.... A multiple time-scales perturbation method is used to construct formal asymptotic approximations of the solutions, and it is shown how different oscillation modes are damped. beams (structures) damping initial value problems perturbation theory springs (mechanical) vibrations To obtain...

... initial value method. Approximate and exact relations are also found. initial value problems shapes (structures) vibrations column vibration heavy support The vibration of a standing column under gravity is important for structures such as poles, antennas, and tall buildings. Systems...