vibration

LAB DYNAMICS OF MACHINE MEC424

Table of Contents

1) Title : Vibration Experiment - Undamped Oscillations of..........................................1

Single Degree Of Freedom System..................................................................1

2) Objective......................................................................................................................1

3) Background..................................................................................................................2

4) Apparatus......................................................................................................................2

5) Theory :........................................................................................................................3

5.1Spring Constant............................................................................................................3

5.2Natural Oscillations.....................................................................................................4

5.3Undamped Oscillation..................................................................................................5

6) Experiment Procedure..................................................................................................5

7) Data and Result............................................................................................................6

7.1Prepared by : Siti Rozemah Bte Abu Bakar (2009443598).........................................8

7.2Prepared By : ...............................................................................................................9

7.3Prepare By : ................................................................................................................9

8) Discussion....................................................................................................................8

8.1Prepared by : Siti Rozemah Bte Abu Bakar (2009443598).........................................8

8.2Prepared By : ...............................................................................................................9

8.3Prepare By : ................................................................................................................9

8.4Prepare By : ..............................................................................................................10

9) Conclusion..................................................................................................................12

9.1Prepared by : Siti Rozemah Bte Abu Bakar (2009443598) ......................................12

9.2Prepare By : ..............................................................................................................13

9.3Prepare By : ..............................................................................................................13

10) Reference....................................................................................................................14

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1) Title : Vibration Experiment - Undamped Oscillations ofSingle Degree Of Freedom System.

2) Objective : In this experiment, we have to: 1) Determine an elastic property (the spring constant) of a spiral

spring2) Determine the natural frequency (f)3) Study the vibration motion of a mass suspended from the

spring.4) Understand the phenomena associated to resonance frequency.5) Determine the spring constant (k).6) Determine the natural frequency (f).

3) Background:

For this experiment, we find that many objects that performed oscillatory motion that is repeating a movement over the same path over the same interval of time. Harmonic motion are those oscillatory motions are repeat itself at regular time interval. Example such motions are oscillation of the pendulum, vibration of loaded spring and oscillation of air molecules when sounds travel through air. The simplest vibratory system can be described by a single mass connected to a spring (and possibly a dashpot). The mass is allowed to travel only along the spring elongation direction. Such systems are called Single Degree-of-Freedom system.

The specific dynamics of this spring-mass system are described mathematically by the simple harmonic oscillator and the regular periodic motion is known as simple harmonic motion. In the spring-mass system, oscillations occur because, at the static equilibrium displacement, the mass has kinetic energy which is converted into potential energy stored in the spring at the extremes of its path. The spring-mass system illustrates some common features of oscillation, called the existence of equilibrium and the presence of a restoring force which grows stronger the further the system deviates from equilibrium.

Simple harmonic motion is a special kind of motion that is produced, when the force acting on a body is proportional to the displacement and directed towards for simple al the equilibrium positions. A physical example for simple harmonic motion of a block which is attached to the end of the spring, with the block free to move on the horizontal frictionless track.

The simplest mechanical oscillating system is a mass attached to a linear spring subject to no other forces. The system is in equilibrium state when the spring is outstretched. If the system displaced from the equilibrium, here is a net restoring force of the mass, tending to bring back to the equilibrium. If a constants force such as gravity is added to the system, the point of equilibrium is shifted. The time taken for an oscillation to occur is often referred to as the oscillatory period.

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http://en.wikipedia.org/wiki/Potential_energy

http://en.wikipedia.org/wiki/Kinetic_energy

http://en.wikipedia.org/wiki/Statics

http://en.wikipedia.org/wiki/Simple_harmonic_motion

http://en.wikipedia.org/wiki/Simple_harmonic_motion

http://en.wikipedia.org/wiki/Period_(physics)

http://en.wikipedia.org/wiki/Harmonic_oscillator#Simple_harmonic_oscillator

http://en.wikipedia.org/wiki/Dynamics_(mechanics)


The maximum displacement of the simple harmonic motion is called the amplitude. The time interval for the block to complete full cycles called the period, T of the simple harmonic motion. The inverse of the period which represents the number of oscillation per unit time is called its frequency, f.4) Apparatus :

Figure 1 : Vibration Apparatus Set 3

4

1

7 6

8 5

2

6

1) Carriage 7. Guide columns2) Base 6. Additional mass3) Adjuster 8. Mechanical recorder4) Helical spring 9. ruler5) Guide roller

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5) Theory :

In vibrations of any system, it is subjected to Hooke’s Law. Hooke’s Law states that the extension of a spring is in direct proportion with the load added, as long as it does not exceed the spring’s elastic limit. It is applicable to linear-elastic materials

1.1 Spring Constant

The spring constant is a characteristic indicating the relationship between spring load and the deflection of the spring. Normally, this relationship is linear.The following equation gives the relationship between spring load ‘F’ and spring deflection ‘x’.

