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8/10/2019 Application of Geometrically Exact Beam Formulation for Modeling Advanced Geometry Rotor Blades - M.Tech The

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Application of Geometrically ExactBeam Formulation for Modeling

Advanced Geometry Rotor Blades

A thesis submitted in partial fulfilment of the requirements

for the degree of

Master of Technology

Author:

Palash Jain

Supervisor:

Dr. Abhishek

Department of Aerospace Engineering

Indian Institute of Technology, Kanpur

August 2014
http://department%20or%20school%20web%20site%20url%20here%20%28include%20http//www.iitk.ac.in/aero/home/)http://www.iitk.ac.in/http://www.iitk.ac.in/http://department%20or%20school%20web%20site%20url%20here%20%28include%20http//www.iitk.ac.in/aero/home/)


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Certificate


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AbstractApplication of Geometrically Exact Beam Formulation for Modeling

Advanced Geometry Rotor Blades

by Palash Jain

Slender beams undergoing dynamic loading are a feature in many structural applica-

tions like high aspect ratio aircraft wings, bridges, helicopter blades, space structures,

propellers and wind turbines to name a few. When these beams are subjected to large

deformations, linear beam theories like that of Euler-Bernoulli and Timoshenko no longer

remain valid. Nonlinearities in beam modeling arise from: (i) geometry and / or (ii)

material. The focus is on the former. Earlier approaches for beam analyses used approx-

imation of nonlinear strains and curvatures by truncated Taylor series expansion, and

involved use of ordering scheme for retaining terms up to desired level of accuracy. This

technique is satisfactory for moderate deformations, however for large deformations and

those having coupling, the errors associated with approximations are significant.

Geometrically Exact Beam Theory (GEBT) based on Hodges Mixed Variational For-

mulation provides a more accurate description of dynamics of beams by involving all

irreducible parameters to solve the energy conserving Hamiltons Equation. This results

in a set of implicit nonlinear partial differential equations involving intrinsic or natural

parameters for beam characterization viz. sectional forces, moments, displacements, ro-

tations and linear and angular velocities. These equations are numerically discretized by

finite element methods and solved iteratively, very often by Newton-Raphson method

and time marching for dynamic cases. The eigenvalue analysis of dynamics equations

obtained by superposing small perturbations over the steady state gives the naturalfrequencies of the system.

The present thesis discusses the implementation of GEBT for modeling advance geome-

try rotor blades. The algorithms for implementing GEBT for static and dynamic cases

are discussed. Thereafter, a verification and validation of the results obtained from

the present analysis is performed with benchmark analytical and experimental results

demonstrating a good correlation and the superiority of the present approach over ear-

lier methods. Finally, ways to further improve these results and the future uses of the

developed codes and GEBT in general are discussed.


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Acknowledgements

I would like to express my sincere gratitude to my advisor, Dr. Abhishek, for his able

guidance and encouragement throughout this thesis work. The positivity and patience

shown by him during our meetings is highly appreciated. His experience in rotorcraft

dynamics has proved greatly helpful in dealing with this challenging area.

I am indebted to the faculty at the Department of Aerospace Engineering for helping

me develop the learning aptitude essential for solving complex engineering problems.

The academic environment and research facilities provided by the Institute is highly

appreciated.

I would also like to thank the people at my lab for their support in making my task easier.

My stay at IITK couldnt have been more memorable without friends and colleagues

especially my wing-mates Ashutosh, Kumar, Tanmay, Pratyush, Ayush, Anirudh and

Pratik.

My words of acknowledgement would never be enough for my Parents unconditional

support and affection which provided me the courage to walk the difficult paths in life.

Finally, I am grateful to everyone who could not be mentioned here but had constructive

influence on me.

iii


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Contents

Certificate i

Abstract ii

Acknowledgements iii

Contents iv

List of Figures vi

List of Tables vii

Symbols viii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review of Previous Work . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Earlier Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Historical Insight on GEBT . . . . . . . . . . . . . . . . . . . . . . 3

1.2.3 Theoretical Development . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.4 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.5 Benchmark Results. . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Theory 7

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Hamiltons Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Strain Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.3 Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.4 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Mixed Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Finite Element Implementation . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Finite Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Beam Statics: Verification and Validation 18

3.1 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

iv


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Contents v

3.2 Benchmark Static Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Bending moment applied at the tip of an elastica . . . . . . . . . . 21

3.2.2 Dead (non-follower) force applied at the tip of an elastica . . . . . 22

3.2.3 Princeton beam experiment . . . . . . . . . . . . . . . . . . . . . . 26

4 Beam Dynamics: Verification and Validation 32

4.1 Estimation of Natural Frequencies . . . . . . . . . . . . . . . . . . . . . . 33

4.1.1 Steady State Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1.2 Perturbation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Results and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.1 Cantilevered Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.2 Princeton Beam Experiment . . . . . . . . . . . . . . . . . . . . . 40

4.2.3 Maryland Beam Experiment . . . . . . . . . . . . . . . . . . . . . 43

5 Conclusion and Recommendation 48

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2 Scope and Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Bibliography 52


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List of Figures

2.1 Beam Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 Elastica Deflection with Low Tip Moment . . . . . . . . . . . . . . . . . . 23

3.2 Elastica Deflection with High Tip Moment. . . . . . . . . . . . . . . . . . 24

3.3 Tip Angle of Rotation under Tip Load . . . . . . . . . . . . . . . . . . . . 25

3.4 Vertical Tip Deflection under Tip Load . . . . . . . . . . . . . . . . . . . 253.5 Horizontal Tip Deflection under Tip Load . . . . . . . . . . . . . . . . . . 26

3.6 Princeton Beam Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.7 Princeton Beam Load Deformation . . . . . . . . . . . . . . . . . . . . . . 27

3.8 Princeton Beam Axes System . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.9 Princeton Beam Root Setting Angle . . . . . . . . . . . . . . . . . . . . . 29

3.10 Horizontal Deflections of Princeton Beam . . . . . . . . . . . . . . . . . . 30

3.11 Vertical Deflections of Princeton Beam . . . . . . . . . . . . . . . . . . . . 30

3.12 Twist Deformations of Princeton Beam . . . . . . . . . . . . . . . . . . . 31

4.1 Schematic of Princeton Beam Frequency Experiment . . . . . . . . . . . . 40

4.2 Princeton Beam Tip Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Princeton Beam Frequency Modes . . . . . . . . . . . . . . . . . . . . . . 41

4.4 Twist Deformations of Princeton Beam . . . . . . . . . . . . . . . . . . . 42

4.5 Maryland Beam Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.6 Maryland Beam Finite Element Mesh . . . . . . . . . . . . . . . . . . . . 44

