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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 91 records · Page 5
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NASA NTRS2007

2nd ISS Treadmill Development "T2 Project"

An overview of the development of a treadmill for the International Space Station is presented. Topics discussed include: flight certification of a Commercial Off the Shelf (COTS) Woodway Path treadmill; development and certificaiton of a crew interface to optimize use of the COTS design and/or existing NASA design (such as the ARED Pacebook); development and certification of a power supply to provide power from the ISS Vehicle to the treadmill system (crew interfaces, motor, controller and subject loading devices).

MacNeill, Kevin↗
NASA NTRS2017

Computationally Efficient Motion Planning Algorithms for Agile Autonomous Vehicles in Cluttered Environments

Fast, real-time motion planning of an agile, autonomous vehicle in a cluttered environment, with many geometrically-fixed obstacles, is a very complex problem, especially because of the vehicle dynamics constraints and resource constrained computational capabilities onboard the vehicle. In this paper, we present computationally-efficient versions of our novel motion planning algorithm called the Spherical Expansion and Sequential Convex Programming (SE–SCP) algorithm. The SE–SCP algorithm first uses a spherical-expansion-based randomized sampling algorithm to explore the workspace. Oncea path is found from the start position to the goal position, the algorithm computes a locally optimal trajectory, within its homotopy class for a desired cost function, by solving a sequence of convex optimization problems. Thus, the SE–SCP algorithm is anytime locally optimal and the trajectory is globally optimal if the number of samples tends to infinity. In this paper, we further enhance the computational efficiency of the SE–SCP algorithm using uni-directional and bi-directional rewiring techniques. We also present a detailed proof of the local optimality characteristics of the new SE–SCP algorithms for aspecial case of vehicle dynamics. Simulation examples involving quadrotor and spacecraft help demonstrate the effectiveness of our new algorithms.

Bandyopadhyay, Saptarshi↗
NASA NTRS1993

A hypersonic vehicle approach to planetary exploration

An enhanced Mars network class mission using a lifting hypersonic entry vehicle is proposed. The basic vehicle, derived from a mature hypersonic flight system called SWERVE, offers several advantages over more conventional low L/D or ballistic entry systems. The proposed vehicle has greatly improved lateral and cross range capability (e.g., it is capable of reaching the polar regions during less than optimal mission opportunities), is not limited to surface target areas of low elevation, and is less susceptible to problems caused by Martian dust storms. Further, the integrated vehicle has attractive deployment features and allows for a much improved evolutionary path to larger vehicles with greater science capability. Analysis of the vehicle is aided by the development of a Mars Hypersonic Flight Simulator from which flight trajectories are obtained. Atmospheric entry performance of the baseline vehicle is improved by a deceleration skirt and transpiration cooling system which significantly reduce TPS (Thermal Protection System) and flight battery mass. The use of the vehicle is also attractive in that the maturity of the flight systems make it cost-competitive with the development of a conventional low L/D entry system. Finally, the potential application of similar vehicles to other planetary missions is discussed.

Murbach, Marcus S.↗
NASA NTRS2008

Rapid Trajectory Optimization for the ARES I Launch Vehicle

A simplified ascent trajectory optimization procedure has been developed with application to NASA's proposed Ares I launch vehicle. In the interest of minimizing bending loads and ensuring safe separation of the first-stage solid rocket motor, the vehicle is con- strained to follow a gravity-turn trajectory. This reduces the design space to just two free parameters, the pitch rate after a short vertical rise phase to clear the launch pad, and initial launch azimuth. The pitch rate primarily controls the in-plane parameters (altitude, speed, flight path angle) of the trajectory whereas the launch azimuth primarily controls the out-of-plane portion (velocity heading.) Thus, the optimization can be mechanized as two one-dimensional searches that converge quickly and reliably. The method is compared with POST-optimized trajectories to verify its optimality.