F = kx

where ‘k’ is the spring constant.

k is determine by fitting weight ‘m’ and measured the extension ‘x’

k = (m 1 - m 0 )g x1 - x0

Where,

x0 = initial extension

x1 = final extension

m0 = initial mass

m1 = final mass

As a check on linearity, the measured values can be plot on a load – extension curve

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1.2 Natural Oscillations

The term natural oscillation refers to the normal vibration of an uninfluenced oscillatory system. It is initially deflected out of its equilibrium position and then oscillates about this until it is brought to rest by any external or internal damping.

1.3 Undamped Oscillation

Equation of motion

The equation of motion involves establishing equilibrium of force at the mass

ΣF = -Fk + mg = mx

The spring load Fk is calculated from deflection ‘x’ and spring constant ‘k’.

Fk = k (xsta + x) = kxsta + kx

Where, xsta = initial deflection

kxsta = mg

This result in the following 2nd order differential equation as equation of motion

X + k x =0 m

Solving the equation gives harmonic oscillations with natural angular frequency ‘ω’ or natural frequency ‘f’.

x(t) = x0 cos (ω0 t)

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ω0² = k , f = 1 √(k/m) m 2∏

The equation for periodic time is:

T = 2∏ √ (m/k)

6) Experiment Procedure

PART A : Procedure to determine the spring constant, k.

1 .The apparatus was set up.2 .The datum was set up at 20mm from the top of graph paper using adjuster with no additional mass.3. The mechanical recorder’s button was pressed for a second.4. Then, 2kg additional mass was added to the carriage weight.5 .The button was press again for a second.6 .The steps 4 to 5 were repeated with 4kg, 6kg, 8kg, and 10kg additional mass.7 .The data was taken from the graph paper.

PART B : Procedure to determine the natural frequency, f.

1. The mechanical recorder was set up at the center of the graph using adjuster.2. The carriage weight without additional mass was pressed till the end. 3. Then, released it at same time the button was pressed until at least 10 oscillations recorded at the graph paper.4. Step 2 to 3 was repeated with 2kg, 4kg, 6kg, 8kg, and 10kg.5. The data was taken from the graph paper.

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7) DATA AND CALCULATIONS

Spring Constant:

Experiment value for spring constant, k = 1.80 N/mm

Load (N)

Mass (kg) Additional Mass (kg)

Deflection reading (mm)

Extension reading (mm)

12.26 1.25 0 20 031.88 3.25 2 31 1151.50 5.25 4 42 2271.12 7.25 6 53 3390.74 9.25 8 64 44110.36 11.25 10 75 55

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Undamped Oscillation:

Theoretical value for spring constant, k = 1.78 N/mmMass of carriage, Mc = 1.25 KgRecorder speed is 20 mm/s

Mass (kg)

Additional Mass (kg)

Experimental TheoreticalDifference

(%)Displacement of T10 (mm)

Frequency (Hz)

Frequency (Hz)

1.25 0 35 5.714 6.005 4.83.25 2 55 3.636 3.724 2.35.25 4 70 2.857 2.930 2.47.25 6 80 2.500 2.493 0.39.25 8 95 2.105 2.207 4.611.25 10 105 2.041 2.001 4.0

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SAMPLE OF CALCULATION

Prepared By : SITI ROZEMAH BTE ABU BAKAR (2009443598)

Experiment 1: Determine the spring constant.

There 2 method that can be used to calculate the value of k where;

1. By equation.

The following equation gives the relationship between spring load ‘F’ and spring

displacement ‘x’.

From equation: F = k · X

and

k =

Where F is spring load

X is spring displacement and

k is spring constant

Total weight, F = Total mass (m) × gravity (g)

= (2 + 2+ 2 + 2+ 2) × 9.81

= 98.10 N

Total displacement, x = 11+11+11+11+11

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= 55 mm

k experimental = and k theoretical =

=

= 1.80

Percentage difference = × 100 %

=

= 2.0%

2. By the gradient of the graph.

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Figure 2: Deflection graph

Gradient =

= 1.80 Nmm-1

From the graph above, the value of the gradient is 1.80 Nmm-1.

