MEC424 2B1 G1 Experiment No. 1 PDF

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UNIVERSITI TEKNOLOGI MARA FAKULTI KEJURUTERAAN MEKANIKAL Program : Bachelor of Engineering (Hons) Mechanical (EM220/EM221) Course : Applied Mechanics Lab Code : MEC 424 Lecturer : Dr. Valliyappan David Natarajan Group : MEC 424 - LABORATORY REPORTTITLE : Free Vibration Experiment – Natural Frequency...

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Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

UNIVERSITI TEKNOLOGI MARA FAKULTI KEJURUTERAAN MEKANIKAL ___________________________________________________________________________ Program : Bachelor of Engineering (Hons) Mechanical (EM220/EM221) Course : Applied Mechanics Lab Code : MEC 424 Lecturer : Dr. Valliyappan David Natarajan Group : ___________________________________________________________________________

MEC 424 - LABORATORY REPORT TITLE

:

Free Vibration Experiment – Natural Frequency Of Spring Mass System Without Damping

No

NAME

STUDENT ID

LABORATORY SESSION

:

30 April 2020

REPORT SUBMISSION

:

06 May 2020

SIGNATURE

*By signing above you attest that you have contributed to this submission and confirm that all work you have contributed to this submission is your own work. Any suspicion of copying or plagiarism in this work will result in an investigation of academic msconduct and may result in a “0” on the work, an “F” in the course, or possibly more severe penalties.

Marking Scheme No 1 2 3 4 5

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Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

FACULTY OF MECHANICAL ENGINEERING

Teamwork Assessment Form Scale Level

1 Poor

2

3 Acceptable

4

5 Excellent

You will rate yourself and your team’s member on the following criteria

Element

Self

Earned Assessment Members 1

I was ready to work with my team I did my assigned work well and always on time I was fair to my teammates and myself I listened to others appreciatively and was supportive I was very committed and focused in my team I put extra efforts to finish or accomplish our task I encouraged others in my team and was helpful I managed and coordinated team efforts effectively I was able to lead discussions and provide solutions Overall, I was very satisfied and enjoyed my work Total Comment Self : Ready to give commitment to the group Member 1 : Happy to work together Member 2 : Happy to work together Member 3 : Happy to work together Member 4 : Happy to work together

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5

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5 5 5 5

5 5 5 5

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5 5 5

5 5 50

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5 5 50

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

ABSTRACT The aim of this experiment is to study the free vibration in a system. In this experiment, different masses were used to determine the spring constant and the natural frequencies. After calibrating the apparatus and measure the dimensions needed, the experiment then was conducted exactly as the procedure stated. Then, the data were retrieved after the spring subjected to different loads. The data were in the form of graph, so proper technique to read the graph was required to get a precise data. The tabulated data was then plotted into a graph to determine the spring constant. After that, a few calculations were made to determine the value of natural frequencies for the theoretical and experimental. The result was then compared to validate the data taken. In conclusion, the experiment was successful as the percentage error were in acceptable range and the objective were completed.

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

Table of Contents TITLE

: Free Vibration Experiment – Natural Frequency Of Spring Mass System Without Damping.....................1

Teamwork Assessment Form................................................................................................................................................... 2 ABSTRACT....................................................................................................................................................................................... 3 LIST OF TABLE.............................................................................................................................................................................. 5 LIST OF GRAPH............................................................................................................................................................................. 5 LIST OF FIGURES.......................................................................................................................................................................... 6 INTRODUCTION............................................................................................................................................................................ 7 THEORY............................................................................................................................................................................................ 8 EXPERIMENTAL PROCEDURE............................................................................................................................................. 11 Apparatus ............................................................................................................................................................................... 11 Procedure 1(To determine the spring constant, K)............................................................................................... 12 Procedure 2(To determine natural frequency)....................................................................................................... 12 RESULT........................................................................................................................................................................................... 13 DISCUSSION................................................................................................................................................................................. 18 CONCLUSION............................................................................................................................................................................... 19 REFERENCES.............................................................................................................................................................................. 20

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

LIST OF TABLE Table 1 : Data Spring Constant, k..............................................................................................................................13 Table 2 : Data of Respective Length for Oscillation, Period, T(s) & Natural Frequency.........................................15 Table 3 : Data Percentage Error for Natural Frequency............................................................................................17

LIST OF GRAPH Graph 1 : Graph Load against Extension..................................................................................................................13

