Body-fixed Coordinate System Research Articles

Suspensions of fibers in viscoelastic fluids: Rheology

Suspensions of fibers in polymeric fluids are regarded as two-component fluids. One component (the fibers) is a rigid body fluid; the second component (polymeric fluid) is a fluid composed of dumbbells. States of the suspension are chosen to be characterized by overall momentum field, configuration space distribution functions of fibers and macromolecules (or alternatively fields of conformation tensors of fibers and macromolecules), and the field of angular momentum (in the body-fixed coordinate system) of the fibers. The time evolution equations (that include expressions for the stress tensor) for these fields are formulated by following Hamiltonian modeling (GENERIC). In the process of solving the governing equations a simplified set of governing equations is derived. This simplified set of equations extends Jeffery’s equations describing suspensions in simple fluids to equations describing suspensions in polymeric fluids. The time evolution and the corresponding expression for the stress tensor are discussed for both distribution functions and conformation tensors.

The stick insect, Obrimus asperrimus (Phasmida, Bacillidae) walking on different surfaces

Unrestrained adult stick insects ( Obrimus asperrimus) walked below a tread wheel 4 or 30 mm wide in the upside-down position or above a ‘bridge’ 30 or 60 cm wide in the upright position. They were recorded on video and the positions of reference points on the legs and on the body were measured on still frames. Step parameters such as step amplitude, step duration, swing duration, body height and ground width are given for broad and narrow footing as well as for leg trajectories and the course of leg joint angles. Joint angles were calculated directly between the leg segment vectors. Walking with a narrow footing, the stick insect used the same slow metachronal gait as on a broad footing. Adjustment to the narrow footing was accomplished by narrowing the ground base, raising the body distance, depressing the femora (by 40–45 °) and flexing the tibiae (20–25 ° in the front and middle legs). The upper turningpoint of the swing movement seems to be determined by a constant amplitude of the vertical movement component rather than a given position in a body-fixed coordinate system. Walking on a horizontal rod of circular cross-section, the insects preferred to walk upside down below the rod for small diameters, but preferred to walk upright above the rod if its diameter was great enough.

Numerical simulation of a rigid rotating body by Obrechkoff integration

It is first supposed that Euler's differential equations for the rotational dynamics of a rigid body have been solved, so that the angular velocity vector for the body and the derivative thereof are known vector-valued functions of time in a body-fixed coordinate system. A method is discussed by which a set of ordinary differential equations (ODEs) may be numerically integrated to deduce the orientation of the body in an inertial frame. The method employs a quaternion formulation for the orientation of the body, exhibits a local truncation error which is typically of fifth order, adheres to the boundary of the region of absolute stability, and requires only one angular velocity evaluation and one angular acceleration evaluation each step. It is then supposed that the solution to Euler's equations for a given body is not known. An interative process is obtained for the integration of Euler's equations and the ODEs for body orientation simultaneously.

Dynamic equations of motion for a rigid or deformable body in an arbitrary non-uniform potential flow field

In this paper we present a general method for calculating the hydrodynamic loads (forces and moments) acting on a deformable body moving with six degrees of freedom in a non-uniform ambient potential flow field. The corresponding expressions for the force and moment are given in a moving (body-fixed) coordinate system. The newly derived system of nonlinear differential equations of motion is shown to possess an important antisymmetry property. As a consequence of this special property, it is demonstrated that the motion of a rigid body embedded into a stationary flow field always renders a first integral. In a similar manner, we show that the motion of a deformable body in the presence of an arbitrary ambient flow field is Hamiltonian. A few practical applications of the proposed formulation for quadratic shapes and for weakly non-uniform external fields are presented. Also discussed is the self-propulsion mechanism of a deformable body moving in a non-uniform stationary flow field. It leads to a new parametric resonance phenomenon.

