Aeronautical engineering - structural dynamics and aeroelasticity

Completed notes of the course

Complete course

POLITECNICO DI MILANO Filippo D’AmbrosioSTRUCTURAL DYNAMICS AND AEROELASTICITY Anno accademico 2024/2025 Docente: Giuseppe Quaranta 1 Preface These handouts are based on my personal notes taken in the academic year 2024/2025, on the slides of Professor G. Quaranta and on the handouts made by Jacopo Alberti, who wrote another version of it the year before (credits to you J). I don’t take any responsibility about its accuracy and correctness. It is just a tool to help you guys pass the exam, i hope you’ll find it useful! Last but not least, it has not been written to be sold, students MUST be able to study from it charge-free. For any possible mistake on it or any suggestion to make it better, write an email to [email protected] Contents 1 Introduction8 1.1 Divergence and Flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91.1.1 Aeroelasticity through time . . . . . . . . . . . . . . . . . . . . . . . . . . .9 2 Static Aeroelasticity 112.1 Static aeroelasticity of the typical section . . . . . . . . . . . . . . . . . . . . . . .112.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 2.1.2 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 2.2 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112.2.1 Comparison between rigid and elastic lift . . . . . . . . . . . . . . . . . . .13 2.2.2 Torsional divergence as a stability problem . . . . . . . . . . . . . . . . . .13 2.2.3 Torsional divergence as an eigenvalue problem . . . . . . . . . . . . . . . . .14 2.2.4 Effects of Mach number on torsional divergence . . . . . . . . . . . . . . . .14 2.3 Control surface effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .152.3.1 Plot of control elastic efficiency . . . . . . . . . . . . . . . . . . . . . . . . .16 2.3.2 Matrix form of control reversal problem . . . . . . . . . . . . . . . . . . . .17 2.4 Addition of control surface stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . .18 2.5 Rolling of a straight wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 2.5.1 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 3 Multi Degrees of Freedom (MDoF) model for torsional divergence 243.1 Virtual work principle (VWP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 3.2 MDOF model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 3.3 Power method (for eigenvalues) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 3.3.1 Application to the MDOF case . . . . . . . . . . . . . . . . . . . . . . . . .26 3.4 Equivalent stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26 3.5 Integral formulation - Flexibility influence functions . . . . . . . . . . . . . . . . .27 3.6 Slender straight wing: analytical solution . . . . . . . . . . . . . . . . . . . . . . .293.6.1 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 3.6.2 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . .30 3.6.3 Solution of the homogeneous problem . . . . . . . . . . . . . . . . . . . . .31 3.6.4 Solution of the forced problem . . . . . . . . . . . . . . . . . . . . . . . . .32 3.7 Approximate approach: Ritz method . . . . . . . . . . . . . . . . . . . . . . . . . .343.7.1 Collocation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35 3.7.2 Weighted integrals - Ritz-Galerkin approximation . . . . . . . . . . . . . . .35 3.7.3 Requirements for Ritz approximation in the weak formulation . . . . . . . .36 4 Swept wings37 4.1 Characterization of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .384.1.1 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 4.1.2 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 4.1.3 Total change of angle of attack . . . . . . . . . . . . . . . . . . . . . . . . .39 4.1.4 Forces and moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 4.2 Divergence - typical section model . . . . . . . . . . . . . . . . . . . . . . . . . . .404.2.1 Lift effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42 4.3 Control reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 4.4 Aeroelastic tailoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44 5 Dynamic Aeroelasticity: introduction to Flutter 475.1 The homogeneous problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .485.1.1 Determination of the associated eigenvectors . . . . . . . . . . . . . . . . .49 5.2 Inclusion of steady aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . .50 5.2.1 Logarithmic decrement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51 5.2.2 Stability of a typical section with steady aerodynamics . . . . . . . . . . . .52 5.2.3 Effect of the phase shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53 5.3 Quasi-steady aerodynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . . .555.3.1 Quasi-steady aerodynamics + effect of pitch rate . . . . . . . . . . . . . . .55 5.4 Unsteady aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56 3 5.4.1 Wave length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56 6 Structural dynamics 576.0.1 Continuous deformable structure: Hypothesis . . . . . . . . . . . . . . . . .57 6.1 Elements of structural mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . .576.1.1 Constitutive law: linear, isotropic, elastic material . . . . . . . . . . . . . .58 6.1.2 Principle of virtual works . . . . . . . . . . . . . . . . . . . . . . . . . . . .58 6.2 Beams: introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .596.2.1 Stress resultant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59 6.3 Euler-Bernoulli’s beam model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .596.3.1 Strain in the beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60 6.3.2 Pure beam bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60 6.3.3 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61 6.3.4 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61 6.3.5 Summary of Euler-Bernoulli beam model . . . . . . . . . . . . . . . . . . .62 6.4 Timoshenko’s beam model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .636.4.1 Bending about z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 6.5 Free vibrations for the beam bending . . . . . . . . . . . . . . . . . . . . . . . . . .646.5.1 Problem in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 6.5.2 Problem in time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 6.6 Free vibrations for beam torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 6.7 Properties of modal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .666.7.1 Orthogonality with respect to the mass distribution . . . . . . . . . . . . .66 6.7.2 Orthogonality with respect to the stiffness distribution . . . . . . . . . . . .67 6.7.3 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67 6.8 Modal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67 6.9 Ritz-Galerkin approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68 6.10 Finite Element Method (FEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .696.10.1 Ritz-Galerkin VS FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69 7 Modal Analysis 707.0.1 State-space approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70 7.0.2 Normalization of eigenvectors or modal shapes . . . . . . . . . . . . . . . .70 7.1 Decoupling through eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .707.1.1 Transformation to first order state space form . . . . . . . . . . . . . . . . .71 7.2 Coincident eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71 7.3 Rigid body modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72 7.3.1 Gram-Schmidt orthogonalization . . . . . . . . . . . . . . . . . . . . . . . .72 7.4 Computation of the solution to initial conditions . . . . . . . . . . . . . . . . . . .72 7.5 Rayleigh quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .737.5.1 Error on the eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73 7.6 Computation of a group of eigenvalues and eigenvectors . . . . . . . . . . . . . . .747.6.1 Bloch-Stodola block iteration . . . . . . . . . . . . . . . . . . . . . . . . . .74 8 Modeling of structural damping 758.1 Visco-elastic behavior: Kelvin-Voigt . . . . . . . . . . . . . . . . . . . . . . . . . .75 8.2 Visco-elastic behavior: Maxwell . