An Introduction to Rotorcraft DynamicsOutline of the CourseIntroductionTheory of ResonanceFinite Element Based Formulation for Nonlinear Multibody SystemsRotor as a Nonlinear Multibody SystemTransmission as a Nonlinear Multibody SystemSimulation of Rotor on Ship BoardElement Library: Structural ElementsElement Library: Beam ElementsElement Library: Shell ElementsThe Six Lower PairsBlade DynamicsDYMORE Rotor ModelFan Plot of Frequencies for the RotorDynamic Responses - DisplacementsDynamic Responses - RotationsDynamic Responses - ForcesDynamic Responses - MomentsPitch-lag InstabilityPitch-flap InstabilityFlap-lag InstabilityConclusionsAn Introduction to Rotorcraft DynamicsDr. Wenbin YuSchool of Aerospace EngineeringGeorgia Institute of TechnologyEmail: [email protected]: www.ae.gatech.edu/~wyuOutline of the Course•Introductions•Theory of resonance•Introduction to DYMORE•Blade dynamics•The rotor as a filter, airframe dynamic response and coupled blade-fuselage response•Vibration control devices•Typical instabilities–Ground resonance–Pitch-lag instability–Pitch-flap instability–Flap-lag instabilityIntroduction•Rotorcraft are dynamic machinery. The dynamic problem are very important•Some dynamic problem are detrimental to the vehicle performance. If not dealt properly, they could cause catastrophic tragedies•Three categories of rotorcraft vibration–Vibrations due to rotor excitation. The frequencies are integral multiples of the rotor rotation speed–Vibrations due to random aerodynamic excitation. The frequencies are the natural frequencies of the structure–Self-excited vibrations, such as flutter and ground resonances. Negative damping could cause divergent oscillationsTheory of Resonance•A single DOF dynamic system•Natural frequency •Forced vibration of the system without damping•The importance of natural frequency for design•Vibration with damping •Mathematica example•Flapping blade•Lagging blade0.5 1 1.5 2246810Finite Element Based Formulation for Nonlinear Multibody Systems •Model configurations of arbitrary topology:–Assemble basic components chosen from an extensive library of structural and constraint elements•Avoids modal expansion•This approach is that of the finite element method which has enjoyed, for this very reason, an explosive growth•This analysis concept leads to simulation software tools that are modular and expandable•Elements of the library can be validated independentlyRotor as a Nonlinear Multibody SystemTransmission as a Nonlinear Multibody SystemSimulation of Rotor on Ship Board•The complete model involves:–17 beam elements,–5 prescribed displacements,–1 prismatic joint,–1 relative displacement,–21 rigid bodies,–12 revolute joints,–12 relative rotation,–3 spherical joints,–1 universal joints,•For a total of 950 degrees of freedom.Element Library: Structural Elements•Rigid bodies•Flexible joints: linear and torsional springs and damper•Cable element•Beam elements: geometrically exact, shear deformable. Capable of modeling all the elastic coupling effects arising from the use of advanced laminated composite materials•Shell elements: geometrically exact, shear deformable, modeling of composite material effectsThe finite element formulation is used for all elements, no modal reduction is performedElement Library: Beam Elements•Geometrically exact beam elements. Six degrees of freedom (three displacements, three rotations) per node•Accounts for –Shearing deformation effects–Offsets of the center of mass, shear center, and centroid–All elastic couplings that can arise from the use of laminated composite materials (Fully coupled 6x6 stiffness matrix)–Material viscous dissipationElement Library: Shell Elements•Geometrically exact shell elements. Five degrees of freedom (three displacements, two rotations) per node. Locking free element is achieved using the mixed interpolations of strains tensorial components•Accounts for –Shearing deformation effects–Offsets of the center of mass–All elastic couplings that can arise from the use of laminated composite materials (Fully coupled 8x8 stiffness matrix)–Material viscous dissipationThe Six Lower PairsBlade Dynamics•Blade dynamics is important because–High blade vibratory response results in high stresses–High blade vibratory response leads to high fuselage vibration levels–Blade resonances and mode shapes are important in stability analysis of rotor systems•DYMORE example for a single blade•DYMORE example for a complete rotor (ITU LCH)•FAN plot for ITU LCH•Changing frequencies by playing with weightDYMORE Rotor ModelThe DYMORE model for the ITU LCH RotorFan Plot of Frequencies for the RotorFan plot in Vacuum for the ITU LCH Rotor (Verifying the Auto-Trim concept)Dynamic Responses - Displacements0 0.5 1 1.5 2 2.5 3TimeSec-0.8-0.6-0.4-0.200.20.4edalB pit tnemecalpsidteeFTime history of blade tip displacementred: axial displacement; green: in-plane displacement; blue: out-of-plane displacementDynamic Responses - RotationsTime history of blade tip rotationsred: pitching; green: flapwise direction; blue: chordwise direction0 0.5 1 1.5 2 2.5 3Time Sec -0.0200.020.040.060.08edalB pit noitator  snaidaR Dynamic Responses - ForcesTime history of forces at different locationsred: at flex root; green: at flex tip0 0.5 1 1.5 2 2.5 3TimeSec05001000150020002500ecroF ni gnippalf noitcerid bL Dynamic Responses - MomentsTime history of moments at different locationsred: at flex root; green: at flex tip0 0.5 1 1.5 2 2.5 3Time Sec -50005001000gnippalF tnemomPitch-lag InstabilityPitch-flap InstabilityFlap-lag Instability–Ground resonanceConclusions•Dynamic problem are very important for rotorcraft. A good design must come from a good understanding to dynamic behavior of the vehicle•Locating the natural frequencies of the system is the key to avoid resonance•DYMORE is a handy tool to deal with rotorcraft dynamics•Either passive (damping) or active devices (vibration absorbers) can be used to reduce the resonance or shift the natural frequencies•Dynamic instabilities should and can be avoided by design