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Damped Oscillation.

Extracts from this document...

Introduction

MECHANICS 4

Coursework

Work based on a combination of the Modelling and Experimental cycles

Damped Oscillation

By Jian Qin Lu

  • Introduction

Simple Harmonic Motion (SHM) is a very interesting motion. In the ideal situation the acceleration of the moving object is proportional to the distance between the object and the origin (O), and the time period of the oscillation is constant. However, in a real situation the motion doesn’t exactly behave like this. Because there is damping, it makes the amplitude of the motion decrease and finally the motion will be stopped. The whole system continues lose energy due to against the resistance (i.e. air resistance).

        Simple pendulum motion can be approximated as a SHM at a small angle (less than 170 or 0.3 radius at 2 decimal places accuracy level). Therefore it can be modelled as SHM (with the damping term).

(NB: all the time measurements in this coursework are accurate to 0.01 second.)

  • Aim

In this coursework I am going to use differential equation to model the damping of the simple pendulum motion in a thin liquid. And find the general solution of the differential equation. Also I will give the particular solution of this situation.

  • Simplifying the situation and setting up the model

Here I will list the basic data of the experiment.

  1. The mass of the pendulum bob (m) = 1kg  
  2. The length of the string (l) = 1m

3.  

...read more.

Middle

image26.png.

Dividing both sides by m and using the small angle approximationimage27.png, the equation of motion is

image28.png.

This can alternatively be written as

image29.png  or   image30.png

        In this case because image31.png is the only factor which can vary x which is the displacement from centre to the current pendulum bob location (length of the string is constant). Hence I can replace image31.png by x. The equation will looks like this: image33.png or image34.png. From this equation, we know that image35.png. In this particular coursework the length of the string I use is 1 meter long, hence image36.png where g is 9.8.

        The damped term image37.png in fact is the resistance. I did a separate experiment in order to find the resistance.

        What I did is that dropped the pendulum bob into a long tube full of water from the surface of water. I made a mark on the tube whose position is 0.2 meter from the surface of water. I would like to record the time which the ball took to cover this distance. In this experiment I assumed that the acceleration in this period is constant and the ball did not reach its terminal velocity in the 0.2 meter distance. The left hand side diagram shows the set of the experiment.image38.png

                                               In order to reduce the error, I just took the reading from the first 0.2 meter and I repeated this experiment twice.

...read more.

Conclusion

Anyhow in the experiment I did, my model worked. It can represent the motion pretty well. Therefore I consider that this differential equation is the model of this situation.

  • Assessment of the improvement

       Since my reaction time is the biggest factor which effect on the experiment, I could use some better equipment to record the time for me. The reading will be more accurate than that I gained.

       Also I can consider more factors rather than make assumptions. The model should be able to represent the real situation better.

  • Conclusion

       Through this coursework I have obtained the differential equation which models the motion of a 1kg bob with 1 meter string in the water. The differential equation is image47.png. The general solution of the differential equation is image51.png and the particular solution is image57.png. The whole is system is overdamping.

       The biggest variability in my coursework is the time measurements. It may change the parameter of the damped term. Therefore the type of damping may be changed as well. However according to the experiment, the whole system seems cannot be underdamping.

  • Reference

1. Differential Equations by Mike Jones and Roger Porkess

...read more.

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