Dukeman, Greg A.↗
NASA NTRS2010

Cabin Noise Studies for the Orion Spacecraft Crew Module

Controlling cabin acoustic noise levels in the Crew Module (CM) of the Orion spacecraft is critical for adequate speech intelligibility, to avoid fatigue and to prevent any possibility of temporary and permanent hearing loss. A vibroacoustic model of the Orion CM cabin has been developed using Statistical Energy Analysis (SEA) to assess compliance with acoustic Constellation Human Systems Integration Requirements (HSIR) for the on-orbit mission phase. Cabin noise in the Orion CM needs to be analyzed at the vehicle-level to assess the cumulative acoustic effect of various Orion systems at the crewmember's ear. The SEA model includes all major structural and acoustic subsystems inside the CM including the Environmental Control and Life Support System (ECLSS), which is the primary noise contributor in the cabin during the on-orbit phase. The ECLSS noise sources used to excite the vehicle acoustic model were derived using a combination of established empirical predictions and fan development acoustic testing. Baseline noise predictions were compared against acoustic HSIR requirements. Key noise offenders and paths were identified and ranked using noise transfer path analysis. Parametric studies were conducted with various acoustic treatment packages in the cabin to reduce the noise levels and define vehicle-level mass impacts. An acoustic test mockup of the CM cabin has also been developed and noise treatment optimization tests were conducted to validate the results of the analyses.

Dandaroy, Indranil↗
NASA NTRSDate not supplied

Physics and Modeling of Quasi-Steady and Forced Shock Trains

The dual-mode scramjet is a propulsive concept for contemporary hypersonic vehicles promising flight cruise speeds upwards of Mach 5. Despite decades of development and a recent resurgence in interest, such engines are not routinely deployed for commercial or military purposes. Instead, they have been confined to the realm of ground-test research and limited one-off flight test missions. One of the key stumbling blocks on the critical path to deployment of a dual-mode scramjet is the optimal design and stable operation of the isolator, a typically constant area duct that separates the scramjet inlet from the combustor. During ramjet mode at Mach numbers from 3 to 5 and during the transition to scramjet mode at higher Mach numbers, the isolator contains the pseudo-shock. Necessary to meet the back-pressure in the combustor, the pseudo-shock is typically divided into a shock dominated shock train region followed by a turbulence driven mixing region. Fluctuations in combustor pressure due to vehicle maneuvering, throttling, or combustion instabilities drive the shock train to shift streamwise within the isolator. In response to sharp pressure fluctuations, the shock train is unable to stabilize within the isolator, and the scramjet unstarts by ejecting the shock train out the inlet. This work addresses three key gaps in understanding of the isolator pseudo-shock. The phenomenon is studied in three experimental facilities to investigate the quasi-steady and unsteady fluid dynamics of the shock train in fueled and un-fueled test conditions. The strong link between boundary layer morphology and shock train structure is studied in detail and used as a tool to evaluate the state-of-the-art definitions and models of the pseudo-shock. Identifying the limitations of these models, a new process driven flux-conserved pseudo-shock model is proposed. This provides a computationally efficient model grounded in the fluid dynamic mechanisms of the pseudo-shock with better predictive capability than the current state-of-the-art across all facilities tested. The unsteady behavior of the shock train is further investigated to identify and understand two distinct forced dynamic modes: mechanically induced, low-rate forcing and high-rate combustion driven unsteadiness that sheds upstream propagating shocks. The proposed process driven model is found to be effective in the the limiting case of high-rate combustion driven unsteadiness by evaluating the model in a shock stationary reference frame. Further analysis identifies the approximately rigid chain motion of the shock train structure to mechanically induced forcing. The time delay between the onset of forced dynamics is found to saturate to a constant value and maximum shock train speed along the isolator with respect to the perturbation rate of change. The result of this work provides dual-mode scramjet designers with improved tools and physical understanding to develop more efficient, robust, and capable propulsion systems for hypersonic vehicles.

JAXA↗
NASA NTRSDate not supplied

NASA's Space Launch System Begins Integration, Stacking in Preparation for Artemis I Launch