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75 8.3 Equivalent damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .768.3.1 Modal damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76 8.3.2 Rayleigh damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76 8.3.3 Equivalent damping: identification . . . . . . . . . . . . . . . . . . . . . . .76 8.3.4 Coulomb friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78 8.3.5 Hysteretic damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78 9 Dynamic response 799.1 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .799.1.1 State space format of the Laplace transform . . . . . . . . . . . . . . . . . .80 9.2 Frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .819.2.1 Dynamic stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81 9.2.2 Modal computation of the response . . . . . . . . . . . . . . . . . . . . . . .82 4 9.2.3 Contribution of the different terms to the solution . . . . . . . . . . . . . .83 9.3 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .839.3.1 Non periodic but periodic-sizable signals . . . . . . . . . . . . . . . . . . . .84 9.4 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .849.4.1 Solution using Fourier transform . . . . . . . . . . . . . . . . . . . . . . . .84 9.5 Limited bandwidth excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85 9.6 Inertia Relief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86 9.7 Anti-resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87 9.8 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88 9.8.1 Solution through the convolution . . . . . . . . . . . . . . . . . . . . . . . .88 9.8.2 Convolution and impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . .88 9.8.3 Response using the step response . . . . . . . . . . . . . . . . . . . . . . . .89 9.8.4 Response using the indicial function . . . . . . . . . . . . . . . . . . . . . .89 9.9 Truncation of the modal basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89 9.10 Recovery of internal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90 9.10.1 Direct recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90 9.10.2 Acceleration modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91 10 Random vibrations 9210.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92 10.1.1 Examples of cases where the input should be considered stochastic . . . . .92 10.2 Typical statistical quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9210.2.1 Probability distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92 10.2.2 Cumulative Probability Function (CPF) . . . . . . . . . . . . . . . . . . . .93 10.2.3 Probability Density Function (PDF) . . . . . . . . . . . . . . . . . . . . . .94 10.2.4 Expected values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94 10.2.5 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94 10.2.6 Standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95 10.3 Gaussian process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9510.3.1 Joint (multivariate) distributions . . . . . . . . . . . . . . . . . . . . . . . .96 10.4 Process function of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9610.4.1 Autocorrelation and cross-correlation . . . . . . . . . . . . . . . . . . . . . .96 10.4.2 Stationary process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97 10.5 Ergodic process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9710.5.1 Auto-Covariance and Cross-Covariance . . . . . . . . . . . . . . . . . . . . .98 10.5.2 Autocorrelation as a means to de-noise a signal . . . . . . . . . . . . . . . .99 10.6 Power spectral density (PSD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99 10.7 White noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100 10.7.1 Band-limited white noise (colored noise) . . . . . . . . . . . . . . . . . . . .101 10.8 Process with exponential correlation . . . . . . . . . . . . . . . . . . . . . . . . . .101 11 Random response 10211.1 Computation of the response of a SISO system using impulse response . . . . . . .102 11.1.1 Computation of the mean of the output knowing the mean of the input . .102 11.1.2 Computation of the mean using the second order formulation . . . . . . . .102 11.1.3 Computation of the mean using the state space formulation . . . . . . . . .103 11.2 Computation of the variance of the output using the impulse response . . . . . . .103 11.3 Computation of the response in frequency domain . . . . . . . . . . . . . . . . . .104 11.4 White noise approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104 11.4.1 Linear oscillator sub ject to a white noise input . . . . . . . . . . . . . . . .104 11.4.2 Applicability of the white noise approximation . . . . . . . . . . . . . . . .105 11.5 State-space formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106 11.6 Lyapunov equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106 11.7 Shape filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107 11.8 Risk assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108 11.9 Rice formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11011.9.1 Probability of crossing a threshold level b in a finite time interval T . . . .110 5 12 Unsteady aerodynamics 111 12.1 Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111 12.2 Unsteady aerodynamics phenomena . . . . . . . . . . . . . . . . . . . . . . . . . .111 12.2.1 Simplified physical models . . . . . . . . . . . . . . . . . . . . . . . . . . . .112 12.2.2 Spatial filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112 12.