FREE VIBRATION EXPERIMENT-NATURAL FREQUENCY OF SPRING MASS SYSTEM WITHOUT DAMPING PDF

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UNIVERSITI TEKNOLOGI MARA FAKULTI KEJURUTERAAN MEKANIKALProgram : Bachelor of Engineering (Hons) Mechanical Course : Applied Mechanics Lab Code : MEC 424 Lecturer : Group :MEC 424 - LABORATORY REPORTTITLE : FREE VIBRATION EXPERIMENT-NATURAL FREQUENCY OF SPRING MASS SYSTEM WITHOUT DAMPINGNo NAME STUD...

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UNIVERSITI TEKNOLOGI MARA FAKULTI KEJURUTERAAN MEKANIKAL

Program Course Code Lecturer Group

: : : : :

Bachelor of Engineering (Hons) Mechanical Applied Mechanics Lab MEC 424

MEC 424 - LABORATORY REPORT TITLE

:

FREE VIBRATION EXPERIMENT-NATURAL FREQUENCY OF SPRING MASS SYSTEM WITHOUT DAMPING

No 1 2 3 4 5

NAME

LABORATORY SESSION

STUDENT ID

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REPORT SUBMISSION

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Marking Scheme No

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Total

ABSTRACT

The conducted experiment of free vibration was to obtain the value of the spring constant (k) and the natural frequency of the spring (f). The experiment was conducted by following the procedures from the given lab sheet. The group managed to plot the graph of the spring constant,k from the results obtained. The spring constant and natural frequency are also managed to obtained by theoretically and experimentally. The group has observed the effect of mass towards the natural frequency of spring mass system and from the calculation, the group is able to obtain the percentage error value by comparing between theoretical and experimental. The experiment went well.

TABLE OF CONTENTS

No. 1 2 3 4 5 6 7 8 9 10 11

Contents List of Tables List of Figures Introduction Fundamental Theory Apparatus Procedure Result Sample Calculation Discussion Conclusion References

Page 1 2 3 4-7 8 9 10-12 13-14 15-19 20-21 22

LIST OF TABLES

No. 1 2 3 4 5

Tables Table 1: Spring Constant Experimental Table 2: Natural Frequency of Spring Table 3: Natural Frequency Experimental Table 4: Natural Frequency Theoretical Table 5: Percentage Error

LIST OF FIGURES

1

Page 10 10 10 11 11

No. 1 2 3 4

Figures Figure 1: Extension Of Spring Figure 2: Free Body Diagram and Kinetic Diagram

Figure 3: Graph Force against Extension Figure 4: Graph Experimental against Theoretical Frequency

INTRODUCTION

2

Page 4 5 12 12

Vibration is a phenomenon where oscillations take place about at an equilibrium point. Free vibration means there are no external forces acted on the system. The system oscillates based on the action of force in the system due the initial disruption that is affected from the surroundings by time and causes the amplitude to reduce to zero. They are a discrete and continuous systems. The physical properties are discrete quantities. The system has finite number of natural frequencies. The degree of freedom is important in locating the exact location and orientation in the space of a body. Vibration can be seen or proven from the motion of a tuning fork, instrument or harmonica, a mobile phone, or the cone of a loudspeaker. A free vibration is used to measure the properties of a system which in this experiment is to calculate the spring constant, k and the natural spring frequency, f.

FUNDAMENTAL THEORY

3

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. The stiffness and strong of spring can be measured by spring constant (k ) . The distance of the spring,

x

is stretched or compressed

away from its equilibrium or rest position. The force exerted by a spring is called a restoring force. It always acts to restore the spring toward equilibrium.

F = force k = spring constant x = the length of the spring deflection

Figure 2: Extension Of Spring

4

Figure 2: Free Body Diagram and Kinetic Diagram

For this experiment, our objective is to find the value of spring constant (k ) . The equation between spring constant with force and deflection is called Hooke’s law. We used this formula to find the spring constant for the theoretical value.

Where are:

m 0 = mass initial m 1 = mass final

x0

= initial extension

x1

= final extension

5

Then we can plot the graph between force, (N) against extension, (mm). So the gradient of the straight line of this graph is same with the spring constant.

All objects have the natural frequency or set of frequency at which they vibrate. Natural frequency is the frequency at which a system tends to oscillate in the absence of any driving or damping force.

