Results

Displacement (mm)

Strain

-20

230

-15

175

-10

120

-5

60

0

0

5

-60

10

-110

15

-140

20

-170

Table 1 : The Values of Displacement and Corresponding Strain

Figure 1: Graph of Displacement vs Strain

The relationship between the displacement and the strain obtained from spreadsheet can be expressed with equation:

The equation can then be used to obtain the value of x when recording the value of strain in 30mm and 50mm cantilever displacement on both free-undamped by water and damped by water experiments.Example of calculation of the displacement of the cantilever:

If the strain value = 58, then the corresponding x will be =mm

If the strain value = -238, then the corresponding x will be =mm

The values of strain obtained from step 7 to step 11 are then used to obtain the displacement of the cantilever to obtain the graph of displacement vs time in free-undamped 30mm, free-undamped 50mm, water damped 30mm and water damped 50mm. The graphs obtained are at below:Figure 2 : Graph of Displacement vs Time in 30mm Free-undamped Cantilever

Figure 3 : Graph of Displacement vs Time of Water Damped Cantilever Figure 4 : Graph of Displacement vs Time of Free-undamped Cantilever

Figure 5 : Graph of Displacement vs Time of Water Damped Cantilever

Mass of cantilever beam = 295g

Equivalent mass of the beam,

Stiffness of cantilever, k =

Theoretical natural frequency of the cantilever beam,

From the free-undamped 30mm displacement of the cantilever beam:

5 oscillations completed at 0.84s, = 0.84s/5 = 0.168s

From the free-undamped 50mm displacement of the cantilever beam:

5 oscillations completed at (0.92-0.08)s, = (0.92-0.08)s/5 = 0.168s

Experimental natural frequency of the cantilever beam:

Theoretical frequency of viscously damped cantilever with damped mass (122g):

For the water damped 30mm displacement of cantilever beam:

5 oscillations completed in (1.52-0.14) s, = (1.52-0.14) s/5 = 0.276 s

For the water damped 50mm displacement of cantilever beam:

3 oscillations completed in (1.1-0.28) s, = (1.1-0.28) s/3 = 0.273 s

Experimental frequency of viscously damped and 30mm displacement of the cantilever beam:

Experimental frequency of viscously damped and 50mm displacement of the cantilever beam:

Discussion

1. The theoretical natural frequency of the cantilever beam can be calculated using the following equation:

Where: k is the stiffness of the systemis the period of the oscillationfn is the frequency of the oscillationm is the mass of the cantilever, in this case the equivalent mass of the cantilever,

Substituting the k and the m with will yield the theoretical natural frequency of the cantilever.

k can be obtained by using the following formula:

Where: E = modulus of elasticity, for aluminium, E = 70GPaL = length of the beamI = moment of inertia, for rectangular area, I = b = width of the beamh = height of the beam

Thus k =

Theoretical natural frequency, fn =

From Graph of Strain vs Time obtained from 30mm and 50mm with no water, it can be observed that the experimental periods of oscillation are:

For 30mm and 50mm displacement of free-undamped cantilever beam:

It can be observed from the graph that 5 oscillations completed in 0.84s, thus experimental natural frequency of the cantilever is equal to:

The difference between the experimental values and the theoretical value occurs because the cantilever experience slight damping in the experiment due to friction that is cause by the air when vibrating, according to the equation:

Where is the damped circular frequency and the is the natural circular frequency that can be represented by the equation:

And the < 1 for under-damped system, thus the < , the experimental natural frequency, < , experimental values is lower that the theoretical value .

The accuracy of the experiment of the experiment for free-undamped and free-damped increase as the initial displacement of the cantilever increases from 30mm to 50mm. When the experimental results were compared with the theoretical natural frequency, it can be seen that the 50mm case yields result that is closer to the theoretical result.

From 30mm free-damped cantilever, the experimental damped frequency is 3.623 while for 50mm free-damped cantilever the result is 3.663 in which the latter is closer to the theoretical damped frequency that is 3.768

The accuracy of the data can also be determined by the percentage error, the smaller the percentage error when compare with theoretical result the more accurate the data is.

From discussion No.1 the error percentage for 30mm and 50mm displacement of free-undamped cantilever beam is 4.83%,

For 30mm displacement and free-damped cantilever beam:

Error percentage =

For 50mm displacement and free-damped cantilever beam

Error percentage

Other findings

Graphs plotted by the spreadsheet are not smooth enough, the graph can be further smoother by reduces the softwares time interval of data recording from 5ms to time intervals lower than 5ms, which will record more strains value and can then be used to plot smoother graphs.

The period of oscillations obtained from the graphs are approximated based on the values at the x-axis intercept as the value are approximated due to limiting number of smaller graphs grid line.

Initial displacements of cantilever beam are approximated to 30mm and 50mm due to the limitation of the resolution of the ruler.

Graph of Displacement vs Strain

Displacement = -0.095(Strain) + 1.1082

230175120600-60-110-140-170-20-15-10-505101520

Strain

Displacement (mm)

Results

Displacement (mm)

Strain

-

20

230

-

15

175

-

10

120

-

5

60

0

0

5

-

60

10

-

110

15

-

140

20

-

170

Table

1

: The Values of Displacement and Corresponding Strain

Figure

1

: Graph of Displacement vs

Strain

The relationship between the displacement and the strain obtained from spreadsheet can be

expressed with equation:

????????????????????????

,

??

=

-

0

.

095

?

??????????

????

????????????

?

+

1

.

1082

The equation can then be used to obtain the value of x when r

ecording the value of strain in

30mm and 50mm cantilever

displacement

on both

f

ree

-

undamped

by water and damped by

Displacement =

-

0.095(Strain) + 1.1082

-25

-20

-15

-10

-5

0

5

10

15

20

25

-200

-150

-100

-50

0

50

100

150

200

250

D

i

s

p

l

a

c

e

m

e

n

t

(

m

m

)

Strain

Graph of Displacement vs Strain

Results

Displacement (mm) Strain

-20 230

-15 175

-10 120

-5 60

0 0

5 -60

10 -110

15 -140

20 -170

Table 1 : The Values of Displacement and Corresponding Strain

Figure 1: Graph of Displacement vs Strain

The relationship between the displacement and the strain obtained from spreadsheet can be

expressed with equation:

????????????????????????,??= -0.095?????????? ???? ????????????+1.1082

The equation can then be used to obtain the value of x when recording the value of strain in

30mm and 50mm cantilever displacement on both free-undamped by water and damped by

Displacement = -0.095(Strain) + 1.1082

-25

-20

-15

-10

-5

0

5

10

15

20

25

-200-150-100-50050100150200250

D

i

s

p

l

a

c

e

m

e

n

t

(

m

m

)

Strain

Graph of Displacement vs Strain