Free Vibration Of A Cantilever Objective Biology Essay

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The purpose of this experiment is to determine the natural frequency of a cantilever beam study both undamped and damped free vibration motion of a cantilever beam.

Vibration is the periodic motion of a body or system of connected bodies displaced from a position of equilibrium. In general, there are two types of vibration, free and forced. Free vibration is maintained by gravitational or elastic restoring force. Forced vibration is caused by an external periodic or intermittent force applied to the system. Both of these types of vibration may be either damped or undamped. Undamped vibrations can continue indefinitely because frictional effects are neglected in the analysis. Basically, if a system that is subjected to an initial disturbance and is left to vibrate on its own, the subsequent vibration is known as the free vibration. Vibrations without damping would result in a continuous vibration of the particular oscillatory body. As a matter of fact, it will produce a displacement-time graph of such nature as shown in the following figure. This graph is commonly referred to as the simple harmonic motion. http://upload.wikimedia.org/wikipedia/commons/4/44/Simple_harmonic_motion.png

Figure 1: Simple Harmonic Motion (Displacement-Time Graph)

However, in reality, just as with many other scientific theories, this is impossible because friction and other forces are present both internally and externally. As the system is subjected to these forces, this phenomena is called damping. The principle effect of damping is to reduce the amplitude of an oscillation, not to change its frequency. So, the graph of the amplitude of a normal damped oscillation might look like the following:

http://www.efunda.com/formulae/vibrations/sdof_images/SDOF_UnderDamped_Response.gif

Figure 2: Graph of Damped Oscillation (Displacement-Time Graph)

Apparatus and Materials:

1. Cantilever beam apparatus

-Modulus of elasticity of aluminium(E) : 70GPa

-Dimension of the cantilever beam : 927mm (L) Ã- 19.09mm (W) Ã- 6.35mm (H)

-Mass of the cantilever beam : 292.59g

- Mass of the damper : 122 g

2. Strain gauge

3. Strain recorder

4. Viscous damper

Experimental Procedures:

Figure 3: Experiment Setup without Viscous Damper

Figure 4: Experiment Setup with Viscous Damper

The computer and the strain recorder were switched on.

The strain recorder application software was started by double clicking on the "DC104REng" shortcut icon on the computer desktop.

The experiment setup was shown in Figure 3. The operation of the strain recorder and the recorder application software were referred to the operational manual.

The viscous damper was removed if it was attached to the beam.

The beam was held and displaced by, ymax, -20mm, -15mm, -10m, -5mm, 0, 5mm, 10mm, 15mm,and 20mm. The strain recorder reading for each displacement value from the "Numerical Monitor" screen of the application software was recorded manually.

The relationship of the displacement (of the free end of the beam) and the strain recorder reading was obtained by plotting an appropriate graph using a spreadsheet.

The beam is displaced by 30mm and the beam is left to vibrate on its own. The strain recorder reading was recorded by clicking on the "Play" and "Stop" button.

The recorded file was retrieved by clicking on the "Read USB" button.

The graph of the beam displacement versus time, t was plotted.

The experiment was repeated by using beam displacement of 50mm.

The viscous damper was connected as shown in Figure 3. Steps 7 and 10 were repeated by using beam displacement of 30mm and 50mm respectively.

Results:

Beam Displacement(mm)

-20

-15

-10

-5

0

5

10

15

20

Strain

350

200

145

80

0

-70

-115

-170

-240

Table 1 Relationship of the beam displacement and the strain recorder reading

Figure 5 Graph of beam displacement against strain

From the graph plotted, Y= mX + C equation is obtained :

y= -0.0703x

Gradient, m = -0.0703mm

a) Theoretical Calculations

As given in the experiment:

Modulus of elasticity of aluminium,E = 70 GPa

Length of the cantilever beam, L = 0.927m

Width of the cantilever beam,b = 0.019m

Thickness of the cantilever beam, h = 0.006m

Mass of the cantilever beam, mcantilever = 0.293 kg

Mass of the damper, mdamper = 0.122 kg

b) Experimental Results and Calculations

Free Vibration of Cantilever Beam at 30mm Displacement

Sample calculations:

y = -0.0703x

at t=0.1; strain=88

y= -(0.0703)(88) = -6.18564mm

This is to obtain the experimental value of displacement during the oscillations.

