DEN 108

Dynamics Laboratory ReportDynamics

Nusrath4/29/2010

ContentsContents..............................................................................................................................................2

Introduction.........................................................................................................................................3

Background and Theory.......................................................................................................................3

Apparatus............................................................................................................................................4

Experimental procedure......................................................................................................................5

Results.................................................................................................................................................6

Calculations.........................................................................................................................................7

Discussion............................................................................................................................................9

Conclusion...........................................................................................................................................9

References...........................................................................................................................................9

Introduction

The aim of this experiment is to measure the natural frequencies of different types of oscillating systems. These are: Rotary swinging pulley, the pulley pendulum and pulley-load-spring system. The frequency of the oscillations can be determined by finding the time taken for the number of oscillations, and then the theoretical frequency can be found. Therefore, we can compare the frequency of oscillations found from the experiment against the theoretical.

Background and TheoryThe oscillating system will vibrate in an equilibrium position if it is displaced by a certain distance. For example, a swinging pendulum will return to its original position due to the tension in the string and the force of gravity. Therefore, the oscillating system will experience a force which is dependent on the displacement from the equilibrium position. The frequency is the number of oscillations per second.

f=1/T

This is the equation that will be used to calculate the frequency from the laboratory experiment.The natural frequency of free vibration of the assembled system is given by:

f1 =

Where k is the spring stiffnessm1 is the mass of the pulleym2 is the mass of the stirrupm3 is the mass of the springm4 is the mass of the bodyIc is the moment of inertia of the pulley about its axis of rotationr is the effective radius of the pulley

When a body is simply suspended from a spring, the natural frequency of free vibration is given by:

f2=

In the bifilar suspension the two light cords of length, L, are separated throughout by 2a. For small rotary displacement of the suspended body (the pulley), the natural frequency of free vibration is given by:

f3=

Where a is half the length from one pin to the otherL is the length of the string from the ceiling

Apparatus Stop watch Light string Weighing machine Pulley Stirrup Spring Body

Fig 1.1 Apparatus to measure natural frequency of the complete pulley-load-spring

system

Fig 1.2 Apparatus to measure natural frequency of Rotary swinging pulley

Experimental procedurePreliminary measurements

Using the balance the masses (m1, m2, m3 and m4) were measured. The smallest degree of accuracy was 1 gram.The radius of the pulley was measured from the centre line of the cord.

Measuring the spring constant, k

The apparatus was setup in order to collect data for measuring the spring constant. The body was setup in order to oscillated up and down (vertically) by hand this was done by pulley the body down and then releasing the body. Using a stop watch the time taken for 20 oscillations were recorded from spring release, this was used to calculate one oscillation (to reduce errors). This procedure was repeated three times to calculate the average.

Measuring the second moment of inertia of the pulley, Ic

The apparatus was shown in figure 1.2. The distance 2a and L were measured, using a measuring tape. The pulley was twisted and then was released to oscillate in the horizontal axis. Using a stop watch the time taken for 20 oscillations were recorded from pulley release, this was used to calculate one oscillation (to reduce errors). This procedure was repeated three times to calculate the average.

Measuring the natural frequency of the complete spring-mass-pulley system

The apparatus was setup as shown in figure 1.1. The body was carefully setup in order to move vertically up and down oscillation by hand, this was done by pulling the body down and then releasing. Using a stop watch the time taken for 20 oscillations were recorded from spring release, this was used to calculate one oscillation (to reduce errors). This procedure was repeated three times to calculate the average.

d

Resultsk = ? (Find out using equation f2)m1 = 0.23kgm2 = 0.055kgm3 = 0.002kgm4 = 0.35kgIc = ? (Find out using equation f3)r = 0.075m

a = = 0.0705m

L = 2.13m

Time of 20 oscillations

Mass and spring: 8.12s

7.02s Average Time =

7.14s

Pulley and string: 41.80s

42.02s Average Time =

41.92s

Pulley, mass and spring: 16.41s

16.62s Average Time =

16.73s

Therefore, the time for one oscillation is:

Mass and spring: Time = = 0.3715s

Pulley and string: Time = = 2.0955s

Pulley, mass and spring: Time = = 0.8295s

CalculationsFrequency of one oscillation

Mass and spring: f2 = = 2.6918 Hz

Pulley and string: f3 = = 0.4772 Hz

Pulley, mass and spring: f1 = = 1.2055 Hz

Finding k using equation f 2:

f2=

Substituting the values into f2:

2.6918 =

k = 100.31 N/m

Finding Ic using f3:

f3=

Substituting the values into f3:

0.4772 =

Ic = 5.849 kg m2

Finding f1 using the equation:

f1 =

Substituting the values into f1:

f1 =

f1 = 1.1953 Hz

Error percentage:

Discussion

Throughout the experiment one of the large source of uncertainty probably lies in number of readings, for example, the weighing balance scale was not the most accurate. The minimum counter weight was one gram. Therefore, there could have been slight errors in the mass readings. This concludes an error of +/- 0.0005kg in the masses. This will have an effect on the frequencies. For calculating the spring constant, while doing the experiment a lot of care was taken. Firstly, for all the readings the initial positions were the same. Secondly, each experiment was recorded three times and averaged to improve the accuracy of the results. While calculating the oscillation time human error (human response starting and stopping the stop watch and the oscillations of the initial and final positions). It was difficult to make the suspension cord to be the same length as the rotational pulley. It was also difficult to make the pulley exactly horizontal; this was due to the rotational oscillation for not being completely planar (pulley quivered slightly).

In order to reduce multi-nodal effect care was taken for the oscillation of load-pulley system. This was done to maintain vertical oscillation. Each experiment was recorded three times and averaged to improve the accuracy of the results.

Conclusion

The experiment successfully met the aim which was to determine the experimental frequency compared to the theoretical frequency. The experimental frequency was calculated 1.2058 and the theoretical frequency was 1.1953. It is clearly showing that the values are close to each other. It was concluded that the theoretical simple harmonic motion model were idealistic for this experiment (did not take energy loss and damping effects into account). There were minimal impact errors in the measurements of length and mass while calculating values of experimental frequency. The value obtained from the experimental frequency is more accurate compared to the theoretical frequency as experimental is a real-life situation.

References

Hibbler, R.C. (2010). Engineering Mechanics Dynamics: SI Version, 12th ed. USA

Meriam, J.L. (2003). Engineering Mechanics-Dynamics: SI Version, 5th ed. New York: Wiley.

Briggs, A. (2009), Elementary Vibrations Lab Handout, DEN108 Module, Queen Mary University London

Briggs, A. (2009), DEN108 Course Notes, DEN108 Module, Queen Mary University London