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

LIST OF FIGURES Figure 1 : Load On Spring...........................................................................................................................................8 Figure 2 : Force vs Displacement................................................................................................................................8 Figure 3 : FBD.............................................................................................................................................................9 Figure 4 : Dynamics Mode..........................................................................................................................................9 Figure 5 : FBD...........................................................................................................................................................10 Figure 6: Mass Spring Vibration TM160...................................................................................................................11 Figure 7 : Ball pen.....................................................................................................................................................11 Figure 8 : Wavelength of Spring Stiffness.................................................................................................................14 Figure 9 : Wavelength of Spring Mass Under for Total Mass of 1.25kg, 3.25kg & 5.25kg.....................................15 Figure 10 : Wavelength of Spring Mass Under for Total Mass of 7.25kg, 9.25kg & 11.25kg..................................16

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

INTRODUCTION

Free vibration occur when a mechanical system is set off with an initial input and then allowed to vibrate freely, and the action of forces inherent in the system itself. The mechanical system will then vibrate at one or more of its natural frequencies and damp down to zero, which are properties of the dynamic system established by its mass and stiffness distribution.

Vibration can be defined directly from the dictionary as a regular oscillatory or periodic motion which repeats itself after a definite interval. But, from the physics definition it can be define as the oscillating, reciprocating, or other periodic motion of a rigid or elastic body or medium forced from a position or state of equilibrium. In this experiment, Newton's second law is the first basis for examining the motion of the system. Newton’s second law state F = ma, that the acceleration, is proportional to the net force and is inversely proportional to the mass. Free vibration is a sort of vibration in which a force is acted on the object once and the object or part can vibrate at its natural frequency. For examples of free vibration is hitting a tuning fork and let it ring. The mechanical system vibrates at one or more of its natural frequencies and damps down to motionlessness.

This is mostly related to the principle of Hooke’s law which is a principle of physics that states that the force (F) needed to extend or compress a spring by some distance X is proportional to that distance. That is F = kx, where k is a constant factor characteristic of the spring which is its stiffness, and X is small compared to the total possible deformation of the spring.

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

THEORY

Regarding the first part of the experiment, determining the value of spring constant can be obtained through Hooke’s Law formula which defined as force needed to extend or compress a spring by some distance. Hooke’s law states that the applied force F equals a constant k times the displacement or change in length x, or F = kx. The value of k depends not only on the kind of elastic material under consideration but also on its dimensions and shape. The illustration is as shown:

Figure 1 : Load On Spring.

the displacement of the deformation is directly proportional to the deforming force or load. Hence, when plotted a graph of force vs displacement, the slope of the graph represents the spring constant, k.

Figure 2 : Force vs Displacement

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

For the 2nd part, which is to determine the natural frequency of vibration. N atural frequency is the frequency or rate that an object vibrates naturally when disturbed. For this particular experiment, the property happens to be free vibration since there is no external vibration load act on it and the friction is neglected. Thus, it is called undamped natural frequency. This experiment is conducted in two ways in obtaining the natural frequency which are by static mode and dynamics mode. From these two, both equation that came from the free body diagram (FBD) will be compared. i.

Static mode (FBD from figure 1)

Figure 3 : FBD

↓+∑ F y =0 mg−k δ ST =0 ∴mg=k δ ST ii.

Dynamics mode

Equation 1

Figure 4 : Dynamics Mode.

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

Figure 5 : FBD

↓+∑ F y =ma mg−k (X + δST )=m ´x mg− kX −k δ ST −m x´ =0 -

Substitute

k δ ST =mg into eqn 2. Then subs eqn 2 into eqnEquation 1. 2

mg− kX −mg −m ´x =0 kX +m ´x =0 k ´x + X=0 m

( )

-

Noted that

( mk )=ω

2 n

Also, ω n=2 πf -

and frequency, f =

Natural Frequency:

√

ω n=

k 2π = m T

1 T

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

EXPERIMENTAL PROCEDURE

Apparatus : 1. Mass Spring Vibration TM160 i. Base ii. Carriage iii. Adjuster iv. Helical spring v. Guide roller vi. Additional mass vii. Guide columns viii. Mechanical recorder ix. Ruler

Figure 6: Mass Spring Vibration TM160

2. Ball Pen

Figure 7 : Ball pen

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

Procedure 1(To determine the spring constant, K) 1. Switch on the recorder 2. Put the graph paper in the slot recorder and make sure the graph paper stick to the recorder one rotation 3. Put ball pen in the slot pen and secure it tightly 4. Set the scale of graph paper to 20mm. 5. Loosen the adjuster and lock back to make sure that we get 20mm on the scale 6. Open the grip and put the mass slowly 7. Turn the recorder to record the deflection and then turn off 8. Repeat steps 6 and 7 until the mass reach 10kg