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

∑ F=−kx+ mg=mx 

 F ma

 mg  Fs mx  mg  k (se  x ) mx  mg  kse  kx mx Before we get the value of frequency “F” we need used this equation to get the value of f,

m ´x +kx=0

By using Standard second order,

´x + Wn2 x=0

The result will be obtained,

x+

k x=0 m

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

6

Where x = 0, It defines the equilibrium position of the mass,

x  A sin

Where the term is

k k t  B cos t m m

k m the angular natural frequency defined by,

n 

τ=

k m Rad/sec

2π ωn

APPARATUS

7

Adjuster Helical Spring

Guide Constant

Mechanical Recorder

Carrier

Guide Roller Damper Load

Base

PROCEDURE 8

SPRING CONSTANT PROCEDURE

1) The plotter pen was fitted to the chart paper. 2) All weight was removed from the carriage. 3) The carriage was adjusted to ensure the plotter pen is on 20mm line one the chart paper. 4) The load was placed to the carriage. 5) The extension of the spring was recorded. 6) The stepped curve was obtained by repeating step 4 and step 5 by applying 2kg, 4kg, 6kg, 8kg and 10kg of load to the carriage.

NATURAL FREQUENCY PROCEDURE

1) The plotter pen was fitted to the chart paper. 2) The carriage was recorded with mass of 1.25kg. 3) The carriage was adjusted to ensure the plotter pen is on the middle of the chart paper. 4) The recorder was started. 5) The carriage was deflected downwards by hand and released. 6) The recorder was stopped. 7) Step 3 to 7 was repeated by adding 2kg, 4kg, 6kg, 8kg, and 10kg of load to the cart. 8) The distance for 5 oscillations, x was recorded for each loads.

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RESULTS

TO DETERMINED SPRING CONSTANT (EXPERIMENTAL) NO 1 2 3 4 5

MASS (kg) 2 4 6 8 10

LOAD DEFLECTION (mm) (N) 19.62 32 39.24 44 58.86 56 78.48 68 98.1 80 Table 1 : Spring Constant Experimental

EXTENSION (-20)(mm) 12 24 36 48 60

TO DETERMINED SPRING CONSTANT (THEORETICAL) NO 1 2 3 4 5

MASS (kg) 2 4 6 8 10

LOAD DEFLECTION (mm) (N) 19.62 31 39.24 42 58.86 53 78.48 64 98.1 75 Table 2 : Spring Constant Theoretical

EXTENSION (-20)(mm) 11 22 33 44 55

TO DETERMINED NATURAL FREQUENCY (EXPERIMENTAL)

NO

MASS (kg)

1 2 3 4 5 6

1.25 2 4 6 8 10

LENGTH OF Natural OSCILLATIONS Frequency (5 oscillation) (theory) (mm) (Hz) 1.25 19 6.00 3.25 29 3.72 5.25 33 2.93 7.25 43 2.49 9.25 49 2.20 11.25 54 2.00 Table 3 : Natural Frequency Experimental

TOTAL MASS (kg)

10

Natural Frequency (experiment) (Hz) 5.26 3.45 3.03 2.32 2.04 1.85

TO DETERMINED NATURAL FREQUENCY (THEORETICAL)

NO

MASS (kg)

1 2 3 4 5 6

1.25 2 4 6 8 10

LENGTH OF Natural OSCILLATIONS Frequency (5 oscillation) (theory) (mm) (Hz) 1.25 19 5.89 3.25 29 3.65 5.25 33 2.87 7.25 43 2.44 9.25 49 2.16 11.25 54 1.96 Table 4 : Natural Frequency Theoretical

TOTAL MASS (kg)

Natural Frequency (experiment) (Hz) 5.56 3.57 2.86 2.38 2.12 1.92

PERCENTAGE ERROR BETWEEN NATURAL FREQUENCY

Number 1 2 3 4 5 6

Natural Frequency Natural Frequency (theory) (experiment) (Hz) (Hz) 6.00 5.26 3.72 3.45 2.93 3.03 2.49 2.32 2.20 2.04 2.00 1.85 Table 5 : Percentage Error

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Percentage Error (%) 12.3 7.25 3.4 6.8 7.27 7.5

