Background to decay with time. This eventually leads to

Background information

 

Damped
oscillations occur when a resistive force, usually friction causes the
amplitude of the oscillation to decay with time. This eventually leads to the
dissipation of energy from a system, whereby this oscillation is taking place.
The damping effect is known to be especially useful in the automotive industry
where shock absorbers make use of certain viscous fluids like oil to improve
the ride quality of vehicles when they travel on bumpy roads. Vehicle
manufacturers often strive to make sure that their vehicles undergo critical
damping whereby the oscillating material returns to the equilibrium position as
quickly as possible without overshooting and producing more oscillations with
time. The term, ‘critical damping’, is used to describe this motion. However in
reality, it is often hard to achieve critical damping as the damping ratio,
denoted by

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(zeta) has to reach the ideal value of 1. Thus, in
most cases, the shock absorbers in the vehicle suspension system are slightly
underdamped, whereby

 will be
slightly less than 1 as the system oscillates at most for a few times, before
the amplitude reaches zero.1

 

The damping force
is mostly influenced by the viscosity of the liquid in shock absorbers which is
commonly referred to as hydraulic fluid. Hydraulic fluids used in recent times,
are usually mineral based oils which are favoured on the account of their
relatively high viscous damping coefficient, across a wide range of
temperatures. A high damping coefficient, ? which is a theoretical
parameter that explains the energy dissipation due to friction that occurs in a
fluid as a system oscillates in it,2
would lead the damping ratio to also be of a high value, as they are both
directly proportional. However, the mineral based oils come with their own set
of limitations as they are usually non-regenerative, scarce and harmful for the
environment.3 As
the world becomes more environmentally conscious, there is a pressing need to
find other alternatives which are widely available, inexpensive and more
environmentally friendly. At this juncture, the viability of vegetable oils as
fluids which can be used to dissipate energy in oscillating systems is
explored.

 

 

This in turn,
leads us to form our research question,

 

“How
effective is vegetable oil in dissipating the energy of an oscillating
mass-spring system at different temperatures?”

 

Theory

                                                  

When the amplitude
of the oscillation of a mass-spring system suspended in a fluid, like vegetable
oil gradually decays with time, the energy is lost due to viscous damping.

Even as the fluid drag forces
involved in viscous damping are complicated as they act differently in relation
to velocity, there is a general model
for linear drag that accurately describes damped harmonic motion, in
most cases. The model states that the damping force which is proportional to,
but in a direction opposite to that of the velocity of the oscillating system
is,

 

 

When the model is
applied to a mass-spring system in order to
illustrate its motion, the Newton’s Second Law of Motion can be used to
arrive at,

 

 

 

 

This can be expressed
as a differential equation in the form,

 

 

The equation above can be solved to obtain,

,

 

In the preceding equation,

Let the value inside the square root be known as J.

When J> 0, it can be
gathered through simple manipulation that b2 4mk as there is overdamping, in which the
damping coefficient is larger than the other quantities like m and k. Finally when J = 0 , b2 equals 4mk and there is critical damping,
which is essentially the midpoint between underdamping and overdamping, whereby
the system would return to its equilibrium position in the shortest amount of
time possible without oscillating.

Investigation

A preliminary trial was first
conducted to verify if the oscillation of the mass spring system was indeed
underdamped when suspended in vegetable oil. This was crucial as the purpose of
the main investigation was to test the viability of using vegetable oil as the
base oil for dashpots in shock absorbers that are usually either critically
damped or underdamped. In the preliminary trial, the mass-spring system indeed
went past the equilibrium position and oscillated for a few times about it,
which meant that the oil was a suitable candidate for the experiments that were
to be conducted later on.   

 

Palm oil was chosen to be used in
the experiment as it is affordable and an easily obtainable oil in most
developing countries where mineral based oils are in short supply due to their high
costs. The damped oscillatory motion of the mass spring system in the palm oil
at different temperatures from  25 degree
Celsius to 95 degree Celsius, at 10 degree intervals were investigated to, find
out if this liquid can retain its ability to be a damping fluid even when there
is a huge temperature change in the surrounding environment.

