Integrating 3D Stress Evaluation With Rotorcraft Comprehensive Analysis

Integrating 3D Stress Evaluation With Rotorcraft Comprehensive Analysis Olivier A. Bauchau Department of Aerospace Engineering University of Maryland National Institute of Aerospace November 19th, 2015 Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 1 / 60

Outline of the presentation 1 Introduction 2 Multibody Dynamics Formulations 3 Finite Element Based Formulation and Element Library 4 Coupling Structural and Fluid Dynamics 5 Examples of Application 6 Evaluation of 3D Stresses Weakest Link Principle The computational challenge Effect of distributed loading 7 Conclusions and future work Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 2 / 60

Rotorcraft Simulation Challenges Transonic, 3D,viscous flows Aero-servo-elasticity Flap Nonlinear multibody dynamics Complex topologies HHC Ground resonance Unsteady aerodynamics A.V.S. Composite materials Coupled rotor/fuselage dynamics-aerodynamics Dynamic stall Blade-vortex interaction Complex vortex wake Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 4 / 60

Types of Multibody Systems 1 Rigid multibody systems: an assemblage of rigid bodies in arbitrary motion with respect to each other. Many efficient formulations are available. Commercial packages available. 2 Linearly elastic multibody systems: elastic displacements and rotations remain very small at all times. Elasticity is treated by modal expansion. Commercial packages available (ADAMS, DADS). 3 Nonlinear multibody systems: displacements and rotations due to deformation are inherently nonlinear. Modal expansion is questionable. Finite element formulation seems appropriate. Commercial packages available (MECANO). Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 6 / 60

Classical Approach to Modeling 1 Derive the equations of motion of the system. Take advantage of topological features: equations are derived in a rotating system; Recursive formulations for tree topologies; Simplify the equations: ordering schemes are used to decrease the number of nonlinear terms. Obtain minimum set of equations (avoid constraints). 2 Perform modal reduction for the elastic components. 3 Solve the nonlinear modal equations Static or dynamic response. Stability and trim analysis. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 7 / 60

Shortcomings of the Approach 1 Topology of the system is hard wired into the solution process. 2 If specific design details are modified, the equations of motion must be re-derived. 3 Different codes are derived for different systems: Flexible Robots, Road vehicles, Rotorcraft, Wind-turbines, Space systems. 4 Modal reduction might not be accurate and reliable. 5 Software validation is a difficult issue. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 8 / 60

Shortcomings of the Approach The present approaches put severe limitations on the generality and flexibility of the resulting codes. Designers are ahead of the analysts. A more general and flexible paradigm for modeling of nonlinear multibody systems is needed. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 9 / 60

Finite Element Based Formulation 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 independently. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 11 / 60

Rotorcraft System Blade Bearingless rotor { { Flexbeam Snubber Torsion cuff Pitch-link ~ Swash plate: rotating non-rotating Hub Rigid body Beam Revolute joint Sliding joint Articulated rotor Flap, Lag, and pitch hinges Shaft Scissors Blade Pitch-horn Pitch-link Actuators Flexible joint Spherical joint Universal joint Ground clamp The various elements of the model are selected from a library of structural elements. Various topologies are readily modeled. kinematic constraints are also formulated finite elements. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 12 / 60

Tail Rotor Drive System i 2 S R Gear box 1 Shaft 1 Fuselage Shaft 2 G Tail M i 1 Rotor Q T Gear box 2 Rigid body Beam Flexible joint Revolute joint Ground clamp Q T Side view of tail rotor H Plane of rotor Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 13 / 60

Wind Turbine System S T R Bed plate rvj yaw Tower rvj tilt S Nacelle U rvj shaft Shaft Revolute joints Beam elements Blade 2 H Blade 1 ~ Hub ~ E rvj teeter E Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 14 / 60

Wing with a Trailing Edge Flap Connecting brackets Reference line Spherical joint Prismatic joint Universal joint Revolute joint (prescribing flap rotation) Revolute joint (spring and damper) Structural model of the flap. Brackets: beam elements. Flap: Beam elements. The various joints are required to decouple the flap deformations from those of the blade. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 15 / 60

