Laplace transform
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In mathematics, the Laplace transform, named after Pierre-Simon Laplace, is an integral transform that converts a function of a real variable to a function of a complex variable s. The functions are o
Pierre-Simon, marquis de Laplace
Complex frequency s domain negative
Hooke's law
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In physics, Hooke's law is an empirical law which states that the force needed to extend or compress a spring by some distance scales linearly with respect to that distance—that is, Fs = kx, where k i
Laplace transform
…equations, and by replacing convolution with multiplication. For example, through the Laplace transform, the equation of the simple harmonic oscillator (Hooke's law) x ″ ( t ) + k x ( t ) = 0 is converted into the algebraic equation s 2 X ( s ) − s x ( 0 ) − x ′ ( 0 ) + k X ( s ) = 0 , which incorporates the initial…
(a) Schematic of a polymer nanospring. The coil radius, R, pitch, P, length of the spring, L, and the number of turns, N, are 2.5 μm, 2.0 μm, 13 μm, and 4, respectively. Electron micrographs of the nanospring, before loading (b-e), stretched (f), compressed (g), bent (h), and recovered (i). All scale bars are 2 μm. The spring followed a linear response against applied force, demonstrating the validity of Hooke's law at the nanoscale.