Search NASA⌕ Search

SEARCH · Search NASA

Results for “Lagrange”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 145 records · Page 8
All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Connection forces in deformable multibody dynamics

In the dynamic formulation of holonomic and nonholonomic systems based on D'Alembert-Lagrange equation, the forces of constraints are maintained in the dynamic equations by introducing auxiliary variables, called Lagrange multipliers. This approach introduces a set of generalized reaction forces associated with the system generalized coordinates. Different sets of variables can be used as generalized coordinates and accordingly, the generalized reactions associated with these generalized coordinates may not be the actual reaction forces at the joints. In rigid body dynamics, the generalized reaction forces and the actual reaction forces at the joints represent equipollent systems of forces since they produce the same total forces and moments at and about any point on the rigid body. This is not, however, the case in deformable body analyses wherein the generalized reaction forces depend on the system generalized reference and elastic coordinates. In this paper, a method for determining the actual reaction forces at the joints from the generalized reaction forces in deformable multibody systems is presented.

Shabana, A. A.↗
NASA NTRS1990

Simulation and analysis of flexibly jointed manipulators

Modeling, simulation, and analysis of robot manipulators with non-negligible joint flexibility are studied. A recursive Newton-Euler model of the flexibly jointed manipulator is developed with many advantages over the traditional Lagrange-Euler methods. The Newton-Euler approach leads to a method for the simulation of a flexibly jointed manipulator in which the number of computations grows linearly with the number of links. Additionally, any function for the flexibility between the motor and link may be used permitting the simulation of nonlinear effects, such as backlash, in a uniform manner for all joints. An analysis of the control problems for flexibly jointed manipulators is presented by converting the Newton-Euler model to a Lagrange-Euler form. The detailed structure available in the model is used to examine linearizing controllers and shows the dependency of the control on the choice of flexible model and structure of the manipulator.

Murphy, Steve H.↗
NASA NTRS1992

Application of Newton modified barrier method (NMBM) to structural optimization

This paper presents the application of the NMBM to obtain a minimum weight structure with constraints on displacements and minimum sizes. The solution to the problem is obtained via minimizing the Modified Barrier Function (MBF) at each step by using the Newton Method and updating Lagrange multipliers. The Lagrange multipliers are updated by using the value of the constraints at the minimum of the MBF. Three truss problems with a different number of design variables are solved. The convergence to the minimum weight design was found to be monotonic and the algorithm is potentially robust for solving problems with a large number of design variables.

Khot, N. S.↗
NASA NTRS1984

Equations of motion for control of the SCOLE laboratory experiment

The objectives of this study are listed as follows: (1) to develop Lagrange's equations of motion for the shuttle antenna configuration in orbit; (2) to modify equations using the Lagrange multiplier method to develop equations of motion for the laboratory experiment; and (3) to discuss methods for simulation and control. The equations are presented in graph form.

Meirovitch, L.↗
NASA NTRS1994

Eliminating Computational Instability In Multibody Simulations

TWOBODY implements improved version of Lagrange multiplier method. Program ultilizes programming technique eliminating computational instability in multibody simulations in which Lagrange multipliers used. In technique, one uses constraint equations, instead of integration, to determine coordinates that are not independent. To illustrate technique, it includes simple mathematical model of solid rocket booster and parachute connected by frictionless swivel. Written in FORTRAN 77.

Watts, Gaines L.↗
NASA NTRS1995

Sensitivity analysis of frictional contact response of axisymmetric composite structures

A computational procedure is presented for evaluating the sensitivity coefficients of the static frictional contact response of axisymmetric composite structures. The structures are assumed to consist of an arbitrary number of perfectly bonded homogeneous anisotropic layers. The material of each layer is assumed to be hyperelastic, and the effect of geometric nonlinearity is included. The sensitivity coefficients measure the sensitivity of the response variations in different material, lamination and geometric parameters of the structure. A displacement finite element model is used for the discretization. The normal contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with the fundamental unknowns consisting of nodal displacements, and Lagrange multipliers associated with the contact conditions. The Lagrange multipliers are allowed to be discontinuous at interelement boundaries. Tangential contact conditions are incorporated by using a penalty method in conjunction with the classical Coulomb's friction model. The Newton-Raphson iterative scheme is used for the solution of the resulting nonlinear algebraic equations, and for the determination of the contact region, contact conditions (sliding or sticking), and the contact pressures. The sensitivity coefficients are evaluated by using a direct differentiation approach. Numerical results are presented for the frictional contact of a composite spherical cap pressed against a rigid plate.

