A model diatomic bond obeys with , ( )

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

Modes, Packets and Media, Ch Driven and pumped nonlinear oscillators and the route to chaos

Problem

A model diatomic bond obeys x¨+2βx˙+ω02x+ηx2+εx3=(F0/m)cos⁡ωt\ddot x + 2\beta\dot x + \omega_0^2 x + \eta x^2 + \varepsilon x^3 = (F_0/m)\cos\omega t with ω0=1.00×1014 rad s−1\omega_0 = 1.00\times10^{14}\,\text{rad s}^{-1}, β=5.0×1010 s−1\beta = 5.0\times10^{10}\,\text{s}^{-1} (Q=1000Q = 1000), η=−9.8×1037 m−1s−2\eta = -9.8\times10^{37}\,\text{m}^{-1}\text{s}^{-2} and ε=3.0×1047 m−2s−2\varepsilon = 3.0\times10^{47}\,\text{m}^{-2}\text{s}^{-2}. For a free oscillation with both a quadratic and a cubic force term, the combined backbone shift is ω(A)=ω0+κ′A2,κ′=3ε8ω0−5η212ω03.\omega(A) = \omega_0 + \kappa' A^2, \qquad \kappa' = \frac{3\varepsilon}{8\omega_0} - \frac{5\eta^2}{12\omega_0^3}. Assume, as for the cubic term alone, that the driven resonance peak rides this backbone. ((a)) Compute κ′\kappa' for this bond. ((b)) At a vibrational amplitude A=5.0×10−12 mA = 5.0\times10^{-12}\,\text{m} (5 pm), find the fractional lean of the resonance peak, κ′A2/ω0\kappa'A^2/\omega_0, and state whether the bond behaves as a hardening or a softening oscillator at this amplitude.

Browse the problem library