Nonlinear Analysis of a Rigid Rotor on Magnetic Bearings

1995 ◽  
Author(s):  
C. Nataraj
Keyword(s):  
Nonlinear Analysis ◽  
Equations Of Motion ◽  
Forced Response ◽  
Magnetic Bearings ◽  
Rigid Rotor ◽  
Stability Criteria ◽  
Safe Operation ◽  
Qualitative And Quantitative ◽  
Quantitative Results ◽  
Nonlinear Equations Of Motion

A simple model of a rigid rotor supported on magnetic bearings is considered. A proportional control architecture is assumed, the nonlinear equations of motion are derived and some essential nondimensional parameters are identified. The free and forced response of the system is analyzed using techniques of nonlinear analysis. Both qualitative and quantitative results are obtained and stability criteria are derived for safe operation of the system.

A Numerical Study on the Effect of Unbalance and Misalignment Fault Parameters in a Rigid Rotor Levitated by Active Magnetic Bearings

2019 ◽  
Author(s):  
Prabhat Kumar ◽  
Rajiv Tiwari
Keyword(s):  
Numerical Study ◽  
Equations Of Motion ◽  
Rotor System ◽  
Magnetic Bearings ◽  
Inertia Force ◽  
Dimensional System ◽  
Rigid Rotor ◽  
Active Magnetic Bearings ◽  
Fault Parameters ◽  
Misalignment Fault

Abstract This paper focusses on analysing the vibration behaviour of a rigid rotor levitated by active magnetic bearings (AMB) under the influence of unbalance and misalignment parameters. Unbalance in rotor and misalignment between rotor and both supported AMBs are key fault parameters in the rotor system. To demonstrate this dynamic analysis, an unbalanced rigid rotor with a disc at the middle levitated by two misaligned active magnetic bearings has been mathematically modelled. One of the novel concepts is also described as how the force due to active magnetic bearings on the rigid rotor is modified when the rotor is parallel misaligned with AMBs. With inclusion of inertia force, unbalance force and force due to misaligned AMBs, the equations of motion of the rigid rotor system are derived and converted into dimensionless form in terms of various non-dimensional system and fault parameters. Numerical simulations have been performed to yield the dimensionless rotor displacement and controlling current responses at AMBs. The prime intention of the present paper is to study the effect on the displacement response of the rigid rotor system and the current consumption of AMBs for different ranges of disc eccentricities and rotor-AMB misalignments.

A Reliable Pre-Processing for the Simulation of Friction Joints in Turbomachineries and its Validation: A Case Study With Policontact

2019 ◽  
Author(s):  
Christian M. Firrone ◽  
Giuseppe Battiato
Keyword(s):  
Equations Of Motion ◽  
Good Practice ◽  
Forced Response ◽  
Mathematical Basis ◽  
Reduction Techniques ◽  
Safety Margins ◽  
Pass Through ◽  
Nonlinear Equations Of Motion ◽  
Physical Phenomena

Abstract Industry and University collaborates to develop methods to simulate the nonlinear dynamics of components in rotating assemblies like turbine or compressor modules in the presence of friction joints. This collaboration produced fruitful results providing a family of numerical solvers with the goal of foreseeing the safety margins against High Cycle Fatigue failure. Softwares are therefore intended as design tools to exploit the damping effect of joints by controlling geometrical features, materials and contact loads. Contact models, reduction techniques to handle complex blade geometries modeled by Finite Element softwares, and numerical techniques to solve the nonlinear equations of motion are refined to provide the level of vibration amplitude as fast as possible by keeping the representativeness of the physical phenomena that are involved. A reliable compromise between speed and accuracy must be confirmed by several ‘gates’ to pass through during all the simulation process, in particular during pre-processing phase. The objective of this paper is to propose a good practice made of a list of actions to check the goodness of the mathematical basis to obtain reliable results from simulation. Experience gained thanks to the long-lasting collaboration between Politecnico di Torino and GE Avio for the development of the software Policontact provides a case study of an effective synthesis between two requirements that are often opposed to each other: complex mathematical models to simulate the nonlinear forced response of rotating components on one side and a robust, confident implementation of an easy-to-use tool intended for industrial staff with complementary background on the other side.