Time(s)

0

0.1

0.2

0.3

0.4

0.5

0.6

Beam Displacement (mm)

20.81

-6.19

-13.64

21.65

-14.62

-4.36

-18.56

Table 2 Displacement of the beam with respect to time of free vibration of 30mm.

Figure 6 Graph of frequency against time of free vibration of 30mm.

Period of Oscillation 1 (Ï„1) Natural frequency , fn

= 0.225-0.07

= 0.16s

Period of Oscillation 2 (Ï„2)

= 0.38-0.225

= 0.16s

Period of Oscillation 3 (Ï„3)

= 0.535-0.38

= 0.16s

Average =

Damped Circular Natural Frequency, Damping ratio,

(2Ï€) =

= (2Ï€)/0.16s =

= 39.27 rad/s = ∞ (Over-Damped)

Free Vibration of Cantilever Beam at 50mm Displacement

Time(s)

0

0.1

0.2

0.3

0.4

0.5

0.6

Beam Displacement (mm)

-7.87

40.21

-40.63

10.55

27.14

-43.59

25.73

Table 3 Displacement of the beam with respect to time of free vibration of 50mm.

Figure 7 Graph of frequency against time of free vibration of 50mm.

Period of Oscillation 1 (Ï„1) Natural frequency , fn

= 0.19-0.035

= 0.16s

Period of Oscillation 2 (Ï„2)

= 0.345-0.19

= 0.16s

Period of Oscillation 3 (Ï„3)

= 0.500-0.345

= 0.16s

Average =

Damped Circular Natural Frequency, Damping ratio,

(2Ï€) =

= (2Ï€)/0.16s =

= 39.27 rad/s = ∞ (Over-Damped)

Viscously Damped Vibration of Cantilever Beam at 30mm Displacement

Time(s)

0

0.1

0.2

0.3

0.4

0.5

0.6

Beam Displacement (mm)

-4.50

-0.56

3.37

-3.80

0.98

1.69

-2.95

Table 4 Displacement of the beam with respect to time of viciously damped vibration of 30mm.

Figure 8 Graph of frequency against time of viciously damped vibration of 30mm

Period of Oscillation 1 (Ï„1) Natural frequency , fn

= 0.320-0.040

= 0.280s

Period of Oscillation 2 (Ï„2)

= 0.600-0.320

= 0.28s

Period of Oscillation 3 (Ï„3)

= 0.880-0.600

= 0.280s

Average =

Damped Circular Natural Frequency,

(2Ï€)

= (2Ï€)/0.28s

= 22.44 rad/s

Damping ratio,

=

=

= ∞ (Over-Damped)

Viscously Damped Vibration of Cantilever Beam at 50mm Displacement

Time(s)

0

0.1

0.2

0.3

0.4

0.5

0.6

Beam Displacement (mm)

4.92

-10.26

5.34

1.12

-5.34

4.22

0.70

Table 5 Displacement of the beam with respect to time of viciously damped vibration of 50mm.

Figure 9 Graph of frequency against time of viciously damped vibration of 50mm

Period of Oscillation 1 (Ï„1) Natural frequency , fn

= 1.23-0.945

= 0.29s

Period of Oscillation 2 (Ï„2)

= 0.665-0.385

= 0.28s

Period of Oscillation 3 (Ï„3)

= 0.945-0.665

= 0.280s

Average =

Damped Circular Natural Frequency,

(2Ï€)

= (2Ï€)/0.28s

= 22.44rad/s

Damping ratio,

=

=

= ∞ (Over-Damped)

Discussion:

Moment of inertia, I

I =

= 3.42 x 10-10 m4

Equivalent mass of the beam, me

Theoretical Natural Frequency,

Theoretical Damped Natural

Frequency,

Stiffness of the beam, k

=

= 90.16 N/m

Total Equivalent Mass of Beam with Viscous Damper , meq

Natural Circular Frequency for Damped Free Vibration,

= 36.15rad/s

Natural Circular Frequency of Beam with Viscous Damper,

The free vibration of the theoretical natural frequency of the cantilever beam in this experiment is 5.75Hz while the experimental natural frequency of the cantilever beam is 6.25Hz for amplitude of 30mm and 6.25Hz for amplitude of 50mm.

