Development of a Set of Equations for Incorporating Disk Flexibility Effects in Rotordynamic Analyses

1993 ◽  
Vol 115 (2) ◽  
pp. 227-233 ◽  
Author(s):  
G. T. Flowers ◽  
S. G. Ryan
Keyword(s):  
Dynamic Behavior ◽  
Equations Of Motion ◽  
Rotor System ◽  
Rotor Speed ◽  
Disk System ◽  
Rigid Disk ◽  
First Order ◽  
Transverse Vibrations ◽  
Rotor Dynamic ◽  
Sample System

Rotordynamic equations that account for disk flexibility are developed. These equations employ free-free rotor modes to model the rotor system. Only transverse vibrations of the disks are considered, with the shaft/disk system considered to be torsionally rigid. Second-order elastic foreshortening effects that couple with the rotor speed to produce first-order terms in the equations of motion are included. The approach developed in this study is readily adaptable for usage in many of the codes that are currently used in rotordynamic simulations. The equations are similar to those used in standard rigid disk analyses but with additional terms that include the effects of disk flexibility. An example case is presented to demonstrate the use of the equations and to show the influence of disk flexibility on the rotor dynamic behavior of a sample system.

Development of a Set of Equations for Incorporating Disk Flexibility Effects in Rotordynamical Analyses

1991 ◽  
Author(s):  
George T. Flowers ◽  
Stephen G. Ryan
Keyword(s):  
Equations Of Motion ◽  
Second Order ◽  
Rotor System ◽  
Rotor Speed ◽  
Disk System ◽  
Rigid Disk ◽  
First Order ◽  
Transverse Vibrations ◽  
Sample System

Rotordynamical equations that account for disk flexibility are developed. These equations employ free – free rotor modes to model the rotor system. Only transverse vibrations of the disks are considered, with the shaft/disk system considered to be torsionally rigid. Second order elastic foreshortening effects that couple with the rotor speed to produce first order terms in the equations of motion are included. The approach developed in this study is readily adaptable for usage in many of the codes that are current used in rotordynamical simulations. The equations are similar to those used in standard rigid disk analyses but with additional terms that include the effects of disk flexibility. An example case is presented to demonstrate the use of the equations and to show the influence of disk flexibility on the rotordynamical behavior of a sample system.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

Dynamic Analysis of Elastic Beam-Like Mechanism Systems

1989 ◽  
Vol 111 (4) ◽  
pp. 626-629
Author(s):  
W. Ying ◽  
R. L. Huston
Keyword(s):  
Dynamic Analysis ◽  
Dynamic Behavior ◽  
Cantilever Beam ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Order Perturbation ◽  
First Order ◽  
Geometric Stiffness ◽  
Stiffness Matrices ◽  
First Order Perturbation

In this paper the dynamic behavior of beam-like mechanism systems is investigated. The elastic beam is modeled by finite rigid segments connected by joint springs and dampers. The equations of motion are derived using Kane’s equations. The nonlinear terms are linearized by first order perturbation about a system balanced configuration state leading to geometric stiffness matrices. A simple numerical example of a rotating cantilever beam is presented.

Nonlinear Dynamic Study on Effects of Rub-Impact Caused by Oil-Rupture in a Multi-Shafts Turbine With a Squeeze Film Damper

2014 ◽  
Author(s):  
T. N. Shiau ◽  
C. R. Wang ◽  
D. S. Liu ◽  
W. C. Hsu ◽  
T. H. Young
Keyword(s):  
Dynamic Behavior ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Rotor System ◽  
Squeeze Film ◽  
Speed Ratio ◽  
Frequency Spectra ◽  
Squeeze Film Damper ◽  
Assumed Mode Method ◽  
Nonlinear Dynamic Behavior

An investigation is carried out the analysis of nonlinear dynamic behavior on effects of rub-impact caused by oil-rupture in a multi-shafts turbine system with a squeeze film damper. Main components of a multi-shafts turbine system includes an outer shaft, an inner shaft, an impeller shaft, ball bearings and a squeeze film damper. In the squeeze film damper, oil forces can be derived from the short bearing approximation and cavitated film assumption. The system equations of motion are formulated by the global assumed mode method (GAMM) and Lagrange’s approach. The nonlinear behavior of a multi-shafts turbine system which includes the trajectories in time domain, frequency spectra, Poincaré maps, and bifurcation diagrams are investigated. Numerical results show that large vibration amplitude is observed in steady state at rotating speed ratio adjacent to the first natural frequency when there is no squeeze film damper. The nonlinear dynamic behavior of a multi-shafts turbine system goes in its way into aperiodic motion due to oil-rupture and it is unlike the usual way (1T = >2T = >4T = >8T etc) as compared to one shaft rotor system. The typical routes of bifurcation to aperiodic motion are observed in a multi-shafts turbine rotor system and they suddenly turn into aperiodic motion from the periodic motion without any transition. Consequently, the increasing of geometric or oil parameters such as clearance or lubricant viscosity will improve the performance of SFD bearing.

scholarly journals Dynamics of Flexible Rotor Systems with an Interim Mass Unbalanced Disk Using a Spectral Element Model