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11312.3.1 Reynolds Transport theorem . . . . . . . . . . . . . . . . . . . . . . . . . .113 12.4 Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11312.4.1 Mass conservation (continuity equation) . . . . . . . . . . . . . . . . . . . .113 12.4.2 Balance of momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114 12.4.3 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114 12.4.4 Final system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114 12.5 Rate of change of kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .115 12.6 Entropy (II principle of thermodynamics) . . . . . . . . . . . . . . . . . . . . . . .115 12.7 Acceleration and Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115 12.8 Circulation and Kelvin’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .11612.8.1 Sudden start of an airfoil in 2D . . . . . . . . . . . . . . . . . . . . . . . . .116 12.9 Velocity potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117 12.10Unsteady Bernoulli equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117 12.11Unsteady Bernoulli equation: compressible adiabatic flow . . . . . . . . . . . . . .11712.11.1 Pressure coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118 12.11.2 Pressure coefficient for an incompressible, adiabatic flow . . . . . . . . . . .118 12.12Full potential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118 12.13Kutta condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119 12.14Linearized potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120 12.15Linearized boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121 12.16Solution of the incompressible problem: Biot-Savart . . . . . . . . . . . . . . . . .12312.16.1 Straight vortex in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123 13 2D unsteady airfoil theory 12513.1 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125 13.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125 13.3 Wake vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12613.3.1 Wake vorticity in frequency domain . . . . . . . . . . . . . . . . . . . . . .126 13.4 Method of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12613.4.1 Change of coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126 13.4.2 Decomposition of bound vorticity . . . . . . . . . . . . . . . . . . . . . . . .127 13.5 Computation of loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128 13.5.1 Jump of pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128 13.5.2 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128 13.6 Solution in frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12913.6.1 Wake vorticity in frequency domain . . . . . . . . . . . . . . . . . . . . . .129 13.6.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129 13.7 Theodorsen: pitch and plunge harmonic oscillation . . . . . . . . . . . . . . . . . .13013.7.1 Solution: Theodorsen for harmonic motion . . . . . . . . . . . . . . . . . .131 13.7.2 Theodorsen function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131 13.7.3 Lift variation due to an harmonic oscillation in pitch . . . . . . . . . . . . .132 13.8 Cicala, Küssner-Schwarz approach for arbitrary deformations . . . . . . . . . . . .133 13.9 Wagner approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133 13.10Lumped vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13313.10.1 Lumped vortexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134 13.11Frozen gust front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13513.11.1 Gust speed in frequency domain . . . . . . . . . . . . . . . . . . . . . . . .135 13.11.2 Gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136 13.11.3 Gust: Sears function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136 13.11.4 Final result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .137 6 14 Morino’s method - Panel method for unsteady compressible flows 138 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138 14.2 Green’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138 14.2.1 Fundamental Solution of Laplace Equation . . . . . . . . . . . . . . . . . .139 14.3 Morino’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139 14.4 Incompressible, steady flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13914.4.1 Analysis of the wake (steady case) . . . . . . . . . . . . . . . . . . . . . . .140 14.4.2 Discretization: panel method . . . . . . . . . . . . . . . . . . . . . . . . . .141 14.4.3 Computation of loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142 14.5 Incompressible, Unsteady Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14214.5.1 Wake analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142 14.5.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143 14.6 Compressible case: steady flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14314.6.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144 14.7 Unsteady supersonic flow: fundamental solution . . . . . . . . . . . . . . . . . . . .144 14.7.1 Explanation ofτ(for the fluid at rest) . . . . . . . . . . . . . . . . . . . . .145 14.7.2 Subsonic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146 14.7.3 Supersonic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146 14.7.4 Supersonic flow panelization . . . . . . . . . . . . . . . . . . . . . . . . . . .146 14.7.5 Indicial response: compressible vs incompressible flow . . . . . . . . . . . .147 14.8 Piston theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148 15 Formulations of the aeroelastic stability problem 14915.1 Computation of flutter speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149 15.2 Assembly of the aeroelastic system . . . . . . . . . . . . . . . . . . . . . . . . . . .149 15.3 The aeroelastic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .150 15.4 Quasi-steady approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15115.4.1 Computation of the QS approximation from the frequency response . . . .151 15.4.2 Meaning of QS approximation in time domain . . . . . . . . . . . . . . . . .152 15.4.3 Direct solution for flutter speed . . . . . . . . . . . . . . . . . . . . . . . . .152 15.4.4 Flutter V-f V-g diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . .152 15.5 Flutter P-K method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153 15.5.1 Flutter P-K method algorithm . . . . . . . . . . . . . . . . . . . . . . . . .153 15.5.2 State-space representation of unsteady aerodynamics . . . . . . . . . . . . .154 15.6 Flutter speed: the continuation approach . . . . . . . . . . . . . . . . . . . . . . .155 15.7 Types of flutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15615.7.1 Dependency of flutter on main parameters . . . . . . . . . . . . . . . . . . .157 15.8 Transonic dip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157 16 The aeroelastic problem: formulation of static and dynamic response problems 15916.1 Equilibrium maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .159 16.2 Dynamic maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16016.2.1 Modeling of a control surface . . . . . . . . . . . . . . . . . . . . . . . . . .160 16.2.2 Control chain and servo-actuators . . . . . . . . . . . . . . . . . . . . . . .161 16.2.3 Modeling of a dynamic compliance surface . . . . . . . . . . . . . . . . . . .162 16.2.4 Complete model of a servo-controlled surface . . . . . . . . . . . . . . . . .162 16.3 Gust response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16216.3.1 Sharp gust and "1-cosine" gust . . . . . . . . . . . . . . . . . . . . . . . . .163 16.3.2 Gust penetration effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .164 16.3.3 Stochastic gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .164 7 1 Introduction Structural dynamics is concerned with vibrations and dynamic response of the elements of a struc- ture. Aeroelasticity is the study of structures typical of aerospace industry considering their interaction with the fluid flows surrounding them. Aerospace structures need to