Laboratory Report: Changes in Natural Frequency

Abstract

Experiment AM2.1, "Free and Forced Vibrations," aimed to enhance the understanding of the relationship between natural frequency and added mass of a Rigid Beam and Spring and a Simply Supported Beam. Dunkerley's Theory was also employed to calculate the natural frequency of the Simply Supported Beam without added mass. The TM1016 and TecQuipment's Versatile Data Acquisition System (VDAS) were utilized to carry out this experiment. In the first part, loads ranging from 400 g to 2.2 kg were added onto an exciter attached to a Rigid Beam and Spring System, and time period readings were recorded using the VDAS.

The natural frequencies were then calculated and compared to theoretical values. In the second part, masses were added to a Simply Supported Beam System, and the same procedure was followed. Dunkerley's theory was applied to determine the natural frequency of the beam without any added mass. The results for natural frequencies were in line with expectations, with the Rigid Beam and Spring System producing relatively small errors, and the Simply Supported Beam yielding minor errors.

Both sets of errors can be attributed to a combination of systematic and human errors. However, a significant error was observed when comparing the natural frequency of the Simply Supported Beam to the theoretical value, likely due to certain assumptions made during the experiment.

Introduction

Natural frequency is the frequency at which a body or system vibrates when released from an initial displacement from its equilibrium position, without the influence of repeated external forces [1]. This natural frequency varies with changes in the mass of the system [2].

Understanding the natural frequencies of a system is crucial, especially when other moving or vibrating components are present within the system. If these components start vibrating at the same frequency as the system's natural frequency, resonance can occur [3]. Resonance leads to significant oscillations with larger amplitudes, which can have disastrous consequences in structures such as buildings and bridges [4].

This experiment consists of two parts, both aimed at determining the effect of added mass on natural frequency but on two different systems. The first system involves a Rigid Beam and Spring. In this part of the experiment, it is assumed that the beam remains "rigid" without bending, and the exciter is treated as a single point mass at a specific distance along the beam [5]. Natural frequencies are then analyzed based on these assumptions. The second system is a Simply Supported Beam, where the beam is supported at both ends, and the center of the beam oscillates. In this part, it is assumed that the mass of the beam is negligible, and the exciter is considered as a single point mass, allowing the application of Rayleigh's theory to calculate the natural frequency [5]. The results from this second part are used to validate Dunkerley's theory, which states that the addition of the reciprocals of individual elements' angular velocity of a whirling shaft equates to the reciprocal total angular velocity of the shaft. This theory can also be applied to calculate the natural frequencies of a beam, as both systems follow the laws of simple harmonic motion [5].

Aim & Objectives

The aim of this laboratory experiment is to enhance the fundamental understanding of free vibrations and the impact of added mass on a system, along with the associated calculations. The objectives of this laboratory experiment are as follows:

1. Learn how to set up and safely use the apparatus by following the laboratory instructions.
2. Investigate the influence of added mass on the natural frequency of a Rigid Beam and Spring and a Simply Supported Beam.
3. Analyze the results by plotting graphs and identifying potential sources of errors.
4. Compare and understand the disparities between experimental values and theoretical values, and present them in tables and graphs.

Methods

The main method for both parts of the experiment was identical, with adjustments made for the Rigid Beam and Spring setup and the Simply Supported Beam setup.

Rigid Beam and Spring Setup

The "Free and Forced Vibrations TM1016" apparatus was used to conduct the AM2.1 Free and Forced Vibrations experiment.

The apparatus was used in conjunction with a Versatile Data Acquisition System (VDAS) configured with Channel 1 Displacement (y-axis) set to 2 millimeters and Timebase (x-axis) set to 50 milliseconds. The SET ZERO controls were employed to ensure that the VDAS trace was centered on the displayed chart. Initially, there was no added mass, and the VDAS was started. The top of the exciter was struck with a flat hand, causing the beam to oscillate with an initial amplitude of a few millimeters. After a few seconds, the VDAS was stopped, and the resulting waveform, as shown in Figure 3, was displayed on the screen.

The difference in the x-coordinates on the graph within VDAS was then determined, and Equation 1 was used to calculate the time period of the oscillation.

₍₁₎
^{T = 2π}⁄_Δx

Equation 2 was subsequently employed to calculate the natural frequency, and this value was recorded.