Wing with a Trailing Edge Flap Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 16 / 60

Element Library: Structural Elements 1 Rigid bodies: concentrated mass properties. 2 Flexible joints: rectilinear and torsional springs and damper. F = Dd + Dḋ. 3 Cable element. 4 Beam elements: geometrically exact, shear deformable. Capable of modeling all the elastic coupling effects arising from the use of advanced laminated composite materials. 5 Shell elements: geometrically exact, shear deformable, modeling of composite material effects. 6 Modal elements: import elastic structure from NASTRAN, based on Herting s component mode synthesis formulation. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 17 / 60

Element Library: Joint Elements 1 The six lower pairs: the revolute, prismatic, screw, cylindrical, planar and spherical joints. 2 Sliding joints and sliding screw joints. 3 Backlash behavior in all joints. 4 Unilateral contact elements between rigid bodies of the model including local deformation, friction and rolling. 5 Wheel elements with tire flexibility and shimmy model. 6 Clearance joints with lubrication model. 7 Contact elements with contact and friction force models. Holonomic and non-holonomic constraints are enforced via the Lagrange multiplier technique. A joint is a finite element generating constraint forces. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 18 / 60

Element Library: the Six Lower Pairs Cylindrical Prismatic Screw Revolute Spherical Planar Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 19 / 60

CFD/CSD Coupling: Overall Model Rotor Lifting lines Airstations Airstation Lifting line Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 21 / 60

CFD/CSD Coupling Aerodynamic forces and moments Structural dynamics Displacements and velocities Airloads model Circulations Inflow model Inflows The structural model predicts the dynamic response of the system, given the externally applied loads. The aerodynamic model predicts the aerodynamic forces and moments given the structural displacement, velocities, and inflow velocities. The inflow model predicts the inflow at a point in the flow given the circulation distribution. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 22 / 60

CFD/CSD Coupling CSD Wake models: - Dynamic inflow model - prescribed wake, - free wake 2-D unsteady aerodynamic models: - Peters model, - Leishman-Beddoes model. Airstation Lifting line Forces and moments data flow Displacements and velocities data flow Inflow data flow When using the internal, simplified aerodynamic models, 2D unsteady aerodynamic model combine an analytical model with empirical data, tables of lift, moment, and drag coefficients. 3D inflow models: dynamic inflow model, prescribed or free wake. Semi empirical dynamic stall models: ONERA model. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 23 / 60

CFD/CSD Coupling Structural model Displacements and velocities Aerodynamic model Aerodynamic forces and moments The structural model predicts the dynamic response of the system, displacements and velocities, given the externally applied loads. The aerodynamic model predicts the aerodynamic forces and moments given the structural displacement, velocities. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 24 / 60

CFD/CSD Coupling CSD Loads interface Kinematics interface Wake capturing CFD model Airstation Lifting line Forces and moments data flow Displacements and velocities data flow When using an external, Computational Fluid Dynamics code, Could use full potential, Euler, or Navier-Stokes codes. If the entire flow around the rotor is captured by the CFD grid, there is no need for an additional wake model. Could use loose or tight coupling strategy. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 25 / 60

The Kinematics Interface Airstations CFD grid point Displacement and velocity of grid point Relative position of grid point Displacement and velocity of airstation For an external CFD code, An entire slice of the CFD near-body grid is moved to reflect the motion of the airfoil. The near-body grid undergoes a rigid body motion with the airfoil. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 26 / 60

The Loads Interface Airstations Aerodynamic forces FEM node point FEM nodal forces beam element Nodal forces computed from work principles using FEM shape functions The airloads computed by the aerodynamics code are applied to the structural model, Grids used by CSD and CFD models must remain completely independent. Energy considerations must be used to ensure consistent application of the loads. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 27 / 60

Examples of Application Mode shapes of the UH-60 rotor Comprehensive model of the Wrats Co-axial rotor Wing with a Trailing Edge Flap Solar panel deployment Wind turbine Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 29 / 60