Karaoglan, I.↗
NASA NTRS1995

Dynamic sensitivity analysis of frictional contact/impact response of axisymmetric composite structures

A computational procedure is presented for evaluating the sensitivity coefficients of the dynamic frictional contact/impact response of axisymmetric composite structures. The structures are assumed to consist of an arbitrary number of perfectly bonded homogeneous anisotropic layers. The material of each layer is assumed to be hyperelastic, and the effect of geometric nonlinearity is included. The sensitivity coefficients measure the sensitivity of the response to variations in different material, lamination and geometric parameters of the structure. A displacement finite element model is used for the discretization. The normal contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with the fundamental unknowns consisting of the nodal displacements, and the Lagrange multipliers associated with the contact conditions. The Lagrange multipliers are allowed to be discontinuous at interelement boundaries. Tangential contact conditions are incorporated by using a penalty method in conjunction with the classical Coulomb's friction model. Temporal integration is performed by using Newmark method. The Newton-Raphson iterative scheme is used for the solution of the resulting nonlinear algebraic equations, and for the determination of the contact region, contact conditions (sliding or sticking), and the contact pressures. The sensitivity coefficients are evaluated by using a direct differentiation approach. Numerical results are presented from the frictional contact/impact response of a composite spherical cap impacting on a rigid plate.

Karaoglan, Levent↗
NASA NTRS1998

Using Solar Radiation Pressure to Control L2 Orbits

The main perturbations at the Sun-Earth Lagrange points L1 and L2 are from solar radiation pressure (SRP), the Moon and the planets. Traditional approaches to trajectory design for Lagrange-point orbits use maneuvers every few months to correct for these perturbations. The gravitational effects of the Moon and the planets are small and periodic. However, they cannot be neglected because small perturbations in the direction of the unstable eigenvector are enough to cause exponential growth within a few months. The main effect of a constant SRP is to shift the center of the orbit by a small distance. For spacecraft with large sun-shields like the Microwave Anisotropy Probe (MAP) and the Next Generation Space Telescope (NGST), the SRP effect is larger than all other perturbations and depends mostly on spacecraft attitude. Small variations in the spacecraft attitude are large enough to excite or control the exponential eigenvector. A closed-loop linear controller based on the SRP variations would eliminate one of the largest errors to the orbit and provide a continuous acceleration for use in controlling other disturbances. It is possible to design reference trajectories that account for the periodic lunar and planetary perturbations and still satisfy mission requirements. When such trajectories are used the acceleration required to control the unstable eigenvector is well within the capabilities of a continuous linear controller. Initial estimates show that by using attitude control it should be possible to minimize and even eliminate thruster maneuvers for station keeping.

Tene, Noam↗
NASA NTRS1998

A Posteriori Bounds for Linear-Functional Outputs of Crouzeix-Raviart Finite Element Discretizations of the Incompressible Stokes Problem

We present a finite element technique for the efficient generation of lower and upper bounds to outputs which are linear functionals of the solutions to the incompressible Stokes equations in two space dimensions; the finite element discretization is effected by Crouzeix-Raviart elements, the discontinuous pressure approximation of which is central to our approach. The bounds are based upon the construction of an augmented Lagrangian: the objective is a quadratic "energy" reformulation of the desired output; the constraints are the finite element equilibrium equations (including the incompressibility constraint), and the intersubdomain continuity conditions on velocity. Appeal to the dual max-min problem for appropriately chosen candidate Lagrange multipliers then yields inexpensive bounds for the output associated with a fine-mesh discretization; the Lagrange multipliers are generated by exploiting an associated coarse-mesh approximation. In addition to the requisite coarse-mesh calculations, the bound technique requires solution only of local subdomain Stokes problems on the fine-mesh. The method is illustrated for the Stokes equations, in which the outputs of interest are the flowrate past, and the lift force on, a body immersed in a channel.

