Steady-State Unbalance Response of Continuous Rotors on Anisotropic Supports

An analytical investigation of the steady-state unbalance response of axisymmetric rotor systems with anisotropic, flexible and damped bearings is presented. According to the exact approach of the dynamic stiffness method, the rotor is modelled by means of continuous beam elements. In this work, the expression of the 8 × 8 dynamic stiffness matrix of a rotating Timoshenko beam is derived and it is shown that it is related, by means of a simple law, to the previously published 4 × 4 dynamic stiffness matrix, which holds for the isotropic bearings case. The effects of concentrated disks and bearings are included into the formulation; in particular, each bearing is described by eight constant coefficients, according to the well-known linearized model of the bearing forces. The unbalance response of a typical rotor system taken from the literature is analyzed. A comparison is presented with the finite element results reported by other authors.

Whirling of Composite-Material Driveshafts Including Bending-Twisting Coupling and Transverse Shear Deformation

A simplified theory for predicting the first-order critical speed of a shear deformable, composite-material driveshaft is presented. The shaft is modeled as a Bresse-Timoshenko beam generalized to include bending-twisting coupling. Numerical results are compared with those for both thin and thick walled shell theories and generalized Bernoulli-Euler theory.

High Speed Rotating Shafts Design Algorithm

This paper reveals an invention for the first time to the public about the fundamental concepts for the design of high speed rotating shafts based on the wave spectrum analysis of the vibrational modes of elastic circular rods and shells that was completed in 1967, even though the Pochhammer-Chree-Love solution of the transversal (n = 1) vibration of a solid elastic rod was originated by Pochhammer in 1876.

The Effect of Inlet Grooves and Swirl on the Dynamics of Long Annular Seals in Centrifugal Pumps

This paper describes additional results from a continuing research program which aims to identify the dynamics of long annular seals in centrifugal pumps. A seal test rig designed to experimentally identify dynamic coefficients using a least-squares technique based on the singular value decomposition method. The analysis is carried out in the time domain using a multifrequency forcing function. The experimental method relies on the forced excitation of a flexibly supported stator by two hydraulic shakers. A rigid rotor supported in rolling element bearings runs through the stator. The only physical connection between shaft and stator is a pair of annular gaps filled with pressurised water discharged axially. The experimental coefficients obtained from the tests are compared with theoretical values.

Modelling Fluid Inertia Forces of Short Journal Bearings for Rotordynamic Applications

It has been recently suggested that fluid inertia may play an important role in the dynamic behavior of rotors supported on journal bearings. This paper presents a model for fluid inertia forces in short cylindrical journal bearings based on an energy approximation. The inertialess velocity profiles predicted by the solution of Reynolds’ equation are inserted in the axial momentum equation multiplied by the axial velocity profile and integrated across the film thickness, to obtain the pressure in short journal bearings including the fluid inertia effect. The pressure is then integrated to obtain the fluid inertia forces. It is shown that the inertia forces thus obtained are proportional to the usual radial, centripetal, tangential and coriolis accelerations of the journal, in addition to a nonlinear radial acceleration. Moreover, it is shown that the inertia forces contribute to the stiffness and damping characteristics of the journal bearings. The inertia coefficients of the bearings are obtained in cartezian and cylindrical coordinates, for both uncavitated and cavitated bearings, and are plotted versus the eccentricity ratio. The model thus obtained is an analytical closed form model for fluid inertia forces in short journal bearings. Such a model is the most suitable for rotordynamic applications, particularly for time transient rotordynamic simulations.

Internal Rotor Friction Induced Instability in High-Speed Rotating Machinery

Results of a program to investigate internal rotor friction destabilizing effects are presented. Internal-friction-producing joints were shown to excite the rotor system first natural frequency, when operating either below or above the first critical speed. The analytical models used to predict the subsynchronous instability were also confirmed. The axial spline joint demonstrated the most severe subsynchronous instability. The interference fit joint also caused subsynchronous vibrations at the first natural frequency but these were bounded and generally smaller than the synchronous vibrations. Comparison of data from the two test joints showed that supersynchronous vibration amplitudes at the first natural frequency were generally larger for the interference fit joint than for the axial spline joint. The effects of changes in imbalance levels and side loads were not distinguishable during testing because amplitude-limiting bumpers were required to restrict orbits.