The viscously damped vibration of the theoretical natural frequency of the cantilever beam is 3.45Hz and the experimental natural frequency of the cantilever beam for amplitude of 30mm and 50mm are 3.57Hz and 3.57Hz.

Percentage error, % x 100%

Free Vibration of Cantilever Beam at 30mm Displacement

Percentage error, %= x 100%

=8.70%

Free Vibration of Cantilever Beam at 50mm Displacement

Percentage error, %= x 100%

=8.70%

Viscously Damped Vibration of Cantilever Beam at 30mm Displacement

Percentage error, %= x 100%

=3.48%

Viscously Damped Vibration of Cantilever Beam at 50mm Displacement

Percentage error, %= x 100%

=3.48%

In this experiment, it is calculated that the percentage error for the free vibration is both 8.70% for 30mm and 50mm. For the viciously damped vibration, the percentage error for the 30mm and 50mm were both 3.48%. The results of the experiment were slightly inaccurate. This may be caused by the external force which is the air resistance as the beam oscillates. Another factor is caused by parallax error which occurred during the measuring of displacement before the beam was released as our eye level was not perpendicular to the scale of the metre rule. When the beam is released, it slightly hit the bottom of the container which decreases the original force released drastically which affects the amplitude of oscillation. Furthermore, the Modulus of Elasticity(Young's Modulus) was given, which might not be accurate. All these source of error may affect the results of the experiment to be inaccurate.

The results of the experiment can be improved by measuring the Modulus of Elasticity. Using a deeper container would also avoid the damper to hit the bottom of the container. Furthermore, the experiment can be done in a vacuum box to avoid air resistance. Eye level should be adjusted until it is perpendicular to the metre rule scale. This steps can increase the accuracy of the results.

The damped period, damped natural frequency and the damping ratio of the system of free vibration is:

Displacement of the beam(mm)

Damped period, (s)

Damped natural frequency, (s-1)

Damping ratio,

30

0.16

6.25

∞ (Over-damped)

50

0.16

6.25

∞ (Over-damped)

The damped period, damped natural frequency and the damping ratio of the system of viscous damped is:

Displacement of the beam(mm)

Damped period, (s)

Damped natural frequency, (s-1)

Damping ratio,

30

0.28

3.57

∞ (Over-damped)

50

0.28

3.57

∞ (Over-damped)

When the amplitude is 30mm or 50mm for both cases, they have the same damped period, damped natural frequency and damped ratio. The percentage error for the free vibration of 30mm and 50mm were both 8.70% while for the viciously damped of 30mm and 50mm were both 3.48%. This indicates that the difference in amplitudes do not affect the frequency of the oscillation. From the general equation of frequency :

, where c=speed of wave

λ= wavelength

This formula proves that amplitude or displacement does not affect the frequency of the oscillation.

If the strain gauge is mounted on the other end of the cantilever beam, the results would be not accurate as the gauge is very sensitive to changes. At the other end, there is not much difference in the change of length which affects the strain. In Figure 3, the strain can be detected more easily as the change in length is very obvious. This is because when the free end is free to vibrates, the vibration will be strong but the compression and tension that results on the surface of pressure sensor is not that strong.

Conclusion :

The theoretical natural frequency , for this experiment is 5.75Hz for free vibration cantilever. The value of both 30mm and 50mm frequency obtained were 6.25Hz which have percentage error of 8.70%. Whereas, the theoretical damped natural frequency , is 3.45 Hz. The value of both 30mm and 50mm frequency obtained were 3.57Hz which have percentage error of 3.48%.

Furthermore, the results proves that the displacement does not affect the vibration(frequency) of the oscillation of the cantilever beam.

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