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Introduction to rotordynamics
Mathias Legrand
McGill University
Structural Dynamics and Vibration Laboratory
October 27, 2009
Introduction Equations of motion Structural analysis Case studies References
Outline
2 / 27
1 IntroductionStructures of interestMechanical componentsSelected topicsHistory and scientists
2 Equations of motionInertial and moving framesDisplacements and velocitiesStrains and stressesEnergies and virtual worksDisplacement discretization
3 Structural analysisStatic equilibriumModal analysis
4 Case studiesCase 1: flexible shaft bearing systemCase 2: bladed disks
Introduction Equations of motion Structural analysis Case studies References
Outline
3 / 27
1 IntroductionStructures of interestMechanical componentsSelected topicsHistory and scientists
2 Equations of motionInertial and moving framesDisplacements and velocitiesStrains and stressesEnergies and virtual worksDisplacement discretization
3 Structural analysisStatic equilibriumModal analysis
4 Case studiesCase 1: flexible shaft bearing systemCase 2: bladed disks
Introduction Equations of motion Structural analysis Case studies References
Structures of interest
Structures of interest
4 / 27
Jet engines
Electricity power plants
Machine tools
Gyroscopes
Introduction Equations of motion Structural analysis Case studies References
Mechanical components
Mechanical components
5 / 27
Terminology
• Supporting components
• Driving components
• Operating components
supporting components
operatingcomponents
Ω
driving components
Definitions
• Supporting components: journal bearings, seals, magnetic bearings
• Driving components: shaft
• Operating components: bladed disk, machine tools, gears
Introduction Equations of motion Structural analysis Case studies References
Selected topics
Selected topics
6 / 27
1 Driving components
linear vs. nonlinear formulations constant vs. non-constant
rotational velocities Ω Ω-dependent eigenfrequencies isotropic vs. anisotropic
cross-sections rotational vs. reference frames stationary and rotating dampings gyroscopic terms external forcings and imbalances critical rotational velocities,
Campbell diagrams, stability reduced-order models and modal
techniques fluid-structure coupling bifurcation analyses,chaos
2 Supporting components
linear vs. nonlinear formulations constant vs. non-constant
rotational velocities gyroscopic terms fluid-structure coupling
3 Operating components
linear vs. nonlinear formulations concept of axi-symmetry reduced-order models fluid-structure coupling critical rotational velocities,
Campbell diagrams, stability fixed vs. moving axis of rotation centrifugal stiffening
Introduction Equations of motion Structural analysis Case studies References
History and scientists
History and scientists
7 / 27
• 1869 – Rankine – On the centrifugal force on rotating shafts
steam turbines notion of critical speed
• 1895 – Föppl, 1905 – Belluzo, Stodola
notion of supercritical speed
• 1919 – Jeffcott – The lateral vibration of loaded shafts in the neighborhod
of a whirling speed
• Before WWII: Myklestadt-Prohl method lumped systems cantilever aircraft wings precise computation of critical speeds
• Contemporary tools finite element method nonlinear analysis
Introduction Equations of motion Structural analysis Case studies References
Outline
8 / 27
1 IntroductionStructures of interestMechanical componentsSelected topicsHistory and scientists
2 Equations of motionInertial and moving framesDisplacements and velocitiesStrains and stressesEnergies and virtual worksDisplacement discretization
3 Structural analysisStatic equilibriumModal analysis