Patera, Anthony T.↗
NASA NTRS2003

Materials trade study for lunar/gateway missions

The National Aeronautics and Space Administration (NASA) administrator has identified protection from radiation hazards as one of the two biggest problems of the agency with respect to human deep space missions. The intensity and strength of cosmic radiation in deep space makes this a 'must solve' problem for space missions. The Moon and two Earth-Moon Lagrange points near Moon are being proposed as hubs for deep space missions. The focus of this study is to identify approaches to protecting astronauts and habitats from adverse effects from space radiation both for single missions and multiple missions for career astronauts to these destinations. As the great cost of added radiation shielding is a potential limiting factor in deep space missions, reduction of mass, without compromising safety, is of paramount importance. The choice of material and selection of the crew profile play major roles in design and mission operations. Material trade studies in shield design over multi-segmented missions involving multiple work and living areas in the transport and duty phase of space mission's to two Earth-Moon co-linear Lagrange points (L1) between Earth and the Moon and (L2) on back side of the moon as seen from Earth, and to the Moon have been studied. It is found that, for single missions, current state-of-the-art knowledge of material provides adequate shielding. On the other hand, the choice of shield material is absolutely critical for career astronauts and revolutionary materials need to be developed for these missions. This study also provides a guide to the effectiveness of multifunctional materials in preparation for more detailed geometry studies in progress. c2003 COSPAR. Published by Elsevier Ltd. All rights reserved.

NASA Discipline Radiation Health↗
NASA NTRS2005

Retrieving Storm Electric Fields From Aircraft Field Mill Data. Part I: Theory

It is shown that the problem of retrieving storm electric fields from an aircraft instrumented with several electric field mill sensors can be expressed in terms of a standard Lagrange multiplier optimization problem. The method naturally removes aircraft charge from the retrieval process without having to use a high voltage stinger and linearly combined mill data values. It also allows a variety of user-supplied physical constraints (the so-called side constraints in the theory of Lagrange multipliers). Additionally, this paper introduces a novel way of performing the absolute calibration of an aircraft that has several benefits over conventional analyses. In the new approach, absolute calibration is completed by inspecting the time derivatives of mill and pitch data for a pitch down maneuver performed at high (greater than 1 km) altitude. In Part II of this study, the above methods are tested and then applied to complete a full calibration of a Citation aircraft.

Koshak, W. J.↗
NASA NTRS2004

Dynamics of Earth orbiting formations

In this paper the equations of motion of a formation consisting of n spacecraft in Earth orbit are derived via Lagrange's equations. The equations of motion of the formation are developed with respect to both (1) a bound Keplerian reference orbit, and (2) a specific spacecraft in the formation. The major orbital perturbations acting on a formation in low Earth orbit are also included in the analysis. In contrast to the traditional approach based on the balance of linear momentum, the use of Lagrange's equations leads to a high-level matrix derivation of the formation equations of motion. The matrix form of the nonlinear motion equations is then linearized about a bound Keplerian reference orbit. Next, it is demonstrated that under the assumption of a circular reference orbit, the linearized equations of motion reduce to the well-known Hill-Clohessy-Wiltshire equations. The resulting linear and nonlinear dynamic equations lead to maximal physical insight into the structure of formation dynamics, and are ideally suited for use in the design and validation of formation guidance and control laws.

formation flying↗
NASA NTRS2006

Retrieving Storm Electric Fields from Aircrfaft Field Mill Data: Part II: Applications

The Lagrange multiplier theory developed in Part I of this study is applied to complete a relative calibration of a Citation aircraft that is instrumented with six field mill sensors. When side constraints related to average fields are used, the Lagrange multiplier method performs well in computer simulations. For mill measurement errors of 1 V m(sup -1) and a 5 V m(sup -1) error in the mean fair-weather field function, the 3D storm electric field is retrieved to within an error of about 12%. A side constraint that involves estimating the detailed structure of the fair-weather field was also tested using computer simulations. For mill measurement errors of 1 V m(sup -l), the method retrieves the 3D storm field to within an error of about 8% if the fair-weather field estimate is typically within 1 V m(sup -1) of the true fair-weather field. Using this type of side constraint and data from fair-weather field maneuvers taken on 29 June 2001, the Citation aircraft was calibrated. Absolute calibration was completed using the pitch down method developed in Part I, and conventional analyses. The resulting calibration matrices were then used to retrieve storm electric fields during a Citation flight on 2 June 2001. The storm field results are encouraging and agree favorably in many respects with results derived from earlier (iterative) techniques of calibration.

