geometrically exact beam theory-presentation by palash jain

8/10/2019 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

Palash Jain (Supervisor: Dr. Abhishek) Geometrically Exact Beam Theory July 30th , 2014 1 / 43


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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


Th


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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.


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)


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)


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)


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)


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)


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)


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


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.


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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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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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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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


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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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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Results

0 1 2 3 4 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


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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Results

0 1 2 3 4 5

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0


2/EI

2

Norm

alizedHorizonta

lDeflectionu

2(R

)/R

Analytical

RCAS

Current

Figure 6: Horizontal deflection of elastica tip under transverse load


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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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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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



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Results

0 15 30 45 60 75 90

0

0.05

0.1

0.15

0.2

0.25


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



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Results

0 15 30 45

0

0.5

1

1.5

2

2.5

3

3.5

4


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.


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.


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.


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


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


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).


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.



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



R lt


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Results

Figure 18: Effect of tip sweep and RPM on 1st flap-bending frequency ofMaryland Beam



Results


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Results

Figure 19: Effect of tip sweep and RPM on 2nd flap-bending frequency ofMaryland Beam



Results


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Results

Figure 20: Effect of tip sweep and RPM on 3rd flap-bending frequency ofMaryland Beam



Results


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Results

Figure 21: Effect of tip sweep and RPM on 4th flap-bending frequency ofMaryland Beam



Results


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Results

Figure 22: Effect of tip sweep and RPM on 5th flap-bending frequency ofMaryland Beam


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.


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.


geometrically exact beam theory-presentation by palash jain

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