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

(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 . This occurs only in the case of underdamping
in which the damping coefficient is relatively smaller than the mass and the
spring constant. Conversely, when J < 0,
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.