APPENDIX 2

Routh-Hurwitz Stability Criterion

For the following two-degree-of-freedom system:

M 00 M

xxyy

Dxx Dxy

Dyx Dyy

_xx_yy

Kxx Kxy

Kyx Kyy

xy

0

0

A2:1

with the solution

x x0 est, y y0 est A2:2

where s is an eigenvalue and x0, y0 are constants of integration, the correspondingcharacteristic equation to calculate four values of s can be given by:

a0s4 a1s 3 a2s 2 a3s a4 0 A2:3

where

a0 M2, a1 M Dxx Dyy

, a2 M Kxx Kyy

Dxx Dyy Dxy Dyx,

a3 Dxx Kyy Kxx Dyy Dxy Kyx Dyx Kxy, a4 Kxx Kyy Kxy KyxA2:4

For the polynomial of the nth order, the Routh-Hurwitz matrix, RH, is given by:

RH

a1 a3 a5 . . . 0a0 a2 a4 . . . 00 a1 a3 . . . 00 a0 a2 . . . 0

. . . . . . . . . . . . . . .0 0 0 . . . an

26666664

37777775

993

2005 by Taylor & Francis Group, LLC

The Routh-Hurwitz stability criterion is based on positive values of partial discriminantsformed from the main matrix (A2.5) as follows:

D1 a140, D2 a1 a3a0 a2

40, D3

a1 a2 a5a0 a2 a40 a1 a3

40, D4

a1 a3 a5 a7a0 a2 a4 a60 a1 a3 a50 a0 a2 a4

, . . .A2:6

For the particular case of the fourth order polynomial, Eq. (A2.1), since a5 a6 a7 0,the stability conditions are as follows:

D1 a14 0, D2 a1a2 a3a04 0

D3 a3 a1a2 a0a3 a21a44 0, D4 a4D34 0 A2:7

Since values of ai (i 1,2) are usually positive for the considered system, the systemstability is based on the second to fourth inequalities (A2.7). Most often it is the fourthinequality, D44 0, thus, a44 0, which provides the boundary of the unstable motion,when D4 0.

994 ROTORDYNAMICS

2005 by Taylor & Francis Group, LLC

Table of ContentsAPPENDIX 2: Routh-Hurwitz Stability CriterionAPPENDIX 1: Introduction to Complex NumbersAPPENDIX 3: Rotor Lateral Motion Forced SolutionsAPPENDIX 4: Relations Between Bearing Dynamic Coefficients In Two Fixed FramesAPPENDIX 5: Gyroscopic Rotor Responses to Synchronous and Nonsynchronous Forward and Backward PerturbationAPPENDIX 6: Basic Trigonometric RelationshipsAPPENDIX 7: Couette FlowAPPENDIX 8: Matrix Calculation ReviewAPPENDIX 9: Numerical Data for Rotor Lateral/Torsional Free VibrationsAPPENDIX 10: Fluid Circumferential Average Velocity Ratio as a Journal Eccentricity Function Based on Lubrication TheoryGlossary