Effects Of Gyroscopic Moments Research Articles

Whirling analysis of unbalanced rotor using the vectors of solution coefficients

A prevalent issue in rotors is mass unbalance, which leads to synchronous lateral vibrations. In this paper, the critical whirling speeds and global responses of an unbalanced rotor-bearing system with viscoelastic boundary condition are computed using the relationships between the vectors of solution coefficients of differential equations governing the transverse displacements. The mathematical formulation is based on the matrix form of the aforementioned relationships and boundary conditions in the resonance case. The Timoshenko beam model, including the effects of gyroscopic moments is employed. It was found that the presented approach in comparison with the transfer matrix method reduces the matrix product when shaft segments have the same geometric and mechanical properties. A good agreement is found between the results of this study and the findings of studies available in the literature. Furthermore, the eigenvectors for free vibrations of the system were calculated by combining Natanson’s technique with the presented approach while considering the derived rotor’s characteristic equation after removing the forced term in the equation that describes the forced vibrations. Each obtained forward whirling mode shape describes perfectly the global response in the vicinity of the corresponding forward critical speed. The presented approach can, therefore, be applied to obtain the unbalance responses, whirling speeds, and associated mode shapes of rotor-bearing systems with complex boundary conditions.

Effect of gyroscopic moments on the attitude stability of a satellite in an elliptical orbit

Effect of gyroscopic moments on the attitude stability of a satellite in an elliptical orbit

Effects of Rub-Impact on Vibration Response of a Dual-Rotor System-Theoretical and Experimental Investigation

In order to understand the mechanism of fixed-point rub-impact in aero-engine, a dual-rotor system with inter-shaft bearing able to describe the mechanical vibration caused by unbalance and fixed-point rub-impact is developed in this paper. Firstly, a finite element (FE) model of the dual-rotor based on Timoshenko beam element and disk element is proposed, in which the effects of gyroscopic moments, rotary inertias, bending and shear deformations are taken into account. A rub-impact model is developed to derive the rubbing force between the rotor and fixed limiter, in which the softening characteristics of coating painted in casing is fully taken into account. Moreover, the Coulomb model is applied to describe the frictional behaviors. Secondly, the governing equations to describe the motion of the dual-rotor system with rub-impact are solved numerically by the Runge-Kutta method, and the dynamic characteristics are investigated by 3D water-fall plots and frequency spectrum. Combined harmonic frequency, whirling frequency and rotational frequency components of dual rotors are observed. Finally, the experiments of fixed-point rub-impact are performed on a dual-rotor test bench. Good agreement between the theoretical and experimental results shows the accuracy of the dual-rotor FE model and rub-impact model. The results indicate that (1) first backward whirling and forward whirling are excited under the influence of unbalance force; (2) several combination frequency components and second backward whirling and forward whirling frequency components are presented under fixed-point rubbing condition.

Finite element based stability analysis of a rotor-bearing system having a functionally graded shaft with transverse breathing cracks

In spite of gaining practical importance of functionally graded (FG) shafts, stability analysis of such FG rotor-bearing system has not been reported yet. This paper presents the development of a finite element (FE) procedure and code for stability analysis of cracked FG rotor-bearing system under thermal environment. Two-noded Timoshenko beam elements are used to model the FG shaft considering the effects of gyroscopic moments, translational and rotary inertia, bending and shear deformation and material damping. Zirconia (ZrO2) and stainless steel (SS) are considered as constituents of the radially graded FG shaft with temperature dependent material properties. Considering breathing crack behaviour, local flexibility coefficients (LFCs) are derived using the Paris's equation and the Castigliano's theorem. Results show that while the depth, orientation and locations of cracks, thermal gradient and material damping affect the stability threshold speed, it is important to choose the material gradient index judiciously. Thus, even when surface cracks appear on the FG shaft, threshold speed could still be in the desired range, which is important for damage tolerant design in high temperature applications.

