Page 38 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.2 brieF history oF the study oF Vibration 35
infinite, the resulting frequencies were found to be the same as the harmonic frequencies
of the stretched string. The method of setting up the differential equation of the motion of a
string (called the wave equation), presented in most modern books on vibration theory, was
first developed by D’Alembert in his memoir published by the Berlin Academy in 1750.
The vibration of thin beams supported and clamped in different ways was first studied by
Euler in 1744 and Daniel Bernoulli in 1751. Their approach has become known as the
Euler-Bernoulli or thin beam theory.
Charles Coulomb did both theoretical and experimental studies in 1784 on the tor-
sional oscillations of a metal cylinder suspended by a wire (Fig. 1.4). By assuming that
the resisting torque of the twisted wire is proportional to the angle of twist, he derived the
equation of motion for the torsional vibration of the suspended cylinder. By integrating
the equation of motion, he found that the period of oscillation is independent of the angle
of twist.
There is an interesting story related to the development of the theory of vibration
of plates [1.8]. In 1802 the German scientist, E. F. F. Chladni (1756–1824) developed
the method of placing sand on a vibrating plate to find its mode shapes and observed
the beauty and intricacy of the modal patterns of the vibrating plates. In 1809 the
French Academy invited Chladni to give a demonstration of his experiments. Napoléon
Bonaparte, who attended the meeting, was very impressed and presented a sum of 3000
francs to the academy, to be awarded to the first person to give a satisfactory mathemati-
cal theory of the vibration of plates. By the closing date of the competition in October

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FiGure 1.4 Schematic diagram of
Coulomb’s device for torsional vibration tests.