Influence Coefficient Matrices Research Articles

Study on the Aerodynamic Characteristics of Wings Flying Over the Nonplanar Ground Surface

Aerodynamic analysis of NACA wings moving with a constant speed over guideways are performed using an indirect boundary element method (potential-based panel method). An integral equation is obtained by applying Green's theorem on all surfaces of the fluid domain. The surfaces over the wing and the guideways are discretized as rectangular panel elements. Constant strength singularities are distributed over the panel elements. The viscous shear layer behind the wing is represented by constant strength dipoles. The unknown strengths of potentials are determined by inverting the aerodynamic influence coefficient matrices constructed by using the no penetration conditions on the surfaces and the Kutta condition at the trailing edge of the wing. The aerodynamic characteristics for the wings flying over nonplanar ground surfaces are investigated for several ground heights.

Evaluation of Computational Algorithms Suitable for Fluid-Structure Interactions

The objective was to identify and mathematically evaluate suitable methods to transfer information between nonlinear computational fluid dynamics (CFD) and computational structural dynamics (CSD) grids. This data transfer is vital in the field of computational aeroelasticity, where the interpolation method between the two grids can easily be the limiting factor in the accuracy of an aeroelastic simulation. The data to be transferred can include a variety of field variables, such as deflections, loads, pressure, and temperature. For a method to be suitable, it is important that it provide a smooth, yet accurate transfer of data for a wide variety of functional forms that the data may represent. An extensive literature survey was completed that identified current algorithms in use, as well as other candidate algorithms from different implementations, such as mapping and CAD/CAM. The performance of the various methods was assessed on a number of analytical functions, followed by a series of applications that have been or are currently being studied using nonlinear CFD methods coupled with linear representations of the CSD equations (equivalent plate/shell mode shapes and influence coefficient matrices). Two methods, multiquadric-biharmonic and thin-plate spline, are shown to be the most robust, cost-effective, and accurate of all of the methods tested.

Kinematics of a five degree-of-freedom prosthetic arm

The kinematics of a prosthetic arm consisting of a two-DOF multiloop spatial mechanism and a three-DOF series mechanism is described in this paper. In the beginning, the position analysis is discussed. After that the velocity and acceleration are analyzed. The forward and inverse formulas of velocity and acceleration are deduced using kinematic influence coefficient by means of hypothetic mechanism method and all formulas are obtained by explicit expressions of influence coefficient matrices that are the function of dimensional and positional parameters. Finally, a numerical example is calculated and the results of the calculation are verified by an equivalent plane mechanism. The method proposed in this paper may be expanded to the kinematic analysis of any complex series–parallel mechanism.

Analytical integration in the 2D boundary element method

The integrals required in the computation of influence coefficient matrices of the boundary element method (BEM) depend on the distance r(x,x′) from the collocation point or field point x to the source or load point x′. As a consequence, a distinction must be made between the case where the collocation point does not belong to the integration domain (proper integrals) and the case where the collocation point does belong to the integration domain (improper integrals). Moreover, situations arise where x comes close to x′ and the integrals, albeit of a regular character, behave almost as improper, this case being referred to as nearly singular integration. Analytical integration captures best the singular or nearly singular kernel behavior, but this technique can only be carried out in very simple situations as, for instance, boundary integrals over straight elements. In the present paper a set of useful analytical integration formulas for the 2D BEM with curved elements is deduced, employing a symbolic computational algebra system. © 1997 John Wiley & Sons, Ltd.

Analytical integration in the 2D boundary element method

The integrals required in the computation of influence coefficient matrices of the boundary element method (BEM) depend on the distance r(x,x′) from the collocation point or field point x to the source or load point x′. As a consequence, a distinction must be made between the case where the collocation point does not belong to the integration domain (proper integrals) and the case where the collocation point does belong to the integration domain (improper integrals). Moreover, situations arise where x comes close to x′ and the integrals, albeit of a regular character, behave almost as improper, this case being referred to as nearly singular integration. Analytical integration captures best the singular or nearly singular kernel behavior, but this technique can only be carried out in very simple situations as, for instance, boundary integrals over straight elements. In the present paper a set of useful analytical integration formulas for the 2D BEM with curved elements is deduced, employing a symbolic computational algebra system. © 1997 John Wiley & Sons, Ltd.

Prediction of Strip Profile in Rolling Process Using Influence Coefficients and Boussinesq’s Equations

A simplified numerical method is proposed in this article to predict strip profile. The mill is simulated by beams (rolls) on the infinite elastic half space (between rolls and strip). The roll deflection and indentation are calculated using the influence coefficients and Boussinesq’s equations respectively. The roll contours in the contact zone can be expressed in terms of the influence coefficient and indentation matrices. Compatibility equations are derived by matching two surface profiles in the contact zone. Equilibrium equations ensure force and moment balance of the strip and each roll. A complete linear system is established by assembling compatibility and equilibrium equations. The contact rolling pressure distribution and the rigid motion of the roll chocks can be solved. Roll surface contours and strip profiles can be further calculated accordingly.