Dynamic characteristics analysis of a dual-rotor system with inter-shaft bearing

The dual-rotor structure is susceptible to vibration, causing the malfunction of the entire operating system. In order to avoid the resonance during operation, it is significant and necessary to conduct modal analysis of such rotors. The dynamic analysis of the full dual-rotor system under operating conditions is also necessary to study dynamic characteristics of the rotating system. In this paper, one-dimension Timoshenko beam-type model, including the effects of gyroscopic moments, rotary inertias, bending and shear deformations, and three-dimension model for dual-rotor system with inter-shaft bearing are developed. Critical speed tests of dual-rotor are carried out to verify the analytical results. Based on the finite element models, the first critical speed excited by inner rotor and the first two critical speeds excited by outer rotor are calculated. The comparisons between both finite element models indicate that 1D model costs less time, which can be used to predict the critical speeds. Good agreement between the theoretical and experimental results shows the accuracy of the FE models. The Campbell diagram, critical speeds, operational deflection shapes and unbalance response of the dual-rotor are obtained to fully study the dynamic characteristics of the dual-rotor system.

The effect of wheel set gyroscopic action on the hunting stability of high-speed trains

ABSTRACTThis study explores the effects of wheel set gyroscopic action on hunting stability by calculating linear and nonlinear critical speeds. First, a dynamic model for a high-speed vehicle with 23 degrees of freedom is developed by considering wheel set gyroscopic action. The linear and nonlinear critical speeds are calculated by eigenvalue analysis and drawing a bifurcation map, respectively. Two computer programs for linear and nonlinear stability analysis are developed. Second, based on an actual high-speed vehicle in China, the effects of wheel set gyroscopic action on hunting stability are quantitatively investigated using computer simulation. Furthermore, the difference between the effects of gyroscopic moments about the x-axis and z-axis is discussed. The results show that the moment about the x-axis is harmful to hunting stability, but the moment about the z-axis is beneficial to hunting stability. However, the integrated effect of these two moments can enhance the critical speeds and suppress the hunting motion.

Identification of bearing dynamics under operational conditions for chatter stability prediction in high speed machining operations

Chatter is a major problem causing poor surface finish, low material removal rate, machine tool failure, increased tool wear, excessive noise and thus increased cost for machining applications. Chatter vibrations can be avoided using stability diagrams for which tool point frequency response function (FRF) must be determined accurately. During cutting operations, due to gyroscopic moments, centrifugal forces and thermal expansions bearing dynamics change resulting in tool point FRF variations. In addition, gyroscopic moments on spindle–holder–tool assembly cause separation of modes in tool point FRF into backward and forward modes which will lead to variations in tool point FRF. Therefore, for accurate stability predictions of machining operations, effects of operational conditions on machine tool dynamics should be considered in calculations. In this study, spindle bearing dynamics are identified for various spindle rotational speeds and cutting forces. Then, for a real machining center, tool point FRFs under operating conditions are determined using the identified speed dependent bearing dynamics and the mathematical model proposed. Moreover, effects of gyroscopic moments and bearing dynamics variations on tool point FRF are examined separately. Finally, computationally determined tool point FRFs using revised bearing parameters are verified through chatter tests.

Analytical modeling of asymmetric multi-segment rotor – bearing systems with Timoshenko beam model including gyroscopic moments

In this study, analytical modeling and an analysis approach for asymmetric multi-segment rotor – bearing systems are presented. Timoshenko beam model which includes the effect of gyroscopic moments is employed for modeling rotor segments. Instead of applying FEM, sub-segment Frequency Response Functions (FRFs) are obtained analytically, and sub-segment FRFs obtained are coupled by using receptance coupling method. Bearing properties are included into system dynamics by employing structural modification techniques. The proposed analytical model is verified by using FEM approach. It is shown that using analytical model and receptance coupling, compared with FEM, reduces computational time drastically without losing accuracy.

