Flexural vibrations of a thin clamped steel bar

Abstract

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.

The mechanics of bending a strip of paper

“…One way is to investigate the heavy elastica bending, when the strip straight orientation is horizontally aligned, for doubly-clamped or doubly-hinged boundary conditions as these may present a spontaneous break in symmetry. Another possibility is the investigation of the effects of mechanical noise on the displacement fluctuations of the strip near its tip, such as on the pre-buckling standing column configuration using a laser CCD sensor [28]. Finally, we note that the sequence of hinged-clamped profiles shown here could be used in designing a new method of soft-robot locomotion [29].…”

Stochastic resonance and amplification in the ac driven Duffing oscillator with added noise

“…Using the first normal-mode solution, Eq. (B4), we find that the effective mass for the fundamental normal mode at height z [33] is…”

Impact of a ball with a flexible beam

“…The equation of motion for bending waves on a uniform, flexible beam of density ρ, crosssectional area A and Young's modulus E is given in its simplest form by [3][4][5][6][7][8][9][10][11][12] ρA…”

“…For a rectangular cross-section beam of thickness a and width b, I = ba 3 /12, assuming that the beam bends in the thickness direction. Equation ( 12) has been solved many times in the past with F 0 = 0 to determine the vibration frequencies and shapes of a beam [6][7][8][9][10][11][12], but less frequently with a forcing term, F 0 . Nevertheless, it is relatively straightforward to find numerical solutions by dividing the beam into many equal mass, equal length segments, with an impact force applied to one or more segments.…”

“…The experiment could be conducted in an undergraduate laboratory and could be analysed either with a rigid beam or a flexible beam model. The vibration frequencies of a flexible beam have been analysed previously by many authors [6][7][8][9][10][11][12], so the additional effort of calculating the rebound speed of an incident ball is not beyond undergraduate students.…”