Generalized Kirchhoff equations for a deformable body moving in a weakly non-uniform flow field

The classical Kirchhoff’s method provides an efficient way of calculating the hydrodynamical loads (forces and moments) acting on a rigid body moving with six-degrees of freedom in an otherwise quiescent ideal fluid in terms of the body’s added-mass tensor. In this paper we provide a versatile extension of such a formulation to account for both the presence of an imposed ambient non-uniform flow field and the effect of surface deformation of a non-rigid body. The flow inhomogeneity is assumed to be weak when compared against the size of the body. The corresponding expressions for the force and moment are given in a moving body-fixed coordinate system and are obtained using the Lagally theorem. The newly derived system of nonlinear differential equations of motion is shown to possess a first integral. This can be interpreted as an energy-type conservation law and is a consequence of an anti-symmetry property of the coefficient matrix reported here for the first time. A few applications of the proposed formulation are presented including comparison with some existing limiting cases.

Use of the classical centrifugal-sudden approximation for transport and relaxation cross-sections

The classical centrifugal-sudden approximation (cCSA) in molecular collision dynamics is studied. Following the traditional quantal approach, the problem is first formulated in the body-fixed (BF) coordinate system exactly. Attempts to deduce the cCSA from the BF Hamiltonian, however, fail to provide proper directional information for the relative motion. Instead, the cCSA can be derived from the uncoupled angular momentum representation in the space-fixed coordinate system, and directional information can then be obtained. The cCSA is applied to the study of transport and relaxation cross-sections of the He-N2 and Ar-N2 systems. For both systems the dynamical error is investigated by comparing the cCSA results with exact classical trajectory results, while for the He-N2 system the quantization error is obtained by comparing the cCSA and quantal centrifugal-sudden approximation results. The cCSA is found to be accurate for the viscosity and diffusion cross-sections. Encouraging results were found for the relaxation cross-sections for depolarized Rayleigh light scattering, the viscomagnetic effect and the field effect on the thermal conductivity. When comparisons are possible, the quantization error is usually found to be larger than the dynamical error.

Microprocessor Implementation of the Inverse Dynamical System for Industrial Robot Control

The inverse system problem of robot control is resurveyed. It is found, that the body-fixed coordinate system also plays an important role in computational efficiency of inverse dynamical formulations. The driving-axis coordinate system, whose origin locates on the driving axis of each link, is proposed to derive the recursive Newton-Euler formulation of manipulator inverse dynamics. The new algorithm with considering trivial calculations is implemented using the Assembly language on an INTEL 8086/8087 microprocessor (5 MHz clock). The measurement of computational efficiency shows that this software package marginally satisties the sampling rate criterion of robot control even with a 16-bit microprocessor

Automatic localization of anatomical landmarks on the back surface and construction of a body-fixed coordinate system

A method for automatic measurement of anatomical landmarks on the back surface is presented. The landmarks correspond to the verteba prominens, the dimples of the posterior superior iliac spines and the sacrum point (beginning of rima ani), which are characterized by distinct surface curvature. The surface curvatures are calculated from rasterstereographic surface measurements. The procedure of isolating a region of interest for each landmark (surface segmentation) and the calculation of the landmark coordinates are described in detail. The accuracy of landmark localization was tested with serial rasterstereographs of 28 patients (with moderate idiopathic scoliosis). From the results the intrinsic accuracy of the method is estimated to be little more than 1 mm (depending on the sampling density of the surface measurement). Therefore, the landmarks may well be used for the objective definition of a body-fixed reference coordinate system. The accuracy is, however, dependent on the specific landmark and a minor influence of posture variations is observed.

The effect of reagent rotation on steric requirements of elementary exchange reactions

The body-fixed coordinate system appears to be particularly suited for discussing the role of reagent rotation. The simple picture provided by the j z -conserving approximation accords with the available evidence and with specially performed trajectory computations for the O + HCl( v= 0, j) and O + DCl( v= 0, j) reactions. An even simpler, Line-of-centers-type model is also discussed. Both the possible decline in reactivity at low rotational excitation and its increase at higher levels of excitation can be accounted for.

