Undamped Vibration

Mechanical Vibration Laboratory Report: Free Undamped Vibration

Name & ID: Ahmad syabil ahmad zorin 0309110 Group Members: Hidayat Hamzah Alex Laimon Aisyah Hanis Low Heng Yan

Date of Experiment:24-09-2014 Report due date:10-08-2014 Report submission date: 10-082014 Checked by: Dr. Housseini

School of Engineering

Item/marks

Taylor’s University

Format/10

Malaysia 10th October 2014

Abstract and Introduction/10 Figures and Diagrams/15 Materials and Method/10 Results Discussions/45 References/10 Total

CONTENTS

Abstract……………………………………………………………………………………………………..2

1.0 Introduction……………………………………………………………….……………………....…2

2.0 Experiment Design…………………………………………………………..……………………4

2.1Materials ……………………………………………………………….………………………........4

2.2Methods ………………………………………………………………………………………....….4

2.3Procedures …………………………………………………………..………………….……….….5

3.0Results & Discussions.……………………………………………..…………………….…..…..6

4.0Conclusions & Recommendations..………………………………..…………………….….7

References……………………………………………………………………………………….….…….7 Appendices………………………………………………………………………………………….…….10

ABSTRACT The experiment is carry to investigate the frequencies of undamped oscillation of springs related to certain variables that will affect the system. Variables that are available in this experiment are the thickness of the spring and also the distance of the beam attached to the spring. The experiment ran for 30 minutes in a laboratory environment with less random errors, such as wind, rain or other factor that can affect the result. Further details of the experiment will be explain as below 1.0 INTRODUCTION Vibratory system mainly consists of a system which is able to store potential energy, kinetic energy and also gradually losing energy. In vibration system, the kinetic energy and potential energy are alternating between one another until gradually it lost energy due to dampener. The simplest vibratory system consists of a spring with, k coefficient of spring, and a mass. In additional, this system, by means the spring, is attached to an immovable support. This is to ensure a single translation motion of direction is obtain from this. Therefore, the spring will only move in either X direction or only Y direction. Due to this, it is called a single degree of freedom system. Due to un additional of external force during the vibration of the spring due to the mass is displaced, this is said that the system has free vibration. The machine that will be used in this experiment is called the free and forced vibration apparatus. This machine is made of aluminium frames which consist of a beam/ bar which is mounted at the left end of the frame. Helical spring is attached to the beam to enable the beam to oscillate. The springs can be change depend on the user. Other than that, it has exciters which are control by the control unit. The exciter control unit responsible in moving the graph paper during this experiment. This machine is widely used in the laboratory to investigate natural vibration damped vibration, forced vibration, resonance and amplitude and phase response. Therefore this machine is suitable to be used for our experiment. In this experiment, few equations are used to calculate responding variables. The equations are as below: Equation of motion is formed with the availability of the moment equilibrium at the pivot point of the beam ∑ 𝑀𝑜 = 𝐽𝑜 𝜑̈ = −𝐹𝑎 As we know the formula for spring force, it consists of the deflection of spring, x and also spring constant, k. Due to small angle, equation will be as below: 𝐹 = 𝐾𝑥 = 𝐾𝜑𝑎 The mass moment of inertia of the beam is: 𝑚𝐿2 𝐽𝑜 = 3

The equation of motion is thus, as below: 𝜑̈ +

3𝐾𝑎2 𝜑=0 𝑚𝐿2

The frequency, f due to the harmonic oscillation above is as below: 𝑓=

1 3𝐾𝑎2 √ 2𝜋 𝑚𝐿2

The natural angular velocity is as below 𝜔𝑜2 =

3𝐾𝑎2 𝑚𝐿2

The periodic time is as below:

𝑇 = 2𝜋√

3𝐾𝑎2 𝑚𝐿2

Figure 1: experimental set up

2.0 EXPERIMENTAL DESIGN

Figure 2:apparatus set up

2.1 materials    

pen graph paper free and force vibration apparatus 3 springs with different stiffness, k = 0.84N/mm, 1.44N/mm, 3.09N/mm

2.2 Method Osculating graph is obtained by realising the cantilever beam from a certain position at the same time, the switch for the paper graph roller is switched on. When the osculation is stop, the switched is switched off. This is repeated three times using the same variable to reduce error in the experiment. After that the spring is changed with another spring that has different stiffness value.