Percentage difference = × 100 %

=

= 1.12%

Experiment 2: Determine the natural frequency.

The data collected in table 2 show the time taken of 5 oscillations and also the time

taken to complete an oscillation.

To determine period of 10 peaks,

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To determine a period of 1 peak,

Data calculation from table 3

To determine the natural frequency using experimental results

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To determine the natural frequency, f using theoretical spring constant, k = 1780N/m

8) Discussion

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Prepared By : SITI ROZEMAH BTE ABU BAKAR (2009443598)From

the vibration experiment that we have done, graph load versus extension was

obtain for part A. Based on the graph that we plotted, load was directly

proportional to the extension. The gradient that calculated from the graph is

1.80 N/mm, which is the value of constant k for experimental value. While,

for the given theoretical value for spring constant given is 1.78 N/mm. Both

value for experimental and theoretical has percentage error around 1.12%,

which is not too bad. The result was not exactly the same as theory given

because of some error during conducted the experiment. The error occurs

maybe because the external force that exist in the environment. Then, the

spring also does not have constant deflection and this will affect the result we

get.For the experiment part B, to find the natural frequency the result that we

have, also not too far from the theoretical value. For example, data no 4 the

natural frequency that we get is 2.500Hz and the theoretical value is 2.493Hz .

So, the average percentage error is about 0.3%, which is very close to

theoretical value. We find that the error occur might be because during we

conduct the experiment, we assume that the oscillations is a free oscillation

and was not disturbed by the external forces and the total energy was

conversed, but in actual there was a few external forces that effect the

experiment. The initial disturbance happened was maybe the stiffness of the

spring was not good or do not have the constant extension. Then, the motor

speed at the mechanical recorder also does not obtain right oscillation if it has

no constant speed. The error also will increase. Besides, other reason is we do

not lock the additional mass during the experiment and this is causing them to

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move when we press the carriage weight. The mechanical recorder press

button and the carriage weight was not release at the same time. Here will

cause the error to our data record at the graph paper. The oscillation that

produced was not really accurate. Other than that, the carriage weight

probably was not pressed till the end as mentioned. Then, center that we made

on the graph paper not really accurate and also effect the oscillation produced.

Lastly, we do not familiar with the apparatus and cannot handle it very well.

Conclusion

9.1 Prepared By : SITI ROZEMAHBTE ABU BAKAR (2009443598)

The conclusion that we can made from this experiment was the experiment not really success because the experimental value that we get do not really accurate with the theoretical value but we achieved the objective given. For part A, the spring constant for constant theoretical value is 1.78N/mm and experimental is 1.80N/mm and the calculated percentage error is about 1.12% and the graph that we obtain is the linear graph ( load directly propotional to the extension ).Then, for part B, the average percentage error for natural frequency around 3.06%. The differences were not too big but the objective was achieved and has to be improved. The result was different because the error that discussed before, which is do not lock the additional mass, the button and the carriage was not release simultaneously, and do not press the carriage weight till the end. External forces also can be the reason for the different result that we get. All this error happened assume happened to both part A and B because the experimental and theoretical value for both part was different.

From the error that we find, here was little recommendation to overcome this problem. First of all, we may be able to be more thorough during conduct the experiment. Make sure we lock the additional mass that we put to avoid the error. Try to control the adjuster accurately to the datum or the center that mentioned in the procedure. We also have to press the carriage till the end and press it simultaneously with the button for mechanical recorder. Then, make sure the apparatus was in a good condition before conduct the experiment. Avoid from the environmental disturbance and any external forces to reduce the error or overcome them.

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9) Reference :

9.3 All about Hooke’s Law

http://asms.k12.ar.us/classes/physics/GENERAL/KENNETH

/HOOKE.HTM

9.4 Spring: Hooke’s Law

http://dev.physicslab.org/Document.aspx?

doctype=3&filename=Dynamics_HookesLawSprings.xml

9.5 Error Analysis

http://www.lhup.edu/~dsimanek/errors.htm

9.6 Simple Harmonic Motion

http://www.physics.ucla.edu/demoweb/demomanual/

harmonic_motion_and_waves/simple_harmonic_motion/

simple_harmonic_motion.html

9.7 Forced Oscillation

http://cnx.org/content/m15880/latest/

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