Page 37 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

34 Chapter 1 Fundamentals oF Vibration
experiments to find a relation between the pitch and frequency of vibration of a string.
However, it was Joseph Sauveur (1653–1716) who investigated these experiments thor-
oughly and coined the word “acoustics” for the science of sound [1.6]. Sauveur in France
and John Wallis (1616–1703) in England observed, independently, the phenomenon of
mode shapes, and they found that a vibrating stretched string can have no motion at certain
points and violent motion at intermediate points. Sauveur called the former points nodes
and the latter ones loops. It was found that such vibrations had higher frequencies than
that associated with the simple vibration of the string with no nodes. In fact, the higher
frequencies were found to be integral multiples of the frequency of simple vibration, and
Sauveur called the higher frequencies harmonics and the frequency of simple vibration
the fundamental frequency. Sauveur also found that a string can vibrate with several of
its harmonics present at the same time. In addition, he observed the phenomenon of beats
when two organ pipes of slightly different pitches are sounded together. In 1700, Sauveur
calculated, by a somewhat dubious method, the frequency of a stretched string from the
measured sag of its middle point.
Sir Isaac Newton (1642–1727) published his monumental work, Philosophiae
Naturalis Principia Mathematica, in 1686, describing the law of universal gravitation as
well as the three laws of motion and other discoveries. Newton’s second law of motion
is routinely used in modern books on vibrations to derive the equations of motion of
a vibrating body. The theoretical (dynamical) solution of the problem of the vibrating
string was found in 1713 by the English mathematician Brook Taylor (1685–1731), who
also presented the famous Taylor’s theorem on infinite series. The natural frequency of
vibration obtained from the equation of motion derived by Taylor agreed with the exper-
imental values observed by Galileo and Mersenne. The procedure adopted by Taylor
was perfected through the introduction of partial derivatives in the equations of motion
by Daniel Bernoulli (1700–1782), Jean D’Alembert (1717–1783), and Leonard Euler
(1707–1783).
The possibility of a string vibrating with several of its harmonics present at the
same time (with displacement of any point at any instant being equal to the algebraic
sum of displacements for each harmonic) was proved through the dynamic equations of
Daniel Bernoulli in his memoir, published by the Berlin Academy in 1755 [1.7]. This
characteristic was referred to as the principle of the coexistence of small oscillations,
which, in present-day terminology, is the principle of superposition. This principle was
proved to be most valuable in the development of the theory of vibrations and led to
the possibility of expressing any arbitrary function (i.e., any initial shape of the string)
using an infinite series of sines and cosines. Because of this implication, D’Alembert
and Euler doubted the validity of this principle. However, the validity of this type of
expansion was proved by J. B. J. Fourier (1768–1830) in his Analytical Theory of
Heat in 1822.
The analytical solution of the vibrating string was presented by Joseph Lagrange
(1736–1813) in his memoir published by the Turin Academy in 1759. In his study,
Lagrange assumed that the string was made up of a finite number of equally spaced identi-
cal mass particles, and he established the existence of a number of independent frequencies
equal to the number of mass particles. When the number of particles was allowed to be