{'fraction_non_alphanumeric': 0.015171503957783642, 'fraction_numerical': 0.0006596306068601583, 'mean_word_length': 4.97244094488189, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Flexural vibrations of a thin clamped steel bar

We present experimental results of damped and forced lateral vibrations on a 6″ steel ruler in which the displacement measurements were made by an optical sensor. The experimental apparatus is straightforward to assemble and the phenomenon of resonance is easily observable and measurable. We verified that the Newtonian dynamics of the harmonic oscillator with one degree-of-freedom is sufficient to describe the oscillations of the bar. We obtained precise theoretical adjustments for time series data of damped oscillations, from which we found the dissipation constant and the resonance frequency of the resonator. With these parameters, we were able to adjust the amplitude and phase curves of forced oscillations around the resonance. In addition, we show that this system can also be used to measure small ac forces at resonance and as a sensitive mass sensor. Furthermore, we verified that the shapes of the first and second normal modes of vibration of the bar follow closely to the predicted shapes of the Euler–Bernoulli beam theory (EBBT). Moreover, we also measured the Young’s modulus of the bar and verified that the resonant frequency of the first normal mode depends on the length of the clamped bar according to the prediction by this theory. Finally, we proposed a theoretical model for calculating the effective mass of normal modes of the bar based on the EBBT. Very good agreement with experimental results for the fundamental mode was obtained.