Showing papers on "Angular velocity published in 1974"

Numerical Modelling of Instantaneous Plate Tectonics

Abstract: Assuming lithospheric plates to be rigid, we systematically invert 68 spreading rates, 62 fracture zones trends and 10^6 earthquake slip vectors simultaneously to obtain a self-consistent model of instantaneous relative motions for eleven major plates. The inverse problem is linearized and solved iteratively by a maximum likelihood procedure. Because the uncertainties in the data are small, Gaussian statistics are shown to be adequate. The use of a linear theory permits (1) the calculation of the uncertainties in the various angular velocity vectors caused by uncertainties in the data, and (2) quantitative examination of the distribution of information within the data set. The existence of a self-consistent model satisfying all the data is strong justification of the rigid plate assumption. Slow movement between North and South America is shown to be resolvable. We then invert the trends of 20 linear island chains and aseismic ridges under the assumptions that they represent the directions of plate motions over a set of hot spots fixed with respect to each other. We conclude that these hot spots have had no significant relative motions in the last 10 My.

Dynamic properties of bacterial flagellar motors

Abstract: TETHERED bacteria1 rotate at the angular velocity at which the torque generated by the flagellar motor2 is balanced by the torque due to the viscous drag In general, M = bηΩ, where M is the torque, η is the viscosity, Ω is the angular velocity, and b is a coefficient which depends on the size and the shape of the cell, the position of the axis of rotation, and the distance between the cell and the wall For a sphere of radius a (not too close to the wall) M = 8πη a3Ω (ref 3) Viscous forces are so large in comparison with inertial forces4 that Ω will change with M virtually instantaneously; any discontinuities in the one will be evident in the other Consider a cell of radius a and uniform density ρ rotating at an angular velocity Ω0; if its motor is suddenly disengaged, Ω will decay exponentially to 0 with a time constant ρ a2/15η, and the cell will stop in Ωρ a2/15η radians For Escherichia coli this is less than a millionth of a revolution The cell also is subject to rotational diffusion, but this will be evident only if the coupling between the flagellum and the body of the cell is fluid The root-mean-square deviation in the angular position is (2Dt)½, where D is the rotational diffusion constant and t is the time For a cell which can rotate freely, D = kT/bη, where k is Boltzmann's constant and T is the absolute temperature

Hydromechanics of low-Reynolds-number flow. Part 1. Rotation of axisymmetric prolate bodies

Abstract: The present series of studies is concerned with low-Reynolds-number flow in general; the main objective is to develop an effective method of solution for arbitrary body shapes. In this first part, consideration is given to the viscous flow generated by pure rotation of an axisymmetric body having an arbitrary prolate form, the inertia forces being assumed to have a negligible effect on the flow. The method of solution explored here is based on a spatial distribution of singular torques, called rotlets, by which the rotational motion of a given body can be represented. Exact solutions are determined in closed form for a number of body shapes, including the dumbbell profile, elongated rods and some prolate forms. In the special case of prolate spheroids, the present exact solution agrees with that of Jeffery (1922), this being one of very few cases where previous exact solutions are available for comparison. The velocity field and the total torque are derived, and their salient features discussed for several representative and limiting cases. The moment coefficient C[sub]M = M/(8[pi][mu][omega sub 0]ab^2) (M being the torque of an axisymmetric body of length 2a and maximum radius b rotating at angular velocity [omega], about its axis in a fluid of viscosity [mu]) of various body shapes so far investigated is found to lie between 2/3 and 1, usually very near unity for not extremely slender bodies. For slender bodies, an asymptotic relationship is found between the nose curvature and the rotlet strength near the end of its axial distribution. It is also found that the theory, when applied to slender bodies, remains valid at higher Reynolds numbers than was originally intended, so long as they are small compared with the (large) aspect ratio of the body, before the inertia effects become significant.