UH60 Flight Test Database 1 The UH60 flight test database High-quality, comprehensive flight test database Wealth of data for correlation and validation Aerodynamic and structural data are both available 2 Important observations If aerodynamic loading is used to excite the rotor, good correlation with strain gage data is recovered CFD analysis with rigid blades cannot predict measured airloads 3 Fluid-structure interaction algorithms were developed Coupling of legacy CFD and CSD tools Dramatic improvement in accuracy of predictions resulted 4 Much work remains to be done Still very far from being a design tool, but All helicopter companies are assimilating the technology Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 31 / 60

Weakest Link Principle Assume that a comprehensive analysis tool comprises the following disciplines: dynamics, aerodynamics, structures,.... Assume that errors are incurred by each discipline E dyn = α d0 +α d1 O(ε)+α d2 O(ε 2 )+h.o.t. E aero = α a0 +α a1 O(ε)+α a2 O(ε 2 )+h.o.t. E elas = α e0 +α e1 O(ε)+α e2 O(ε 2 )+h.o.t., The error incurred by the comprehensive analysis is then E comp = (Σα 0s)+(Σα 1s)O(ε)+(Σα 2s)O(ε 2 )+h.o.t. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 33 / 60

Weakest Link Principle Consider the following hypothetical case E dyn = α d1 O(ε)+α d2 O(ε 2 )+h.o.t. E aero = E elas = α a2 O(ε 2 )+h.o.t. α e2 O(ε 2 )+h.o.t. The following error results for the comprehensive analysis Weakest Link Principle E comp = α d1 O(ε)+(Σα 2 s)o(ε2 )+h.o.t. The error incurred by a comprehensive analysis equals that incurred by the weakest link Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 34 / 60

How to Evaluate 3D stresses? Local Model σ 1 (x 1, x 2, x 3 ) = Eǫ(x 1, x 2, x 3 ) σ 1 (x 1, x 2, x 3 ) = x 3 M 2 /H 22 Global Model M 2 (x 1 ) = H 22 κ 2 (x 1 ) Bending stiffness H 22 = A Ex2 3 da Euler-Bernoulli governing equations H 22 ū 2 = p 2 (x 1 ) M 2 (x 1 ) = H 22 ū 2 (x 1) Euler-Bernoulli beam theory is based on kinematic assumptions: plane sections remain plane and normal to the deformed axis of the beam Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 35 / 60

How to Evaluate 3D stresses? 1 Perform a comprehensive analysis of the rotor 2 Identify critical areas: Maximum bending moment, shear force Trunnion, pin attachment area, etc. 3 Apply the load in a quasi-static manner on a 3D FE model (Ansys, Nastran) 4 Obtain the 3D, ply-by-ply, stress field in all components Apply failure criterion Perform fatigue analysis Study crack propagation, etc. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 36 / 60

Saint-Venant s principle Extremity solutions Central solutions The presence of central solutions and extremity solutions is a direct consequence of Saint-Venant s principle. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 37 / 60

Saint-Venant s principle = F_ c M_ c Propagating polynomial central solution + Exponentially decaying extremity solution Arbitrary stress distribution Sectional force & moment resultants Self-equilibrating stress distribution An arbitrary stress distribution applied over the cross-section can be decomposed unequivocally into 1 Six stress resultants excite the central solutions, polynomial solutions propagating over the beam s entire span. 2 Self-equilibrating stress distribution excite the extremity solutions, which are exponentially decaying solutions. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 38 / 60

Shortcomings of the approach 1 How do we apply the loads to the FE model? Apply stress distribution is equipollent to the stress resultants computed by the comprehensive analysis Stress diffusion process? 2 Loads are applied in a quasi-static manner: local inertial force are neglected 3 Local aerodynamic forces are neglected 4 Although based on a rigorous 3D FEM, this approach is very primitive Neglects available data Raises many more issues than it solves Gives a false impression of accuracy Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 39 / 60

Shortcomings of the approach 1 A well designed blade should act as its own vibration absorber ( Conformable rotor ) 2 D Alembert s Principle: f inertial + f aero + f elastic = 0 3 Equilibrium requires all the loads to be applied on the model 4 Why neglect the local aerodynamic loads? They have been computed Their distribution is not uniform at all 5 Why neglect local inertial forces? They are easy to evaluate Their distribution is not uniform at all Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 40 / 60