be lightweight and really performing. These requirements lead to the necessity of adopting thin-walled structures, which is possible thank to the implementation of composite materials into their design. This way they get extremely flexible. The aerodynamic loads generated by a fluid moving around an ob ject depend on the shape of the ob ject itself. By changing the shape of the structure, the aerodynamic loads generated change as well. Moreover, every time the structure deforms, accelerations are present, causing inertia forces that tend to modify the shape of the structure as well, resulting in a further acceleration. Aeroelasticity studies the interaction between inertial, elastic and aerodynamic forces occurring while an elastic body is exposed to a fluid flow. The main phenomenons are generally dynamic, but static aeroelasticity can have a strong impact on the structure’s design. Aeroelastic and structural dynamics phenomena can have a severe effect on structures by creating dangerous loading conditions and instabilities. The most common instabilities encountered are divergence and flutter.(a) Col lar triangle, to show the multidisciplinary aspect of aeroelasticity(b) Feedback loop between Inertial, Elastic and Aerody- namic forces Figure 1.0.1 A more complete version of the Collar triangle is the Aero-Servo-Thermo-Elasticity pyramid (figure 1.0.2), which shows the interaction with controls and the thermal effects (they induce changes in shape, additional stresses, taking part, this way, in the feedback loop structure) as well.Figure 1.0.2: Aero-servo-thermo-elasticity pyramid 8 1.1 Divergence and Flutter By schematizing the aircraft as a rigid fuselage and a flexible cantilever wing, the latter shows two possible modes of motion: a bending mode, associated to a relatively low frequency, and a twisting mode, related to a higher frequency. If the aircraft is flying inside its nominal flight envelope, these two modes of motion are almost irrelevant. By increasing the flight speed, instabilities may occur: the bending and twisting modes may increase their amplitude until the enlarged and abnormal motion of the wing causes a fatal damage on it. Divergence is related to torsional stiffness, i.e. to the cross sectional area of the wing. It’s a really fast phenomenon. Flutter is a type of instability characterized by an oscillatory behaviour, while divergence shows an exponential behaviour. 1.1.1 Aeroelasticity through time The first planes were double-wing aircrafts, featuring both a wide cross sectional area and a large frontal cross area. The first was needed in order to have satisfying torsional stiffness and avoid divergence phenomena. Later on, while trying to increase the cruise speed of the plane, the first mono-wing aircrafts were introduced: the aim was to reduce the drag resulting from such large frontal cross section, typical of the previous double-wing planes; this was achieved by reducing the cross sectional area of the wing and, thus, its torsional stiffness, making these planes easily sub ject to divergence. By employing metallic alloys in the design, mono-wing aircrafts gained a sufficient torsional stiff- ness to safely fly at higher speeds. To control the aircrafts, the Wright brothers applied wing warping: they were able to bend the wing tips oppositely in order to generate a rolling moment due to the uneven distribution of lift. Later on movable surfaces were introduced. Soon another type of aeroelastic instability started to arise:control surface flutter. The stabilizers were independently rotating, creating a twisting moment on the backwards part of the fuselage. The solution was connecting left and right stabilizer rigidly. Flutter was affecting the main wing as well: whenever the aerodynamic loads on the wing changed, a twisting moment was created. Moving through time, faster and faster aircrafts were developed and other aeroelastic phenomena started to become relevant: e.g. in transonic region, by rotating the movable surfaces, the shock waves start to oscillate and, possibly, interact with the boundary layer, giving birth to the so called buzz aileron phenomena. Flutteris an oscillatory instability characterized by a combination of bending and torsion; it is not limited to the lifting surfaces but it affects with the whole structure. Aileron (control) reversal: Normally, lift increases with the square of the speed. When a mov- able surface is deflected, lift is increased, but this also generates a nose-down pitching moment. This moment tends to reduce the angle of attack, counteracting the aileron’s intended function of locally increasing the lift coefficient. At high speed this effect becomes dominant and, thus, the twist of the wing creates a negative∆α: this way, by deflecting downwards the right aileron, the plane will turn right instead of left. The verge condition is called control reversal and occurs at control reversal speed: it’s the condition for which the aircraft looses controllability. To increase the control reversal speed typically multiple movable surfaces are implemented: at high speed, inner ailerons are used instead of the usual outer ones in order to reduce the twisting moment. Nowadays, inner ailerons have left room to spoilers, which are flutter free. Panel flutter:it is an aeroelastic instability that arises at really high velocities (high super- sonic/hypersonic regime). It is due to the massive values of dynamic pressures that lead to panels oscillations (because of aeroelastic interactions) until they break. To prevent flutter, it is needed a detailed numerical analysis of the limiting cases, ground vibration 9 testings, wind tunnel testing and, finally, flight flutter testing. To practically improve flutter performances, it is possible to employ either flutter control and flut- ter suppression systems or wisely use composite materials in order to get an aeroelastic tailoring effect. 10 2 Static Aeroelasticity 2.1 Static aeroelasticity of the typical sectionFigure 2.1.1: Scheme of a wing employing typical section model A typical section is an experimental tool as well as an abstraction to model aeroelastic phe- nomena. This model represents a flexible wing through a rigid wing connected to a clamped torsional spring. The rigid wing is considered to be straight, with span perpendicular to the air flow and constant section over the span. All the deformability is concentrated into the torsional spring at the root. For the aerodynamics, 2D strip theory is con- sidered valid as well as the linear lift coefficient approximation (the flow is considered attached to the wing on all the range of angles of attack considered). 2.1.1 Definitions Shear center (SC):For a given 2D section, the shear center is the point with respect to which, if a shear load is applied, there is no variation in the twist of the section. It is strictly a property of the cross section. Elastic axis (EA):It is the axis about which the wing rotates, i.e. the locus of all the shear centers along the span of the wing. The torsional spring is lumped in correspondence of the elastic axis. Aerodynamic center (AC):for a given 2D section, it is the point about which the aerodynamic moment coefficient is constant independently from the angle of attack, i.e.C mACα“ 0. The aero- dynamic center is approximately at one quarter of the chord from the leading edge of the airfoil (considering valid the thin airfoil approximation, the AC is exactly atc{4). 