Vibrations Practical Lab Report 2

VIBRATIONS LAB REPORT: SIMPLE PENDULUM

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VIBRATIONS PRACTICAL LAB REPORT SIMPLE PENDULUM BY AICHA EUGENE EN292-0662/2009

SCHOOL OF MECHANICAL, MANUFACTURING AND MATERIALS ENGINEERING COLLEGE OF ENGINEERING th Presented on 24 July 2014 th Commenced on 10 July 2014


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ABSTRACT

This lab focuses on experimentally obtaining the relationships between variables involved with torsional load to circular cross-section rod of mild steel. The variables that were focused on included applied torque, angular deflection, and length of rod, polar moment of inertia and the shear modulus. The apparatus that were used to get the data from the torsion testing machine setup were the steel rule and vernier calipers. By measuring the applied torque with respect to the angle of twist, the shear modulus, shear stress at the limit of proportionality, and failure conditions were found. The results found within this experiment were somewhat inaccurate. The shear modulus for mild steel specimen was determined to be 80.3 G Pa, which contains nearly 2.95% error when compared against the standard 78 G Pa. There are a multitude of reasons for the discrepancy between the theoretical and calculated values of the modulus of rigidity for the mild steel specimen this experiment. One was that perhaps the test specimen wasn’t of a consistent diameter. A varying diameter throughout the length of the test specimen would interfere with an accurate calculation by varying the polar moment of inertia; J. Instances of a varying J value would yield unique G values. Other sources of error present in this experiment included the fact that the indicated angles were only read to the nearest half degree. This limitation prevented an accurate measurement for the angle of twist of a specimen under torsion According to the graphs however, the experiment was very successful. A good correlation could be seen between the values of TL and J ф. Overally, Overally, this lab was very helpful in visualizing such a difficult topic like torsion. The apparatus helped with the understanding of the relationship between the force and angle of twist, modulus of rigidity and polar moment.

Table of Contents


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Title Page .......................................................................................................................... 1 Abstract ............................................................................................................................. 2 Table of Contents .............................................. ................................................................ 3 List of Figures ................................................................................................ ................... 3 1

Objectives of Experiment ......................................................................................... 4

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Introduction ............................................... ................................................................ 5 Background information ........................................................................................... 5

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Materials and Methods ................................................. ............................................. 9 4.1 4.2 4.3 4.4 4.5 4.6

Torsion Test Specimen ................................................ ..................................... 9 Equipment and Apparatus ................................................................................. 9 Set up diagram .................................................................................................. 9 Test Sample Installation............................................... ................................... 10 Procedure ................................................... ..................................................... 11 Precautions ................................................. ..................................................... 11

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Results .............................................. ....................................................................... 12

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Analysis of Results ................................................................................................. 13

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Discussion ................................................. .............................................................. 13 7.1 7.2 7.3 7.4

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Calculations........................................................... .................................. 14 Comments ................................................... ........................................... 14 Torsion failures ................................................... ................................... 15 Error analysis and Recommendations .................................................... 14

Conclusion ................................................ .............................................................. 17

References ................................................ ....................................................................... 18

List of Figures F ig. 1: Torsion in cylindrical bar[Ref. 2] ……………………………………..…………...5 F ig. 2: Relationship between torque and angle of twist [Ref. 2]………….………....6 : Torsion of a solid bar [Ref.4] ……………………………………..................6 F ig. 3 : Relationship between modular shear stress and shear strain[Ref. 6]…….….8 F ig. 4 Fig. 5: Torsion specimen……………………………………………………………….…….…9 F ig. 6: Details of torsion testing machine .................................................................. .. .10 : Types of failure in torsion ................................................ ................................... 16 F ig. 7

List of Tables Table 1: Specimen Results ………………………………………………………..………....12 Table 2: Torsion testing machine readings………………………………….………….6


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1. Objectives In this experiment, the objectives are: To examine the relationship between the period of a simple pendulum and its amplitude. To determine the relationship between the pendulum period and the length of the pendulum if the amplitude is small enough that the variation with amplitude is negligible. To measure the gravitational field strength

