CADE10002 – Lab 4

Signals and Dynamics

ay24421

Abstract

According to the CADE10002 Lab Guidelines and Dynamics Lab Circuit Step-by-Step Building Guide from the University of Bristol, the experimental purpose, experimental equipment, experimental methods and calculation formulas were determined. The aim of this experiment is to  estimate Young's modulus of a metal ruler by using its natural vibration frequency. The experiment involved constructing a microphone circuit with a Raspberry Pi Pico to capture sound signals, processing these signals with Python to determine frequency, and applying a theoretical model to calculate material properties. Key findings indicate that the ruler's Young's modulus approximates that of steel, around 257.305 GPa, though limitations such as errors, noise and measurement precision introduce uncertainty. The experiment successfully demonstrates a low-cost method for material property estimation, applicable in resource-limited scenarios like disaster relief.

1. Introduction

The determination of material properties, such as Young's  modulus, is critical in engineering applications, particularly in scenarios where traditional testing facilities are unavailable, such as disaster relief efforts  requiring emergency repairs. This investigation explores a cost-effective method to estimate the Young's modulus of a metal ruler by measuring its natural frequency as a cantilever oscillator. The approach leverages basic electronics and programming skills to capture and analyze time-varying sound signals produced by the vibrating ruler. Drawing from concepts introduced in Session 17, including vibrational dynamics and signal processing, the experiment aims to calculate the Young's modulus using the fundamental frequency and assess its suitability as a substitute material, for instance, in an ambulance suspension repair. The specific objectives are to construct a functional microphone circuit, acquire and process sound signals accurately, and derive a reliable estimate of the material's Young's modulus.

2. Methods

As shown in figure 1, the microphone circuit was assembled on the breadboard. Key components included a condenser microphone (CMC-9745-44P), an LM741 op-amp, resistors, capacitors, and a Raspberry Pi Pico for data acquisition. The circuit amplified the microphone's output, with a high-pass filter (C1) removing DC offset and an op-amp providing gain, while Schottky diodes (1N5817) protected the Pico's ADC pin.

Figure 1: the microphone circuit

The ruler was clamped to a table edge using a hardback book, creating a cantilever with overhang lengths of 80 mm, 100 mm, and 120 mm, as shown in figure 2. The microphone was positioned 5 cm from the ruler’s free end. DataAcquisition.py was executed in Thonny to record sound signals at 1 kHz when the ruler was 'twanged' three times per length.

Figure 2: Experimental setup for 100 mm overhang

Signal data was exported as .csv files and analyzed in Jupyterlab. The python code extracted the larger amplitude portion of the ruler vibration signal to eliminate the effect of background noise to accurately analyze the frequency.

Implementation steps:

1.    Use  Jupyterlab  to  calculate  the  mean  by  reading the  initial  noise file  (0.csv)  Estimate the amplitude of the static portion of the signal as the noise level

2.    The processed data and the initial mean value were subtracted from the parameters of the steel rule vibration respectively to obtain the data with the transverse axis of 0

3.    Set 30% of the  maximum amplitude to the threshold, retaining only signal segments whose amplitude exceeds the threshold.

4.    Find the start and end points in the signal with significant amplitude and crop the signal to extract valid data.

5.    Plot images and estimate frequencies using Jupyterlab.

Then the basic frequency of the ruler was estimated by counting the number of times the signal crosses the "zero point". Finally, equation 1, equation 2 and deformation equation 3 were used to calculate Young's modulus of steel rule.

                       [1]

where f is the fundamental frequency in Hz, E is the Young’s Modulus, I is the second moment of area, q is the mass per unit length, L the free overhang length of the cantilever.

                                 [2]

where I is the second moment of area, b is the width and h is the thickness.

                             [3]

where E is the Young’s Modulus, f is the fundamental frequency in Hz, I is the second moment of area, q is the mass per unit length, L the free overhang length of the cantilever.

3. Results

Measurements of the ruler yields a width (b) of 0.0254 m and thickness (h) of 0.0007 m. According to equation 2, the second moment of area  I  = 7.26 × 10-13 m4 . According to Davis’s paper entitled Metals Handbook Desk Edition. ASM, the density of steel is 7850 kg/m^3. And due to equation 4, the mass per unit length  q = 0.1396kg/m

[4]

where q is the mass per unit length,P   is the density, b is the width and h is the thickness

According to Jupyterlab's calculations, the initial mean is 35437.4305 (0mm overhang).

When the overhang length of the steel rule is 80mm, the original vibration signal curve is shown in Figure 3.