4.7 First Flap Frequency of Maryland Beam . . . . . . . . . . . . . . . . . . . 45

4.8 Second Flap Frequency of Maryland Beam. . . . . . . . . . . . . . . . . . 45

4.9 Third Flap Frequency of Maryland Beam . . . . . . . . . . . . . . . . . . 46

4.10 Fourth Flap Frequency of Maryland Beam . . . . . . . . . . . . . . . . . . 46

4.11 Fifth Flap Frequency of Maryland Beam . . . . . . . . . . . . . . . . . . . 47

5.1 Comprehensive Analyses Tools . . . . . . . . . . . . . . . . . . . . . . . . 50

vi


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List of Tables

2.1 Finite rotation parameters to be used in finite element equations. . . . . . 17

3.1 Simulation Parameters for Elastica with Tip Moment. . . . . . . . . . . . 22

3.2 Simulation Parameters for Elastica with Tip Load . . . . . . . . . . . . . 24

3.3 Princeton Beam Simulation Parameters . . . . . . . . . . . . . . . . . . . 28

4.1 Simulation Parameters for Frequency estimation of Cantilevered Beam . . 39

4.2 Frequency data (in Hz) for Cantilevered Beam . . . . . . . . . . . . . . . 40

4.3 Simulation Parameters for Frequency Estimation of Princeton Beam . . . 41

4.4 Properties of tip mass used in Princeton Beam experiment. . . . . . . . . 42

4.5 Maryland Beam Simulation Parameters . . . . . . . . . . . . . . . . . . . 44

vii


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Symbols

Ai unit vectors fixed in reference frame U strain energy per unit length

Biunit vectors fixed in cross-sectionalframe of deformed beam

u displacement vector of reference line

biunit vectors fixed in cross-sectionalframe of undeformed beam

V inertial velocity of reference line

C direction cosine matrix (Section-2.5) v velocity vector in reference frame

c finite rotation parameter (Rodrigues

or Wiener-Milenkovic) W

work done by external forces andmoments

e1 [100]T w width of the undeformed beam

E Youngs modulus x1 beam coordinate along the referenceline in b frame

F external applied load vectorx2,x3

position of CG wrt undeformed beamreference line

Ficross-sectional stress resultant forcevector

X complete set of unknown beam

variables

f distributed applied force vector per

unit length along beam qBvirtual displacement vector

G complete set of geometrically exactbeam equations Bvirtual rotation vector

G shear modulus (33) identity matrix

H cross-sectional inertial angular

momentum vector

small perturbation for approximationof derivatives

I cross-sectional (33) inertia vector strain vector [11, 212, 213]T


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Symbols ix

i2, i3, i23

cross-sectional mass moments andproduct of inertia

= (K k) elastic curvature

K deformed beam curvature and twist

vector i Lagrange multipliers

K kinetic energy per unit length tip sweep angle

k undeformed beam curvature and twist

vector mass per unit length

L length of undeformed beam inertial angular velocity of deformed

beam cross-sectional vector

M cross-sectional stress resultant

momentum vector a

angular velocity vector in referenceframe

m distributed applied momentum vector

per unit length along beam material mass density

N number of finite elements,

Princeton Beam pitch (root-setting)angle

P vross-sectional inertial linear

momentum

non-dimensional lengthwisecoordinate

R,S,Tcross-sectional (33) flexibilitycoefficient matrices

() ()/t

s deformed beam coordinate system () ()/x1

t thickness of undeformed beam () cross product matrixt1, t2

arbitrary instants in time(), ()

known and unknown boundaryconditions


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Dedicated

to

my Parents

x


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Chapter 1

Introduction

The aim of this thesis is to implement the formulation for modeling helicopter rotor

blades using Geometrically Exact Beam Theory (henceforth abbreviated as GEBT). The

GEBT contains equations describing the overall dynamics of beam members undergoing

arbitrary motions. This thesis is meant to be a complete showcase of building up and

implementation of GEBT to perform dynamic analysis of rotor blades with advanced

geometry and composite materials.

1.1 Motivation

Beam members with one of the dimensions significantly higher than the other two are

building blocks of many structures. They form the subject area of research in Aerospace,

Civil, Mechanical and Biomedical Engineering. Many of these structures like bridges

and buildings undergo relatively small deformations. Linear beam theories like those

of Euler-Bernoulli and Timoshenko provides quite accurate results for these structures.

However, beams undergoing large deformations like helicopter blades, aircraft wings,

propeller blades, shafts and axles require nonlinear analysis.

The thrust for lightweight materials in structural applications have led to extensive use

of composites. These materials not only allow higher out of plane deformations but also

usually have coupling between bending and torsional modes. While 3-D finite element

method may seem to be the ultimate solution for all structural modeling needs, it may

1


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Chapter 1. Introduction 2

neither be always be computationally feasible nor altogether necessary. It is at this point

that Geometrically Exact Beam Theory (GEBT) provides dimensional reduction with

sufficient accuracy for modeling of highly deformable structures such as those mentioned

above.

GEBT finds its applications in modeling dynamic response of rotating beam structures,

particularly helicopter blades for a wide range of loading and boundary conditions. It

can be seamlessly integrated with aerodynamic models for comprehensive analysis of

helicopter rotor. Presence of accurate modeling capabilities could save a lot of time

and money in simulating the behavior of rotating blades under different flight condi-

tions. Simplicity and comprehensiveness of this theory in solving a wide range of beam

modeling problems with surprising accuracy has been the motivation behind this thesis.

1.2 Literature Review of Previous Work

The present thesis has been built upon following the literature which can be classified

into four broad types:

Earlier Approaches

Historical Insight on GEBT

Theoretical manifestation/formulations of GEBT

Implementation of mixed variational formulation of GEBT

Benchmark results to validate the implemented algorithms

1.2.1 Earlier Approaches

With the advent of flexible rotor systems in helicopters several attempts were made to

develop the theory governing the dynamics of rotor blades. Hodges and Dowell [1] formu-

lates approximate theory with ordering schemes for analysis of helicopter blades, which

are still the standard equations for nonlinear beam dynamics. To avoid inaccuracies due


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to large torsional rotations, Hodges et al. [2] developed beam theory by involving an

explicit torsional parameter along with the usual displacement variables.

To make this kind of theory work for moderate deformations Rosen and Friedmann

[3] and Johnson [4] terms upto second order were accounted for in developing beam

equations. With each addition in the order of the approximation, the beam equations

becomes complex, so much so that one equation could last several pages. This complexity

and approximate nature of these theories inspired researchers to focus on other beam

models.

One solution to deal with cases involving large deformations and hence associated finite

rotations is to have a frame attached with each deformed finite element of the beam.

Bauchau and Kang [5] and Johnson [4] have developed the theory which is known as

Multibody Formulation. Saberi et al. [6] documents the Rotorcraft Comprehensive Anal-

ysis System which was earlier based on this methodology. This approach produced quite

accurate results for even larger deformations allowed by previous theories. Hopkins and

Ormiston[7] have verified and validated some of the benchmark problems using RCAS

multibody formulation with good correlation. The problems solved in this reference is

the inspiration behind the models built in this thesis.

1.2.2 Historical Insight on GEBT

With the multibody formulation, the beam equations were still approximate ones, with

the exception of an attached frame with them. To model large deformation using this

method more finite elements are required meaning more computational expenditure. To

do away with the approximations altogether, Geometrically Exact Beam Theory was

proposed.

One of the earliest formulation for static elastica problem exist in the form of Kirchhoff-

Clebsch equations described by Love[8]. These equations are obtained by equilibrium

analysis of elastica under general static loading conditions. The resulting set of nonlinear

equations are the functions of 3-dimensional forces, moments and curvatures. Chapter

XIX describes an interesting result known as the Kirchhoff Kinetic Analogue, which


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states that the equations of equilibrium of a thin, prismatic rod applied with tip load

and moment is identical to the equations of motion of a rigid body turning about a fixed

point. This analogy is used to derive the kinematic relations for elastica with different

loading and boundary conditions.

Reissner [9] presented first truely geometrically exact formulation for finite strain (i.e.

not negligible) deformations occuring in a plane. This work was expanded again by

Reissner[10] to include deformations not necessarily restricted to a plane. The mixed

variational formulation as described in 2 simplifies to the equations presented in this

reference for static cases.

The geometrically exact analysis of beam dynamics that are deformable in 3-D is pro-

posed by Simo [11]. In addition, Simo and Vu-Quoc [12] also provides solution and

incremental algorithms for dynamic analysis of beam structures.

A number of alternate formulations for geometrically exact beam theories exist by various

authors who have approached the same problem in different ways. All of these works

have built on the works of Reissner and Simo. Hence, GEBT is also referred to as

Reissners Beam Theory or Reissner-Simo Beam Theory.

1.2.3 Theoretical Development

Hodges[13] describes the derivation of the mixed variational formulation of GEBT from

Hamiltons Principle in detail. It also contains references about some of the earlier works

that helped shape this theory. It also states the assumptions used in this theory and

some of those from the earlier ones that were relaxed.

While the earlier reference contained formulation containing displacement and rotation

terms,Hodges[14] states an alternate formulation which can be used for dynamic analy-

sis, especially for perturbation analysis of slender beams to calculate natural frequencies.