1) To determine the relation between the period (T) of oscillation of a simple pendulum with verify 3 types of different specimens. 2) To determine the center of gravity of a connecting rod, as well as the radius of gyration about the center of gravity by using compound pendulum. 3) Determine the mass moment of inertia, (I g & Io ) by oscillation and manual calculation. This experiment aims at studying the behaviour of both simple and compound pendulums, in order to realise the following objectives: 1) 2) 3)

The independence of the period of oscillation of the simple pendulum from its mass. The relationship between the period of oscillation and its length. The determination of the value of the gravitational acceleration g , to be compared with t he known standard value.

2. Introduction

The length of a pendulum is the distance between the pivot point and the center of the bob. The amplitude is the angle that is formed by the length when the bob has been raised. Gravitational forces accelerate the bob downwards. The mass of the bob is measured in grams, it does not affect the period of the pendulum. The period of the pendulum is the total amount of time it takes the pendulum to complete one complete cycle, which is to return to the point where it started. The displacement in this case would be zero

I- Introduction: Simple pendulum is simply a concentrated mass m attached to one of the ends of a mass-less cord of length l , while the other end is fitted as a point of oscillation, such that t he mass is free to oscillate about that fixed point in the vertical plane. The compound pendulum differs from the simple one in that it has a mass di stribution along its length -that is its mass i s not concentrated at a given point-, therefore it has a mass moment of inertia I about its mass centre. Any rigid body that has a mass m, and mass moment of inertia I and suspended at a given distance h from its centre of gravity represents a compound pendulum.

It should be realised in the derivation of the governing equations, that the angle of oscillation of the pendulum, simple or compound, should be small.


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3. Background Information

The schematic representation of the simple pendulum is shown in Figure-1.1-a, which consists of a small ball of mass m suspended by a mass-less cord of length l . The system is given an initial small angular displacement

 ,

pendulum oscillates in the vertical plane by a time varying angle

and as a result the  (t) with

the vertical

direction.

Schematic representation of the Figure-1.1 (a)simple pendulum

The dynamic equilibrium equation (equation of motion) corresponding to the tangential direction of motion of the concentrated mass yields:

  mg sin θ  0 ml θ

(1)


Assuming small magnitude for the angle

 ,

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so that sin   , and simplifying

eqn-1

leads to the equation:

  

g l

  0

(2)

Let the motion defined by the function  (t )   sin  n t ,

where

 n is

eqn-2 and simplifying gives

 n

g 

l

 (t) be

a simple harmonic motion defined as

the natural frequency of the pendulum. Substituting for

 in

 n as:

(3)

The period of oscillation , is defined as the time required to complete one full cycle of motion or one oscillation. By observing the function  (t), the period  is given as:

 

2  n



2

l g


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Physical pendulum In this case, a rigid body – instead of point mass - is pivoted to oscillate as shown in the figure. There is no requirement of string. As a result, there is no tension involved in this case. Besides these physical ramifications, the working of compound pendulum is essentially same as that of simple pendulum except in two important aspects:



Gravity acts through center of mass of the rigid body. Hence, length of pendulum used in equation is equal to linear distance between pivot and center of mass

(“h”). 

The moment of inertia of the rigid body about point suspension is not equal to

“mL2 ”

* as in the case of simple pendulum. The time period of compound

pendulum, therefore, is given by :



In case we know MI of the rigid body, we can evaluate above expression of time period for the physical pendulum. For illustration, let us consider a uniform rigid rod, pivoted from a frame as shown in the figure. Clearly, center of mass is at a distance “L/2” from the point of suspension :


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h= L/2 Now, MI of the rigid rod about its center is:

We are, however, required to evaluate MI of the rod about the point of suspension, i.e. “O”. Applying parallel axes theorem,

Putting in the equation of time period, we have:

The important thing to note about this relation is that time period is still independent of mass of the rigid body. However, time period is not independent of mass distribution of the rigid body. A change in shape or size or change in mass distribution will change MI of the rigid body about point of suspension. This, in turn, will change time period.

