Arbitrary Mass Distribution Research Articles

On the gravitational field of an arbitrary axisymmetric mass possessing a magnetic dipole moment

An exact asymptotically flat solution of the static Einstein-Maxwell equations describing the gravitational field of an arbitrary axisymmetric mass distribution endowed with a magnetic dipole moment is presented.

Dynamic modelling of skeletal systems with arbitrary mass distributions

AbstractThis study develops and illustrates a dynamic model for vibration analysis of rod and beam elements with arbitrary mass distributions. This arbitrary mass model represents the element shape function using a frequency dependent series that becomes exact as the number of terms in the series approaches infinity. The arbitrary mass model yields the natural frequencies and modes of elements with mass distributions which vary as continuous functions of the spanwise co‐ordinate.The arbitrary mass model, implemented using the unsymmetric formulation for vibration analysis, is illustrated in examples and compared with other dynamic models which use stepped mass representations to model the arbitrary mass distributions. Studies show that the arbitrary mass model is more computationally efficient than the stepped mass representation models, particularly when accuracy requirements are high.

Dynamics of an elastic multibody chain: part a—body motion equations

In this paper—the first of a series describing the dynamics of an arbitrary multibody system—motion equations governing a set of individual bodies in a chain configuration are discussed. A chain consisting entirely of rigid bodies is considered first. Motion equations for a typical body of arbitrary shape and arbitrary mass distribution are then briefly summarized. Finally, the geometrical constraints necessary to connect the individual bodies into a chain are derived. Large translational and rotational motions are permitted at the joints connecting contiguous bodies. In other words, both prismatic and revolute joints are included, alone and in combination. As well, the interbody force constraints required to ensure that equal but opposite forces and torques exist at each joint are developed. The resulting expressions are amenable to the introduction of constraint and control forces at the chain joints. This permits the number of actively controlled degrees of freedom at any specific joint to be arbitrari...

An analytically soluble problem in fully nonlinear statistical gravitational lensing

The amplification probability distribution p(I)dI for a point source behind a random star field which acts as the deflector exhibits a I exp-3 behavior for large amplification, as can be shown from the universality of the lens equation near critical lines. In this paper it is shown that the amplitude of the I exp-3 tail can be derived exactly for arbitrary mass distribution of the stars, surface mass density of stars and smoothly distributed matter, and large-scale shear. This is then compared with the corresponding linear result.

The determination of the gravitational potential of arbitrary mass distribution

Numerical computation of the gravitational potential for arbitrary mass distribution in spherical coordinates is considered. It is possible to determine the potential from the spherical expansion of the Poisson integral with 1/2n 2 NML operations — i.e., with a number proportional to the mesh numberNML multiplied by the square of the Legendre functions of indexn. The computation of the potential based on the solution of the Poisson differential equation is discussed.

Exact solutions for spectra and Green's functions in random one-dimensional systems

For chains of harmonic oscillators with random masses a set of equations is derived, which determine the spatial Fourier components of the average one-particle Green's function. These equations are valid for complex values of the frequency. A relation between the spectral density and functions introduced by Schmidt is discussed. Exact solutions for this Green's function and the less complicated characteristics function-the analytic continuation into the complex frequency plane of the accumulated spectral density and the inverse localization length of the eigenfunctions-are derived for exponential distributions of the masses. For some cases the characteristic function is calculated numerically. For gamma distributions the equations are cast in the form of ordinary, higher order differential equations; these have been solved numerically for determining the characteristic function. For arbitrary mass distributions a cumulant expansion and a peculiar symmetry of the Green's function are discussed. The method is also applied to chains where the spring constants and/or the masses have random values. Also for these systems exact solutions are discussed; for exponential distributions, e.g., of both masses and spring constants the characteristic function is expressed in Bessel functions. The relation with certain random relaxation models is shown. Finally, X-Y Hamiltonians with random exchange constants and/or magnetic fields-or, equivalently, tight-binding electron models with diagonal and/or off-diagonal disorder-are considered. Here the Green's function does not depend on the wave number if the distribution of exchange constants is symmetric around the origin. New solutions for the characteristic function and Green's function are derived for a number of cases, including exponentially distributed magnetic fields and power law distributed exchange constants.