Here the formulation contains reduced number of intrinsic parameters and does not con-

tain any displacement or rotation terms, hence the name Fully Intrinsic Formulation.

The Fully Intrinsic Formulation exists as a set of nonlinear PDEs in space and time andcan be solved by finite difference methods.


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An alternate implementation of GEBT is suggested by Jelenic and Saje [15]in incremen-

tal, iterative and invariant finite elements formulation for statics of the beam. Jelenic

and Crisfield [16]have expanded the same formulation for both statics and dynamics.

Pai [17] provides GEBT for three different forms of deformations beams can undergo.

Finite element implementation method for a few cases and results for the same are also

given.

The major textbook on this subject is by Hodges [18]. Here he sums up the history,

precursors, references, formulations, implementation and applications of GEBT in detail.

1.2.4 Solution Procedure

Hodges et al.[19] derive the finite element equations from the mixed variational formu-

lation for solving beam dynamics problem. A hint for solving eigenvalue problems is

also provided and the formulation is validated for Maryland Beam experiment.

More cases for validation of GEBT are provided by Wang et al. [20] who use Wiener-

Milenkovic parameters for defining finite rotations. Here the finite element equations

are presented in complete detail. The use of global frame facilitates application of dead

forces like gravity. This paper also has a few static and dynamic response examples for

validation.

Dynamic response problems require time marching schemes for solving initial value prob-

lem. Yu and Blair [21]presents the applications of the GEBT finite element equations.

The application of Newmark time marching method is also explained in detail.

Many optimization strategies are developed for solving GEBT equations .Patil and

Hodges [22] proposes one such improved strategy for solution of intrinsic, mixed vari-

ational beam equations by variable order hp-finite elements. Since this is the most

efficient method in terms of CPU time required, it is highly useful for dynamic response

analysis of complex structures where iterations are to be performed over a large number

of time steps.


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1.2.5 Benchmark Results

The following section comprises of references which provides results of experimental

benchmarks used to validate nonlinear beam theories and also previous attempts at

modeling the experimental results.

Benchmarks for fixed fram statics and dynamics were established by Dowell and Traybar

[23] and Dowell and Traybar[24]. The results described in these references are famously

referred to as the Princeton Beam static and frequency measurement experiments.

Epps and Chandra [25]provide experimental results for frequency measurement of swept-

tip aluminum and composite beam aka Maryland Beam experiments. The beam di-

mension, material properties, composite ply direction and angular velocities are also

provided.

1.3 Organization of Thesis

The thesis document comprises of front matter, five chapters and bibliography section.

1. Chapter-1includes introduction and review of references relevant to the thesis.

2. Chapter-2contains the derivation of the theoretical formulation of GEBT and the

finite elements equations to be used for beam structural analysis.

3. Chapter-3conatins solution procedure for static problems. Verification and vali-

dation of the results is also presented with benchmarks.

4. Chapter-4presents formulation for dynamic analysis of beams. Solution procedure

for steady state and perturbation analysis for determining natural frequencies for

fixed and rotating frame structures is also stated.

5. Chapter-5closes this thesis with a summary and a note on the future scope and

recommendation for further developments


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Chapter 2

Theory

2.1 Preliminaries

Some of the definitions from earlier works that will be used as axioms and theorems

from previous works are presented here. The origin of these results are given in [13].

A vector quantity Z can be expressed in any basis such that,

Z= ZAiAi = ZA={ZA1, ZA2, ZA3}T (2.1)

=Zbi bi =Zb ={Zb1, Zb2, Zb3}T (2.2)

=ZBiBi= ZB ={ZB1, ZB2, ZB3}T (2.3)

They are tranformed from one basis to another multiplying with a transformation matrix

ZB= CBb

Zb (2.4)

The expression for the transformation matrix is given in Section-2.5.

7


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Chapter 2. Theory 8

The cross product operator () is defined as

Z=

0 Z3 Z2

Z3 0 Z1Z2 Z1 0

,and (2.5)

Z Y= Z Y (2.6)Figure-2.1 shows the states of the beam before and after deformation. Measurements

are made relative to a point in the fixed frame A. The undeformed and deformed states

have frames b and B attached to the reference cross section. Note thatB1 may not in

general be tangent to R due to shear deformations.

Figure 2.1: Undeformed and deformed states of beam showing reference line andarbitrary cross section.

The following results for stain and curvatures are derived in detail in Danielson and

Hodges [26]. These relations obtained by frame tranformations are exact and are defining


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Chapter 2. Theory 9

feature of GEBT.

= C(e1+ ub+

kbub) e1 (2.7)

= KB kb, where (2.8)KB =CCT + CkbCT (2.9)

2.2 Hamiltons Principle

The derivation of the intrinsic beam equations start from the Hamiltons Principle which

is stated below in its weakest form.

t2t1

l0

[(KU) + W] dx1dt= A (2.10)

Heret1 and t2 are known times, l is length of the beam, KandUare kinetic and strain

energy densities, respectively and W is the virtual work of the applied loads. The

integration is carried over spanwise coordinate, x1 and time coordinate, t. A is the

virtual action at the end of time interval.

2.2.1 Strain Energy

The strain energy is a nonlinear function of strain and curvature which, in turn, are

nonlinear functions of displacements and rotations.

U =U(, ) (2.11)

Carrying out the variation as follows

l0Udx1=

l0

T

U

T+ T

U

Tdx1 (2.12)


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Chapter 2. Theory 10

Defining the virtual displacement and virtual rotation parameters as follows:

qB :=Cub (2.13)

B :=CCT (2.14)() indicates that these virtual quantities are variations of their respective variable inundeformed bbasis but not in deformed Bbasis.

Substituting the equations (2.13)-(2.14) in equations (2.7)-(2.9) and carrying out varia-

tions, we get

= q

B+ (e1+ )B (2.15) =

B+KBB (2.16)

Thus, the expression for virtual strain energy comes out to be:

U= [(q

B)T qTB

KB TB(

e1+

)]FB+ [(

B)T TB

KB]MB (2.17)

2.2.2 Kinetic Energy

The sectional kinetic energy is a function of sectional linear and angular velocities.

K=K(V, ) (2.18)

In the variational form it can be written as,

l0Kdx1=

l0

VTB

K

VB

T+ TB

K

B

Tdx1 (2.19)


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Similar to equation (2.7)-(2.9), the expressions for sectional linear and angular velocities

are given by Kane and Levinson[27]

VB =C(vb+ ub+ bub) (2.20)B =CCT +CbCT (2.21)Carrying out variation for getting virtual linear velocity expression:

VB =C(vb+ ub+

bub) + C(ub+

bub) (2.22)

Substituting the result from equation-(2.13), the final expression for virtual linear ve-

locity becomes

VB =qB+BqB+VBB (2.23)

Similarly, carrying out variation of equation-(2.21) to get virtual angular velocity:

B =CCT CCT + CbCT +CbCT (2.24)Substituting equation-(2.14) and simplifying the expression gives virtual anguar velocity:

B = B+BBBB (2.25)Using the property of cross product,

YZ=ZY + Y Zthe final expression for virtual angular velocity becomes:

B = B+BB (2.26)


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The momentum-velocity relations are given as

PB =

K

VB

T=m(VB

BB) (2.27)

HB = KB

T=iBB+ mBVB (2.28)

where, m and iB are sectional mass and moment of inertia tensors and

B = [0, x2, x3]T

is the location of CG with respect to the beam reference line.

