A bicycle fork's suspension is modelled as a mass-spring-damper with mass

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations · Simple, damped and driven oscillations

Mathematical Methods for Physics, Ch Second-order linear equations: damping, driving and resonance

Problem

A bicycle fork's suspension is modelled as a mass-spring-damper with mass m=8.0 kgm = 8.0\,\mathrm{kg}, spring stiffness k=3.2×103 N m−1k = 3.2\times10^3\,\mathrm{N\,m^{-1}} and damping coefficient b=64 kg s−1b = 64\,\mathrm{kg\,s^{-1}}, so its compression xx from equilibrium obeys mx¨+bx˙+kx=0m\ddot x + b\dot x + kx = 0. After a bump the fork is compressed x(0)=3.0 cmx(0) = 3.0\,\mathrm{cm} and is already moving back towards equilibrium at x˙(0)=−0.50 m s−1\dot x(0) = -0.50\,\mathrm{m\,s^{-1}}. ((a)) Find ω0\omega_0 and γ\gamma, and confirm the motion is underdamped. ((b)) Fit the two constants of the general solution to the stated initial conditions, and write x(t)x(t) in the amplitude-phase form x(t)=Ae−γt/2cos⁡(ωdt−ϕ)x(t) = Ae^{-\gamma t/2}\cos(\omega_{\mathrm{d}}t - \phi). ((c)) Find the fork's compression at t=0.10 st = 0.10\,\mathrm{s}.

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