₍₂₎
^{f = 1}⁄_T

The mass holder, weighing 200 g, was then added to the system. The "Added Mass" field in VDAS was updated, and the procedure was repeated to calculate the new natural frequency, which was recorded. Subsequently, five 400 g masses were individually added to the mass holder, updating the "Added Mass" field each time, and the natural frequencies were calculated and recorded after each addition.

Simply Supported Beam Setup

For the Simply Supported Beam setup, the following values were relevant to the system:

The total mass moment of inertia of the system could be calculated using Equations 3 and 4, as shown below:

₍₃₎
^{ΣI = I_b + I_e}

₍₄₎
^{I_e = m_e * L_e2}

Using Equation 5, the theoretical natural frequency was then calculated.

₍₅₎
^{f_theory = 1}⁄_2π * √_ΣI⁄_k

A graph of natural frequency against added mass was plotted, and the theoretical and experimental values were compared and analyzed by calculating the percentage errors using Equation 6.

₍₆₎
^{Percentage Error (%) = |(f_theory - f_exp)| * 100}⁄_ftheory

Simply Supported Beam

For the Simply Supported Beam setup, the following values were relevant to the system:

Using Rayleigh's theory that provides a corrected beam mass, the effective mass was calculated using Equation 7 below.

₍₇₎
^{m_eff = m_c + m_e}

Since the exciter effectively acted as a clamp, the beam could be treated as two cantilevers. This allowed the use of Equation 8 to calculate the theoretical natural frequencies.

₍₈₎
^{f_theory = 1}⁄_2π * √_meff⁄_k

After calculating the theoretical natural frequencies, Equation 9 was applied.

₍₉₎
^{f_beam = 2 * f_theory - f_exciter}

Once these values were computed, a graph of f_beam against total exciter mass was plotted. The trend line was extended until it crossed the vertical axis, and this value was used to calculate the theoretical natural frequency of the beam itself.

Equation 10 was then employed to calculate the natural frequency of the beam from the theory.

₍₁₀₎
^{f_beam = 1}⁄_2π * √_mbeam⁄_k

The two values were subsequently compared and analyzed using Equation 6 above, and the values of f_beam were further analyzed through regression analysis using Equation 11.

₍₁₁₎
^{[1/f2 = _a·x + _b]}

Results

Rigid Beam and Spring

The results obtained in the experiment with the Rigid Beam and Spring are presented in Table 1 below.

Table 1: Results for Natural Frequency with Added Mass for a Rigid Beam and Spring

Added Mass (kg)	Total Exciter Mass (kg)	Iexciter	Itotal	Natural Frequency f (Hz)	Experimental	Theoretical
0.00	4.20	0.67	1.16	6.45	6.83
0.20	4.40	0.70	1.19	6.25	6.74
0.60	4.80	0.77	1.26	6.06	6.57
1.00	5.20	0.83	1.32	5.97	6.40
1.40	5.60	0.90	1.38	5.88	6.25
1.80	6.00	0.96	1.45	5.80	6.11
2.20	6.40	1.02	1.51	5.63	5.98

Table 1 illustrates the variations in the natural frequency of the beam as mass is added to the exciter. It is evident that as the total exciter mass increases, the natural frequency of the beam decreases. Additionally, the results in Table 1 align with the same trend observed in the theoretical values for the natural frequency.

Error Analysis

Percentage errors were calculated using Equation 6, and the results are presented in Table 2 below.

Table 2: Percentage Errors for Rigid Beam and Spring

Natural Frequency f (Hz)	% Error	Experimental	Theoretical
6.45	-5.53%	6.83
6.25	-7.27%	6.74
6.06	-7.75%	6.57
5.97	-6.72%	6.40
5.88	-5.89%	6.25
5.80	-5.12%	6.11
5.63	-5.79%	5.98

Table 2 displays the percentage errors for the Rigid Beam and Spring experiment. These errors range from -5.12% to -7.75%. While these values may seem relatively large, it is important to note that they fall within the range of less than 10%. As a result, these errors can be considered reliable for the given experiment.

Simply Supported Beam

Table 3 below presents the results obtained in the experiment using the simply supported beam.

Table 3: Results for Natural Frequency with Added Mass for Simply Supported Beam

Added Mass (kg)	Total Exciter Mass (kg)	Effective Mass (kg)	Natural Frequency f (Hz) (x10-3)	Experimental	Theoretical
0.00	4.20	5.00	15.15	15.50	4.36
0.20	4.40	5.20	14.70	15.19	4.63
0.60	4.80	5.60	14.50	14.64	4.76
1.00	5.20	6.00	13.90	14.15	5.18
1.40	5.60	6.40	13.50	13.70	5.49
1.80	6.00	6.80	13.20	13.29	5.74
2.20	6.40	7.20	12.80	12.91	6.10

Table 3 clearly demonstrates a change in natural frequency as mass is added to the simply supported beam system. It highlights the relationship between added mass and natural frequency, with the experimental values closely following the same trend as the theoretical values.