4 Case studiesCase 1: flexible shaft bearing systemCase 2: bladed disks
Introduction Equations of motion Structural analysis Case studies References
Inertial and moving frames
Inertial and moving frames
9 / 27
ex
ey
ez
e1
e2
e3 s
u
O
O′
M
Rm
Rg
y
Full 3D rotating motion of the dynamic frame• Fixed (galilean, inertial) reference frame Rg
• Moving (non-inertial) reference frame Rm
• Rigid body large displacements Rm/Rg
• Small displacements and strains in Rm
Coordinates• x: position vector of M in Rm
• u: displacement vector of M/Rm in Rm
• s: displacement vector of O′/Rg in Rg
• y: displacement vector of M/Rg in Rg
Introduction Equations of motion Structural analysis Case studies References
Displacements and velocities
Displacements and velocities
10 / 27
ex
ey
ez
e1
e2
e3
eX
eY
eZ
θz
θ1
θY
Rotation matrices
R1 =
[
cos θz − sin θz 0
sin θz cos θz 0
0 0 1
]
; R2 =
[
cos θY 0 − sin θY
0 1 0
sin θY 0 cos θY
]
; R3 =
[
1 0 0
0 cos θ1 − sin θ1
0 sin θ1 cos θ1
]
⇒ R = R3R2R1
Coordinates transformation• absolute displacement y = s + R(x + u)• angular velocity matrix Ω | R = RΩ
Velocities• Time derivatives of positions in Rg
• y = s + Ru + R(x + u) = s + Ru + RΩ(x + u)
Ω =
0 −Ω3(t) Ω2(t)Ω3(t) 0 −Ω1(t)−Ω2(t) Ω1(t) 0
Introduction Equations of motion Structural analysis Case studies References
Strains and stresses
Strains and stresses
11 / 27
Quantities of interest
• Stresses σ
• Strains ε: Green measure
εij =12
(
∂ui
∂xj
+∂uj
∂xi
+∂um
∂xi
∂um
∂xj
)
= εlnij + εnl
ij
• Pre-stressed configuration and centrifugal stiffening
Constitutive law
• Generalized Hooke law σ = Cε
• C→ linear elasticity tensor
Introduction Equations of motion Structural analysis Case studies References
Energies and virtual works
Energies and virtual works
12 / 27
Kinetic energy
EK =12
∫
V
ρyT ydV =12
∫
V
ρuT udV +
∫
V
ρuTΩudV +
12
∫
V
ρuTΩ
2udV
−
∫
V
ρ(uTΩ− uT )(RT s +Ωx)dV +
12
∫
V
ρ(
sT s + 2sT RΩx − xΩ2x)
dV
Strain energy
U =12
∫
V
εT CεdV
External virtual works
Wext =
∫
V
uT RfdV +
∫
S
uT RtdS ← (f, t) expressed in Rm
Introduction Equations of motion Structural analysis Case studies References
Displacement discretization
Displacement discretization
13 / 27
Discretized displacement field
• Rayleigh-Ritz formulation
• Finite Element formulation⇒ u(x, t) =
n∑
i=1
Φi(x)ui (t)
Introduction Equations of motion Structural analysis Case studies References
Displacement discretization
Displacement discretization
13 / 27
Generic equations of motion
Mu + (D + G(Ω)) u +(
K(u) + P(Ω) + N(Ω))
u = R(Ω) + F
Structural matrices• mass matrix M =
∫
VρΦT
ΦdV
• gyroscopic matrix G =∫
V2ρΦT
ΩΦdV
• centrifugal stiffening matrix N =∫
VρΦT
Ω2ΦdV
• angular acceleration stiffening matrix P =∫
VρΦT
ΩΦdV
• stiffness matrix K =∫
Vσ
TεdV =
∫
Vε
T (u)Cε(u)dV ← nonlinear in u
• damping matrix D (several different definitions)
External force vectors• inertial forces R = −
∫
VρΦT (RT s + Ωx +Ω
2x)dV
• external forces F =∫
VΦ
T fdV +∫
SFΦ
T tdS
Introduction Equations of motion Structural analysis Case studies References
Displacement discretization
Displacement discretization
13 / 27
Generic equations of motion
Mu + (D + G(Ω)) u +(
K(u) + P(Ω) + N(Ω))
u = R(Ω) + F
Choice space functions Φ and time contributions u• order-reduced models or modal reductions• 1D, 2D or 3D elasticity• beam and shell theories (Euler-Bernoulli, Timoshenko, Reissner-Mindlin. . . )• complex geometries: bladed-disks
Other simplifying assumptions• constant velocity along a single axis of rotation• no gyroscopic terms• no centrifugal stiffening• localized vs. distributed imbalance• isotropy or anisotropy of the shaft cross-section