 

An increase in
the temperature would cause the molecules to move faster as they gain kinetic
energy. Following which, the relatively stronger forces of attraction between
them initially will weaken and therefore would be a fall in the viscosity of
the fluid. In a case like this, the mass spring system will experience less
viscous drag, which means that the damping coefficient would be lower and
therefore the oscillation might decay at a slower rate.

 

As such, it can
be hypothesised that at lower temperatures, the energy from the mass spring system
will be dissipated at a faster rate, since the oscillations would come to a
stop, more quickly.

 

Variables

 

Independent: Temperature
of vegetable oil (Palm oil)

 

Dependent: Quality
factor of the system (Parameter used to describe the rate of energy loss)

 

Controlled: Mass of
mass-spring system, Type of spring (spring constant), Volume of oil

 

Materials and
Equipment

 

1.    
Vernier
Dual-Range Force Sensor

2.    
Labquest
Mini connected to the LoggerPro software on laptop

3.    
500 ml
beaker

4.    
Spring

5.    
Slotted
masses with weight hanger (250g)

6.    
Retort
stand

7.    
Laboratory
thermometer

8.    
Vernier
Temperature Sensor

 

Method

 

In order to investigate the oscillation of
the mass spring system, a Vernier Dual-Range Force Sensor was used. It was hung
from a retort stand at a particular height, so as to ensure that the
equilibrium position of the suspended mass spring system was in the middle of
the 500 ml beaker, where there was ample space for the system to oscillate
without surfacing out of the oil. The force sensor was then connected to the
LoggerPro graphing software on the laptop with the help of Labquest Mini, which
is a sensor interface.

 

The spring constant (k) of the spring used
was firstly determined with the help of the software, as it has to be used
later in calculations. To do this, a mass of 100 g was hung from the spring.
The resulting spring extension, x was measured using a ruler with an
uncertainty of

 . This was then repeated with a 200g mass and
a 300 g mass to ensure accuracy. The values for force of the spring which
varied for the extensions were also taken down and then used to calculate k.

 

Based on the equation for the Hooke’s Law,

, k was
determined separately for each of the three measurements with different masses
and then the final value of k was taken to be the average of the three. The
value of the k was used later to obtain the graph of displacement over time for
the oscillation as

  can be taken to be the displacement of the
system from its equilibrium position, when the force sensor is zeroed at that
point (0 N).

 

Subsequently, oil which had earlier been heated
up to a temperature of a 95 degree Celsius was used as the medium in which the
oscillation took place. The mass spring system was then released from a height
of 0.02 m above the equilibrium point, to trigger oscillation. The damped
oscillatory motion that was then represented as a Force – time graph on the
graphing software was converted to a Displacement-time graph by dividing the
value of Force by a factor of 90.909 (to 5 s.f.) as the spring constant was
90.909 N/m.

 

This crucial step was then repeated for the
other readings that were decreasing in intervals of 10 degree Celsius. In the
end, all the displacement-time graphs produced for oil at different temperatures
were analysed to find out the quality factor, which is measure for the rate of
energy loss in a damped system.

Raw Data

 

 

Figure
1.1 – Graph of damped oscillatory motion in veg. oil at

 

 

 

Figure 1.2 –
Graph of damped oscillatory motion in veg. oil at

 

 

 

 

Figure
1.3 – Graph of damped oscillatory motion in veg. oil at

 

 

 

 

Figure 1.4 – Graph of damped oscillatory motion in veg.
oil at

 

 

Figure 1.5 – Graph of damped oscillatory motion in veg.
oil at

 

 

 

 

 

Figure 1.6 – Graph of damped oscillatory motion in veg.
oil at

 

 

Figure 1.8 – Graph of damped oscillatory motion in veg.
oil at

 

 

As the amplitude of oscillation in all of the graphs above, underwent
exponential decay, as seen from the general shape, a curve fit for the damped
harmonic motion model was applied to all of the data points.

