Rotordynamic Analysis in the Design of Rotating Machinery

摘要: Rotordynamic analysis is an important step in the design of any rotating machine. To go beyond very simple models yielding a good qualitative insight but cannot predict details dynamic behaviour rotors, it necessary to resort numerical methods and among them Finite Element Method without doubt most suited for implementation context computer aided engineering. Instead resorting general purpose codes, particular characteristics rotordynamic make expedient use specialised tools like DYNROT, FEM code which allows perform complete study rotors. Although initially designed solve basic linear problems (Campbell diagram damped or undamped systems, unbalance response, critical speeds, static loading), has been extended nonstationary motions nonlinear systems [1] torsional axial rotors reciprocating machines. Its distinctive features Guyan reduction extensively using complex co-ordinates both isotropic non symmetric allow reduce time large number computations at reasonable cost. The can thus be used as routine called by optimisation procedures aimed including performances into definition optimum INTRODUCTION considerations play role machine elements, particularly when rotational speeds are high. Dynamic analysis, past mostly only computation currently many instances obtain whole picture (e.g., Campbell response), must accompany stress all other related working conditions machines (e.g. fluid turbomachinery, electrical generators motors, etc.). There cases deeply influences parts machine: example case composite material transmission shafts vehicular applications diameter shaft orientation reinforcing fibres determined need avoiding presence speed within range [2], high turbomolecular pumps on magnetic bearing rising third speed, one deformations rotor, dictates details. In this cases, if attempted, enters process. Traditionally, was performed either much simplified through similar Myklestat-Prohl method, based transfer matrices approach. finite element method (FEM) presently gaining popularity also field rotordynamics, mainly its ability modelling intricate geometries way, least from point view user, possibility writing codes. However, although standard commercial codes structural sometimes possible, compels some sort “tricks” take account effects rotation system, have strong influence flexural vibration, affecting natural frequencies coupling such way that more correct speak whirling motion than vibrations. instance, possible force gyroscopic matrix, affects mass matrix devised not handy procedure disadvantage allowing synchronous whirling. It then diagram, i.e. plot whirl against spin tool understanding rotor. Even problematic accelerating Only purposely written code, takes correctly circulatory linked with damping perhaps centrifugal stiffening effect, due bladed disks, fulfil adequately task. Starting end seventies, development specifically undertaken Department Mechanics Politecnico di Torino. evolved various versions twenty years [3]. original HPL HP-BASIC language desktop HP 9800 computers present MATLAB (MATLAB trademark MathWorks, Inc.) interactive software package PC, workstation mainframe system. THEORETICAL BACKGROUND Usually modelled beam-like structures form beam theory, Euler-Bernoulli Timoshenko approach [4, 5] used. Numerical solutions Myklestadt-Prohl [6, 7] were widespread. Also consider composed elements [8, 9]; types found literature [10]. beams even springs. Under above mentioned assumptions, model rotor straight, their axes aligned axis centre gravity shear cross sections lay axis, axial, behaviours uncoupled. small couple does modify substantially feature. Clearly assumptions leading uncoupling do hold crankshafts, vibrations customary so-called equivalent essentially straight structure and, consequence, restored [11]. further linearity, displacements equation motion; however, discretised axially symmetrical about rotates constant ω, linearised type ) ( t f x H K G C M = + & , (1) where vector containing generalised co-ordinates, referred inertial frame, skew-symmetric (it usually linearily dependent stiffness may contain part proportional ω), ω time-dependant forcing functions listed. One these residual which, small, neglected. Unbalance forces harmonic time, amplitude frequency equal ω. Equation non-natural, system hence differs typical equations encountered dynamics structures, symmetric. noted that, tends zero, terms vanish reduces structure. obtained assuming symmetrical, runs stator symmetry properties. If, contrary, considered becomes complicated, unless assumption made nonrotating latter case, rotor-fixed reference angular velocity (1), non-inertial frame obtained. If non-isotropic respect coefficients periodic 2ω. No closed solution available reach approximated far complicated symmetrical. When uncoupled ones, holds first one, while those natural, non-circulatory best choice what concerned co-ordinates. As assumed beam-like, each node four real degrees freedom, namely two lateral rotations. Assuming coincides z-axis orthogonal xyz displacement expressed