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Transcript of Geometrically Exact Beam Theory-Presentation by Palash Jain
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Application of Geometrically Exact Beam Formulation
for Modeling Advanced Geometry Rotor Blades
Palash Jain
Supervisor: Dr. Abhishek
July 30th
, 2014
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Outline
1 Outline
2 Introduction
3 Theory
4 Static Load Deformation AnalysisEquationsProcedureElastica with Tip MomentElastica with Tip Load
Princeton Beam Experiment5 Dynamic Frequency Analysis
EquationsSteady State Analysis
Perturbation AnalysisCantilevered ElasticaPrinceton Beam ExperimentMaryland Beam Experiment
6 Conclusion
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Introduction
Introduction
Aim: To implement the formulation for modeling helicopter rotorblades using Geometrically Exact Beam Theory (henceforth
abbreviated as GEBT).GEBT: contains equations describing the overall dynamics of beammembers undergoing arbitrary motions.
This presentation will showcase theoretical formulation and
implementation of GEBT to perform static and dynamic analysis ofrotor blades with advanced geometry and composite materials.
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Introduction
Introduction
Nonlinearities in beam modeling arise from: (i) geometry or (ii) material.
Earlier Approaches
Approximation of nonlinear strain-displacement relation by truncatedTaylor series expansion
Multibody methods with an additional frame attached to the finite
elements while the above approximation is still followed
Geometrically Exact Beam Theory (GEBT)
GEBT provides dimensional reduction with sufficient accuracy for
modeling of highly deformable structuresGEBT is easier to implement compared to approximate theories whichbecome more complicated as order of approximation is raised
It can be seamlessly integrated with aerodynamic models forcomprehensive analysis of helicopter rotor.
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Theory
Coordinate Systems: Definition
Figure 1: Coordinate Systems
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Theory
Coordinate Systems: Transformation
Z= ZAiAi=ZA=
{ZA1,ZA2,ZA3
}T (1)
They are tranformed from one basis to another multiplying with atransformation matrix
ZB=CBb Zb (2)
Rodrigues Wiener-Milenkovic
c 2tan2 4tan4
c0 1 + cTc
4 2 cTc8
C [(1 1
4cTc)+ 1
2ccTc]
c0
[(c20cTc)2c0c+2ccT]
(4c0)2
Q 1c0
[ c/2] 1(4c0)2 [(4 14cTc) 2c+ 12ccT]Table 1: Finite rotation parameters to be used in finite element equations.
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Th
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Theory
Hamiltons Principle
Hamiltons Principle
t2
t1 l
0[(K U) +W] dx1 dt=A (3)
U=U(, ) (4)
K =K(V, ) (5)
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Theory
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Theory
Dynamics Equations
Euler Lagrange Equations
FB+
KBFB+fB= PB+
BPB (6)
MB+KBMB+ (e1+ )FB+mB= HB+ BHB+VBPB (7)Kinematic Relations
u =CT(e1+)
e1
ku (8)u=CTV v u (9)c =Q1(+k Ck) (10)c=Q1( C) (11)
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Theory
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Theory
System Characteristics
Constitutive Relations:
=
R S
ST T
F
M
(12)
Momentum-Velocity Relations:P
H
=
T I
V
(13)
where =
{0, x2, x3
}T.
Linear-Angular Velocity Relations:
va =v0+
a
d1d2d3
(14)
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Theory
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Theory
Final Equations
Starting Node
fu1 F1 =0 (15)f1 M1 =0 (16)
fF1 u1 =0 (17)
f
M1 c1 =0 (18)
Intermediate Points (i = 1 to N-1)
f+ui + fui+1
=0 (19)
f+i + f
i+1=0 (20)
f+Fi + f
Fi+1=0 (21)
f+Mi + fMi+1 =0 (22)
Ending Node
f+uN FN+1 =0 (23)f+N MN+1 =0 (24)
f+FN + uN+1 =0 (25)
f+MN + cN+1 =0 (26)
In Each of the Elements (for i = 1 toN)
fPi =0 (27)
fHi =0 (28)
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Theory
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Theory
Final Equations
The fmatrices for each element are presented below:
fui = CTCabFi fi +Li
2 [aCTCabPi+ CTCabPi] (29)
fi =
CTCabMi
mi +
Li
2
[aCTCabHi+
CTCabHi (30)
+CTCab(e1+ i)Fi)] (31)fFi = ui
Li2
[CTCab(e1+i) Cabe1)] (32)
f
Mi = ci Li
2 Q
1
a C
ab
i (33)fPi =C
TCabVi vi aui ui (34)fHi =i CbaCa CbaQaci (35)
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Theory
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y
Final Equations
fi =
10
(1 )faLid (36)
f+i = 1
0faLid (37)
mi =
10
(1 )maLid (38)
m+i = 10
maLid (39)
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Static Load Deformation Analysis Equations
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y
Statics Equations
The equations set can be represented as G(X, F) =0 and is a function of:
12(N+ 1) unknownsX: F1, M1,u1, c1,F1,M1, ...,uN, cN, FN, MN, uN+1, cN+1
12 boundary conditions F:u1, c1,F
N+1,M
N+1
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Static Load Deformation Analysis Procedure
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Procedure
(i) Assume an initial guess for unknowns X. An ideal guess would beundeformed state X=0.
