Inertial Equations Research Articles

Theoretical and experimental study of low-temperature, capacitively coupled, radio-frequency helium plasmas

A comparison is made of one-dimensional theoretical simulations with measured densities of electrons and He metastables in a Gaseous Electronics Conference Reference Cell. The theory attempts to be as ab initio and complete as necessary for describing a He plasma. The basic theoretical description incudes the time-dependent, electric-field-consistent Boltzmann equation for the electrons, inertial fluid equations for the heavy particles, and a multilevel model of the He excitations. The novel numerical algorithms are rapidly convergent. The experimental and theoretical absolute number densities are generally within a factor of 2 for both the electron plasma density and the metastable densities. In addition the spatial variation of the metastable populations (singlet and triplet) between the electrodes is well reproduced.

Asymptotic inertial motion and Lorentz-Dirac equation

Using mathematical arguments, it is proved that a scattering process of a system of elementary classical point charges obeying the Lorentz-Dirac equation is free of runaway solutions and of causality-violating properties.

The vector-network method for the modelling of mechanical systems

This paper presents an extension of the vector-network approach to the problem of motion prediction. The entire procedure is a basic application of concepts of graph theory in which laws of vector dynamics have been combined. A comprehensive mathematical model for the systematic formulation of the equations of motion of dynamic three-dimensional constrained multi-body systems is derived. The method embodies simultaneously the three-dimensional inertial equations associated with each rigid body and the kinematic constraints into a symmetrical format yielding the differential equations governing the response of the system. The modelling technique is thoroughly described in this work and its validity is impartially established.

The stochastic, one-dimensional inertial equation, the inviscid Burger’s model, and the surface intersection problem

A collisionless (rarefied) gas with an initial velocity in one dimension is discussed. It is shown that this inertial problem, the problem of the multiple intersections of a line of sight with a random surface, and the inviscid Burger’s model all have the same solutions. Kahng earlier used the full Wiener–Hermite expansion and guessed a solution to the stochastic problem. It is shown, using a constructive functional derivative procedure, that this solution is a linear combination of the occurring multiple solutions, with alternating signs. It is also shown that such an alternating sum of any moment is conserved, so that the initial, marginal distribution is invariant for this forceless problem. Kahng’s later-time energy spectrum for one example, which for large k was found to be k−3, is shown to be the correct result because of square root irregularities appearing at later times for solutions of the inertial equation. Further it is shown that any Gaussian process with arbitrary initial spectrum approaches k−3 for k large, with the initial dissipation length as its characteristics length. Finally a new technique is used to obtain the Wiener–Hermite expansion for physically reasonable fields in the collisionless gas problem.

Aeroservoelasticity in the time domain

The objective of this investigation was to develop and evaluate a nonlinear technique for the stability assessment of high-performa nce aircraft in the time domain. The technique employed is compatible with popular digital flutter and/or vibration mathematical models and is adaptable to various aircraft control systems arid/or modal information. The aircraft may possess servocontrol systems of high complexity and may dictate the use of aerodynamic schemes of the user's choice. The method of approach involves the use of the Advanced Continuous Simulation Language (ACSL), chosen because of its hybrid computer-like features and its conversational language. All modules of the simulation are solved in the time domain as the vehicle is subjected to disturbances of the user's choice. The vehicle motion rates are then output in the time domain for the engineer's concluding assessment of stability. Instabilities experienced by an early version of the F-16 aircraft are used as check cases. The stability boundary obtained is unconservativ e in its agreement with that determined in flight test, whereas the frequencies obtained are in excellent agreement. Conclusions include the following: 1) the simultaneous solution of large systems of aerodynamic, inertial, elastic, and servocontrol equations can be obtained using advanced simulation languages with mnemonic features that aid in their implementation and use; and 2) some improvements, when compared to frequency domain results, can be obtained using this nonlinear time domain simulation approach, especially when studying the behavior of phenomena with nonlinear effects or response.

Nonlinear aerodynamic modeling of flap oscillations in transonic flow - A numerical validation

The regime of validity of a nonlinear aerodynamic force and moment formulation, based on concepts from nonlinear functional analysis and applicable to a transonic airfoil with a deflecting flap, is investigated. A time-dependent finite difference technique is used to evaluate the aerodynamic data of the formulation in terms of specified, characteristic motions. Flap-motion histories are generated from the flap inertial equations of motion, with aerodynamic reactions specified by the moment formulation. The motion histories depicting the cases of decaying and growing flap oscillations are compared with histories generated through simultaneous, coupled solution of the fluid-dynamic equations and flap inertial equations of motion. The range of applicability of the formulation is discussed.

Meandering of equatorial flows

The periods, wavelengths, and amplitudes of meanders in equatorial flows and particle trajectories therein are computed by using the inertial motion equations on the beta plane and a sphere. Data from the fifth voyage of the scientific-research vessel “Dmitrii Mendeleev” are presented, which indicate the asymmetry of the Cromwell flow relative to the equator and its possible meandering.

Gravito-inertial fields and relativity

Gravitational and inertial field equations of a Maxwellian form are established which lead to a consistent theory of gravitation. The new approach to dynamics arising from the field theory is in agreement with the theory of special relativity, while producing a fresh insight into some of the relativistic phenomena.

Computation of the Effects of Impulsive and Short-Term Perturbing Forces on an Orbiting Body

This paper provides a simple yet accurate method for computing the long-term effects of impulsive and short-term perturbing forces on a body in elliptical orbit. The solutions are cast in a form which requires integration over a maximum of one orbit period, and thus avoids the high cost and low accuracy normally associated with deriving perturbations over many orbit periods. The basic equations solved are the nondimensionalized Tschauner-Hempel Equations, and their correspondence to Inertial Guidance Position and Velocity Error Equations for systems in orbit is pointed out.