Showing papers on "Rotation published in 1971"

Mental Rotation of Three-Dimensional Objects

Abstract: The time required to recognize that two perspective drawings portray objects of the same three-dimensional shape is found to be (i) a linearly increasing function of the angular difference in the portrayed orientations of the two objects and (ii) no shorter for differences corresponding simply to a rigid rotation of one of the two-dimensional drawings in its own picture plane than for differences corresponding to a rotation of the three-dimensional object in depth.

Lateral and rocking vibration of footings

Abstract: Numerical data are presented for the steady-state response of a rigid circular disk, or footing, supported at the surface of an elastic halfspace and excited by a harmonically varying horizontal force and a harmonically varying overturning moment. The disk is considered to be massless. The response quantities evaluated include the displacements of the disk in the directions of the exciting forces, the rotation due to the horizontal force, and the distributions of the contact stresses beneath the disk. Data are also presented for the stiffness and damping coefficients in an equivalent spring-dashpot representation of the disk-foundation system. The major displacement components are finally compared with those determined from two previously reported approximate solutions, including one based on an assumed distribution of the contact pressure.

Structure of a Line Vortex in an Imposed Strain

Abstract: The velocity of a vortex line depends on its structure, i.e. the shape of the cross-section and the detailed vorticity distri­bution. As a first step towards an understanding of how the struc­ture depends on the motion and the construction of a valid approxi­mation for the motion of vortex lines in general flow fields, we consider the structure of straight line vortices in a uniform two-dimensional straining field. Two cases are considered in detail, irrotational strain and simple shear. In the first case, it is shown that steady exact solutions of the inviscid equations exist, in which the boundary of the vortex is an ellipse with principal axes at 45° to the principal axes of strain. There are two possible axis ratios provided e/ωo 0.15, and it is believed from some numerical work that in this case the strain field will cause the vortex to break up. For simple shear, there is one steady shape of elliptical form if the shear rotation and vorticity are in the same sense and e′<ωo, where e′ is the rate of shear. The major axis is parallel to the streamlines and the shape is stable to two-dimensional deformations. For shear rotation and vorticity in opposite senses, there are two steady elliptical shapes if e′/ωo<0.21, with major axes perpendicular to the streamlines. The more elongated form is unstable, and the less elongated one is stable. Disturbances of three-dimensional form are also considered in the limit of extremely large axial wavelength.

Vibration analysis of rotating cantilever plates

Abstract: A finite element technique is used to determine the natural frequencies and the mode shapes of a cantilever plate mounted on the periphery of a rotating disc. The plane of the plate is assumed to make any arbitrary angle with the plane of rotation of the disc. The distributed centrifugal force is resolved into two components—one acting in the plane of the plate and the other normal to the plate. The stresses produced in the middle surface of the plate due to the in-plane forces are first determined. The increase in the bending stiffness of the plate elements due to these in-plane stresses is obtained in a manner similar to that used in the stability analysis of plates. The component of the distributed centrifugal force normal to the plate surface is added to the inertia force. From the results of computations carried out for various values of the aspect ratio, the speed of rotation, the disc radius and the setting angle, empirical formulae are derived giving the effect of these parameters on the natural frequencies. These empirical formulae are observed to be in agreement with the corresponding known formulae for rotating cantilever beams, when the aspect ratio is high.

Motion of a Vortex Filament and its Relation to Elastica

Abstract: Steady rotation of a very thin plane vortex filament with uniform angular velocity \(-\varOmega\) in the direction opposite to the circulatory rotation around it is discussed on the basis of the localized induction equation. It is shown that the possible form of the filament is that of the plane elastic filament of flexural rigidity B under the action of thrust F applied at its ends provided that \(G/\varOmega{=}B/F\), where G is the coefficient of local induction proportional to the circulation of the filament.

Strong cylindrical shocks in a rotating gas

Abstract: Similarity solutions describing the flow behind a diverging strong cylindrical shock wave, advancing into a nonuniform gas having solid body rotation, are studied. The effects of the angular velocity variation on the shock velocity are shown graphically. It is found that an increase in the initial angular velocity leads to a decrease in the shock velocity.

On hydromagnetic precession in a cylinder

Abstract: The hydrodynamic theory of the resonant cylinder (Gans 1970) is extended to include the effects of a magnetic field parallel to the rotation axis. The linear response is modified by a change in boundary-layer suction and a change in the resonant length. These effects are of equal importance. The theory is valid for small container conductivity and for amplitudes such that the cube of the amplitude is less than the dimensionless precession rate. The importance of container conductivity is assessed. The free modes of the system are given in an appendix. These modes move both east and west.Experimental apparatus capable of producing magnetic Reynolds numbers of the order of 20 for indefinite lengths of time is described. The apparatus was used to assess the linear theory, though not designed for this purpose. Experiments beyond the range of linear theory are described. The results show finite amplitude effects similar to those previously observed in precessing spheroids in the absence of magnetic effects (Malkus 1968). Additional structure attributable to magnetic effects is observed.

