Enhanced Modal Solutions for Structural Dynamics in Aerothermoelastic Analysis

Abstract

Hypersonic vehicle design and simulation require models that are of low order. Modeling of hypersonic vehicles is complicated due to complex interactions between aerodynamic heating, heat transfer, structural dynamics, and aerodynamics in the hypersonic regime. This work focuses on the development of efficient modal solutions for structural dynamics of hypersonic vehicle structures under transient thermal loads. The problem is outlined and aerothermoelastic coupling mechanisms are described. A previously developed reducedorder, time-domain aerothermoelastic simulation framework is used as the starting point for this study. This paper focuses on three main modeling areas: 1) The effect of aerodynamic heating on the evolution of free vibration modes shapes and frequencies is examined, 2) A surrogate modeling technique is employed for directly updating the generalized stiffness matrix and thermal loads based on the transient temperature distribution and 3) Basis augmentation techniques are employed in order to obtain more accurate solutions for the structural dynamic response. The techniques to be studied are described and applied to a representative hypersonic vehicle elevator structure. Nomenclature= vector of inputs to function C(b, X) = kriging correlation matrix C = correlation matrix c = vector of POD modal coordinates c = vector of POD modal coordinates for transformed system c p = specific heat at constant pressure d = structural modal coordinates, kriging sample point E = modulus of elasticity F T = thermal load vector of full system in physical space F S = structural load vector of full system in physical space Full = solution vector of full-order model f T = generalized thermal load vector of reduced system in modal space f S = generalized structural load vector of reduced system in modal space G(X (i) , X (j) ) = Gaussian correlation function for kriging model H i = coefficient matrices in numerical integration for structural response h = altitude h i = thickness of i-th layer of thermal protection system K T = thermal conductivity matrix of full system in physical space K G = geometric stiffness matrix K S = structural stiffness matrix * Ph.D. Candidate, K * S = modified structural stiffness matrix k * S = generalized stiffness matrix of reduced system in modal space k T = generalized thermal conductivity matrix of reduced system in modal space L = lower triangular factor in decomposition of K * S L ∞ = L ∞ error l = number of augmented mode shapes lb i = lower bound for i-th POD modal coordinate M = Mach number M T = thermal capacitance matrix of full system in physical space M S = structural mass matrix of full system in physical space m T = generalized thermal capacitance matrix of reduced system in modal space m S = generalized mass matrix of reduced system in modal space MAC i,j = modal assurance criterion value corresponding to modes i and j max i = maximum value of i-th POD modal coordinate min i = minimum value of i-th POD modal coordinate MS = margin of safety with respect to POD bounds NRMSE =...

Error Quantification and Confidence Assessment of Aerothermal Model Predictions for Hypersonic Aircraft

“…This is primarily a result of the errors in p 4 and Q 4 predictions being more easily scaled as defined in Eqs. (12) and (13). Thus, calibrating the errors did result in a shift in the mean values, as seen in Fig.…”

“…2,3,[6][7][8][9][10][11][12] Neglecting these interactions can lead to gross errors in model predictions. [10][11][12][13][14][15] Aerothermoelastic aircraft structures can be modeled at multiple levels of fidelity for structural and thermal effects. However, there are limitations in computational resources, which make the degree of model fidelity and the level of coupling necessary for a particular problem play a key role in the computational tractability of the model used.…”

“…For example, aerodynamic pressure and heating could be calculated using a computational fluid dynamics (CFD) model, or the decision could be made to use reduced-order models or the simpler piston theory and Eckert's reference enthalpy method, respectively. 3,4,12,16 Substantial research has been performed on investigating the model components for the physics of a coupled aerothermoelastic panel and the solution procedures for both quasi-static and dynamic solutions. 2,3,6-12 However, the current state of the art focuses on deterministic calculations with limited uncertainty analysis.…”

Reduced-Order Aerothermoelastic Framework for Hypersonic Vehicle Control Simulation

“…This is the subject of ongoing investigation by Falkiewicz and Cesnik [74] and is not discussed further in the present work. The ROMs are now used within the aerothermoelastic framework to assess the effect of aerothermoelasticity on the total lift and drag forces acting on the control surface.…”

“…There exists potential for significant improvement in overall computational cost by using reduced-order techniques to update the generalized stiffness matrix directly based on the temperature distribution. A methodology for accomplishing this is the subject of a recent work by Falkiewicz and Cesnik [74]. …”