Figure 3: 80mm original vibration signal

The adjusted vibration signal curve of the 80mm suspension is shown in Figure 4.

Figure 4: 80mm adjusted vibration signal

The cropped effective vibration signal curve of 80mm overhang is shown in Figure 5.

Figure 5: 80mm cropped effective vibration signal

By clipping and calculating the zero-intersection using python code, the final frequency is estimated to be 104.52 Hz. Finally, using formula 3, Young's modulus is 274.31 GPa.

When the overhang length of the steel rule is 100mm, the original vibration signal curve is shown in Figure 6.

Figure 6: 100mm original vibration signal

The adjusted vibration signal curve of the 100mm suspension is shown in Figure 7.

Figure 7: 100mm adjusted vibration signal

The cropped effective vibration signal curve of 100mm overhang is shown in Figure 8.

Figure 8: 100mm cropped effective vibration signal

By clipping and calculating the zero-intersection using python code, the final frequency is estimated to be 33.51Hz. Finally, using formula 3, Young's modulus is 68.82 GPa.

When the overhang length of the steel rule is 120mm, the original vibration signal curve is shown in Figure 9.

Figure 9: 120mm original vibration signal

The adjusted vibration signal curve of the 100mm suspension is shown in Figure 10.

Figure 10: Figure 7: 120mm adjusted vibration signal

The cropped effective vibration signal curve of 100mm overhang is shown in Figure 11.

Figure 11: 120mm cropped effective vibration signal

By clipping and calculating the zero-intersection using python code, the final frequency is estimated to be 43.48Hz. Finally, using formula 3, Young's modulus is 240.30 GPa.

4. Discussion

In the second experiment, when the steel rod is overhung at 100 mm, the estimated frequency and Young's modulus are much lower than normal. The possible cause is that the force applied by the  hand is not consistent with the other times, which may be caused by too little force. For a longer overhang length (100mm), the vibration of the steel ruler may also be doped with some other vibrations, which are frequencies of other modes, so that the calculated Young's modulus is low. It is also possible that the clamping conditions are unstable, and the clamping point between the steel rule and the book may slip or loosen slightly, affecting the accurate measurement of the frequency. It is also possible that when the signal is clipped, there is a large noise interference in   the signal, or the improper selection of the clipping threshold, resulting in inaccurate zero crossing detection, making the frequency estimation wrong. Therefore, this set of experimental data will be invalid.

Taking the data of 80mm and 120mm overhang, the average Young's modulus is 257.305GPa and the standard deviation is 17.005GPa. This average of 257.31 GPa is close to the typical Young's modulus of steel (190-210 GPa), but slightly higher, possibly due to measurement errors or differences in the material properties of the steel ruler. Measurement errors may include the difficulty in accurately controlling the magnitude and direction of the force when applying it manually; The steel rule and the hardcover book clamping place are not completely fixed, there will be slight loosening; The method of estimating the frequency through the zero crossing may not be  very accurate, the signal will mutate as if by interference, and the error is still significant after slight  removal of the mutation in the horizontal axis of the signal. For material properties, the steel ruler may not be pure steel, or there may be manufacturing inhomogeneity, resulting in the actual Young's modulus not exactly consistent with the theoretical value.

However, improvements can include the use of standard weights instead of manual force to ensure the same force and direction for each excitation vibration; The use of professional clamps instead of hardcover books to ensure that the steel ruler is completely fixed at the clamp, reducing the change of boundary conditions; The sampling accuracy can also be improved to improve the resolution and accuracy of the frequency measurement. Thus, more accurate measurement of Young's modulus of steel rule material characteristics can be achieved.

5. Conclusion

The experiment successfully uses the low-cost microphone circuit and Python signal processing technology to estimate the Young's modulus of the steel ruler by measuring its natural vibration frequency. After excluding the abnormal  100mm overhang data, the average Young's  modulus based on the 80mm and 120mm overhang results is 257.31 GPa, close to the typical value of steel (190-210 GPa). Despite some errors, this method demonstrates the feasibility of material property estimation in resource-limited environments such as disaster relief. In the future, the measurement accuracy can be further improved through standardized force application, professional fixation, higher sampling rate and more accurate signal processing technology, and the scheme can also be used to estimate Young's modulus of other materials.

References

University of Bristol, CADE10002 Lab Guidelines, 2024.

University of Bristol, Dynamics Lab Circuit Step-by-Step Building Guide, 2024.

Davis, J.R. Metals Handbook Desk Edition. ASM, 1998.