The analysis presented was executed in a program called Nonlinear Aeroelastic Trim and Stability for HALE Aircraft (NATASHA). NATASHA is a capability built on broad body of scientific research. Research directly related to the development of NATASHA is reviewed in the first section of this chapter. Additional efforts in the area of coupled flight dynamics and aeroelasticity are reviewed in the following section. Previous joined-wing work has established the need to capture geometric nonlinearities in the aeroelastic analysis. Literature leading to this realization is reviewed in the final section of the chapter.
The equations governing the structure in NATASHA are Hodges’ intrinsic (i.e., without displacement and rotational variables) equations for the dynamics of initially curved and twisted beams in a moving frame. Ref. Provided kinematical relations based on displacement and rotation. Hodges  later derived intrinsic kinematical equations. Because they are geometrically exact, it is the intrinsic kinematical equations that are implemented in NATASHA. This robust geometrically-exact intrinsic beam formulation allows NATASHA to handle geometric nonlinearities, including large deformations, follower forces, and buckling. Peters and Johnson presented a 2-D finite-state airloads model. Later Peters et al. Derived an inflow model based on linear potential flow. The resulting closed-form equations for the inflow and states give excellent correlation with Theodorsen and Wagner functions. Peters’ airloads and inflow model are implemented in NATASHA. In 2000, Patil et al. Published results from a precursor analysis based on the same principles. Flutter results were validated through comparison to the analytical Goland wing solution. Examples of different nonlinear aeroelastic effects were also presented. Later Patil and Hodges  developed another precursor code that helped quantify the importance of capturing geometric nonlinearities in aeroelasitic analysis of HALE aircraft. This code had the option to use either the aformentioned Peters’ aerodynamics, or a nonplanar, unsteady, fixed-wake doublet-lattice method to allow for geometrically exact representation of the aerodynamics. There was a negligible difference between the airloads calculated using the exact nonplanar wing geometry as compared to loads calculated assuming a planar wing. It was demonstrated that the dominant nonlinearity for long slender wings is dependence of the structural dynamic properties on the wing deformation. Patil and Hodges  published the baseline NATASHA formulation in 2006. Flight dynamic results where presented for the flying wing HALE model that is used in this work. An extreme sensitivity to loading (alluded to in the Helios mishap report) was confirmed. The classical phugoid mode was shown to be unstable at a sufficiently high payload. Patil and Taylor added gust functionality to NATASHA. Frequency domain continuous gust analysis was performed. Different models for the spanwise gust cross correlation were used. The spanwise non-uniform gust model predicted response many times that predicted by the spanwise uniform gust model. Patil later added a transient gust analysis capability to NATASHA. The nonlinear continuous gust response from transient analysis was compared to the continuous response from linearized frequency analysis. The results matched well for a case with relatively small disturbances. Ricciardi et al. Examined the utility of Pratt’s quasi-static gust response methodology for the Patil and Hodges developed HALE model and for a newly developed joined-wing model. A parametrized 1-cosine gust profile was used as input for the transient analysis. Results from the transient analysis were used as a baseline for evaluating the quasi-static method. It was concluded that quasi-static analysis was useful for preliminary analysis of the joined-wing model, but not useful for the flying wing HALE model. Although this work has similar conclusions, an improved transient gust analysis was used for more accurate results.
Coupled Flight Dynamics and Aeroelasticity
Milne was among the first to study flight dynamics of elastic aircraft. General equations of motion for a flexible aircraft were developed. The analysis was applied to study the static stability of an example configuration. The paper was the first to introduce the concept of mean axis for an elastic system. This concept was expanded. Longitudinal and lateral dynamics of elastic airplanes were modeled and validated experimentally by Swaim et al. The state space model expressed aerodynamic forces from elastic vibration in terms of stability derivatives based on the rigid body modes. The model showed reasonably good agreement with flight data for the B-1 bomber. Waszak and Schmidt derived the equations of motion for an elastic aircraft using Lagrange’s equation and the principle of virtual work. Equations were derived in the mean axis system assuming small deformations and a constant inertia matrix. Quasi-steady strip theory aerodynamics were used. A rigid model and simplified flexible model was derived by including residualized modes. The higher fidelity full aeroelastic model was used to quantify adequacies and deficiencies of simplifying assumptions that were used to derive lower order models. It was shown formulating with assumed elastic modes could account for static effects of elastic coupling and better approximate the dynamics of a moderately flexible vehicle. However, the residualized-mode model had significant errors if flexibility was sufficiently increased. It was also shown that the models obtained under the rigid-body assumption may not be accurate, even when the form of the desired