3-WindTurbineAeroelasticity

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Short Course on Wind Energy

- Wind Turbine Aeroelasticity -

Carlo L. Bottasso Politecnico di Milano

November 2011

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POLITECNICO di MILANO POLI-Wind Research Lab

Contents

• Blade dynamics - Rigid flapping and lagging blade - Elastic blade - Simplified blade element aerodynamics

• Basic concepts in aeroelasticity - The rotor as a filter - Aerodynamic damping - Divergence - Flutter

• Stability - Concepts of static stability - Stability of LTI systems (with and without analytical model) - Stability of LTP systems (with and without analytical model)

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Blade Dynamics

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The Flapping Equation

Consider a rigid flapping blade (simplest possible approximation of beam flapwise bending):

Acceleration at hinge H

Hinge offset

Gravity

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Flap stiffness due to centrifugal loading: Flap stiffness due to gravity: Restoring moments proportional to flap angle


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Kinematic quantities:

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Hinge moment:

Aerodynamic Gravity Hinge spring

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Equations of dynamic equilibrium wrt an accelerating moving frame centered in H: Moments of inertia wrt hinge H:

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Equations of motion (in components): Feathering (twist), flap and lag dynamic equilibrium:

Gravity Spring

Centrifugal load

No twisting

Lag moment due to flapping (Coriolis)

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Flap dynamic equilibrium: Hinge offset for a uniform blade:


Hinge offset Non-rotating natural frequency

Centrifugal stiffness Gravity induced stiffness (pulsating)

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In a vacuum and neglecting gravity, harmonic oscillator (no damping): Fundamental frequency increased by centrifugal stiffening and hinge offset Campbell (Southwell, fan) diagram:


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Forced spring-mass-damper system: Set Solution:

Response of Second Order Systems

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Frequency ratio (forcing/natural): Amplitude: Phase:

Response of Second Order Systems

(From Bramwell 2001)

Forcing at resonance results in 90 deg delay

Peak response obtained at freq ratios smaller than resonance

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Typically for a wind turbine blade: For a hinged blade (null spring):



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Flapping Equation for an Elastic Blade

Equilibrium of blade segment:

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Differential equations of equilibrium: From beam theory: Equations of motion:

Flapping Equation for an Elastic Blade

Centrifugal stiffening

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Rigid Blade Dynamics

Considering lag (simplest possible approximation of blade edgewise bending):

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Considering yaw:

Acceleration at hinge H due to Ω

Acceleration at hinge H due to q

Yaw rate


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Kinematic quantities: (having dropped small terms )

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Kinematic quantities:

Neglect (small if q<<Ω)

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Feathering (twist), flap and lag dynamic equilibrium:

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Feathering equilibrium:

Gyroscopic blade twisting due to yawing

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Flapping equilibrium:

Non-rotating freq Flap/lag coupling

Centrifugal stiffness Gravity Yaw (Coriolis)

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The flap and feathering gyroscopic moments can be used to explain also gyroscopic effects on the whole rotor:

Flap Feather

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Computing the net effects of all B blades: and the rotor is subjected to a tilting moment when it yaws This is the same result obtained using the Principal Theorem of the Gyroscope:

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Equilibrium in lag:

Non-rotating frequency

Centrifugal stiffness Lag due to flap (Coriolis)

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Centrifugal stiffening: notice different behavior in flap and lag Flap: Lag:

Centrifugal force changes direction with lag angle

Centrifugal force does not change direction with flap angle

Larger stiffening effect

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Blade Element Aerodynamics

Simplified blade element aerodynamics (for development of analytical models):

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Linear vertical wind shear

Crossflow

View from above

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Wind+inflow Yaw Vertical shear

Flap damping Cross-flow

Out-of-plane local wind:

View from above

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Rotor speed

Cross-flow

In-plane local wind:

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Flapping moment:

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Flapping moment: In terms of non-dimensional quantities:

Non-dimensional quantities:

Lock number (ratio of aerodynamic and

inertial forces)