The Artemis era of human lunar exploration is nearing take-off as NASA’s new super heavy-lift launch vehicle, the Space Launch System (SLS), begins stack-ing and integration operations in mid-2020 at Kennedy Space Center (KSC) in Florida. With a planned upgrade path to progressively more powerful vehicles and availability in crew and cargo configurations, SLS provides a unique and flexible launch solution to send crew, large-scale infrastructure and robotic probes to deep space. The SLS Block 1 vehicle, the initial variant to fly, is optimized for lunar missions with a proven propulsion system consisting of four liquid hydrogen (LH2)/liquid oxygen (LOX)-fed RS-25 engines and twin five-segment solid rocket boosters (SRBs). The Block 1 vehicle can also be outfitted with an industry-standard 5 m-class payload fairing (the “cargo” configuration) and will launch at least 27 metric tons (t) of mass to trans-lunar injection (TLI). SLS is the backbone of NASA’s Artemis program, which will return the agency’s human spaceflight program to the Moon for the first time since 1972. For the Artemis I mission, SLS will send an uncrewed Orion spacecraft to TLI, where it will enter a distant retrograde lunar orbit and fly 38,000 nmi past the Moon – farther than any spacecraft built for humans has ever traveled. The SLS Block 1 vehicle for Artemis I completed manufacturing in 2019. Several elements, including the upper stage, have been delivered to the Exploration Ground Systems (EGS) program at KSC and are being prepped for integration and stack-ing. The five-segment solid rocket boosters – the largest and most powerful ever built for flight – are also complete. The booster motor segments for the Artemis I flight are scheduled to ship from prime contractor Northrop Grumman’s Utah facilities and begin stacking and integration at KSC in June 2020. The SLS core stage is the largest rocket stage NASA has ever built in terms of volume and height, and includes the avionics and the tanks that feed cryogenic propellant to the four RS-25s (formerly Space Shuttle Main Engines [SSMEs]). They have been modified with an updated controller and nozzle insulation to protect them from the hotter launch environment. The SLS core stage is currently being test-ed at NASA’s Stennis Space Center (SSC) in a series of “green run” tests to verify it meets design and performance requirements. Following the green run test series, which is scheduled to culminate with a full-duration hot-fire of the four RS-25 engines, the core stage will ship to KSC and be stacked between the sol-id rocket boosters in the Vehicle Assembly Building (VAB). Integration of the vehicle will continue with the upper stage, known as the Interim Cryogenic Propulsion Stage (ICPS) and the Launch Vehicle Stage Adapter (LVSA) on the core stage. Another adapter, the Orion Stage Adapter (OSA), connects SLS to Orion and provides housing for 13 6U CubeSat payloads manifested on Artemis I. The CubeSats will be released in deep space after Orion separates from the vehicle, and the flight marks the first ride share opportunity for independent small-sats to deep space. The second major SLS variant to come online, Block 1B, replaces the single-engine ICPS with a four-engine LH2/LOX Exploration Upper Stage (EUS). This more powerful upper stage, along with other vehicle up-grades, will enable the Block 1B vehicle to launch 38-42 t to TLI, depending on crew or cargo configuration. The final evolution of the vehicle, Block 2, will onramp evolved solid rocket boosters to increase mass to TLI to 43-46 t, de-pending on crew or cargo configuration. The Block 1B/Block 2 vehicles can be outfitted with an 8.4 m-diameter payload fairing in 19.1 m or 27.4 m lengths, to provide unprecedented volume for payloads. With the initial Block 1 vehicle completely manufactured and the core stage in final testing before shipping to KSC, the SLS Program and its industry partners have made significant progress manufacturing subsequent vehicles. For the second Block 1 vehicle, the solid rocket motor segments are complete, as are the RS-25 engines with controllers. All five major components of the Artemis II core stage – the forward skirt, LOX and LH2 tanks, intertank and engine section – are manufactured and technicians are installing subsystems at NASA’s rocket factory, Michoud Assembly Facility. The RL-10 engine for the Artemis II ICPS is complete and panels have been machined for its LH2 tank. In addition, panels are machined for the vehicle’s two adapters, with welding scheduled to begin in summer 2020. Flight hard-ware is also in production for the third SLS vehicle, with several booster motor segments cast. The pace of development on the EUS has increased, with the goal to complete Critical Design Review (CDR) in December 2020. Several EUS test rings have been machined at Michoud. The EUS is designed to exe-cute a variety of missions – human spaceflight, deployment of deep-space infra-structure, or high-C3 missions to the outer solar system – with crew and cargo configurations available beginning in the mid-2020s. The near-term goal for the nation’s powerful new space exploration asset, however, is to launch the Arte-mis program, and send the first woman and the next man to the lunar surface. At the Astrodynamics Specialist Conference, the SLS program will update the community on the progress of the initial Block 1 vehicle in final green run test-ing, integration and stacking. In addition, this paper will provide an update to the community on the manufacturing status of subsequent Block 1 and Block 1B vehicles.

