Group Members
Experiment No
Date
: 2A
:13/10/2024
Subject name
: Dynamics
Lecture
:
1
Contents
No table of contents entries found.
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1. Introduction
This practical is about providing insight into a system's natural frequency. Practical two
involves determining the disc-shaft and beam natural frequency. Theoretical methods
and experimental methods will be used in practical two. On disc-shaft practical, the
shaft is made from aluminum and hangs from a rigid support, and it is attached to the
disc at the end.
The natural frequency of the system will be determined whereby there will be angular
displacement applied to the disc thus twisting the shaft and when the disc is released
there will be oscillations in the system which will be counted while the timer is on. These
results will be compared to the calculated results.
On the beam practical there will be a force or input frequency applied to the beam
through a hammer and the vibrations induced in the beam will be recorded and
analyzed to perform calculations thus determining the natural frequency. The theoretical
and experimental results will be compared, and percentage errors will be determined.
2. Objective
To calculate the system’s natural frequency and evaluate the analytical results against
the experimental data.
3. Apparatus





Steel disc
Stop- watch
Aluminum shaft
Vernier caliper or micrometer
Measurement ruler
4. Theoretical Consideration
The continuous rotation of the aluminum shaft, caused by the disc’s angular movement,
could potentially influence the steel shaft’s diameter.
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5. Experimental Procedure
1. Experiment instructions
The one end of the shaft is fixed to the rigid support, as shown in Figure 1 below, while
the disc is attached using a setscrew on the opposite end of the wire.
Figure 1: Experiment Setup
4
Figure 2: Sectional view of the Disc
Figure 3: View of some of the apparatus
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The disc must be slightly rotated from its resting position by approximately 13 mm as
indicated in Figure 3 and 4, and then released to oscillate on its own for forty-five (45)
oscillations. The time taken to make 45 oscillations must be recorded. This must be
repeated. The average frequency must be used.
2. Theoretical Calculations
Formula’s Used
𝐺𝐼
𝜔
𝜔𝑛 = √ 𝐿𝐽𝑃 and 𝑓𝑛 = 2𝜋𝑛
Where,
𝜔�𝑛� – is the natural frequency of the system in rads/s.
𝐺� – shear modulus of shaft (take as 26.9 GPa – From Table A-5 Shingley’s Mechanical
Engineering Design).
𝑙� – is the length of the shaft.
𝐼�𝑝� – is the polar moment of inertia for the cross-section of the shaft.
𝐽� – is the polar mass moment of inertia of the disc.
𝑓𝑛 – is the natural frequency
Polar moment of inertia for the cross-section of the shaft
𝜋𝐷 4
𝜋(5×10−3 )4
𝐼𝑝 = 32 =�
32
= 61.359 × 10−12 𝑚4
Polar mass moment of inertia of the disc:
𝐽=
𝑚𝑅 2
3.314(0.08)2
�=
2
2
= 0.0106048�𝑘𝑔. 𝑚2
Natural Frequencies
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𝑓𝑛 for length 440mm:
26.9×109 (61.359×10−12 )
𝐺𝐼𝑝
𝜔𝑛 = √ 𝑙𝐽 = √
𝜔
𝑓𝑛 = 2𝜋𝑛 =
=18.808rad/s
0.44(0.0106048)
18.808
2𝜋
= 2.993𝐻𝑧��
𝑓𝑛 for length 880mm:
26.9×109 (61.359×10−12 )
𝐺𝐼𝑝
𝜔𝑛 = √ 𝑙𝐽 = √
𝜔
𝑓𝑛 = 2𝜋𝑛 =
13.299
2𝜋
=13.299rad/s
0.88(0.0106048)
= 2.117𝐻𝑧
Experimental Calculations
For both lengths, the duration for 45 oscillations was measured. Three trials were
conducted for each length, and the average result are to be used to calculate the
natural frequencies and periodic time.
Time average for length 440 mm:
𝑡𝑎𝑣𝑒 =
13.97 + 14.27 + 14.47
= 14.24�𝑠𝑒𝑐
3
Time average for length 880 mm:
𝑡𝑎𝑣𝑒 =
22.40 + 21.23 + 20.38
= 21.34�𝑠𝑒𝑐
3
Periodic time for length 440 mm and 880 mm:
𝑇𝑃 =
𝑇𝑃(440𝑚𝑚) =
𝑇𝑃(880𝑚𝑚) =
14.24
45
𝑇𝑖𝑚𝑒�𝑓𝑜𝑟�𝑜𝑠𝑐𝑖𝑙𝑙𝑎𝑡𝑖𝑜𝑛𝑠
𝑁𝑢𝑚𝑏𝑒𝑟�𝑜𝑓�𝑜𝑠𝑐𝑖𝑙𝑙𝑎𝑡𝑖𝑜𝑛𝑠
= 0.316 sec
21.34
45
�= 0.474 sec
Experimental Natural frequencies:
For 440 mm:
1
1
𝑓𝑛 = 𝑇 = 0.316 = 3.165𝐻𝑧
𝑃
For 880 mm:
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1
1
𝑓𝑛 = 𝑇 = 0.474 = 2.110𝐻𝑧
𝑃
3. Discussion
The experiment is based on determining the natural frequency of a disc attached to a
slender aluminium shaft using both analytical and experimental approaches. The close
agreement between the analytical and experimental values suggests that the theoretical
model effectively captured the key dynamics of the system. However, some important
factors should be considered when interpreting the results.
