Equations for Runge-kutta Formulas Through the Eighth Order
Tables for elementary weights of Runge-Kutta formulas of first eight orders and for relations of explicit formulas through order seven
WEIGHTING /MATH/↗
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Strength of synchronous rotation of the moon.
An expression for the strength of the synchronous rotation of the moon in terms of upper and lower limits on the rotational period is derived. The rotational periods for locked-in motion are expected to lie between 1.025T and 0.975T, where T is the period of revolution.
SATELLITE ROTATION↗
Laminar film condensation on the inside of slender, rotating truncated cones.
A solution for film condensation on the inside of slender rotating truncated cones is established in the region where the half cone angle phi is close to zero and where the equation of Dhir and Lienhard (1971) is no longer valid. All the familiar assumptions used in Nusselt's theory of laminar film condensation are assumed valid, including negligible sub-cooling and momentum changes in the condensate film, no interfacial shear between the condensate and the vapor, and pure conduction within the condensate film. A Runge-Kutta numerical integration method using a digital computer was used. Results for a specific geometry are given for water condensing at 212 F on a condenser wall maintained at 70 F.
Marto, P. J.↗
Runge-kutta type formulas of high-order accuracy and their application to the numerical integration of the restricted problem of three bodies.
Runge-Kutta type formulas applied to restricted three-body problem achieving high order accuracy by m-fold differentiation and simple transformation
THREE-BODY PROBLEM↗
Computer program for analysis of imperfection sensitivity of ring stiffened shells of revolution
A FORTRAN 4 digital computer program is presented for the initial postbuckling and imperfection sensitivity analysis of bifurcation buckling modes for ring-stiffened orthotropic multilayered shells of revolution. The boundary value problem for the second-order contribution to the buckled state was solved by the forward integration technique using the Runge-Kutta method. The effects of nonlinear prebuckling states and live pressure loadings are included.
Cohen, G. A.↗
SENS-5D trajectory and wind-sensitivity calculations for unguided rockets
A computational procedure is described which numerically integrates the equations of motion of an unguided rocket. Three translational and two angular (roll discarded) degrees of freedom are integrated through the final burnout; and then, through impact, only three translational motions are considered. Input to the routine is: initial time, altitude and velocity, vehicle characteristics, and other defined options. Input format has a wide range of flexibility for special calculations. Output is geared mainly to the wind-weighting procedure, and includes summary of trajectory at burnout, apogee and impact, summary of spent-stage trajectories, detailed position and vehicle data, unit-wind effects for head, tail and cross winds, coriolis deflections, range derivative, and the sensitivity curves (the so called F(Z) and DF(Z) curves). The numerical integration procedure is a fourth-order, modified Adams-Bashforth Predictor-Corrector method. This method is supplemented by a fourth-order Runge-Kutta method to start the integration at t=0 and whenever error criteria demand a change in step size.
Singh, R. P.↗
Tables for Supersonic Flow of Helium Around Right Circular Cones at Zero Angle of Attack
The results of the calculation of supersonic flow of helium about right circular cones at zero angle of attack are presented in tabular form. The calculations were performed using the Taylor-Maccoll theory. Numerical integrations were performed using a Runge-Kutta method for second-order differential equations. Results were obtained for cone angles from 2.5 to 30 degrees in regular increments of 2.5 degrees. In all calculations the desired free-stream Mach number was obtained to five or more significant figures.
Sims, J. L.↗
A study of numerical methods of solution of the equations of motion of a controlled satellite under the influence of gravity gradient torque
Numerical methods of integration of the equations of motion of a controlled satellite under the influence of gravity-gradient torque are considered. The results of computer experimentation using a number of Runge-Kutta, multi-step, and extrapolation methods for the numerical integration of this differential system are presented, and particularly efficient methods are noted. A large bibliography of numerical methods for initial value problems for ordinary differential equations is presented, and a compilation of Runge-Kutta and multistep formulas is given. Less common numerical integration techniques from the literature are noted for further consideration.
