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 Lab Manual Engineering Dynamics

Experiment no 6

Objective:

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.

Apparatus:

Hooke’s coupling Apparatus.

     

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 element which transmit rotation from one axis to an other. A universal join can accommodate larger angles between the shafts. An arbitrary and accepted lower limit is 3°. In general if the angularity between two shafts is less than 3° 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.

This configuration uses two U-joints joined by an intermediate shaft, with the second U-joint phased in relation to the first U-joint to cancel the changing angular velocity. In this configuration, the angular velocity of the driven shaft will match that of the driving shaft, provided that both the driving shaft and the driven shaft are at equal angles with respect to the intermediate shaft (but not necessarily in the same plane) and that the two universal joints are 90 degrees out of phase. This assembly is commonly employed in rear wheel drive vehicles, where it is known as a drive shaft or propeller (prop) shaft.

                                      

Even when the driving and driven shafts are at equal angles with respect to the intermediate shaft, if these angles are greater than zero, oscillating moments are applied to the three shafts as they rotate. These tend to bend them in a direction perpendicular to the common plane of the shafts. This applies forces to the support bearings and can cause "launch shudder" in rear wheel drive vehicles. The intermediate shaft will also have a sinusoidal component to its angular velocity, which contributes to vibration and stresses.

The following analysis will show that the angle as the angle between the shaft 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 coupling introduces equal and opposite fluctuation, 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 separate pieces to which the shaft are attached. Thus standard can be simply fixed to any length of shaft.

A hooke’s coupling is shown in schematically below

                                   

The driven shaft OB is inclined at angle to the shaft in plan view. The axes of the shaft are on the same horizontal plane. In the figure below the spider arm CD moves though an angle q to C1D1.

                                          

P p’ is the plan view of rotation of CD and Q Q’ is the plan view of the plane rotation of EF. If we now draw RO at right-angle to C1D1 and project R ro 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 though by the arm EF in the plane of rotation where p P’ is the plan view

Consider,

                                                  Tanq=RS/SO                                        Tanf=TS/SO

So;

                                 Tanf=TS/SOTanq ----------------------------------(1)

                                 Tanf=OT1/OR1Tanq

                                 Tanf=OT1/OR2Tanq

                                 \Tanq=Tanf.Cosα---------------------------------(2)

The above equation gives the displacement. The velocity equation may be obtained by differentiating equation (2)

Velocity;

                                       df/wdt=cosα/1-sin²α.cos²q

the graph below shows the relationship between input and output angle from 0 to 90° for displacement where the joint angle is 10° to 50°

Procedure:

1-                   Set both a1 and a2 equal at 30-, both bend at the same side.

2-                   For one revolution of the input shaft(at 10 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 30.

5-                   Tabulate results in the table given.

Graph and result:

1-      Plot the output shaft angle versus the input shaft angle for both readings, parallel and same side reading in the same graph.

 Result sheet:

For alpha 1, a1= alpha2, a2=30 degrees (same / parallel)

Input, q	Output, j

Conclusion:

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