Driven Harmonic Oscillator

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CURTIN UNIVERSITY OF TECHNOLOGY
DEPARTMENT OF APPLIED PHYSICS

Physical Measurement 201

DRIVEN HARMONIC OSCILLATOR

Abstract
The purpose of this experiment is to investigate the various aspects of harmonic motion using a Driven Harmonic Motion Analyser (DHMA). The aspects that are investigated in this experiment is the spring constant k of two springs, the resonant frequency, the damping factor, and the relationship between phase and resonant frequency. The value of k of the first string by calculating the mean is (12.51 ± 0.001) N/m and the value of k by using linear regression is (12.7 ± 0.3) N/m. The value of k of the second string by calculating the mean is (9.79 ± 0.001) N/m and the value of k calculated using linear regression is (9.1 ± 0.5) N/m. The damping coefficient caused by magnet is (0.021 ± 0.001) Ns/m. The value of resonant frequency using spring 2 with no mass added is (2.1 ± 0.1) Hz, this value is obtained from looking at the maximum amplitude, the other values obtained from calculation and looking at the phase are 2.05 Hz and (2.5 ± 0.1) Hz.

Experiment Conducted:
27th April 2004

AIM
General:
- Investigate various aspects of harmonic motion using a driven harmonic motion analyser

Specific:
- Measure the spring constant k for two springs
- Measure the resonant frequency
- Investigate the effect of damping on the period of the oscillation, and to determine the damping coefficient b - To investigate the relationship between the phase and resonant frequency.

EQUIPMENT
- Driven harmonic motion analyser
- Computer interface
- Additional masses (0.01 kg each)
- Magnetic damping unit
- 2 Springs
- Meter ruler

INTRODUCTION
Consider a mass attached to a spring, damped and oscillated with external force as seen on the following diagram:

[pic]
Figure 1. Mass attached to a spring with constant k and it is damped with a damping factor of b. F is force that is changing with time.

The equation of motion for the mass is:
[pic]

Now consider the following cases:

1. Undamped (b = 0), Undriven (F0 = 0) harmonic oscillator The equation of motion become:
[pic]
The general solution for this second order differential equation is:
[pic]
Where
[pic]
2. Underdamped, undriven (F0 = 0) harmonic oscillator
The equation of motion for this case is:
[pic]
The general solution to this equation is:
[pic]
The new frequency of the oscillation is ω’, where:
[pic]

3. Underdamped, driven harmonic oscillator
The equation of motion for this case is:
[pic]
The solution to this differential equation can be divided into two parts, which are the complementary and the particular solution. The complementary solution will decay with time and will become very small compared with the particular solution which is the steady state solution, which is:

[pic]
[pic]

The resonant frequency is when the mechanical impedance is at the minimum value, that is when mω = k/ω.

EXPERIMENTAL SETUP

[pic]
Figure 2. The experimental setup. This is the overview of DHMA assembly

EXPERIMENTAL PROCEDURE
Taken from the lab sheet

1. Measuring the Spring Constant
• Adjust the length of the Drive Cord so that the 17 cm mark on the mass bar is aligned with the top edge of the Upper Mass Guide. • Add 10g mass to the hanger, and record the position of the Mass Bar scale. Repeat the procedure for each additional mass added to the hanger. Using the balance check that the accuracy of the nominal values on the added masses. • Determine the displacement caused by the addition of each mass. • Two approaches can be taken to calculate the spring constant 1. Calculate the spring constant using Hooke’s law: F = - kx 2. A better approach is to plot the...
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