This document summarizes a student's modeling of the dynamics of a gyroscopic system - a flexible rotating shaft. The student models the shaft as a mass connected to ground by springs, rotating at an angular velocity. Equations of motion are derived in a rotating coordinate system. Results show the system becomes unstable when the angular velocity reaches certain critical speeds, due to poles of the characteristic equation becoming purely imaginary. Both internal and external damping are analyzed and shown to potentially destabilize the system.

1. 2011 ME Graduate Student Conference
April 30, 2011
MODELING OF DYNAMICS OF GYROSCOPIC SYSTEMS
Siddharth Sharma
M.S. Candidate
Faculty Advisor: Yitshak M. Ram
ABSTRACT
Critical Speed of Flexible Rotating Shaft
The stability of a flexible shaft rotating in bearings is kυ and k ζ which is further divided equally between the two
investigated to avoid large vibrations at the stability
springs of the same set. The stiffness of the shaft is
boundaries of the system. A stability boundary is
considered to be acting in two mutually perpendicular
encountered when the poles of the system becomes purely
imaginary. Subsequently, the effects of external damping directions. The origin A represents the centre of the cross
provided by the bearings and internal damping offered by section of the shaft in clamped roller bearings. The point B
the shaft are also analyzed for their impact on the stability locates the centre of mass of the rotor which is deflected
of shaft. The well known counter-intuitive phenomenon, from its equilibrium position when the shaft is rotating at
e.g. [1], [2] and [3], that internal damping destabilizes the constant angular velocity ω . The equilibrium position of the
system is demonstrated. rotor coincides with point A .
Modeling, Transformation of Coordinates and z
ζ
Equations of Motion
kζ
The simplest system consists of a rotor mounted on a 2
kυ
flexible shaft which rotates at an angular velocity ω in kυ 2
clamped roller bearing. 2 B υ
Rotor θ
z A y
ζ υ kζ
2
ω
θ = ωt y
χ
A
x
Figure 2 Model of shaft rotating in clamped roller
bearing
Clamped Roller Bearing Shaft
The motion of shaft in the χυζ rotating coordinates is
governed by the following set of equations, see e.g., [3]
Figure 1 A system of shaft rotating in clamped roller
bearing  & (
mυ& − 2mωζ& + kυ − mω 2 υ = 0 ) (2)
 &&
The rotating coordinate system χυζ sweeps an angle
( 2
mζ + 2mωυ + k ζ − mω ζ = 0
 & )
θ = ωt from the stationary coordinate system xyz . The
Symmetrical Shaft, Symmetrical Bearings (SSSB) with
transformation of coordinates is governed by
displacement of shaft at bearing
υ   cos ωt sin ωt  y  (1) A symmetrical shaft has equal stiffness in two mutually
 ς  − sin ωt cos ωt  z 
 =  
     perpendicular directions and the same is true for
symmetrical bearings. The shaft has finite displacement at
The model represents the stiffness of shaft by two sets of bearing.
two springs each. The stiffness constant of the two sets is

2. Im(s )
z z
υ ς
ς B
B s-plane
ω =0 ω = 10
υ
A A(υ A ς A ) A ωt * Our poles
O x χ O(0 0 ) O y . Smith’s poles
m = 10
σ ′′(ς − ς A )
& & σ ′ = 0. 2
σ ′′(ς& − ς& A )
κ ′′(ς − ς A ) σ ′′ = 0.3
κ ′′(υ − υ A ) κ ′′(υ − υ A )
z κ ′′(ς − ς A ) ς z
ς κ ′ = 50
B B σ ′′(υ − υ A )
& &
σ ′′(υ − υ A )
& & κ ′′ = 100
υ υ 0 ≤ ω ≤ 10
κ ′y A
A ωt ωt
σ ′y A
& y y
O O
κ ′z A σ ′z A
& Re(s )
Free Body diagram for massless Free Body diagram for rotor
shaft (moments not shown)
Root locus
(a) (b)
Figure 3 (a) Free body diagram for massless shaft Im(s )
(moments not shown), and (b) Free body diagram for s-plane
rotor
* Our poles
ω = 400
. Smith’s poles ω=0
The free body diagrams results equations of the form
m = 10
M&& + Cx + (K + ωG )x = 0
x & (3) σ ′ = 0. 2 Re(s )
σ ′′ = 0.3
κ ′ = 50
where M , C , K , G are the mass, damping, stiffness and κ ′′ = 100
0 ≤ ω ≤ 400
gyroscopic matrices respectively, each of order 4 and
T
x = ( yA zA y z) . (4)
The First-Order realization of the above equation gives the Root Locus ∆ω=10
roots s of the characteristic equation φ (ω , s ) of the system.
Acknowledgements
We denote
6
φ (ω , s ) = ∑ α k s k . (5) I would like to mention regards to my advisor Dr. Yitshak
k =0 M Ram for his guidance. I convey my sincere thanks to
Siemens Energy and Automation, Inc., for sponsoring the
The characteristic polynomial corresponding to Smith’s project.
formulation, see e.g., [2] is defined as
References
4
φ (ω , s ) = ∑ α k s k .
ˆ ˆ (6)
k =0
1. S. H. Crandall. Physical explanations of the
destabilizing effect of damping in rotating parts,
Results Texas A & M University Instability Workshop,
369-383 (1980).
For our system, eight real solutions appear 2. D. M. Smith. The motion of a rotor carried by a
flexible shaft in flexible bearings, Proceedings of
{ω = 6.69551004 s = ±1.82583925} (7) the Royal Society (London) A 142, 92-118 (1933).
3. H. Zeigler. Principles of structural stability, 1968
{ω = −6.69551004 s = ±1.82583925} (8)
(Blaisdell Publishing Company, Waltham).
{ω = 18632.40800 s = ±2.23597851i} (9)
{ω = −18632.40800 s = ±2.23597851i} (10)
The system is unstable for 6.695510048 < ω < 18632.40800
and stable for other positive values of ω .
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