Thus, the expression for the virtual kinetic energy turns out to be:

K= (qB+BqB+VBB)PB+ ( B+BB)HB (2.29)2.2.3 Virtual Work

The virtual work expression in terms of externally applied distributed forces and mo-

mentum can be written as,

W =

i0

(qT

BfB+ T

BmB)dx1 (2.30)

2.2.4 Variational Formulation

Substituting equation-(2.17), (2.29) and (2.30) into Hamiltons Principle equation-(2.10),

the variational form of equation comes out to be:

t2t1

l0{(qTB qTBB TB VB)PB+ ( TB TBB)HB

[(qB)TqTBKBTB(e1+)]FB+[(B)TTBKB]MB+qTBfB+TBmB}dx1dt=

l0

(qT

BPB+

T

BHB)

t2

t1

dx1 t2t1

(qT

BFB+ T

BMB)

l

0

dt (2.31)


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Performing integration by parts to remove time derivatives of virtual quantities, we get

the final equation as:

t2t1

l0{qTB(FB+KBFB+ fB PB BPB) + TB[MB+KBMB+ (e1 +)FB+mB

HBBHBVBPB]}dx1dt= l0

[qT

B(PBPB) + TB(HBHB)]t2t1

dx1

t2t1

[qT

B(FBFB) + TB(MBMB)]l0

dt (2.32)

The Euler-Lagrange equations can be obtained from the variational equation as

FB+KBFB+ fB = PB+BPB (2.33)MB+KBMB+ (e1+ )FB+ mB = HB+ BHB+VBPB (2.34)

These equations satisfy the condition of equilibrium for the general motion of beam under

external forces and moments. To obtain displacement and rotation, we need kinematic

relations. Although obtained from energy principles, these equations have Newtonian

character, which confirms the universality of these equations.

2.3 Mixed Variational Formulation

The kinematical relations are obtained by rearranging the terms of equations-(2.7), (2.8),

(2.20) and (2.21) as follows:

u

=CT

(e1+ ) e1 ku (2.35)u= CTV v u (2.36)c =Q1( + k Ck) (2.37)

c= Q1(C) (2.38)

Q1 is defined in Section-2.5


25/66


Writing the mixed variationsl formulation by combining variational and kinematical

equations by means of lagrange multipliers, we obtain:

t2t1 l0{[VTP+ TH TF TM+ q

T

f+ T

m]

+ [1(u CT(e1+ ) + e1+ ku)]

+ [2(uCTV + v+ u)]+ [3(c

Q1( + k Ck))]

+ [4(cQ1(C))]}dx1dt

(2.39)

The Lagrange multipliers are obtained by satisfying the independent variation of virtual

quantities. The final expression for mixed variational formulation which is the governing

equation describing overall dynamics of slender beams is given below:

l0{uTa Fa+ Ta Ma+ Ta [ Ha+ aHa+VaPa CTCab(e1+ )FB]+ uTa (

Pa+ aPa) FTa [CTcab(e1+ )Cabe1] FTa ua MTa ca MTaQ1a Cab + PTa (Va va aua ua)+ H

T

a (BBCbaQaca) uTa Tama}dx1= (uTa

Fa+ T

aMa FTaua MTa ca)

l0

(2.40)

2.4 Finite Element Implementation

The beam is discretized into N elements of length Li having two nodes i and j (for

simplicity beams with no joints will have j =i+1). Leti be the lengthwise coordinatewithin a beam such that,

i =x1 Li

Li(2.41)

dx1= Lidi (2.42)

Z = 1

L1

Z

i(2.43)

where,Li is the coordinate of the node i of the element and Zis an arbitrary vector.


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Since the Mixed Variatonal Formulation is in its weakest form, the finite elements are

discretized using linear shape functions as:

ua= (1 )ui+ uj a= (1 )i+ j (2.44)Fa= (1 )Fi+ Fj Ma= (1 )Mi+ Mj (2.45)

Performing the quadrature of equation-(2.40) numerically gives:

Ni=1

{uTi fui + uTi+1f+ui + T

i f

i+

T

i+1f+i

+ FT

i f

Fi+ F

T

i+1f+Fi

+

MT

i fMi + MT

i+1f+Mi + PT

i fPi + HT

i fHi}

(2.46)

The fmatrices for each element are presented below:

fui =CTCabFi fi +Li

2 [aCTCabPi+ CTCabPi] (2.47)

fi=CTCabMi mi +

Li2

[aCTCabHi+ CTCabHi+ CTCab(e1+ i)Fi)](2.48)

fFi

=ui Li2

[CTCab(e1+ i) Cabe1)] (2.49)

fMi

=ciLi2

Q1a Cabi (2.50)

fPi = CTCabVi vi aui ui (2.51)

fHi = i CbaCa CbaQaci (2.52)

fi = 10

(1 )faLid (2.53)

f+i =

10faLid (2.54)

mi =

10

(1 )maLid (2.55)

m+i =

10maLid (2.56)

Here, fa and ma are distributed force and moment vectors. Similarly, va and a are

elemental linear and angular velocities respectively in global frame such that


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Also in each of the elements (for i = 1 to N)

fPi =0, (unknown) (2.70)

fHi =0, (unknown) (2.71)

In addition to these set of equations, the formulation requires momnetum-velocity rela-

tions of Eq.-(2.27) and (2.28). Also, the following set of constitutive relations is required

to complete the formulation.

=

R SST T

FM

(2.72)

2.5 Finite Rotation

Two popular methods for handling finite rotations are Rodrigues parameters and Wiener-

Milenkovic parameters. The table below provides the expressions for parameters to be

used in finite elements analysis for rotation angle of (deg or rad). Rodrigues param-

eters are a preferred choice for smaller angular rotations (< 90 deg) after which tan2

becomes infinite. For larger rotations (i.e., upto 180 deg) Wiener-Milenkovic parameters

are opted for.

Rodrigues Wiener-Milenkovic

c 2tan2 4tan4

c0 1 + cTc4 2 c

Tc8

C 1c0 [(1 14cTc) + 12ccT c] 1(4c0)2 [(c20 cTc)2c0c + 2ccT]Q 1c0 [

c/2] 1(4c0)2 [(4

14c

Tc)2

c+ 12cc

T]

Table 2.1: Finite rotation parameters to be used in finite element equations.


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Chapter 3

Beam Statics: Verification and

Validation

Static analysis for beams can be performed in a number of ways. The Euler-Bernoulli

and Timishenko Beam theories provide sufficiently accurate results for very small strains.

Nonlinear Taylor Series expansion methods as described in Hodges and Dowell[1] work

well for small to moderate deformations. Even multibody methods such as the one

described in Bauchau and Kang [5] are not exact as per analytical models. Therefore, to

model arbitrarily large deformations, Geometrically Exact Beam Theory is the preferred

choice.

The equations 2.58-2.71 described in Chapter-2 could be modified for usage in static

analysis of highly deformable beam elements. Due to involvement of intrinsic rotation

term, the prediction of load-deformation characteristics shows visible improvement over

nonlinear finite element methods as shown in Hodges and Patil [28] and multibody

formulations used in Hopkins and Ormiston [7]. The modified implementation of the

GEBT code for static analysis was used to solve the following benchmark problems:

Tip moment applied on elastica

Tip load applied on elastica

Princeton beam static large deformation experiment

18


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Chapter 3. Beam Statics: Verification and Validation 19

3.1 Solution Procedure

The generalized dynamic analysis equations (2.58)-(2.71) described in Chapter-2can be

specialized for static analysis. The terms involving time derivatives would be eliminated.