2.1.2 Aerodynamics Slope of lift coefficient: the slope of the lift coefficient is the derivative ofC Lwith respect to the angle of attack, i.e.C Lα“ a“B C LB α. Thin airfoil theory provides us with models to evaluate the lift coefficient’s slope case by case; in particular: •2D incompressible: C2 D Lα“ a 0« 2π, thin airfoil theory; •3D incompressible: C3 D Lα“a 01 `a 0πλ p 1`τq, where λ“b2 {Sis the aspect ratio of the wing andτis the non-elliptic lift distribution coefficient; •Compressible flow: CC Lα«a 0? 1 ´M2, Prandtl-Glauert correction (For subsonic, compressible flows, up toM“0.9) 2.2 Divergence Referencing to the figure 2.2.1,θis theelastic twist(around the elastic axis due to the change of pressure distribution),eis the static margin, defined as the distance between the elastic axis and the aerodynamic center,α 0is the angle between the velocity Uand the zero lift line (ZLL),K αis the stiffness of the clamped torsional spring andαis the angle of attack, defined asα“α 0` θ. On top of these considerations, we are focusing on static aeroelasticity, thus no inertia contributions will be considered. 11 Figure 2.2.1: 2D view of the Typical section Our aim is to find an expression for the elastic twistθ: assuming an equilibrium condition, it is possible to write an equilibrium equation with respect to the elastic axis as follows: Kαθ “e¨L`M AC(2.1) Using aerodynamics to define lift and moment (whereqis the dynamic pressure), it yields to: Kαθ “qeS C Lαα `qcS C MAC(2.2) By exploiting the previously introduced definition forαand the definition of the aerodynamic center, by whichC MAC“ constant with respect toα, θpK α´ qeS C Lαq “ qeS C Lαα 0` qcS C MAC(2.3) θ“qS peC Lαα 0` cC MACqK α´ qeS C Lα(2.4) It is immediate to notice how the elastic twist depends on aerodynamic and elastic properties of the section as well as from the dynamic pressure, i.e. the speed of the aircraft.Figure 2.2.2: feedback structure for the typical sectionThe numerator of the previous expression is no other than the initial momentM 0applied at the EA to the airfoil through the spring. M0“ qSpeC L0` cC MACq (2.5) The denominator in the expression ofθ(equa- tion 2.4) represents the equivalent stiffness of the system, depending both on structural and aero- dynamic properties. Referring to figure 2.2.2, it is possible to formalize the following definitions (somehow already intro- duced): Aerodynamic additional moment:M Aθ, is the additional moment generated by the system due to the interaction between aerodynamics and the deformable structure. Aerodynamic stiffness:K A“ eS C Lα Structural stiffness:K α Effective stiffness:˜ K“K α´ qK A Let’s now suppose that the initial moment is constant. The elastic twist can be expressed as M0K α´ qK A“ M 0K αˆ 11 ´qK A{ K α˙ (2.6) The latter expression can be plotted as a function of the dynamic pressureq, keeping in mind that, forqÑK α{ K A, θÑ 8. At that specific dynamic pressureθdiverges: the wing starts twisting around the elastic axis exponentially without nothing that can stop this process. Such dynamic pressure is called 12 Dynamic pressure of Divergence: q D“K αK AFigure 2.2.3: Deformation of the aeroe- lastic system in terms of performance index This dynamic pressure of divergence makes sense if eą0, i.e. if the aerodynamic center is in front of the elastic axis. Otherwise,q Dă 0which doesn’t make sense: it’s a no divergence condition. The smaller ise, the larger is the dynamic pressure of divergence: the duty of the aeroelastician is to makeeas small as possible. By manipulating slightly the latter expression, is possible to define the Speed of Divergence:U D“d2 K αρeS C LαIt’s possible to define a performance index, θ{θ 0as fol- lows:θθ 0“ 11 ´qq D(2.7) The closer isqtoq D, the higher will be the incidence of aeroelastic effects.θ 0“ M 0{ K αis the initial deforma- tion. Aeroelastic effect start to be relevant wheneverq«q D( q{q Dą 0.5,U{U Dą 0.7). 2.2.1 Comparison between rigid and elastic lift Let’s introduce the rigid liftL R, i.e. the lift produced by the same structure if it was rigid, and the elastic liftL, i.e. the actual lift produced by the structure.Figure 2.2.4: Comparison between rigid and elas- tic lift Using basic aerodynamics definitions, we can define both as LR“ qS C L0“ qS C Lαα 0; L“qS C Lαp α 0` θq (2.8) To compare the incidence of aeroelastic effects over the total lift, it’s possible to evaluate the following ratio, representing the percentage of total lift due to elastic effect (remember the definition of the performance index, introduced in equation 2.7): L´L RL R“ θα 0“ θ 0α 0ˆ 11 ´q{q D˙ (2.9) In typical applications, the parenthesis is close to 1; however, increasingq{q D, the incidence of aeroelastic effects on the total lift increases significantly (as shown in figure 2.2.4; keep in mind it’s a semilogarithmic plot!). 2.2.2 Torsional divergence as a stability problem Static stability:consider a system that is in equilibrium; the equilibrium is saidstatically stableif, given a slight perturbation to the equilibrium, there is a tendency to return to the equilibrium condition. In order to have equilibrium, the total moment applied to a structure must be null. In practice, the equilibrium condition can be written as: eL`M AC´ K αθ “0ÑM AE“ fpθ T, q q “0(2.10) 13 Note: The equilibrium condition is typically known as trim condition (subscript "T"). To evaluate the stability of the system, a small perturbation is added to the twist angle and the response of the system is analyzed. Consider θ“θ 0` ∆θ(2.11) It’s possible to linearize the system around the equilibrium condition: MAEp θ, qq “M AEp θ T, q q `B M AEB θ∆ θ`...(2.12) As shown in equation 2.10, the overall moment evaluated in the trimmed condition is null; thus, the first term of the previous expression can be canceled out. To achieve an asymptotically stable equilibrium condition, the system must react to any pertur- bation∆θwith a decrease of the twist angle: this yields to the condition to identify the verge between a stable and an unstable equilibrium condition. •Asymptotically stable equilibrium:B M AEB θă 0Ñqăq D. The system will react to a perturbation by returning to the equilibrium condition. •Stable equilibrium:B M AEB θ“ 0Ñq“q D. This is the limiting case for stability: in response to a perturbation of the equilibrium condition, the system doesn’t diverges neither returns to the original equilibrium condition; it simply updates the equilibrium condition as θTi`1“ θ Ti` ∆θ. This condition allows us to find the dynamic pressure of divergence. •Unstable equilibrium:B M AEB θą 0Ñqąq D. If the system is perturbed, it diverges in the direction that increase furthermore the perturbation. MAE“ qS eC Lαp α 0` θq `qS cC MAC´ K αθ (2.13) To find the dynamic pressure of divergence we impose the limit condition, i.e. BM AEB θ“ 0Ñ peqS C Lα´ K αq ∆θ“0Ñq D“K αeS C Lα(2.14) 2.2.3 Torsional divergence as an eigenvalue problem The equilibrium condition already introduced states that MAE“ fpθ T, q q ` pK α´ qS eC Lαq ∆θ“0Ñ pK α´ qS eC Lαq ∆θ“0(2.15) We can find a trivial solution,∆θ“0, or a non trivial one through the resolution of the