June 24, 2017
June 24, 2017
June 28, 2017
A topic routinely covered in an elementary physics or (to a greater extent) mechanical vibrations course is the case of an object exhibiting small-amplitude oscillations with no significant energy dissipation (at least over many cycles of oscillation). This situation closely approximates simple harmonic motion, and it enables a gentler introduction to the study of vibratory systems. Since no appreciable damping mechanisms are present, only the concepts/characteristics of amplitude, period, and frequency are considered.
An experiment has been devised for a mechanical engineering course (taught numerous times by the author) which is utilized to demonstrate a vibratory system. The object considered consists of a stable aluminum half disk that rocks/rolls back and forth when given an initial displacement from its equilibrium position and released from an initial state of rest (see reference diagrams). Prior to performing this experiment, the students enrolled in this course are initially led through a derivation of the governing equation of motion (an ordinary non-linear differential equation) for this system. However, because the derivation is based upon an energy (i.e., Rayleigh’s) method, the essential parameter characterizing the vibration can be determined from this equation without the need to formally solve the differential equation.
This derivation also provides an excellent opportunity for students to review and reinforce their understanding of, as well as proficiency in calculating, such kinetic properties as the mass center and the mass-moment of inertia for a nontrivial geometric solid object. The students must utilize these results directly within the equations involved in order to obtain the theoretical formula for the undamped natural frequency omega-n of the rocking/rolling half disk:
[Equation for Omega-n]
From this formula, the theoretical period can be evaluated as well, and the experiment essentially consists of comparing the actual period with the theoretical period over a time interval of about ten oscillations for the half disk. The effect of the initial angular displacement for the half disk (larger values of which produce increasingly noticeable non-linear behavior) is also investigated.
The experiment itself can be conveniently conducted in a classroom with minimal apparatus: the aluminum half disk, a dry clean table surface, a chronometer, and a printable/erectable reference protractor that facilitates the release of the half disk from several predetermined initial angular displacements. It is anticipated that this experiment can (or will) be briefly demonstrated during the formal presentation of this article.
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