DEVELOPMENT AND MODIFICATION OF A UNIFIED BALANCING METHOD FOR UNSYMMETRICAL ROTOR-BEARING SYSTEMS

On the basis of theoretical developments, this study proposes procedures for a modified unified balancing method for unsymmetrical rotor-bearing systems. A formulation of modal influence coefficient matrices is derived from the motion equations for unsymmetrical rotors, using a complex co-ordinate representation and the finite element method. Due to unequal properties in two principal directions, two sets of modal influence coefficients are presented. This formulation indicates that two trial masses in different directions are required in the two trial operations for each balancing plane. Also, the modal influence coefficients are found to be correlated with forward precession and unbalanced forces when asymmetry of bearings is considered. Therefore, forward precessions instead of measured displacements are required to calculate the unbalance distribution. Several examples are presented to verify the validity of the present work.

A MODIFIED INFLUENCE COEFFICIENT METHOD FOR BALANCING UNSYMMETRICAL ROTOR-BEARING SYSTEMS

A formulation of influence coefficient matrices is derived from the equations of motion for unsymmetrical rotors by using a complex co-ordinate representation and the finite element method. Due to the unequal properties in the two principal directions, the present formulation results in two sets of modified influence coefficients. The formulation indicates that two trial masses place in different directions are required in the two trial operations for each balancing plane. Also, from the analysis, the modified influence coefficients are found to be correlated to forward precession and imbalanced forces when the asymmetry of the bearings is considered. Therefore one requires forward precessions instead of measured displacements to calculate the influence coefficients and imbalance distribution. Several examples are presented to verify the validity of the present work.

Constraint embedding in kinematics and dynamics of hybrid manipulator systems

An approach to kinematics and dynamics based on kinematic influence coefficient matrices is proposed for applications to the analysis and control of motion controlled multibody systems, e.g., hybrid robotic manipulator systems. The scheme is unique in a sense that all the kinematic constraints are completely embedded into the formulations at the kinematics level and equation of motion of the system is obtained in a closed form with respect to the minimal set of independent joint coordinates. Furthermore, all kinematic and dynamic formulations are expressed compactly in the same format by using two special algebraic operators. This isomorphic formalism allows systematic transformations of kinematic and dynamic informations between different sets of coordinates in a purely algebraic way.

Impact Vibration Analysis of Group of Hexagonal Bars.

A simulation method was studied to calculate the vibration response during seismic excitation of a group of hexagonal bars installed in a restraint. Since impact phenomena take place between adjacent bars, or between a bar and a restraint, nonlinear force caused by the impact should be considered in the response calculation. The simulation method is a combination of the mode superposition method and the pseudo external force method. Time integration is carried out using the method proposed by Nigam. The nonlinear forces at the next time step can be determined through matrix calculation by introducing influence coefficient matrices which express the influence of impact forces on intersections at impact points. In this report, the influence coefficient matrices are expanded for hexagonal bars in a two-dimensional arrangement, by taking the impact direction of each impact point into consideration. In order to verify the above method, the calculated results were compared with experimental ones. The two results for maximum impact force were similar.

A Numerical Analysis for Piston Skirts in Mixed Lubrication: Part II—Deformation Considerations

This paper presents a mathematical model for piston skirts in mixed lubrication. It takes into account the effects of surface waviness, roughness, piston skirt surface profile, bulk elastic deformation and thermal distortion of both piston skirts and cylinder bore on piston motion, lubrication and friction. The corresponding computer program developed can be used to calculate the entire piston trajectory and the hydrodynamic and contact friction forces as functions of crank angle under engine running conditions. Complete distributions of the oil film thickness and elastic deformation as well as the hydrodynamic and contact pressures can also be given at any crank angle if needed. This paper is the second part of a series of two papers. The first part (Basic Modeling), presented earlier by Zhu et al. (1991), gave the basic formulation and some preliminary results without bulk deformation considerations. In the present part, the three-dimensional finite element method is used to calculate so-called influence coefficient matrices. These matrices are repeatedly used to compute bulk elastic deformations of piston skirts. Results for 12 different cases are presented, and discussions are given focusing on the influences of elastic and thermal deformations on piston motion, lubrication and friction. An attempt to compare the calculated friction with experimental data is made, and agreement appears good for the two available cases. The computer program presented should be a useful tool for piston design and development.

Impact Vibration Analysis of Bar Group in Single Row.

A simulation method was studied to calculate vibration response of a bar group installed between restraints in a single row under seismic excitation. Because impact phenomena take place between adjacent bars or a bar and a restraint, nonlinear force caused by the impact should be considered in the response calculation. In this study, a new method for determining the nonlinear force without convergence iteration was proposed. The simulation method is basically a combination of the mode superposition method and the pseudo external force method. The method proposed by Nigam is applied for time integration. An impact point is modeled as a set of a gap, a linear spring and a linear damper. The nonlinear forces at the next time step can be determined with matrix calculation by introducing influence coefficient matrices, which express the influence of impact forces on intersections at impact points. In order to verify the above method, the calculation results were compared with experimental ones, which showed that the method can simulate the experimental results well.

