The nonlinear normal modes of a horizontally supported Jeffcott rotor are investigated. In contrast with a vertically supported rotor, there are localized and nonlocalized nonlinear normal modes because the linear natural frequencies in the horizontal and vertical directions are slightly different due to both gravity and the nonlinearity of restoring force. Reflecting such nonlinear normal modes, the frequency response curves are characterized in the primary resonance. In the case where the eccentricity is small, i.e., the response amplitude is small, the whirling motion is localized in the horizontal or vertical direction in the resonance. On the other hand, when the eccentricity is large, two kinds of whirling motion, which are localized in the vertical direction and nonlocalized in any direction, coexist simultaneously in a region of rotational speed. Experiments are conducted, and the theoretically predicted nonlinear responses based on localized and nonlocalized nonlinear normal modes are observed.H. Yabuno ( )

The effect of Coulomb friction on the nonlinear dynamics of a van der Pol oscillator is presented. A map from the magnitude of a peak to that of the succeeding valley in the time history is analytically described by considering both the exponential growth due to negative viscous damping and the switching condition due to Coulomb friction, which is a function of the sign of the velocity of the system. The steady states and their stability are clarified and the difference from those in the case without Coulomb friction is revealed. The addition of Coulomb friction makes the trivial equilibrium, which is an unstable focus in the system without friction, into a locally asymptotically stable equilibrium set. The branch of stable nontrivial steady states is not bifurcated from the trivial steady state by the effect of Coulomb friction and is different from the branch in the case without Coulomb friction, which is bifurcated from the trivial steady state through Hopf bifurcation. Furthermore, experiments are conducted and the theoretically predicted dynamics due to Coulomb friction is confirmed.

No abstract

This study deals with nonlinear dynamics of a horizontally supp ⊂ )rted Jeffcott rotor ． The equation of motiol ／ is derived by considering the gravity e 飩 ct and the cubic Iionlinearity of restoring forces、The linear natural f 艶 quencies in the ver し ical and hori70ntal directions are different due to し he gravity effect and they are independently changed depending on the mass of the rigid disl (． Also， the equiva − lent cubic nQnlinearity ， which depends on the mass o £ the rigid disk ， determines the nonlinear spring characteristics of backbone curves ， It is theoreticaly predicted that the decrease of the mass makes the nonllnear characteristics hardening and the critic 副 magnitudes of the mass ， at which the nonhnear spring char 乱 cter 重 stic i＄ trans缶rmed from soft to hard ， are different f6r 玉 ateral and vertica 至 directions ．The experimental results by a simple apparatus quahtatively co 面 rm the theoretically predicted non − ］ inear characteristics of the horizontally supPor

The 1 ／ 2−order subharmonic resonance occurs wher)the rotational speed is hl the vicillity of twice the na し ural frequency ， Jeffcott Rotor used basica ユ ly and widely in し he analysis of resonancies is a tw ( γ degree − of − freedom model with a desk at the midspan of a massless − shaft ． For the above condition on the rotational speed ， we darjfy the nonlinear characteristics of the resonance in horizontal直 y supported Jeffcott Rotor、 Applying the method of multiple scales ， we directly derive 出 e amplitude eq 岨 t孟 ons fbr the forward and backward whirlings and depict the relationsh 三 p between the rotational speed and the response amplitude ．恥 rthermore ， experiments are perfc ) rmed and the resu1 も s are compared with the theoretical ones ．