Spin-up of a stratified fluid: theory and experiment

Abstract: Stratified spin-up, the process of adjustment of a uniformly rotating stratified fluid to an abrupt change in the rotation of the container, is important in many geophysical contexts. An experimental study of this process is presented here for the case where a linearly stratified salt solution is enclosed in a cylindrical container whose rotation rate is changed by a small amount. Results are presented for a limited range of values of B, the internal Froude number, which measures the ratio of the frequencies due to buoyancy and rotation. The experimental study is augmented by a theoretical treatment of idealized models which clarify the more fundamental physical processes that occur. The response of a stratified fluid is faster than that of a homogeneous fluid but the adjustment is limited to layers near the bottom and top boundaries the thickness of which is determined by the value of B. A comparison of the experimental results with the theories of Holton, Walin and Sakurai is also made and it is shown that for the present physical arrangement (insulated side walls) the theories of the latter two authors agree much more closely with experiment than does the theory of Holton. However, all three theories tend to over-estimate the azimuthal displacement in the regions near the upper and lower boundaries where the spin-up is most rapid. The Sweet-Eddington circulation, which accompanies the ideal state of rigid-body rotation, can be significant under normal laboratory conditions and it was necessary to correct some of the spin-up results for this effect. The circulation in the vertical plane is described qualitatively.

The large-scale velocity fields of the solar atmosphere

Abstract: Magnetograph velocity data are studied for evidence of large-scale velocity fields. It is established that there exist on the surface of the sun regions of more or less coherent downward motion with dimensions of the order of a solar radius. Velocity amplitudes in these regions are in the range 50–75 m/sec. Downward-moving large-scale features are observed to live for at least several days in general and to rotate at least approximately with the solar rotation rate. Horizontal east-west motions appears to have lifetimes of at least many months. The extent in longitude of these horizontal features is about 25°. There is no evidence for meridional motions from these data, with an upper limit to the line-of-sight velocity of about 30 m/sec. Active regions, as reported previously, are areas of generally downward motion. Some features in the autocorrelation of the rotational velocity of the sun remain unexplained.

Magneto‐Optical Rotation and Ellipticity Measurements with a Spinning Analyzer

Abstract: An apparatus employing the spinning analyzer technique is described for measurement of Faraday and Kerr rotation in fields up to 20 kOe and at temperatures down to 3 K. Rotation angles are measurable to an accuracy better than 0.01° and are continuously recorded. The technique is extended to include measurement of Faraday and Kerr ellipticity.

A Velocity Field for Some Passive Earth Pressure Problems

Abstract: Synopsis A simple velocity field is developed for the passive earth pressure problem of the displacement of a rough plane wall into a mass of dry sand. Three modes of wall displacement are considered: wall rotation about the top, wall rotation about the toe and wall translation. The validity of the velocity field for the three cases is discussed. The pattern of the velocity field and the velocity boundary conditions enables predictions to be made of the strain fields associated with the three modes of wall movement and these predictions are compared with experimental observations. The velocity field also predicts the location and sequence of the rupture surfaces that occur in dense sand and leads to an understanding of the differences in the experimentally observed strain fields and wall stress distributions. This Paper follows on from an earlier paper (James and Bransby, 1970) in which the boundary stresses and the predicted principal stress directions in the mass were compared with measured boundary str...

The development of concentrated vortices: a numerical study

Abstract: Amongst the more important laboratory experiments which have produced concentrated vortices in rotating tanks are the sink experiments of Long and the bubble convection experiments of Turner & Lilly. This paper describes a numerical experiment which draws from the laboratory experiments those features which are believed to be most relevant to atmospheric vortices such as tornadoes and waterspouts.In the numerical model the mechanism driving the vortices is represented by an externally specified vertical body force field defined in a narrow neighbourhood of the axis of rotation. The body force field is applied to a tank of fluid initially in a state of rigid rotation and the subsequent flow development is obtained by solving the Navier–Stokes equations as an initial-value problem.Earlier investigations have revealed that concentrated vortices will form only for a restricted range of flow parameters, and for the numerical experiment this range was selected using an order-of-magnitude analysis of the steady Navier–Stokes equations for sink vortices performed by Morton. With values of the flow parameters obtained in this way, concentrated vortices with angular velocities up to 30 times that of the tank are generated, whereas only much weaker vortices are formed at other parametric states. The numerical solutions are also used to investigate the comparative effect of a free upper surface and a no-slip lid.The concentrated vortices produced in the numerical experiment grow downwards from near the top of the tank until they reach the bottom plate whereupon they strengthen rapidly before reaching a quasi-steady state. In the quasi-steady state the flow in the tank typically consists of the vortex at the axis of rotation, strong inflow and outflow boundary layers at the bottom and top plates respectively, and a region of slowly-rotating descending flow over the remainder of the tank. The flow is cyclonic (i.e. in the same sense as the tank) in the vortex core and over most of the bottom half of the tank and is anticyclonic over the upper half of the tank away from the axis of rotation.

