intro rotor dynamics

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Introduction to rotordynamics

Mathias Legrand

McGill University

Structural Dynamics and Vibration Laboratory

October 27, 2009


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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


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Structures of interest

Structures of interest

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Jet engines

Electricity power plants

Machine tools

Gyroscopes

I d i E i f i S l l i C di R f


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Mechanical components

Mechanical components

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Terminology

Supporting components

Driving components

Operating components

supporting components

operatingcomponents

driving components

Definitions

Supporting components: journal bearings, seals, magnetic bearings

Driving components: shaft

Operating components: bladed disk, machine tools, gears

I t d ti E ti f ti St t l l i C t di R f


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Selected topics

Selected topics

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1 Driving components linear vs. nonlinear formulations constant vs. non-constant

rotational velocities -dependent eigenfrequencies

isotropic vs. anisotropiccross-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 coupling3 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



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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 Fppl, 1905 Belluzo, Stodola notion of supercritical speed

1919 Jeffcott The lateral vibration of loaded shafts in the neighborhodof a whirling speed

Before WWII: Myklestadt-Prohlmethod lumped systems cantilever aircraft wings precise computation of critical speeds

Contemporary tools finite element method

nonlinear analysis



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Outline

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2

Equations of motionInertial and moving framesDisplacements and velocitiesStrains and stressesEnergies and virtual worksDisplacement discretization

3

Structural analysisStatic equilibriumModal analysis




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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 ofM in Rm u: displacement vector ofM/Rm in Rm s: displacement vector ofO/Rg in Rg y: displacement vector ofM/Rg in Rg



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q y

Displacements and velocities

Displacements and velocities

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ex

ey

ez

e1

e2

e3

eX

eY

eZ

z

1Y

Rotation matricesR1 =

cos z sin z 0

sin z cos z 0

0 0 1

; R2 =

cos Y 0 sin Y

0 1 0

sin Y 0 cos Y

; R3 =

1 0 0

0 cos 1 sin 10 sin 1 cos 1

R= R3R2R1

Coordinates transformation absolute displacement y= s+ R(x+ u) angular velocity matrix | R= R

Velocities Time derivatives of positions in Rg y= s+ Ru + R(x + u) =s + Ru+ R(x + u)

=

0 3(t) 2(t)3(t) 0 1(t)

2(t) 1(t) 0


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Energies and virtual works

Energies and virtual works

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Kinetic energy

EK=12

V

yTydV =12

V

uTudV+

V

uTudV+12

V

uT2udV

V

(uT uT)(RTs+ x)dV+12

V

sTs +2sTRx x2x dV

Strain energy

U=12

V

TCdV

External virtual works

Wext = VuTRfdV+

SuTRtdS (f, t)expressed in Rm



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Displacement discretization


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Discretized displacement field

Rayleigh-Ritz formulation

Finite Element formulation u(x, t) =

ni=1

i(x)ui(t)



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Generic equations of motion

Mu + (D + G())u +

K(u) + P() + N()

u= R() + F

Structural matrices mass matrix M= V

TdV

gyroscopic matrix G=V2

T

dV

centrifugal stiffening matrix N=VT2dV

angular acceleration stiffening matrix P=VTdV

stiffness matrix K= V TdV = V

T(u)C(u)dV nonlinear in u

damping matrix D(several different definitions)

External force vectors inertial forces R=

VT(RTs+ x + 2x)dV

external forces F= VTfdV+ SF TtdS



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Generic equations of motion

Mu + (D + G())u +

K(u) + P() + N()

u= R() + F

Choice space functions 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



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Outline

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Static equilibrium

Static equilibrium

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Solution and pre-stressed configuration

Find displacementu0, solution of:K(u0) + N()

u0 =R() + F

V0 V V(t)

u0i,

0ij,

0ij u

i ,

ij,

ij

initialconfiguration

pre-stressedconfiguration

additional dynamicconfiguration

Applications slender structures plates, shells shaft with axial loading radial traction and blade untwist



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Modal analysis

Modal analysis

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Eigensolutions

Defined with respect to the pre-stressed configuration V

State-space formulation

M D + G()0 M

u

u + 0 K(u0) + N()

M 0 u

u= 0

0 (1) Consider:

y=

u

u

= Aet and find (y, )solution of (1)

