agusta using finite element multibody tilt-rotor … · agusta tilt-rotor analysis and design using...

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a TiltTilt--Rotor Analysis and Design Rotor Analysis and Design Using Finite Element Using Finite Element MultibodyMultibody

ProceduresProceduresCarlo L. BottassoCarlo L. Bottasso, LorenzoLorenzo TrainelliTrainelli

Politecnico di Milano, ItalyPierre Pierre AbdelAbdel--NourNour, GianlucaGianluca LabòLabò

Agusta S.p.A., Italy

28th European Rotorcraft ForumBristol, September 17–20, 2002

Politecnico di Milano

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g Presentation OutlinePresentation Outline

• Finite element multibody procedures for rotorcraft modeling;

• Tilt-rotor virtual model;

• Three different constant speed joint realizations: universal joint, Agusta design, ideal joint.

• Structural and aerodynamic model validation;

• Whirl-flutter analysis;

• Results;

• Conclusions and future work.


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g Structural Dynamics ModelingStructural Dynamics Modeling

A modern approachmodern approach to rotorcraft modeling:

• Vehicle is viewed as a complex flexible mechanismcomplex flexible mechanism.

• Model novel configurations of arbitrary topology by assembling assembling basic componentsbasic components chosen from an extensive library of elementslibrary of elements.

This approach is that of the finite element methodfinite element method which has enjoyed, for this very reason, an explosive growth.

This analysis concept leads to simulation software tools that are modularmodular and expandableexpandable.

Simulation tools are applicable to configurations with arbitrary arbitrary topologiestopologies, including those not yet foreseen.


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gStructural Dynamics ModelingStructural Dynamics Modeling

Definition of multibodymultibody: a finite element modelfinite element model, where the elements idealize rigid and deformable bodies (beams, shells, etc.) and mechanical constraints.

Systems with complexcomplex topologies, where each body undergoes large large displacements and finite rotationsdisplacements and finite rotations (but only small strains).

Idealization process:

Virtual prototype.


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g Structural Dynamics ModelingStructural Dynamics Modeling

Lower pairsLower pairs:

• Sensors;

• Actuators, controls.

Other modelsOther models:

• Flexible joints;

• Unilateral contacts;

Body modelsBody models: geometrically exact, composite ready beams and shells; rigid bodies.


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g Numerical ChallengesNumerical Challenges

• Highly nonnon--linearlinear equations.

• Differential-algebraicalgebraic nature (Lagrangemultipliers), highly stiffstiff.

• Dynamic invariantsinvariants (energy, momenta).

Solutions that satisfy the constraints

Solutions that satisfy the invariants Constraint

manifold

Manifold of the invariants

System manifold

Drifting solution

Classical approachClassical approach: derive the equations and apply an off-the-shelf general-purpose DAE integrator.

Pros: easy.

Cons: the integrator knows nothing about the problemknows nothing about the problem being solved.

èè Invariants are not preserved, only linear notions of stability.

èè Lack of robustness, failure for particularly difficult problems.


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g Geometric IntegrationGeometric IntegrationDesign (backwardDesign (backward--engineer)engineer) integrators that incorporate specific knowledge of the equations being solved:

• Exact treatment of geometric nongeometric non--linearitieslinearities.

• Exact satisfaction of the constraintsconstraints (no drift).

• NonNon--linear unconditional stabilitylinear unconditional stability and preservation of invariants: bound on the total energy of deformable bodies + vanishing of the work of the forces of constraint + conservation of momenta.

The numerical procedure inherits qualitative featuresqualitative features of the true solution èè greatly improved robustness, reliability.

Energy methodsEnergy methods: discrete energy conservation imply unconditional stability in the non-linear regime (Simo & Wong 1991: rigid bodies; Simo & Tarnow 1994: shells; Simo et al. 1995: beams).

Tf=Ti (T=K+V)


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g Beyond Geometric IntegrationBeyond Geometric IntegrationEnergy decaying methods Energy decaying methods (Bottasso & Borri 1997: beams; Bottasso & Borri 1998: multibodies; Bottasso & Bauchau 1999: multibodies and shells):

Tf=Ti-TD TD≥≥0 dissipated total energy.

