Abstract
Vibration theory belongs to well-developed branches of mechanics and physics. It cannot be understood without a good command of the fundamentals of mathematics. A large body of literature exists that is devoted to the theory of vibrations of discrete and continuous systems; it is not cited here in full; we mention only a few works [1–16], where an extensive bibliography covering the field can be found. This book will give certain basic information concerning the vibrations of discrete (or lumped) systems from the viewpoint of “mechanics.” The vibrations of lumped mechanical systems are described by ordinary differential equations. We dealt with such equations in Chaps. 1–3.
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Notes
- 1.
Brook Taylor (1685–1731), English mathematician known for Taylor’s theorem and Taylor series.
- 2.
Colin Maclaurin (1698–1746), Scottish mathematician working mainly in Edinburgh.
- 3.
John Rayleigh (1842–1919), English physicist awarded the Nobel prize in 1904.
- 4.
Guillame de l’Hospital (1661–1704), French mathematician, taught by Leibnitz and Johann Bernoulli, who published a l’Hospital rule that was, in fact, discovered by J. Bernoulli.
- 5.
Pierre-Simon Laplace (1749–1827), French mathematician and astronomer.
- 6.
Ernst Mach (1838–1916), Czech/Austrian physicist and philosopher; the Mach number \(M = \frac{{v}_{0}} {{v}_{{_\ast}}}\), where v 0 is the velocity of an object and v ∗ the velocity of sound in the considered medium.
- 7.
Wilhelm Nusselt (1882–1957), German engineer; the Nusselt number \(N = \frac{hL} {{k}_{f}}\), where L is the characteristic length, h the convective heat transfer coefficient, and k f the thermal conductivity of a liquid.
- 8.
Osborne Reynolds (1842–1912), Irish professor who studied fluid dynamics; the Reynolds number \(\mathrm{Re} = \frac{{v}_{{_\ast}}L} {\nu }\), where v ∗ is the mean fluid velocity, L the characteristic length, and ν the kinematic viscosity of a fluid.
- 9.
Vincent Strouhal (1850–1922), Czech physicist; the Strouhal number \(Sr = \frac{f{c}^{3}} {U}\), where f is the frequency, c the coefficient of expansion, and U the flow rate.
- 10.
William Froude (1810–1879), English engineer; the Froude number (dimensionless) \(Fr = \frac{v} {c}\), where v is the characteristic velocity and c the characteristic velocity of water wave propagation.
- 11.
Jean-Baptiste Biot (1774–1862), French physicist, astronomer, and mathematician; the Biot number \(Bi = \frac{hl} {k}\), where h is the heat transfer coefficient, l the characteristic length of a body, and k the coefficient of thermal conductivity of the body.
References
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Awrejcewicz, J. (2012). Vibrations of Mechanical Systems. In: Classical Mechanics. Advances in Mechanics and Mathematics, vol 29. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3740-6_6
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