On the parametrization of the three-dimensional rotation group.
Parameterization of group of rotations of Euclidean three-dimensional space applied to integration of rigid body motion equations
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Parameterization of group of rotations of Euclidean three-dimensional space applied to integration of rigid body motion equations
The inviscid and viscid effects existing within the passages of a three bladed axial flow inducer operating at a flow coefficient of 0.065 are investigated. The blade static pressure and blade limiting streamline angle distributions were determined and the three components of mean velocity, turbulence intensities, and turbulence stresses were measured at locations inside the inducer blade passage utilizing a rotating three sensor hotwire probe. Applicable equations were derived for the hotwire data reduction analysis and solved numerically to obtain the appropriate flow parameters. The three dimensional inviscid flow in the inducer was predicted by numerically solving the exact equations of motion, and the three dimensional viscid flow was predicted by incorporating the dominant viscous terms into the exact equations. The analytical results are compared with the experimental measurements and design values where appropriate. Radial velocities are found to be of the same order as axial velocities within the inducer passage, confirming the highly three dimensional characteristic of inducer flow. Total relative velocity distribution indicate a substantial velocity deficiency near the tip at mid-passage which expands significantly as the flow proceeds toward the inducer trailing edge. High turbulence intensities and turbulence stresses are concentrated within this core region. Considerable wake diffusion occurs immediately downstream of the inducer trailing edge to decay this loss core. Evidence of boundary layer interactions, blade blockage effects, radially inward flows, annulus wall effects, and backflows are all found to exist within the long, narrow passages of the inducer.
A simplified theory of the dynamic motion of aerofoils of finite thickness in transonic flow is presented which excludes the effect of shock waves on the aerofoil itself and, thus, is restricted to free stream Mach numbers equal to unity or above. Numerical examples are analyzed for two-dimensional steady and unsteady (including transient) aerofoil motion and three-dimensional steady and unsteady flow over delta wings. The effects of flow separation and improvements in Bernoulli's equation and the surface boundary condition are briefly discussed.
The three-dimensional, Reynolds-averaged, Navier-Stokes (RANS) equations are used to numerically simulate nonsteady vortical flow about a 65-deg sweep delta wing at 30-deg angle of attack. Two large-amplitude, high-rate, forced-roll motions, and a damped free-to-roll motion are presented. The free-to-roll motion is computed by coupling the time-dependent RANS equations to the flight dynamic equation of motion. The computed results are in good agreement with the forces, moments, and roll-angle time histories. Vortex breakdown is present in each case. Significant time lags in the vortex breakdown motions relative to the body motions strongly influence the dynamic forces and moments.
Nonlinear analysis of earth-moon system motion stability in three dimensions near L4 libration point when perturbed by sun
The problem of determining the trajectories, partially or wholly contained in the atmosphere of a spherical, nonrotating planet, is considered. The exact equations of motion for three-dimensional, aerodynamically affected flight are derived. Modified Chapman variables are introduced and the equations are transformed into a set suitable for analytic integration using asymptotic expansions. The trajectory is solved in two regions: the outer region, where the force may be considered a gravitational field with aerodynamic perturbations, and the inner region, where the force is predominantly aerodynamic, with gravity as a perturbation. The two solutions are matched directly. A composite solution, valid everywhere, is constructed by additive composition. This approach of directly matched asymptotic expansions applied to the exact equations of motion couched in terms of modified Chapman variables yields an analytical solution which should prove to be a powerful tool for aerodynamic orbit calculations.
The dynamics and stability of large orbiting flexible beams, and platforms and dish type structures oriented along the local horizontal are treated both analytically and numerically. It is assumed that such structures could be gravitationally stabilized by attaching a rigid light-weight dumbbell at the center of mass by a spring loaded hinge which also could provide viscous damping. For the beam, the small amplitude inplane pitch motion, dumbbell librational motion, and the anti-symmetric elastic modes are all coupled. The three dimensional equations of motion for a circular flat plate and shallow spherical shell in orbit with a two-degree-of freedom gimballed dumbbell are also developed and show that only those elastic modes described by a single nodal diameter line are influenced by the dumbbell motion. Stability criteria are developed for all the examples and a sensitivity study of the system response characteristics to the key system parameters is carried out.
The dynamics and stability of large orbiting flexible beams and platforms oriented along the local horizontal are treated both analytically and numerically. It is assumed that such structures could be gravitationally stabilized by attaching a rigid lightweight dumbbell at the center of mass by a spring loaded hinge which also could provide viscous damping. For the beam it is seen that the small amplitude inplane pitch motion, dumbbell librational motion, and the anti-symmetric elastic modes are all coupled. The three dimensional equations of motion for a circular flat plate in orbit with a two-degree-of-freedom gimballed dumbbell are also developed and show that only those elastic modes described by a single nodal diameter line are influenced by the dumbbell motion. Stability criteria are developed for both examples and a parametric study of the least damped mode characteristics together with numerically simulated transient responses are carried out.