An Adaptive Squeeze-Film Bearing

This paper examines the effect of controlling the oil supply pressure to squeeze-film bearings in applications where these elements are used to provide damping for a light flexible transmission shaft having an arbitrary unbalance mass distribution. The shaft length and diameter selected for the study are typical of those used for helicopter tail rotor transmissions. A computer simulation is undertaken to study the effect of a squeeze-film damper located at: 1) The end supports. 2) Mid-span with undamped end supports. 3) Mid-span with damped end supports. The simulation shows that in this type of application, good vibration control can be achieved by using a squeeze-film damper which is capable of switching between two levels of damping. The feasibility of attaining such a characteristic is examined experimentally.

Dynamic Force and Torque Analysis of Spherical Four Link Mechanisms

In this paper we present a dynamic analysis of a general spherical four-link mechanism whose links have arbitrary mass distribution. Results, which are in explicit analytical expressions in terms of inertia-induced forces and moments in links, are useful for optimum design of the mechanism under high-speed operation.

Numerical investigation of two-phase flow with coagulation and fragmentation of particles in axisymmetric laval nozzles

The method and results of a calculation of the parameters of a polydis-perse two-phase flow with coagulation and fragmentation of particles in collisions are described. The problem is considered in a two-dimensional formulation. The secondary particles (fragments) formed by fragmentation are assumed to have arbitrary mass and velocity distributions.

On stability of steady motions of a heavy solid body on an absolutely smooth horizontal plane

Stability of steady motions of a heavy solid on an absolutely smooth horizontal plane is investigated. Conditions of existence and stability of permanent rotations are obtained in the case of arbitrary distribution of mass and arbitrary surface of the body, and those of existence and stability of regular precessions of a dynamically symmetric body bounded by a surface of revolution. A definite analogy of the problem investigated here to that of stability of permanent rotations and regular precessions, respectively, of an arbitrary and a dynamically symmetric body with a fixed point, as well as the essential differences between them, are pointed out.

On a result of Rubin and Vere-Jones concerning subcritical branching processes

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.

On a result of Rubin and Vere-Jones concerning subcritical branching processes

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.

On the stability of permanent rotation of a heavy solid body about a fixed point: PMM vol. 40, n≗ 3, 1976, pp. 408–416

The permanent rotation of a heavy solid body about its principal axis of inertia with a fixed point is considered. Stability is investigated with the use of the theorem on the stability of Hamiltonian systems with two degrees of freedom in the general elliptic case. It is shown that in the absence of certain resonance relationships in the region of necessary stability conditions, which does not coincide with the region of known sufficient conditions, the first approximation indicates the existence of stability, except possibly, in the case when the parameters of the problem lie on some specific manifolds of the parameter space. Subregions that are free of such exceptional manifolds are indicated in each region of necessary stability conditions. Necessary stability conditions for permanent rotation about principal axes of inertia of a solid body were investigated by Grammel [3]. Sufficient conditions that matched necessary conditions were obtained by Chetaev in the case of Lagrange integrability [4], and by Rumiantsev in that of Kowalewska integrability [5]. Permanent rotation of a body with arbitrary mass distribution about its principal axis of inertia was considered in [6–8], where sufficient stability conditions were established by Chetaev's method and on the basis of the Routh theorem. Bifurcation of permanent rotations and changes of stability were investigated in [9].

Analysis of the nonequilibrium two-phase flow with coagulation and crushing of condensate particles for arbitrary mass and velocity distributions of the secondary drops

The problem of the nonequilibrium flow of a two-phase gas-polydisperse condensate drop mixture is considered in a one-dimensional approximation taking account of coagulation and crushing of particles of different size during collisions. A closed system of equations in the flow characteristics is obtained in application to the general case of arbitrary size, velocity, and temperature distributions of the secondary drops formed during crushing. Regularities in the interaction of freely moving drops during collisions are investigated experimentally.