Koshak, William↗
NASA NTRS2006

Retrieving Storm Electric Fields from Aircraft Field Mill Data: Theory - Part 1

It is shown that the problem of retrieving storm electric fields from an aircraft instrumented with several electric field mill sensors can be expressed in terms of a standard Lagrange multiplier optimization problem. The method naturally removes aircraft charge from the retrieval process without having to use a high voltage stinger and linearly combined mill data values. It allows a variety of user-supplied physical constraints (the so-called side constraints in the theory of Lagrange multipliers) and also helps improve absolute calibration. Additionally, this paper introduces an alternate way of performing the absolute calibration of an aircraft that has some benefits over conventional analyses. It is accomplished by using the time derivatives of mill and pitch data for a pitch down maneuver performed at high (greater than 1 km) altitude. In Part II of this study, the above methods are tested and then applied to complete a full calibration of a Citation aircraft.

Koshak, W. J.↗
NASA NTRS2001

The Lunar L1 Gateway: Portal to the Stars and Beyond

Our Solar System is interconnected by a vast system of winding tunnels generated by the Lagrange Points of all the planets and their moons. These passageways are identified by portals around L1 and L2, the halo orbits. By passing through a halo orbit portal, one enters the ancient and colossal labyrinth of the Sun. This natural Interplanetary Supher highway System (IPS) provides ultra-low energy transport throughout the Earth's Neighborhood, the region between Earth's L1 and L2. This is enabled by an accident: the current energy levels of the Earth L1 and L2 Lagrange points differ from that of the Earth-Moon by only about 50 rn/s (as measured by AV). The significance of this happy coincidence to the development of space cannot be overstated. For example, this implies that Lunar L1 halo orbits are connected to halo orbits around Earth's L1 or L2 via low energy pathways...

lunar gateways↗
NASA NTRS2010

Habitation Concepts and Tools for Asteroid Missions and Commercial Applications

In 2009 studies were initiated in response to the Augustine Commission s review of the Human Spaceflight Program to examine the feasibility of additional options for space exploration beyond the lunar missions planned in the Constellation Program. One approach called a Flexible Path option included possible human missions to near-Earth asteroids. This paper presents an overview of possible asteroid missions with emphasis on the habitation options and vehicle configurations conceived for the crew excursion vehicles. One launch vehicle concept investigated for the Flexible Path option was to use a dual launch architecture that could serve a wide variety of exploration goals. The dual launch concept used two medium sized heavy lift launch vehicles for lunar missions as opposed to the single Saturn V architecture used for the Apollo Program, or the one-and-a-half vehicle Ares I / Ares V architecture proposed for the Constellation Program. This dual launch approach was studied as a Flexible Path option for lunar missions and for possible excursions to other destinations like geosynchronous earth orbiting satellites, Lagrange points, and as presented in this paper, asteroid rendezvous. New habitation and exploration systems for the crew are presented that permit crew sizes from 2 to 4, and mission durations from 100 to 360 days. Vehicle configurations are presented that include habitation systems and tools derived from International Space Station (ISS) experience and new extra-vehicular activity tools for asteroid exploration, Figure 1. Findings from these studies and as presented in this paper indicate that missions to near-Earth asteroids appear feasible in the near future using the dual launch architecture, the technologies under development from the Constellation Program, and systems derived from the current ISS Program. In addition, the capabilities derived from this approach that are particularly beneficial to the commercial sector include human access to geosynchronous orbit and the Lagrange points with new tools for satellite servicing and in-space assembly.

