scholarly journals Dynamics of Bohr almost periodic motions of topological abelian semigroups

2016 ◽  
Vol 95 (2) ◽  
pp. 303-313 ◽  
Author(s):  
Xiongping Dai
Keyword(s):  
Almost Periodic ◽  
Periodic Motions

Internal Resonance of the Jeffcott Rotor With Nonlinear Spring Characteristics

2001 ◽  
Author(s):  
Yukio Ishida ◽  
Tsuyoshi Inoue
Keyword(s):  
Internal Resonance ◽  
Critical Speed ◽  
Nonlinear Phenomena ◽  
Almost Periodic ◽  
Periodic Motions ◽  
Nonlinear Analyses ◽  
Rotor Systems ◽  
Gyroscopic Moment ◽  
Jeffcott Rotor ◽  
Two Degree Of Freedom

Abstract The Jeffcott rotor is a two-degree-of-freedom linear model with a disk at the midspan of a massless elastic shaft. This model executing lateral whirling motions has been widely used in the linear analyses of rotor vibrations. In the Jeffcott rotor, the natural frequency of a forward whirling mode pf and that of a backward whirling mode pb have the relation of internal resonance pf : pb = 1 : (−1). Recently, many researchers analyzed nonlinear phenomena by using the Jeffcott rotor with nonlinear elements. However, they did not take this internal resonance relationship into account. While, in many cases of the practical rotating machinery, such a relationship holds apprximately due to small gyroscopic moment. In this paper, nonlinear phenomena in the vicinity of the major critical speed and the rotational speeds of twice and three times the major critical speed are investigated in the Jeffcott rotor and rotor systems with small gyroscopic moment. Especially, the influences of internal resonance on the nonlinear resonances are studied in detail. The following were clarified theoretically and experimentally: (a) the shape of resonance curves becomes far more complex than that of a single resonance, (b) almost-periodic motions occur, (c) these phenomena are influenced remarkably by the asymmetrical nonlinearity and gyroscopic moment, and (d) the internal resonance phenomena are strongly influenced by the degree of the discrepancies among critical speeds. The results teach us the usage of the Jeffcott rotor in nonlinear analyses of rotor systems may induce incrrect results.

scholarly journals Nonlinear Dynamics of a Blade Rotor with Coupled Rubbing of Labyrinth Seal and Tip Seal

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Huabiao Zhang ◽  
Xinye Li ◽  
Dongai Wang ◽  
Tingting Liu
Keyword(s):  
Shooting Method ◽  
Rotation Velocity ◽  
Labyrinth Seal ◽  
Dynamic Equations ◽  
Almost Periodic ◽  
Contact Deformation ◽  
Periodic Motions ◽  
Bifurcation Diagrams ◽  
Hertz Contact ◽  
Chaotic Motions

The dynamic response and its stability of a blade rotor with coupled rubbing in the labyrinth seal and tip seal are investigated. The dynamic equations are established based on the Hertz contact rubbing force at the labyrinth seal and the tip rubbing force considering both the contact deformation of the tip seal and the bending deformation of the blade. Numerical simulations show that the coupled rubbing response includes periodic motions, almost periodic motions, and chaotic motions. Compared with the single rubbing fault, coupled rubbing increases the range of rotation velocity of contact. A new continuation shooting method is used in the solution and stability analysis of the periodic response to give the bifurcation diagrams. The paths of the system for entering and exiting chaos are analyzed.

Birkhoff’s conjecture and almost periodic motions on torusT 2

1997 ◽  
Vol 40 (3) ◽  
pp. 279-284
Author(s):  
Tiecheng Li ◽  
Wenzao Huang
Keyword(s):  
Almost Periodic ◽  
Periodic Motions

Vibration Suppression of Rotating Machinery Utilizing an Automatic Ball Balancer and Discontinuous Spring Characteristics

2009 ◽  
Vol 131 (4) ◽  
Author(s):  
Jun Liu ◽  
Yukio Ishida
Keyword(s):  
Rotational Speed ◽  
Large Amplitude ◽  
Small Amplitude ◽  
Critical Speed ◽  
Vibration Suppression ◽  
Rotating Machinery ◽  
Almost Periodic ◽  
Periodic Motions ◽  
Speed Range ◽  
Suppression Method

Automatic ball balancer is a balancing device where two balls inside a hollow rotor move to optimal rest positions automatically to eliminate unbalance. As a result, vibrations are suppressed to the zero amplitude in the rotational speed range higher than the major critical speed. However, it has the following defects. The amplitude of vibration increases in the rotational speed range lower than the major critical speed. In addition, almost periodic motions with large amplitude occur in the vicinity of the major critical speed due to the rolling of balls inside the rotor. Because of these defects, an automatic ball balancer has not been used widely. This paper proposes the vibration suppression method utilizing the discontinuous spring characteristics together with an automatic ball balancer to overcome these defects and to suppress vibration. The validity of the proposed method is confirmed theoretically, numerically, and experimentally. The results show that amplitude of vibration can be suppressed to a small amplitude in the vicinity of the major critical speed and the zero amplitude in the range higher than the major critical speed.

Analytical and Numerical Study on a Parametric Pendulum with the Step-Wave Modulation of Length and Forcing

2019 ◽  
Vol 19 (05) ◽  
pp. 1941006
Author(s):  
Paweł Olejnik ◽  
Michal Fečkan ◽  
Jan Awrejcewicz
Keyword(s):  
Numerical Modeling ◽  
Numerical Solutions ◽  
Numerical Study ◽  
Time History ◽  
Step Function ◽  
Stable Limit ◽  
Almost Periodic ◽  
Periodic Motions ◽  
Parametric Pendulum ◽  
Stable Limit Cycles

A parametric pendulum excited by a discrete wave-modulated step function of length is subjected to a mathematical analysis and numerical modeling. We observe an existence of almost periodic solutions of ordinary differential equations with linear boundary value conditions. An exemplary oscillator subject to both an almost periodic step elongation and forcing synchronizes with the forcing, tending to almost periodic motions like stable limit cycles. Conditions for that synchronization as well as trajectories of numerical solutions on time history plots and phase planes are shown to confirm correctness of the analytical derivations and dedicated numerical modeling.

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