Similarly, the linear and angular velocities and their corresponding momenta would

become zero. The last two equations would be trivially satisfied leaving the formulation

with 12(N + 1) equations and the same number of variables. The modified equations

for static analysis with cantilevered boundary conditions are presented below[20]:

At the starting node,

fu1F1= 0 (unknown) (3.1)

f1 M1= 0 (unknown) (3.2)

fF1

u1= 0 (known) (3.3)

fM1

c1= 0 (known) (3.4)

At the ending node,

f+uN FN+1= 0 (known) (3.5)

f+N MN+1= 0 (known) (3.6)

f+FN

+ uN+1= 0 (unknown) (3.7)

f+MN

+ cN+1= 0 (unknown) (3.8)

At the intermediate points of the elements (for i = 1 to N-1),

f+ui+ fui+1

=0 (unknown) (3.9)

f+i+ fi+1 =0 (unknown) (3.10)

f+Fi

+ fFi+1

=0 (unknown) (3.11)

f+Mi

+ fMi+1

=0 (unknown) (3.12)


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The modified fmatrices for the static case are presented below:

fui =CTCabFi fi (3.13)

fi =CTCabMi mi Li2 [CTCab(e1+ i)Fi)] (3.14)

fFi

=ui Li2

[CTCab(e1+ i) Cabe1)] (3.15)

fMi

=ciLi2

Q1a Cabi (3.16)

Here,

f

i = 10 (1 )faLid (3.17)f+i =

10faLid (3.18)

mi =

10

(1 )maLid (3.19)

m+i =

10maLid (3.20)

where,

=x1 Li

Li(3.21)

The equations could be represented in compact notation as G(X, F) =0 where Grep-

resents the set of vector equations (3.1)-(3.12) as functions 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,FN+1,MN+1

The set G represents a set of nonlinear equations which cannot be solved explicitly,

hence, solved iteratively by Newton-Raphson method. The procedure for implementa-

tion of the same is described below.

(i) Assume an initial guess for unknowns X. An ideal guess would be undeformed

state X= 0.


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(ii) CalculateG(X, F), where Fare known boundary conditions.

(iii) Populate Jacobian matrix by perturbing previousX such thatj = 1 to 12(N+1);B(:,j) = [G(X(j)=X(j)+, F)

G(X(j), F)]/, where epsilon is small relative to

anticipated solution of these equations. Consequently, B is a square matrix of size

12(N+1).

(iv) Calculate the updated X as X= X B1G.

(v) Repeat steps (ii) to (iv) until norm ofGis smaller than the required tolerance.

Newton-Raphson method requires inversion of 12(N+ 1) square matrix makes it slow,

however, the convergence is achieved in 2-5 steps for well posed boundary conditions.

3.2 Benchmark Static Problems

GEBT can be used for a large number of cases involving slender beams without any

restriction on cross-section, material variety, boundary conditions or load distribution,

provided the finite elements are appropriately chosen to take these variations into ac-

count. GEBT is essentially based on the concept of dimensional reduction of complex

beam structures like large aspect ratio aircraft wings and helicopter blades. These struc-

tures are analyzed, initially, along a reference line and then in the deformed state over

the cross-section to get the complete picture. The following cases considered in the

present thesis for static analysis involve prismatic beams.

3.2.1 Bending moment applied at the tip of an elastica

This case is of significance as it serves as the basis for verification of any theory with

the analytical solution. An elastica is defined as the shape of the curve into which the

central-line of a thin, prismatic rod is bent on application of forces and moments, applied

at its tip [8]. It is more of a theoretical concept, with no limit on the permissible loading.

The governing differential equation for this case is given by

EId

ds =M(s) (3.22)


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where is the slope, M(s) is the bending moment of the segment at distance s in local

deformed coordinate system. Solving this equation for cantilevered boundary conditions,

the expression for displacements are found to be

(s) =TS

EI (3.23)

u(s) =EI

T sin(

Ts

EI) s (3.24)

v(s) =EI

T [1 cos(Ts

EI)] (3.25)

The present work attempts to solve the elastica problem of Hopkins and Ormiston [7].

Relevant properties for modeling are provided in Table - 3.1 . Figure -3.1 shows the

excellent confirmation of the current work with analytical solution. The RCAS solution

of Hopkins and Ormiston[7] are not shown here due to indistinguishable overlap.

Dimension Material

Properties Accuracy Boundary Conditions

L 6.096 m E 71.6 GPa Node,N 20 M2= NEI/L,

w 15.24 cm G 26.9 GPa Perturbation, 104 where, = 0.05556

t 9.5 mm 2800 kgm3 Tolerance, 109 and 0.2778

Table 3.1: Simulation Parameters for Elastica with Tip Moment

When the tip moment is increased toML = 0.2778 NEIL the superiority of the currentapproach over the solution from RCAS1 is clearly visible in Figure - 3.2. Here the finite

rotations were modelled by Wiener-Milenkovic parameters[20] to avoid singularity, as

described in section-2.5, encountered when beam rotations nears 2. GEBT is thus

suitable for flexible beams undergoing large deformation where approximate methods

tend to fail due to accumulated errors.

3.2.2 Dead (non-follower) force applied at the tip of an elastica

Analytical solution exists for tip loaded elastica in line with that of section 3.2.1. This

case is of practical significance and many experiments are designed to verify the theories

1RCAS (Rotorcraft Comprehensive Analysis System) is a rotorcraft comprehensive analysis code (see

Section-5.2)that uses second order FEM coupled to multibody formulation for beam analysis.


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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

Ve

rticalPosition,2r3

Analytical

Current

Figure 3.1: Spanwise position of elastica deformed by tip moment (when 2R/N =

0.05556).

developed to model the deformations in this case. The governing differential equation

for this case is given by Doyle [29](page-176).

EId2

ds2 =P sinsin Pcoscos=Pcos( + ) (3.26)

Solving equation for transverse loading and cantilevered boundary conditions (= 0),

the tip displacements are obtained by numerically evaluating the integrals:

J1(L) =

L0

dsin(L+ ) sin(+ )

=

2 (3.27)

J2(L) =

L0

sind

sin(L+ ) sin(+ )

=

2

LvL (3.28)

J3(L) = L0

(cos1) dsin(L+ ) sin(+ )

= 2LuL (3.29)


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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

RCAS

Current

Figure 3.2: Spanwise position of elastica deformed by tip moment (when 2R/N =0.2778).

Relevant properties for modeling are provided in Table - 3.2. The present work obtains

excellent correlation with analytical solution as evident in Figs. 3.3, 3.4and3.5. The

superiority of the GEBT over multibody finite element formulation (used in RCAS)

results is also visible as the former unlike the latter not only follows the trend but alsomakes accurate prediction of tip deformation and rotation.

Dimension Material

Properties Accuracy Boundary Conditions

L 6.096 m E 71.6 GPa Node,N 20 F3= EI/L2,

w 15.24 cm G 26.9 GPa Perturbation, 104 where, = 0 to 5

t 9.5 mm 2800 kgm3 Tolerance, 109

Table 3.2: Simulation Parameters for Elastica with Tip Load


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0 1 2 3 4 5

0

10

20

30

40

50

60

70

Normalized Load, P3R

2/EI

2

TipA

ngleofRotation,

|

2(R)|[d

eg]

Analytical

RCAS

Current

Figure 3.3: Angle rotated by elastica tip under transverse load.

0 1 2 3 4 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


2/EI

2

NormalizedTipV

erticalDeflection,

u3(R)/R

Analytical

RCAS

Current

Figure 3.4: Vertical deflection of elastica tip under transverse load.


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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

NormalizedT

ipVerticalDeflection,u3(R)/R

Analytical

RCAS

Current

Figure 3.5: Horizontal deflection of elastica tip under transverse load

3.2.3 Princeton beam experiment

Princeton Beam experiments were a series of benchmark results published by Dowell and

Traybar[23, 24] to study the static nonlinear deformations of slender beams under tip

load. The deformations were identical to that of a rotor blade flap, lag and twist motion.

A number of previous attempts have been made to model these deflections. Dowell

et al. [30] employed nonlinear theory for modeling moderate deformations. Hopkins and

Ormiston [7] employed RCAS to model these results using multibody methods. Hodges

and Patil [28]uses GEBT, as does the current work. Figure 3.6 provides a schematic of

the experimental setup used in the Princeton Beam experiments for static deformation

estimation. The setup consists of a milling machine type precision, indexing-chuck with

specially fabricated end fixtures to provide near ideal cantilevered boundary conditions

as well as to get desired pitch angle. Figure3.7shows loading condition and resultant


38/66


deformation for a general pitch angle, . The characteristic nonlinear flap-lag-torsion

coupling is easily visible.

Figure 3.6: Schematic of the experimental setup used in Princeton Beam experiments.

