Multi-body Dynamics for Aeroelastic Stability Analysis of ... · SIMPACK User Meeting 2011,...

SIMPACK User Meeting 2011, 19/05/2011

Multi-body Dynamics for Aeroelastic Stability Analysis of a Helicopter in Forward Flight

Alireza Rezaeian, German Aerospace Center (DLR)

Institut of Aeroelasticity, Göttingen

[email protected]

Folie 2 SIMPACK User Meeting 2011, A. Rezaeian

19/05/2011

Content

Fundament: Simulation Experience with SIMPACK, Stability Analysis,

Multi-blade Coordinates, Multi-blade Coordinates Transformation

Rotor Blade Flapping Motion in Forward Flight

Flapping Stability of Helicopter Rotor in Forward Flight

• Method of Analysis

• Simulation with SIMPACK

• Simulation Results and Comparison with the Analytical Results

Conclusion


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Simulation Experience with SIMPACK

Paper: Dr. Wolf R. Krüger, Multi-body

Analysis of Whirl Flutter Dynamics on a TiltrotorWind Tunnel Model, International Forum of Aeroelasticity and Structural Dynamics (IFASD) 2009

User Routine:Aerodynamic

FEMBS:elasticity


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Simulation Experience with SIMPACK

a: Paper:1- Dr. Stefan Waitz, From FEM to MBS: Stability Analysis of the Elastic H/C-Rotor, European Rotorcraft Forum (ERF) 2010

2- Jürgen Arnold, Using Multi-body Dynamics for the Simulation of Flexible Rotor Blades- Modelling Limits of an Innovative Blade Layout based on Beam Approach, ERF 2010

a

b

b: Paper:1- Alireza Rezaeian, Helicopter Ground Resonance Analysis Using Multi-body Dynamics, European Rotorcraft Forum 2010


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1- Elements of „A“ are constant in time Eigenvalue stability analysis

2- Elements of „A“ are time-periodic Classical eigenvalue analysis is not valid

Stability Analysis of a System with Linear Differential

Equation

linear system matrix


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Stability Analysis of non-linear Time-periodic

Systems

Nonlinear rotating system with time-periodic parameters

linear time-periodic system

some periodic terms change to constant

differential equation with constant coefficients

Stability analysis

Flo

qu

et

theo

ry

multi-blade transformation, non-rotating reference frame

characteristic equation (polynomial)

calculation of equilibrium state

making time average of remained periodic terms

Equilibrium state: static or dynamic

linearization MATLAB

SIMPACK

Utilized Simulation Tools


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Multi-blade Coordinates (eg. Flapping)

rotor coning angle differential flappinglongitudinal flapping

( )i

diSiCi 1sincos110

−+++= βψβψβββ

animation


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Multi-blade Coordinates Transformation

[ ]

matrixsystem

A

−

[ ]

⋅=

4

3

2

1

4

3

2

1

4

3

2

1

4

3

2

1

β

β

β

β

β

β

β

β

β

β

β

β

β

β

β

β

&

&

&

&

&

&

&&

&&

&&

&&

&&

&&

&

&

&

&

&

&

y

x

y

x

A

y

x

y

x

[ ]

⋅=

d

s

c

d

s

c

d

s

c

d

s

c

y

x

y

x

T

y

x

y

x

β

β

β

β

β

β

β

β

β

β

β

β

β

β

β

β

&

&

&

&

&

&

&&

&&

&&

&&

&&

&&

&

&

&

&

&

&

1

1

0

1

1

0

1

1

0

1

1

0

multiblade transformation:

non-orthogonal transformation

time dependent

[ ]

matrixsystemdtransforme

T

−−

state variables

rotating coordinates

DOFs of rotating parts: Blade

multi-blade coordinates

non-rotating coordinates


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Content








Conclusion


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Equation of Flapping Motion of a Rigid Rotor Blade in

Forward Flight

centrifugal force

aerodynamic force

inertia force

( )∫ ∫ ∫ =−Ω+R R R

Z rdrFdrrrmrdrmr0 0 0

20ββ&&

r

ββββ: flap angle


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Forward Flight

iiiiii BBI

kBB βψ

γµψ

γµβψ

γµ

γβ

β

++

Ω++

++ 2sin

8cos

61sin

68

223

2

34 &&&

γ: blade lock number

µ: rotor advance ratio

B: tip loss factor Κβ: flapping spring stiffness

I: blade flapping inertia ψ: azimuth angle

Blade: rigid, spring restrained, centrally hinged blade,

without reverse flow

I

acR4ρ

γ = a: blade lift curve slope


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Forward Flight

( ) ( )2

13602

1180π

ψπ

−−<<−− ii i

iiiiiiiii BBI

kBB βψψµ

γψ

γµψ

γµβψµ

γψ

γµ

γβ

β

−++

Ω++

+++ 34223

2

4434sincos

62sin

8cos

61sin

12sin

68

&&&

Blade: rigid, spring restrained, centrally hinged blade,

with reverse flow

M-betaM-beta-dot


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Stiffness Coefficient of the Flapping Motion Equation

with and without Reverse Flow

µ = 0

µ = 0.5

µ = 0.9

B = 0.97

γ = 12Kβ = 0


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Damping Coefficient of the Flapping Motion Equation

with and without Reverse Flow

µ = 0

µ = 0.5

µ = 0.9

B = 0.97

γ = 12


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Content








Conclusion


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Flapping Stability of Helicopter Rotor in Forward

Flight: Method of Analysis

SIMPACK

1- Equilibrium state

2- Linearization about the equilibriumstate

3- Obtaining the linear system matricesfor the different azimuth angles

MATLAB as postprocessor

4- Multi-blade transformation of the linear system matrices and thencalculation of the time average matrix

5- Eigenvalues of the calculated timeaveraged matrix

6- Sorting the eigenforms

7- Plotting the results


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Flight: SIMPACK Model

Model parameters

( )Ω,,,, βµγ kB

SIMPACK Expressions

SIMPACK Force

Elements


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Flight: Results and Validation

SIMPACK Linear System Matrix


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Multiblade Transformation of SIMPACK

Linear System Matrix


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[ ]⇒

∑n

Ti

π

ψ

2

0

time average: analytical result



SIMPACK model result


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nutation/ progressive mode: higher frequency mode

precession/ regressive mode: lower frequency mode


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Content








Conclusion


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Conclusion

Existing SIMPACK modeling options like „Substitution Variables“ and „Expression“ allows creating parametric model of a complex mathematical formulation, which can be used for example as force or time excitation inside SIMPACK model.

SIMPACK interfaces with other codes like „MATLAB“ is a great option for multidisciplinary simulation to use the computational advantages of different codes.

Simulation and stability analysis presented here is just a simple example to introduce the method, which can be used for a detailed helicopter model created inside SIMPACK.

Multi-body Dynamics for Aeroelastic Stability Analysis of ... · SIMPACK User Meeting 2011,...

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