Dynamics Study of a Rigid Rotor ElasticallySupported in Axial and Lateral Directions

Term ProjectSubmitted by

Nitin SinghME 562 Advanced Dynamics, Fall 2012

School of Mechanical EngineeringPurdue University

Abstract

The stability of semi-trivial solution of a rotor system which is harmonically excited in the axial directionis analyzed. A rigid rotor elastically mounted in the radial and axial directions is assumed. The axialthrust bearing is supposed to act as a joint. Lagrangian formulation using Euler angles is used to obtainEquations of Motions. Amplitude of the oscillations in upright position is taken as the small parameter.The asymptotic expansion about the vertical oscillation leads to Mathieu like coupled ode’s. Stabilityanalysis of these Mathieu Equations shows that model parameters as well as frequency of oscillation canproduce parametric resonance in the system. For one of these parameters numerical investigation isdone which supports the analytical results.

Introduction

Rotating machinery has applications in many spheres of our daily life. These include machine tools,power stations, turbo-machinery, aircraft jets, automobiles and marine propulsion. Studying dynamicsand stability of rotating machinery is of interest as it plays an important role in improving the safety andperformance of the systems they are part of. A number of these rotating machines can be modeled by arigid rotor whose axis of rotation is vertical and is elastically mounted in axial and lateral directions [1].In the present study, the axial thrust bearing is modeled as a joint which rests on elastic support, hencethe elastic mounting in the axial direction. Rotor considered to be rigid is assumed to be perfectlybalanced and only the stability of small oscillations (of the upright position) due to external excitation isconsidered. Furthermore system was assumed to be conservative with no damping.

Formulation

The following formulation is inspired by the works of Tondl [1] and Ruijgork et al. [3]. We assume therigid rotor to be a heavy disk of mass M which rotates around an axis (Figure 1). The axis of rotation iselastically mounted on the supporting foundation with a joint at A. The connection holding the rotor inupright position is also assumed to be elastic hence the lateral elasticity. To describe the position of therotor three generalized coordinates (Figure 2) are used viz. the axial displacement u in the verticaldirection (considered positive upwards), the precession angle (the angle of the axis of rotation withthe z axis) and the angle of nutation (the angle of rotation around the z axis). The distance betweenthe centre of gravity of the disc and the point A is R. The respective moments of inertia with respect tothe principal axis of the disc are I1, I2 and I3. Due to the symmetry of the disc I1 = I2.

I1 = I2 = MR2/2

To derive the equations of motion we use Lagrange’s Equations with Euler angles. Consider first, thekinetic energy of the system.

Stability Analysis

Equations (5) are nonlinearly coupled second order differential equations. Thus the primary system (of 1 DOF)being the elastic support in axial direction and the secondary system (of 2 DOF) being the elastic supports in lateraldirection constitute an autoparametric mechanical system. The secondary system can be at rest while the primarysystem is vibrating. This state referred as semi-trivial solution is the normal mode of the system response [1] [2].This semi-trivial solution becomes unstable in certain intervals of frequency of excitation. This interval is called asinstability interval. Instability intervals of the semi-trivial solution exhibit autoparametric resonance and hencebecome topic of interest while carrying out dynamic analysis of the system. The loss of stability of semi-trivialsolution depends on the system parameters and coupling between the primary and secondary system. Hence theautoparametric vibrations occur only in a limited region of the system parameters. This property assumesprofound interest in engineering applications.

In the present study only the stability analysis of the semi-trivial solution of the system is considered. Rotorfoundation is harmonically excited. Hence the semi-trivial solution for our system becomes:

x0(t) = 0

y0(t) = 0

u0(t) = a cos2t – g/422 = a cos2t – Mg/2k0

This semi-trivial solution corresponds to the vertical oscillations.

Results

Numerical investigation for the obtained analysis result was done using MATLAB. The script and function file isattached in appendix. Files contain the data assumed for parameters for the instability region for the abovementioned semi-trivial solution.

It was observed that the vertical oscillations in the upright position are unstable for the above found conditions.Hence the numerical investigations support the analytical solution.

Graph showing the instability of the upright oscillation is presented below:

References

[1] Tondl, A., On the stability of a rotor system, Acta Tech. Cesk. Akad. Ved 36, 331-338, 1991a.[2] Tondl, A., Parametric resonance vibration in a rotor system, Acta Tech. Cesk. Akad. Ved 37, 185-194, 1992a.[3] Ruijgork, M., Tondl, A., and Verhulst, F., Resonance in a rigid rotor with elastic support, Z. Angew. Math. Mech.