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Flapping dynamic equilibrium including aerodynamics:

Lock number <0 if blade stalled (Clα <0)

Damping

Flap Cross-flow

Centrifugal Nat. freq. Gravity Cross-flow

Cross-flow

Stiffness

Coning

Vertical shear Yaw

Gyroscopic Aerodynamic

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Flapping-induced aerodynamic damping: Flap damping ratio: For a typical blade: i.e. rapid damping of flapping blade motions:

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Analytical solution of the flapping equation: 1. Assume the solution is of the form (dropping higher harmonics) 2. Insert into 3. Collect terms to match harmonic coefficients (dropping higher harmonics)


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Assumed flapping solution:


View from above

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Solving system in matrix form: where:

Flap freq including centrifugal effect

with offset

Axisymmetric flow term


Gravity

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FG

CF

VS

Y


FG: flow+gravity CF: cross-flow VS: vertical wind shear Y: yaw : determinant of solving system

Solution organized by contributors to response:

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Consider solution to flow and gravity: Cyclic sharing (ratio of sine and cosine harmonic amplitudes): Hinged blade: Mostly yawing Stiff blade: Mostly tilting This can also be explained in terms of phase lag: gravity is a cosine input


FG

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The Lagging Equation

Equilibrium in lag:

Non-rotating frequency

Centrifugal stiffness Lag due to flap (Coriolis)

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Lagging dynamic equilibrium including aerodynamics: Stiffness

Centrif. Gravity Nat. freq.

Yaw

Lag-flap coupling

Steady lag

Vertical shear Cross-flow

Gravity

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Undamped oscillator: Remarks: - In reality, there is small damping term due to changes in drag (neglected in the present derivations) - In any case, damping in lag is much smaller than in flap (see later on)

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Analytical solution of the lagging equation: 1. Assume the solution is of the form (dropping higher harmonics) 2. Insert into 3. Collect terms to match harmonic coefficients (dropping higher harmonics)


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Lag motion solution - steady lag angle

where:

Denominator is null if:


Lag freq including centrifugal effect

with offset



Gravity

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Lag motion solution – cosine response

Lag motion solution – sine response

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Basic Concepts in Aeroelasticity

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The Rotor as a Filter

Periodic trimmed condition in non-turbulent wind: all blades have same motion and loads Fore-aft tower force: • Rotating frame: complex Fourier series expansion of blade shear • Non-rotating frame: total fore-aft force on tower

Only pB/rev harmonics are transmitted to the tower

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Rotor torque: Rotating frame: complex Fourier series expansion of in-plane blade shear and bending moment • Non-rotating frame: total torque

Transmission of the sole pB/rev harmonics


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Side-side tower force: Rotating frame: complex Fourier series expansion of in-plane blade shear and axial force • Non-rotating frame: total side force

pB/rev harmonics caused by rotating pB±1/rev harmonics


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This concept leads to the definition of multiblade (or Coleman) coordinates From blade coordinates to fixed-frame coordinates (assuming 3 blades): From fixed-frame coordinates to blade coordinates:


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Coleman coordinates can be used for transforming equations of motion written in terms of blade coordinates into equations written in terms of fixed-frame coordinates This does not completely remove the periodicity, simply filters out all harmonics which are not multiples of the number of blades (see IPC control later on for details) Interpretation: “Collective” average: Horizontal tilting: (i.e. yaw) Vertical tilting:


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Whirling Rotor Modes

CG of K-th blade: Assuming small lag angle:

CG of whole rotor: (compare with Coleman

transform)

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Non-dim. lag frequency: (see lag eqs.) Lag motion at the lag frequency: Insert into expression for CG, to get whirling rotor CG motion: Typically for wind turbines , hence (2 progressive) Coefficients:

Frequency ωζ+Ω Same direction as Ω (progressive)

Frequency ωζ-Ω Direction depends on sign (regressive when negative)

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Whirling CG motion excites side-side tower motions Possible resonant conditions (Campbell diagram):

