Natural Frequency Of Spring Mass System Without Damping PDF

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UNIVERSITI TEKNOLOGI MARAFAKULTI KEJURUTERAAN MEKANIKAL___________________________________________________________________________Program : Bachelor of Engineering (Hons) Mechanical (EM220/EM221) Course : Applied Mechanics Lab Code : MEC 424 Lecturer : PROFESOR MADYA IR.TS BULAN ABDULLAH Group : EMD...

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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 : PROFESOR MADYA IR.TS.DR BULAN ABDULLAH Group : EMD4M1A1 / G2 ___________________________________________________________________________

MEC 424 - LABORATORY REPORT TITLE: Free Vibration Experiment – Natural Frequency Of Spring Mass System Without Damping

No

NAME

STUDENT ID

GROUP

1 2 3 4

AZIB SYAHMI BIN MOKHTAR ILYASA’ HAKIM BIN FAUZI HARIZ SAUFI BIN MOHD SUMARI MOHD RASYDAN BIN RUSLIB

2020898836 2020816538 2020483594 2020859824

EMD4M1A1 EMD4M1A1 EMD4M1A1 EMD4M1A1

*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 misconduct 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

1

2

3

4

5

6

7

8

9

Total

1.0 ABSTRACT The experiment demonstrates Newton's Second Law, which is the fundamental concept of free vibration. In this experiment, we must define the theoretical and experimental spring constants and natural frequencies. The spring constant is a measurement of how rigid and solid a spring is. Natural frequency can be damped or undamped depending on whether the device has significant damping. The normal frequency of the spring mass mechanism is not dampened in this experiment.

Table of Contents Course

: Applied Mechanics Lab...................................................................................................1

1.0 ABSTRACT.........................................................................................................................................2 2.0 INTRODUCTION................................................................................................................................6 3.0 THEORY..............................................................................................................................................7 4.0 PROCEDURE...................................................................................................................................10 4.1 APPARATUS/EXPERIMENTAL SETUP....................................................................................10 5.0 RESULTS..........................................................................................................................................17 5.1 AZIB SYAHMI BIN MOKHTAR (2020898836)..........................................................................17 5.2 ILYASA’ HAKIM BIN FAUZI (2020816538)...............................................................................22 5.3 HARIZ SAUFI BIN MOHD SUMARI..........................................................................................27 5.4 MOHD RASYDAN BIN RUSLIB 2020859824..........................................................................32 6.0 DISCUSSION...................................................................................................................................37 6.1 HARIZ SAUFI BIN MOHD SUMARI (2020483594).................................................................37 6.2 AZIB SYAHMI BIN MOKHTAR (2020898836)..........................................................................38 6.3 ILYASA’ HAKIM BIN FAUZI (2020816538)...............................................................................39 6.4 MOHD RASYDAN BIN RUSLIB (2020859824).......................................................................40 7.0 CONCLUSION..................................................................................................................................41 7.1 HARIZ SAUFI BIN MOHD SUMARI (2020483594).................................................................41 7.2 AZIB SYAHMI BIN MOKHTAR (2020898836)..........................................................................42 7.3 ILYASA’ HAKIM BIN FAUZI (2020816538)...............................................................................43 7.4 MOHD RASYDAN BIN RUSLIB (2020859824).......................................................................44 8.0 REFERENCES.................................................................................................................................45 9.0 APPENDICES...................................................................................................................................46 10.0 EVIDENCE OF MEETING............................................................................................................47

List of Tables Table 1 Calculation of the result of the extension of the spring without deflection......................18 Table 2 : Comparison of Theoretical and Experimental values of Spring Constant, k................19 Table 3 : Calculation of Natural Frequency of Spring after Deflection........................................19 Table 4 : Comparison of Theoretical and Experimental values of Natural Frequency, f..............19 Table 5. Experimental results to determine spring constant.......................................................22 Table 6. Experimental results to determine natural frequency....................................................25 Table 7: Result of deflection from experiment............................................................................28 Table 8: The percentage error of the spring constant obtained..................................................28 Table 9: Tabulated result from the oscillation chart.....................................................................29 Table 10: Calculation of the result of the extension of the spring...............................................33 Table 11: Graph of Load (N) vs Extension (mm)........................................................................34 Table 12: Experimental results to determine natural frequency..................................................34 Table 13: Appendix Table...........................................................................................................46

