Abstract: Dynamic Analysis Of Structural Members Can Be An

Abstractthe Dynamic Analysis Of Structural Members Can Be An Intensive

The dynamic analysis of structural members can be an intensive process heavily dependent on the analysis of first and second-order systems. The primary objectives of frequency analysis for structural members is determining the resonant and damping frequencies, as well as the damping coefficients of the system under different conditions. In this experiment a cantilever beam was excited in such a way that the resultant vibration could be analyzed using signal analyses including as FFT. The damping coefficients, damping frequencies, and natural frequencies of two systems with unique weights acting as dampeners were determined through experimentation and subsequent calculation.

Introduction Differential equations can be used to solve many natural problems that happen in the world. One of the real life problems that it helps with is with a cantilever beam. There will be a beam that will be set up with a Wheatstone bridge to try to find the natural frequency of the beam. This is important in terms of vibration analysis. From here there will be different masses placed upon it to see the effects of damping on the system. With the Wheatstone bridge, the damping and natural frequencies, as well as the damping coefficients, can be determined through experimentation and subsequent calculations.

This damping coefficient will change depending on the different type mass that was added to the cantilever beam. Theory Using a Wheatstone bridge connected to strain gauges and a data acquisition system one can find out the different natural and damping frequencies of a cantilever beam with different dampeners. The Wheatstone bridge circuit is utilized by using the circuit to record the changes in material that is being tested. Since they are very sensitive it works by measuring the local strain in the beam using a strain gauge. This will then be measured and recorded into a frequency produced by the beam once the mass and beam are “excited." From here one can use the recorded data to find the frequency and damping coefficient of the harmonic reaction.

The function can be used to find the log decrement. The ln(y) is from the first peak as the 2nd ln(y) is from a certain number n of peaks away. This then is used to find the damping coefficient through the equation . Once our damping coefficient is found we can therefore find the natural frequency of the system through . The damping frequency can be found by creating an Amplitude-Response plot. Solving for provides the natural frequency of the system.

Apparatus and Procedure Refer to instructions titled “ME 335 Lab Experiment #3 – Static and Dynamic Response of a Cantilever Beam Scale” in the lab manual, “ME-335 Handout,” pgs 48-50. Results Table 3.1 - Summary of results for 105 g weight 105 g Weight Results δ Damping Coefficient, C Ringing frequency, ωR Natural frequency, ωn 0...677 Hz 14.677 Hz Table 3.2 - Summary of results for 457 g weight 457 g Weight Results δ Damping Coefficient, C Ringing frequency, ωR Natural frequency, ωn 0...75 Hz 6.75 Hz 13.50 Hz

Paper For Above instruction

The dynamic analysis of structural members, particularly cantilever beams, is an essential aspect of structural engineering and vibration analysis. This study focuses on experimentally determining the natural and damping frequencies, along with damping coefficients, for a cantilever beam subjected to varying loads. Understanding these dynamic properties is critical for predicting the response of structures under dynamic loads, which can influence design choices and safety measures.

Introduction and Theoretical Background

Structural members exhibit complex vibrational behaviors that can be modeled using differential equations. These equations describe the response of beam systems to external excitations, accounting for their stiffness, mass distribution, and damping characteristics. A cantilever beam, fixed at one end and free at the other, serves as an ideal model for studying vibrational properties because of its relevance in construction and mechanical systems. Identifying the natural frequency (ωn), damping frequency (ωR), and damping coefficient (ξ) is essential for characterizing how the structure responds to dynamic forces.

The application of experimental methods, such as using strain gauges connected via a Wheatstone bridge and signal processing tools like the Fast Fourier Transform (FFT), provides practical means for analyzing vibrations. The Wheatstone bridge circuit is used to detect minute strain variations in the beam, which are then converted into voltage signals. These signals provide data on the vibrational response when the beam is excited with different masses. By analyzing the frequency components of the recorded vibrations through FFT, the fundamental frequencies, damping effects, and decay rates can be accurately determined.

Methodology and Experimental Setup

The experimental setup involves attaching strain gauges to the cantilever beam, which are connected to a Wheatstone bridge configuration to enhance sensitivity. The beam is excited mechanically, often using a controlled impulse or harmonic excitation, to induce vibrations. Different masses, such as 105 g and 457 g weights, are placed at the free end of the beam to alter the vibrational dynamics. The system's response is recorded, focusing on the amplitude decay and frequency content of the vibrations.

Data collection involves measuring the voltage variations across the strain gauges, which correlates directly to strain in the beam. These signals are digitized and analyzed using FFT to extract the vibrational frequencies. The damping coefficient (C) can be derived from the logarithmic decrement, calculated from successive peak amplitudes, using the relation:

λ = \frac{1}{n} \ln \frac{A_0}{A_n}

where λ is the logarithmic decrement, A₀ and Aₙ are successive peak amplitudes separated by n cycles. The damping coefficient is then calculated based on the relation between the logarithmic decrement and damping ratio.

Results and Analysis

The experimental results reveal that the natural frequency decreases as the added mass increases, which aligns with classical vibration theory. For instance, the 105 g weight produced a damping frequency (ωR) of approximately 14.677 Hz, while the 457 g weight resulted in a damping frequency of around 6.75 Hz. These frequencies were obtained through FFT analysis of the recorded vibrations. The damping coefficients, which quantify energy dissipation in the system, increased with increasing mass, with values of approximately 0.0084 for the 105 g weight and 0.01455 for the 457 g weight.

This increase in damping coefficient with added mass may indicate enhanced energy absorption or internal friction due to the additional mass. The damping coefficients, natural frequencies, and damping frequencies were consistent with theoretical predictions based on classical mechanics and damping models.

Discussion and Implications

The experiment demonstrates the effectiveness of using FFT and strain gauge measurements in vibration analysis of structural members. The decrease in natural and damping frequencies with increased mass is consistent with the inverse proportionality predicted by beam theory:

ψn = \frac{1}{2\pi} \sqrt{\frac{k}{m}}

where k is the stiffness and m is the mass. Higher mass results in lower frequencies, confirming the theoretical models. The damping coefficients' increase suggests that adding weight contributes to greater energy dissipation, potentially through internal damping mechanisms.

Errors and Limitations

Experimental uncertainties include calibration errors, instrument resolution limitations, and systematic errors during the measurement process. The discrepancy between calculated and measured weights could be attributed to calibration inaccuracies, which could influence the precision of frequency and damping values. Moreover, environmental factors and the inherent damping of the measurement devices can also impact results, emphasizing the need for careful experimental control.

Conclusion

This study successfully demonstrated how vibration analysis techniques, particularly FFT analysis combined with strain gauge measurements, can be employed to determine the dynamic properties of a cantilever beam. The experimental findings support classical theories of vibrations, showing that the natural frequency decreases with mass, and damping coefficients increase in relation to added dampers. These results underscore the importance of dynamic analysis in structural engineering, especially for the design and assessment of lightweight and flexible systems prone to vibrational issues.

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