Rotation-vibration separability in the classical motion of triatomics

The intramolecular vibrational-rotational energy transfer is followed in time along classical trajectories of non-rigid (Ar 3clusters) and semi-rigid (H 2O) triatomics, where the exact amounts of rotational and vibrational energies are calculated for three body-fixed coordinate systems (Eckart, PAS, IMAS). It turns out that the Eckart system is allways a reasonable choice to partition such energies, although significative differences appear in comparisson with the IMAS for highly rotationally excited molecules, while the PAS differs considerably from the other two. The mechanism for rotation-vibration energy transfer proceeds via centrifugal interactions in the cases studied.

Free-falling autorotating plate - A coupled fluid and flight mechanics problem

A computational method coupling the three-degrees-of-freedom flight mechanics equations and the twodimensional, time-dependent Navier-Stokes equations was developed which could be used to predict the autorotating characteristics of a free-falling, two-dimensional flat plate. The two-dimensional incompressible Navier-Stokes equations were cast in a body-fixed coordinate system. The corresponding velocities were cast in an inertial reference system. The Navier-Stokes equations were represented by implicit finite differences and solved using a successive-overrelaxation iteration technique. The resulting aerodynamic coefficients were entered into the three-degrees-of-freedom flight mechanics equations. The system of ordinary differential equations describing the flight mechanics was solved using an Adams-Bashforth explicit method to predict the movement of the plate. New values of the boundary conditions for the Navier-Stokes equations solver were obtained from this movement of the plate. The process was repeated to advance the solution in time. The flight path of a freefalling autorotating plate was predicted using the computational procedure and the validity of the overall approach demonstrated by comparison with experiment.

Molecular reorientation from NMR relaxation times in planar molecules

The spectral density J(0) for the autocorrelation function referred to an arbitrarily chosen body-fixed coordinate system has been rederived for a planar asymmetric rotor undergoing rotational diffusive motions. The angular dependence of J(0) was found to be more complicated than in the case of coincidence of the body-fixed frame with the principal axes of rotation. Therefore, the orientation of the principal axes of the rotational diffusion tensor can be determined from four different J(0) or 1/T1 values. This result disagrees with previous conclusions based upon the assumption that the same angular dependence should be expected irrespective of the relative orientation of the diffusion tensor axes. Likewise, the values of the three diffusion coefficients can be in principle determined from more than two (or three) different J(0) values for molecules with (or without) a binary symmetry axis. In practice, however, only two relations between them can usually be obtained due to the uncertainty in T1 measurements and to the relative insensitivity of J(0) to some correlated changes in the diffusion coefficients for moderate anisotropic molecular rotations.

Hamiltonian model for the Brownian motion of a rigid rotor

Abstract We derive stochastic equations for the motion of a rigid rotor in a linear heat bath starting from a fully dynamical (Hamiltonian) description. We obtain a generalized Langevin equation for the angular velocity of the rotor in which the fluctuations in the torque arise from the initial conditions of the heat bath. The dissipation is in general nonlinear and is related to the fluctuations via a fluctuation-dissipation relation that is a natural consequence of our description. In the Langevin limit we show that our equations are equivalent to the phenomenological Euler-Langevin equations. However, whereas the latter are necessarily expressed in components referred to a body-fixed coordinate system, our representation is valid in an arbitrary system of axes and is in that sense more general. Finally, we also examine the reduction of our model to the rotational diffusion equation in the high viscosity limit. We show that this reduction can be carried out via standard methods of adiabatic elimination if our representation of the rotor equations of motion is used.

Finite element analysis of rotating disks

A finite element formulation is presented for the analysis of rotating disks in either a body-fixed or a space-fixed co-ordinate system. The in-plane stress distribution resulting from the in-plane body force due to rotation is determined first by a plane stress finite element analysis. This stress distribution is then used in calculating the out-of-plane geometric stiffness which in turn is added to the linear bending stiffness. In the space-fixed co-ordinate system, inertia and a viscous type damping also contribute to the out-of-plane stiffness, even in the steady state case. The formulation presented here places no restrictions on the disk geometry if the problem is solved in a body-fixed co-ordinate system, although only disks of axisymmetric geometry may be considered in the space-fixed co-ordinate system. A direct method of determining the critical speeds through an eigenvalue analysis in space-fixed co-ordinates is presented. Then the undamped steady state response to a space-fixed transverse point load is examined. The effects of a viscous type damping are also presented.