Analysis of Rotors Supported upon Many Bearings

This paper presents a technique for setting up a mathematical model of a rotor supported by many bearings. The technique takes into account the coupling between horizontal and vertical motions at the bearings, as well as inter-bearing couplings. The rotor, which is isotropic and undamped, can be of arbitrary stiffness and mass distribution. The economy and accuracy of the programme is demonstrated for the very complex case of a large turbo-set line-out in which the rotor is supported upon twelve bearings.

A New Method for Completely Force Balancing Simple Linkages

A new method, herein referred to as the “Method of Linearly Independent Vectors,” is shown to permit the complete force balancing of certain planar linkages. This method consists of writing the equation describing the position of the total mechanism center of mass in such a way that the coefficients of the time-dependent terms may be set equal to zero. In this way, the total center of mass can be made stationary, and the shaking force vanishes. Derivations as well as practical applications are shown for four-bar and six-bar linkages with arbitrary link mass distributions.

Gravity Gradiometer Computer Model for Simulated Gradient Contour Mapping

Considerations for gravity gradiometer application have established the need for predicting gradiometer response to mass distributions of particular interest. A digital computer program has been developed to simulate the rotating gravitational mass sensor, and to map the gradient contours of the gravitational field created by an arbitrary mass distribution. This analysis demonstrates the interaction of the gradiometer with seond and higher order gravitational gradients. The information about the mass distribution of an object was found to increase with the gradient order. Considered in this study is the ``cruciform'' mass sensor, now being developed by Hughes Research Laboratories. This rotating gradiometer is theoretically capable of measuring the second, sixth, tenth, etc., order gradient. We are presently engaged in laboratory experiments which combine with these computer results in the understanding of gradient-sensor interaction. This paper gives a basic introduction to gravitational tensors, followed by a mathematical formulation of the gradiometer model. Computer results are included which demonstrate the gravitational gradient contours associated with some selected mass distributions.

Influence of Coupled Asymmetric Bearings on the Motion of a Massive Flexible Rotor

The rotors of a majority of rotating machines are supported upon bearings whose flexibility has a significant effect on the dynamics of the system. Supports incorporating hydrodynamically maintained oil films in particular, in addition to possessing different stiffness and damping in mutually perpendicular planes, exhibit non-reciprocal coupling between motions in these planes. The author develops the equations of motion for a rotor-bearing system of this type, and describes how both the response to an out-of-balance forcing and the unstable behaviour of such a system may be predicted by evaluating separately the dynamical characteristics of the rotor and the bearings. The method is applicable to systems in which the rotor is axisymmetric and has an arbitrary mass and stiffness distribution symmetrically disposed about its centre of span. Some new aspects of the resonant and unstable behaviour of rotating shafts are brought out due to the nature of the method of solution. Theoretical predictions are compared and agree very well with the results of tests on a 1380 lb model rotor supported on two 4 in diameter cylindrical journal bearings.

Radial oscillations of a plasma cylinder with arbitrary density distribution

This research note discusses the radial hydromagnetic oscillations of a plasma cylinder confined by an axial magnetic field for the general case of an arbitrary distribution of mass. The oscillation frequency is expressed in terms of the value for an annular distribution multiplied by a correction factor, g. Values of g are calculated analytically for several simple distributions and the results compared with those obtained by a numerical method and using a variational principle. The numerical and variational methods are used to calculate g-values for other distributions, including examples from the Hain-Roberts theta-pinch code and from a theta-pinch experiment. The value of g varies from 1 for an annulus to about 1.4 for a distribution peaked on the axis. These simple calculations show that an accurate value of the plasma mass can be obtained when the oscillation and density distribution are known.

Balancing of Flexible Rotors Having Arbitrary Mass and Stiffness Distribution

A method for balancing a flexible rotor is developed which is based upon the dynamic response of the rotor in its normal modes as excited by the distributed mass unbalance. The balance corrections are determined by equating the modal components of the corrections to the modal components of the unbalance of the rotor. Methods of balancing individual modes as well as simultaneous balancing in any number of modes are presented. The suggested procedure requires calculation of the normal modes and critical speeds, and measurement of the dynamic amplitude at selected points along the rotor axis.