 

Function for curve fit: 

 

 

Sample
calculation:

 

 

Figure 2.1 – LoggerPro curve-fitting tool
for damped harmonic motion in oil at

 

By fitting the data points to the damped
harmonic model, the coefficients B and C that correspond to

 and the angular frequency ? respectively as
seen from the general equation for damped oscillation below, can be determined
through the LoggerPro program that calculates them accordingly.

 

 

Figure 2.2 – Curve-fit model in relation to
the actual equation

 

For mass-spring system suspended in oil with a temperature of

 

 

 

 

 

The same calculation was then repeated for all the other temperature
points, in order to obtain the other damping coefficients.

 

 
Temperature (

)

 
Coefficient B in the curve fit equation

 
Damping coefficient
(b)

25.0

0.6226

0.3113

35.0

0.4500

0.2250

45.0

0.4045

0.2023

55.0

0.3664

0.1832

65.0

0.3297

0.1649

75.0

0.3186

0.1593

95.0

0.2724

0.1362

 

 

 

Figure
2.4 – Graph of Damping coefficient vs Temperature for palm oil

 

Conclusion

 

From Figure 2.4
above, it can be concluded that the damping coefficient of the oil has a negative
exponential relationship with temperature. This is observed from the best-fit
line that was plotted for the data points on the graph.

 

Therefore, this fits
the theory that the increase in resistive forces which occurs as the
temperature decreases, would cause the damping coefficient to be greater for
more viscous fluids and concurrently the exponential decay of the peaks of the
amplitude of the oscillation as time passes would occur at a faster rate.

 

However, such a
steep drop in the value of the damping coefficient to that extent, was not
expected based on the results obtained through experimentation by scientists in
the field like Awogbemi of Ekiti State University, who in fact suggested that
palm oil was a suitable alternative to mineral-based oils.

 

The fact that
there is a steep drop in the damping coefficient over the temperature range might
prove to be a challenge, if vegetable oil is considered in the future to be one
of the base oils for fluids used in shock absorbers. This is the case as the
oil will not be that effective at high temperatures in dissipating the energy
of the oscillatory motion that occurs as vehicles hit bumps along the road,
thus causing rides to be extremely uncomfortable.

 

 

Evaluation

 

In the graphs
obtained for displacement against time, it can be observed that there are many
minor turning points for each major turning point (crest or trough), which
suggests that the interference of the waves created by the oscillation of the
mass-spring system, is causing the readings of the amplitude of the oscillation
at different points in time to be inaccurate.

 

As a result of
this, the damping coefficient is affected, as the uncertainties of the
amplitude propagate. In the future, a larger tank could be used to minimize
this case of systematic error. Moreover, the spring used for the experiment was
wobbly and a bit worn out, which meant that the oscillation was not always
vertical, as there was also circular motion due to the nature of the spring.
This issue can be rectified with the use of a short and firm spring that would
indeed generate more accurate values of amplitude as the system undergoes
damping.

 

As a whole, this
investigation can be improved with the use of more oils over an ever greater
temperature range in the future, as the damping coefficient of oil in
particular is known to be more sensitive to small changes in temperature at
certain values.

1 Robert
G. Brown. Properties of the Damped Oscillator, 4 Dec. 2004,
webhome.phy.duke.edu/~rgb/Class/phy51/phy51/node25.html.

 

2 JE
Escalante-Martínez.”Experimental evaluation of viscous damping coefficient in
the fractional underdamped oscillator.”  Advances in Mechanical
Engineering, 13 Apr.
2016, journals.sagepub.com/doi/full/10.1177/1687814016643068#articleCitationDownloadContainer.

 

3 Binfa
Bongfa. “POSSIBILITIES OF VEGETABLE OILS AS BASE OILS FOR AUTOMOBILE SHOCK
ABSORBER FLUIDS.” International Journal of Scientific & Engineering
Research, vol. 6, no. 4, Apr. 2015,
www.ijser.org/researchpaper/POSSIBILITIES-OF-VEGETABLE-OILS-AS-BASE-OILS-FOR-AUTOMOBILE-SHOCK-ABSORBER-FLUIDS.pdf.

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