(ii) Calculate G(X, F), where F are known boundary conditions.
(iii) Populate Jacobian matrix by perturbing previous X such thatj= 1to 12(N+1); B(:,j) = [G(X|X(j)=X(j)+,
F) G(X,
F)]/. B is asquare matrix of size 12(N+1).
(iv) Calculate the updated X as X= X B1G.(v) Repeat steps (ii) to (iv) until norm ofG is smaller than the required
tolerance.
Newton-Raphson method requires inversion of 12(N+ 1) square matrixmaking it slow, however, the convergence is achieved in 2-5 steps for wellposed boundary conditions.
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Static Load Deformation Analysis Elastica with Tip Moment
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Elastica with Tip Moment
Dimension
Material
Properties Accuracy
Boundary
ConditionsL 6.096 m E 71.6 GPa N 20 M2 =NEI/L,
w 15.24 cm G 26.9 GPa 104 where, = 0.05556
t 9.5 mm 2800 kgm3 109 and 0.2778
Table 2: Simulation Parameters for Elastica with Tip Moment
Analytical Solution:
(s) =TS
EI
(40)
u(s) =EI
Tsin(
Ts
EI) s (41)
v(s) =EI
T[1 cos(Ts
EI)] (42)
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Static Load Deformation Analysis Elastica with Tip Moment
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Results
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
Axial Position, 2r1
VerticalPos
ition,2r3
AnalyticalCurrent
Figure 2: Elastica deflection under low tip Moment (2L/N= 0.05556).
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Static Load Deformation Analysis Elastica with Tip Moment
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Results
1 0.5 0 0.5 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Axial Position, 2r1
VerticalPosition,2r3
Analytical
RCASCurrent
Figure 3: Elastica deflection under high tip Moment (2L/N= 0.2778).
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Static Load Deformation Analysis Elastica with Tip Moment
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Elastica with Tip Load
Dimension Material
Properties Accuracy Boundary Conditions
L 6.096 m E 71.6 GPa N 20 F3 =EI/L2,
w 15.24 cm G 26.9 GPa 104 where, = 0 to 5
t 9.5 mm 2800 kgm3 109
Table 3: Simulation Parameters for Elastica with Tip Load
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Static Load Deformation Analysis Elastica with Tip Load
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Analytical Solution
Governing Differential Equation
EId2
ds2 =Psinsin Pcoscos= Pcos(+) (43)
Solution involves numerical evaluation of the integrals:
J1(L) =
L0
dsin(L+) sin(+)
=
2 (44)
J2(L) =
L
0
sin dsin(L+) sin(+) =
2
LvL (45)
J3(L) =
L0
(cos 1) d
sin(L+) sin(+)
=
2
LuL (46)
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Static Load Deformation Analysis Elastica with Tip Load
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Results
0 1 2 3 4 5
0
10
20
30
40
50
60
70
Normalized Load, P3R
2/EI
2
T
ip
AngleofRotation,
|2
(R)|[deg]
Analytical
RCAS
Current
Figure 4: Angle rotated by elastica tip under transverse load.
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Static Load Deformation Analysis Elastica with Tip Load
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Results
0 1 2 3 4 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Normalized Load, P3R
2/EI
2
Norm
alizedTipVertic
alDeflection,u3(R
)/R
Analytical
RCAS
Current
Figure 5: Vertical deflection of elastica tip under transverse load.