Angular motions of freely falling spheroidal hailstone models

Abstract: The free fall behavior of rotating oblate spheroids with Reynolds numbers of 4 × 104 to 4 × 105 is studied by solving semiempirical Eulerian equations of motion which use a quasistatic approximation for aerodynamic forces and torques. The angular motions are classified in terms of solutions to the restricted set of equations in which the center of mass falls uniformly. One class of solutions, named symmetric gyration, is relevant for studies of the symmetric growth of hailstones because symmetrically equivalent points on the surface of a spheroid are equally exposed to the flow. For large hailstones with horizontal total angular momentum, critical frequencies sufficient for symmetric gyration are in the range of 2‐6 Hz (Strouhal numbers 0.005 to 0.01), they increase with increasing axis ratio, decrease with size, and are independent of the density ratio spheroid/air. Solutions to the full equations of motion show that the rotational energy of a symmetrically gyrating spheroid increases with time. This results in an increase in nutational amplitude which depends on the spheroid, its spin rate and its nutation amplitude. Such an effect occurs because the horizontal velocity, which tends to align with the horizontal major axis, causes the aerodynamic torque to have an oscillating vertical component in phase with the oscillating angular velocity about the vertical. A similar interaction prevents a steady helical motion.

Variable shear rate, wide dynamic range true indicating viscometer

Abstract: A viscosity measuring system of the rotational viscometer type, in which a spindle is secured to one end of a shaft the other end of which is rigidly attached to a drive motor. A precision electronic control system maintains the angular speed of rotation of the motor, shaft and spindle at a desired value, which may be varied manually or automatically over a wide range of speeds. When the spindle is immersed in liquid whose viscosity is to be determined, the electronic control system supplies sufficient drive current to the motor to overcome the viscous torque exerted on the spindle by the liquid, plus a residual current to overcome friction, windage and other losses not associated with the liquid. A sensing circuit provides an output signal proportional to the motor current, and a compensation circuit subtracts from this output signal a value corresponding to the residual current, the resulting signal corresponding to the viscous torque. A permanent magnet DC motor having a non-magnetic core is employed, so that the aforementioned resulting signal varies linearly with viscous torque, and therefore is directly proportional to the viscosity of the liquid at the angular speed at which the shaft is rotated. A dividing circuit provides an output voltage to a display or recording instrument corresponding to the ratio of viscous torque to angular shaft speed. For Newtonian liquids this ratio is independent of speed and provides a true indication of liquid viscosity at any desired speed. The Newtonian or nonNewtonian character of a liquid can be easily and rapidly determined merely by varying the motor speed setting and observing whether there is any change in the true viscosity reading of the display or recording instrument.

Method and apparatus for measuring small variations in the speed of rotating shafts

Abstract: Method and apparatus for measuring torsional vibrations in a rotating shaft. A detection device is connected to provide output pulses at a rate proportional to the instantaneous angular velocity of the shaft. The pulses are compared in a phase detector with pulses produced by a voltage controlled oscillator which oscillator is adjusted to produce pulses at a frequency corresponding to the average angular velocity of the shaft. The phase detector generates an output signal proportional to the instantaneous phase difference between the two pulse inputs which output signal varies as a function of variations in the frequency of pulses from the detection device. The output signal is therefore representative of torsional vibrations in the shaft. The output signal may be filtered to produce a signal having an instantaneous amplitude and frequency corresponding to the torsional vibrations.

Strapdown Guidance Error Analysis

Abstract: A crucial part of so-called ?strapdown? techniques for inertial guidance systems is the generation of the matrix of direction cosines relating the body axes to the reference axes. This is generally done by direct integration of a set of differential equations having the body angular velocities as inputs. This paper is devoted to an error analysis of two commonly used integration schemes, namely, direction cosines and quaternions. Scale, skew, and drift errors are defined, and the susceptibility of the integration schemes to these types of error is examined. It is concluded that the quaternion scheme offers an advantage because it intrinsically yields zero skew error. The paper is presented in two parts, for convenience.