Helicopter rotor blade cross-section Typical helicopter rotor blade cross-section. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 42 / 60

Wind turbine blade cross-section Typical wind turbine rotor blade cross-section. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 43 / 60

The computational challenge 1 Consider the UH-60 (8 ton helicopter) all composite rotor blade. Length, 8.6 m, chord, 0.72 m. Main D-spar: 60 plies of Graphite/Epoxy material, t p = 125 µm 2 Mesh the blade with 20 node brick elements. Assume: one element per ply, 1/10 aspect ratio. Number of elements: 6,880 (length), 1,152 (cross-section), 60 (through the thickness), for a total of 475 million brick elements. Number of nodes: 12,760 (length), 2,304 (cross-section), 120 (through the thickness), for a total of 3.8 billion nodes. Number of dofs: 3.8 billion 3 = 11.5 billion dofs. 3 The UH-60 helicopter was designed using modal methods, i.e., about 20 modes per blade. 4 Computational saving due the dimensional reduction: 11.5 10 9 / 20 = 575 10 6. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 44 / 60

Why Do We Neglect Local Loads? Cantilevered beam under uniform transverse pressure, σ a Mid-span maximum axial stress: σ 11max = 3/4σ a (L/h) 2 Mid-span maximum shear stress: τ 12max = 3/4σ a (L/h) Effect of local applied stress: σ 22max = σ a (σ 22max σ 11max ) Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 46 / 60

These Problems Are Not New 1 Beam loaded at its ends only: Saint-Venant s Problem He gave exact solution for torsion (1855) and bending (1856) Exact solutions of 3D elasticity 2 For distributed loading: Almansi-Michell s Problem They gave exact solutions for simple problems (1901) Exact solutions of 3D elasticity 3 Exact solutions of 3D elasticity are now available Blade of arbitrary cross-sectional shape made of heterogeneous, anisotropic materials 4 Based on 2D cross-sectional discretization Three to four order of magnitude faster than 3D FEM Remains exact for large motion, small strain problems Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 47 / 60

Planar Rectangular Beam _ b 2 _ O b 1 p t2 p b2 L p t1 p b1 2h A planar rectangular beam under distributed load Planar rectangular beam subjected to lateral surface tractions Simply supported beam of length L and height 2h = 0.2L is made of homogenous, isotropic material case (a) p b1 = q(x 1 /L) 2 /2, p t1 = p t2 = p b2 = 0, case (b) p t2 = q sin(x 1 /L), p t1 = p b1 = p b2 = 0. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 48 / 60

Planar Rectangular Beam 4.0 0.03 3.0 0 0 /q σ 11 2.0 1.0 /q 0.01 0.0 0.00-1.0-1.0-0.5 0.0 0.5 1.0 α 2 /h -0.01-1.0-0.5 0.0 0.5 1.0 α /h Distribution of stress component σ 11 /q, case (a). Distribution of stress component σ 22 /q, case (a). Solid blue: exact solution. Red circle: proposed approach. Dashed: ignore local applied loads Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 49 / 60

Planar Rectangular Beam 1 0.0-0.1 8.0 /q /q 11 0.0 - -8.0-0.3-1.0-0.5 0.0 0.5 1.0 α /h -1.0-0.5 0.0 0.5 1.0 α /h Distribution of stress component σ 12 /q, case (a). Distribution of stress component σ 11 /q, case (b). Solid blue: exact solution. Red circle: proposed approach. Dashed: ignore local applied loads Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 50 / 60

Planar Rectangular Beam 0.2 8.0 σ 22 0.0-0.2-0.4 /(q α 1 ) σ 12 6 4.0 2.0 0.0-1.0-1.0-0.5 0.0 0.5 1.0 α 2 /h -2.0-1.0-0.5 0.0 0.5 1.0 α 2 h Distribution of stress component σ 22 /q, case (b). Distribution of stress component gradient σ 12 /(q α 1 ), case (b). Solid blue: exact solution. Red circle: proposed approach. Dashed: ignore local applied loads Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 51 / 60