Fur ther, we should note that physical pendulum is an effective device to measure “g”. As a matter of fact, this device is used extensively in gravity surveys around the world. We only need to determine time period or frequency to determine the value of “g”. Squaring and rearranging,

Point of oscillation


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We can think of physical pendulum as if it were a simple pendulum. For this, we can consider the mass of the rigid body to be concentrated at a single point as in the case of simple pendulum such that time periods of two pendulums are same. Let this point be at a linear distance “Lo “from the point of suspension. Here,

The point defined by the vertical distance, "L o “, from the point of suspension is called point of oscillation of the physical pendulum. Clearly, point of oscillation will change if point of suspension is changed.

4. Materials and Methods:

4.1 Torsion test specimen

F igure 5: Torsion specimen

4.2 Equipment and apparatus Suggested Equipment List

The Lab Kit included Electronic balance Thread or dental floss T-bracket or corner brace Metric measuring tape C-clamp Corrugated cardboard for protecting door

VIBRATIONS LAB REPORT: SIMPLE PENDULUM Lead fishing weight Metre stick Protractor Stop watch or watch with second hand.

1) Wooden pendulum 2) Vee support, cylindrical support 3) Ruler 4) Stopwatch

The apparatus used for the experiment.

4.3 Set-up diagram

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F igure 6: Schematic diagram showing details of torsion testing machine [2]

The SM1 testing machine could apply torque up to 30Nm. The torque was applied manually via a 60:1 reduction gear box. The torque is reacted to by a torsion bar whose displacement relative to the displacement arm is measured by a linear potentiometer connected to a TQ digital Torque meter. Angular rotation was applied to one end of the specimen by means of a reduction gear, protractor being provided to measure this rotation and torque developed in the specimen was balanced and measured on the other side of the specimen. Protractor scales were then used to provide an accurate measurement of the angle of torque. 4.4. Test specimen installation The lower clamp on the right side of the machine allows for the lateral movement of the input shaft to accommodate samples of various lengths. With this clamp loosened, the input shaft was slid to the right enough to allow the sample to fit between the input shaft and the pendulum shaft. The sample was then installed into the socket on the input shaft. After firmly tightening the clamp on the input socket, the input shaft was slid towards the pendulum shaft until the sample was far enough into the other socket. Because the end of the sample was hex shaped, the hand crank had to be rotated to allow the sample to align with the clamp on the pendulum shaft. The lower clamp and the socket clamp were tightened on the pendulum shaft. Normally, for the circular cross section rods, the length is the length of the circular area while for the hexagonal cross section rods, the length is the distance between the clamps of the apparatus, which needed to be measured and recorded. 4.5 Procedure


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1. The dimension for the wooden pendulum is taken ( thickness, width and length ) 2. Vee support is inserted into the hole of the wooden pendulum. 3. The wooden pendulum is hanging at the testing apparatus to swing it. 4. The wooden pendulum was swing from right at 15 o , and time for 10 complete cycle was taken. 5. Step 4 was repeated for swinging the pendulum from left. 6. Step 2 to 5 was repeated for cylindrical support for another hole of wooden pendulum.

Steel and plastic balls are used separately in this experiment as follows:

1) Attach the cord to the steel ball at one end, and attach the other end to the main frame. Record the length of the cord l . 2) Displace the ball form its neutral position by a small amount, and then release it to oscillate freely. Measure and record the time T required to complete ten oscillations. 3) Adjust the cord length to a new value and repeat step-2. 4) Repeat Step-3 six more times so that eight pairs of l and T are recorded. 5) Replace the steel ball with the plastic ball and repeat steps-1 through 4.

