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 ( )

Accurate mechanical models of elastic beams undergoing large in-plane motions are discussed theoretically and experimentally. Employing the geometrically exact theory of rods with appropriate kinematic assumptions and asymptotic arguments, two approximate models are obtained-a relaxed model and its constrained version-that describe extensional and bending motions and neglect shear deformations. These models are shown to be suitable to predict, via an asymptotic approach, closed-form nonlinear motions of beams with general boundary conditions and, in particular, with boundary conditions that longitudinally constrain the motions. On the other hand, for axially unrestrained or weakly restrained beams, an inextensible and unshearable model is presented that describes bending motions only. The perturbations about the reference configuration up to third order are consistently derived for all beam models. Closed-form solutions of the responses to primary-resonance excitations are obtained via an asymptotic treatment of the governing equations of motion for two different beam configurations; namely, hinged-hinged (axially restrained) and simply supported (axially unrestrained) beams. In particular, considering the present theory and the existing theories, variations of the frequency-response curves with the beam slenderness or the relative boundary mass are investigated for the lowest modes. The fidelity of the proposed nonlinear models is ascertained comparing the theoretically obtained frequency-response curves of the first mode with those experimentally obtained. (c) 2005 Elsevier Ltd. All rights reserved

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We pr （ ，pose a van der Po1 − type self − excited cantilever beam わy positive velocity feedback and nonli 【 〕 ear feedback ． Self − excited oscillation keeps the resonant condition indeper 】dent of the mQdulation of system p 乱 rameters and the resonance chaTacteristic o 正 self − excited oscillation is suitable to raalize aut 〔 ， ． resonance machines such as AFM micro − cautilever proble ． Because the arnp 巨tude of self − excited oscMation grows w 圭 th tirl ユe ， a special control method i8 required for the amplitude controL To this end ， we propose the application of the nonlineur dynamics of van der Pol osci11ator ． Making ug． e of thc characteristic of a stacked − typc piezoelectricactuator ， we demeilstrate that the alnplitude control ｛ユ f a czntilevcr beam by using only integril controller without d正 f〔 erential controller ． We show the equati ⊂ 旧 of motiDn il ユ whlch the nonhnear effect is taken jnto account alld the averaged equation 、 vhich is derived by applying the method of mul 七 iple scales ． The bifurcation d壬 agram is theorctically descrlbed， Then ， it isclarihcd that the amplitude of the cantilever beam can be controlled by setting the nonlinear feedback guin ． Furthermore ， a van der Po 卜 type self excited 〔 ： antilever beam is rea ！ 正 zed by using a simple apParatus and the validity of amplitude cDntrol method

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