OBJECTIVES 1.
To fami famili liari arize ze with with the the techn techniq ique uess in in mea measu suri ring ng dyna dynami micc qua quant ntit ities ies..
2.
To dete determ rmin inee the the reso resona nant nt freq freque uenc ncie iess and and the the corre corresp spon ondi ding ng mode mode-s -sha hape pess of the vibrating beam with several different techniques.
R ESULTS ESULTS Natural frequency
where
f i
EI 1 ( ) 2 , i = 1, 2, 3, .... 2 m 2π L
E
= Young’s modulus
I
= Area moment of inertia of beam about neutral axis
L
= Span of beam
m
= mass per unit length
λ
Given
=
2 λ i
i
= 1.875
i
=1
= 4.694
i
=2
= 7.855
i
=3
= 10.996
i
=4
b = 0.03 m t = 1.2 x 10-3 m L = 0.475 m E = 220 GPa ρ
= 7903 kg / m 3
I = bt3 / 12 m = ρ bt /unit length
Experimental and Theoretical Values Table
Mode
Theoretical Pos of nodes (m)
Expt Pos of nodes (m)
CRO frequency (Hz)
Stroboscope frequency (Hz)
Theoretical Natural frequency (Hz)
1
-
-
-
-
4.55
2
0.075
0.102
25.51
25.37
28.41
3
0.0475, 0.2375
0.065, 0.235
80.97
70.60
79.5
4
0.033, 0.171, 0.310
0.044, 0.169, 0.304
148.6
149.67
155.4
Position of node is taken from the free end of the ruler.
Sample Calculation: Taking data from Mode 1, Theoretical Natural frequency is
f i
=
=
2
EI 21 ( ) 2 2π L m λ i
1.875
2
2(0.475 )π
= 4.55 Hz
220 ×10
2
(
9
×
1
×
0.03 (1.2 ×10
12 7903 × 0.03 ×1.2 ×10
3
−
3
−
)
3
1
)2
X(m) 0 0.0475 0.095 0.1425 0.19 0.2375 0.285 0.3325 0.38 0.4275 0.475
X/L 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Mode 2
Y Mode 3
Mode 4
0 0.186 0.607 1.07 1.41 1.51 1.33 0.883 0.265 -0.400 -0.972
0 0.456 1.208 1.509 1.043 0.019 -0.992 -1.410 -0.997 0.002 1.001
0 0.770 1.508 0.868 -0.630 -1.410 -0.640 0.832 1.397 0.437 -1.000
Graphs for the above modes 2-4: Mode 2
2 1.5 1 0.5 Y
0 -0.5
0
0.1
0.2
0.3
-1 -1.5 X
0.4
0.5
Mode 3
2 1.5 1 0.5 0
Y
-0.5 0
0.1
0.2
0.3
0.4
0.5
-1 -1.5 -2 X
Mode 4
2 1.5 1 0.5
Y
0 -0.5
0
0.1
0.2
0.3
0.4
0.5
-1 -1.5 -2 X
Analysis of Results 1. Position of nodes The theoretical mode shape is given as follows: y
≈ cosh
where a1 = 0.734 ai ≈ 1.00, i > 1
λix
L
− cos
λ i x
L
λ x λ x − ai sinh i − sin i L L
The graphs of all 4 theoretical modes were plotted. The experimental values obtained on the positions of the nodes were included also for comparison. The comparisons are as follow:
Mode 2: The experimental position of the node is greater than the theoretical position slightly. Mode 3: The first node of experimental positions of the nodes is greater than the theoretical position while the 2 nd node was lesser. Mode 4: The 1 st node occurred greater than the theoretical position while the other 2 were fairly close to the theoretical position.
In all the modes recorded, the experimental occurred fairly close to the theoretical position with a slight margin of error due to certain errors in the experiment. This could be inaccuracy in finding the node as it occurred over a range where the centre had to be found.
2. Frequency of excitement Comparing both the experimental data from the CRO, Stroboscope and the theoretical data calculated, the experimental values tend to be smaller than the theoretical one. The mass of the accelerometer played a part in the deviation of the results. The end effect is a lowered natural frequency as a result. The mode shape also changes as a result but this only affects the amplitude and not the natural frequency.
The beam is also assumed to be smooth in the experiment. In reality, there are tiny vibrations along the beam due to the fact it is not a smooth beam.
DISCUSSION 1. Significance of Resonant Frequencies
The resonant frequency of an object is the frequency at which it will vibrate with increasing amplitude which theoretically will reach infinity. However, this amplitude is limited to a large number due to damping of the system by the surroundings. At this resonant frequency, resonance occurs and dangerously large oscillations may result.
For a system vibrating at it resonant frequencies, much damage can be caused due to the large amplitude at which it vibrates violently. Many engineering disasters have occurred due to the structure vibrating at its resonant frequency which eventually led to catastrophic failure. Because of this, it is important to measure the natural frequency of an object before constructing it so methods can be employed to damp the oscillations at the peaks.
In the experiment, there are infinite resonant frequencies at which the ruler will vibrate at. However, we are only interested in the first 4 resonant frequencies. This is because the peaks decrease as the frequencies increase, thus making the 1 st resonant frequency the most dangerous. After the 2 nd resonant frequency, the peaks became much lower and this would mean that the potential damage it can cause is much lesser.
2. Significance of mode shapes
Looking at the mode shapes from the graphs, we are able to see that there are points where the amplitude is 0. These points are known as nodes. Also the distance between two similar points on the same graph can be used to calculate the time for each period and thus the resonant frequency can be found.
To reduce the vibrations of a system, damping methods can be employed at the points of maximum amplitude on the mode shape. Supports can be built at the nodes points to prevent them from getting damaged due to the vibrations.
Sensitive equipment can also be place at the nodes in order to eliminate any vibrations which may cause such expensive equipment to get damaged or lose accuracy.
3. Effect of the accelerometer’s mass on these quantities.
The accelerometer’s mass will have an effect on the natural frequency of the beam. Because of the additional weight, the natural frequency measured occurred at a slightly lower frequency as a result. In actual fact, the natural frequency measured is for the beam and accelerometer combined and not just the beam alone which would be that of the theoretical value derived. The mass of the accelerator also affects the mode shape, the overall shape is the same but the amplitude of the vibrations is affected.
CONCLUSION 1. A better understanding of resonant frequencies and how we can make use of them to our advantage was achieved. The principles behind how we can use the resonant frequency to improve designs as well as prevent disasters became clear. 2. The mode shapes of vibrations gave a better understanding of how we can use that knowledge to place damping devices as well as sensitive equipment at the right locations for the best effect. 3. We got a better understanding of techniques used to measure dynamic quantities.