following eigenvalue problem (where the eigenvalues are the dynamic pressures): pK α´ qK Aq ∆θ“0(2.16) 2.2.4 Effects of Mach number on torsional divergence If an incompressible aerodynamic model is considered, the slope of theC L´ αcurve, i.e.C Lα, is constant. Whenever the compressibility effects need to be considered, the Prandtl-Glauert correction is implemented to evaluate theC Lα. Mach number is a function of local velocity that is a function of dynamic pressure; the slope of the CL´ αcurve is dependent on the dynamic pressure as well. Finally, the static marginedepends on the dynamic pressure since, in the transonic region, the aerodynamic center moves slightly forward and then backwards up to half of the chord; this movement is ruled by the changing Mach number. The ob jective of the static aeroelastic problem is still then same but the problem itself is highly non-linear. There is a limit Mach number for which the Prandtl-Glauert correction is valid; the latter depends on the thickness of the airfoil: the thinner, the larger the range of Mach number for which the approximation is valid. 14 If we consider the case of a thin airfoil and Mă0.6,C Lα‰ C Lαp Mq. The dynamic pressure of divergence can be written as qD“12 ρU 2 D“12 ρc 2 M2 D, wherec2 “γ RT“γpρ for an ideal gas (2.17) qD“12 γ p 8M2 D“ AM2 D(2.18) The condition to have divergence isK α´ qeS C Lα“ 0. So Kα“ q DS eC M “0 Lαa 1 ´M2 DÑ q D“K αeS C M “0 Lαb1 ´M2 D“ qM “0 Db1 ´M2 D(2.19) Elaborating the previous equation, keeping in mind that A is a function of altitude, we get M4 D`ˆ qM “0 DA ˙ 2 M2 D´ˆ qM “0 DA ˙ 2 “0(2.20) 2.3 Control surface effectivenessFigure 2.3.1: Typical section with a movable surface Let’s add another degree of freedom by the inclusion of a movable surface (a flap) in the typical section model. Referring to figure 2.3.1, the param- eterErepresents the percentage of the chord dedicated to the flap so, the productE¨cgives the length of the movable surface. From now on, equilibrium will be considered satisfied so, whenever an angle is involved, it represents a per- turbation of the trimmed condition. It is possible to expand the aerody- namic model considered up to now to the case of the airfoil with a movable surface as follows: #L“qS` CL0` C Lαθ `C Lββ˘ MAC“ qS c` CmAC` C mββ˘ (2.21) WhereC mβand C Lβrepresent respectively the variation of moment coefficient and the variation of lift coefficient due to the rotation of the movable surface of an angleβ. TypicallyC mβă 0and CLβą 0. Cmβis typically negative since, for a downwards rotation of the movable surface, a pitching down moment is generated. There are experimentally extrapolated expressions that link the ratiosC Lβ{ C Lαand C mβ{ C Lαto the previously introduced parameterE. Let’s now try to evaluate the twist generated by the rotation of the movable surface of an angleβ: Kαθ “e¨L`M ACÑ K αθ “qS e` CLαθ `C Lββ˘ `qS cC mββ (2.22) Remembering the definition of the aerodynamic stiffness,K A“ eS C Lα, the twist angle can be found as θ“qSC Lβe `C mβcK α´ qK Aβ(2.23) It is trivial to see that the twist angle will tend to infinity whenever the denominator goes to zero, i.e. atq“q D: the wing will still be affected by divergence. However, the numerator has an effect on the twist angle as well, making it linearly dependent from the dynamic pressure. We can now compare elastic and rigid lift in order to understand how the elasticity of the wing 15 affects the aerodynamic forces. Let’s define the rigid lift coefficient derived with respect toβasC Lβp E , Mqand the elastic lift coefficient derived with respect toβaspC Lβq e. Referring to the expression of θin equation 2.23, pC Lβq e“LqS “ C Lαθ `C Lββ “C LαqSC Lβe `C mβcK α´ qK A` C Lβ(2.24) pC Lβq e“qS C LαeC Lβ` qS cC mβC Lα` C Lβp K α´ qS eC LαqK α´ qK A(2.25) Let’s introduce theControl elastic efficiency,E c, a parameter that shows the ratio between the elastic and a rigid lift: Ec“qS eC mβC LαC LβK α´ 1´qq D¯ `K αK α´ 1´qq D¯ (2.26) Ec“1 `qS cC LαC m βC L βK α1 ´qq D(2.27) By increasing the dynamic pressure, the denominator gets smaller and smaller and the second term of the numerator, since it is already negative, gets more and more negative, yielding to a lower and lower control efficiency. It is possible to find a particular value ofqsuch that the numerator goes to zero: the meaning of it is a loss of effectiveness of the movable surface in terms of controllability of the aircraft (by rotating the flap, the plane doesn’t roll because there is no change in the distribution of lift). This dangerous condition is calledControl reversal conditionand it is encountered when the dynamic pressure value reaches the control reversal one,q RØ E c“ 0: qR“ ´K αC LβS cC LαC mβ(2.28) The condition is critical when flying close toq Rbecause the control surface becomes ineffective, compromising the aircraft controllability. Passing the control reversal condition, the effect obtained by the rotation of a movable surface is the opposite of what is expected: by deflecting downwards the right aileron, the aircraft will roll to the right instead of the left. 2.3.1 Plot of control elastic efficiencyFigure 2.3.2: Control surface elastic efficiency By re-writing the control elastic efficiency as follow, it is possible to plotE cin terms of the ratio betweenqandq Rfor various values of q D{ q R(figure 2.3.2): Ec“1 `qq R1 ´qq Rq Rq D(2.29) 16 Typically, the dynamic pressure of divergence is larger than the control reversal one: this way, the control efficiency monotonically decreases for an increasing dynamic pressure. Ifq D“ q R, the two effects are equal and opposite and cancel out, giving back a control efficiency of 1. In the unreal case where the dynamic pressure of divergence is smaller than the control reversal one, the control efficiency increases diverging towards infinity as we increaseq. Then, after reaching qD, it starts decreasing. This aeroelastic effect by which the control efficiency increases with the speed of the aircraft, can be exploited in designing aircrafts that are more and more maneuverable the higher the cruise speed (useful in military applications). When rotating a movable surface, the pressure coefficient changes along the wing: the area under its curve expands causing its increment. The deflection of the movable surface increases theC pbut causes the center of pressure to move backwards chord-wise, generating a pitching aerodynamic moment that reduces the angle of attack of the wing, reducing the pressure coefficient due to decrement of the lift distribution. 