Article ID: CBB271814830

Euler first theory of resonance (2022)

unapi

We examine a publication by Euler, De novo genere oscillationum, written in 1739 and published in 1750, in which he derived for the first time, the differential equation of the (undamped) simple harmonic oscillator under harmonic excitation, namely the motion of an object acted on by two forces, one proportional to the distance traveled, the other varying sinusoidally with time. He then developed a general solution, using two different methods of integration, making extensive use of direct and inverse sine and cosine functions. After much manipulation of the resulting equations, he proceeded to an analysis of the periodicity of the solutions by varying the relation between two parameters, $$a$$and $$b$$, eventually identifying the phenomenon of resonance in the case where $$2b=a$$. This is shown to be nothing more than the equality between the driving frequency and the natural frequency of the oscillator, which, indeed, characterizes the phenomenon of resonance. Graphical representations of the behavior of the oscillator for different relations between these parameters are given. Despite having been a brilliant discovery, Euler’s publication was not influential and has been neglected by scholars and by specialized publications alike.

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Authors & Contributors
Chen, Jiang-Ping Jeff
H Bradford Hawley
Warwar, Ronald E.
Bullock, John D.
Unger, J. Marshall
Éric Chassefière
Concepts
Mathematics
Trigonometry
Number theory; number concept
Calculus
Functions (mathematics)
Data collection; methods
Time Periods
18th century
19th century
17th century
Renaissance
21st century
Places
Russia
China
Japan
Germany
Finland
Europe
Institutions
St. Petersburg Academy of Sciences
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