Thompson, J. F.↗
Apollo experience report: Onboard navigational and alignment software
The onboard navigational and alignment routines used during the nonthrusting phases of an Apollo mission are discussed as to their limitations, and alternate approaches that have more desirable capabilities are presented. A more efficient procedure for solving Kepler's equation, which is used in the calculation of Kepler's problem and Lambert's problem is included, and a sixth-order predictor scheme with a Runge-Kutta starter is recommended for numerical integration. The extension of the rendezvous navigation state to include angle biases and the use of a fixed coordinate system is also evaluated.
Savely, R. T.↗
On bilateral approximation in the numerical integration of Cauchy's problem for ordinary differential equations
Runge-Kutta formulas for ordinary differential equations
Zarubych, S. Z.↗
Computer program for investigating effects of nonlinear suspension-system elastic properties on parachute inflation loads and motions
A computer program is presented by which the effects of nonlinear suspension-system elastic characteristics on parachute inflation loads and motions can be investigated. A mathematical elastic model of suspension-system geometry is coupled to the planar equations of motion of a general vehicle and canopy. Canopy geometry and aerodynamic drag characteristics and suspension-system elastic properties are tabular inputs. The equations of motion are numerically integrated by use of an equivalent fifth-order Runge-Kutta technique.
Poole, L. R.↗
Effect of support flexibilty and damping on the dynamic response of a single mass flexible rotor in elastic bearings
A steady state analysis of the shaft and the bearing housing motion was made by assuming synchronous precession of the system. The conditions under which the support system would act as a dynamic vibration absorber at the rotor critical speed were studied; plots of the rotor and support amplitudes, phase angles, and forces transmitted were evaluated by the computer, and the performance curves were automatically plotted by a CalComp plotter unit. Curves are presented on the optimization of the support housing characteristics to attenuate the rotor unbalance response over the entire rotor speed range. The complete transient motion including rotor unbalance was examined by integrating the equations of motion numerically using a modified fourth order Runge-Kutta procedure, and the resulting whirl orbits were plotted by the CalComp plotter unit. The results of the transient analysis are discussed with regards to the design optimization procedure derived from the steady-state analysis.
Kirk, R. G.↗
Comparison of numerical integration techniques for orbital applications
The present work gives a brief comparison of the performance of programs for integrating differential equations for orbital applications. The evaluation criteria and the method of testing are described, and the results of the test problem set are included. Integration methods that were chosen for comparison include high-order Runge-Kutta methods; a rational extrapolation method (Bulirsch and Stoer, 1966); a variable step, variable order, multistep method (Krogh, 1969); classical multistep methods of Adams and Cowell; and modified multistep methods. The high-order Runge-Kutta methods used in the comparison include RKF 7(8) and RKF 8(9) (Fehlberg, 1968), and RKS 8-10 (Shanks, 1966).
Moore, H.↗
Numerical integration of second order differential equations
Performance characteristics of higher order approximations of Runge-Kutta type are analyzed, and performance predictors for time required on machine and for error size are developed. Technique is useful in evaluating system performance, analyzing material characteristics, and designing inertial guidance and nuclear instrumentation and materials.
Shanks, E. B.↗
Conservation properties of numerical integration methods for systems of ordinary differential equations
If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.
Rosenbaum, J. S.↗
Some numerical methods for integrating systems of first-order ordinary differential equations
Report on numerical methods of integration includes the extrapolation methods of Bulirsch-Stoer and Neville. A comparison is made nith the Runge-Kutta and Adams-Moulton methods, and circumstances are discussed under which the extrapolation method may be preferred.
Clark, N. W.↗
On the stability of numerical integration routines.
Numerical integration methods for the solution of initial value problems for ordinary vector differential equations may be modelled as discrete time feedback systems. The stability criteria discovered in modern control theory are applied to these systems and criteria involving the routine, the step size and the differential equation are derived. Linear multistep, Runge-Kutta, and predictor-corrector methods are all investigated.
Glover, K.↗
On the stability of numerical integration routines for ordinary differential equations.
Numerical integration methods for the solution of initial value problems for ordinary vector differential equations may be modelled as discrete time feedback systems. The stability criteria discovered in modern control theory are applied to these systems and criteria involving the routine, the step size and the differential equation are derived. Linear multistep, Runge-Kutta, and predictor-corrector methods are all investigated.
Glover, K.↗