2.3.2 Matrix form of control reversal problem Consider the two degrees of freedom system introduced before and write the equilibrium equation together with the expression of lift. #pK α´ qK Aq θ“qSpeC Lβ` cC mβq β L“qSpC Lαθ `C Lββ q(2.30) The system 2.30 can be written in matrix form, consideringβfixed and known, and taking as unknowns the twist angle and the distribution of lift, yielding to: „Kα´ qK A0 ´qS C Lα1ȷ " θ L* “" qS eCLβ` qS cC mβ qS CLβ* β(2.31) By a trial and error method (an iterative process of guesses and checks ofq R), once βis assigned, the control reversal dynamic pressure can be found. There are 2 possible approaches in order to find the control reversal dynamic pressure. Lift increment approach:the unknowns of the system 2.31 areθandL; we are looking for the lift increment due to a given increment ofβ(qis considered fixed). Thus, "L θ* “qS„ 0K α´ qK A 1´qS C Lαȷ ´1" eCLβ` cC mβ CLββ* (2.32) To identify the control reversal dynamic pressure,q R, we need to find qsuch that, for any deflection of the movable surface, the resulting lift increment is null. Eigenvalue problem:the unknowns are the twist angle and the deflection angle of the movable surface, while the lift is given. It yields to the following system of equations: „´qSpeC Lβ` cC mβq K α´ qK A ´qS C Lβ´ qS C Lαȷ " β θ* “" 0 ´1* L(2.33) The matrix characterizing the previous system can be re-written dividing the structural terms from the aerodynamic related ones: ˜„ 0K α 0 0ȷ ´q„ SpeC Lβ` cC mβq K A S CLβS C Lαȷ ¸ " β θ* “" 0 ´1* LÑA" β θ* “" 0 ´1* L(2.34) When looking for the control reversal dynamic pressure,L“0. This makes the system 2.34 an eigenvalue problem (where the eigenvalues are the particular values of dynamic pressure for which, a variation ofβdoesn’t change the lift distribution) thus, to findq R, Amatrix must be singular, which yields to detpAq “0Ñq2 RS2 peC Lβ` cC mβq C Lα` q RS C Lβp K α´ q RK Aq “ 0(2.35) 17 It’s a quadratic equation, we expect two different control reversal dynamic pressures: $ & %q R“ 0 qR“ ´K αC LβcS C mβC Lα(2.36) The null dynamic pressure is a control reversal condition since, when the relative speed of the wing is null, by rotating the movable surface of anyβ, the variation of lift over the wing is null (definition of control reversal condition). 2.4 Addition of control surface stiffnessFigure 2.4.1: Addition of control surface stiffness through a rotational spring K β Since the control chain employed to deflect the movable surface is flexible, a more accurate model consists in considering a torsional spring that connects the hinge of the movable surface to the fixed aircraft. The spring represents the flexibility of the whole control chain. When applying a moment to the movable surface, the pressure distribution over the wing changes in such a way that an aerodynamic moment is generated in order to bring back the movable surface to the neutral position. This moment is typically calledaerodynamic hinge momentand is defined as (considering as always perturbations of the trim condition, i.e.α“θ) H“qS fc fp C Hαθ `C Hββ q(2.37) The system has two degrees of freedom (βandθ): to solve the system we need to write two equilibrium equations, one about the elastic axis and the other one about the flap hinge. #qS epC Lαθ `C Lββ q `qS cC mββ ´K αθ “0 qSfc fp C Hαθ `C Hββ q ´K βp β´β 0q “ 0(2.38) Whereβ 0is the ideal position for the control surface required by the control chain (imposed by the pilot) andβis the actual position of the control surface considering the elasticity of the control chain (real angle reached at equilibrium). The equilibrium system can be re-written in matrix form as follows: ˜„ Kα0 0K βȷ ´q„ S eCLαS eC Lβ` S cC mβ Sfc fC HαS fc fC Hβȷ ¸ " θ β* “" 0 Kβ* β0(2.39) Calling the big matrixA, the unknown vectorxand the forcing termF, the system can be solved inverting the matrixA: x“A´ 1 F,A´ 1 “1det pAq„ a22´ a 12 ´a 21a 11ȷ (2.40) detp Aq “K αK β„ˆ 1´aK AK α˙ˆ 1´qS fc fC HβK β˙ ´qS eK αqS fc fK βC Hαˆ CLβ`ce C mβ˙ȷ (2.41) 18 The solution of the system is $ ’ & ’ %θ “qSk βdet Ap eC Lβ` cC mβq β 0 β“k βdet Ap k α` qS eC Lαq β 0(2.42) In order to find the control reversal dynamic pressure and understand how much efficiency is lost by increasing more and more the dynamic pressure, we need to evaluate the control elastic efficiency, i.e. the ratio between elastic and rigid lift: L“qSk βk αdet A´ CLβ`qS ck αC mβC Lα¯ β0, L R“ qS C Lββ 0(2.43) Ec“LL R“ k αk βdet Aˆ 1`qS ck αC mβC LαC Lβ˙ (2.44) The reversal condition is the same as for the infinite flap hinge stiffness:q R“C Lβk αS cC mβC Lα. Control reversal is not affected by the flexibility of the control chain. If we define the divergence dynamic pressure of the typical section without the flap asq D0“k αS eC Lα andx“C HαC Hβˆ CLβC Lα` ce C mβC Lα˙ , the first term of equation 2.44 can be re-written as kαk βdet A“ 1ˆ 1´qq D0˙ˆ 1`qq Df˙ `xqq D0qq Df(2.45) The divergence dynamic pressure is the one that makes the denominator of equation 2.45 go to zero. There are two different divergence dynamic pressures since the system has two degrees of freedom.Figure 2.4.2: Plot of the ratio q Dq D0 q D“ q D0´ 1´q D fq D 0¯ ˘c´ 1´q D fq D 0¯ 2 `4p1´xqq D fq D 02 p1´xq (2.46) There are two particular conditions to be analyzed:1.If the flap is unrestrained, meaning in has null hingestiffness (k β“ 0), the dynamic pressure of diver- gence becomesq D“q D01 ´x; 2.Ifx“0,q D“ q D0, q D“ ´ q Df. Ifxă0,q Dą q D0: the divergence dynamic pressure of the system will be higher than its flap-free version. This case makes modeling of the system easier because, from a divergence assessment point of view, considering the flap-free typical section, leads to a conservative esti- mation ofq D, meaning that the dynamic pressure of divergence, considering the typical section with the flap, will be larger than the predicted one: the system will still be able to withstand any load required by its flight envelope (with a certain margin). Ifxą0,q Dă q D0, which leads to a non conservative estimation of the dynamic pressure of divergence in the flap-free case with respect to the one considering it. 2.5 Rolling of a straight wing This model is introduced in order to analyze the aeroelastic behavior of a typical section wing undergoing a rolling maneuver due to a deflection of the ailerons. The rolling motion results in an acceleration and, thus, inertia forces applied to the center of 19 Figure 2.5.1: Rolling typical section wing with ailerone gravity of the wing rising. Let’s allow the typical section model to freely rotate around the roll axis withroll angular speed ϕ. Inertia forces can be generally expressed as the product between the mass of the section and its acceleration (with a minus sign in front: they develop in the opposite direction of the acceleration of the mass). In this model acceleration of the section is a combination of angular acceleration due to the rolling movement and the one due to deformation of the section. To analyze this problem some hypothesis have to be made: •The motion of the wing is considered to be the superimposition of a rigid movement and the deformation of the wing structure; •The deformation of the structure behaves statically, i.e. –The characteristic time required by the deformable structure to adapt to a change in loading is much smaller than the characteristic time of the rigid movement; –The structure is deformed as if the load is applied statically, i.e. the different loading conditions during the maneuver are equivalent to a sequence of static deformations; •Aerodynamics behaves statically too. This model can still be considered part of static aeroelasticity since the only inertia forces considered are those caused by the rigid motion. 