4.6 Precautions 1. Readings were noted at a perpendicular view to eliminate parallax error. 2. Dimensions of the specimen were measured very carefully 3. The Angle of twist was accurately measured for the corresponding value of Torque. 5. Results 6. VI - Coll ected Data:

7. 8. Part One- Sim ple Pendul um :


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9. 10. Table-1.1 Collected data for the simple pendulum part

Tr ial

Steel Ball l (cm)

T (second)

1 2 3 4 5 6 7 8

11. 12. 13.

14.Par t Two- Compound Pendul um: 15. 16. l = …………cm

Plastic Ball l (cm)

T (second)


17. 18. Table-1.2 Collected data for the compound pendulum part

Trial 1 2 3 4 5 6 7 8 9 10 19. 20.

21. 22. 23.

(cm) h

(second) T

14


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24. 25. 26. VI I - Data Processin g:

27. 28.Part One- Sim ple Pendul um : 29. Use eqn-4:





l

2

g

Square both sides of eqn-4 to get:

2

τ 

2

4π

Evaluate the theoretical

The values of  Theor are to be

period  Theor corresponding

compared with the

to each length l .

experimental values

Draw  2 versus l as shown

Slope =

4

 Exper .

2

g

in Figure-1.2. g is found and compared



l g

to the standard value.

Analysis of results Table-1.3 Data processing analysis for the simple pendulum part

Steel Bal l

VIBRATIONS LAB REPORT: SIMPLE PENDULUM Trial

l (cm)

16 2

Exper

(second)

Theor

(second)

( Exper. )

Percent

2

Error ( )

(second)

1 2 3 4 5 6 7 8

Table-1.4 Data processing analysis for the simple pendulum part

Plastic Ball Trial

l (cm)

1

2

Exper

(second)

Theor

(second)

( Exper. )

Percent

2

Error ( )

(second)


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2 3 4 5 6 7 8

Data processing results for the simple pendulum part. Table-1.5 Quantity

Slope fr om Figure-1.2:

Steel Bal l Plastic Ball

2

g (m/s )

Percentage Err or of g ( )


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30. Discussion of results 30.1

Calculations

SAMPLE OF CALCULATIONS: Experimental Result Point

Point

For Point 1 2 3 Average

T1 14.65 14.5 14.5 14.53

1 period of oscillation , T 1 T2 Point 1 T  2

= = = =

T2 14.1 14.0 14.1 14.07

14.53 / 10 1.453 s 14.07 / 10 1.407 s

L1

g (1.453) = (2 Л)2 ( L1 / 9.81 ) L1= 0.5246 m 2

Point 2 T 2

 2

L2

g (1.459) 2 = (2 Л)2 ( L2 / 9.81 ) L2 = 0.529 m

Component 1 Component 1

Component 2 Component

Component 3


Length Height Width Volum e

80cm 1cm 8cm 640cm

3 1cm 45cm 1cm 45cm

Diameter Height Volume

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Component 2 2.5cm 1cm 4.9cm3

So, Total volume, V T = V1 – ( V2 + V3 ) = 640 – ( 45 + 4.9 ) = 590.1 cm3 Given mass, m = 0.6 kg

Density, ϼ = m / v total = 0.6 kg / 590.1 × 10 -6 = 1016.78 kg/m 3 To obtain the mass for each component, m1= ϼ×v m -6 = 1016.78 × (640×10 ) = 0.6507 kg

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=ϼ×v = 1016.78 × (45×10-6) = 0.0458 kg

m 2= ϼ × v = 1016.78 × (4.9×10 -6) = 4.982 × 10 -3 kg

X= 73 cm

 = 15

r G

= x (L2 – x) / (L 1 + L2 - 2x) = 0.73 ( 0.529 – 0.73 ) / ( 0.5203 + 0.529 - 1.46 ) = -0.1467 / -0.4107 = 0.3572 m

to obtain I0 :

At point 1 (T1)2 = (2 Л) 2 ( I01 / mgr G )


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(1.453) 2 = (2 Л)2 ( I01 / 0.6 × 9.81 × 0.3572) I01 = 0.1124 kg.m 2