2. 3 Procedure 1. the apparatus is set up as figure 2 above 2. Pen is placed in the calliper and graph paper is placed. Graph paper is checked to insure that the ball pen is not to near to the graph paper. This can cause the graph paper to tear and damaged the whole paper 3. beam is adjusted so that it is 90 degree with the side frame of the vibration apparatus 4. Spring with stiffness of 0.84 N/mm is screwed at the bar and the upper part frame of the vibration apparatus machine. 5. Distance of the spring from the pivot point O,a is taken and recorded 6. The stiffness of the spring, k is taken and recorded 7. The beam is then pull down to a certain extend and is released, at the same time the oscillator machine is switched on 8. After the beam has stop osculating, the oscillator is switch off and the graph is recorded 9. Step 5 to 8 is then repeat for 3 times to reduce the error in this experiment. The result is than recorded 10. Step 9 is repeated using 3 different distance, a. the result is recorded 11. Step 10 is then repeated using different stiffness of spring, k 12. Result is then recorded

3.0 RESULT AND DISCUSSION

Graph 1: spring stiffness of 840 N/m

Graph 2:spring stiffness of 1440 N/m

Graph 3: spring stiffness of 3090N/m

Table 1:free undamped vibration data tabulation set spring stiffness, k (N/mm) length of a (m) natural frequency, Wo experimental period,T(s) experimental frequency,(Hz) theoretical frequency (Hz) percentage error (%)

0.2 19.99 0.35 2.86 3.24 11.82

1 3090 0.35 34.98 0.2 5.00 5.67 11.82

0.5 49.97 0.15 6.67 8.1 0.18

0.2 13.64 0.45 2.22 2.21 -0.55

2 1440 0.35 23.88 0.25 4.00 3.87 -3.36

0.5 0.2 34.11 10.42 0.175 no data 5.71 no data 5.53 1.69 -3.33 no data

3 840 0.35 18.24 0.45 2.22 2.96 24.92

0.5 26.05 0.25 4.00 4.22 5.21

Based from the above table, we could see the values for experimental period,T(s), which we calculated the boxes from the graph paper. One small box represent 0.05 second. Therefore to calculate the experimental period, the boxes from one peak to another peak (one complete cycle) is calculate and multiply by 0.05s. To calculate the experimental frequency, 1 will be divided by the experimental period,T. Experimental frequency= 1/T As for the theoretical frequency, the formula that is given before is used. As we seen from the graph, we could see that as spring stiffness is increase the experimental period is getting shorter when we compare with the same a.840N/m spring stiffness has a period of 0.25second. As for 1440 N/m spring stiffness has 0.175 s period and lastly 3090N/m spring stiffness, which are the highest among all has a period of 0.15 second. This is because as the spring is stiffer, it will tend to oscillate faster, therefore produce shorter period, thus bigger frequency. Other than that, distance ‘a’ is one of the factor that related to the frequency and the period of the graph. For this experiment, distance ‘a’ has been set for all three springs. The distance ‘a’ are 0.2m, 0.35m, 0.5m. As we picked spring stiffness of 1440N/m, we compared both values of theoretical and experimental values. We can see that for the distance of ‘a’ from 0.2m, 0.35m to 0.5m, the experimental frequency give a values of 2.22Hz,4.00Hz, and 5.71Hz while for the theoretical frequency, it showed a slightly different values, which are 2.21Hz, 3.87Hz, and 5.53Hz. To see the difference more clearly percentage error has been calculated so that differentiate and comparison can be made. As we see the percentage errors, we could see that there are few that are 10% or more and this may be due to energy loss such as the kinetic energy during the process of oscillation. This is maybe due to the free force vibration machine itself not being properly maintain. This will cause the machine to have less lubrication thus losing kinetic energy in that sense. Other than that, it may due to the friction happen between the pen and the graph paper. The nearer the graph paper is with the pen, the higher the pressure the pen will give to the paper. Due to this it might even cause the paper to be torn. During this moment, kinetic energy is lost. As we can see the percentage errors in the table, we could see that there are negative values. This shows that human error might occur during the experiment session, for example, the initial position is too low or too high that might cause the displacement to exceed the required value. Therefore, the experimental result will be more that it should, thus giving negative percentage of error. Finally, we could see that the graph of the free undamped vibration is gradually diminished even though there was no damper used in this experiment. As we know free undamped vibration osculate due to its initial disturbance without any external force acting on it. However, in real application, the amplitude will gradually diminished in time. This is due to other factors such as the air or the temperature that might affect the oscillation of the vibration.

4.0 CALCULATION AND RECOMENDATION In conclusion, we can conclude that by changing the distance ‘a’ and the stiffness of the spring, it will affect the frequency of the vibration. However, the theoretical values might be slightly different compare to the experimental values we obtain in this experiment. This is due to the fact that errors might occurred during the procedure of the experiment. Errors such as friction that happen during the experiment, might reduce the kinetic energy in time, thus giving different result compared to the theoretical experiment. Other than that, the experiment was not done in a vacuum and close environment. Therefore random errors might occur. Other than that the force and vibration apparatus might not be well maintain. Thus lack of lubrication at the beam would cause the vibration to loss energy due to friction Therefore few recommendations are needed so that in the future, this random errors can be minimised. First of all, the force and vibration machine apparatus need to be well maintained to give a better reading. Before experiment is started, everything need to be calibrated. This is to ensure there are no systematic errors occur. Finally, to reduce error, repeat the experiment for several time so that the average can be calculated, by doing this, the result will be more accurate.

REFERENCE  

Ganesh, R. (2010). Control Engineering. Pearson Education India Damped Vibration, Indian Institute of Technology Delhi. Retrieved 2 August 2014, from (http://www.iitr.ac.in/outreach/web/CIRCIS/PG/AVN/RC/Revision%20of%20concep ts3_Damped.pdf)