Multi-body Dynamics for Aeroelastic Stability Analysis of ... · SIMPACK User Meeting 2011,...
Transcript of 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
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
Folie 3 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
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
Folie 4 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
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
Folie 5 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
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
Folie 6 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
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
Folie 7 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
Multi-blade Coordinates (eg. Flapping)
rotor coning angle differential flappinglongitudinal flapping
( )i
diSiCi 1sincos110
−+++= βψβψβββ
animation
Folie 8 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
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
Folie 9 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
Folie 10 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
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
Folie 11 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
Equation of Flapping Motion of a Rigid Rotor Blade in
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
Folie 12 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
Equation of Flapping Motion of a Rigid Rotor Blade in
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
Folie 13 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
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
Folie 14 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
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
Folie 15 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
Folie 16 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
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
Folie 17 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
Flapping Stability of Helicopter Rotor in Forward
Flight: SIMPACK Model
Model parameters
( )Ω,,,, βµγ kB
SIMPACK Expressions
SIMPACK Force
Elements
Folie 18 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
Flapping Stability of Helicopter Rotor in Forward
Flight: Results and Validation
SIMPACK Linear System Matrix
Folie 19 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
Flapping Stability of Helicopter Rotor in Forward
Flight: Results and Validation
Multiblade Transformation of SIMPACK
Linear System Matrix
Folie 20 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
[ ]⇒
∑n
Ti
π
ψ
2
0
time average: analytical result
Flapping Stability of Helicopter Rotor in Forward
Flight: Results and Validation
SIMPACK model result
Folie 21 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
Flapping Stability of Helicopter Rotor in Forward
Flight: Results and Validation
nutation/ progressive mode: higher frequency mode
precession/ regressive mode: lower frequency mode
Folie 22 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
Folie 23 SIMPACK User Meeting 2011, A. Rezaeian
19/05/2011
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.