Figure 3.7: Load applied to a cantilevered beam: undeformed and deformed states[23].

For modeling the beam used in Princeton Beam experiments, the axis system like that

shown in Figure 3.8 was used. X-Y-Z represents the inertial coordinate system, while

U-V-W represents body fixed coordinate system post deformation. Figure3.9shows the

end view (tip to root) of the blade showing the reference pitch angle as well as the sense

of measurement of the same.


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Figure 3.8: Beam cross-section showing space-fixed and body-fixed axes system.

Dimension Material

Properties Accuracy

BoundaryConditions

L 0.508 m E 71.6 GPa N 8 F3 13.345 N

w 12.7 cm G 26.9 GPa 109 1 0to 90

t 3.2 mm 2800 kgm3 109

Table 3.3: Princeton Beam Simulation Parameters

These experiments are testimonial of the fact that large deformations cannot be modeled

by linear methods which will simply ignore the coupling in bending and torsion modes.

Hence, a number of attempts have been made to model these deformations by nonlinear

methods, multibody formulation, GEBT etc. The present thesis uses geometrically exact

beam modeling to reproduce results in correlation with the experimental benchmarks.

The initial pitch deflections were taken into account by the rotation matrix, Cab which

aligns the rotated coordinate system U-V-W with undeformed beam. The distributed

gravitational load as well as the tip load acted as dead load and were in the space-fixed

coordinate system X-Y-Z. The resultant deformations and twist are to be expressed in

the same space-fixed coordinate system for validation with the previous results.


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Figure 3.9: Princeton beam end view showing variation of the pitch angle, usingindexing chuck[24].

The beam is rectangular, prismatic with thickness and chord much smaller than the

length. The material of the beam is 7075-T651 Aluminum alloy. The input parame-

ters for the Princeton Beam problem are given in Table - 3.3. Figure - 3.10shows the

vertical deflections for the cantilevered beam as obtained experimentally and modelledusing GEBT. Figure - 3.11 provides a comparison of the horizontal deflections of the

cantilevered beam under vertically applied loads with excellent prediction of the experi-

mental outcome. This shows coupling between flap and lag modes in the cases when the

beam coordinate system is not aligned with the space-fixed coordinate system. Figure -

3.12depicts the twist in the beam due to beams own weight and the applied tip load.

Once again good agreement (within experimental limitations) is demonstrated.

A slight discrepancy is observed in twist modeling which is not unique and is observed

in other methods also [7]. This can be attributed to the fact that twist deformation is

sensitive to tip twisting moment, which can arise due to even slight displacement of load

pane from centre of the chord creating a moment arm between beam vertical reference

line and the tip load.

These results show that GEBT is highly suitable for modeling slender beams undergoing


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0 15 30 45 60 75 90

0

0.05

0.1

0.15

0.2

0.25

Pitch Angle, ||[deg]

NormalizedHorizontalDeflection,

|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 = .25

Exp. s/R = .50

Exp. s/R = .75

Exp. s/R = 1.0

Figure 3.10: Horizontal deflection vs. pitch angle of Princeton Beam forTip load = 3 lb (13.3 N).

0 15 30 45 60 75 90

0

0.05

0.1

0.15

0.2

0.25


Normalized

VerticalDeflection,

|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 = .25

Exp. s/R = .50

Exp. s/R = .75Exp. s/R = 1.0

Figure 3.11: Vertical deflection vs. pitch angle of Princeton Beam forTip load = 3 lb (13.3 N).


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0 15 30 45

0

0.51

1.5

2

2.5

3

3.5

4


Twist,|1

(s,P3,

)1

(s,0,

)|[deg]

GEBT s/R = .25

GEBT s/R = .50

GEBT s/R = .75

GEBT s/R = 1.0Exp. s/R = .25

Exp. s/R = .50

Exp. s/R = .75

Exp. s/R = 1.0

Figure 3.12: Twist deformation vs. pitch angle of Princeton Beam for

Tip load = 3 lb (13.3 N).

arbitrary loading and deformations. The only condition is the linear constitutive rela-

tions should hold good and the material should remain in elastic limits. Having said that,

it covers a large range of applications as mentioned earlier. Not only this, the beams

with initial curvatures, sweep or non-uniformity can be modeled using this method. A

particular utility of GEBT lies in modeling slender composite beams with any kind of

nonlinear coupling due to various ply arrangements. However, static deformation is just

a special case of the greater purpose that GEBT can serve beam dynamics.


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Chapter 4

Beam Dynamics: Verification and

Validation

The dynamics of moving beams are of particular interest for various Aerospace and

Mechanical applications including flying wings, rotating blades, driving shafts, high

temperature gas turbines and wind turbines to name a few. In the previous Chapter a

specialized case of the mixed variational formulation of GEBT was used to demonstrate

its utility in static load-deformation characterization. In addition to that, as mentioned

in Chapter-2, GEBT can be used to predict the steady-state and time-dependent re-

sponse for slender beams. Also, this formulation can be used to predict the natural

frequencies of static and rotating beams by introducing small perturbations about the

steady state and performing eigenvalue analysis.

The complete set of equations (2.58)-(2.71) described in Chapter - 2 is required for

complete description of beam dynamics. The solution procedure for dynamic response,

though implemented, is not validated and hence left for future works to be pursued. The

present thesis validates the frequency estimation of beams for both fixed and rotating

cases. A number of attempts have been made to validate several beam theories with the

frequency results of Princeton Beam for fixed frame and Maryland Beam for dynamic

case. While GEBT is wonderful for response canculations, natural frequency estimation

is not straight forward. If all the subtleties of modeling are taken care of then the

GEBT makes quite accurate estimation of these frequencies as described in Hodges andPatil[28] and Hodges et al. [19]. Only the multibody method adopted by Hopkins and

32


44/66

Chapter 4. Beam Dynamics: Verification and Validation 33

Ormiston[7]provides accuracy of the same order as the present methods. Following is

a list of cases considered for validation of frequency estimations by present work:

Analytical frequencies of a cantilevered beam

Fixed frame case of Princeton beam

Rotating frame case of Maryland beam

It should be noted that the natural frequency of the system decides the structural

dynamic characteristics of a system and hence is considered ahead of the beam response

problem.

4.1 Estimation of Natural Frequencies

The equations described in Chapter-2were represented in nonlinear form as

G(X, X, F) =0 (4.1)

To calculate natural frequencies, these equations need to be linearized in time by intro-

ducing small perturbations about the steady state. This gives a linear equation of the

form[19]

AX + BX= F (4.2)

where

A=G

X

B=G

X

and Frepresents dynamic boundary conditions which is 0 for analysis about the steady

state.


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Frequency estimation is a two step procedure; firstly, determining the steady state of the

system under the given set of loading and boundary conditions, and then introducing

small perturbations to form the basis for eigenvalue analysis. Slender beams with pre-

twist, taper, sweep and different elastic and CG axes can be modelled by this method.

Slight modifications as discussed in Chapter-2can be made to accommodate complicated

boundary conditions like root offset, flap and/or lag hinge, and even joined beams and

flexible root supports with a variety of spring coefficients.

Another method which can be used to predict first fundamental frequencies in all bending

and torsion modes is by obtaining the dynamic response of the system under an impulsive

loading. Then, performing the Fourier transform of this response gives the natural

frequencies of fundamental modes. However, this method is not suitable, as large amount

of time required for sufficient number of time marches. Further, the resolution of the

FFT algorithm used for Fourier analysis depends on the number and size of the time

steps, which are difficult to predetermine.

