The Damped Harmonic Oscillators

This experiment was a study of the effects of damping on structures of harmonic oscillation.
Question for Study
How does the oscillation of a damped harmonic device influence the shift of different parameters, such as spring constant, mass, or even amplitude?
Theory and Formulas
The force resisting the extension of the spring is proportional to the amplitude of the spring and is given by Hooke's law, considering a device in which a body of mass m is suspended on a fixed surface by a spring of spring constant k, by Hooke's law. F= - k x ; where x = 0
If a mass block is added to the spring, it stretches further by the new weight by the relationship given by
-x0= - mg/k ;g is gravitational constant

Applying Newton’s Second Law of motion,

m d2x/dt2 = - k x

Which is the same as

d2x/dt2+ w20 x=0;where ω 20 =k/m

ω0 =

For a Damped Harmonic Oscillator, the displacement from the mean position during oscillation of the mass decreases gradually, due to effects such as friction and air resistance ("Simple Harmonic Motion" 11). Therefore, the equation representing this motion may be written as

m d2x/dt2 = - k x – ßdx/dt;ß is the damping coefficient (constant)

Which may be rewritten as

d2x/dt2 + 2γ dx/dt +ω20 x = 0;γ=ß/2m

A solution to this equation of second order differential is not a simple harmonic oscillation, but one affected by damping one. Given that the oscillation is still sinusoidal with a decreasing amplitude with time, the equation may be written as

x(t) = Ae-γt cos (ωt + ϕ)

Where A gives the amplitude and ϕ is the phase angle.

ω =ω20 – γ2) is the angular frequency.

Variables

Name

Symbol

Unit

Independent variable(s)

time

t

second

Dependent variables(s)

frequency

f

Hz

Controlled variables(s)

Amplitude (displacement)

A

meter

Apparatus and Equipment

Damper Scale

Pendulum

Spring

Objects of known masses

Oscilloscope

Diagram of Set-Up

Fig. 1: Experimental setup

Clearly explain how you would change the independent variable, measure the dependent variable and keep the controlled variable constant.

Variable Type

Variable

Method of controlling variables

Independent:

Time

None

Dependent:

Frequency

Rate of change of displacement

Controlled:

Amplitude

Displacement variation

Experiment Procedure

Part A

Weights of known masses were obtained for use in the experiment.

One of the available masses was attached to the end of the tube which forms the pendulum arm.

The oscilloscope is set to ‘single arm mode.’ After a suitable voltage sensitivity setting had been achieved, the pendulum was made to swing freely against the spring.

The vibration was noted, and the frequency reading was recorded from the slope.

The procedure was repeated thrice, and the average value for the frequency was noted. Similarly, the values of the frequency were determined for using the second mass and the also when attached to no mass.

Part B

With initial apparatus set up in ‘part A’ previously, the large mass was attached and also the damper unit to pendulum arm.

The damper unit was set to 1 on the damper scale, and the scope was adjusted to read zero voltage by moving the vertical cursor and aligning the horizontal cursor on the input signal trace.

The pendulum was made to swing freely against the spring with the resulting trace on the scope fixed using the store facility.

The voltage reading which represented the peak value of the waveforms were noted. This step is repeated for damper scale setting of R3 and R5.

The damper was set to a higher scale, and the corresponding system response was observed.

Raw Data Table

Experimental Values

1

2

3

Average

Frequency FL with Mass ML attached to arm (Hz)

3.68

3.73

3.73

3.71

Frequency FS with Mass MS attached to arm (Hz)

4.31

4.38

4.39

4.36

Frequency F0 without mass

5.68

5.68

5.68

5.68

Table 1: The Frequency obtained when damper was set to a higher scale.

Sources: Experiment Values, Simanek E. Donald, A Laboratory Manual for Introductory Physics.

Eagles Publishers. 2nd Edition. 2012.

More data was collected and documented in spreadsheets that have been attached. Sampled excerpts are shown below.

