Rotordynamic Behavior Research Articles

케이징 구조 유연성을 고려한 터보펌프 임계 속도 해석

케이징 구조 유연성을 고려한 30톤 추력급 터보펌프에 대한 임계 속도 해석이 수행되었다. 로터와 케이징을 포함한 전체 모델에 대한 3차원 유한요소법이 회전체 동역학 특성을 예측하기 위하여 적용되었다. 케이징 구조 유연성의 영향을 고찰하기 위하여 로터 단독 모델과 고정-고정 및 자유-자유 경계조건을 갖는 로터-케이징 연계 모델에 대한 수치 해석이 이루어 졌다. 무부하 하중 조건과 부하 하중 조건에서의 볼 베어링 강성이 각각 진공 조건과 실제 엔진 장착 조건에서의 작동을 모사하기 위하여 사용되었다. 수치해석 결과로부터 케이징 구조 유연성의 영향이 터보펌프의 임계 속도를 감소시킴을 확인하였다. 특히, 높은 베어링 강성을 갖는 부하 로터 조건에서 낮은 베어링 강성을 갖는 무부하 로터 조건에 비하여 더 많은 영향을 받음을 알 수 있었다. A critical speed analysis is performed for a 30 ton thrust turbopump considering the casing structural flexibility. A full three-dimensional finite element method including rotor and casing is used to predict rotordynamic behavior. Rotor alone model and rotor-casing coupled model with fixed-fixed and free-free boundary conditions are calculated to investigate the effects of the casing structural flexibility. The stiffness of ball bearings are applied as unloaded and loaded values to consider rotor operating conditions in vacuum and real engine respectively. From the results of the numerical analyses, it is found that the effect of the casing structural flexibility reduces the critical speeds of the turbopump. Especially, the loaded rotor condition with higher bearing stiffness is affected dramatically rather than the unloaded rotor condition with lower bearing stiffness.

207 破砕機の動力学技術(WS-1 もの造り-技術の伝承の問題点(1),研究発表講演)

This paper describes a dynamic analysis of the rotor bearing system on rotary crushers and shredder which dispose of large sized discarded articles such as refrigerators, washing machines, bicycles and kitchen utensils. In the analytical model, the rotary share type shredder with knifes, the vertical type crusher with breakers and the horizontal type crusher with swing hammers are simplified to mass and spring system on the rotational vibration and on the bending vibration. In order to clear the relation between bearing vibration and rotor speed drop, simulation is carried out by using Runge-Kutta-Gill method. Calculation of results indicate that bearing vibration is related well to the rotor dynamic behavior similar to the rotor speed drop.

Transient Vibration Prediction for Rotors on Ball Bearings Using Load-Dependent Nonlinear Bearing Stiffness

Rolling-element bearing forces vary nonlinearly with bearing deflection. Thus an accurate rotordynamic transient analysis requires bearing forces to be determined at each step of the transient solution. Analyses have been carried out to show the effect of accurate bearing transient forces (accounting for non-linear speed and load dependent bearing stiffness) as compared to conventional use of average rolling-element bearing stiffness. Bearing forces were calculated by COBRA-AHS (Computer Optimized Ball and Roller Bearing Analysis - Advanced High Speed) and supplied to the rotordynamics code ARDS (Analysis of Rotor Dynamic Systems) for accurate simulation of rotor transient behavior. COBRA-AHS is a fast-running 5 degree-of-freedom computer code able to calculate high speed rolling-element bearing load-displacement data for radial and angular contact ball bearings and also for cylindrical and tapered roller beatings. Results show that use of nonlinear bearing characteristics is essential for accurate prediction of rotordynamic behavior.