On the interweaving of partial cross sections of different parity

Partial cross sections of definite parity, calculated for electronic-rotational energy transfer in the F + H 2 collision system, interweave with increasing total angular momentum J. An explanation, in terms of diabatic curve crossings induced by the centrifugal potential in the body-fixed coordinate system, predicts the interweaving to occur only in systems having half-integer J.

On ’’Theory of collisions between an atom and a diatomic molecule in the body-fixed coordinate system’’

The nonparity asymptotic boundary conditions for the S matrix in the body−fixed close-coupling approximation are discussed.

Comment on ’’Theory of collisions between an atom and a diatomic molecule in the body-fixed coordinate system

It is shown that the phase factor iJ+j+l omitted in a theory of atom-diatom monreactive scattering formulated by schatz and kupperman but included in the Choi, Poe, and Tang theory does not lead to errors in the analysis of te (H,H2) system.

On the nature of unsteady three-dimensional laminar boundary-layer separation

Solutions have been obtained for a family of unsteady three-dimensional boundary-layer flows which approach separation as a result of the imposed pressure gradient. These solutions have been obtained in a co-ordinate system which is moving with a constant velocity relative to the body-fixed co-ordinate system. The flows studied are those which are steady in the moving co-ordinate system. The boundary-layer solutions have been obtained in the moving co-ordinate system using the technique of semi-similar solutions. The behaviour of the solutions as separation is approached has been used to infer the physical characteristics of unsteady three-dimensional separation.In the numerical solutions of the three-dimensional unsteady laminar boundary-layer equations, subject to an imposed pressure distribution, the approach to separation is characterized by a rapid increase in the number of iterations required to obtain converged solutions at each station and a corresponding rapid increase in the component of velocity normal to the body surface. The solutions obtained indicate that separation is best observed in a co-ordinate system moving with separation where streamlines turn to form an envelope which is the separation line, as in steady three-dimensional flow, and that this process occurs within the boundary layer (away from the wall) as in the unsteady two-dimensional case. This description of three-dimensional unsteady separation is a generalization of the two-dimensional (Moore-Rott-Sears) model for unsteady separation.

Theory of collisions between an atom and a diatomic molecule in the body-fixed coordinate system.a) I. Coupled differential equation and asymptotic boundary conditions

The body-fixed (BF) formulation for atom–diatom scatterings is developed to the extent that one can use it to perform accurate close-coupling calculation, without introducing further approximation except truncating a finite basis set of the target molecular wave function, on the same ground as one use the space-fixed (SF) formulation. In this formulation, the coupled differential equations are solved an the boundary conditions matched entirely in the BF coordinate system. A unitary transformation is used to obtain both the coupled differential equation and the boundary condition in BF system system from SF system. All properties of the solution with respect to parity are derived entirely from the transformation, without using the parity eignfunctions of the BF frame. Boundary conditions that yield the scattering (S) matrix and the reactance (R) matrix are presented for each parity in both the far asymptotic region (where the interaction and the centrifugal potentials are both negligible) and the near asymptotic region (where the interaction potential is negligible but the centrifugal potential is not). While our differential equations are the same as those derived by others with different methods, our asymptotic boundary conditions disagree with some existing ones. With a given form of the BF coupled differential equations, the acceptable boundary conditions are discussed.

Quantum mechanical theory of a structured atom–diatom collision system: A+BC(1Σ)

The problem of a 2P state atom colliding with a singlet sigma state diatom, which involves multiple potential surfaces, is investigated. Within a diabatic representation for the electronic degrees of freedom (plus spin–orbit interaction), coupled scattering equations are derived in both space-fixed and body–fixed coordinate systems, and coefficients analogous to Percival–Seaton coefficients are exhibited. Approximations to the exact equations, including angular momenta decoupling approximations, are discussed for both the space-fixed and body-fixed formalisms.