Resistive Plasma Rotation and Shock Formation in Toroidal Geometry

Abstract: The rotation of a toroidally confined plasma about its magnetic axis due to resistive diffusion is studied by means of a single fluid model in which the only dissipation occurs through a small scalar resistivity. The time evolution of the rotation is examined by considering the resistivity as inducing first‐order time variation on a dissipationless steady state. The known instability of flows with small rotational velocity to rapidly speed up toward a critical speed is confirmed; more importantly, it is found that a shock which develops at the critical speed stabilizes the flow, so that the ultimate state of the system is characterized by steady rotation with a very weak shock front directed radially inward from the magnetic axis. The outward flux of plasma in this state is found to be essentially the Pfirsch‐Schluter flux enhanced by a factor of the square root of the aspect ratio.

A Model of Eye Movements Induced by Head Rotation

Abstract: It is well known that head rotation will induce eye movements known as rotational nystagmus, the slow phase of which compensates for head rotation fairly well, and the quick phase of which takes place intermittently in the opposite direction to the preceding slow phase. From both frequency and transient responses, it is confirmed that the slow phase velocity is proportional to the output of the semicircular canal, the main transducer of head rotation. The relationship between the canal output and the quick phase is also discussed. A simple model is proposed in which the quick phase and slow phase are separately generated. In cats under controlled ether anesthesia, it is found that both phases of the rotational nystagmus can be decomposed into primary and secondary components, and a new model of the vestibulo-ocular system is proposed which includes the simultaneous influence of these two components. The model is analyzed to find a condition where the summed effect of primary and secondary components of response constituting the slow phase of rotational ocular nystagmus can be made proportional to the canal output. Many simulation results are presented to demonstrate the validity of the model.

Device for sensing the direction and speed of a shaft

Abstract: A device for sensing the direction of rotation and the speed of a shaft includes a two-phase AC generator having a rotor attachable to the shaft and rotatable therewith and first and second spaced apart stator windings. The first and second stator windings are adapted to have first and second sinusoidal voltages established thereacross respectively, in response to rotation of the rotor with the shaft. The first sinusoidal voltage is adapted to lead the second sinusoidal voltage when the shaft rotates in a first direction and the second sinusoidal voltage is adapted to lead the first sinusoidal voltage when the shaft rotates in a direction opposite the first direction. A logic circuitry is provided for determining which of the first and second sinusoidal voltages is leading the other of the first and second sinusoidal voltages to thereby determine the direction of rotation of the shaft. The magnitude of the first and second sinusoidal voltages is dependent upon the speed of rotation of the shaft and means are provided for sensing the magnitude of one of the sinusoidal voltages to provide an output signal indicative of the speed of rotation of the shaft.

Analytical methods for the orbits of artificial satellites of the moon.

Abstract: The motion of a close artificial satellite of the Moon is considered. The principal perturbations taken into account are caused by the nonsphericity of the Moon and the attraction of the Earth and the Sun. To begin with, the expansions of the disturbing functions due to the nonsphericity of the primary body and the action of the disturbing mass-point body have been derived. The second expansion is produced in terms of the Keplerian elements of a satellite and the spherical coordinates of the disturbing body. Both expansions are valid for an arbitrary reference plane. The motion of a satellite of the Moon is studied in the selenocentric coordinate system referred to the Lunar equator and rotating with respect to the fixed ecliptic system. However, the coordinate exes in the equatorial plane are chosen so that the angular speed of rotation of the system is small. The motion of the satellite is described by means of the contact elements which enable one to utilize the conventional Lagrange's planetary equations and may be regarded as the generalization of the notion of the osculating elements to the case of the disturbing function depending not only o the coordinates and the time but on the velocities as well. Two methods are proposed to represent the motion of Lunar satellites over long intervals of time: the von Zeipel method and the Euler method of analytical integration with application of the variation-of-elements technique at every step of integration. The second method is exposed in great detail.