model is that of a rigid vehicle. Crimaldi et al. Formulated a continuous, linear elastic, three dimensional gust model for the B-2 aircraft. Responses from both symmetric and asymmetric gust loading conditions were compared. It was shown that symmetric gusts produced the highest loads for that particular flying wing configuration. The work in was used by Schmidt and Raney to develop a modeling approach that could add flexibility effects to existing rigid body simulations. The required data for modeling a specific vehicle included aerodynamic stability derivatives, aerodynamic influence coefficients, elastic mode shapes, modal frequencies and damping, and generalized masses. The authors presented two case studies that involved the development of motion based simulators. Transient and continuous responses were presented as well as evaluations of elastic effects on handling characteristics. Results showed the strong impact that dynamic aeroelasticity could have on flying qualities. A need was demonstrated for aeroelastically accurate motion-based simulation facilities to facilitate the understanding of flight dynamic and flying quality characteristics of flexible aircraft. Fully nonlinear rigid body equations of motion coupled with linearized flexibility effects were developed by Reschke. The influence of inertial coupling terms on simulation and loads computation was shown using a dynamic simulation of a transport aircraft. Kier applied different aerodynamic theories to Reschke’s work and compared the computational cost and resulting loads. The gust response was calculated for a range of symmetric 1-cosine gust inputs. High sensitivity to the gust frequency was indicated. Nguyen derived aeroelastic flight dynamic equations that account for propulsive and inertial coupling. Structural modes obtained for the wing from a finite element beam model were used as generalized coordinates. Flight dynamic equations were coupled with the structural dynamics through force and moment expressions. The surveyed work up to this point of the section has been aeroelasticlly linear. There has been limited work in the area of flight dynamics coupled with nonlinear aeroelasticity. The previously discussed NATASHA work is in this exclusive category. Three additional efforts are discussed below. Drela developed ASWING, a licensed integrated analysis tool with exceptional documentation. The code uses unsteady lifting line aerodynamics and a structural model based on geometrically nonlinear isotropic beam analysis. Gust field inputs can be specified. Love et al. Used ASWING to model the behavior of a flying wing SensorCraft configuration. Results indicated that body freedom flutter is an issue for that configuration over lower altitude portions of the flight envelope. A case was made for exploring active flutter suppression systems. Recently ASWING has been modified to analyze flapping wing MAVs. Willis et al. describe a multifidelity framework that includes the new ASWING capability. Nonlinear Aeroelastic Simulation Toolbox (NAST) has been developed in a parallel effort. Cesnik and Brown introduced the strain-based structural modeling approach. NAST incorporates the previously reviewed Peters aerodynamic formulation that is also used in NATASHA. Transient gust analysis for a joined-wing configuration was demonstrated. Cesnik and Su extended NAST by adding fuselage flexibility; effects on singlewing and joined-wing SensorCraft responses were studied. It was shown that the joined-wing vehicle is more susceptible to the induced flexibility of the fuselage and tail when considering roll maneuvers. Adding flexibility to both the fuselage and vertical tail reduced the linearized flutter speed of joined-wing configuration, but it did not significantly impact the flutter speed of the single-wing configuration. It was also noted that adding flexibility to the fuselage decreased buckling speed, but adding the flexibility to the vertical tail increased it for the case studied. Shearer and Cesnik used NAST to show that nonlinear flight dynamics coupled with linearized aeroelastic equations could significantly differ from solutions using the fully nonlinear system. Therefore fully nonlinear simulations are required to accurately predict the response for some cases. Su and Cesnik performed gust response analysis on the Patil/Hodges flying wing HALE. A model to capture the change in the wing torsional stiffness due to skin wrinkling arising from large bending curvatures was demonstrated. It was found that skin wrinkling associated with the wing torsional stiffness significantly affected motions of the vehicle only in the lateral direction. Finally, Wang et al. used the previously discussed Hodges beam modeling coupled with unsteady vortex lattice aerodynamics. The formulation was demonstrated for a Goland wing model and the Patil/Hodges HALE model. Unique transient results of the spanwise deformed wake were presented for an extreme HALE case. Trim results were similar to those reported by Patil and Hodges.
1.3 Background – Gust Response Analysis
1.4 Background – Geometric Nonlinearity
2 Literature Review
2.1 NATASHA Foundations
2.2 Coupled Flight Dynamics and Aeroelasticity
2.3 Joined-Wing Geometric Nonlinearities
3.1 Pratt’s Quasi-Static Method
3.2 Evaluation of Error in Pratt’s Curve Fit Equation
3.3 Quasi-Static and Transient Gust Analysis
3.4 Gust Response Analysis Verification
3.5 HALE Applications
3.6 Computational Framework
4 Results and Discussion
4.1 Pratt’s Curve Fit Error
4.2 Joined-Wing Parameter Interpretation
4.3 Application Dependent Quasi-Static Error’s
5 Conclusions and Future Work
5.2 Future Work
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