Comparison of analytical techniques for nonlinear vibrations of a parametrically excited cantilever

This paper is concerned with second-order approximations to the steady-state principal parametric resonance response of a vertically mounted flexible cantilever beam subjected to a vertical harmonic base motion. The unimodal form of the nonlinear equation describing the in-plane large amplitude parametric response of the beam, derived in Krishnamurthy based on the previous analysis in Crespo da Silva and Glynn, is analyzed using the harmonic balance (HB) and the perturbation method of multiple time scales (MMS). Single term HB, two terms HB, and second-order MMS with reconstitution version I and version II approximations to the steady-sate frequency-amplitude curves of the principal parametric resonance for each of the first four natural modes of the cantilever beam are compared with each other and with those obtained by numerically integrating the unimodal equation of motion. The time transformation T = Ωt is used in obtaining these approximations; also detuning is used in obtaining the square of the forcing MMS approximations. The obtained results show that, for the problem under consideration, the MMS version II is, in comparison with MMS version I, simpler to apply and leads to qualitatively more accurate second-order results. These results, however, show that the MMS version II tends to produce appreciable over corrections to the first-order results and may breakdown at relatively low response amplitudes, whereas the two terms HB solutions tend to improve the first-order results and lead to fairly accurate results even for relatively large response amplitudes.