At point 2 (T2)2 = (2 Л)2 ( I02 / mgr G ) (1.407) 2 = (2 Л)2 ( I02 / 0.6 × 9.81 × 0.3572 ) I02 = 0.1054 kg.m2

To obtain IG : L1 = (IG1 + mr G²) / mr G

Thus,

IG = mr G (L – r G)

At point 1 IG1 = mr G (L1 – r G) = (0.6)(0.3572)(0.5203-0.3572) = 0.035 kg.m 2 At point 2 IG2 = mr G (L2 – r G) = (0.6)(0.3572)(0.529-0.3572) = 0.0368 kg.m 2 Theoretical Result

Area (A1) = 73 × 8 = 584 cm2

Total area, A t = A1 – ( A2 + A3 ) 2 = 534.091 cm

Area (A 2) =   = 3.142(1.25)2 = 4.909cm2

Area (A3) = 45 × 1 = 45cm2


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Hanging at point 1 8cm

Y

73cm

*Height = 1cm Y=

 

= 36.5 (A1) – 15(A3).25(A2) - 50. At = 19037.36 / 534.091 = 35.644 cm To obtain Io, Moment of Inertia : 2

I0 = IG + md

For component 1 I01 = 1/12 ( mh 12 ) + md12 2 2 = 1/12 (0.6507×0.8 ) + (0.6507×(0.365) ) = 0.1214 kg.m 3

For component 2 I02 = 1/4 mr 22 ) + md22 = 1/4 (4.982 × 10 -3 ×0.01252) + (4.982 × 10 -3 ×0.0125 2) = 9.7305×10-7 kg.m 3 For component 3 2 2 I03 = 1/12 ( mh 3 ) + md3 = 1/12 ( 0.0458×0.45 2 ) + ( 0.0458×0.505 2 ) = 0.01245 kg.m 3 So, Io,total

= I01 + I02 + I03 -7 = 0.1214 - 9.7305×10 - 0.01245 = 0.10895 kg.m 2

Io,total = IG + md2 Thus, IG can be obtain IG = Io,total - md2 = 0.10895 – (0.6)(0.356442) = 0.03272 kg.m 2 Percentage Error  – 

 %


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At point 1 : I01 = 0.1115 kg.m 2 (experiment) = 0.10895 kg.m 3 (theory) So, Percentage error = 0.1115 – 0.10895 x 100% 0.1115 = 2.287%

7.2 Comments

During the experiment, we found out there are errors identified that affects the results. The value k is different compare to the theoretical value because of errors as below; 

Human error – This experiment is conducted by human so there must be some error in terms of readings and procedure. Parallax error is one of the most common errors in conducting the experiment and then, the handling of stopwatch timing is not accurate.



Device error – The apparatus used is not reliable because one of the parts of the apparatus (protractor) is gone but the angle still can be obtained by using the protractor which is drawn by pencil.



Environment factors –This factor slightly can be taken as a minor cause of the error. The experiment been ran out in a very conducive laboratory but possibilities of the present of blowing wind must be considered.

Analysis

The mass of the pendulum does not affect its period because mass does not affect its acceleration due to gravity. The acceleration due to gravity on the surface of the earth is a constant 9.8 meters per second squared, which is not affected by the mass of the object being accelerated in any way. This has been proven by Galileo’s experiment when he dropped two different masses from a tower, and they fell at the same time. This also holds true for pendulums, the mass does not therefore affect the period, and is not present in the period formula. We had to divide the time by 50 vibrations as to get the average time of one period, due to the fact that our


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experiment included friction from the air. We called o ne cycle a vibration rather than a period because the air friction kept the bob from reaching the same point again, it never completed a true period. In a vacuum chamber, with ideal conditions, we would never need only one vibration in order to calculate the period, because our amplitude would never change. During the trial, the amplitudes fell, due to air resistance. The period o f the pendulum is determined by length of the string and acceleration due to gravity. If we were to go to the moon where the acceleration due to gravity would be smaller the period of the pendulum would increase, there would be less acceleration so it would move slower. To test the effect of different mass on the motion o f the pendulum, I would set up two pendulums. They would be in the same gravitational field, and in a vacuum but would have different masses of bobs. They would be released from the same height, and I would expect them to have the same period, and therefore swinging motion.

