lab2 repaired) repaired) repaired)

8/3/2019 LAB2 Repaired) Repaired) Repaired)

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ABSTRACT

Free vibration is an experiment that been used to determine the value of spring constant and

natural frequency by taking the graph of displacement and the oscillations graph. This

experiment held in the dynamic lab. For the spring constant, the experiment started with no load

and the mechanical recorder will take the reading after that, few masses was added regularlyuntil up to 10 kg. For natural frequency experiment, the process is just the same like the spring

constant test but this time the carriage must be pushed downward to obtain the vibrations

oscillations. The spring constant will obtain a linear graph as the extensions are uniform. Given

here, the theoretical spring constant is 1.7 N/mm and the theoretical value that been calculated is

1.706 N/mm. This experiment can be used for spring in many type of material.


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TABLE OF CONTENT

Page

1. INTRODUCTION 12.

THEORY 23. APPARATUS 6

4. PROCEDURE 75. RESULT & SAMPLE OF CALCULATIONS 86. DISCUSSIONS & CONCLUSIONS

i. Siti Nur Fatin Tarmedi 15ii. Nurul Hasniza Binti Abu Shaari 17

iii. Yusaliza Binti Mohd Yusof 20iv. Wahiddu Rahman Kamarruzaman

7. REFERENCES8. APPENDIX

LIST OF ILLUSTRATIONS

1. FIGURE 1: UNDAMPED FREE VIBRATION SYSTEM 12. FIGURE 2: SPRING DISPLACEMENT 33. FIGURE 3: SPRING OSCILLATIONS 44. FIGURE 4 : VIBRATION APPARATUS SET 65. TABLE 1 : SPRING CONSTANT RESULT 86.

TABLE 2 : NATURAL FREQUENCY RESULT 97. TABLE 3 : PERCENTAGE ERROR RESULT 14

8. GRAPH 1 : GRAPH LOAD VERSUS EXTENSION ( ) 159. GRAPH 2 : GRAPH LOAD VERSUS EXTENSION () 1710.GRAPH 3 : GRAPH LOAD VERSUS EXTENSION () 2011.GRAPH 4 : GRAPH LOAD VERSUS EXTENSION (v)12.APPENDIX 1: RESULT EXTENSION13.APPENDIX 2: OSCILLATION NO LOAD14.APPENDIX 3: OSCILLATION 2KG15.APPENDIX 4: OSCILLATION 4KG16.APPENDIX 5: OSCILLATION 6KG17.APPENDIX 6: OSCILLATION 8KG18.APPENDIX 7: OSCILLATION 10KG


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INTRODUCTION

Vibration is the periodic motion of a body or system of connected bodies displaced from

the position of equilibrium. There are two types of vibration, free and forced. Free vibration

occurs when the motion is maintained by gravitational or elastic restoring forces, such as

swinging motion of pendulum or the vibration of an elastic rod. Every vibrating system got itsown natural frequency. Natural frequency is the frequency at which a system naturally vibrates

once it has been set into motion. In other words, natural frequency is the number of times a

system will oscillate (move back and forth) between its original position and its displaced

position, if there is no outside interference

Here, the simplest type of vibrating motion is undamped free vibration, represented by

the figure below (Figure 1)

FIGURE 1: UNDAMPED FREE VIBRATION SYSTEM

The block has a mass m and it is attached to a spring having a stiffness k. This is similarto the experiment where we would like to find the spring stiffness, k and the natural frequency of

after adding the mass. The block of mass is represented by the carriage and the additional mass.


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THEORY

Generally, free natural vibrations occur in elastic system when a body moves away from

its rest position. The internal forces tend to move the body back to its rest position. The restoring

forces are in proportion to the displacement. The acceleration of the body which is directly

related to the force on the body is therefore always towards the rest position and is proportional

to the displacement of the body from its rest position. The body moves with simple harmonic

motion.

In vibrations of any system, it is subjected to Hookes Law. Hookes Law states that the

extension of a spring is in direct proportion with the load added, as long as it does not exceed the

springs elastic limit. It is applicable to linear-elastic materials.

