the

input shaft (2). Use Table 3 to record your results.

2 2

3 3

4

5

Input Shaft

Intermediate Shaft

Output Shaft(Parallel to input shaft)

Input Shaft

Intermediate Shaft

Output Shaft(Perpendicular to input shaft)

45 45

Figure 2. Output Parallel to Input Figure 3. Output Perpendicular to Input

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REPORT

Single Joint Results:

(1) Complete Table 2 summarizing the results of part (1) of the procedure.

(2) Plot the angle error (err = 3- 2) versus the input angle (in = 2) for each of the fiveangles between the shafts ()found in Table 1. Plot all five curves on the same graph

(Graph 1).

(3) Using equation (1) from the Discussion section, calculate the output angle error for = 30o

and plot these results on Graph 1.

(4) Explain any difference between your results and the theoretical results for err for = 30o.

Two Joint Results:

(5) Complete Table 3 summarizing the results of parts (2) and (3) of the procedure.

(6) Plot the angle errors (err = i- 2, i = 3, 4, 5) versus input angle (in = 2) for each of the

three output angles (3, 4, and 5) for the parallel pin configuration. Plot all three on the

same graph (Graph 2).

(7) Plot the angle errors (err = i- 2, i = 3, 4, 5) versus input angle (in = 2) for each of the

three output angles (3, 4, and 5) for the perpendicular pin configuration. Plot all three on

the same graph (Graph 3).

(8) Discuss the results obtained from using two universal joints. Include comments on output

versus input shaft orientation and on the relationship between the universal joint installation(pin orientations).

Angular Speed Variation Analysis:

(9) Use equation (2) to determine the maximum and minimum angular velocity ratios between

the output and input shafts (3/2) for various angles between them ( = 10, 20, 30, 40

and 50). Hint: determine the angle 2 at which the maximum ratio occurs, and then the

angle 2 at which the minimum ratio occurs. Complete the following table:

Table 1. Angular Velocity Ratio Analysis

Angle between shafts, 10 20 30 40 50

Maximum speed ratio, 3/2

Minimum speed ratio, 3/2

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Table 2. Single Joint Analysis

Input Angle Output Angle Angle Between Shafts,

in = 2 out = 3 10o 20o 30o 40o 50o

0o out

err

30o out

err

60o out

err

90o out

err

120o out

err

150o out

err

180o out

err

210o out

err

240o out

err

270o out

err

300o out

err

330o out

err

Note:

err = out- inerr = 3- 2

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Table 3. Two Joint Analysis

Input Angle Output Angle Parallel Pins Perpendicular Pins

in = 2 out 3 4 5 3 4 5

0o out

err

30o out

err

60o out

err

90o out

err

120o out

err

150o out

err

180o out

err

210o out

err

240o out

err

270o out

err

300o

out

err

330o out

err

Note:

err = out- inerr = i- 2 where i = 3, 4, or 5

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