Progressive ωζ+Ω

Progressive ωζ-Ω

1st side-side tower mode

2nd side-side tower mode

Possible resonant conditions

Not a possible resonant conditions Exchange of frequencies (need a coupled rotor-tower analysis to see this effect)

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Simple airfoil case: consider airfoil motion orthogonal to flow

Aerodynamic Damping

Airfoil speed

Air speed wrt to airfoil

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Aerodynamic Damping

Damping Positive and large in pre-stall conditions

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Simple airfoil case: consider airfoil motion aligned with flow

Aerodynamic Damping

Air speed wrt to airfoil

Airfoil speed

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Aerodynamic Damping

Damping Small compared to CLα

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Blade-like case: consider vibrating blade cross section (neglecting stall, rotor inflow and unsteady aerodynamic effects)

Aerodynamic Damping

Rotor plane

Neglect inflow for simplicity

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Linearizing the aerodynamic force components, one gets the damping terms: For a rough estimate, when

Aerodynamic Damping

The larger of the two

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Consider an elliptical motion of the blade cross section:

Aerodynamic Damping

Rotor plane

Parameter describing shape of elliptical motion (out-of-plane/in-plane displacement)

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Work on blade section (if negative, dissipation and positive damping): Effective damping (unstable motion if negative): Blade in-plane vibration: Blade out-of-plane vibration:

Aerodynamic Damping

Mean damping Damping term affected by direction of vibration

The larger of the two

Stall will reduce mean damping

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Divergence

Structural deflection under aerodynamic loading that enhances further the aerodynamic loading itself It is a static aeroelastic phenomenon (no presence of inertial or unsteady aerodynamics effects) Typical section (simplest model of blade torsional deformation):

Zero lift line Aerodynamic

center

Elastic axis

Structural deformation

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Divergence

Lift: Aerodynamic moment: Structural moment: Equilibrium: Structural torsional deflection: Divergence:

Divergence dynamic pressure

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Flutter

Flutter: instability due to the interaction of aerodynamic, elastic and inertial forces which result in the extraction of energy from the airstream, leading to limit cycles or catastrophic amplification of oscillatory motion Seldom (so far) a problem in wind turbines, but this might change with larger and slender new blades Difficult to draw conclusions with simple analytical models (very complex derivations even for few dofs and simplified equations) Simple models are used for understanding main parameters Then, use:

• Comprehensive models capable of capturing all physical effects (aerodynamic, structural, inertial) and couplings • Tools for damping estimation • Extensive parametric investigations

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Flutter Many possible potential flutter mechanisms (most are typically not very likely in wind turbines):

• Flap-torsion flutter

As the blade flaps, inertial and aerodynamic moments twist the blade, which in turn modify the flap forces Influenced (and cured) by chordwise CG position

• Flap-lag flutter Flap-lag coupling: flap induces lag by Coriolis forces, which induce change in angular velocity, which in turn change centrifugal and aerodynamic forces (including flap forces) Tendency to flutter increases for similar flap and lag frequencies

• Flap-lag-torsion flutter Similar to flap-lag, but further excited by changes in pitch due to torsion

• All the above coupled to tower/drive-train/nacelle modes • Whirl-flutter

Coupling with tower modes (bending and/or torsion) (very unlikely, would require extremely soft support)

• Stall-induced flutter Due to coupling between torsional blade deflection and dynamic stall effects on the airfoil aerodynamic pitching moment Leads to limit cycle oscillation


Torsional stiffness parameter

CG aft movement

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Dynamic stall: Flow remains attached for AOAs exceeding static stall angle, significant increase in maximum lift Separation delay due to: - Kinematic induced camber effect - Influence of shed wake - Unsteady turbulent boundary layer Adverse pressure gradient produces reversed flow at LE forming dynamic stall vortex (DSV) Secondary vortical structure at LE can produce additional lift increase Flow reattachment only for AOAs well below static stall angle, boundary layer separated for most of downstroke

Flutter

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Flutter

(Singh at al., JoA 2006)

