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


Outline

3 / 27






Structures of interest

Structures of interest

4 / 27

Jet engines

Electricity power plants

Machine tools

Gyroscopes


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


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


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


Outline

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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


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


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


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


Displacement discretization


13 / 27

Discretized displacement field

• Rayleigh-Ritz formulation

• Finite Element formulation⇒ u(x, t) =

n∑

i=1

Φi(x)ui (t)




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




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


Outline

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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


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


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


Outline

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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




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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




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




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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)




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




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A few modes• cylindrical modes

• conical modes


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




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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


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

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