7.4 Error Analysis In the torsion testing it was discovered through the use of the sample that ultimate shear stress (USS) was approximately 72.43% of ultimate tensile stress (UTS). It was also found that the small error margin in modulus of rigidity-2.95% was due to some irregularities in the experiment. During the course of the experiment, the torsion machine malfunctioned between data points 17 and 19 and the hand wheel could go back to the opposite direction of rotation adjusting the reading and affecting the subsequent ones. There was slippage for some unknown reason, and the machine immediately began reading roughly higher than it previously had been reading. For this reason, the data from points 17 through 25 have been adjusted downward linearly. The raw data and associated shear graph can be seen in the results section. The experimental modulus of rigidity was reasonably close to its published value. The experimental proportional limit, however, was not as close, and there existed a very large amount of error with the experimental ultimate shear stress. As it is known there was a machine malfunction prior to reaching USS, it is possible that our linear adjustment to these values was not sufficient. Given both experimental USS and the proportional limit were off by more than 20%, it is also possible that the machine had not been functioning properly for a longer period of time than was obviously visible between data points 19 and 24. Also, as noted in the experiment, even with both degree indicators tightened, the 6 degree wheel and the 360 degree wheel did not agree on angular measurement. For example, when 30 degrees had been achieved on the 6 degree wheel, the 360 degree wheel did not read 30 degrees as well.


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Another possible cause for the calculated valu e for the modulus of rigidity, being 2.95 % different from the accepted value for mild steel was perhaps the test rod specimen wasn’t of a consistent diameter. A varying diameter throughout the length of the test specimen would interfere with an accurate calculation by varying the polar moment of inertia; J. Instances of a varying J value would yield unique G values. Other sources of error present in this experiment include the fact that the indicated angles were only read to the nearest half degree. This limitation prevented an accurate measurement for the angle of twist of a specimen under torsion. Improvements for this experiment are in order. From an educational point of view, the experiment would be more interesting if the calculated modulus was used to identify the test material. Another improvement would be to have clearly identified test specimens. As seen with the discrepancy in theoretical and experimental G values for the mild steel, it is difficult to perform this experiment successfully without the proper materials present. As noted, the equipment lacked a torsiometer for accurate angle of twist measurement. One final improvement that should be made to this experiment would be to use a strain gage and Hooke’s Law to calculate the modulus of rigidity. The use of precise, digital instruments would also increase the attractiveness of an experiment Conclusion

The purpose of this lab, which was to measure the period of a pendulum and to measure the acceleration due to gravity that is causing the pendulum’s movement, was achieved because of the low reported present error, of 5%. Some sources of error include air friction, and the friction of the rope, as well as the fact that the string was not attached to the center of the pendulum. During this lab we learned about pendulums, the parts that make them up, and the different things that affect their behavior.

From this experiment, we obtained the experimental value is slightly differs to the theoretical value. From the discussion above, we found out the reading is not accurate. Minimize the parallax error (reading been taken parallel to the eyes). The experiment is best conducted in a vacuum. Precising the value by taking readings more than 2 or 3 times (during handling the stopwatch) and get the average. To obtain a better experiment result, a new set of apparatus must be replaced with the unreliable one. Since the percentage error is less than 15%, we can conclude the experiment is succeed and the objective of the experiment is achieved. REFERENCES

VIBRATIONS LAB REPORT: SIMPLE PENDULUM 

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Engineering Mechanics Dynamics, 11 th Edition In SI Units by R.C Hibbeler

Publisher : Pearson Prentice Hall  http://cnx.org/content/m15585/latest/

Vibrations Practical Lab Report 2

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