List of Figures Figure 1: Example of single degree of freedom system...............................................................7 Figure 2: Graph representative of elasticity region.......................................................................8 Figure 3 : Labelled Diagram of Vibration Apparatus...................................................................10 Figure 4 : The plotter pen is fitted...............................................................................................11 Figure 5 : Weight is removed from carriage...............................................................................12 Figure 6 : The carriage is adjusted.............................................................................................12 Figure 7 : The plotted paper is ensured to land on the 20mm line on the chart paper................13 Figure 8 : The weight is loaded onto the spring.........................................................................13 Figure 9 : The individual steps is read.......................................................................................14 Figure 10: Example of a stepped curve.....................................................................................14 Figure 11 : Plotter pen is fitted...................................................................................................15 Figure 12: Setting the adjuster...................................................................................................15 Figure 13: The recorder is started at the same as the hand is released from the carriage.........16 Figure 14 : The oscillation of the spring caused by the frame and the added load (from 1.25kg to 5.25kg)....................................................................................................................................... 17 Figure 15 : The deflection of the spring caused by the force......................................................17 Figure 16 : The oscillation of the spring caused by the frame and the added load (from 7.25kg to 11.25kg)..................................................................................................................................... 18 Figure 17 : Graph of Load (N) vs Extension (mm).....................................................................18 Figure 18: Deflection of spring with static force..........................................................................22 Figure 19 : Graph Load vs Extension.........................................................................................23 Figure 20: Oscillation of spring with load from 1.25 kg to 5.25 kg..............................................24 Figure 21 : Oscillation of spring with load from 7.25 kg to 11.25 kg...........................................25 Figure 22: Deflection of spring with static force..........................................................................27 Figure 23: Oscillation of spring with load from 1.25 kg to 5.25 kg..............................................27 Figure 24: Oscillation of spring with load from 7.25 kg to 11.25 kg............................................28 Figure 25: Graph for Load vs Extension....................................................................................29 Figure 26: Deflection of spring with static force..........................................................................32 Figure 27: The oscillation of the spring and the added load (from 1.25kg to 5.25kg).................32 Figure 28: The oscillation of the spring and the added load (from 7.25kg to 11.25kg)................33 Figure 29: Graph of Load (N) vs Extension (mm)......................................................................33 Figure 30: Evidence of Meeting.................................................................................................47

2.0 INTRODUCTION When a device oscillates due to forces inherent in the system, it is called free vibration. The device can vibrate at one or more of its natural frequencies under free vibration, which are dynamics system properties determined by the mass and stiffness distribution. This research is important for engineers because in mechanics and construction, a resonance disaster is described as the collapse of a building or a mechanical process caused by induced vibrations at a system's resonance frequency. The behavior is determined by ordinary differential equations, and physical properties are discrete quantities. Natural frequencies in a discrete system are finite, while natural frequencies in a continuous system are infinite. The system has a finite number of degrees of freedom, while in a continuous system, physical properties are a function of spatial coordinates, and system behavior is expressed by partial differential equations, which have an infinite number of degrees of freedom.

3.0 THEORY Vibration happen when a system is trying to reach its equilibrium state after the external forces applied to the system was removed. This cause the system to create a periodic movement. Undamped free one degree of freedom vibration is one of the simplest vibrations to be analyze. Without damping means that with each movement, there are no losses in energy which means that the system can vibrate for a long time unless an additional external force is applied. A single degree of freedom means that the system with mass is considered to move in one direction, in which either x or y axis. The system that have more than one mass and direction of vibration, that system have more than one degree of freedom. In this experiment, the vibration occur in y axis.

Figure 1: Example of single degree of freedom system.

Hooke’s Law is a law used to identify a material elasticity. This law was discovered by Robert Hooke in 1660. The law states that the deformation of an object that cause displacement is directly proportional to the force or load applied to the object. When the force applied to the object is released or removed, the object will return to its original length or shape. This is a condition called elasticity.

Figure 2: Graph representative of elasticity region. With the help of Figure 1, we can deduce that the spring constant can be determined if it the force and extension remains in the elastic region. In this experiment, the theoretical value will be tested with the experimental value. the theoretical value of the spring constant had already been provided, which is k = 1.710 (N/mm). Based on all the theory above, we can deduce that in mathematical perspective:

F=kx Where, F = external force applied to the spring (N) k = spring constant (N/mm) x = length of deflection (mm)

In static mode, the following equation can be derived:

∑ F=0 mg− k( δst )=0 kδst =mg

Based on the equation above, in vibration mode the following equation can be derived:

+↓ ∑F=ma y mg− k( x−δst )=m ´x

m ´x +kx + kδst −mg =0

By substituting the equation from the static mode into the vibration mode:

m ´x +kx + kδst −mg =0 ∴m ´x + kx=0

´x +

k x=0 m

Comparing the equation above with the vibration equation:

´x + ωn x =0 2

Therefore, the angular natural frequency of the system is:

ωn = 2

k m

√

ω n=

( )

k rad m s

4.0 PROCEDURE 4.1 APPARATUS/EXPERIMENTAL SETUP Apparatus Name 1. 2. 3. 4. 5. 6. 7. 8. 9.

Base Carriage Adjuster Helical Spring Guide Roller Additional Mass Guide Columns Mechanical Recorder Ruler

Apparatus Diagram

Figure 3 : Labelled Diagram of Vibration Apparatus

Relevant measurements and calibrations used of items related to apparatus Carriage: Mass 1250g Additional Masses: 2000g Tension / Compression spring with retaining screws: 1.7 N/mm Recorder Speed: 20 mm/s 4.2 Experiment Procedure Procedure 1: The spring constant K is determined 1. The plotter pen & paper are fitted

Figure 4 : The plotter pen is fitted

2. The weight is removed from the carriage Figure 5 : Weight is removed from carriage

3. The adjuster used to set the carriage, such that the plotter pen is on the 20mm line on chart paper.

Figure 6 : The carriage is adjusted

Figure 7 : The plotted paper is ensured to land on the 20mm line on the chart paper.