Multiharmonic Forced Response Analysis of a Turbine Blading Coupled by Nonlinear Contact Forces

2009 ◽  
Author(s):  
Christian Siewert ◽  
Lars Panning ◽  
Jo¨rg Wallaschek ◽  
Christoph Richter
Keyword(s):  
Frequency Domain ◽  
Steam Turbine ◽  
Equations Of Motion ◽  
Forced Vibrations ◽  
Forced Response ◽  
Contact Forces ◽  
Dynamic Loads ◽  
Turbine Stage ◽  
Nonlinear Equations Of Motion ◽  
Turbine Blading

The rotor blades of a low pressure (LP) steam turbine stage are subjected to high static and dynamic loads during operation. The static loads are mainly due to the centrifugal force and thermal strains, whereas the dynamic loads are caused by fluctuating gas forces resulting in forced vibrations of the blades. The forced vibrations can lead to high cycle fatigue (HCF) failures causing substantial damage and high maintenance effort. Thus, one of the main tasks in the design of LP steam turbine blading is the vibration amplitude reduction in order to avoid high dynamic stresses that could damage the blading. The vibration amplitudes of the blades in a LP steam turbine stage can be reduced significantly to a reasonable amount if adjacent blades are coupled by shroud contacts that reinforce the blading, see Fig. 1. Furthermore, in the case of blade vibrations, relative displacements between neighboring blades occur in the contacts and friction forces are generated that provide additional damping to the structure due to the energy dissipation caused by micro- and macroslip effects. Therefore, the coupling of the blades increases the overall mechanical damping. A three-dimensional structural dynamics model including an appropriate spatial contact model is necessary to predict the contact forces generated by the shroud contacts and to describe the vibrational behavior of the blading with sufficient accuracy. To compute the nonlinear forced vibrations of the coupled blading, the nonlinear equations of motion are solved in the frequency domain owing to the high computational efficiency of this approach. The transformation of the nonlinear equations of motion into the frequency domain can be carried out by representing the steady-state displacement in terms of its harmonic components. After that transformation, the nonlinear forced response is computed as a function of the excitation frequency in the frequency domain.

A Time-Varying Stiffness Rotor Active Magnetic Bearings Under Combined Resonance

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
U. H. Hegazy ◽  
M. H. Eissa ◽  
Y. A. Amer
Keyword(s):  
Multiple Scales ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Multiple Scale ◽  
Magnetic Bearings ◽  
Time Varying ◽  
Active Magnetic Bearings ◽  
Gyroscopic Effects ◽  
Nonlinear Equations Of Motion ◽  
Combined Resonance

This paper is concerned with the nonlinear oscillations and dynamic behavior of a rigid disk-rotor supported by active magnetic bearings (AMB), without gyroscopic effects. The nonlinear equations of motion are derived considering a periodically time-varying stiffness. The method of multiple scales is applied to obtain four first-order differential equations that describe the modulation of the amplitudes and the phases of the vibrations in the horizontal and vertical directions. The stability and the steady-state response of the system at a combination resonance for various parameters are studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical nonlinear behaviors, including multiple-valued solutions, jump phenomenon, hardening, and softening nonlinearity. A numerical simulation using a fourth-order Runge-Kutta algorithm is carried out, where different effects of the system parameters on the nonlinear response of the rotor are reported and compared to the results from the multiple scale analysis. Results are compared to available published work.

Investigation of Turbulence Effects on the Nonlinear Vibration of a Rigid Rotor Supported by Finite Length 2-Lobe and Circular Bearings

2019 ◽  
Vol 14 (12) ◽  
Author(s):  
Nuntaphong Koondilogpiboon ◽  
Tsuyoshi Inoue
Keyword(s):  
Nonlinear Vibration ◽  
Limit Cycles ◽  
Reynolds Equation ◽  
Equations Of Motion ◽  
Rigid Rotor ◽  
Reynolds Equations ◽  
Multiplier Analysis ◽  
Turbulence Effects ◽  
The Stability ◽  
Nonlinear Equations Of Motion

Abstract This study investigated the effect of turbulence on the nonlinear vibration of a symmetrical rigid rotor supported by two identical journal bearings. The bearings consisted of various length to diameter (L/D) ratio circular and 2-lobe bearings with differing pad preloads. Two turbulent (Ng–Pan–Elrod and Constantinescu model) and one laminar Reynolds equations were selected for comparison, and they were solved using a finite difference method to obtain nonlinear bearing forces. The nonlinear equations of motion for the rotor-bearing system were solved using a shooting method and arclength continuation to obtain limit cycles for each bearing configuration. Floquet multiplier analysis was then utilized to identify the stability of the obtained limit cycles. For the cases of the circular and 2-lobe bearing without pad preload, the turbulent Reynolds equations yielded a lower onset speed of instability and L/D ratio at which the bifurcation type changed from supercritical to subcritical than the laminar Reynolds equation. However, at higher pad preloads (preloads of 0.25 or 0.5), the turbulence effects increased the onset speed of instability, especially for L/D ratios > 0.7, and only supercritical bifurcation was observed. For all bearing configurations, the ratio of the limit cycle whirl frequency to shaft rotational speed for both turbulence bearing models was higher than that of the laminar bearing model, and the Ng–Pan–Elrod turbulence model always generated lower onset speed of instability than the Constantinescu model.