Hydrostatic Gas Journal Bearings for Micro-Turbomachinery

Several years ago an effort was undertaken at MIT to develop high-speed rotating MEMS (Micro Electro-Mechanical Systems) using computer chip fabrication technology. To enable high-power density the micro-turbomachinery must be run at tip speeds of order 500m∕s, comparable to conventional scale turbomachinery. The high rotating speeds (of order 2 million rpm), the relatively low bearing aspect ratios (L∕D<0.1) due to fabrication constraints, and the laminar flow regime in the bearing gap place the micro-bearing designs to an exotic spot in the design space for hydrostatic gas bearings. This paper presents a new analytical model for axially fed gas journal bearings and reports the experimental testing of micro gas bearings to characterize and to investigate their rotordynamic behavior. The analytical model is capable of dealing with all the elements of, (1) micro-devices, (2) dynamic response characteristics of hydrostatic gas bearings, (3) evaluation of stiffness, natural frequency and damping, (4) evaluation of instability boundaries, and (5) evaluation of effects of imbalance and bearing anisotropy. First, a newly developed analytical model for hydrostatic gas journal bearings is introduced. The model consists of two parts, a fluid dynamic model for axially fed gas journal bearings and a rotordynamic model for micro-devices. Next, the model is used to predict the natural frequency, damping ratio and the instability boundary for the test devices. Experiments are conducted using a high-resolution fiber optic sensor to measure rotor speed, and a data reduction scheme is implemented to obtain imbalance-driven whirl response curves. The model predictions are validated against experimental data and show good agreement with the measured natural frequencies and damping ratios. Last, the new model is successfully used to establish bearing operating protocols and guidelines for high-speed operation.

Contact Dynamic Response With Misalignment in a Flexible Rotor/Magnetic Bearing System

This paper investigates the vibration characteristics of rotor displacement signals in a magnetic bearing system under conditions when rotor contact with auxiliary bearings is possible. Since these signals may be used for feedback control, it is necessary to determine how they may affect the ability of the controller to regain rotor levitation. An experimental system is used to demonstrate the sensitivity of the rotor nonlinear dynamic behavior to unbalance, which is sufficient to cause contact during rotor run-up through rigid-body and flexural mode critical speeds. Complex rotor dynamics may involve contact with more than one auxiliary bearing or bush. Application of appropriate rotating forces to the rotor through a magnetic bearing is also shown to induce similar contact dynamics. Thus, an alternative procedure for assessing the nonlinear rotor dynamic behavior is established with the potential for identification of appropriate control forces. The contact dynamics are also considered in the presence of auxiliary bearing misalignment. Misalignment may arise through physical translation of a housing or through steady-state offset errors in sensor measurements. A misalignment of 50% of the nominal radial clearance is applied at an auxiliary bearing. Various contact modes are evident as the rotor is run up in speed. During rundown, different contact dynamics may be encountered and the level of such hysteresis is assessed.

Transverse Crack Modeling and Validation in Rotor Systems Including Thermal Effects

In this article, a model is described that allows one to simulate the static behavior of a transversal crack in a horizontal rotor, under the action of the weight and other possible static loads and the dynamical behavior of the rotating cracked shaft. The crack “breaths,” i.e., the mechanism of opening and closing of the crack, is ruled by the stress acting on the cracked section due to the external loads; in a rotor the stress is time‐depending with a period equal to the period of rotation, thus the crack “periodically breaths.” An original simplified model is described that allows cracks of different shape to be modeled and thermal stresses to be taken into account, since they may influence the opening and closing mechanism. The proposed method has been validated using two criteria. Firstly, the crack “breathing” mechanism, simulated with the model, has been compared with the results obtained by a nonlinear 3‐D FEM calculation and a good agreement in the results has been observed. Secondly, the proposed model allows the development of the equivalent cracked beam. The results of this model are compared with those obtained by the above‐mentioned 3‐D FEM. There is a good agreement in the results, of this case as well.Therefore, the proposed crack model and equivalent beam model can be inserted in the finite beam element model used for the rotor dynamical behavior simulation—the obtained equations have time‐depending coefficients, but they can be integrated in the frequency domain by using the harmonic balance method. The model is suitable for finite beam elements with six degrees of freedom per node, in order to account also for torsion vibrations and coupling between torsion and flexural vibrations.