73, 255-263, 1993.[4] Hatwal, H., Mallik, A. K., Ghosh, A., Forced Nonlinear Oscillations of an Autoparametric System – Part 1.[5] Simakhina, S. V., Stability Analysis of Hill’s Equation. MS Thesis, University of Illinois, Chicago.

Appendix

Matlab File for numerically solving coupled ODEs:

% Nitin Singh% Term Project% ME 562, Advanced Dynamics% School of Mechanical Engeering, Purdue Univeristy%% This m-file uses the MATLAB function ode45 to solve the following% coupled ODEs:% ======================================================================% x_doubledot - 2*alpha*y_dot + x*(1 - (M*R*u_doubledot/I1)) = 0% y_doubledot - 2*alpha*x_dot + y*(1 - (M*R*u_doubledot/I1)) = 0% u_doubledot + 4*eta^2*u - (1/R)*(x_dot^2 + x*x_doubledot +% y_dot^2 + y*y_doubledot)+ (g/omega^2) = 0% ======================================================================% The solutions x(t), y(t) and u(t) are% ploted for t = [0,35] with a sampling rate of 100 Hz.

clear all; close all; clc;

% Define the Parameters:k = 1; k0 = 1; % lateral and Vertical Stiffnedd CoefficientsM = 1; r = 1; % Mass and radius of DiscR = 1; % Distance of COM from connection point AI1 = (M*r^2)/4; I3 = (M*r^2)/2; % Moments of Inertiaw = 5.634; % Angular Velocity of rotation about Z-axisg = 9.81; % Acceleration due to gravityomega = sqrt((2*k*R^2 - M*g*R)/I1);alpha = (w*I3)/(2*I1*omega);eta = sqrt(k0/(2*omega^2*M));

a = 0.1; % Amplitude of the external excitation

% Create grid of time steppingf = 100;t0 = 0; tf = 35;t = linspace( t0 , tf , (tf-t0)*f+1 );

% Initial conditions:x0 = 0 ; x_dot0 = 0; y0 = 0; y_dot0 = 0; u0 = (a - (M*g)/(2*k0)); u_dot0 = 0;z0 = [x0; x_dot0; y0; y_dot0; u0; u_dot0];

% ODE solver[t,z] = ode45(@project_ode,t,z0,[],R,alpha,eta,g,omega,M,I1);

x = z(:,1);x_dot = z(:,2);y = z(:,3);y_dot = z(:,4);u = z(:,5);

u_dot = z(:,6);

% Plotfigure('Name','Simulation Plot Window','NumberTitle','on');plot(t,x,'r',t,y,'b',t,u,'k');grid on;legend('x(time)','y(time)','u(time)' ,'Location', 'NorthWest');xlabel('time');ylabel('x(time), y(time) & u(time)');title('Term Project: Dynamics Of Rigid Rotor');

% This m-file defines the coupled ODEs:% ================================================================% x_doubledot - 2*alpha*y_dot + x*(1 - (M*R*u_doubledot/I1)) = 0% y_doubledot - 2*alpha*x_dot + y*(1 - (M*R*u_doubledot/I1)) = 0% u_doubledot + 4*eta^2*u - (1/R)*(x_dot^2 + x*x_doubledot +% y_dot^2 + y*y_doubledot)+ (g/omega^2) = 0% ======================================================================% in first order form where z(t) is the state vector:% z(t) = [ x(t) , x'(t) , y(t) , y'(t) , u(t) , u'(t) ]% and z'(t) is the time rate of change of the state vector:% z(t) = [ x'(t) , x''(t) , y'(t) , y''(t) , u'(t) , u''(t) ]

function [z_dot] = project_ode(t,z,R,alpha,eta,g,omega,M,I1)z_dot(6,1) = ((1/R)*( (z(2))^2 + (z(4))^2 - 2*alpha*(z(1)*z(4) - z(2)*z(3))…- ((z(1))^2 + (z(3))^2)) - 4*eta^2*z(5) - (g/omega^2) )/(1 - (((z(1))^2 +(z(3))^2) *M/I1));z_dot(1,1) = z(2);z_dot(2,1) = -2*alpha*z(4) - (1 - (M*R*z_dot(6,1)/I1))*z(1);z_dot(3,1) = z(4);z_dot(4,1) = 2*alpha*z(2) - (1 - (M*R*z_dot(6,1)/I1))*z(3);z_dot(5,1) = z(6);