• Unconditional stabilityUnconditional stability in the non-linear regime from the bound on the total energy.

• Mechanism for controlling the unresolved unresolved frequencies.

Energy preserving solution

Energy manifold

Energy decaying solution

TD


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gReferences on Structural References on Structural

Dynamics ModelingDynamics Modeling

• C.L. Bottasso, O.A. Bauchau, `Multibody Modeling of Engage and Disengage Operations of Helicopter Rotors', J. Amer. Helic. Soc., 46:290-300, 2001.

• O.A. Bauchau, C.L. Bottasso, Y.G. Nikishkov, `Modeling RotorcraftDynamics with Finite Element Multibody Procedures', Math. Comput. Modeling, 33:1113-1137, 2001.

• O.A. Bauchau, J. Rodriguez, C.L. Bottasso, `Modeling of Unilateral Contact Conditions with Application to Aerospace Systems Involving Backlash, Freeplay and Friction', Mech. Res. Comm., 28:571-599, 2001.

• C.L. Bottasso, M. Borri, L. Trainelli, `Integration of Elastic Multibody Systems by Invariant Conserving/Dissipating Schemes. Part I Formulation', Comp. Methods Appl. Mech. Engrg., 190:3669-3699, 2001. Part II Numerical Schemes and Applications', Comp. Methods Appl. Mech. Engrg., 190:3701-3733, 2001.


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gTiltTilt--Rotor WhirlRotor Whirl--FlutterFlutter

GoalGoal: investigate the effects of the design of the constantconstant--speed speed jointjoint on the flutter speed.

Three possible solutions:

1) Universal joint. 2) The Agusta “artichoke”. 3) Ideal constant-speed joint.

ΩΩ3,hub= ΩΩ3,shaft


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g Virtual TiltVirtual Tilt--Rotor ModelRotor Model

• Detailed description of the control linkagescontrol linkages;

• Equivalent mechanism models flexibility of controlsflexibility of controls;

• Aerodynamics based on 2D theory2D theory with unsteady correction and Peter’s inflowinflow model.


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g ConstantConstant--Speed Joint ModelsSpeed Joint Models

ScissorsDriving shaftDriven hubAB

AgustaAgusta “artichoke” joint“artichoke” joint as implemented in the tilt-rotor model:


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AgustaAgusta “artichoke” joint“artichoke” joint as implemented for numerical characterization:ScissorsDriving shaftHubPrescribed rotations

at the universal jointACDEB


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g ConstantConstant--Speed Joint ModelsSpeed Joint ModelsOscillations in a planeOscillations in a plane:

ExactExact constant-speed transmission.

Oscillation amplitude: 20 deg.

Oscillation speed: 4 times driver angular speed.


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g ConstantConstant--Speed Joint ModelsSpeed Joint ModelsCone motionsCone motions: precession speed four times driver angular speed.

Semi-aperture 20 deg. Semi-aperture 5 deg.

Nearly exactNearly exact constant-speed transmission for smallsmall flapping motions.Hub frequency twice of precession frequency.


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Virtual model of the idealideal joint:


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g Model ValidationModel ValidationValidationValidation of:

• Structural model;

• Aerodynamic model.

Validation by comparisoncomparison with alternative solution procedures:

• CAMRAD/JA;

• NASTRAN.


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g

Structural validationStructural validation:

Model ValidationModel Validation

Wing eigenfreq. Rotor eigenfreq. (airplane mode)

Non-rotating. Rotating.


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g Model ValidationModel ValidationStructural validationStructural validation: comparison of tennis-racquet effects.

Flexible controls, stiff blade. Stiff controls, flexible blade.


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g Model ValidationModel ValidationAerodynamic validationAerodynamic validation: airplane mode, axial flow.

Thrust vs. power, CAMRAD/JA trim points.


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g Model ValidationModel ValidationAerodynamic validationAerodynamic validation: quasi-static conversion, off-axial flow.