Stability and Response of Rotors Supported on Active Magnetic Bearings

A PC-based program has been developed which is capable of performing stability analysis and response calculations of rotor-bearing systems. The paper discusses the modeling of rotors supported on active magnetic bearings (AMB) and highlights the advantages in the modeling using the finite element method, over the transfer matrix method. An 8-stage centrifugal compressor supported on AMB was chosen for the case study. The results for the stability analysis, obtained using the finite element program was compared with those obtained by the well established transfer matrix codes. The results of unbalance response, including the effects of sensor non collocation are presented and this demonstrates how an AMB supported rotor can experience a synchronous instability for selected sensor locations and balance distributions.

Theory Versus Experiment for Short (L/D = 1/6) Honeycomb and Smooth Annular Pressure Seals

A study which compares theoretical predictions of experimental rotordynamic and leakage results is presented for short (L/D = 1/6) honeycomb and smooth annular pressure seals. A computer code used in this comparison has been developed from a theory that employs a perturbation analysis of the governing equations flow and uses Moody’s pipe friction relationship for the surface friction of the rotor and stator. This study was undertaken to investigate how well an existing code could predict these characteristics with input provided from recorded test data and independent flat-plate tests. The results examine the effect that the following independent test parameters have on the experimental measurements and theoretical predictions: inlet preswirl, rotor speed, inlet pressure, pressure ratio across seal, seal clearance, and honeycomb cell width. Experimental results show that leakage is reduced by decreasing the honeycomb cell width. Rotordynamically, the short seals are stabilizing over all test parameter ranges. However, the short seals did not perform as favorably as longer (L/D = 1/3) seals. In general, the theory overpredicts rotordynamic coefficients and leakage.

Dynamic Characteristics of TEHD Tilt Pad Journal Bearing Simulation Including Multiple Mode Pad Flexibility Model

An approach for incorporating the heat transfer and elastic deformation effects into dynamic coefficient calculation is presented. A global analysis method is used, which finds the equilibrium pad tilt angles at each eccentricity position and includes cross-film variable viscosity, heat transfer effects in the lubricant, elastic deformation, heat conduction effects in the pads, and elastic deformation effect in the pivots. Deflection modes are used to approximate deformation of the top surface of the pads. The dynamic coefficients of a single pad are calculated at the equilibrium state of the bearing, based on numerical perturbation with respect to the bearing degrees of freedom. These include journal position, pad rotation, pivot deformation, and modal coordinates. The stiffness and damping coefficients are calculated and show very good agreement with experimental and numerical results from the existing literature.

Torsional Vibration Analysis of Shafts With Sections of Tapered Diameter

This paper presents the development of two tapered finite elements for use in torsional vibration analysis of rotor systems. These elements are particularly useful in analysis of systems that have shaft sections with linearly varying diameters. Both elements are defined by two end nodes, and inertia matrices are derived based on a consistent mass formulation. One element assumes a cubic displacement function and has two degrees of freedom at each node: rotation about the shaft’s axis and change in angle of rotation with respect to the axial distance along the shaft. The other element assumes a linear displacement function and has one rotational degree of freedom at each node. The elements are implemented in a computer program. Calculated natural frequencies and mode shapes are compared for both tapered shaft sections and constant diameter sections. These results are compared with results from an available constant diameter element. It is shown that the element derived assuming a cubic displacement function offers much better convergence characteristics in terms of calculated natural frequencies, both for tapered sections and constant diameter sections, than either of the other two elements. The finite element code that was developed for implementation of these elements is specifically designed for torsional vibration analysis of rotor systems. Lumped inertia, lumped stiffness, and gear connection elements necessary for rotor system analysis are also discussed, as well as calculation of natural frequencies, mode shapes, and amplitudes of response due to a harmonic torque input.