According to the Hookes Law, it can be translated into a common mathematical equation, that

is:

[1]Where

F = Restoring force (Newton)

k = Spring constant (Newton/Meter)

x = Displacement (Meter)

The negative (-) sign indicates that the direction of displacement (x) is the opposite of the

direction of force (F).


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FIGURE 2: SPRING DISPLACEMENT

From the diagram above, we can find the spring constant k, by using the values given from the

diagram.

[2]

[3]

By substituting equation [2] and [3] into equation [1], we will get:

[4]Simplified, we get:

[5]Therefore, the spring constant value can be obtained, provided that the value of load anddisplacement are known.


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Oscillation

Oscillation describes the repetitive movement of some measure about a central value or

equilibrium between two or more different states. It is also called a normal vibration, being the

centre as the equilibrium position. A natural oscillation oscillates the distance of the damper. If

there are no damps, oscillation will continue until rest.

FIGURE 3: SPRING OSCILLATION

The simplest form of oscillation can be found in a linear spring with mass attached to it. If there

are no external forces, it will stay at its equilibrium position. When for example a downward

external force is applied, the load will shift its position downward, gaining momentum in the

process. Once it is released, the momentum generated from the displacement shoots the load up.

When it reaches the maximum height, the spring pushes the load back downwards. This process

will go on repeating itself, thus resulting oscillation.

Regarding what we get from the Hookes Law, we can also obtain the periodic time and

frequency of an oscillation. By using the equation of motion, we obtain:

[6]The spring load Fk is calculated from deflection x and spring constant k,


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Where,

= initial deflection = mg

By using 2nd order differential equation on the equation of motion, we obtain:

To solve the equation using harmonic oscillations with natural angular frequency n or natural

frequency fn, we get,

Angular frequency n,

Natural frequency fn,

The equation for periodic time T,


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APPARATUS

FIGURE 4: VIBRATION APPARATUS SET

Guide

Collumn

Mechanical

Recorder

Chart Paper

BasedDamper Additional Mass

Carriage

Helical Spring

Adjuster


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PROCEDURES

a) For spring constant (k) experiment:1) The chart paper was placed at the mechanical recorder2) A pen was placed at a pen holder which attached to the carriage to record the chart.

The zero position was adjusted by using a threaded spindle at the top of the spring.

3) Then, the mechanical recorder was run for a few seconds in order to get a short line atthe chart paper.

4) After that, a 2 kg mass was added. And let the mechanical recorder run for a fewseconds.

5) Step 4 was repeated by adding the additional mass until up to 10 kg.b) For natural frequency (f) experiment:

1) The zero position was adjusted to the middle of the of the chart paper.2) After that, the carriage with no load was pushed and run the mechanical recorder in

the same time.

3) Next, after few oscillations (took five best oscillations), a 2 kg mass was added to thecarriage and step 2 was repeated.

4) Step 1 until 3 was repeated until up to 10 kg


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RESULT

Given that:

1.

Mass of carriage, mc is 1.25 kg.

2. Mechanical recorder velocity, v is 20 mm/s.3. Spring constant theory, kT = 1.7 N/mm4. Extension due to carriage, x0 = 7.17 mm

For spring constant:

Additional

mass (kg)

Total mass,

m (kg)Load, F(N)

Extension, x

(mm)

0 1.25 12.2625 0

2 3.25 31.8825 11.5

4 5.25 51.5025 23.0

6 7.25 71.1225 34.5

8 9.25 90.7425 46.0

10 11.25 110.3625 57.5

TABLE 1: SPRING CONSTANT RESULT


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For natural frequency:

Additional

mass (kg)

Total mass,

m (kg)

Average length of

an oscillation, L

(mm)

Period per

cycle, T (s)

Natural frequency,

fe(Hz)

0 1.25 22.0 5.379 0.1859

2 3.25 29.0 8.673 0.1153

4 5.25 33.0 11.025 0.0907

6 7.25 38.0 12.953 0.0772

8 9.25 37.0 14.620 0.0684

10 11.25 43.0 16.129 0.0620

TABLE 2: NATURAL FREQUENCY RESULT


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SAMPLE OF CALCULATION

SPRING CONSTANT

Taking the result at additional mass of 2 kg:

a. Total mass = Additional mass + mass of carriageSince mass of carriage, mc = 1.25 kg