4.1.1 Steady State Analysis

To perform steady state analysis the set of equations (2.58)-(2.71) from Chapter -2 are

modified by eliminating the time dependent terms. This leaves the formulation with

18N + 12 equations and the same number of variables. Given below are the set of

equations required to perform the steady state analysis using GEBT:



f1 M1= 0 (unknown) (4.4)

fF1

u1= 0 (known) (4.5)

fM1

c1= 0 (known) (4.6)


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Here,

fi =

1

0(1 )faLid (4.21)

f+i = 10faLid (4.22)

mi =

10

(1 )maLid (4.23)

m+i =

10maLid (4.24)

where,

=x1

Li

Li(4.25)

The equations could be represented in compact notation as G(X, F) =0 where Grep-

resents the set of vector equations (3.1)-(3.12) 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,MN+1

Loading conditions: f,m,v,

The set G represents a set of nonlinear equations which cannot be solved explicitly,

hence, solved iteratively by Newton-Raphson method. The procedure for implementa-

tion of the same is described below.

(i) Assume an initial guess for unknowns X. An ideal guess would be static, unde-

formed state X= 0.

(ii) CalculateG(X, F), where Fare known boundary conditions.

(iii) Populate Jacobian matrix by perturbing previousX such thatj = 1 to 12(N+1);B(:,j) = [G(X|X(j)=X(j)+epsilon, F)G(X), F)]/, where epsilon is small rela-

tive to anticipated solution of these equations. Consequently,B

is a square matrixof size 18N+12.


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(iv) Calculate the updated X as X= X B1G.

(v) Repeat steps (ii) to (iv) until norm ofGis smaller than the required tolerance.

(vi) Return the value of the JacobianB and steady state variables X for use in pertur-

bation analysis.

The prediction of correct steady state is very important for accurate frequency pre-

dictions because the deformation in this state affects the tangent stiffness and inertia

matrices highlighting the importance of nonlinear analysis[19].

4.1.2 Perturbation Analysis

The purpose of this analysis is to obtain the A matrix of equation-4.2. The method here

is same as that for calculate the Jacobian matrix Bexcept that it requires no iteration

and convergence. The equations used here are essentially the time derivative terms of

equations2.58-2.71. They are presented below for ready reference.



f1 M1= 0 (unknown) (4.27)

fF1

u1= 0 (known) (4.28)

fM1

c1= 0 (known) (4.29)

At the ending node,

f+uN FN+1= 0 (known) (4.30)

f+N

MN+1= 0 (known) (4.31)

f+FN

+ uN+1= 0 (unknown) (4.32)

f+MN

+ cN+1= 0 (unknown) (4.33)


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At the intermediate points of the elements (for i = 1 to N-1),

f+ui+ fui+1

=0 (unknown) (4.34)

f+i + fi+1

=0 (unknown) (4.35)

f+Fi

+ fFi+1

=0 (unknown) (4.36)

f+Mi

+ fMi+1

=0 (unknown) (4.37)

Also in each of the elements (for i = 1 to N)

fPi =0, (unknown) (4.38)

fHi =0, (unknown) (4.39)

The modified fmatrices for the perturbation case are presented below:

fui =Li

2 [

CTCabPi] (4.40)

fi=

Li2

[

CTCabHi] (4.41)

f

Fi=0 (4.42)

fMi

=0 (4.43)

fPi =ui (4.44)

fHi =CbaQaci (4.45)

The equations could be represented in compact notation as G(X, F) =0. These equa-

tions will be trivially satisfied at the steady state. At this step, iteration is not required

and the A matrix can be determined in the same way as the Jacobian matrix B was

determined in section-4.1.1by following the steps given below.

(i) For each value ofX introduce small perturbations which is assumed to be small.

(ii) Corresponding to each of these perturbed values ofX, populate the A matrix one

column at a time by dividing the returned Gvector with 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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Now that both A and B matrices of equation 4.2 are obtained, a direct use of eig

function of MATLAB will give multiple roots of which imaginary parts are our required

natural frequencies.

4.2 Results and Validation

This section provides the details of intricacies involved in the modeling of fixed and

rotating frame dynamic problems. The frequencies provided here are the result of eigen-

values generated from MATLAB eigfunction in sorted order and are not verified from

eigenvectors for various modes due to lack of detailed literature.

4.2.1 Cantilevered Beam

This case is considered to compare the results of GEBT with analytical solution pre-

sented in Hopkins and Ormiston [7]. The first natural frequency of a clamped-free

uniform, prismatic beam is given by:

= (1.875)2

EI

wtL4 (4.46)

where w is the chord, t is the thickness and R is length of the beam. Table- 4.1provides

the dimensions of the beam and its material properties. Table-4.2provides the flatwise

and edgewise frequency data for analytical. experimental and simulaton results.

Dimension Material

Properties Accuracy

BoundaryConditions

L 0.508 m E 71.6 GPa Node,N 30 Fixed-free with gravity

w 12.7 cm G 26.9 GPa Perturbation, 109 1root = 0and 90

t 3.2 mm 2800 kgm3 Tolerance, 109

Table 4.1: Simulation Parameters for Frequency estimation of Cantilevered Beam


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Method Flatwise Edgewise

Anaytical 10.049 40.196

GEBT 0 10.052 40.134

GEBT 90

10.053 40.125Experimental 10.150 41.143

Table 4.2: Frequency data (in Hz) for Cantilevered Beam

4.2.2 Princeton Beam Experiment

Princeton Beam experiment established benchmark not only for static deflections but

also for natural frequencies for a cantilevered case. The schematic of the experimental

setup is shown in Figure-4.1. While the precision chuck and beam are same as in the

static case, load is applied by tip mass slid over the tip of the beam as shown in Figure-

4.2. The two modes of vibration are shown in Figure-4.3for various root setting angles.

Figure 4.1: Schematic of the experimental setup used for frequency estimation inPrinceton Beam experiment showing tip mass.

The properties of the beam are given in Table-4.3. The properties of tip masses are

provided in Table-4.4. The finite dimension of these masses and their positioning with

each half on the either side of the beam tip has to be taken into account for modeling.

The beam (including tip mass) is divided into 16 finite elements for accurate analysis.

Weight of the beam as well as the tip mass has to be taken into account.

The beauty of GEBT lies in the fact that no separate treatment of the tip mass is

required. It can be considered as a finite element with different area and moment of


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Figure 4.2: Princeton beam showing location of cylindrical tip mass[23].

Figure 4.3: Fundamental modes of vibration of Princeton beam at different rootsetting angles[23].

Dimension Material

Properties Accuracy

BoundaryConditions

L 0.508 m E 71.6 GPa N 8 F3 13.345 N

w 12.7 cm G 26.9 GPa 109 1 0to 90

t 3.2 mm 2800 kgm3 109

Table 4.3: Simulation Parameters for Frequency Estimation of Princeton Beam

inertias acted upon by the same gravitational acceleration as the rest of the beam. The

presence of finite rotations as intrinsic parameters facilitate the construction of trans-

formation matrices from deformed to undeformed frame and vice versa. This further

simplifies the treatment of dead forces like gravity and follower forces and moments like

aerodynamic lift, drag, pitching moment etc.

Figure-4.4shows the excellent correlation of the Princeton Beam experimental data with


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4.2.3 Maryland Beam Experiment

Maryland beam experiments were designed to understand the effect of sweep on the

natural frequencies of a rotating beam made of both aluminum and composites. These

experiments were carried out in a vacuum chamber, hence the aerodynamic forces are

negligible and gravity is the only external force causing out of plane deformations. These

experiments served as the basis to verify the various theoretical models made to pre-

dict these frequencies. Swept tips causes flap-torsion coupling in cantilevered beams.

Previous attempts to model these experimental findings have been made by Epps and

Chandra[25] using nonlinear methods, Hopkins and Ormiston[7]using multibody RCAS

code and Hodges et al. [19]using GEBT with varying degree of accuracy.

The present thesis uses GEBT for modeling the tip-swept beams whose geometry is

shown in Figure-4.5. The properties used in modeling are given in Table-4.5. A root

offset of 2.5 in. was modelled by assuming a rigid element of that size. Tip sweep

was modelled by introducing an equivalent rotation angle in the undeformed coordi-

nate frame and accounting for it in the Cab matrix. The angular velocity, of each

element considered for comparison with experimental data is 0, 500 or 750 RPM. The

corresponding velocity of each element is given by

v(i) =(i) r(i)

where r(i) is the radial location of the centre of element i in undeformed coordinate

system measured from the rotation axis. Figure-4.6provides the details of finite element

mesh used for the analysis and is inspired by Hodges et al. [ 19]. Refining meshes near

the junction takes into account the large changes in intrinsic variables between straight

and swept part.