Dynamic stall:

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Flutter

(Singh at al., JoA 2006)

Dynamic stall:

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Flutter

Stall-induced flutter

Work on airfoil (if negative, dissipation and positive damping) Flutter onset if net damping is negative

Counter-clockwise loop

Clockwise loop

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Flutter


Oscillating motion: Aerodynamic moment: Work: Thus damping depends on out-of-phase component of pitching moment wrt angle of attack

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Flutter


Several effects govern extent of negative damping loops: • Angle of attack (main parameter):

- Light dynamic stall: Minor flow separation from airfoil Small hysteresis, small change in airloads Sensitive to airfoil geometry and frequency of motion

- Deep dynamic stall: Vortex-shedding Large hysteresis, large rapid change in airloads Less sensitive to parameters

• Airfoil type • Frequency of motion

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Stability

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Thrust-wind stability (vibrations in stall-regulated wind turbines, floating wind turbines):

Concepts of Static Stability

Apparent wind on rotor plane

Unstable behavior

Stable behavior

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Torque-TSR stability:

Torque : TSR:

Concepts of Static Stability

Stable behavior

Unstable behavior

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Stability Analysis

Applications in wind turbine design and verification:

• Explaining the causes of observed vibration phenomena

• Assessing the proximity of the flutter boundaries

• Evaluating the efficacy of control laws for low-damped modes

• …

Desirable characteristics of stability analysis tools:

• Closed loop: damping of coupled wind turbine/controller system

• Applicable to arbitrary mathematical models (e.g., finite element multibody models, modal-based models, etc.)

• Applicable to a real wind turbine in the field

• Accounting correctly for underlying physics

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Contents

• Stability analysis for Linear Time Invariant (LTI) systems

• Continuous time analysis

• Continuous to discrete time conversion

• Discrete time analysis

• Stability analysis for Linear Time Periodic (LTP) systems

• Continuous time

• Continuous to discrete time conversion

• Discrete time analysis

• Input/output model in the discrete-time domain

• Auto-Regressive eXogenous (ARX) sequence (LTI)

- Identification / State space realization / Stability analysis

• Periodic Auto-Regressive eXogenous (PARX) sequence (LTP)

- Identification / State space realization / Stability analysis

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LTI Stability Analysis

Linear Time Invariant (LTI) system: Solution: Autonomous problem: Spectral decomposition: Solution is asymptotically stable iff

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LTI Stability Analysis

Eigenvalue matrix:

For each eigenvalue , define the frequency

and damping factor

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LTI Continuous to Discrete Conversion

Sample generic signal (input, output or state) at constant time step

Definition:

Continuous time:

Discrete time:

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Continuous time system:

Integrate (Lagrange formula) from time to time

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• Sample signal with time step :

• ZOH (zero-order-hold, input constant during time step)

Lagrange formula becomes

Introducing we get

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Continuous time solution:

ZOH solution at discrete time instants:

Discrete time state-space form:

And the relationship between continuous and discrete time forms is:

Remark: it is an approximation; it implies constant inputs within the step

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LTI Stability Analysis in Discrete Time

Discrete time solution

Autonomous problem:

Spectral decomposition:

Solution is asymptotically stable iff

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Return to continuous poles to get frequencies and damping factors

where is the j-th discrete pole and the j-th continuous pole

Having , one can compute the associated frequency and damping

Remark: one might use Tustin transformation

LTI Stability Analysis in Discrete Time

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LTP Systems

Wind turbine models are characterized by periodic coefficients Example: rigid blade flapping equation in first order form

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LTP Systems

Linear Time Periodic (LTP) system: Periodicity with period T:

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LTP Systems

Autonomous problem (i.e. ):

State transition matrix:

Remark: notice that, since

and

then the transition matrix obeys the following

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Other important remark: The state transition matrix at t+T is a linear combination of the state transition matrix at t, i.e. In fact, assuming the above holds, then Recalling that which gives and, by the periodicity of , we get which proves the initial statement