Steve Creech↗
NASA NTRS1985

Predictor laws for pictorial flight displays

Two predictor laws are formulated and analyzed: (1) a circular path law based on constant accelerations perpendicular to the path and (2) a predictor law based on state transition matrix computations. It is shown that for both methods the predictor provides the essential lead zeros for the path-following task. However, in contrast to the circular path law, the state transition matrix law furnishes the system with additional zeros that entirely cancel out the higher-frequency poles of the vehicle dynamics. On the other hand, the circular path law yields a zero steady-state error in following a curved trajectory with a constant radius. A combined predictor law is suggested that utilizes the advantages of both methods. A simple analysis shows that the optimal prediction time mainly depends on the level of precision required in the path-following task, and guidelines for determining the optimal prediction time are given.

Grunwald, A. J.↗
NASA NTRS1990

Rapid near-optimal trajectory generation and guidance law development for single-stage-to-orbit airbreathing vehicles

General problems associated with on-board trajectory optimization, propulsion system cycle selection, and with the synthesis of guidance laws were addressed for an ascent to low-earth-orbit of an air-breathing single-stage-to-orbit vehicle. The NASA Generic Hypersonic Aerodynamic Model Example and the Langley Accelerator aerodynamic sets were acquired and implemented. Work related to the development of purely analytic aerodynamic models was also performed at a low level. A generic model of a multi-mode propulsion system was developed that includes turbojet, ramjet, scramjet, and rocket engine cycles. Provisions were made in the dynamic model for a component of thrust normal to the flight path. Computational results, which characterize the nonlinear sensitivity of scramjet performance to changes in vehicle angle of attack, were obtained and incorporated into the engine model. Additional trajectory constraints were introduced: maximum dynamic pressure; maximum aerodynamic heating rate per unit area; angle of attack and lift limits; and limits on acceleration both along and normal to the flight path. The remainder of the effort focused on required modifications to a previously derived algorithm when the model complexity cited above was added. In particular, analytic switching conditions were derived which, under appropriate assumptions, govern optimal transition from one propulsion mode to another for two cases: the case in which engine cycle operations can overlap, and the case in which engine cycle operations are mutually exclusive. The resulting guidance algorithm was implemented in software and exercised extensively. It was found that the approximations associated with the assumed time scale separation employed in this work are reasonable except over the Mach range from roughly 5 to 8. This phenomenon is due to the very large thrust capability of scramjets in this Mach regime when sized to meet the requirement for ascent to orbit. By accounting for flight path angle and flight path angle rate in construction of the flight path over this Mach range, the resulting algorithm provides the means for rapid near-optimal trajectory generation and propulsion cycle selection over the entire Mach range from take-off to orbit.

Calise, A. J.↗
NASA NTRS2018

Auralization of an Unmanned Aerial Vehicle Under Propeller Phase Control

Community noise has been identified as a major barrier to entry into service of unmanned aerial vehicle platforms. Common among such vehicle architectures is a distributed electric propulsion system, in which many propellers and/or rotors are used for thrust and/or lift. A promising noise reduction technique was recently developed to favorably modify the vehicle source noise directivity, by adjusting the relative azimuthal position of the prop/rotor blades between propulsors. Un- der idealized conditions, such phase control can yield noise reductions on the order of 20 dB at a ground observer relative to the average case having a random phase distribution. However, human perception of the resulting noise may not be fully captured by such a simple acoustic measure. To this end, this paper under- takes the auralization of the tonal vehicle noise. Specifically, total propeller tonal noise is optimized over a set of observer zones, e.g., 45 deg. wedges, around the vehicle. Each zone represents a unique set of relative phases. The forward flight of the vehicle adjacent to a noise-sensitive area is simulated by prescribing a series of zones over the flight path having minimum noise in the direction of the ob- server. Auralizations and their associated metrics are compared with a reference condition.