Results for the Natural Frequency of the Beam

Table 4: Results for the Natural Frequency of the Beam

x10-3	fbeam (Hz)	% Error	Experimental (2 Cantilevers)	Calculated
1.14	29.6	-22.92%	38.4

Table 4 provides the values calculated from Graph 3 and Equation 10. There is an 8.8 Hz difference between the two values. This discrepancy is attributed to the variations in the results presented in Table 3 due to experimental errors. Additionally, it should be noted that the experimental f_beam represents the frequency of two cantilevers rather than a single solid beam.

Error and Statistical Analysis

Table 5 below displays the percentage errors calculated using Equation 6.

Table 5: Percentage Errors for Simply Supported Beam

Natural Frequency f (Hz)	% Error	Experimental	Theoretical
15.15	-2.26%	15.50
14.70	-3.23%	15.19
14.50	-0.96%	14.64
13.90	-1.77%	14.15
13.50	-1.46%	13.70
13.20	-0.68%	13.29
12.80	-0.85%	12.91

Table 5 demonstrates the percentage errors for the Simply Supported Beam experiment. These errors are considerably smaller than those observed in the first part of the experiment. Given that their magnitude is below 5%, these results can be considered highly reliable.

Regression Analysis of Graph 3

Table 6: Regression Analysis of Graph 3

Total Exciter Mass (kg)	Experimental values
4.20	4.36	-0.820	0.672	4.384	-0.796	0.633
4.40	4.63	-0.550	0.303	4.539	-0.641	0.411
4.80	4.76	-0.420	0.176	4.847	-0.333	0.111
5.20	5.18	0.000	0.000	5.156	-0.024	0.001
5.60	5.49	0.310	0.096	5.465	0.285	0.081
6.00	5.74	0.560	0.314	5.773	0.593	0.352
6.40	6.10	0.920	0.846	6.082	0.902	0.814
Mean:	5.18	Total:	2.407	Total:	2.403	R2 Value:	0.998

Table 6 presents the results of the regression analysis conducted on Graph 3. The R² value of 0.998 indicates a very strong positive linear correlation among the experimental values, highlighting the reliability and consistency of the data.

Discussion

In the experiment with the Rigid Beam and Spring, the experimental results exhibited variations from the theoretical values, with the largest percentage error being 7.75%. This error might have arisen due to the unfamiliarity of the apparatus among the users, as they may have needed more time to understand and familiarize themselves with it. Human error could also have been introduced during the quick process of starting and stopping the VDAS. Additionally, reading the change in the x-axis coordinates for calculations could have introduced some measurement error. To reduce errors of this magnitude, it is advisable to repeat the experiment, taking multiple readings and calculating an average. In the Simply Supported Beam experiment, the percentage errors were much smaller, with the largest being 3.23%. Since these errors are below 5%, these results can be considered reliable. Nonetheless, errors may still have occurred due to the factors mentioned above. Another potential source of error in both cases is the assumption that damping due to air resistance is negligible. If the VDAS was not stopped quickly enough, it could have influenced the readings.

In both systems, there is a negative relationship between Natural Frequency and Added Mass, although the specific relationship is not evident in the tables. A clearer relationship might become visible on a graph with more readings taken with higher added masses. However, by examining the equations (Equations 3, 5, and 8), it is clear that the relationship is proportional to the square root of the total exciter mass (i.e., ^{√total exciter mass}). Upon closer examination of the graphs, a slight curve in the relationship can also be observed.

When the values of ^{√total exciter mass} were plotted against Total Exciter Mass, a clear positive linear correlation between the two became evident. Regression analysis yielded an R² value of 0.998, indicating a very strong linear pattern in the experimental values, likely due to the increase in Total Exciter Mass. The beam's natural frequency was determined using Dunkerley's Method. Although this method is typically used for whirling shafts, it can also be applied to vibrating beams as they both obey the laws of simple harmonic motion [5]. The value of 1.14x10^-3 was read from the graph, and the natural frequency of the entire beam was calculated as 29.6Hz. When compared to the value of 38.4Hz calculated using Equation 10, this resulted in a percentage error of -22.92%. This substantial error can be attributed to the frequency obtained from the graph representing the natural frequency of two cantilevers rather than a single entire beam. Additionally, errors in the experimental values from the experiment, as seen in Table 5, also contributed to this error.