Force (N) vs Extension (mm) 70 60 50 40 30 20 10 0 10

20

30

40

50

60

70

80

90

100

110

5.5

6

6.5

Figure 3 : Force vs Extension

Experimental vs Theoretical 6 5 4 3 2 1 0 1.5

2

2.5

3

3.5

4

4.5

5

Figure 4 : Natural Frequency Experimental vs Natural Frequency Theoretical

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SAMPLE CALCULATION

To determine spring constant , k ( experimental) k=

∆F ∆X

k= =

Y 2 −Y 1 X 2− X 1

78.48 −58.86 48−36

= 1.66 N/mm To determine spring constant , k (theoretical)

(m 1 −m 0 ) g x 1−x 0 k=

(8−2)( 9 . 81 ) =1. 78 (44−11) k= N/mm

To determine the percentage error between k(experimental) and k (theory) − Experimental |Theoretical |×100 % Theoretical

Percentage Error =

1.78 −1.66 1.78

=

×100 %

= 6.7 % To determine the natural frequency ( theory) and Natural Frequency ( experiment) For total mass, m= 1.25kg Natural Frequency, f (theory) = =

1 2π

√ √

k m

1 1780 2 π 1.25

=6.0 Hz

1 Natural Frequency, f (experimental) =

T CYCLE

13

1 0.19

=

= 5.26 Hz To determine the percentage error between f (experimental) and f (theory)

Percentage Error =

− Experimental |Theoretical |×100 % Theoretical

For total mass, m = 1.25 kg Percentage Error =

−5.26 |6.006.00 |×100 %

= 12.3 %

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DISCUSSION HAKIMI For the first experiment, we need to find the spring constant, k. The graph force vs extension has been plot. From the graph, we can see that force and extension is directly proportional. As the load increase, the extension also increase. From the experimental result, the value of spring constant, k is 1.66 N/mm which is lower than theoretical value which is 1.78 N/mm. From the table of results, we can see that the value of deflection and extension from the experimental is slightly different from the theoretical value. This shows that the experiment is near successful as the difference of the value is small. The percentage error between k (experimental) and k (theoretical) is 6.7%. So,the experiment to find the stiffness of the spring, k is near successful as the percentage error value is acceptable which is below 10%. The value of k obtain was then used to find natural frequency, f. For the next experiment, we need to find the natural frequency, f based on different mass loaded. Based on the results obtained, the value of natural frequency decrease as the total mass increase. While on the graph Natural frequency theory vs Natural frequency experimental, we can see that the graph is increasing. Both of the frequency is directly proportional. Based on the sample calculation, the value of f, (theory) is 6.0 Hz and f, (experimental) is 5.26 Hz. The difference is small. The percentage error between f (theory) and f (experimental) is 12.3%. The error is quite big so, there are some improvement that can be took to get more precise outcome and reduce the error. Improvements that can be take : 1) Place our line of sight perpendicular to the scale to avoid parallax error. 2) Fixed measuring ruler should have been used. 3) The precision of the fixed ruler should be more than 0.1 cm.

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4) A small pointer should be attached to the bottom of the spring so that when the spring extends, it is easier to read the value off from the ruler that it corresponds to. 5) More sets of readings to be taken and the average find out to avoid random errors in the experiment.

DANIAL Based on the experiment conducted, the value of spring constant, k and natural frequency, f is determined. Hooke's Law states that the restoring force of a spring is directly proportional to a small displacement. In equation form, F = -kx where x is the displacement or elongation of the spring. As the displacement is acting downwards, it is considered that it acting in negative direction. Therefore, F = -k (-x) is also F= kx. The proportionality constant k is specific for each spring. For this experiment, the theoretical value of k of the spring is 1.78N/mm. Even though initially there is no additional mass attached to the spring, the mass of the carriage is taken into account, which is 1.25kg. Therefore for each additional mass, their values are added with 1.25kg and the initial mass considered as 1.25kg instead of 0kg. The relation between force, F and spring elongation, x is proportional. As the force acting on the spring increases, the elongation of the spring also increase. It is observed that the value of experimental and theoretical spring constant, k is slightly different that carried 6.7% of error. The different between the values is very small. The error can therefore be assumed to be ignored. Hence, the values of experimental and theoretical spring constant, k are approximately equal. To determine the natural frequency, each mass on the spring is allowed to vibrate to obtain a sinusoidal graph. The length of five oscillations is recorded to obtain the time of five oscillations by dividing the length with the velocity of the mechanical recorder. The experimental natural frequency, f is obtained as the reciprocal of time for one oscillation. Each value of natural frequency, f is different for each mass attached to the spring. The theoretical natural frequency, f of the spring is calculated using the formula given. It is a function of spring constant, k and mass, m. The percentage errors between the experimental and theoretical values of natural frequency, f of the spring are only minor and can be considered as insignificant difference.