Effects of gyroscopic moment on the damage of a tapered roller bearing

This paper investigates the effect of gyroscopic moment on the induction of damage of a tapered roller bearing. High speed and high load are the typical characteristics of nowadays express trains which rely heavily on the tapered roller bearings. However, the phenomena of damage, especially for the side damage, of rollers and ring raceways in tapered roller bearings cannot be neglected with the occurrence of super speed trains in China. It first presents the geometry of a tapered roller bearing and then discusses the angular speed relationship between the rotations around the axis of shaft and self-axes of rollers. Gyroscopic moment is then established with geometry parameters of the bearing and physical parameters of the rollers. Then the effect of gyroscopic moment of rollers is discussed and the criterion that the pose of the tapered rollers should not be affected by the gyroscopic moment is obtained for the first time. This new theoretical result offers a design criterion for high speed tapered roller bearings. Analysis indicates that the gyroscopic moment could not bring any direct damage to the rollers and raceways but it does provide the boundary conditions to induce the side damage of rollers and raceways when the parameters of a bearing are not properly designed.

On the stabilising effect of gyroscopic moments in an automotive turbocharger

Automotive turbochargers, which operate at very high speeds, exceeding 180,000 r/min, exhibit two strong sub-harmonic modes of vibrations due to oil-whirl instability. These are a conical mode and an in-phase whirl mode. The gyroscopic effects can be very important in such a rotor system. This article presents a theoretical investigation into these effects on the conical whirl instability of a turbocharger induced by the angular (tilting) motion of a rigid rotor. A simplified linear model is used to analyse the rotor-bearing system by investigating the effects of the gyroscopic moment on the internal moments. A gyroscopic coefficient, defined by the geometry of the rotor, is shown to govern the stability of the conical whirl motion. A threshold value of ½ is determined for this coefficient to suppress the conical whirl. This value remains unaffected if the rotor is asymmetric and is supported by floating ring bearings, which is the case in a practical turbocharger.

Chaotic Vibration and Internal Resonance Phenomena in Rotor Systems

Rotating machinery has effects of gyroscopic moments, but most of them are small. Then, many kinds of rotor systems satisfy the relation of 1 to (−1) type internal resonance approximately. In this paper, the dynamic characteristics of nonlinear phenomena, especially chaotic vibration, due to the 1 to (−1) type internal resonance at the major critical speed and twice the major critical speed are investigated. The following are clarified theoretically and experimentally: (a) the Hopf bifurcation and consecutive period doubling bifurcations possible route to chaos occur from harmonic resonance at the major critical speed and from subharmonic resonance at twice the major critical speed, (b) another chaotic vibration from the combination resonance occurs at twice the major critical speed. The results demonstrate that chaotic vibration may occur even in the rotor system with weak nonlinearity when the effect of the gyroscopic moment is small.

Internal Resonance Phenomena of the Jeffcott Rotor With Nonlinear Spring Characteristics

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>0 and that of a backward-whirling mode pb<0 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. Furthermore in many practical rotating machines, the effect of gyroscopic moments are relatively small. Therefore, the one-to-one internal resonance relationship holds approximately between forward and backward natural frequencies in such machinery. 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 a small gyroscopic moment. 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 that the usage of the Jeffcott rotor in nonlinear analyses of rotor systems may induce incorrect results.

DYNAMIC MODELLING OF ROTATING DRILLSTRINGS WITH BOREHOLE INTERACTIONS

A dynamic model for the axial and transverse vibrations of a rotating drillstring is presented. The governing non-linear equations, which are obtained by using a Lagrangian approach, are fully coupled with time varying coefficients. They include the effects of gyroscopic moments, contact with the borehole wall, axial excitation due to bit/formation interactions, and hydrodynamic damping due to the presence of drilling mud outside the drillstring. Simulation results show that parametric resonance and whirling may occur simultaneously within the range of operating conditions of drilling. The dynamic behaviour is quite complicated and may become non-periodic, suggesting a chaotic behaviour.

COMPLEX GEOMETRY, ROTARY INERTIA AND GYROSCOPIC MOMENT EFFECTS ON DRILL VIBRATIONS

Increasing demand for better quality products and higher production rates requires a more complete understanding of drill dynamics. A transverse vibration model for drill bits is presented which includes the effects of gyroscopic moments and rotary inertia. The model is used to demonstrate the significance of these effects, and of complex drill geometries, on the natural frequencies of drill bit transverse vibration. Advances in bearing technology are likely to increase spindle speeds to more than 60 krpm; a speed at which gyroscopic moments and rotary inertia were found to have significant effect. The most significant parameter associated with complex drill geometries was found to be the number of flutes. Using a comparable three or four fluted drill versus a two fluted drill can increase the stiffness and, thus, the fundamental frequency of the drill bit by as much as 54%. Other complex drill geometry parameters (material properties, web taper, coolant holes) were not found to be as significant.