LTP Systems

Constant matrix

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Decomposition of the state transition matrix (Floquet normal form): : captures contractivity of solution : captures periodicity of solution The choice of the non-periodic part as implies that is periodic In fact

LTP Stability Analysis

Periodic

Constant matrix

Non-periodic

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Consider the change of coordinates Since is periodic, it is also bounded, and therefore the stability conditions for are the same as the ones for Then and using and we get the LTI whose stability is dictated by the eigenvalues of matrix

( , )


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Definition: monodromy matrix (transition matrix over one period) From the definition: and in turn Defining Dynamic equation of a time invariant discrete-time system Contractivity of solution is contained in matrix


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Compute spectral decomposition:

Solutions are stable iff the eigenvalues of are

(Remark: since , and have the same eigenvectors )

: characteristic exponents : characteristic multipliers

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Since , the relationship between and is

which gives in turn

Remark: arbitrariness in the imaginary part of !!!

The significance of this issue will be more clear later on …

For now ignore the arbitrariness choosing for example

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Rewrite transition matrix as a function of characteristic exponents

Notice that where

Then the state transition matrix becomes

where and is the order of the system

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Remark: is periodic and could be expanded in a Fourier series

The final form of the transition matrix is

The LTP exponents, , theoretically infinite, are present in the system response and matrices determine their relative contribution

Definition: modal participation factor

Measure of the relative strength of the n-th harmonic in the j-th mode

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The LTP exponents are the analogs of the LTI eigenvalues

For each exponent we can compute frequency and damping

, and determine the dynamic behavior of the LTP system

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Problem related to the multiplicity in the solutions of

The arbitrariness in the imaginary part of the is now understood:

• All exponents obtained by adding any integer multiple of 2π/T to the imaginary part of are present in the response of the system and each of them is associated to a specific participation factor

• The arbitrary choice in the multiple solutions of has no consequence since it involves a frequency shift in the harmonic content of such that the triads , and remain the same

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Periodic analysis of a LTI system

LTI: LTP:

Equating the two transition matrices

The closer the participation of a certain harmonic is to 1, the more the mode behaves as invariant (i.e. non periodic)

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Implementation:

1. Compute transition matrix: • Remarks:

- Potentially expensive! To get the transition matrix numerically by perturbation requires integrating the system for one period for each degree of freedom in the model

- Need high accuracy integration (perturbation needs to be small for linearity of response, need to capture effect of perturbation on response)

2. Get the monodromy matrix

3. Compute characteristic multipliers and characteristic exponents

4. Compute periodic eigenvectors

5. Compute modal participation factors

6. Triads describe the behavior of all modes of interest

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Numerical example:

Consider the following 2dof system

Remark: for the system is LTI with one pair of complex poles with frequency 0.7 rad/s and damping factor 0.2


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Consider the invariant case ( )

Response to a non-zero initial condition ( )


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Compute state transition matrix within a period, monodromy matrix and characteristic multipliers

Since , the system is asymptotically stable


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Compute characteristic exponents

5 -0.1400 + 5.3141i -0.1400 + 4.6859i

4 -0.1400 + 4.3141i -0.1400 + 3.6859i

3 -0.1400 + 3.3141i -0.1400 + 2.6859i

2 -0.1400 + 2.3141i -0.1400 + 1.6859i

1 -0.1400 + 1.3141i -0.1400 + 0.6859i

0 -0.1400 + 0.3141i -0.1400 - 0.3141i

-1 -0.1400 - 0.6859i -0.1400 - 1.3141i

-2 -0.1400 - 1.6859i -0.1400 - 2.3141i

-3 -0.1400 - 2.6859i -0.1400 - 3.3141i

-4 -0.1400 - 3.6859i -0.1400 - 4.3141i

-5 -0.1400 - 4.6859i -0.1400 - 5.3141i


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Compute for a given exponent ( e.g. )

Notice that and have the same amplitude but phases that differ by 180 degree


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Compute frequencies, damping factors and participation factors