Comments -dependent complex (conjugate) eigenmodes and eigenvalues

Right and left eigenmodes

Whirl motions

Campbell diagrams and stability conditions



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Modal analysis

Modal analysis

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Campbell diagrams

Coincidence of vibration sources and -dependent natural resonances

Quick visualisation of vibratory and design problems

(Hz)

backward whirl

forward whirl

frequency(Hz)

0 2 4 6 8 10 120

1

2

3

4

5

6

7

8

9

(Hz)

unstable

()

0 2 4 6 8 10 12

-80

-60

-40

-20

0

20

40

60

80

Stability

i | (i)>0 unstable



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Outline

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Case 1: flexible shaft bearing system


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Rotating shaft bearing system

Constant velocity Small deflections Elementary slicerigid disk

bearing 1bearing 2

oil film

Lb

Y

X

Z shaft

L

Quantities of interest

Kinetic Energy

EK =S

2

L0

(u2+ w2)dy+I

2

L0

u

y

2+

w

y

2dy+IL22I

L0

u

y

w

ydy

Strain Energy U= EI

2

L0

2u

y2

2+

2w

y2

2dy

Virtual Work of external forces W =Fu+Gw



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Z

X

journal

bearing

e

ObOj(xj, zj)

h(, t)

Reynolds equation h journal/bearing clearance fluid viscosity angular velocity of the journal Router radius of the beam pfilm pressure

y

h3

6p

y

+

1R2

h3

6p

=

h

+2

h

t

Bearing forces

Hj=1 Zjcos(+) +Xjsin(+) G=Zjsin(+) +Xjcos(+) 2

Zjcos(+) Xjsin(+)

Fx= R Lb

3

2c2

0

Gsin(+)

Hj3 d; Fz=

RLb3

2c2

0

Gcos(+)

Hj3 d



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Linearization Equilibrium position Xe,Ze First order Taylor series of the nonlinear forces

KXX = FXX

e

; KXZ = FXZ

e

; KZX = FZX

e

; KZZ= FZZ

e

CXX= FX

X

e

; CXZ = FX

Z

e

; CZX = FZ

X

e

; CZZ = FZ

Z

e

Field of displacement

U(x, t) =

Ni=1

i(x)ui(t); W(x, t) =

Ni=1

i(x)wi(t) x= (u,w)T

Linear equations of motion

Mx + (G+ Cb)x + (K+ Kb)x= 0


C fl f


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Linear equations of motion

Mx + (G+ Cb())x+ (K+ Kb())x= FNL

M=

A + B 0

0 A + B

; G=

0 2B2B 0

; K=

C 0

0 C

A= S

L0

Tdy; B= I

L0

,Ty ,ydy; C= EI

L0

,Tyy,yydy

Comments

Kbcirculatory matrix destabilizes the system Cbsymmetric damping matrix both matrices act on the shaft at the bearing locations (external virtual work)


C 1 fl ibl h f b i


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Modal analysis

Equilibrium position for each

Cb, Kbfor each

Modal analysis for each

(Hz)

()

0 5 10 15 20 25 30 35 40 45-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(Hz)

()

0 5 10 15 20 25 30 35-15

-10

-5

0

5

10

15

Stability: linear vs. nonlinear models

Frequency(Hz)

(Hz)

Amplitude

0 50

100 150

200 250

0

100

200

3000

1

2

3


C 1 fl ibl h ft b i t


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Modal analysis

Equilibrium position for each

Cb, Kbfor each

Modal analysis for each

(Hz)

(

)

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(Hz)

(

)

0 5 1 0 1 5 2 0 2 5 3 0 3 5-15

-10

-5

0

5

10

15

Stability: linear vs. nonlinear models

Frequency(Hz)

(Hz)

Amplitud

e

0 50

100 150

200 250

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 ofMand K Notion of mistuning Constant




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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 ofMand K Notion of mistuning Constant

Next-generation calculations

Nonlinear terms Supporting+transmission+operating components Damping, gyroscopic terms, stiffening Time-dependent



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References

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[1] Giancarlo GentaDynamics of Rotating SystemsMechanical Engineering Series

Springer, 2005

[2] Michel Lalanne& Guy Ferraris

Rotordynamics Prediction in EngineeringSecond Edition

John Wiley & Sons, 1998

[3] Jean-Pierre LanStructural Dynamics (in French)

Lecture Notescole Centrale de Lyon, 2005

intro rotor dynamics

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