Page 39 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

36 Chapter 1 Fundamentals oF Vibration
1811, only one candidate, Sophie Germain, had entered the contest. But Lagrange, who
was one of the judges, noticed an error in the derivation of her differential equation of
motion. The academy opened the competition again, with a new closing date of October
1813. Sophie Germain again entered the contest, presenting the correct form of the differ-
ential equation. However, the academy did not award the prize to her because the judges
wanted physical justification of the assumptions made in her derivation. The competition
was opened once more. In her third attempt, Sophie Germain was finally awarded the
prize in 1815, although the judges were not completely satisfied with her theory. In fact,
it was later found that her differential equation was correct, but the boundary conditions
were erroneous. The correct boundary conditions for the vibration of plates were given in
1850 by G. R. Kirchhoff (1824–1887).
In the meantime, the problem of vibration of a rectangular flexible membrane, which
is important for the understanding of the sound emitted by drums, was solved for the first
time by Simeon Poisson (1781–1840). The vibration of a circular membrane was studied
by R. F. A. Clebsch (1833–1872) in 1862. After this, vibration studies were done on a num-
ber of practical mechanical and structural systems. In 1877 Lord Baron Rayleigh published
his book on the theory of sound [1.9]; it is considered a classic on the subject of sound and
vibration even today. Notable among the many contributions of Rayleigh is the method of
finding the fundamental frequency of vibration of a conservative system by making use of
the principle of conservation of energy—now known as Rayleigh’s method. This method
proved to be a helpful technique for the solution of difficult vibration problems. An exten-
sion of the method, which can be used to find multiple natural frequencies, is known as the
Rayleigh–Ritz method.

1.2.3 In 1902 Frahm investigated the importance of torsional vibration study in the design of
recent the propeller shafts of steamships. The dynamic vibration absorber, which involves the
Contributions addition of a secondary spring-mass system to eliminate the vibrations of a main system,
was also proposed by Frahm in 1909. Among the modern contributers to the theory of
vibrations, the names of Stodola, De Laval, Timoshenko, and Mindlin are notable. Aurel
Stodola (1859–1943) contributed to the study of vibration of beams, plates, and mem-
branes. He developed a method for analyzing vibrating beams that is also applicable to tur-
bine blades. Noting that every major type of prime mover gives rise to vibration problems,
C. G. P. De Laval (1845–1913) presented a practical solution to the problem of vibration of
an unbalanced rotating disk. After noticing failures of steel shafts in high-speed turbines,
he used a bamboo fishing rod as a shaft to mount the rotor. He observed that this system
not only eliminated the vibration of the unbalanced rotor but also survived up to speeds as
high as 100,000 rpm [1.10].
Stephen Timoshenko (1878–1972), by considering the effects of rotary inertia and
shear deformation, presented an improved theory of vibration of beams, which has
become known as the Timoshenko or thick beam theory. A similar theory was presented
by R. D. Mindlin for the vibration analysis of thick plates by including the effects of rotary
inertia and shear deformation.