Composite Box Beam b _p t p P Q 3 3 b b 3 = i 3 t p b 2 = i 2 Configuration of the composite box cross-section Cantilevered beam with hollow rectangular section. Dimensions: L = 2 m, b = 0.1 m Properties: E L = 181, E T = 10.3, G LT = 7.17 GPa,ν LT = 0.28 andν TN = 0.33; t p = 0.01 m. The beam is subject to surface tractions over the top face, p = (1 α 1 /L) 2 sin(3πα 1 /L) cos(πα 2 /b)ī 3. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 52 / 60

Composite Box Beam (Stressσ 11 ) 0.05 2 0.05 2 0.03 1 0.03 1 0.01 0 0.01 0 α 3 α 3 0.01 1 0.01 1 0.03 2 0.03 2 0.05 0.05 0.03 0.01 0.01 0.03 0.05 3 0.05 0.05 0.03 0.01 0.01 0.03 0.05 3 α 2 α 2 Distribution of the stress componentσ 11. Left: ABAQUS. Right: proposed approach. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 53 / 60

Composite Box Beam (Stressσ 11 ) 0.05 2 0.05 2 0.03 1 0.03 1 0.01 0 0.01 0 α 3 α 3 0.01 1 0.01 1 0.03 2 0.03 2 0.05 0.05 0.03 0.01 0.01 0.03 0.05 3 0.05 0.05 0.03 0.01 0.01 0.03 0.05 3 α 2 α 2 Distribution of the stress componentσ 11. Left: ABAQUS. Right: ignoring local applied loads Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 54 / 60

Composite Box Beam (Stressσ 12 ) 0.05 1.5 0.05 1.5 0.03 1 0.03 1 0.01 0.5 0.01 0.5 α 3 0.01 0 0.5 α 3 0.01 0 0.5 0.03 1 0.03 1 0.05 0.05 0.03 0.01 0.01 0.03 0.05 α 2 0.05 0.05 0.03 0.01 0.01 0.03 0.05 α 2 Distribution of the stress componentσ 12. Left: ABAQUS. Right: proposed approach. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 55 / 60

Composite Box Beam (Stressσ 12 ) 0.05 1.5 0.05 1.5 0.03 1 0.03 1 0.01 0.5 0.01 0.5 α 3 0.01 0 0.5 α 3 0.01 0 0.5 0.03 1 0.03 1 0.05 0.05 0.03 0.01 0.01 0.03 0.05 α 2 0.05 0.05 0.03 0.01 0.01 0.03 0.05 α 2 Distribution of the stress componentσ 12. Left: ABAQUS. Right: ignoring local applied loads Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 56 / 60

( Composite Box Beam!"# 2.0 $"# P 0.0 %"# P 11 #"# 12-2.0 %"# $"# Q -4.0 Q!"# 0 L'% L/2 3L'% L & 1-6.0 0 L/4 L/2 3L/4 L ) 1 Stress distribution along the span. Left: σ 11. Right: σ 12 Solid blue: exact solution. Red cross: proposed approach. Dotted: ignore local applied loads Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 57 / 60

Conclusions 1 A finite element based procedure was proposed for the dynamic analysis of nonlinear multibody systems. 2 It allows the modeling of complex configurations of arbitrary topology, including those not yet foreseen. 3 Proceeds through the assembly of basic elements chosen from an extensive library that includes rigid and deformable bodies as well as joint elements. Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 59 / 60

Conclusions 1 Presently, the evaluation of the three-dimensional stress field in rotor blades is based on a rather primite approach 2 It neglects distributed aerodynamic and inertial loading 3 Tools for high-accuracy, low cost solutions are available 4 Much higher level of integration of the tools is required Graphical definition of sectional shape and material layout Two-dimensional meshing and FE analysis of cross-section Detailed aerodynamic pressure distribution over the blade Rotor comprehensive analysis Olivier Bauchau (UMD) Rotorcraft Comprehensive Analysis National Institute of Aerospace 60 / 60