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Static Load Deformation Analysis Elastica with Tip Load
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Results
0 1 2 3 4 5
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Normalized Load, P3R
2/EI
2
Norm
alizedHorizonta
lDeflectionu
2(R
)/R
Analytical
RCAS
Current
Figure 6: Horizontal deflection of elastica tip under transverse load
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Static Load Deformation Analysis Princeton Beam Experiment
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Princeton Beam Experiment
Figure 7: Schematic of Experimental
Setup
Figure 8: Deformed and UndeformedStates
Figure 9: Coordinate System
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Static Load Deformation Analysis Princeton Beam Experiment
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Simulation Parameters
Dimension Material
Properties Accuracy
BoundaryConditions
L 0.508 m E 71.6 GPa N 8 F3 13.345 Nw 12.7 cm G 26.9 GPa 109 1 0to 90t 3.2 mm 2800 kgm3 109
Table 4: Princeton Beam Simulation Parameters
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Static Load Deformation Analysis Princeton Beam Experiment
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Results
0 15 30 45 60 75 90
0
0.05
0.1
0.15
0.2
0.25
Pitch Angle, ||[deg]
NormalizedHorizontalDeflectio
n,
|u2(s,P3,)u2(s,0,)|/R
GEBT s/R = .25
GEBT s/R = .50GEBT s/R = .75
GEBT s/R = 1.0
Exp. s/R = .25
Exp. s/R = .50
Exp. s/R = .75
Exp. s/R = 1.0
Figure 10: Horizontal deflection vs. pitch angle of Princeton Beam for Tip load
= 3 lb (13.3 N).Palash Jain (Supervisor: Dr. Abhishek) Geometrically Exact Beam Theory July 30th , 2014 26 / 43
Static Load Deformation Analysis Princeton Beam Experiment
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Results
0 15 30 45 60 75 90
0
0.05
0.1
0.15
0.2
0.25
Pitch Angle, ||[deg]
NormalizedVert
icalDeflection
,
|u2(s,P3,)u2(s,0,)|/R
GEBT s/R = .25
GEBT s/R = .50
GEBT s/R = .75
GEBT s/R = 1.0
Exp. s/R = .25Exp. s/R = .50
Exp. s/R = .75
Exp. s/R = 1.0
Figure 11: Vertical deflection vs. pitch angle of Princeton Beam for Tip load =
3 lb (13.3 N).Palash Jain (Supervisor: Dr. Abhishek) Geometrically Exact Beam Theory July 30th , 2014 27 / 43
Static Load Deformation Analysis Princeton Beam Experiment
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Results
0 15 30 45
0
0.5
1
1.5
2
2.5
3
3.5
4
Pitch Angle, ||[deg]
Twist,|1
(s,P3,
)1
(s,0,
)|[de
g]
GEBT s/R = .25
GEBT s/R = .50
GEBT s/R = .75
GEBT s/R = 1.0
Exp. s/R = .25
Exp. s/R = .50
Exp. s/R = .75
Exp. s/R = 1.0
Figure 12: Twist deformation vs. pitch angle of Princeton Beam for Tip load =
3 lb (13.3 N).Palash Jain (Supervisor: Dr. Abhishek) Geometrically Exact Beam Theory July 30th , 2014 28 / 43
Dynamic Frequency Analysis Equations
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Equations
The dynamics equations represented in nonlinear form as
G(X, X, F) =0 (47)
Linearization about the steady state gives equations of the form:
A
X + BX=
F (48)
where
A= G
XB=
G
X
and Frepresents dynamic boundary conditions which is 0 for analysisabout the steady state.
Palash Jain (Supervisor: Dr. Abhishek) Geometrically Exact Beam Theory July 30th , 2014 29 / 43
Dynamic Frequency Analysis Steady State Analysis
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The steady state equations are obtained by dropping the time dependent
terms leaving the set G(X, F) =0 as functions of:18N+ 12 unknowns X:
F1, M1, u1, c1, F1, M1, V1, 1, ...,uN, cN, FN, MN, VN, N, uN+1, cN+1
12 boundary conditions F:u1, c1,F
N+1,M
N+1
Loading conditions: f,m, v,
The solution for steady state is similar to the static state albeit with a
larger set. Here, the Jacobian matrix B and steady state sectional valuesare retained for further analysis.
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Dynamic Frequency Analysis Perturbation Analysis
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The equations could be represented in compact notation as G(X, F) =0.
Steps to determine the A matrix:
(i) For each value ofX introduce small perturbations which is assumedto be small.
(ii) Corresponding to each of these perturbed values ofX, populate theA matrix one column at a time by dividing the returned G vectorwith epsilon. In short,j= 1 to 18N+12;A(:,j) =G(X|X(j)=X(j)+epsilon, F)/. Consequently, A is a squarematrix of size 18N+12.