Fig. 2: Sample data from sheet 3.5CMT1_1 showing mean displacement variance for the experiment.

Fig. 3: Sample data from sheet 30CMT4_1 showing data for showing mean displacement variance for the experiment.

Calculations

Amplitude ratio for the first damping. (R1)

Average value of amplitude ratio for the first damping.

Amplitude ratio for the second damping. (R3)

Average value of amplitude ratio for the second damping.

Amplitude ratio for the third damping. (R5)

Average value of amplitude ratio for the third damping.

A table comparing Damping Coefficient and Damping Ratio.

Damper Setting

Av Amplitude Ratio

Damping Coefficient

Damping Ratio

R1

1.157

0.218

0.023

R3

1.214

0.294

0.031

R5

1.247

0.332

0.035

Sources: Damping Ratio Values obtained from Simanek E. Donald, A Laboratory Manual for Introductory Physics. Eagles Publishers. 2nd Edition. 2012.

Theoretical value for F0 is 5.68 (Blake).

Analysis

A vibratory system is dynamic such that the excitations and responses depend on time. This implies that the system produces force which triggers vibrations in the mechanical equipment. Therefore, this analysis will involve setting up a mathematical model, interpreting results recorded from the experiment as well as the theoretical assumptions.

Data Processing

The data from the experiment was collected and recorded in various spreadsheets, on which graphing has been performed.

The experiment lasted for 30 seconds for each cycle, and numerous values have been taken and documented to come up with a smooth curve that shows a sine wave, as it is expected.

The data found on sheet 3.5CMT1_1, documents the values obtained from the experiment with small displacement. The graphical representation is shown below

Fig. 4: The graphical representation of data showing damping of oscillation for values in sheet 3.5CMT1_1.

The graph above represents harmonic damping, which shows a gradual decrease in amplitude for thirty seconds.

When a mass is given a larger displacement, the decay rate is much higher, thus producing an exponential curve that tends towards the mean position. The data for this experiment was collected and documented in sheet 30CMT4_1, whose sample is shown below

Fig. 5: A perfect representation of damping on harmonic oscillation showing amplitude against time. The graphical representation is based on the values in sheet 30CMT4_1.

The graph shows a diminishing time-variant sinusoidal curve.

Fig. 6: The general sine curve gives the output as below

The outline shows the line of best fit representing the general displacement from the mean position as it tends towards a zero amplitude. The general outline of the graphical representation is the expected curve for damped oscillation for bodies in motion about a mean position.

Limitations

Marginal Error

Comparing the value obtained in the experiment with the theoretical value for F0, it is evident that there was a small marginal error as shown below.

The error produced above is a result of air resistance and effects of weight, which could not be measured or controlled.

Also, the efficiency of the equipment used in the experiment may have been vulnerable to mechanical imperfections.

Suggested Improvements

Use of highly accurate oscilloscope is recommended to obtain smoother curves

Highly sensitive equipment with large enough masses to reduce the resistance factor should be used.

Conclusion

This experiment was conducted with the aim of verifying the effects of damping on harmonic oscillators. The motion of a pendulum or spring action is usually harmonic. It moves from its mean position, with its amplitude being proportional to the amount of energy used in its displacement. It is evident that as a result of varied damper settings, the frequency of the oscillations by the pendulum changes. The energy of the object that acts as the bob loses its energy with time as a result of damping. This effect causes the bob to create an exponential curve graph that tends towards zero displacements. The discovery of the inverse relationship between damper setting and frequency of the oscillations is fascinating. The objectives of the experiment have, thus, been achieved in obtaining a damped exponential graph.

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Works Cited

Blake, R.E. Fundamentals of Physics, 10th Edition. 2002

Simanek E. D. A Laboratory Manual for Introductory Physics. Eagles Publishers. 2nd Edition. 2012.

"Simple Harmonic Motion". Chapter 23 Simple Harmonic Motion. n.d. Online. 10 January 2017. .