2.5.1 AerodynamicsFigure 2.5.2: Relative air speed (black) caused by roll motion (blue velocity vector) Let’s define the angular speed around the roll axis asp. During a rolling maneuver, the velocity of the wing will be modified by the addition of a vertical 20 component due to the motion and the deformation, as shown in figure 2.5.2. The variation of angle of attack caused by the induced vertical velocity is ∆α“tanˆ pyU 8˙ «pyU 8(2.47) The further from the roll axis, the larger the increment of angle of attack (because the roll induced velocity is linearly proportional to the arm with respect to the roll axis). The linearization of the change of AoA (equation 2.47) is due to the fact thatpyăăU 8. The roll aerodynamic moment is defined asL R“ I xx9 p; it can be further expanded as follows: LR“ I xx9 p“qS bˆ CL0´ C LppbU 8` C Lββ˙ (2.48) Where the termI xx9 prepresents the moment of inertia with respect to the roll axis times the angular speed while the right part of the previous equation represents the total aerodynamic moment with respect to the roll axis. The lift and the aerodynamic moment (per unit span) are given by the following expressions $ ’ & ’ %L “qcpyqˆ CLαˆ θ´pyU 8˙ `C Lββ˙ MAC“ qc2 pyqC mββ(2.49) Together with the aerodynamic forces, there is an additional forcing term due to inertia forces generated by the rigid roll movement:F i“ ´ m9 py(9 pis the angular acceleration); this force generates a twisting moment about the elastic axis,M i“ md9 pywheredis the distance between the elastic axis and the center of gravity of the section and it is taken as positive whenever the CG is behind the EA.(a) Damping and inertia forces on a rol ling wing(b) Forces and moments due to displaced aileron Figure 2.5.3 Let’s write the roll equation (equilibrium equation with respect to the roll axis):$ ’ ’ ’ & ’ ’ ’ %ˆ b 0Lydy ´ˆ b 0my 2 dy9 p“0 qcˆ b 0ˆ CLαˆ θ´pyU 8˙ y`C Lββ y˙ dy´I xx9 p“0(2.50) The mass in principle is constant but could be function of the span position. For a rigid system, i.e.θ“0, rigid wing with constant chord, Ixx9 p“qS bˆ CLppbU 8` C Lββ˙ (2.51) The derivatives of the roll moment coefficient are equal toCLp“ ´C Lα3 , C Lβ“C Lβ2 , C Lθ“C Lα2 (2.52) 21 For an elastic system (considering a twist angle different from 0), we get the elastic roll equation: Ixx9 p“qS bˆ CLθθ `C LppbU 8` C Lββ˙ (2.53) By itself, equation 2.53 it’s not solvable: it needs another equation for the elastic twist. We can write the equilibrium around the elastic axis: kαθ “ˆ b 0p Le`M ACq dy`ˆ b 0myd dy 9 p(2.54) The previous equation can be further expanded substituting the expression of lift and aerodynamic moment and integrating over the span of the wing. Collectingθon the left side of the equation, it yields to (the total mass of the wing has been defined asm T“ mb): pk α´ qS eC Lαq θ“qSˆ ´eC Lα2 pbU 8` p eC Lβ` cC mβq β˙ `m Tbd2 9 p(2.55) It is easily recognizable the effective stiffness of the system (the term multiplyingθ). The system containing equation 2.53 and 2.55 can be re-written in matrix form for an easier manipulation.„Ixx´ qS bC Lα2 ´m Tdb2 p k α´ qk Aqȷ " 9 p θ* “qS" bCLβ eCLβ` cC mβ* β´qS" bCLp eCLα2 * pbU 8(2.56) Depending on the value ofd, we have two different cases: •If the center of gravity coincides with the elastic axis, i.e.d“0, the elastic twist becomes θ“qSp eC Lβ` cC mβk α´ qk Aβ ´qSeC Lα2 pk α´ qk AqpbU 8(2.57) •In the most general case, i.e.d‰0, θ“qSp eC Lβ` cC mβqk α´ qk Aβ ´qSeC Lα2 pk α´ qk AqpbU 8` m Tdb2 pk α´ qk Aq9 p(2.58) For sake of simplicity, we will assumed“0from now on. Substituting the expression of the elastic twist inside the equation 2.53, we obtain the following result: Ixx9 p“ ´qS bˆ CLp` qSeC 2 Lα4 pk α´ qk Aq˙ pbU 8` qS bˆ CLβ` qSC Lα2 p eC Lβ` cC mβqk α´ qk A˙ β(2.59) Which can be reduced to the following equation after definingpC Lpq eand pC Lβq e. Ixx9 p“qS bˆ ´pC Lpq epbU 8` p C Lβq eβ˙ (2.60) It is finally possible to formulate the control reversal problem as: find the dynamic pressureq Rfor whichpC Lβq e“ 0. By doing so, we get exactly the same control reversal dynamic pressure as before. The second value gotten forq Rrepresents the dynamic pressure at which, by rotating the movable surface, we get no change in roll moment. The last equation can be treated as algebraic if the derivatives of the unknown variables are considered as unknowns by themselves: the unknowns at hand arep,9 pandβ. By assigning two of them, is possible to find the third one through aconsistent static aeroelastic problem. There are some "standardized" consistent problems often considered when trying to solve equation 2.60: •Compute the angular speedp generated by a rotationβof the movable surface at regime, i.e. when9 p“0:pb U 8“ p C Lβq ep C Lpq eβ (2.61) •Compute the initial acceleration9 p 0“ 9 ppt“0qgenerated by the rotationβof the movable surfaceβ. At the first instantp 0“ ppt“0q “0, so 9 p 0“qS b pC Lβq eI xxβ (2.62) 22 Figure 2.5.4 23 3 Multi Degrees of Freedom (MDoF) model for torsional di- vergenceFigure 3.0.1 Let’s consider the wing as com- posed byNrigid portions con- nected by torsional springs; the first portion is connected to a clamped torsional spring. The elastic axis may not be a straight line if the airfoils of the different portions are different (keep in mind that the elastic axis is the locus of the shear cen- ters of all the sections). Aerodynamic centers axis may not be straight (discontinuous perhaps) either. Let’s define asθ ithe absolute rotation of the i-th section. The hypothesis considered to develop the following model are: •Pseudo-2D aerodynamics approximation is still valid; •No mutual aerodynamic influence between portions is considered. 3.1 Virtual work principle (VWP) Let’s try to write the equilibrium equations using the principle of virtual work (PVW): the virtual work is defined as the work done by forces and moments that work for virtual displacements and rotations respectively. δW“n ÿ i“1F i¨ δr i`m ÿ j“1M j¨ δφ j(3.1) The virtual movementsδr iand δφ jare possible infinitesimal movements compatible with the con- strains of the system at a fixed instant in time. The PVW states that, at a given instant in time, the condition to have equilibrium of the system isδW“0(there is no compatible movement with the constrains considered). riand φ jare both functions of the free generalized coordinates,q k, with kspacing from 1 to N,Nbeing the number of deegrees of freedom of the system. The virtual movements can be written as function of the virtual variation of the generalized coor- dinates, i.e. δr i“N ÿ k“1B r iB q kδq k, δ φ j“N ÿ k“1B φ jB q kδq k(3.2) The virtual work principle can be re-written as follows: δW“N ÿ k“1ˆ n ÿ i“1F i¨B r iB q k˙ δqk`N ÿ k“1ˆ m ÿ j“1M j¨B φ jB q k˙ δqk(3.3) By defining asQ keverything multiplying the virtual variation of the free coordinates, δq k, the equilibrium equations turn out to be δW“N ÿ k“1Q kδq k“ 0ÑQ k“ 0@k“1, ..., N(3.4) The result is computed given the arbitrariness ofδq k. If the forces are conservative, i.e. if it is possible to define aPotential EnergyUsuch that fc“ ´B UB r, the virtual work of the conservative loads is δWC “f c¨ δr“ ´N ÿ k“1B UB rB rB q kδq k“ ´N ÿ k“1B UB q kδq k“ ´ δU(3.5) 24 The virtual work is the sum of the virtual work of conservative forces and the one done by non conservative ones and should be equal to zero in order for the system to be in equilibrium. This leads toδU“δWN C ÑδW i“ δW e. 3.2 MDOF model For the case at hand, we will consider a potential energy given by the springs contributions, which is U“12 N ´1 ÿ i“1k i`1p θ i`1´ θ iq2 `12 k 1θ2 1(3.6) Deriving it with respect to the i-t