Modal Testing Theory of Rotor-Bearing Systems

Since all dynamic characteristics of rotor systems are closely related with rotor rotations, the directivity of modes is very important in rotor dynamics. But the classical modal testing theories which have been applied to nonrotating structures were often applied to rotor systems. Whereas the dynamic characteristics in negative frequency region have no meanings in nonrotating stuctures, the dynamic characteristics of rotor systems in the negative frequency region which are different from those in the positive frequency region have significant physical meanings. Here a new modal testing theory is proposed to separate the rotor vibration into positive and negative frequency regions. In particular, the amplitude and directivity variations of frequency response functions in positive and negative frequency regions are discussed when using complex modal displacement. And a method to identify the directivity of modes such as forward and backward is suggested using the frequency response function obtained by the proposed modal testing theory. The whirl directions of forced responses related with the directivity of modes are also discussed. In addition, even if the anisotropy in bearings and the effects of gyroscopic moments are permitted, the relations between the right and left eigenvectors of general damped anisotropic rotor systems are evaluated under some practical conditions. If the relations between the right and left eigenvectors are given, the necessity of additional modal testing to identify the adjoint modal parameters is relaxed.

Vibration of a Shaft Passing through a Critical Speed : 4th Report, Effect of Gyroscopic Moment

Effects of the gyroscopic moment on the nonstationary vibration, which occurs when a rotor passes through its critical speed at a uniform acceleration rate, are analyzed. The numerical integration results show that the gyroscopic moment makes the maximum amplitude small and makes the rotational speed, where the maximum amplitude occurs, high comparing with the results of a single-degree-of-freedom system. These phenomena may be explained by assuming that the acceleration rate in the gyroscopic system becomes large in appearance. Formulas for the imaginary acceleration rate and the maximum amplitude are derived by considering both the nonstationary vibration of a single-degree-of-freedom system and the change of the natural frequency of the gyroscopic system. The estimated maximum amplitudes are compared with the numerical ones.

Analysis and Experimental Investigation of the Stability of Intershaft Squeeze Film Dampers—Part 2: Control of Instability

The results of an analytical and experimental investigation showing the existence of an intershaft viscous damper instability were presented in reference [1]. In the present investigation, a more comprehensive stability analysis is used to study the stability of the test rig which incorporates a modified intershaft bearing support. The analysis is applicable to large multi-mass, rotor-bearing systems and includes the effects of gyroscopic moments, shear deformation, bearing support flexibility, and damping. The results of the stability analysis are presented in the form of system stability maps which clearly indicate the effectiveness of the modification in improving the instability onset speed of the system. Also presented are the results of an experimental investigation which substantiate the analytical predictions.

Modal analysis of turborotors using planar modes—theory

The generalized dynamic equations of motion have been obtained by the direct stiffness method for multimass flexible rotor bearing systems including the effects of gyroscopic moments, disc skew, and rotor acceleration. A set of undamped critical speed mode shapes calculated from the average horizontal and vertical bearing stiffness is used to transform the equations of motion into a set of coupled modal equations of motion. The modal equations are coupled by the generalized bearing coefficients and the gyroscopic moments. An analysis using only undamped critical speeds or decoupled modal analysis assuming proportional damping may lead to erroneous results. This paper presents a rapid method of calculating rotor resonance speeds with their corresponding amplification factors, stability and unbalance response of turborotors. Examples of the application of this modal approach are presented and results are compared to those of other methods such as matrix transfer analysis.

The Effect of Gyroscopic Moments and Distributed Mass on the Vibration of a Rotating Shaft with a Rotor

The Effect of Gyroscopic Moments and Distributed Mass on the Vibration of a Rotating Shaft with a Rotor

The Effect of Gyroscopic Moment and Distributed Mass on the Vibration of a Rotating Shaft with a Rotor

The Effect of Gyroscopic Moment and Distributed Mass on the Vibration of a Rotating Shaft with a Rotor