Findings of the simulation along with the experimental results are reported in figures 4.7

to4.11. There is a good correlation between the experimental and theoretical results for

lower frequencies. At higher modes due to coupling between flap bending and torsion the

results show some deviation. Nevertheless, the trend is accurately followed. A further

analysis of eigenvectors is required to resolve the issue of coupling for higher frequencies.


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Figure 4.5: Schematic of Maryland beam showing tip sweep and root offset[19].

Figure 4.6: Finite element discretization of Maryland beam.

Dimension Material

Properties Accuracy

BoundaryConditions

L 40 in E 1.06107 psi N 16 Fixed-freew 1 in G 4106 psi 106 Lroot 2.5 int 0.0625 in 2.51104 lb.sec2/in4 104 sweep 0to 45Lswept 6 in t 10

3 RPM 0, 500, 750

Table 4.5: Maryland Beam Simulation Parameters


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Figure 4.9: Effect of tip sweep and RPM on 3rd flap-bending frequency of MarylandBeam

Figure 4.10: Effect of tip sweep and RPM on 4th flap-bending frequency of MarylandBeam


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Chapter 5

Conclusion and Recommendation

This thesis discusses the development of a Geometrically Exact Beam Formulation for

modeling large deformation in advanced geometry rotor blades. Combined with a sec-

tional analysis tool like VABS[31] a complete 3-D analysis of slender beam with even

complex cross sections can be performed. The codes developed under the present thesis

are able to perform high-fidelity analysis of both statics and dynamics. These codes can

be used in various engineering disciplines that require accurate tools to model highly

deformable beams.

5.1 Summary

First, the verification of the present analysis for the deflections of a cantilevered beams

loaded with tip moment and tip load was carried out. The predictions showed very good

correlation for deformation of elastica due to both tip moment and tip force. Next, the

present simulation was verified against the static deformations of the Princeton Beam

theory. Again, good correlation was obtained between the simulation and experimental

results.

The real application of GEBT lies in modeling dynamic response and natural frequencies

of beam structures. While the response part was left to be pursued in future, frequency

predictions were carried out for stationary and then rotating beams to validate the dy-

namics modelling capability. Flatwise and chordwise frequencies of a slender cantilevered

48


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Chapter 5. Conclusion and Recommendation 49

beam were obtained by GEBT and compared with analytical solution. The next step

was to calculate the flap and lag frequencies of a cantilevered beam with varying root

setting angle and a rigid mass slid at the tip of the beam. These results were compared

with the famous Princeton Beam Experiment with excellent agreement. Finally, the pre-

diction for natural frequencies were compared for rotating beams with root offset and

tip sweep with those of the Maryland Beam Experiment. For the first four fundamental

modes the frequencies matched well with the experimental data.

5.2 Scope and Recommendation

This thesis was taken up to develop codes for structural dynamic modeling part of a

bigger comprehensive rotor analysis system. A number of such software exists for com-

mercial, military and research applications. Johnson [32] has talked in detail about

the historical development and progress made in the field of rotorcraft comprehensive

analyses. He defines comprehensive analyses as the tools to predict the aeromechanical

characteristics of rotor and aircraft. These tools involve subsystems to model geome-

try, structure, aerodynamics and control inputs. The aim is to estimate critical traits

like trim, air loads, structural loads, blade response, vibration, noise, stability and per-

formance in an iterative fashion. Figure-5.1 from [32] shows the history of various

comprehensive analysis tools and their corresponding funding agencies.

Since Geometrically Exact Beam Theory or any such predecessor is not a part of normal

curriculum, a lot of time was spent in exploring the literature and understanding the

concept. Being just a couple of decades old (after Hodges [13]), not many books andpapers are available explaining the solution procedure and the details of modeling. This

thesis has attempted to make the job easier by explaining the same in detail for the

GEBT in general in Chapter-2and for the problems solved in particular in chapters3

and4.

At present four types of problems as mentioned in Chapter-2can be solved using GEBT.

The trickiest part of is calculation of dynamic response. Here, convergence is required

at each time step with some values (as initial values) to be carried forward to next


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Figure 5.1: A brief history of comprehensive analyses tools and their developingagencies[32].

steps. The time dependent variables are, by the design of this time-marching scheme,

not continuous between successive time steps. This blows up the time derivative terms

when they are approximated as

X=Xt+tXt

t

Only few time-marching schemes such as backward differencing converge for this case.

However, the response is somewhat dependent on the type of time marching scheme, and

the result from one type may not be as accurate as the other one. To resolve this issue,

a more in-depth insight is required in FEM and time-marching procedures, with specific

reference to the current requirement. The effect of the length of time step also need to

be studies. One method is to employ a space-time kind of shape functions[14] which

performs iteration over the whole time domain in one go. For large beams undergoing


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large deformations, variable order FEM as explained in Patil and Hodges [22] can be

performed for more accuracy and efficiency.

A lot of research groups all over the academic world are using GEBT to solve many

challenging nonlinear problems. Khouli et al. [33] provides details of modeling for artic-

ulated rotor blades with root offset, structural damping and tip sweep. Lee et al. [34]

combines GEBT with aerodynamic model of CAMRAD to model inflow distribution for

helicopter in steady hover. Sotoudeh and Hodges [35] discusses various boundary con-

ditions that need to be used while analyzing structures using GEBT. While the effects

of dynamics and elasticity are taken into account in GEBT, unsteady aerodynamics

and aeroelasticity are to be integrated for complete analysis. Further, GEBT is also

developed for thin elastic plates[36] which could be used to model thin plates and shells

undergoing arbitrary deformations.


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Bibliography

[1] Dewey H Hodges and EH Dowell. Nonlinear equations of motion for the elastic

bending and torsion of twisted nonuniform rotor blades. National Aeronautics and

Space Administration, 1974.

[2] Dewey H Hodges, Robert A Ormiston, and David A Peters. On the nonlinear

deformation geometry of Euler-Bernoulli beams. National Aeronautics and Space

Administration, Scientific and Technical Information Office, 1980.

[3] Aviv Rosen and Peretz P Friedmann. Nonlinear equations of equilibrium for elastic

helicopter or wind turbine blades undergoing moderate deformation. Technical

report, California Univ., Los Angeles (USA), 1978.

[4] Wayne Johnson. Aeroelastic analysis for rotorcraft in flight or in a wind tunnel.

1994.

[5] Olivier A Bauchau and NK Kang. A multibody formulation for helicopter structural

dynamic analysis. Journal of the American Helicopter Society, 38(2):314, 1993.

[6] Hossein Saberi, Maryam Khoshlahjeh, Robert A Ormiston, and Michael J

Rutkowski. Overview of rcas and application to advanced rotorcraft problems.

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[7] A Stewart Hopkins and Robert A Ormiston. An examination of selected problems in

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[8] AEH Love. The mathematical theory of elasticity. 1927.

[9] Eric Reissner. On one-dimensional finite-strain beam theory: the plane problem.

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[10] E Reissner. On one-dimensional large-displacement finite-strain beam theory. Stud-

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[11] JC Simo. A finite strain beam formulation. the three-dimensional dynamic problem.

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[12] Juan Carlos Simo and Loc Vu-Quoc. On the dynamics in space of rods undergo-

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[13] Dewey H Hodges. A mixed variational formulation based on exact intrinsic equa-

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[14] Dewey H Hodges. Geometrically exact, intrinsic theory for dynamics of curved and

twisted anisotropic beams. AIAA journal, 41(6):11311137, 2003.

[15] Gordan Jelenic and Miran Saje. A kinematically exact space finite strain beam

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