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MIN Based on the experiment that have been conducted, we manage to determine the value of the spring constant (k) and also the natural frequency (f). We have been conducted this experiment for a total of 5 oscillations. Hooke's Law states that the restoring force of a spring is directly proportional to a small displacement. The proportionality constant k is specific for each spring. Thus, at the end of the calculation, to obtain the experimental value of spring constant (k), we will calculate the average reading of same oscillations. From the experiment, the value of spring constant (k) can be obtain from the gradient of the graph which is 1.66 N/mm. The theoretical value for spring constant is 1.78 N/mm. So, the percentage error is 6.7% Although there were errors occurred, overall the experiment was successfully conducted. Furthermore, by using the same number of oscillations and steps, we manage to obtain the value of natural frequency (f). The experimental value for natural frequency is 5.26 Hz. While the theoretical value is 6 Hz. The percentage error is 12.3%. Same as the spring constant (k), there were a few errors occurred, but overall the experiment was successfully conducted. The first error that occurred might be parallax error, which is our eyes are not perpendicular to the position of the pen. The pen was supposed to be situated at a certain position. Next, parallax error that might occur was during the recording of the oscillation measurement. Because it was measured by a ruler, our eyes may not be perpendicular to the ruler. Other than that, the error that might occur is when adding the weight. At this process, the spring may not be in a static position when the oscillation recorder already started.

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AFIQ From the experiment conducted, value of spring constant, k and natural frequency, f are determined. For this experiment, the value of spring constant, k for theoretical value is fixed at 1.7N/mm. The initial mass considered as 1.25kg instead of 0kg because of the carriage. From the graph load versus extension, the relation between the load, F and spring extension, x is proportional. As the force acting on the spring increases, the extension of the spring also increases. The value of experimental spring constant, k is obtained from the gradient of the graph. It is observed that the value of experimental and theoretical spring constant, k has slight difference. The difference between the values are very little. Therefore, the errors can assume to be minimum. Hence, the values of the experimental and theoretical spring constant, k are approximately equal. To determine the natural frequency, spring is allowed to vibrate to obtain a graph. Vibration recorded five times with increasing of 2kg of weight. The length of ten oscillations is recorded. The experimental natural frequency, f is obtained as the reciprocal of time for one oscillation. For each mass, the percentage errors between the experimental and theoretical values of natural frequency, f of the spring are very minimum and can be considered as insignificant difference. Slight errors may cause from some factors such as human error. For this experiment, human error can occur when releasing the weight for the spring to vibrate. The weight is pulled down too low so when it is released, the weight jumped. Therefore experiment must be conducted carefully to avoid any errors.

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AQIL

Based on the experiment, we can get the value of spring constant,k and natural frequency,f.. The value of spring constant for experimental is 1.66N/mm while the value of spring constant for theoretical is 1.78N/mm. The value of natural frequency for theoretical is 6.0Hz while the value of natural frequency for experimental is 5.26Hz. The percentage error between value of spring constant for theoretical and experiment is 6.7% while the percentage error between value of natural frequency for theoretical and experimental is 12.3%. From the graph load versus extension, the relation between the load, F and spring extension, x is proportional. As the force acting on the spring increases, the extension of the spring also increases. The value of experimental spring constant, k is obtained from the gradient of the graph. It is observed that the value of experimental and theoretical spring constant, k has slight difference. The theoretical natural frequency, f of the spring is calculated using the formula given. It is a function of spring constant, k and mass, m. The percentage errors between the experimental and theoretical values of natural frequency, f of the spring are only minor and can be considered as insignificant difference.

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CONCLUSION

HAKIMI

As a conclusion, from the first experiment, we can see that the force is directly proportional to the extension. The value of k obtained from the experimental is approximately same to the theoretical value of spring constant, k. For this experiment, the value of p...