4. The spring is loaded onto the carriage by placing weight

Figure 8 : The weight is loaded onto the spring

5. The recorder is started briefly after each weight is added.

6. The individual steps is read from the stepped curve that is obtained Figure 9 : The individual steps is read

Figure 10: Example of a stepped curve

Procedure 2: To determine natural frequency 1. The plotter pen is fitted

Figure 11 : Plotter pen is fitted

2. The chosen additional masses are attached and secured with knurled nut 3. The adjuster is used to set the height of the carriage, so the stylus is centred on paper. Figure 12: Setting the adjuster

4. The recorder is started.

5. The carriage is deflected downwards by hand and allowed to oscillate freely until it finally comes to rest. Figure 13: The recorder is started at the same as the hand is released from the carriage

6. The recorder is stopped. 7. The experiment is repeated with the other additional masses.

5.0 RESULTS 5.1 AZIB SYAHMI BIN MOKHTAR (2020898836) Figure 14 : The deflection of the spring caused by the force

Figure 15 : The oscillation of the spring caused by the frame and the added load (from 1.25kg to 5.25kg)

Figure 16 : The oscillation of the spring caused by the frame and the added load (from 7.25kg to 11.25kg)

Mass of frame = 1.25 kg Theoretical Spring Value Constant, k = 1.710 N/mm Table 1 Calculation of the result of the extension of the spring without deflection Mass (kg) Total Mass (kg) Load (N) 2 3.25 31.883 4 5.25 51.503 6 7.25 71.123 8 9.25 90.743 10 11.25 110.363 Figure 17 : Graph of Load (N) vs Extension (mm)

Deflection (mm) 31.75 42.25 54 64.25 76

Extension (mm) 11.76 22.25 34 44.25 56

120

100

Load (N)

80

60

40

20

0 0

10

20

30

Extension (mm)

40

50

60

Table 2 : Comparison of Theoretical and Experimental values of Spring Constant, k

Spring Value Constant, k

Theoretical (N/mm) 1.710

Experimental (N/mm) 1.774

Percentage Error % 3.743

Table 3 : Calculation of Natural Frequency of Spring after Deflection Mass (kg)

Total Mass (kg)

0 2 4 6 8 10

1.25 3.25 5.25 7.25 9.25 11.25

Natural Frequency (ω n) [rad/f)] Theoretical Experimental 36.986 35.902 22.938 22.176 18.048 18.850 15.358 16.755 13.597 14.362 12.329 13.345

Table 4 : Comparison of Theoretical and Experimental values of Natural Frequency, f Natural Frequency (ω n ) [rad/f)] Theoretical Experimental 36.986 35.902 22.938 22.176 18.048 18.850 15.358 16.755 13.597 14.362 12.329 13.345

Percentage Error % 2.931 3.323 4.444 9.096 5.626 8.241

Sample Calculation Gradient of Load vs Extension Graph, m = Experimental Value of Spring Constant, k

m= k=

y 2− y 1 x 2−x 1

110.363 −31.883 56 −11.76

k =1.774 N /mm Theoretical value of Spring Constant, k = 1.710 N/mm Calculation of Percentage Error, % of Spring Constant

¿ ×100 % Theoretical Value Percentage Error=¿

¿ Theoretical Value−Experimental Value∨

¿ ×100 % 1.710 Percentage Error=¿

¿ 1.710−1.774 ∨

Percentage Error=3.743 %

Determining theoretical natural frequency,

√ √

ω n=

k m

ω n=

1710 1.25

ωn

ω n=36.986 Hz Determining experimental natural frequency,

ωn

Number of oscillations taken for complete cycle = 6

Period ,T n=

Lengthof complete cycle , mm Recorder Speed , mm / s

Period ,T n=

21 20

Period ,T n=1.05 Period ,T 1=

Tn Number of complete cycle

Period ,T 1=

1.05 6

Period ,T 1=0.175 Frequency , f =

1 Period , T 1

Frequency , f =

1 0.175

Frequency , f =5.714

√

ω n=

k =2 πf m

ω n=2 π (5.714) ω n=35.902 Hz

Calculation of Percentage Error, % Natural Frequency

¿ ×100 % Theoretical Value Percentage Error=¿

¿ Theoretical Value−Experimental Value∨

¿ × 100 % 5.887 Percentage Error=¿

¿ 5.887−5.714 ∨

Percentage Error=2.939 %

5.2 ILYASA’ HAKIM BIN FAUZI (2020816538)

Results To determine spring constant Appendix B: (i) Summary of result for M1

Figure 18: Deflection of spring with static force

Experimental results to determine spring constant Mass (kg) 0

Load (N) 0

Deflection (mm) 0

Extension (mm) 0

2

19.62

31

11

4

39.24

42

22

...