Introduction to rotordynamics - McGill...

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

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Jet engines

Electricity power plants

Machine tools

Gyroscopes

Introduction Equations of motion Structural analysis Case studies References

Mechanical components

Mechanical components

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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

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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

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• 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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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[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