Procedure 2(To determine natural frequency) 1. Set scale of graph to 50mm. 2. Loosen the adjuster and lock back to make sure that we get 50mm on the scale graph 3. Pull the carriage downward 4. Quickly switch on the recorder and let the frame go 5. Stop recorder 6. Repeat steps 1 to 5 by adding mass(2kg) to the carriage until 10kg

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

RESULT

Determine the Spring Constant, k

Mass (kg)

Load (N)

Deflection (mm)

Extension (mm)

2

19.62

31.5

11.5

4

39.24

42.5

22.5

6

58.86

54

32

8

78.48

64.5

44.5

10

98.1

76

56

Table 1 : Data Spring Constant, k

Graph 1 : Graph Load against Extension

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

Figure 8 : Wavelength of Spring Stiffness

Calculation for Spring Constant:

Spring Constant , k =

¿

( y 2− y 1 ) ( x 2−x 1 )

(98.1−19.62 ) (56−11.5)

¿ 1.761 N /mm

Percentage Error , %=

value−Experimental value |TheoreticalTheoretical |×100 % value ¿∨

1.710−1.761 ×100 %∨¿ 1.710

¿ 2.89 %

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

Determine the Natural Frequency,

ωn

Mass

Total

Length for 5

Tn (s)

(kg)

Mass (kg)

oscillation

1.25 2.00 4.00 6.00 8.00 10.00

1.25 3.25 5.25 7.25 9.25 11.25

(mm) 17.50 27.00 35.00 42.50 48.00 52.50

0.88 1.35 1.75 2.13 2.40 2.63

T(s)

0.18 0.27 0.35 0.43 0.48 0.53

Frequency

Natural Frequency, ω n

(rad/s)

(Hz)

Theoretical

Experimental

5.71 3.70 2.86 2.35 2.08 1.90

37.53 23.28 18.31 15.59 13.80 12.51

35.90 23.27 17.95 14.78 13.09 11.97

Table 2 : Data of Respective Length for Oscillation, Period, T(s) & Natural Frequency

Figure 9 : Wavelength of Spring Mass Under for Total Mass of 1.25kg, 3.25kg & 5.25kg

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

Figure 10 : Wavelength of Spring Mass Under for Total Mass of 7.25kg, 9.25kg & 11.25kg

Calculation for Natural Frequency : Theoretical :

Natural Frequency ,ωn= ¿

√

√

k m

1761 1.25

¿ 37.53 rad / s Experimental :

Period ,T n=

¿

Length for 5 oscillation( mm ) mm ) Speed ( s

17.5 20

¿ 0.88 Period ,T n= ¿

T n (s) No . of oscillation 0.88 5

¿ 0.18 Natural Frequency ,ωn=2 πf

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

¿ 2 π (5.71) ¿ 35.90 rad / s

Percentage Error, % Natural Frequency, ω n (rad/s) Theoretical Experimental 37.53 23.28 18.31 15.59 13.80 12.51

35.90 23.27 17.95 14.78 13.09 11.97

Percentage of Error (%) 4.34 0.03 1.98 5.14 5.13 4.34

Table 3 : Data Percentage Error for Natural Frequency

Calculation : Percentage Error , %=

Theoretical Value−Experimental Value ×100 % Theoretical Value ¿

37.53 −35.90 × 100 % 37.53

¿ 4.34 %

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

DISCUSSION

From this experiment, we had to find the stiffness of the spring. For the equation, F = -kx . It is considered that x is acting in negative direction because x is going downward. So,we can rewrite the equation as, F = -k (−x) so the equation become F= kx. The proportionality constant k is specific for each spring. The theoretical value of k of the spring is 1.710N/mm and the experimental value of k of the spring is 1.761N/mm which has only slightly different from theoretical value with 2.89%. Even though at the beginning 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. As the force acting on the spring increases, the elongation of the spring also increase.

For the second part of the experiment, we had to find the value of natural frequency based on different mass loaded. The length of five oscillations was recorded to obtain the time of five oscillations by dividing the length with the velocity of the mechanical recorder. Then, the time for one oscillation can be obtained by dividing the time for five oscillations with five. The experimental natural frequency, Each value of natural frequency,

ω n is obtained as the period of time for one oscillation. ω n is not the same for each mass attached to the spring.

There was some disturbance which contributed to get errors while this experiment was carried out. The percentage errors between the experimental and theoretical values of natural frequency,

ω n of the spring are only minor and can be considered as insignificant difference

for the other additional mass.

Applied Mechanics Lab – MEC 424/AHA/MCM Rev. 01-2014

CONCLUSION

In conclusion, the force acting on the spring increases, the elongation of the spring also increases and the value of the spring constant can be obtain by the gradient of the graph. Then, from the result we can conclude that the value is acceptable because the highest percentage error is 4.34%. In the future, we can reduce this er...