Forced Oscillations of a Continuous Rotor With Geometric Nonlinearity: Internal Resonance Phenomena at Harmonic and Subharmonic Resonances

1997 ◽  
Author(s):  
Yukio Ishida ◽  
Imao Nagasaka ◽  
Seongwoo Lee
Keyword(s):  
Geometric Nonlinearity ◽  
Internal Resonance ◽  
Forced Oscillations ◽  
Center Line ◽  
Rotating Shaft ◽  
Restoring Force ◽  
Resonance Phenomena ◽  
Geometric Stiffening ◽  
Resonance Curves ◽  
Subharmonic Resonances

Abstract Harmonic resonances and subharmonic resonances of order 1/2 and order 1/3 in a continuous rotating shaft with distributed mass are discussed. The restoring force of the shaft has geometric stiffening nonlinearity due to the extension of the shaft center line. It is supposed that a distributed bias force, such as the gravity, works. The possibility of their occurrences, the shapes of resonance curves, and internal resonance phenomena are investigated.

Internal Resonance Phenomena of an Asymmetrical Rotating Shaft

2005 ◽  
Vol 11 (9) ◽  
pp. 1173-1193 ◽  
Author(s):  
Yukio Ishida ◽  
Tsuyoshi Inoue
Keyword(s):  
Internal Resonance ◽  
Critical Speed ◽  
Nonlinear Phenomena ◽  
Resonance Case ◽  
Rotating Shaft ◽  
Disk System ◽  
Resonance Phenomena ◽  
Previously Treated ◽  
Unstable Vibration ◽  
Asymmetrical Rotating Shaft

In general, asymmetrical shaft-disk systems have been investigated where unstable vibrations may occur. Most studies have treated a single resonance case for the linear system, and we have previously treated a single resonance case for the nonlinear system. However, when natural frequencies have a simple integer ratio relation in a nonlinear asymmetrical shaft-disk system, an internal resonance may occur and the vibration phenomena change remarkably compared to the characteristics of a single resonance case (the case without internal resonance). In this study, the internal resonance phenomena of an asymmetrical shaft are investigated theoretically and experimentally in the vicinities of the major critical speed, and twice and three times the major critical speed. We clarify that the shape of the resonance curves changes, almost periodic motions occur, and, especially, the occurrence of unstable vibration at the rotational speed of twice the major critical speed is extremely affected by the internal resonance. Further, we show the change of nonlinear phenomena between the systems with and without internal resonance.

Internal Resonance Phenomena of an Asymmetrical Rotating Shaft: Critical Speed of Twice the Major Critical Speed

1999 ◽  
Author(s):  
Yukio Ishida ◽  
Tsuyoshi Inoue
Keyword(s):  
Internal Resonance ◽  
Critical Speed ◽  
Nonlinear Phenomena ◽  
Rotating Shaft ◽  
Disk System ◽  
Rotor Systems ◽  
Resonance Phenomena ◽  
Resonance Relation ◽  
Asymmetric Rotor ◽  
Asymmetrical Rotating Shaft

Abstract Unstable vibrations appear in the vicinities of several critical speeds in asymmetric rotor systems with nonlinear spring characteristics. However, when the natural frequencies satisfy internal resonance relation exactly or approximately, these phenomena may change remarkably. In this paper, such internal resonance phenomena of an asymmetric shaft-disk system are studied theoretically and experimentally. The changes in nonlinear phenomena during the transition from the system with internal resonance to the system with no internal resonance are also investigated.

Nonlinear Forced Oscillations Caused by Quartic Nonlinearity in a Rotating Shaft System

1990 ◽  
Vol 112 (3) ◽  
pp. 288-297 ◽  
Author(s):  
Y. Ishida ◽  
T. Ikeda ◽  
T. Yamamoto
Keyword(s):  
Polar Coordinates ◽  
Forced Oscillations ◽  
Ball Bearings ◽  
Rotating Shaft ◽  
Restoring Force ◽  
Double Row ◽  
Nonlinear Component ◽  
Shaft System ◽  
Nonlinear Components ◽  
Theoretical Results

This paper deals with nonlinear forced oscillations in a rotating shaft system which are caused by quartic nonlinearity in a restoring force. These oscillations are theoretically analyzed by paying attention to the nonlinear components represented by the polar coordinates. It is clarified which kind of nonlinear component has an influence on each oscillation. In experiments it was shown that, when the shaft was supported by double-row angular contact ball bearings, the restoring force had nonlinear spring characteristics involving quartic nonlinearity in addition to quadratic and cubic ones. Experimental results were compared with the theoretical results regarding the probability of occurrence and the shapes of the resonance curves.

Forced Oscillations of a Continuous Asymmetrical Rotor With Geometric Nonlinearity (Major Critical Speed and Secondary Critical Speed)

2008 ◽  
Vol 130 (3) ◽  
Author(s):  
Imao Nagasaka ◽  
Yukio Ishida ◽  
Jun Liu
Keyword(s):  
Geometric Nonlinearity ◽  
Internal Resonance ◽  
Critical Speed ◽  
Longitudinal Direction ◽  
Lateral Force ◽  
Forced Oscillations ◽  
Almost Periodic ◽  
Flexible Rotor ◽  
Double Row ◽  
Stiffening Effect

Forced oscillations in the vicinities of both the major and the secondary critical speeds of a continuous asymmetrical rotor with the geometric nonlinearity are discussed. When the self-aligning double-row ball bearings support the slender flexible rotor at both ends, the geometric nonlinearity appears due to the stiffening effect in elongation of the shaft if the movements of the bearings in the longitudinal direction are restricted. The nonlinearity is symmetric when the rotor is supported vertically, and is asymmetric when it is supported horizontally. Because the rotor is slender, the natural frequency pfn of a forward whirling mode and pbn of a backward whirling mode have the relation of internal resonance pfn:pbn=1:(−1). Due to the influence of the internal resonance, various phenomena occur, such as Hopf bifurcation, an almost periodic motion, the appearance of new branches, and the diminish of unstable region. These phenomena were clarified theoretically and experimentally. Moreover, this paper focuses on the influences of the nonlinearity, the unbalance, the damping, and the lateral force on the vibration characteristics.

Chaotic Response of a Hard Duffing Type Vibration Isolator With Combined Coulomb and Viscous Damping

1995 ◽  
Author(s):  
A. Y. T. Leung ◽  
B. Ravindra ◽  
A. K. Mallik ◽  
C. W. Chan
Keyword(s):  
Chaotic Motion ◽  
Viscous Damping ◽  
Vibration Isolator ◽  
Bifurcation Structure ◽  
Restoring Force ◽  
Coulomb Damping ◽  
Secondary Resonances ◽  
Excited System ◽  
Subharmonic Resonances

Abstract Numerical simulations of the response of a harmonically excited mass on an isolator with a cubic, hard, non-linear restoring force and combined Coulomb and viscous damping are presented. For a base-excited system, the inclusion of a Coulomb damper with a suitable break-loose frequency can suppress the secondary resonances and chaotic motion. However, for a force-excited system, the introduction of Coulomb damping does not alter the bifurcation structure. Transmissibility indices have been defined for the solution obtained by numerical integration and the role of the subharmonic resonances and chaotic motion on the performance of the system is pointed out.

Sign in / Sign up

Export Citation Format

Share Document