R. Whalley

Vibration analysis of distributed-lumped rotor systems

In this paper a distributed-lumped model for the analysis of the flexural vibrations of a rotor-bearing system is considered. A general formula for the determinant of the tri-diagonal partitioned matrix description of the system is derived. This enables the irrational characteristic determinant of the system model to be obtained by the dynamic stiffness matrix method. The results obtained are compared to those acquired from the transfer matrix method. The error source in the computation of the natural frequencies by the dynamic stiffness matrix method is discussed. It is shown that by implementing the transfer matrix method the natural frequencies obtained are of greater accuracy. A numerical example illustrating the two methods, is presented and the results achieved are commented upon.

Automotive gas turbine regulation

A multivariable model of an automotive gas turbine, obtained from the linearized system equations is investigated. To facilitate vehicle speed changes, whilst protecting the system against thermal damage, control of the power turbine inlet gas temperature and gas generator speed is proposed by feedback regulation. Fuel flow and the power turbine nozzle area variations are the selected, manipulatable inputs. Owing to the limited control energy available for regulation purposes a multivariable, optimum, minimum control effort strategy is employed in the inner loop controller design study. Simulated, open and closed loop system responses are presented for purposes of comparison. Significant improvements in the transient response interaction reaction times and low steady state output interaction achieved using passive compensation and output feedback alone. Simplification of the closed loop configuration is proposed in the final implementation without performance penalties.

Machine Tool Axis Dynamics

Abstract The independent axis dynamics of machine tools that employ long slender lead screws, bearings, and workpiece saddles, mounted on supporting slides, will be considered. Distributed-lumped parameter techniques will be used for system analysis and in the model formulation procedures. Realizations that encompass combined torsional and tension loading will be incorporated. Specific, integrated, distributed-lumped machine tool configurations will be derived and described in block diagram form. Simulation studies illustrating the dynamic signature of a machine tool axis drive, when operated at various feed rates, together with the measured results will be presented. The effect of changes in the effective lead-screw length on the workpiece surface finish will be commented on.

Analytical solution of distributed-lumped parameter network models

Abstract System models for distributed-lumped parameter processes are derived. Procedures leading to the Smith normal form and thereafter to the admittance matrix of these models are presented. General results for interconnected distributed-lumped parameter models are provided and a typical application study is included.

Multivariable System Regulation

Abstract The regulation of multivariable systems which are subjected to input set point changes and disturbances is considered. A regulation strategy, for analysis purposes, employing both an inner- and an outer-loop feedback structure is proposed. Prescribed, closed-loop, dynamic behaviour using minimum control effort while confining steady-state output coupling is achieved. Output recovery, following disturbance perturbations, via proportional, outer-loop control is advocated. Elementary procedures enabling the computation of conventional pre and feedback compensators are provided, facilitating thereby cost-effective implementation. Stability assessment via established frequency domain techniques is acknowledged. Application studies, enabling comparisons with existing controller design methods, are outlined.

Matrix Quadratic Models

Process mechanical system models are considered and the techniques for analysis are discussed. A method for the investigation of system models which possess multiple degrees of freedom is outlined. To illustrate this general procedure the dynamical response of a pulp blender-beater, with dual impellers and five degrees of freedom, for paper and board manufacturing is computed.

Flexural vibration of rotating shafts by frequency domain hybrid modelling

Abstract In this paper the flexural vibration of a cantilevered rotating marine propeller is considered. From the transfer matrices of the distributed shaft and the lumped disc model, characteristic determinant of the distributed-lumped model of the system is derived. Direct search optimisation techniques are employed, enabling complex roots of the irrational characteristic determinant to be determined. Frequency response methods are used to determine the flexural response of the system. The discrete inverse Fourier transform technique is employed to compute the transient response from the frequency response data. Both the resonance and gyroscopic effects are identified. It is shown that by using frequency domain hybrid techniques, the system could be represented accurately by a reduced order model.

The torsional response of rotor systems.

Abstract The torsional response of rotor systems comprising bearings, inertia discs, and relatively long, slim shafts is considered. Lumped, finite element and hybrid, distributed-lumped parameter procedures are employed to represent the rotor systems of concern in efforts aimed at increasing accuracy, integrity, and computational efficiency. Rotor, shaft, and bearing elements of arbitrary dimensions, constructed from materials with differing mechanical properties, can be accommodated within the system models formulated. General results for multiple rotor assemblies are derived. Simple computational techniques are employed to obtain the frequency response and time domain characteristics for the models proposed. Analytical validation of the resonance conditions identified is provided. Application studies are presented for purposes of comparison.

Optimum control of a paper making machine headbox

Abstract A multivariable, time invariant model for a Fourdrinier machine headbox is considered. Analysis procedures enabling safe, economical closed loop response regulation are outlined. An optimum, minimum control effort strategy is proposed. Simulated open and closed loop response records are computed. The effectiveness of the design procedure, which minimises the power consumption for regulation purposes improving reliability and reducing refit and maintenance costs thereby, is commented upon.

Error correction in hydrostatic spindles by optimal bearing tuning

In this paper a high precision grinding wheel is considered as a rigid rotor mounted on two hydrostatic bearings. The equations for small perturbations of the wheel on the bearings are derived in the form of a multi-input, multi-output transfer function matrix, enabling the frequency response function of the wheel to be determined. Thereafter an optimisation algorithm is proposed which considers speed, load and dimensions of the spindle, and computes optimal stiffness and damping of the bearings. The dynamic characteristics of the bearings, tuned for minimum radial displacement of the spindle, is achieved maximising thereby the accuracy of the grinding process. Simulation results show that by stiffness coarse adjustment, and fine adjustment of the damping in the bearings, a spindle with 35 μm manufacturing error, can produce components with 3 μm accuracy.