Smitherman, David↗
NASA NTRS2011

Simple, Robust Cryogenic Propellant Depot for Near Term Applications

The ability to refuel cryogenic propulsion stages on-orbit provides an innovative paradigm shift for space transportation supporting National Aeronautics and Space Administration s (NASA) Exploration program as well as deep space robotic, national security and commercial missions. Refueling enables large beyond low Earth orbit (LEO) missions without requiring super heavy lift vehicles that must continuously grow to support increasing mission demands as America s exploration transitions from early Lagrange point missions to near Earth objects (NEO), the lunar surface and eventually Mars. Earth-to-orbit launch can be optimized to provide competitive, cost-effective solutions that allow sustained exploration. This paper describes an experimental platform developed to demonstrate the major technologies required for fuel depot technology. This test bed is capable of transferring residual liquid hydrogen (LH2) or liquid oxygen (LO2) from a Centaur upper stage, and storage in a secondary tank for up to one year on-orbit. A dedicated, flight heritage spacecraft bus is attached to an Evolved Expendable Launch Vehicle (EELV) Secondary Payload Adapter (ESPA) ring supporting experiments and data collection. This platform can be deployed as early as Q1 2013. The propellant depot design described in this paper can be deployed affordably this decade supporting missions to Earth-Moon Lagrange points and lunar fly by. The same depot concept can be scaled up to support more demanding missions and launch capabilities. The enabling depot design features, technologies and concept of operations are described.

McLean, Christopher↗
NASA NTRS2016

Architecture for Mitigating Short-Term Warning Cosmic Threats: READI Project

Earth is being constantly bombarded by a large variety of celestial bodies and has been since its formation 4.5 billion years ago. Among those bodies, mainly asteroids and comets, there are those that have the potential to create large scale destruction upon impact. The only extinction-level impact recorded to date was 65 million years ago, during the era of dinosaurs. The probability of another extinction-level, or even city-killer, impact may be negligible, but the consequences can be severe for the biosphere and for our species. Therefore it is highly imperative for us to be prepared for such a devastating impact in the near future, especially since humanity is at the threshold of wielding technologies that allow us to do so. Majority of scientists, engineers, and policymakers have focused on long-term strategies and warning periods for Earth orbit crossing Near-Earth Objects (NEOs), and have suggested methods and policies to tackle such problems. However, less attention has been paid to short warning period NEO threats. Such NEOs test current technological and international cooperation capabilities in protecting ourselves, and can create unpredictable devastation ranging from local to global scale. The most recent example is the Chelyabinsk incident in Russia. This event has provided a wakeup call for space agencies and governments around the world towards establishing a Planetary Defense Program. The Roadmap for EArth Defense Initiative (READI) is a project by a team of international, intercultural, and interdisciplinary participants of the International Space University's Space Studies Program 2015 hosted by Ohio University, Athens, OH proposing a roadmap for space agencies, governments, and the general public to tackle NEOs with a short warning before impact. Taking READI as a baseline, this paper presents a technical description of methodologies proposed for detection and impact mitigation of a medium-sized comet (up to 800m across) with a short-warning period of two years on a collision course with Earth. The hypothetical comet is on a highly-inclined orbit having a high probability for Earth impact after its perihelion. For detection, we propose a space-based infrared detection system consisting of two satellites located at the Earth-Moon Lagrange points L1 and L2 coupled with space observatories, like the James Webb telescope and the Centennial telescope. These telescopes are supported by ground-based telescopes, like the Arecibo and Green Bank telescope, in the search for NEOs. Upon detection, the comet is tracked constantly using space- and ground-based telescopes. The deflection system is two-pronged, firstly involving the use of a high energy Directed Energy Laser Terminals (DELT) placed at Sun-Earth Lagrange points L4 and L5 so as to initiate and increase the ablation rate of the comet and deviate it from its collision trajectory, and secondly by the Hypervelocity Comet Intercept Vehicle (HCIV), a space-borne system combining a kinetic impactor with a thermonuclear device. The policy and international collaboration aspects to implement these methods are also outlined in the paper. The techniques mentioned could also be applied to mitigate medium-to-large sized asteroids (up to 2km across).

Laser Applications↗