Conclusion

In conclusion, the experimental laboratory has revealed a clear negative relationship between natural frequency and added mass, which follows the relationship ^{√total exciter mass}. The natural frequency of the Simply Supported Beam in the experiment was found to be 29.6 Hz.

Appendix

• By substituting the reading of 3.1 from the VDAS into Equation 1, a value of 0.155 was obtained.
• By substituting the time period into Equation 2, the result is a natural frequency of 6.45 Hz, as shown in the first result in Table 1.
• Substituting the values of the total exciter mass (Table 1) and the exciter length into Equation 3, the moment of inertia of the exciter was calculated as 0.67 kg.m², as shown in Table 1.
• Adding the moment of inertia of the beam, exciter, and spring, as shown in Equation 4, resulted in a total moment of inertia of 1.16 kg.m².
• The theoretical natural frequency for the Rigid Beam and Spring was calculated by substituting the spring constant, the length of the spring, and the total moment of inertia into Equation 5, yielding a value of 6.83 Hz, as shown in Table 1.
• The percentage errors were calculated by substituting the values for the experimental natural frequency and the theoretical natural frequency (Tables 1 and 3) into Equation 6, resulting in a percentage of -5.53% for the first set of readings (Table 1).
• Substituting the value for the mass of the exciter and the corrected mass of the beam into Equation 7 resulted in the effective mass of 5.00 kg when the added mass was 0 kg (Table 3).
• By substituting the values for 6EI_beam, the effective mass, and l³ into Equation 8, a theoretical frequency of 15.50 Hz was obtained (Table 3).
• By substituting the value for the experimental frequency from Table 3 into Equation 9, a value of 4.36 for the first set of readings was obtained.
• The theoretical natural frequency of the beam was determined by substituting the values for EI_beam, the mass of the beam, and l³ into Equation 10, resulting in a value of 38.4 Hz.
• By substituting the values for expected values, mean value, and the experimental values of frequency (Table 6) into Equation 11, a value of 0.998 was obtained for the regression analysis.

References

1. R. C. Hibbeler, Engineering Mechanics Dynamics, 14th Edition, England, Pearson Education Limited, 2016, p. 646.
2. Newport.com, 2019, Newport.com, Online, 2 January 2019, Available from: https://www.newport.com/t/fundamentals-of-vibration
3. Halliday & Resnick, Fundamentals of Physics, 9th Edition, New Jersey, Jearl Walker, 2011, p. 402.
4. Schaeffler, Technical Pocket Guide, 1st Edition, Germany, Schaeffler Technologies, 2014, p. 157.
5. University of Birmingham, Lecture Notes for Year 2 Labs, University of Birmingham, Technical Details for AM2.1: Free and Forced Vibrations.

Experimental and Theoretical Natural Frequencies Against Added Mass for a Rigid Beam and Spring

Table 7: Experimental and Theoretical Natural Frequencies Against Added Mass for a Rigid Beam and Spring

Added Mass (kg)	Natural Frequency (Hz) Experimental	Natural Frequency (Hz) Theoretical
0	6.452	6.8319
0.2	6.25	6.7396
0.6	6.061	6.5657
1	5.97	6.4045
1.4	5.882	6.2547
1.8	5.797	6.1149
2.2	5.634	5.9841

Experimental and Theoretical Natural Frequencies Against Added Mass for a Simply Supported Beam

Table 8: Experimental and Theoretical Natural Frequencies Against Added Mass for a Simply Supported Beam

Added Mass (kg)	Natural Frequency (Hz) Experimental	Natural Frequency (Hz) Theoretical
0	15.15	15.4952
0.2	14.7	15.1943
0.6	14.5	14.6418
1	13.9	14.1454
1.4	13.5	13.6963
1.8	13.2	13.2875
2.2	12.8	12.9132

Dunkerley's Theory - 1/f^2 Against Total Exciter Mass

Table 9: Dunkerley's Theory - 1/f^2 Against Total Exciter Mass

Total Exciter Mass (kg)	1/f^2 x10^-3
4.2	4.3569
4.4	4.6277
4.8	4.7562
5.2	5.1757
5.6	5.4870
6	5.7392
6.4	6.1035