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Dynamic Frequency Analysis Cantilevered Elastica
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Cantilevered Elastica
Dimension Material
Properties
Accuracy Boundary
ConditionsL 0.508 m E 71.6 GPa N 30 Fixed-free with gravity
w 12.7 cm G 26.9 GPa 109 1root = 0and 90
t 3.2 mm 2800 kgm3 109
Table 5: Simulation Parameters for Frequency estimation of Cantilevered Beam
Method Flatwise Edgewise
Anaytical 10.049 40.196
GEBT 0 10.052 40.134
GEBT 90 10.053 40.125
Experimental 10.150 41.143
Table 6: Frequency data (in Hz) for Cantilevered Beam
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Dynamic Frequency Analysis Princeton Beam Experiment
P i B E i
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Princeton Beam Experiment
Figure 13: Schematic of theexperimental Setup Figure 14: Fundamental vibrationmodes
Dimension Material
Properties Accuracy
BoundaryConditions
L 0.508 m E 71.6 GPa N 8 F3 13.345 Nw 12.7 cm G 26.9 GPa 109 1 0to 90
t 3.2 mm 2800 kgm3 109
Table 7: Simulation Parameters for Frequency Estimation of Princeton Beam
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Dynamic Frequency Analysis Princeton Beam Experiment
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0 30 60 90
1
2
3
4
5
Pitch Angle, || [deg]
Natural
Frequency[Hz]
Current Flapwise (2 lb)
Current Chordwise (2 lb)
Exp. Flapwise (2 lb)
Exp. Chordwise (2 lb)
Current Flapwise (3 lb)
Current Chordwise (3 lb)
Exp. Flapwise (3 lb)
Exp. Chordwise (3 lb)
Figure 15: Twist deformation vs. pitch angle of Princeton Beam for Tip load =3 lb (13.3 N).
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Dynamic Frequency Analysis Maryland Beam Experiment
M l d B E i
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Maryland Beam Experiment
Figure 16: Schematic of Maryland beam showing tip sweep and root offset
Figure 17: Finite element discretization of Maryland beam.
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Dynamic Frequency Analysis Maryland Beam Experiment
Si l ti P t
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Simulation Parameters
Dimension Material
Properties Accuracy
BoundaryConditions
L 40 in E 107 psi N 16 Fixed-free
w 1 in G 4 106 psi 106 Lroot 2.5 int 0.0625 in 2.51 104 104 sweep 0to 45Ltip 6 in lb.sec
2/in4 t 103 RPM 0, 500, 750
Table 8: Maryland Beam Simulation Parameters
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Dynamic Frequency Analysis Maryland Beam Experiment
R lt
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Results
Figure 18: Effect of tip sweep and RPM on 1st flap-bending frequency ofMaryland Beam
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Dynamic Frequency Analysis Maryland Beam Experiment
Results
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Results
Figure 19: Effect of tip sweep and RPM on 2nd flap-bending frequency ofMaryland Beam
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Dynamic Frequency Analysis Maryland Beam Experiment
Results
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Results
Figure 20: Effect of tip sweep and RPM on 3rd flap-bending frequency ofMaryland Beam
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Dynamic Frequency Analysis Maryland Beam Experiment
Results
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Results
Figure 21: Effect of tip sweep and RPM on 4th flap-bending frequency ofMaryland Beam
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Dynamic Frequency Analysis Maryland Beam Experiment
Results
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Results
Figure 22: Effect of tip sweep and RPM on 5th flap-bending frequency ofMaryland Beam
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Conclusion
Summary
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Summary
A set of nonlinear coupled finite element equations was solved usingNewton-Raphson iterative scheme
Frequency estimation involves calculation of eigenvalues from beamequations perturbed about the steady state
A good agreement between experimental and GEBT results seen
Visible improvement over approximate methods in prediction of theoutcome of the benchmark results mentioned above
Effectiveness of GEBT in modeling large and coupled deformations= ideally suitable for rotorcraft applications.
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Conclusion
Future Scope
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Future Scope
To develop the structural dynamic modeling codes for comprehensiverotor analysis system
To predict the aeromechanical characteristics of rotor and aircraft.
To integrate structural dynamics with geometry, aerodynamics andcontrol inputs.
Aim is to estimate critical traits like trim, air loads, structural loads,
blade response, vibration, noise, stability and performance in aniterative fashion.
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