Pascioni, Kyle A.↗
NASA NTRSDate not supplied

Adaptive Optimization for System Performance and Combined Bernstein Polynomial, Optimal Reciprocal Collision Avoidance, Differential Dynamic Programming for Trajectory Replanning and Collision Avoidance for UAM Vehicles

The emerging urban air mobility (UAM) sector in aerospace is driving development of unconventional multi-modal vehicle configurations and autonomous flight. The combination of multi-modal vehicle dynamics, complex environment, requirements to deal with flight contingencies in an efficient and safe manner, as well as necessity for precise trajectory following and performance, are the driving influence behind adaptive optimization for system performance. We are interested in trajectory optimization algorithm that would system parameter estimation and identifying the optimal switching time between modes of hybrid dynamical systems. This presentation discusses a parameterized optimal control trajectory optimization algorithm that is an extended and generalized version of Differential Dynamic Programming (DDP), titled Parameterized Differential Dynamic Programming (PDDP). DDP is an efficient trajectory optimization algorithm relying on second order approximations of a system’s dynamics and cost function and has recently been applied to optimize systems with time invariant parameters. Experiments are presented applying PDDP to solve model predictive control (MPC) and moving horizon estimation (MHE) tasks simultaneously. In particular, PDDP is used to determine the optimal transition point between flight regimes of a complex urban air mobility (UAM) class vehicle exhibiting multiple phases of flight and to identify and compensate for actuation faults.

optimization↗
NASA NTRS2011

Multidisciplinary Modeling Software for Analysis, Design, and Optimization of HRRLS Vehicles

The concept for Highly Reliable Reusable Launch Systems (HRRLS) under the NASA Hypersonics project is a two-stage-to-orbit, horizontal-take-off / horizontal-landing, (HTHL) architecture with an air-breathing first stage. The first stage vehicle is a slender body with an air-breathing propulsion system that is highly integrated with the airframe. The light weight slender body will deflect significantly during flight. This global deflection affects the flow over the vehicle and into the engine and thus the loads and moments on the vehicle. High-fidelity multi-disciplinary analyses that accounts for these fluid-structures-thermal interactions are required to accurately predict the vehicle loads and resultant response. These predictions of vehicle response to multi physics loads, calculated with fluid-structural-thermal interaction, are required in order to optimize the vehicle design over its full operating range. This contract with ResearchSouth addresses one of the primary objectives of the Vehicle Technology Integration (VTI) discipline: the development of high-fidelity multi-disciplinary analysis and optimization methods and tools for HRRLS vehicles. The primary goal of this effort is the development of an integrated software system that can be used for full-vehicle optimization. This goal was accomplished by: 1) integrating the master code, FEMAP, into the multidiscipline software network to direct the coupling to assure accurate fluid-structure-thermal interaction solutions; 2) loosely-coupling the Euler flow solver FEFLO to the available and proven aeroelasticity and large deformation (FEAP) code; 3) providing a coupled Euler-boundary layer capability for rapid viscous flow simulation; 4) developing and implementing improved Euler/RANS algorithms into the FEFLO CFD code to provide accurate shock capturing, skin friction, and heat-transfer predictions for HRRLS vehicles in hypersonic flow, 5) performing a Reynolds-averaged Navier-Stokes computation on an HRRLS configuration; 6) integrating the RANS solver with the FEAP code for coupled fluid-structure-thermal capability; and 7) integrating the existing NASA SRGULL propulsion flow path prediction software with the FEFLO software for quasi-3D propulsion flow path predictions, 8) improving and integrating into the network, an existing adjoint-based design optimization code.

Spradley, Lawrence W.↗
NASA NTRS1983

Minimum-fuel aeroassisted coplanar orbit transfer using lift-modulation

Minimum-fuel trajectories and lift controls are computed for aeroassisted coplanar transfer from high orbit to low orbit. The optimal aeroassisted transfer requires less fuel than the all-propulsive Hohmann transfer for a wide range of high orbit to low orbit transfers. The optimal control program for the atmospheric portion of the transfer is to fly at maximum positive L/D initially to recover from the downward plunge, and then, to fly at negative L/D to level off the flight, such that the vehicle skips out of the atmosphere with a flight path angle near zero degrees. To avoid excessive heating rates, the vehicle flies initially at high angle of attack in order to slow down higher in the atmosphere, allowing recovery from the downward plunge, which occurs subsequently using the maximum positive L/D, to take place at a lower atmospheric density, or equivalently, at a higher altitude.