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Sangkyu Choi ◽  
Usik Lee
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Element Model ◽  
Spectral Element ◽  
Rotor System ◽  
Dynamic Responses ◽  
Disk System ◽  
Unbalanced Mass ◽  
Rotor Systems ◽  
Blade System

A frequency domain spectral element model is developed for a rotor system that consists of two spinning shafts and an interim disk or blade system. In this study, the shafts are represented by spinning Timoshenko beam models, and the interim disk system is represented by a uniform thick rigid disk with an unbalanced mass. In our derivation of the governing equations of motion of the disk system, the disk is considered to be wobbling about the geometric center of the disk at which the spinning shafts are attached. The high accuracy of the proposed spectral element model is evaluated by comparison with the natural frequencies obtained using the conventional finite element method (FEM). The spectral element model is then used to investigate the effects of the unbalanced mass on the natural frequencies and dynamic responses of an example rotor system.

scholarly journals Dynamical Behavior Analysis of Rubbing Rotor System under Asymmetric Oil Film Force

2019 ◽  
Vol 2019 ◽  
pp. 1-16 ◽  
Author(s):  
Youfu Tang ◽  
Feng Lin ◽  
Qian Zou
Keyword(s):  
Dynamic Behavior ◽  
Dynamic Models ◽  
Journal Bearing ◽  
Dynamic Theory ◽  
Dynamical Behavior ◽  
Rotor System ◽  
System Response ◽  
Oil Film ◽  
Rotor Structure ◽  
Rotor Dynamic

Rubbing is one of the most common and significant faults that exist in the rotor system. It exhibits extremely complicated dynamic behavior. Existing dynamic models are primarily based on the symmetrical rotor structure, in which the oil film forces at both ends of support are considered identical. However, in practice, the oil film force may be differently affected by multiple factors (e.g., viscosity of lubricants, the thickness of oil film, and clearance of journal bearing). In this study, a novel dynamic model of the rubbing rotor system under asymmetric oil film force was developed. Furthermore, based on the nonlinear rotor dynamic theory, its dynamic behavior with different parameters was analyzed, and the corresponding chaotic features were extracted. The results indicated that the evolution law of chaotic motion was more complicated, and the chaotic region of system response was obviously wider under the asymmetric oil film force than the symmetrical oil film force.

scholarly journals Dynamic Behavior Analysis of Intermediate Bearing-Squeeze Film Dampers-Rotor System under Constant Maneuvering Overload

2021 ◽  
Vol 2021 ◽  
pp. 1-24
Author(s):  
Nan Zheng ◽  
Mo-li Chen ◽  
Gui-Huo Luo ◽  
Zhi-Feng Ye
Keyword(s):  
Dynamic Behavior ◽  
Dynamic Characteristics ◽  
Rotor System ◽  
Squeeze Film ◽  
Nonlinear Dynamic Model ◽  
Maneuvering Flight ◽  
Squeeze Film Dampers ◽  
Frequency Components ◽  
Simulation Results ◽  
Rotor Dynamic

Under the flight maneuvering of an aircraft, the maneuvering load on the rotor is generated, which may induce the change of dynamic behavior of aeroengine rotor system. To study the influence on the rotor dynamic behavior of constant maneuvering overload, a nonlinear dynamic model of bearing-rotor system under arbitrary maneuver flight conditions is presented by finite element method. The numerical integral method is used to investigate the dynamic characteristics of the rotor model under constant maneuvering overload, and the simulation results are verified by experimental works. Based on this, the dynamic characteristics of a complex intermediate bearing-squeeze film dampers- (SFD-) rotor system during maneuvering flight are analyzed. The simulation results indicate that the subharmonic components are amplified under constant maneuvering overload. The amplitude of the combined frequency components induced by the coupling of the inner and outer rotors is weakened. The static displacements of the rotor caused by the additional excitation force are observed. Besides, the period stability of the movement of the rotor deteriorates during maneuver flight. The design of counterrotation of the inner and outer rotors can effectively reduce the amplitude of subharmonic under constant maneuvering overload.

On Parametric Stability of a Nonconstant Axially Moving String Near Resonances

2016 ◽  
Vol 139 (1) ◽  
Author(s):  
Rajab A. Malookani ◽  
Wim T. van Horssen
Keyword(s):  
Order Approximation ◽  
Equations Of Motion ◽  
First Order ◽  
Axially Moving String ◽  
Transverse Vibrations ◽  
Fluctuation Frequency ◽  
Axially Moving ◽  
The Stability ◽  
First Order Approximation ◽  
Speed Fluctuations

The stability of an axially moving string system subjected to parametric excitation resulting from speed fluctuations has been examined in this paper. The time-dependent velocity is assumed to be a harmonically varying function around a (low) constant mean speed. The method of characteristic coordinates in combination with the two timescales perturbation method is used to compute the first-order approximation of the solutions of the equations of motion that governs the transverse vibrations of an axially moving string. It turns out that the system can give rise to resonances when the velocity fluctuation frequency is equal (or close) to an odd multiple of the natural frequency of the system. The stability conditions are investigated analytically in terms of the displacement-response and the energy of the system near the resonances. The effects of the detuning parameter on the amplitudes of vibrations and on the energy of the system are also presented through numerical simulations.

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

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