Introduction Equations of motion Structural analysis Case studies References
Outline
14 / 27
1 IntroductionStructures of interestMechanical componentsSelected topicsHistory and scientists
2 Equations of motionInertial and moving framesDisplacements and velocitiesStrains and stressesEnergies and virtual worksDisplacement discretization
3 Structural analysisStatic equilibriumModal analysis
4 Case studiesCase 1: flexible shaft bearing systemCase 2: bladed disks
Introduction Equations of motion Structural analysis Case studies References
Static equilibrium
Static equilibrium
15 / 27
Solution and pre-stressed configuration
Find displacement u0, solution of:(
K(u0) + N(Ω))
u0 = R(Ω) + F
V0 V∗
V (t)u
0i , ε0
ij , σ0ij u
∗
i , ε∗
ij , σ∗
ij
initialconfiguration
pre-stressedconfiguration
additional dynamicconfiguration
Applications• slender structures• plates, shells• shaft with axial loading• radial traction and blade untwist
Introduction Equations of motion Structural analysis Case studies References
Modal analysis
Modal analysis
16 / 27
Eigensolutions
• Defined with respect to the pre-stressed configuration V ∗
• State-space formulation[
M D + G(Ω)0 −M
] (
u
u
)
+
[
0 K(u0) + N(Ω)M 0
] (
u
u
)
=
(
0
0
)
(1)
• Consider:
y =
(
u
u
)
= Aeλt and find (y, λ) solution of (1)
Comments• Ω-dependent complex (conjugate) eigenmodes and eigenvalues
• Right and left eigenmodes
• Whirl motions
• Campbell diagrams and stability conditions
Introduction Equations of motion Structural analysis Case studies References
Modal analysis
Modal analysis
17 / 27
Campbell diagrams
• Coincidence of vibration sources and ω-dependent natural resonances
• Quick visualisation of vibratory and design problems
Ω (Hz)
Ω
backward whirl
forward whirl
freq
uen
cy(H
z)
0 2 4 6 8 10 120
1
2
3
4
5
6
7
8
9
Ω (Hz)
unstable
ℜ(λ
)
0 2 4 6 8 10 12
-80
-60
-40
-20
0
20
40
60
80
Stability
∃λi | ℜ(λi ) > 0 ⇒ unstable
Introduction Equations of motion Structural analysis Case studies References
Outline
18 / 27
1 IntroductionStructures of interestMechanical componentsSelected topicsHistory and scientists
2 Equations of motionInertial and moving framesDisplacements and velocitiesStrains and stressesEnergies and virtual worksDisplacement discretization
3 Structural analysisStatic equilibriumModal analysis
4 Case studiesCase 1: flexible shaft bearing systemCase 2: bladed disks
Introduction Equations of motion Structural analysis Case studies References
Case 1: flexible shaft bearing system
Case 1: flexible shaft bearing system
19 / 27
Rotating shaft bearing system
• Constant velocity• Small deflections• Elementary slice ⇔ rigid disk
bearing 1
bearing 2
oil film
Lb
Y
X
Z shaft
L
Quantities of interest
• Kinetic Energy
EK =ρS
2
L∫
0
(u2+w 2)dy+ρI
2
L∫
0
[
(
∂u
∂y
)2
+
(
∂w
∂y
)2]
dy+ρILΩ2−2ρIΩ
L∫
0
∂u
∂y
∂w
∂ydy
• Strain Energy U =EI
2
L∫
0
[
(
∂2u
∂y2
)2
+
(
∂2w
∂y2
)2]
dy
• Virtual Work of external forces δW = Fδu + Gδw
Introduction Equations of motion Structural analysis Case studies References
Case 1: flexible shaft bearing system
Case 1: flexible shaft bearing system
20 / 27
θ
φZ
X
journal
bearing
e
Ω
Ob
Oj (xj , zj )
h(θ, t)
Reynolds’ equation• h journal/bearing clearance• µ fluid viscosity• Ω angular velocity of the journal• R outer radius of the beam• p film pressure
∂
∂y
(
h3
6µ
∂p
∂y
)
+1
R2
∂
∂θ
(
h3
6µ
∂p
∂θ
)
= Ω∂h
∂θ+ 2
∂h
∂t
Bearing forces
• Hj = 1− Zj cos(θ + φ) + Xj sin(θ + φ)
• G = Zj sin(θ + φ) + Xj cos(θ + φ)− 2(
Zj cos(θ + φ)− Xj sin(θ + φ))
Fx = −µRLb
3Ω
2c2
π∫
0
G sin(θ + φ)
Hj3 dθ; Fz =
µRLb3Ω
2c2
π∫
0
G cos(θ + φ)
Hj3 dθ
Introduction Equations of motion Structural analysis Case studies References
Case 1: flexible shaft bearing system