Distinctive patterns on the surface of slowly rotating stars whose oscillations are nonlinearly coupled

Abstract: Slowly rotating stars which are oscillating at small amplitude in a broad spectrum of g-modes should display strong surface nonuniformities if even weak nonlinear coupling exists between the modes. Oscillatory power will be concentrated into distinctive patterns which rotate rigidly in spite of differential rotation in the outer stellar layers. Each pattern rotates at a constant rate slower than the star as a whole according to a very simple law of rotation. Virtually all the rotation rates are within 9 per cent of the stellar rate. Evidence is cited that the sun may be oscillating, so other stars along the main sequence may be oscillating, also. If zones obeying the predicted rotation law can be detected in a star, then the rotation rate of the stellar interior becomes known, and differential rotation is negligible over most of the stellar mass.

Structure, Dynamics and Statistical Properties of Galaxies

Abstract: The following empirical or physical parameters describing individual and statistical proper­ties of galaxies are reviewed: morphological type including new or revised types; intrinsic luminosity distribution functions of two main components (spheroid and disk); true ellipticities of different types; characteristic scale parameters (effective diameter and luminosity density) of spheroidal and flat systems; masses and densities of spheroidal, disk and mixed systems from velocity dispersion and rotation velocities; rotation periods, maximum rotation velocities, angular velocities and momenta of different types; neutral hydrogen masses and densities; spectral energy distributions and colour in­dices; number of independent parameters from principal component (factor) analysis; luminosity functions and selection effects; clustering and space distribution; maximum density-radius and velocity dispersion-radius relations in systems of galaxies.

The differential rotation of the solar surface

Abstract: The large-scale flow in the solar convection zone is discussed. The objective is to deduce from observation the principal physical balances in the governing equations. The simplest set of equations that seem potentially realistic are utilized. It is assumed that magnetic fields are negligible, that mixing-length theory gives an accurate representation of the mean structure, and that rigid-body rotation exists at the base. It is deduced that the zonal glow is geostrophic, the meridional flow is controlled by friction, and diffusive heating balances advective cooling due to vertical motion. Next, a detailed calculation of the latitude profile of surface angular velocity is performed, based on approximate equations containing only the principal balances, and agrees well with observation. Finally, it is demonstrated that the amplitude of the flow may be determined by the constraint that the net equatorward angular momentum flux vanish. The conclusions are consistent with the hypothesis that the flow is primarily a single large axisymmetric convection cell in each hemisphere.

Slow Rotation of Two Touching Spheres in Viscous Fluid

Abstract: The Stokes flow for the case in which two solid spheres in contact are steadily rotating with different angular velocities about their line of centers is studied. In particular, it is shown that the couple experienced by a sphere in contact with a fixed infinite plane perpendicular to the axis of rotation is increased by a factor \(\sum\limits_{n=1}^{\infty}1/n^{3}{=}1.202\cdot\cdot\cdot\) as compared with its value in the absence of the plane. It is also found that, for the case of two equal spheres, one of which is kept rotating with angular velocity ω while the other is left free, the latter will rotate with angular velocity ω/7.

Liapunov Stability Analysis of Hybrid Dynamical Systems in the Neighborhood of Nontrivial Equilibrium

Abstract: This paper is concerned with the stability of a hybrid dynamical system in the neighborhood of a nontrivial equilibrium, where the system consists of one rigid part and n elastic members. The body moves in a central-force field with its mass center describing a circular orbit. The nontrivial equilibrium is defined by steady rotation of the system at an angular velocity equal to the orbital velocity, with the elastic members being in deformed state. A Liapunov stability analysis is performed by assuming small perturbations about the nontrivial equilibrium, where the latter is generally defined by nonlinear differential equations. The theory is applied to a gravity-gradient stabilized satellite with flexible appendages.

On differential rotation

Abstract: In a first order approximation the influence of meridional circulations in a spherical shell on the radial dependence of the angular velocity is studied. Due to stationarity the flux of angular momentum which is transported through any sphere by the circulations must be cancelled by the flux of angular momentum due to turbulent friction. If the circulation goes equatorward at the outer surface the law of rotation must be such that angular momentum is transported in outward direction through the sphere.