Nonsmooth Bifurcations in a Cracked Rotor-Active Magnetic Bearings System

2014 ◽  
Vol 989-994 ◽  
pp. 2825-2828 ◽  
Author(s):  
Feng Hong Yang ◽  
Hong Zhi Tong
Keyword(s):  
Equations Of Motion ◽  
Magnetic Bearings ◽  
Active Magnetic Bearings ◽  
Periodic Motions ◽  
Cracked Rotor ◽  
Rotating Shaft ◽  
Chaotic Motions ◽  
Amb System ◽  
Nonlinear Equations Of Motion ◽  
Smooth System

A cracked rotor-active magnetic bearings (AMB) system with the time-varying stiffness is modeled by a piecewise smooth system due to the breath of crack in a rotating shaft. The governing nonlinear equations of motion for the nonsmooth system are established and solved with the numerical method. The simulation results show that a grazing bifurcation, period-double bifurcation and chaotic motions exist in the response. These nonsmooth bifurcations can give rise to jumps between periodic motions, quasi-periodic motions and chaos.

Dynamics and Stability of Non-Planar Rigid Rotor Equipped with Two Ball-Spring Autobalancers

2019 ◽  
Vol 19 (02) ◽  
pp. 1950001 ◽  
Author(s):  
Mousa Rezaee ◽  
Mir Mohammad Ettefagh ◽  
Reza Fathi
Keyword(s):  
Mathematical Model ◽  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Rigid Rotor ◽  
Out Of Plane ◽  
New Type ◽  
Nonlinear Equations Of Motion ◽  
The Mathematical Model ◽  
System Time

Recently, a new type of automatic ball balancer (ABB), called the ball-spring autobalancer (AB), has been proposed, which substantially eliminates the drawbacks of the traditional ABBs. In previous studies, the dynamics of the Jeffcott planar rotor equipped with ball-spring AB has been investigated. In the Jeffcott model, it is assumed that the ABB is located on the plane of the unbalance disk. However, for the non-planar rigid rotor with distributed imbalances, out-of-plane motions may occur, and the Jeffcott model becomes unreliable as the tilting motion cannot be explained. To this end, the aim of this paper is to analyze the capability of the ball-spring AB in balancing non-planar rotors and to reconfirm its pre-claimed advantages over the traditional ABBs for balancing non-planar rotors. To start, the mathematical model of the rigid rotor with two ball-spring ABs is established, based on which the nonlinear equations of motion are derived. Then, the system time responses are computed numerically and the balanced stable regions are acquired by the Lyapunov’s first method. The results of this study show that the ball-spring ABs can balance the non-planar rotors and the tilting motion does not impair the pre-claimed advantages of the ball-spring AB.

Influence of Blade Flexibility on the Dynamic Response Simulation of a Turbomolecular Pump on Magnetic Bearings

2020 ◽  
Vol 142 (4) ◽  
Author(s):  
Alysson Bruno Barbosa Moreira ◽  
Fabrice Thouverez
Keyword(s):  
Mechanical Model ◽  
Equations Of Motion ◽  
Low Frequency ◽  
Coupled System ◽  
Forced Response ◽  
Second Order ◽  
Magnetic Bearings ◽  
Flexible System ◽  
Turbomolecular Pump ◽  
Rigid Body Modes

Abstract This paper proposes the simulation of a complete mechanical model of a turbomolecular pump rotor, including rotor and blades flexibility, suspended by controlled active magnetic bearings. The mechanical model is composed of an eight stage blisk, attached to a shaft. Magnetic forces are linearized by the first-order Taylor expansion around a given point. Including blades and rotor flexibility makes the mechanical system asymmetric, so the equations of motion for the coupled system have periodic terms. A modal controller was designed to control rigid body modes, since the number of sensors is limited and no state observer is implemented. PID controllers are used for low frequency modes combined with the second order filters to damp high frequency modes. First of all, stability analysis was carried out for the axisymmetric case. Second, blades flexibility was included. Forced response of the whole system to an impulsive force was studied. Divergent responses for the system in rotation were obtained as a second order filter pole possibly interacting with blades modes. Taking the second order filters off the control loop allowed the system to be stable. These results show that the analysis method developed here is efficient to evaluate the performance of a controller in closed loop with the complete flexible system. This method may be used in industrial design processes as computation times for the complete system are very short.

