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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Radelet-de Grave, Patricia;
Speiser, David;
Chemla, Karine;
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Old and new approaches to Euler's trigonometric expansions
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Re-examining Dai Zhen's Gougu geyuan ji in Terms of Construction and Mathematical Principles
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Women and Mathematics in the Time of Euler
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Coates, John;
(2008)
Euler's Work on Zeta and L-Functions and Their Special Values
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Ferraro, Giovanni;
(2004)
Differentials and Differential Coefficients in the Eulerian Foundations of the Calculus
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Bullynck, Maarten;
(2010)
Factor Tables 1657--1817, with Notes on the Birth of Number Theory
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Brummelen, Glen Van;
Kinyon, Michael;
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Mathematics and the Historian's Craft: The Kenneth O. May Lectures
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Sandifer, C. Edward;
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The Early Mathematics of Leonhard Euler
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E. Chassefière;
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John D Bullock;
Ronald E Warwar;
H Bradford Hawley;
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Why was Leonhard Euler blind?
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Heeffer, Albrecht;
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The Genesis of the Algebra Textbook: From Pacioli to Euler
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Ut Pictura Poesis: Drawing into Space
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Stén, Johan C. E.;
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A Comet of the Enlightenment: Anders Johan Lexell's Life and Discoveries
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Consideracions al voltant de la Funció Beta a l'obra de Leonhard Euler (1707--1783)
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