Mease, K. D.↗
NASA NTRS2011

Launch Vehicle Demonstrator Using Shuttle Assets

The Advanced Concepts Office at NASA's George C. Marshall Space Flight Center undertook a study to define candidate early heavy lift demonstration launch vehicle concepts derived from existing space shuttle assets. The objective was to determine the performance capabilities of these vehicles and characterize potential early demonstration test flights. Given the anticipated budgetary constraints that may affect America's civil space program, and a lapse in U.S. heavy launch capability with the retirement of the space shuttle, an early heavy lift launch vehicle demonstration flight would not only demonstrate capabilities that could be utilized for future space exploration missions, but also serve as a building block for the development of our nation s next heavy lift launch system. An early heavy lift demonstration could be utilized as a test platform, demonstrating capabilities of future space exploration systems such as the Multi Purpose Crew Vehicle. By using existing shuttle assets, including the RS-25D engine inventory, the shuttle equipment manufacturing and tooling base, and the segmented solid rocket booster industry, a demonstrator concept could expedite the design-to-flight schedule while retaining critical human skills and capital. In this study two types of vehicle designs are examined. The first utilizes a high margin/safety factor battleship structural design in order to minimize development time as well as monetary investment. Structural design optimization is performed on the second, as if an operational vehicle. Results indicate low earth orbit payload capability is more than sufficient to support various vehicle and vehicle systems test programs including Multi-Purpose Crew Vehicle articles. Furthermore, a shuttle-derived, hydrogen core vehicle configuration offers performance benefits when trading evolutionary paths to maximum capability.

Creech, Dennis M.↗
NASA NTRS1973

A model based technique for the design of flight directors

A new technique for designing flight directors is discussed. This technique uses the optimal-control pilot/vehicle model to determine the appropriate control strategy. The dynamics of this control strategy are then incorporated into the director control laws, thereby enabling the pilot to operate at a significantly lower workload. A preliminary design of a control director for maintaining a STOL vehicle on the approach path in the presence of random air turbulence is evaluated. By selecting model parameters in terms of allowable path deviations and pilot workload levels, a set of director laws is achieved which allows improved system performance at reduced workload levels. The pilot acts essentially as a proportional controller with regard to the director signals, and control motions are compatible with those appropriate to status-only displays.

Levison, W. H.↗
NASA NTRS2015

Systems Engineering Management Plan NASA Traffic Aware Planner Integration Into P-180 Airborne Test-Bed

NASA's Traffic Aware Planner (TAP) is a cockpit decision support tool that provides aircrew with vertical and lateral flight-path optimizations with the intent of achieving significant fuel and time savings, while automatically avoiding traffic, weather, and restricted airspace conflicts. A key step towards the maturation and deployment of TAP concerned its operational evaluation in a representative flight environment. This Systems Engineering Management Plan (SEMP) addresses the test-vehicle design, systems integration, and flight-test planning for the first TAP operational flight evaluations, which were successfully completed in November 2013. The trial outcomes are documented in the Traffic Aware Planner (TAP) flight evaluation paper presented at the 14th AIAA Aviation Technology, Integration, and Operations Conference, Atlanta, GA. (AIAA-2014-2166, Maris, J. M., Haynes, M. A., Wing, D. J., Burke, K. A., Henderson, J., & Woods, S. E., 2014).

Maris, John↗
NASA NTRS1989

Analysis of a turning point problem in flight trajectory optimization

The optimal control policy for the aeroglide portion of the minimum fuel, orbital plane change problem for maneuvering entry vehicles is reduced to the solution of a turning point problem for the bank angle control. For this problem a turning point occurs at the minimum altitude of the flight, when the flight path angle equals zero. The turning point separates the bank angle control into two outer solutions that are valid away from the turning point. In a neighborhood of the turning point, where the bank angle changes rapidly, an inner solution is developed and matched with the two outer solutions. An asymptotic analysis of the turning point problem is given, and an analytic example is provided to illustrate the construction of the bank angle control.

Gracey, C.↗
NASA NTRS1988

An approximate atmospheric guidance law for aeroassisted plane change maneuvers

An approximate optimal guidance law for the aeroassisted plane change problem is presented which is based upon an expansion of the Hamilton-Jacobi-Bellman equation with respect to the small parameter of Breakwell et al. (1985). The present law maximizes the final velocity of the reentry vehicle while meeting terminal constraints on altitude, flight path angle, and heading angle. The integrable zeroth-order solution found when the small parameter is set to zero corresponds to a solution of the problem where the aerodynamic forces dominate the inertial forces. Higher order solutions in the expansion are obtained from the solution of linear partial differential equations requiring only quadrature integration.

Speyer, Jason L.↗