Rotordynamic characteristics of a multilobe hybrid journal bearing in turbulent lubrication

This paper presents an investigation of the rotordynamic behaviour of a four-lobe orifice compensated hybrid bearing. It aims to show that multilobe hybrid journal bearings prove to be a better alternative for high-speed turbo pump applications.

Transient Vibration Prediction for Rotors on Ball Bearings Using Load‐Dependent Nonlinear Bearing Stiffness

Rolling‐element bearing forces vary nonlinearly with bearing deflection. Thus an accurate rotordynamic transient analysis requires bearing forces to be determined at each step of the transient solution. Analyses have been carried out to show the effect of accurate bearing transient forces (accounting for nonlinear speed and load‐dependent bearing stiffness) as compared to conventional use of average rolling‐element bearing stiffness. Bearing forces were calculated by COBRA‐AHS (Computer Optimized Ball and Roller Bearing Analysis—Advanced High Speed) and supplied to the rotordynamics code ARDS (Analysis of Rotor Dynamic Systems) for accurate simulation of rotor transient behavior. COBRA‐AHS is a fast‐running five degree‐of‐freedom computer code able to calculate high speed rolling‐element bearing load‐displacement data for radial and angular contact ball bearings and also for cylindrical and tapered roller bearings. Results show that use of nonlinear bearing characteristics is essential for accurate prediction of rotordynamic behavior.

Transverse Crack Modeling and Validation in Rotor Systems, Including Thermal Effects

This article describes a model that allows the simulation of the static behavior of a transverse crack in a horizontal rotor under the action of weight and other possible static loads and the dynamic behavior of cracked rotating shaft. The crack breathes—that is, the mechanism of the crack′s opening and closing is ruled by the stress on the cracked section exerted by the external loads. In a rotor, the stresses are time‐dependent and have a period equal to the period of rotation; thus, the crack periodically breathes. An original, simplified model allows cracks of various shapes to be modeled and thermal stresses to be taken into account, as they may influence the opening and closing mechanism. The proposed method was validated by using two criteria. First the crack′s breathing mechanism, simulated by the model, was compared with the results obtained by a nonlinear, threedimensional finite element model calculation, and a good agreement in the results was observed. Then the proposed model allowed the development of the equivalent cracked beam. The results of this model were compared with those obtained by the three‐dimensional finite element model. Also in this case, there was a good agreement in the results.Therefore, the proposed models of the crack and the equivalent model of the beam can be inserted into the finite element model of the beam used for the rotor′s dynamicbehavior simulation; the obtained equations have timedependent coefficients, but they can be integrated into the frequency domain by using the harmonic balance method.

HTS magnetic bearings

Radial HTS magnetic bearings (SMB) up to 200 mm size are developed and tested in prototype fast rotating machines to demonstrate the potential to replace conventional bearings. The individual rotational bearing components HTS and PM, their physical interaction and technology is reviewed. Characterisation experiments are conducted to understand the rotor dynamic behaviour. In terms of unbalance and critical speeds the suspended wheels and rotors compare favourably with conventional bearing devices. The rationale of our present bearing technology lies in the assembling of both low-speed magnetic bearings for centrifugal and wafer processing units up to 20,000 rpm as well as a high-speed optical mirror accelerated to rim speed of more than 500 m/s (174,000 rpm) confirming stable low-drag and low energy operation. Two new-type U shaped semicircle HTS bearings coupled each with a 6 W/80 K cryocooler of the Stirling type allow the contact–free operation of a Si wafer carrier in semiconductor wet processes.