Apparatus for accurate die-cutting

Abstract: Inaccurate cuts often occur in paperboard blanks passing between a pair of cooperating die and anvil cylinders because of anvil cylinder wear, irregular blank velocity, and other factors. Such inaccuracies are reduced by driving the die cylinder at a preselected angular velocity and driving the anvil cylinder at an angular velocity proportional to the angular velocity of the die cylinder with the preselected proportion being maintained during changes in angular velocity of the die cylinder. The preferred apparatus includes a first mechanical transmission having a primary input driven by the die cylinder and a secondary input driven by a second mechanical variable-ratio transmission for driving the anvil cylinder at an angular velocity corresponding to the angular velocity of the die cylinder but with the anvil cylinder velocity being selectively variable to provide an anvil cylinder velocity selectively proportional to the velocity of the die cylinder. Additionally, a slip clutch is preferably interposed between the drive for the die cylinder and the first transmission to permit the anvil cylinder to change its rotational position relative to the rotational position of the die cylinder upon the occurence of excessive torque.

On the Sun's Differential Rotation. Implications of the Difference in Angular Velocity between the Sunspots and Photosphere

Abstract: It is assumed that the meridional motions (U) and angular velocity (Ω) in the surface layers of the convection zone are given by simple expressions of the form: Ur= 2ψ(r) P2(cosθ)/ϱr2, U0 = −ψ′(r) sinθ cosθ/ϱr, and Ω = Ω0[(1 + ω0(r) + ω2(r) P2(cosθ)] Here ψ(r) is the stream function, P2(cosθ) the second order Legendre polynomial, and θ the polar angle Allowance is made for a possible difference in the rate of momentum exchange between the directions parallel and perpendicular to gravity by introducing an anisotropic turbulent viscosity coefficient, μ, which is assumed furthermore to be proportional to the density, ϱ;μ = ϱν, and νθθ= νφφ= sνrr It is shown that if the sunspots give an indication of the Sun's angular velocity at a depth h(∽ 3 × 104 km) then the turbulent viscosity is necessarily anisotropic The radial variation introduced by this anisotropy seems to explain well the sunspot data if we assume that the sunspots act as tracers of the Sun's angular velocity

Rotational Distortion of the Stars of Arbitrary Structure: Fourth Approximation

Abstract: Clairaut's theory of the rotational distortion of self-gravitating configurations of arbitrary structure, arising from axial rotation with constant angular velocity, previously developed (cf. Kopal, 1973) to quantities of third order in superficial distortion, has now been extended to terms of fourth order. The differential equations governing the form and exterior potential of stars so rotating have been set up by the method followed in our previous paper (Kopal, 1973) together with their boundary conditions; but their applications to practical cases are being postponed for a subsequent investigation.

Effects of rotation on vibrations of circular cylindrical shells

Abstract: The effects of rotation due to centrifugal and Corliolis forces on the frequencies are analyzed on the basis of the Herrmann‐Armenakas bending theory. Closed‐form expressions for the frequency and the nodal rotational velocity of the flexural modes are discussed, and their accuracy is compared with previously published results.

Perturbation formulations for satellite attitude dynamics

Abstract: Two analytical developments for the arbitrarily torqued motion of an asymmetric rigid body, both of which utilize a new torque-free solution as the reference motion, are presented. The first is an Encke-type perturbation formulation in which differential equations for the angular velocity and orientation departures from Poinsot motion are derived. The second technique is a variation of parameters scheme in which an analogue of Herrick's two-body perturbative differentiation technique is employed. The torque-free motion constants selected for variation are initial orientation and initial angular velocity; differential equations which specify the time variation of these parameters are developed, so that the torque-free solution is then instantaneously valid in the presence of arbitrary torques. Both developments are motivated by classical perturbation theories in orbital mechanics. Extensive use is made of the Euler parameter description of body orientation and kinematics rather than the more conventional Euler angles in order to avoid the geometrical singularities implicit in the latter.