Case 1: flexible shaft bearing system
21 / 27
Linearization• Equilibrium position Xe , Ze
• First order Taylor series of the nonlinear forces
KXX =∂FX
∂X
∣
∣
∣
∣
e
; KXZ =∂FX
∂Z
∣
∣
∣
∣
e
; KZX =∂FZ
∂X
∣
∣
∣
∣
e
; KZZ =∂FZ
∂Z
∣
∣
∣
∣
e
CXX =∂FX
∂X
∣
∣
∣
∣
e
; CXZ =∂FX
∂Z
∣
∣
∣
∣
e
; CZX =∂FZ
∂X
∣
∣
∣
∣
e
; CZZ =∂FZ
∂Z
∣
∣
∣
∣
e
Field of displacement
U(x , t) =
N∑
i=1
φi (x)ui(t); W (x , t) =
N∑
i=1
φi (x)wi(t) ⇒ x = (u, w)T
Linear equations of motion
Mx + (G + Cb)x + (K + Kb)x = 0
Introduction Equations of motion Structural analysis Case studies References
Case 1: flexible shaft bearing system
Case 1: flexible shaft bearing system
22 / 27
Linear equations of motion
Mx + (ΩG + Cb(Ω))x + (K + Kb(Ω))x = FNL
M =
[
A + B 0
0 A + B
]
; G =
[
0 −2B
2B 0
]
; K =
[
C 0
0 C
]
A = ρS
L∫
0
ΦTΦdy ; B = ρI
L∫
0
Φ,Ty Φ,y dy ; C = EI
L∫
0
Φ,Tyy Φ,yy dy
Comments
• Kb circulatory matrix → destabilizes the system• Cb symmetric damping matrix• both matrices act on the shaft at the bearing locations (external virtual work)
Introduction Equations of motion Structural analysis Case studies References
Case 1: flexible shaft bearing system
Case 1: flexible shaft bearing system
23 / 27
Modal analysis
• Equilibrium position for each Ω
• Cb , Kb for each Ω
• Modal analysis for each Ω
Ω (Hz)
ℜ(λ
)
0 5 10 15 20 25 30 35 40 45-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Ω (Hz)
ℑ(λ
)
0 5 10 15 20 25 30 35-15
-10
-5
0
5
10
15
Stability: linear vs. nonlinear models
Frequency(Hz)
Ω(Hz)
Am
plitu
de
050
100150
200250
0
100
200
3000
1
2
3
Introduction Equations of motion Structural analysis Case studies References
Case 1: flexible shaft bearing system
Case 1: flexible shaft bearing system
23 / 27
Modal analysis
• Equilibrium position for each Ω
• Cb , Kb for each Ω
• Modal analysis for each Ω
Ω (Hz)
ℜ(λ
)
0 5 10 15 20 25 30 35 40 45-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Ω (Hz)
ℑ(λ
)
0 5 10 15 20 25 30 35-15
-10
-5
0
5
10
15
Stability: linear vs. nonlinear models
Frequency(Hz)
Ω(Hz)
Am
plitu
de
050
100150
200250
0
100
200
3000
1
2
3
Introduction Equations of motion Structural analysis Case studies References
Case 1: flexible shaft bearing system
Case 1: flexible shaft bearing system
24 / 27
A few modes• cylindrical modes
• conical modes
Introduction Equations of motion Structural analysis Case studies References
Case 2: bladed disks
Case 2: bladed disks
25 / 27
Definition
• Closed replication of a sector• Cyclic-symmetry
Mathematical properties
• Invariant with respect to the axis of rotation• Block-circulant mass and stiffness matrices• Pairs of orthogonal modes with identical frequency
Numerical issues
• No damping and gyroscopic matrices• Determination of M and K
• Notion of mistuning• Constant Ω
Introduction Equations of motion Structural analysis Case studies References
Case 2: bladed disks
Case 2: bladed disks
26 / 27
Multi-stage configuration
• Closed replication of a "slice"• Different number of blades• No cyclic-symmetry
Numerical issues
• No damping and gyroscopic matrices• Determination of M and K• Notion of mistuning• Constant Ω
Next-generation calculations
• Nonlinear terms• Supporting+transmission+operating components• Damping, gyroscopic terms, stiffening• Time-dependent Ω
Introduction Equations of motion Structural analysis Case studies References
References
27 / 27
[1] Giancarlo Genta
“Dynamics of Rotating Systems”Mechanical Engineering Series
Springer, 2005
[2] Michel Lalanne & Guy Ferraris
“Rotordynamics Prediction in Engineering”Second Edition
John Wiley & Sons, 1998
[3] Jean-Pierre Laîné
“Structural Dynamics” (in French)Lecture Notes
École Centrale de Lyon, 2005