A Bulk-Flow Model of Angled Injection Lomakin Bearings

Abstract This paper introduces a bulk-flow model for prediction of the static and dynamic force coefficients of angled injection Lomakin bearings. The analysis accounts for the flow interaction between the injection orifices, the supply circumferential groove, and the thin film lands. A one control-volume model in the groove is coupled to a bulk-flow model within the film lands of the bearing. Bernoulli-type relationships provide closure at the flow interfaces. Flow turbulence is accounted for with shear stress parameters and Moody’s friction factors. The flow equations are solved numerically using a robust computational method. Comparisons between predictions and experimental results for a tangential-against-rotation injection water Lomakin bearing show that novel model well predicts the leakage and direct stiffness and damping coefficients. Computed cross-coupled stiffness coefficients follow the experimental trends for increasing rotor speeds and supply pressures, but quantitative agreement remains poor. A parameter investigation shows evidence of the effects of the groove and land geometries on the Lomakin bearing flowrate and force coefficients. The orifice injection angle does not influence the bearing static performance, although it largely affects its stability characteristics through the evolution of the cross-coupled stiffnesses. The predictions confirm the promising stabilizing effect of the tangential-against-rotation injection configuration. Two design parameters, comprised of the feed orifices area and groove geometry, define the static and dynamic performance of Lomakin bearing. The analysis also shows that the film land clearance and length have a larger impact on the Lomakin bearing rotordynamic behavior than its groove depth and length.

Dynamic rotor eccentricity analysis by coupling electromagnetic and structural time stepping FEM

The influence of unbalance forces in rotary electric machines on the rotor vibration behavior is investigated in this paper. The dynamic response of a Switched Reluctance motor (SRM) rotor is analyzed by the coupled structural and electromagnetic Finite Element Method (FEM) in the transient state solved in a step-by-step procedure with respect to time. The proposed procedure allows solving the coupled field. The dynamic behavior of rotor caused by the coupled mechanical and magnetic forces is presented, according to the variation of unbalance mass and bearing stiffness.

SOME CONSIDERATIONS ON THE BASIC ASSUMPTIONS IN ROTORDYNAMICS

The dynamic study of rotors is usually performed under a number of assumptions, namely small displacements and rotations, small unbalance and constant angular velocity. The latter assumption can be substituted by a known time history of the spin speed. The present paper develops a general non-linear model which can be used to study the rotordynamic behaviour of both fixed and free rotors without resorting to the mentioned assumptions and compares the results obtained from a number of non-linear numerical simulations with those computed through the usual linearized approach. It is so possible to verify that the validity of the rotordynamic models extends to situations in which fairly large unbalances and whirling motions are present and, above all, it is shown that the doubts forwarded about the application of a model which is based on constant spin speed to the case of free rotors in which the angular momentum is constant have no ground. Rotordynamic models can thus be used to study the stability in the small of spinning spacecrafts and the insight obtained from the study of rotors is useful to understand their attitude dynamics and its interactions with the vibration dynamics.

Interaction Dynamics Between a Flexible Rotor and an Auxiliary Clearance Bearing

This study investigates the application of synchronous interaction dynamics methodology to the design of auxiliary bearing systems. The technique is applied to a flexible rotor system and comparisons are made between the behavior predicted by this analysis method and the observed simulation response characteristics. Of particular interest is the influence of coupled shaft/bearing vibration modes on rotordynamical behavior. Experimental studies are also performed to validate the simulation results and provide insight into the expected behavior of such a system.

DYNAMIC PROBLEMS CONCERNING THE SPEED OF ROTATION INCREASE OF A TURBINE–BLOWER ASSEMBLY

This paper investigates the dynamic problems of a turbine–blower assembly when its speed of rotation is increased by roughly ten per cent to obtain better efficiency. Changes concerning the dynamic behaviour of rotors in torsion and in bending, the dynamic behaviour of blades and the stresses in blades, have been estimated. It is shown that some modifications are necessary in order to allow the required increased speed.

