The University of the South Pacific

MM313 Dynamic Systems

Experiment 2- Crank Mechanism

Aim:

To investigate the relationship between piston displacement and crank angle for different ratios between the connecting rod and the crank. Also to look at the relationship between the turning moment on the crank shaft and crank angle for a given force on the piston.

Equipment and Instrument:

Introduction:

A crank is an arm attached at right angles to a rotating shaft by which reciprocating motion is imparted to or received from the shaft. It is used to convert circular motion into reciprocating motion, or vice-versa. The arm may be a bent portion of the shaft, or a separate arm attached to it. Attached to the end of the crank by a pivot is a rod, usually called a connecting rod. The end of the rod attached to the crank moves in a circular motion, while the other end is usually constrained to move in a linear sliding motion.

Theory:

Figure 1.0: Slider crank mechanism

The slider crank mechanism as shown in figure 1.0 is a kinematic mechanism. The piston displacement from the top dead centre, x, can be determined from the geometry of the mechanism, in terms of the lengths of the connecting rod, L, and crank, R, and the crank angle, θ, can be expressed as x=L+R-(Lcosφ-Rcosϑ)

Also from the geometry, it can be seen that

Rsinϑ=Lsinφ

And

sinφ=sinϑn

Hence

cosφ=[1+sinϑn2]1/2

Where n is a ratio:

n=LR

Procedure:

Part A:

1) No weights and hangers required, the unit initial starting position 0 in the protractor is setup and 90⁰ and 270⁰ protractor positions to be in line with the level lines in each side. 2) The unit is to be setup in its highest point, Top dead centre point was used to work out the displacement value 3) The mounted disc was turned 30⁰ and the displacement was noted on the results table, this step was again repeated for different angles and different crank positions. Part B:

Results:

PART A

Table 1: Results of Piston Displacement

Crank angle| Displacement|

| P1 (mm) experiment| P1 (mm) theory| P2 (mm) experiment| P2 (mm) theory| P3 (mm) experiment| P3 (mm) theory| 0| 0| 0| 0| 0| 0| 0|

30| 3| 3.180748214| 5| 4.252344481| 7| 5.324742758|

45| 7| 6.86291501| 10| 9.20565874| 13| 11.55001055|

60| 12| 11.51142198| 17| 15.51081741| 20| 19.51263112| 90| 22| 22.02041029| 31| 30.01960212| 39| 38.02202662| 120| 31| 31.51142198| 45| 43.51081741| 53| 55.51263112| 135| 35| 35.14718626| 50| 48.80363849| 63| 62.4616988| 150| 38| 37.82176437| 53| 52.74976709| 68| 67.67857183| 180| 39| 40| 56| 56| 71| 72|

Table 2: calculation of the angle φ

Crank angle| φ|

0| 0|

30| 5.73917|

45| 8.130102|

60| 9.974222|

90| 11.53696|

120| 9.974222|

135| 8.130102|

150| 5.73917|

180| 1.40E-15|

Graph of Displacement (mm) vs. Crank angle position (⁰)

Sample Calculation:

For Displacement P1 at 30⁰ crank angle.

To find, φ, n = 5

sinφ=sinθn

φ=sin-1sinθn=sin-1sin305=5.73917⁰

To calculate the theoretical displacement, x:

x=r1-cosθ+nr(1-cosφ)

x=201-cos30+nr1-cos5.73917=3.180748214 mm

Discussion:

1. After plotting the graph of Displacement versus the crank angle position, the graph show that the experimental values and the theoretical displacement can be compared, the experimental plot and the theoretical plot are almost same. 2. From the results graph the graph show that the measured displacement follows the theoretical curve very well. The maximum difference between the experimental and theoretical displacement is 2 mm. 3. For full rotation i.e. 360⁰ the motion of the piston is close to simple harmonic, after 180⁰ the displacement will gradually decrease to 0, it will form a cosine graph.

PART B: Piston Balance and Forces

Table 3: Piston balance and forces

Angle (⁰)| No added Piston Weight P3 (N)| 4N...