By Steinke G.F.
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Extra info for 2-dimensional minkowski planes and desarguesian derived affine planes
101) √ where ω0 = k/m. We are interested in the absolute value of the amplitude A of the stationary solution x(t) = A exp(iωt). 6]. 103) with various values of δ. 106). with f0 = f˜0 /m, which then gives A= −ω2 f0 . 42): f0 |A| = . 17 for several values of δ. For damping 0 ≤ δ ≤ 1, the maxima of the resonance amplitudes lie on the curve f0 Amax = . e. an undamped system excited at the resonance frequency ω = ω0 would be destroyed, due to unlimited growth of the vibration amplitude. 3. 101) contains only a t-dependence, so it is an ‘external’ force; terms with dependence on x only are the ‘internal’ forces of the system.
This means that the relative maxima of the positions and velocities lie along a line, and likewise for the minima, so that after a finite time, the velocity v(t) becomes zero and the amplitude x(t) becomes constant. 98). 16 one sees that the velocity, and with it the kinetic energy, goes to zero in finite time, so Coulomb friction is much more effective in damping out energies or vibrations than is velocity-dependent friction, especially at small velocities. This effect has various applications.
For the linear oscillator with viscous damping (proportional to velocity), the differential equation is mx¨ + 2δ x˙ + kx = 0. 95) Physically, it corresponds to a linear spring which is damped in a fluid, so that the damping stays proportional to the velocity. If there were inertia effects in the fluid, the resulting ✐ ✐ ✐ ✐ ✐ ✐ “Matuttis-Driv-1” — 2014/3/24 — 19:18 — page 33 — #33 ✐ Mechanics ✐ 33 ‘Newtonian friction’ would be proportional to the square of the velocity. The solution to the viscously damped linear oscillator equation is x(t) = A exp(−δt) exp ±i ω02 − δ 2 t where ω0 = b k , δ= .
2-dimensional minkowski planes and desarguesian derived affine planes by Steinke G.F.