Case Study: Vibration analysis for the design of a high-speed generator for a turbo-electric hybrid vehicle

The project relates to the design and development of a prototype high-speed turbo-generator as the thermal engine in a series hybrid vehicle. The substantial benefit of the turbo-generator against a diesel generator lies in the very high power-weight, power-volume ratio and renders itself particularly attractive for use in hybrid vehicle applications. However, to achieve a 50 kW power output, the turbo-generator has to have an operating speed of 60 000 r/min and thus important mechanical problems have to be solved. The core of this study addresses the requirement for an adequate understanding of rotor-dynamic behaviour by combining the results from both analytical and practical techniques. The assessment of modal testing, finite element analysis and vibration-condition monitoring, their feedback within the design-make-and-test procedure and the practical compromises and design constraints are presented and a design methodology is formulated. It is concluded that, under certain conditions, the prototype generator can be directly coupled to a small gas turbine, can operate safely and can produce the required power output.

Rotordynamic analysis of a vertical pump with tilting-pad journal bearings

The non-linear effects in journal bearings can dominate the rotordynamic behaviour of a vertical machine. This paper describes both linear and non-linear analyses of a twelve stage multiphase pump. For the linear case, the radial tilting-pad journal bearings are characterised by stiffness and damping coefficients. For the non-linear simulation, Reynolds equation is solved to give the non-linear bearing forces, and the time history of the response is derived for an unbalance excitation. The simulation shows that the bearing clearances are important parameters for controlling the dynamic behaviour. With small clearances the non-linear and linear simulations show similar response amplitudes. With a large clearance the damping decreases, and the non-linear results indicate that the rotor becomes more sensitive to sudden excitations which can be important in fluid handling machines. The simulation predicts that with a large bearing clearance a critical speed is introduced in the operating range, showing that the bearing stiffness also decreases. Measurements on the full scale machine with a small nominal bearing clearance show good agreement with calculated results.

Synchronous Dynamics of a Coupled Shaft/Bearing/Housing System With Auxiliary Support From a Clearance Bearing: Analysis and Experiment

This study examines the response of a flexible rotor supported by load sharing between linear bearings and an auxiliary clearance bearing. The objective of the work is to develop a better understanding of the dynamic behavior of a magnetic bearing supported rotor system interacting with auxiliary bearings during a critical operating condition. Of particular interest is the effect of coupling between the bearing/housing and shaft vibration on the rotor-dynamic responses. A simulation model is developed and a number of studies are performed for various parametric configurations. An experimental investigation is also conducted to compare and verify the rotor-dynamic behavior predicted by the simulation studies. A strategy for reducing sychronous shaft vibration through appropriate design of coupled shaft/bearing/housing vibration modes is identified. The results are presented and discussed.

DYNROT: a finite element code for rotordynamic analysis based on complex co‐ordinates

DYNROT is a code based on the finite element method which is intended to perform a complete study of the dynamic behaviour of rotors. Although initially designed to solve the basic linear rotordynamic problems (Campbell diagram for damped or undamped systems, unbalance response, critical speeds, static loading), it can be used for the study of non‐stationary motions of nonlinear rotating systems and for the torsional analysis of rotors and reciprocating machines. Explains that one of the distinctive features of the code is the use of complex co‐ordinates, both for isotropic and non‐symmetric systems. Makes extensive use of complex arithmetics in all parts of the analysis. Applies the modal approach in some of the solution routines to increase the efficiency of the computation or to compute an equivalent viscous damping in those cases where hysteretic damping cannot be introduced directly to the model. The dynamics of bladed discs is included in the code.

COMPOSITE SHAFT ROTORDYNAMIC ANALYSIS USING A LAYERWISE THEORY

Formulations are presented for the rotordynamic analysis of a composite rotor supported on general, eight coefficient bearings, obtained by using conventional equivalent modulus beam theory (EMBT), and a layerwise beam theory (LBT) derived from an existing layerwise shell theory, respectively. The formulations are based on the principle of virtual work. Rayleigh-Ritz displacements are used for deriving the solution equations. The results indicate that the difference between the two theories is not large, but for unsymmetric stacking sequences with bending stretching couplings present, the EMBT may result in inaccurate predictions of rotordynamic behaviour. Furthermore the layerwise theory produces a more realistic stress field in the laminate.