1.2 Brief History of Vibrations
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a prize for the problem of deriving a mathematical theory of plate vibrations. In
response to it, Sophie Germain, a French lady proposed a solution, but was denied
the prize due to an incorrect computation of the variation of a particular integral.
The Academy proposed the subject again in 1813. Though the error previously
mentioned was rectified by now, but Germain failed to give a satisfactory explanation
of an assumption. So the prize again eluded her. When the Academy proposed the
subject once more, Germain did finally win the prize in the third attempt in 1816,
though the judges comprising of Legendre, Laplace and Poisson were still far from
being happy. Poisson treated her as an inferior novice in the company of giants.
Though the differential equation of plate vibration was correctly derived by Germain,
but she made mistakes in boundary conditions which were corrected by Kirchhoff
in 1850.
Lord Rayleigh wrote his classic book on the theory of sound [5] which is still being
treated with great respect. Rayleigh introduced the concepts of generalised forces and
generalised coordinates, the treatment of which proved advantageous to engineers.
In his method, an approximation is made of the suitable form of the type of motion
for obtaining natural frequencies of complicated systems. The famous Rayleigh’s
method is based on the principle of conservation of energy for conservative systems.
This idea of calculating natural frequencies directly from the energy consideration,
without taking recourse to the differential equations of vibration, was later utilised
by Ritz. The so-called Rayleigh–Ritz method is now a very popular approach for
studying not only vibrations, but in solving problems in elasticity, theory of structures,
nonlinear mechanics and other branches of physics.
Euler and Bernoulli derived the differential equation of a vibrating beam undergoing small deflections. Kirchhoff investigated the vibration of bars of variable cross
section and found ‘exact’ solution for certain cases. Duhamel proposed a general
method for analysing the forced vibration of elastic plates. This method was also
used by Saint Venant in studying lateral forced vibration of beams.
One of the first problems in which the importance of studying vibrations was
recognised by engineers was that of torsional vibration of propeller shafts of
steamships. Frahm was the first to investigate the problem theoretically and experimentally. Approximate methods have been introduced for studying beam problems
of non-prismatic sections. One such idea of successively approximating the integration of differential equations was introduced by Vianello, who used it for calculating
buckling load of struts. Its extension to vibration problems was made by Stodola.
The whirling of a shaft carrying a disc was first investigated by Foppl. Investigation
on two-span beam had been carried out theoretically and experimentally by Ayre,
Ford and Jacobsen. Timoshenko included the shear deformation and rotary inertia in
his beam vibration problem to yield an improved theory of beam vibration. Similarly,
Mindlin contributed greatly to propose an improved theory of plate vibrations.
Mathematical theory of nonlinear vibration was developed towards the end of last
century [6]. Duffing and Van der Pol were the first to propose definite solutions in
nonlinear vibrations.
Taylor introduced the concept of correlation function in 1920 and Wiener and
Khinchin the spectral density in 1930s [7, 8]. These opened new vistas for laying
3
a prize for the problem of deriving a mathematical theory of plate vibrations. In
response to it, Sophie Germain, a French lady proposed a solution, but was denied
the prize due to an incorrect computation of the variation of a particular integral.
The Academy proposed the subject again in 1813. Though the error previously
mentioned was rectified by now, but Germain failed to give a satisfactory explanation
of an assumption. So the prize again eluded her. When the Academy proposed the
subject once more, Germain did finally win the prize in the third attempt in 1816,
though the judges comprising of Legendre, Laplace and Poisson were still far from
being happy. Poisson treated her as an inferior novice in the company of giants.
Though the differential equation of plate vibration was correctly derived by Germain,
but she made mistakes in boundary conditions which were corrected by Kirchhoff
in 1850.
Lord Rayleigh wrote his classic book on the theory of sound [5] which is still being
treated with great respect. Rayleigh introduced the concepts of generalised forces and
generalised coordinates, the treatment of which proved advantageous to engineers.
In his method, an approximation is made of the suitable form of the type of motion
for obtaining natural frequencies of complicated systems. The famous Rayleigh’s
method is based on the principle of conservation of energy for conservative systems.
This idea of calculating natural frequencies directly from the energy consideration,
without taking recourse to the differential equations of vibration, was later utilised
by Ritz. The so-called Rayleigh–Ritz method is now a very popular approach for
studying not only vibrations, but in solving problems in elasticity, theory of structures,
nonlinear mechanics and other branches of physics.
Euler and Bernoulli derived the differential equation of a vibrating beam undergoing small deflections. Kirchhoff investigated the vibration of bars of variable cross
section and found ‘exact’ solution for certain cases. Duhamel proposed a general
method for analysing the forced vibration of elastic plates. This method was also
used by Saint Venant in studying lateral forced vibration of beams.
One of the first problems in which the importance of studying vibrations was
recognised by engineers was that of torsional vibration of propeller shafts of
steamships. Frahm was the first to investigate the problem theoretically and experimentally. Approximate methods have been introduced for studying beam problems
of non-prismatic sections. One such idea of successively approximating the integration of differential equations was introduced by Vianello, who used it for calculating
buckling load of struts. Its extension to vibration problems was made by Stodola.
The whirling of a shaft carrying a disc was first investigated by Foppl. Investigation
on two-span beam had been carried out theoretically and experimentally by Ayre,
Ford and Jacobsen. Timoshenko included the shear deformation and rotary inertia in
his beam vibration problem to yield an improved theory of beam vibration. Similarly,
Mindlin contributed greatly to propose an improved theory of plate vibrations.
Mathematical theory of nonlinear vibration was developed towards the end of last
century [6]. Duffing and Van der Pol were the first to propose definite solutions in
nonlinear vibrations.
Taylor introduced the concept of correlation function in 1920 and Wiener and
Khinchin the spectral density in 1930s [7, 8]. These opened new vistas for laying