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Three dimensional resonance effects on particle motion stability near earth-moon equilateral libration points
A third-order analytical solution for halo-type periodic motion about the collinear points of the circular-restricted problem is presented. The three-dimensional equations of motion are obtained by a Lagrangian formulation. The solution is constructed using the method of successive approximations in conjunction with a technique similar to the Lindstedt-Poincare method. The theory is applied to the sun-earth system.
Derivation of three-dimensional three-degree-of- freedom reentry equations of motion in terrestrial navigational coordinates
Here, a novel approach is presented for computing general rigid body motion based on a few known linear accelerations. This method utilizes linear acceleration data obtained from three distinct points on the body, all within a body-fixed reference frame. The only requirement is that the three chosen points must not be collinear. A system of differential-algebraic equations is derived, combining principles of rigid body kinematics with theory of the rotation group SO(3). These equations provide a framework for numerically computing various motion parameters, including angular velocity, angular acceleration, body orientation, velocity field, acceleration field, and displacement field. By numerically solving this system of equations, we can fully characterize rigid body motion in three-dimensional space. A numerical example is provided to demonstrate the practical implementation and efficacy of the proposed technique, illustrating its potential for accurate motion computation in various applications.
A two-dimensional-dependent calculation of a forced turbulent mixing layer is reported. The approach employs a k-epsilon turbulence model to represent the effects of motions which are three dimensional or out of phase with the forcing cycle. Calculations that correspond to conditions of the measurements of Weisbrot (1984) and Weisbrot and Wygnanski (1988) indicate that the main features of the mixing layer development are captured by the computation. The effect of grid refinement on the computed result indicates the importance of adequate resolution of the near field in correctly reproducing the flow evolution. The inlet boundary condition, similarly, is shown to play a decisive role in determining subsequent development.
Stereoscopy and motion provide clues to outlines of objects. Digital image-processing system acts as "intelligent" automatic machine-vision system by processing views from stereoscopic television cameras into three-dimensional coordinates of moving object in view. Epipolar-line technique used to find corresponding points in stereoscopic views. Robotic vision system analyzes views from two television cameras to detect rigid three-dimensional objects and reconstruct numerically in terms of coordinates of corner points. Stereoscopy and effects of motion on two images complement each other in providing image-analyzing subsystem with clues to natures and locations of principal features.
To study the eolian mechanics of saltating particles, both an experimental investigation of the flow field around a model crater in an atmospheric boundary layer wind tunnel and numerical solutions of the two- and three-dimensional equations of motion of a single particle under the influence of a turbulent boundary layer were conducted. Two-dimensional particle motion was calculated for flow near the surfaces of both Earth and Mars. For the case of Earth both a turbulent boundary layer with a viscous sublayer and one without were calculated. For the case of Mars it was only necessary to calculate turbulent boundary layer flow with a laminar sublayer because of the low values of friction Reynolds number; however, it was necessary to include the effects of slip flow on a particle caused by the rarefied Martian atmosphere. In the equations of motion the lift force functions were developed to act on a single particle only in the laminar sublayer or a corresponding small region of high shear near the surface for a fully turbulent boundary layer. The lift force functions were developed from the analytical work by Saffman concerning the lift force acting on a particle in simple shear flow.
Reconfigurable neutral-atom arrays are a promising platform for quantum computing, quantum simulation, and quantum metrology, but atom transport using frequency-chirped acousto-optic deflectors (AODs) is limited by chirp-induced acoustic lensing and trajectory distortion. We address these limitations using a three-dimensional acousto-optic deflector lens (3D-AODL), a design predicted to reduce long-range transport times by more than a factor of two. We further introduce fading-Shepard waveforms that circumvent finite AOD bandwidth, enabling sustained axial displacement. We demonstrate unrestricted three-dimensional optical-tweezer motion over a 200 μm × 200 μm × 136 μm volume with velocities exceeding 4.2 m/s. Arbitrary three-dimensional control of optical-tweezer trajectories enables rapid atom rearrangement and dynamical engineering of optical potentials in tweezer arrays and optical lattices. This capability advances quantum control and atom manipulation in neutral-atom quantum processors by enabling faster rearrangement, higher clock rates, and scalable sorting in complex geometries.
Cosmic rays interact with mesoscale configurations of the interplanetary magnetic field. A technique is presented for calculating such configurations in the inner solar system, which are due to streams and source conditions near the sun, and maps of magnetic field are constructed for some plausible stream and source conditions. One effect of these mesoscale configurations on galactic cosmic rays is shown to be an out-of-the-ecliptic gradient drift sufficient to explain Forbush decreases. The effects on solar energetic particles include small polar drifts due to the field gradients and a possibly large modification of the time-intensity profiles and anisotropy characteristics due to the formation of mirror configurations in space. If a diffusion model is applicable to solar particles, the true diffusion coefficient will be masked by the effects of streams. A conceptual model which incorporates these ideas and those of several other models is presented.