Eli Livne

Equivalent plate structural modeling for wing shape optimization including transverse shear

A new technique for structural modeling of airplane wings is presented taking transverse shear effects into account. The kinematic assumptions of first-order shear deformation plate theory in combination with numerical analysis, where simple polynomials are used to define geometry, construction, and displacement approximations, lead to analytical expressions for elements of the stiffness and mass matrices and load vector. Contributions from the cover skins, spar and rib caps, and spar and rib webs are included as well as concentrated springs and concentrated masses. Limitations of wing modeling techniques based on classical plate theory are discussed, and the improved accuracy of the new equivalent plate technique is demonstrated through comparison with finite element analysis and test results. Expressions for analytical derivatives of stiffness, mass, and load terms with respect to wing shape are given. Based on these, it is possible to obtain analytic sensitivities of displacements , stresses, and natural frequencies with respect to planform shape and depth distribution. This makes the new capability an effective structural tool for wing shape optimization.

Aeroservoelastic aspects of wing/control surface planform shape optimization

Equivalent plate structural modeling and doublet point lifting surface unsteady aerodynamics are used to obtain analytic sensitivities of aeroelastic and aeroservoelastic response with respect to wing and control surface planform shape parameters. Rational function approximations for unsteady aerodynamic forces, their shape sensitivities, and the resulting linear time invariant state space models of aeroservoelastic systems and their shape sensitivities are examined. The goal is to develop effective and numerically efficient approximation techniques for wing shape optimization for use with nonlinear programming and approximation concepts as a multidisciplinary optimization strategy. Effects of structural and unsteady aerodynamic modeling errors are studied. Examination of approximation accuracy using alternative approximation techniques (and the resulting move limits) provide insight and experience on the way to realistic wing/control surface shape optimization with active controls and aeroservoelastic constraints.

Alternative Approximations for Integrated Control/Structure Aeroservoelastic Synthesis

A discussion of alternative complex pole and gust response approximations is presented, motivated by the need to extend nonlinear programming/approximation concepts optimization methodology from structural synthesis to the integrated control/structure synthesis. Two new approximations are presented for the high-order, weakly damped linear time-invariant systems representing actively controlled airplanes in high-speed flight. They are compared with Taylor series, differential equations, and Rayleigh quotient approximations

Accurate calculation of control-augmented structural eigenvalue sensitivities using reduced-order models

A method is presented for generating mode shapes for model order reduction in a way that leads to accurate calculation of eigenvalue derivatives and eigenvalues for a class of control augmented structures. The method is based on treating degrees of freedom where control forces act or masses are changed in a manner analogous to that used for boundary degrees of freedom in component mode synthesis. It is especially suited for structures controlled by a small number of actuators and/or tuned by a small number of concentrated masses whose positions are predetermined. A control augmented multispan beam with closely spaced natural frequencies is used for numerical experimentation. A comparison with reduced-order eigenvalue sensitivity calculations based on the normal modes of the structure shows that the method presented produces significant improvements in accuracy.

Design-Oriented Thermostructural Analysis with Internal and External Radiation, Part 1: Steady State

This paper presents a steady-state sensitivity analysis formulation covering configuration shape and structural sizing design variables for coupled linear structural analysis and nonlinear thermal analysis including the effects of radiation: in external and internal cavities. A new coupled design-oriented thermoelastic analysis capability is first presented and validated by using an available commercial analysis code. It is then used to calculate shape sensitivities and assess a number of standard approximation techniques in an effort to gain insight regarding the functional relations between design variables and thermoelastic responses. A novel approximation scheme is presented for the category of problems discussed that lead to substantial savings in central processing unit time. A hypersonic wing structure similar to a space shuttle wing is used as a test case.

Reduced-Order Design-Oriented Stress Analysis Using Combined Direct and Adjoint Solutions

A new method for extracting accurate stress information from reduced-order structural and aeroelastic models is presented. The method has second-order accuracy when approximate reduced-order direct and adjoint solutions (based on different reduced-order bases) are used simultaneously to obtain approximate stresses. The method is applicable to both static and dynamic linear analysis. A review of four common methods for structural model order reduction [two variants of the mode displacement method (standard mode displacement and the fictitious mass method), the mode acceleration method, and the Ritz vector method] identifies sources of difficulty and causes of errors in stress behavior sensitivity calculations. Considerations used for selection of the reduced-order direct and adjoint bases are discussed. A series of static and dynamic test cases is used to assess accuracy of the new method in an analysis mode. Accuracy studies of sensitivity calculations follow. We hope to contribute to the field of design-oriented structural dynamics in terms of both insight and practice.

Design-Oriented Thermostructural Analysis with External and Internal Radiation, Part 2: Transient Response

This is the second paper in a two-part series presenting a sensitivity-analysis formulation covering configuration shape and structural sizing design variables for nonlinear thermostructural analysis including the effects of radiation: external and in internal cavities. Part 1 presented the steady-state coupled thermal-structural formulation with results from a new coupled design-oriented thermoelastic analysis capability. The present paper considers transient heat transfer analysis problems with temperature-dependent material properties. Structural response is assumed to be quasi-steady. Because structural-thermal integration had been demonstrated earlier, the focus here is on design-oriented analysis aspects of the conduction-radiation problem. An approximation scheme is presented for the category of problems discussed that leads to a reduction of CPU cost from the order of N 3 operations to N 2 for cavity radiation analysis. A hypersonic wing structure similar to a space shuttle wing is used as a test case.

Structural Dynamic Frequency Response Using Combined Direct and Adjoint Reduced-Order Approximations

A second-order combined approximate-direct/approximate-adjoint method for efficient reduced-order calculation of frequency response in structural dynamics is presented. The method is design oriented. It focuses not only on the response itself (in the form of displacements and, especially, stresses) but also on sensitivities of the response with respect to structural design variables. Comparisons of approximate results obtained by mode-displacement order reduction, reduced-order adjoint solutions based on Ritz vectors, and the second-order approximate method demonstrate the accuracy of the second-order method, and provide lessons that suggest directions for future research.

Thermoelastic Buckling and Vibration Analysis for Shape Optimization of Thin Walled Aerospace Structures

A design oriented buckling and natural frequency analysis of thermoelastic thin walled aerospace structures is presented. This includes sensitivity analysis and assessment of the performance of a number of standard approximation techniques. The eect of thermal stresses on buckling and vibration eigenvalues is taken into account through geometricstiness and material property variation. Thermal analysis includes the eects of convective cooling and radiation: external and in internal cavities. Test cases include aerospace structures with temperature dependent material properties. Nomenclature qi Heat flux in the xi direction due to conduction qv Internal volume heat generation qs