Tacoma Narrows Bridge (1940) Billah and Scanlan (1991) reported that in fact, many physics textbooks (for example Resnick et al. and Tipler et al.) wrongly explain that the cause of the failure of the Tacoma Narrows bridge was externally forced mechanical resonance. Resonance is the tendency of a system to oscillate at larger amplitudes at certain frequencies, known as the system's natural frequencies. At these frequencies, even relatively small periodic driving forces can produce large amplitude vibrations, because the system stores energy. For example, a child using a swing realizes that if the pushes are properly timed, the swing can move with a very
Tune shift with amplitude by a Hamiltonian, or a symplectic map. For each initial position, we follow the particle as it traces out its orbit. After transformation into action-angle coordinates, one compute the tune formula_1 and the action formula_2. The tune shift with amplitude is then given by formula_3. The transformation to action-angle variables out of which the tune may be derived may be considered as a transformation to normal form. The tune shift with amplitude is important as a measure of non-linearity of a system. A linear system will have no tune shift with amplitude. Further, it can be important regarding the stability