Hence, total mass = additional mass + 1.25

= 2 + 1.25

= 3.25 kg

b. Load, F = (total mass) x (acceleration of gravity) (g = 9.81 m/s2)= mg

= 3.25 x 9.81

= 31.8825 N

To determine value of spring constant, k

k = =

= 1.706 N/mm

c. Spring constant, ke is the slope of the graph load (F) versus spring extension (x).Fromthe graph, the value of ke is 1.706 N/mm.

d. Percentage of error can be calculated by comparing the value of k obtained with theactual value of the spring constant. Knowing that the actual spring constant is 1.7N/mm :


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Percentage of error = x 100%= x 100%

= 0.35 %


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NATURAL FREQUENCY

Taking the result at additional mass of 2 kg:

a. The calculation of total mass and load for 2 kg additional mass were already shownabove.

b. Length of one complete cycle, L is obtained by calculating the average of 5oscillations.

Length of one completecycle, L

= (Length of 5 oscillation) / 5

= 29.0 / 5

= 5.8 mm

c. Speed is generally equals to distance travelled per unit time. In this experiment, thespeed of the mechanical recorder is the length of complete cycles travelled in one

second. Since the mechanical speed was given (20 mm/s), the period (time) of one

complete cycle can be obtained.

Period of one complete

cycle, T

=

= = 0.29 s

d. The natural frequency of the system when a 110.3625 N of load was added:Natural frequency, fe =

= = 3.45 HZ


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e. Theoretically, the value of natural frequency is:

ft =[

][

]

=[

][]= 0.1151 HZ

f. Percentage of error of the experimental value obtained is :Percentage of error = x 100%

= x 100%= 0.174%


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g. Overall, the result obtained:

TABLE 3: PERCENTAGE ERROR RESULT

Total mass,

m (kg)

Load(N)

Natural frequency,

fe(Hz)

(experimental)

Natural frequency,

ft(Hz) (theoretical)

Percentage of error

(%)

1.25 12.2625 0.1859 0.1856 0.16

3.25 31.8825 0.1153 0.1151 0.17

5.25 51.5025 0.0907 0.0906 0.11

7.25 71.1225 0.0772 0.0771 0.13

9.25 90.7425 0.0684 0.0682 0.29

11.25 110.3625 0.0620 0.0619 0.16


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GRAPH 1: GRAPH LOAD, F (N) VERSUS EXTENSION, x (mm)

DISCUSSION

This experiment required to determine:

1) Spring constant (k)2) Natural frequency

The way to find the spring constant is by measuring the value of extension of the spring when

additional masses added on the carriage. Then after some calculations, a graph of load (F) versus

extension, m (mm) was create

Before that, review the graph from the chart paper where it showed stairs graph where every

displacement is the same after adding additional mass (refer appendix 1). So, a linear graph was

achieved. Therefore, from the graph the spring constant can be determined by taking the slope of

the graph. From there also, percentage error of spring constant can be found. As the theoretical

value of the spring constant is 1.7 N/mm and the percentage was 0.35%.

For the natural frequency, five best oscillations were took to be the reading of the result by

taking peak to peak as one oscillation on the chart paper. Given, the speed of mechanical

recorder is 20 mm/s. From there natural frequency can be determined. Similar to spring constant,

percentage error can be obtained. The errors are small and acceptable. And they are approaching

the theoretical value.

0

20

40

60

80

100

120

0 11.5 23 34.5 46 57.5

Load,

F(N)

Extension, x(mm)

GRAPH LOAD,F (N) VS EXTENSION, X (mm)


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The error occurs because of two factors. One is measurement apparatus as not accurate ruler was

used to measure the extension and the other is the pen that been used to take the graph. So as the

solutions, an accurate ruler should be used in order to get precise measurement and also a darker

slim shape pen also should be used to get the obvious kind of graph.

CONCLUSION

These two experiments obviously can help to determine the spring constant (k) and natural

frequency of any material manually. Certain research can be done to improve the experiment.

lab2 repaired) repaired) repaired)

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