Wide band angular displacement and velocity sensor and method

Abstract: An angular displacement transducer having a mounting base for mounting an inertial element sensitive to low frequency angular displacements and an inertial element sensitive to high frequency angular displacements. The low frequency sensitive inertial mass is a solid rotor device having a pickoff for sensing angular displacement between the solid inertial mass and the mounting base. A feedback centering circuit responsive to the pickoff output maintains the solid inertial mass in a substantially neutral position relative to the mounting base for displacement frequencies below the band of interest and allows the solid mass to move in an open loop fashion within its band of interest. The inertial mass sensitive to high frequency angular displacements is a fluid mass having a pickoff for sensing displacement of the mounting base relative to the fluid mass. A feedback centering circuit responsive to the fluid mass displacement sensor output is disposed to impart the motion of the mounting base to the fluid mass for displacement frequencies below the band of interest and allows the fluid mass to operate open loop within its band of interest. The solid inertial mass pickoff output is directed to a low pass filter and the fluid inertial mass pickoff output is directed to a high pass filter. The response time of the feedback centering circuit for the solid inertial mass is long enough in the band of interest to produce pickoff output related to angular displacement in the low frequency range and the response time of the feedback centering circuit for the fluid inertial mass, while shorter, is long enough in the band of interest to produce pickoff output related to angular displacement in the high frequency range. The outputs from the high and low pass filters are summed and presented as angular displacement sensor output over the wide band of high and low frequencies. Adjustment of feedback centering circuit gain and viscosity of fluid inertial mass viscosity may provide a wide band angular velocity sensor.

Adiabatic pulsations and convective instability of rotating gaseous masses. II. Differential rotation and toroidal magnetic fields

Abstract: Third order virial equations have been used to investigate the oscillations and the stability of the sequence of differentially rotating, compressible Maclaurin spheroids in the presence of toroidal magnetic fields. It is shown that the neutral point occurring at eccentricitye=0.731 13, which is the analogue of the first point of bifurcation along the Dedekind sequence, remains unaffected by the presence of differential rotation or a toroidal magnetic field. The point of onset of dynamical instability corresponding to the third harmonic deformations does, however, depend upon the magnetic field. It is shifted to values higher thane=0.966 96, the value that obtains in the case of uniform rotation; and a sufficiently large magnetic field can suppress this point. Complete frequency spectra (‘Kelvin’ modes belonging to the harmonicsl=3 and compressible modes belonging tol=1) are obtained in two cases of interest: when the equilibrium state is one of equipartition, and when toroidal magnetic and velocity fields (vanishing at the surface) are present in a configuration rotating with a constant angular velocity.

Alignment of the Earth's Magnetic Field with the Axis of Rotation and Reversals of Polarity: Laboratory Experiments on a Mechanism

Abstract: A mechanism that can cause the earth's external magnetic field to be aligned with the axis of rotation and to reverse at random times is described. It depends upon two arbitrary assumptions: (a) A dipole magnetic source, of unspecified nature, deep within the core, wanders randomly in direction. (b) The conducting fluid at the outer boundary of the core circulates in a pattern that is symmetrical with respect to the earth's axis of rotation. It is shown that such a circulating layer will act as an anisotropic screen, which will suppress the field of the transverse component of the source dipole. The field observed outside the core will be mainly that of the axial component of the source, and it will reverse abruptly whenever the direction of the source crosses the equatorial plane. Quantitative experimental studies, made on small-scale models, of the effects and their properties are described. The only datum that even suggests a value that may be used for the angular velocity of the circulating outer layer with respect to the source is the angular velocity of the westward drift of the earth's nondipolar field. If that value is used, the anisotropic screening effect comes out to be strong enough to give alignment and reversal characteristics that are similar to those found from paleomagnetic studies.