5 5.3160 0.0263 0.00

4 4.3164 0.0324 0.00

3 3.3171 0.0422 0.00

2 2.3184 0.0604 0.00

1 1.3216 0.1059 0.00

0 0.3639 0.4071 0.00

-1 0.7000 0.2000 1.00

-2 1.6917 0.0828 0.00

-3 2.6895 0.0521 0.00

-4 3.6885 0.0380 0.00

-5 4.6879 0.0299 0.00

5 4.6879 0.0299 0.00

4 3.6885 0.0380 0.00

3 2.6895 0.0521 0.00

2 1.6917 0.0828 0.00

1 0.7000 0.2000 1.00

0 0.3639 0.4071 0.00

-1 1.3216 0.1059 0.00

-2 2.3184 0.0604 0.00

-3 3.3171 0.0422 0.00

-4 4.3164 0.0324 0.00

-5 5.3160 0.0263 0.00


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Consider the non-invariant case ( )

Response to a non-zero initial condition ( )


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Compute state transition matrix within a period, monodromy matrix and characteristic multipliers

Since , the system is asymptotically stable.


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Compute the characteristic exponents

5 -0.1400 + 0.3579i -0.1400 + 4.6421i

4 -0.1400 + 0.3579i -0.1400 + 3.6421i

3 -0.1400 + 0.3579i -0.1400 + 2.6421i

2 -0.1400 + 0.3579i -0.1400 + 1.6421i

1 -0.1400 + 0.3579i -0.1400 + 0. 6421i

0 -0.1400 + 0.3579i -0.1400 - 0.3579i

-1 -0.1400 - 0.6421i -0.1400 - 1.3579i

-2 -0.1400 – 1.6421i -0.1400 - 2.3579i

-3 -0.1400 - 2.6421i -0.1400 - 3.3579i

-4 -0.1400 - 3.6421i -0.1400 - 4.3579i

-5 -0.1400 - 4.6421i -0.1400 - 5.3579i


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Compute for a given exponent (e.g. )

Notice that and have the same amplitude but phases that differ by 180 degree


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Compute frequencies, damping factors and participation factors

5 5.3597 0.0261 0.0000

4 4.3602 0.0321 0.0002

3 3.3608 0.0417 0.0019

2 2.3621 0.0593 0.0179

1 1.3651 0.1026 0.0892

0 0.3843 0.3643 0.2802

-1 0.6572 0.2131 0.4769

-2 1.6480 0.0850 0.1121

-3 2.6458 0.0529 0.0196

-4 3.6448 0.0384 0.0018

-5 4.6442 0.0301 0.0001

5 4.6442 0.0301 0.0001

4 3.6448 0.0384 0.0018

3 2.6458 0.0529 0.0196

2 1.6480 0.0850 0.1121

1 0.6572 0.2131 0.4769

0 0.3843 0.3643 0.2802

-1 1.3651 0.1026 0.0892

-2 2.3621 0.0593 0.0179

-3 3.3608 0.0417 0.0019

-4 4.3602 0.0321 0.0002

-5 5.3597 0.0261 0.0000


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Fourier transform of state


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LTP Continuous to Discrete Conversion

The approach is similar to the LTI one and it is based on the comparison between the continuous time solution and the ZOH solution between two consecutive time instants

Continuous system:

Discrete system:

Resulting C/D equations

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LTP Stability Analysis in Discrete Time

Linear Time Periodic (LTP) system: Periodicity with period K:

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Autonomous problem (i.e. ):

State transition matrix:

Remark: notice that, since

and

then the transition matrix obeys the following


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Other important remark: The state transition matrix at k+K is a linear combination of the state transition matrix at k, i.e. In fact, assuming the above holds, then Recalling that which gives and, by the periodicity of , we get which proves the initial statement

Constant matrix


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Decomposition of the state transition matrix (Floquet normal form): : captures contractivity of solution : captures periodicity of solution The choice of the non-periodic part as implies that is periodic In fact