Influence of Blade Flexibility on the Dynamic Response Simulation of a Turbomolecular Pump on Magnetic Bearings

2019 ◽  
Author(s):  
Alysson Bruno Barbosa Moreira ◽  
Fabrice Thouverez
Keyword(s):  
Mechanical Model ◽  
Equations Of Motion ◽  
Low Frequency ◽  
Coupled System ◽  
Forced Response ◽  
Second Order ◽  
Magnetic Bearings ◽  
Flexible System ◽  
Turbomolecular Pump ◽  
Rigid Body Modes

Abstract This paper proposes the simulation of a complete mechanical model of a turbomolecular pump rotor, including rotor and blades flexibility, suspended by controlled active magnetic bearings. The mechanical model is composed of an eight stage blisk, attached to a shaft. Magnetic forces are linearized by first order Taylor expansion around a given point. Including blades and rotor flexibility makes the mechanical system asymmetric, so the equations of motion for the coupled system have periodic terms. A modal controller was designed to control rigid body modes, since the number of sensors is limited and no state observer is implemented. PID controllers are used for low frequency modes combined with second order filters to damp high frequency modes. First of all, stability analysis was carried out for the axisymmetric case. Secondly, blades flexibility was included. Forced response of the whole system to an impulsive force was studied. Divergent responses for the system in rotation were obtained as a second order filter pole possibly interacts with blades modes. Taking second order filters off the control loop allowed the system to be stable. These results show that the analysis method developed here is efficient to evaluate the performance of a controller in closed loop with the complete flexible system. This method may be used in industrial design processes as computation times for the complete system are very short.

Test Response and Nonlinear Analysis of a Turbocharger Supported on Floating Ring Bearings

2005 ◽  
Vol 127 (2) ◽  
pp. 107-115 ◽  
Author(s):  
Chris Holt ◽  
Luis San Andre´s ◽  
Sunil Sahay ◽  
Peter Tang ◽  
Gerry La Rue ◽  
...  
Keyword(s):  
Nonlinear Analysis ◽  
Equations Of Motion ◽  
Forced Response ◽  
Experimental Results ◽  
Rotor Speed ◽  
Rotor Imbalance ◽  
Feed Pressure ◽  
Nonlinear Responses ◽  
Reaction Forces ◽  
Bearing Force

Measurements of casing acceleration on an automotive turbocharger running to a top speed of 115 krpm and driven by ambient temperature pressurized air are reported. Waterfall acceleration spectra versus rotor speed show the effects of increasing lubricant inlet pressure and temperature on turbocharger rotordynamic response. A comprehensive analysis of the test data shows regimes of speed operation with two subsynchronous whirl motions (rotordynamic instabilities). Increasing the lubricant feed pressure delays the onset speed of instability for the most severe subsynchronous motion. However, increasing the lubricant feed pressure also produces larger synchronous displacements. The effect of lubricant feed temperature is minimal on the onset and end speeds of rotordynamic instability. Nevertheless, operation with a cold lubricant exhibits lower amplitudes of motion, synchronous and subsynchronous. The experimental results show the subsynchronous frequencies of motion do not lock (whip) at system natural frequencies but continuously track the rotor speed. No instabilities (subsynchronous whirl) remain for operating speeds above 90 krpm. Linear and nonlinear analysis results for the operation of a small automotive turbocharger supported on floating ring bearings are presented. A comprehensive fluid film bearing model predicting the forced response of floating ring bearings is also described. The linear rotordynamic model predicts well the rotor free–free modes and onset speed of instability using linearized bearing force coefficients. The nonlinear model incorporating instantaneous bearing reaction forces in the numerical integration of the rotor equations of motion predicts the limit cycle amplitudes with two fundamental subsynchronous whirl frequencies. Comparisons of both models to experimental results follow. The predictions evidence two unstable whirl ratios at approximately 12 ring speed and 12 ring speed plus 12 journal speed. The transient nonlinear responses reveal the importance of rotor imbalance in suppressing the subsynchronous instabilities at large rotor speeds as also observed in the experiments.

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