The accuracy of the analytical model, the analytical approach is mainly based on the
ideal assumptions, which include perfect material properties, perfect material condition
and a rigid support. Now based on reality, the materials often exhibit a slight variation in
density which affects the Young’s modulus and other properties. This might result in
minor errors in the calculated natural frequency.
We should also consider the sources of discrepancy which include the damping effects,
connection imperfections and the measurement uncertainty. These sources will affect
the analytical model of this experiment as we know that in real life there’s no process
that occurs in perfect conditions.
Boundary conditions, the assumption of a perfectly rigid support is crucial for accurate
analytical results. In practice, any slight flexibility in the support or the way the shaft is
mounted could influence the experimental frequency. This can alter the systems
behaviour, if the support allows for even a small amount of movement or damping.
In conclusion the experiment was successfully demonstrated, the relationship between
theoretical and experiment natural frequencies, with only minor differences observed.
These differences were likely due to teal-world factors like damping and connection
imperfections, which could be minimized with refined experimental techniques,
The small percentage of errors shows that the experiment was performed accurately.
The error for the 440 mm shaft could be due to minor deviations in setup, measurement
inaccuracies such as shaft diameter. And the improper counting of oscillations can have
an effect of percentage error.
The increase in the shaft length reduces the system's natural frequency. This could be
because the shaft is now more flexible, reducing the system's stiffness.
4. Conclusion
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In Practical 2A, we explored the identification of natural frequencies in a disc-shaft
system using both analytical and experimental approaches. The objective was to
determine the system's natural frequency and compare the results from theoretical
calculations with those obtained experimentally.
The analytical approach involved calculating the natural frequency based on the
material properties of the shaft (shear modulus, moment of inertia) and the mass of the
disc. This method required precise measurement of the shaft length, diameter, and disc
dimensions. Using Equation 4, we derived the natural frequency values.
On the other hand, the experimental method involved observing the system's response
when the disc was slightly displaced from its resting position. We recorded the time
taken for 45 oscillations and used this data to calculate the experimental frequency.
Equations used for the theoretical values are shown below.
𝐺𝐼
𝜔
𝜔𝑛 = √ 𝐿𝐽𝑃 and 𝑓𝑛 = 2𝜋𝑛
Comparing the two results showed some percentage error, which was likely due to a
combination of factors. Small variations in the experimental setup, such as friction at the
support or minor inaccuracies in timing the oscillations, contributed to the observed
discrepancy. Another potential source of error could be assumptions made during the
analytical calculations, where idealized conditions (such as no damping or perfect
rigidity of the support) might differ from the actual experimental environment.
Despite the differences, both methods provided reasonably close results, demonstrating
the effectiveness of the theoretical model in predicting natural frequencies. However,
the experiment highlighted the importance of careful measurement and observation to
minimize errors in real-world applications.
This practical reinforced the understanding of mechanical vibrations and the
significance of natural frequency in engineering systems. It was particularly insightful to
observe the twisting of the aluminum shaft and its impact on the disc's oscillatory
behavior. The ability to predict and measure natural frequencies is crucial in various
engineering applications, such as preventing resonance in structures and machinery.
This practical provided a hands-on demonstration of how theoretical concepts apply to
real-life systems, enhancing our appreciation of mechanical dynamics.
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10.Appendix
Question 1
Percentage error:
For length 440 mm
𝑇ℎ𝑒𝑜𝑟𝑖𝑡𝑖𝑐𝑎𝑙(𝑓𝑛 )−𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙(𝑓𝑛 )
%error=
𝑇ℎ𝑒𝑜𝑟𝑖𝑡𝑖𝑐𝑎𝑙(𝑓𝑛 )
× 100=
2.993−3.165
2.993
× 100 = −5.75%
For length 880 mm
𝑇ℎ𝑒𝑜𝑟𝑖𝑡𝑖𝑐𝑎𝑙(𝑓𝑛 )−𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙(𝑓𝑛 )
%error=
𝑇ℎ𝑒𝑜𝑟𝑖𝑡𝑖𝑐𝑎𝑙(𝑓𝑛 )
× 100=
2.117−2.110
2.117
× 100 = 0.33%
Completion of tables:
Wire
Disc
𝑑𝑚 = 5𝑚𝑚
𝐷 = 160𝑚𝑚
𝑙1 = 440𝑚𝑚
𝑚 = 3.314𝑘𝑔
𝑙2 = 880𝑚𝑚
ℎ = 19𝑚𝑚
Table: General natural frequencies for both lengths
General
𝐺 = 26.9𝐺𝑃𝑎
𝑓𝑛 = 2.993𝐻𝑧
𝑓𝑛 = 2.117𝐻𝑧
Length of shaft(m)
Time for 45
Periodic Time
oscillations(sec)
(𝑇𝑝 )(sec)
14.24
0.316
0.44
21.34
0.474
0.88
Table: Experimental time and periodic for 45 oscillations
𝑓𝑛 (𝑇ℎ𝑒𝑜𝑟𝑖𝑡𝑖𝑐𝑎𝑙)(𝐻𝑧) 𝑓𝑛 (𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙)(𝐻𝑧) Percentage error
(%)
𝑙1 = 440𝑚𝑚
2.993
3.165
−5.75
𝑙2 = 880𝑚𝑚
2.117
2.110
0.33
Table: Natural frequencies and Percentage error
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