Periodic Constant matrix

Non-periodic


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Consider the change of coordinates Since is periodic, it is also bounded, and therefore the stability conditions for are the same as the ones for Then and using and we get the LTI whose stability is dictated by the eigenvalues of matrix

( , )

Stability Analysis of LTPs in Discrete Time

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Definition: monodromy matrix (transition matrix over one period) From the definition: and in turn Defining Dynamic equation of a time invariant discrete-time system Contractivity of solution is contained in matrix


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Compute spectral decomposition:

Solutions are stable iff the eigenvalues of are

(Remark: since , and have the same eigenvectors )

: characteristic exponents : characteristic multipliers


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Since , the relationship between and is

which gives in turn

Remark: arbitrariness in the imaginary part of !!!

As for the continuous time case, this arbitrariness has no effect …

For now ignore the arbitrariness choosing for example


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Rewrite the transition matrix as a function of characteristic exponents

Notice that where

Than the state transition matrix becomes

where and is the order of the system


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Remark: is periodic and could be expanded in a Fourier series

The final form of the transition matrix is

All LTP exponents, , are present in the system response, and matrices determine their relative contribution

Definition: modal participation factor

Measure of relative strength of the n-th harmonic in j-th mode


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The LTP exponents are the analogs of the LTI eigenvalues:

• For each exponent we can compute frequency and damping converting the discrete exponent into the continuous one, and then computing frequencies and damping factors as in the continuous time case

• , and determine the dynamic behavior of LTP system


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Stability through System Identification

Important remark:

One might not have the analytical expression of a model

Examples:

• Experimental observations (only input-output data)

• Non-linear comprehensive models (FEM multibody+aerodynamics)

Solution: use system identification techniques using input-output sequences (applicable to experimental observations and “virtual” experiments conducted with numerical models)

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From discrete time state-space model to linear input-output model

State-space model:

State –space observer model (add and subtract ):

Reordering:

LTI Input-Output Model

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Solving for the output with null initial conditions:

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ARX sequence:

Neglecting

Remark: the AR and X part could have different orders (i.e. a different number of coefficients)


AR part (Auto-Regressive)

X part (eXogeneous)

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Example: consider the following discrete time system:

Compute response:

ARX sequence of order 2 for the AR-part and order 1 for the X-part:

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LTI State-Space Realization

State-space realization: find a suitable state-space system which has the given input-output behavior

The problem has infinite possible solutions: find “convenient” solution!

Canonical realization (minimal, reachable and observable):

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ARX sequence in matrix form:

where

Least-squares estimation of ARX system parameters:

LTI Input-Output Model Identification

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LTI Stability from Input-Output Model

Approach 1:

• Identify ARX sequence:

• Compute matrix with canonical realization

• Compute eigenvalues, then frequencies and damping factors of interest

Approach 2:

• Introduce the backward shift operator s.t.

• Write ARX sequence as

where

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LTI Stability from Input-Output Model

(Approach 2 continued)

• Compute roots of polynomial , discrete poles

• Compute continuous poles

• Compute frequencies and damping factors

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LTP Input-Output Model

PARX Periodic Auto-Regressive model with eXogeneous inputs:

with , the number of harmonics in the model

and

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LTP Input-Output Model Identification

Consider PARX sequence and the Fourier expansion of its coefficients:

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The PARX sequence can be written as

The unknown coefficients can be identified by least squares:

where and is the total number of samples

LTP Input-Output Model Identification

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LTP State-Space Realization

The realization of a periodic state-space model from a periodic input-output model is more complicated than in the LTI case (need to ensure that simulation with realized model gives same input-output sequence)

• For stability analysis, one needs only the autonomous response

• Rigorous (same input-output) realization of the sole AR-part of the system is as follows

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LTP Stability from Input-Output Model

• Once the periodic matrix is identified, compute the state transition matrix

• Define monodromy matrix

• Compute multipliers, exponents, and participation factors as previous explained

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Approaches for wind turbine stability analysis:

1. Direct LTP analysis

• Provide suitable excitation of modes of interest

• Identify LTP input-output model from wind turbine response

• Perform stability analysis

2. LTI reformulation of LTP problem

• Provide suitable excitation of modes of interest

• Reformulate LTP into an approximate LTI

• Identify LTI input-output model from wind turbine response

• Perform stability analysis

LTP Stability from Input-Output Model

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LTI Reformulation of a LTP Problem LTP system:

= closed-loop matrix (accounts for pitch-torque controller)

= exogenous input (wind), constant in steady conditions

Fourier reformulation (Bittanti & Colaneri 2000):

1. Approximate state matrix:

2. Transfer periodicity to input term (see later on for details)

Obtain linear time invariant (LTI) system:

where is the exogenous periodic input

Remark: no need for model generality, just good fit with measures

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LTI Reformulation of a LTP Problem

Given reformulated LTI system

use standard Prony’s method (Hauer 1990; Trudnowski 1999):

1. Trim and perturb with doublet (or similar, e.g. 3-2-1-1) input

2. Identify discrete time ARX model (using Least Squares or Output Error method) with harmonic inputs

3. Compute discrete poles, and transform to continuous time (Tustin transformation)

4. Obtain frequencies and damping factors

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Transferring periodicity to input term

Expanding the periodic terms:

In matrix form:

Constant matrix Dummy periodic exogenous input

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Illustrative example: lag equation neglecting flap-coupling terms

where the term accounts for drag-induced and structural damping

Objective: compute lag-damping factor through identification

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Lag equation in state-space form (remark: fixed)

where

Lag freq including centrifugal effect

with offset

Steady lag term

Gravity

Yaw term

Vertical shear term

Cross-flow term

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Fourier reformulation:

Approximation of the state matrix:

Remark: neglecting contribution of gravity to stiffness (negligible)

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Transferring periodicity to input terms:

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Collecting all terms

Resulting LTI system:

Hence, a LTP has been reformulated as a LTI subjected to exogenous periodic inputs

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Excitations (inputs)

Applications and Results

Response (outputs)

Definition of best practices for the identification of modes of interest:

For each mode:

• Consider possible excitations (applied loads, pitch and/or torque inputs) and outputs (blade, shaft, tower internal reactions)

• Verify presence of modes in response (FFT)

• Verify linearity of response

• Perform model identification

• Verify quality of identification (compare measured response with predicted one)

Compiled library of mode id procedures:

In this presentation:

• Tower fore-aft mode

• Rotor in-plane, blade first edge modes

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Excitation: doublet of hub force in fore-aft direction

Example: Damping Estimation of Fore-Aft Tower Modes

Output: tower root fore-aft bending

moment

Verification of linearity of response

Doublets of varying intensity to verify linearity

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First tower mode

Second tower mode

1P

Verification of linearity of response and presence of modes

3P

6P

9P

12P

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Direct LTP analysis

◀ Time domain ▼ Frequency domain

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Exponent at 0.33 Hz with a participation close to 1: first tower mode is nearly invariant

Participation factor implies non-negligible periodic effects

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Direct LTP vsReformulated LTI analysis:

First tower mode

Second tower mode

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◀ Time domain ▼ Frequency domain

• Good quality of identified models

(supports hypothesis A(ψ) ≈ A0)

• Necessary for reliable estimation

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Estimated damping ratios for varying wind speed


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Excitation: doublet of • In-plane blade tip force • Generator torque

Example: Damping Estimation of Blade Edge and Rotor In-Plane Modes

First blade

edgewise mode

Quality of identified model, using blade root bending

Rotor in-plane

mode

Rotor in-plane

mode

Quality of identified model, using shaft torque Outputs: • Blade root bending moment • Shaft torque

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Example: Damping Estimation of Blade Edge and Rotor In-Plane Modes

◀ Little sensitivity to used output (blade bending or shaft torque)

Rotor in-plane mode

Blade edge mode

Documents

3-WindTurbineAeroelasticity