H-force vs. tilt angle.


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g WhirlWhirl--Flutter AnalysisFlutter AnalysisTrim procedureTrim procedure:

1.1. StaticStatic solution identifies equilibrated condition in axial flow at constant speed rotation.

Loads: constant speed rotation, aerodynamic forces.

Wing tip is clampedclamped to prevent off-axial flow.

2.2. TransientTransient dynamic solution uses static solution as initial initial conditioncondition, and leads to periodic trim conditionperiodic trim condition.

Wing tip clamp is removed, flow is off-axial.

If unstable regime, structural dampingstructural damping added to the wingwing to artificially stabilize and reach periodic trim condition.


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g WhirlWhirl--Flutter AnalysisFlutter AnalysisWhirlWhirl--flutter solution proceduresflutter solution procedures:

1.1. Implicit Implicit FloquetFloquet analysisanalysis.

Evaluates dominant dominant eigenvalueseigenvalues using the Arnoldi algorithm without explicitly computing the transition matrix.

Ideal for systems denoted by many dofs.

2.2. Excitation Excitation followed by transient dynamic simulation. Time history shows stable or unstable behavior.

a. Excitation by removing the wing tip clamp after static solution (note: this way the system is not perturbed about the periodic trim condition);

b. Excitation by force (beam direction) triangular pulse at wing tip, starting from periodic trim.


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gWhirlWhirl--Flutter AnalysisFlutter Analysis

Implicit Implicit FloquetFloquet analysis resultsanalysis results (stable: spectral radius < 1; unstable: spectral radius > 1):

Spectral radius of transition matrix vs. flight speed.


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Animation of the unstable eigenvectorunstable eigenvector of the Floquet transition matrix.

Implicit Implicit FloquetFloquet analysis analysis resultsresults:


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Procedure 2.aProcedure 2.a

Time history of wing tip wing tip displacementsdisplacements (350 kts):

Universal joint.

“Artichoke” joint.

Ideal joint.


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Procedure 2.aProcedure 2.a

Time history of wing tip wing tip forcesforces (350 kts):

Universal joint.

“Artichoke” joint.

Ideal joint.

NoteNote: trends of Floquet solution are confirmed

(ideal > artichoke > universal).


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Procedure 2.bProcedure 2.b

Time history of wing tip beam forceswing tip beam forces (350 kts):

NoteNote: no apparent strong dependence on excitation level, trends similar to procedure 2.a.

All joints, max excitation 6000 N. Universal joint, excitations of 3000 N and 6000 N.


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Transient simulation in the unstable regime.


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gConclusionsConclusions

Tested three joint models:

1.1. UniversalUniversal, representative of the level of detail allowed by non-multibody procedures;

2.2. “Artichoke”“Artichoke”, representative of possible actual hardware implementations;

3.3. IdealIdeal, perfectly constant transmission but not realizable in practice.

Basic (preliminary) conclusion: more accuratemore accurate constant speed transmission of the angular velocity to the hub implies a progressive increaseprogressive increase in flutter speed.


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Tested two basic solution procedures:

1.1. Implicit Implicit FloquetFloquet analysisanalysis;

2.2. System excitationSystem excitation.

The basic conclusions found are similarbasic conclusions found are similar (ideal better than artichoke, better than universal), but different procedures yield somewhat somewhat different resultsdifferent results (e.g., damping levels).

Issues:

• Linearized (Floquet) vs. non-linear (excitation) approaches;

• Effects of reference (trimmed) condition.


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However, it is clear that :

1. Modeling assumptions and simplificationsassumptions and simplifications might severely undermineundermine the accuracy of the computed results;

2. The finite element multibody approach offers the potential for enhanced modelingenhanced modeling, through a direct numerical simulationdirect numerical simulation of the system components.

Plans for future workfuture work: assess impact of constant speed joint design on

a. Drive train loads and vibrations;

b. Nacelle loads and vibrations.

agusta using finite element multibody tilt-rotor … · agusta tilt-rotor analysis and design using...

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