A boundary integral formulation for natural convection flows

A novel solver for natural convection flows based on the boundary integral equation method is presented here. The solver can be used to obtain flow solutions in arbitrary 2–D geometries with modest computational effort. Both the vorticity and thermal transport equations are treated as modified Helmholtz equations and solved using boundary integral methods. The non-linear inertial terms partially manifest themselves as volume sources which are computed iteratively using a method of characteristics approach. The buoyancy term is treated as a volume vorticity source. Rapid computation of the volume fields at each iteration level is achieved by precomputing the influence coefficient matrices. Results are presented for flow in a cavity driven by both shear and buoyancy.

Simulation and visualization of 3D particle cloud electrodynamics

A numerical algorithm simulating the electrodynamics of many-particle systems is presented. This algorithm requires solution of the 3-D Poission equation for charge conservation using the boundary integral equation method, and integration of the dynamical equations of motion for discrete interacting particles. Motion of the particles is due to both Coulomb forces and Stokes's drag. Particles suffer elastic collisions with walls and each other. A scatter-gather method together with the use of influence coefficient matrices allowed two-order-of-magnitude speedups in computation compared to direct integration methods. The simulation is implemented within a user interface on a minisupercomputer to enable interactive steering of solution parameters and dynamic visualization of results. Sample solutions for xerographic development of image pixels are discussed. >

A boundary integral equation method for Navier‐Stokes equations—application to flow in annulus of eccentric cylinders

AbstractA novel Navier‐Stokes solver based on the boundary integral equation method is presented. The solver can be used to obtain flow solutions in arbitrary 2D geometries with modest computational effort. The vorticity transport equation is modelled as a modified Helmholtz equation with the wave number dependent on the flow Reynolds number. The non‐linear inertial terms partly manifest themselves as volume vorticity sources which are computed iteratively by tracking flow trajectories. The integral equation representations of the Helmholtz equation for vorticity and Poisson equation for streamfunction are solved directly for the unknown vorticity boundary conditions. Rapid computation of the flow and vorticity field in the volume at each iteration level is achieved by precomputing the influence coefficient matrices. The pressure field can be extracted from the converged streamfunction and vorticity fields. The solver is validated by considering flow in a converging channel (Hamel flow). The solver is then applied to flow in the annulus of eccentric cylinders. Results are presented for various Reynolds numbers and compared with the literature.

Integrated aerodynamic-structural design of a transport wing

The integrated aerodynamic-structural design of a subsonic transport wing for minimum weight subject to required range is formulated and solved. The problem requires large computational resources, and two methods are used to alleviate the computational burden. First, a modular sensitivity method that permits the usage of black-box disciplinary software packages, is used to reduce the cost of sensitivity derivatives. In particular, it is shown that derivatives of the aeroelastic response and divergence speed can be calculated without the costly computation of derivatives of aerodynamic influence coefficient and structural stiffness matrices. A sequential approximate optimization is used to further reduce computational cost. The optimization procedure is shown to require a relatively small number of analysis and sensitivity calculations.

Electrical Nonuniformities in Commercial-Scale Slagging Magnetohydrodynamic Channels

The development of nonuniformities in commercial-scale linear magnetohydrodynamic (MHD) generators with slag layers along the electrode walls has been studied using a computer simulation. The model is based on the influence coefficient matrices, which define the electrical characteristics of the MHD plasma, together with nonlinear elements to describe the near-wall electrical behavior. The nonlinear effects included are slag polarization and burnout, and arcing at the anodes and cathodes. Time-dependent calculations have been carried out for an isolated 50-electrode-pair section of a commercial generator. To avoid difficulties with end effects, a model with repeating 50-electrode-pair sections has also been developed. The predicted steady-state patterns of nonuniformities are in good agreement with experiment. The effect of various model parameters on the properties of the nonuniformities is discussed.

Charge transport in Navier-Stokes flow

The physics of charge transport in steady Navier-Stokes flow is modeled using a novel algorithm that considers full coupling of the electrostatic and fluid equations and is applicable to arbitrary geometries. The hybrid boundary element method-method of characteristics approach is used to solve the nonlinear set of charge transport equations. The fluid problem, inclusive of electrohydrodynamic effects, is solved using a vorticity-stream function boundary-integral-equation formulation that does not require vorticity boundary conditions to be specified a priori. A rapid iterative solution is afforded by the use of influence coefficient matrices that simplify reanalysis and recalculation of fields to the product and sum of matrices. The algorithm is fairly robust, and convergence to less than 2% maximum change is attained, with underrelaxation, in less than 15 iterations for moderately high Reynold's numbers. >

Electrohydrodynamics of charge transport in stoke's flow

The physics of charge transport in steady Stoke's flow is modeled using a novel algorithm that considers full coupling of the electrostatic and fluid equations and is applicable to fairly arbitrary geometries. The hybrid boundary element method - method of characteristics (BEM-MOC) approach is used to solve the charge transport equations. The fluid equations, inclusive of electrohydrodynamic effects (EHD), are solved using a novel vorticity-stream function boundary integral equation formulation (BIEM) for Stoke's flow. An important advantage is that vorticity boundary conditions need not be specified a priori. Rapid iterative solution is afforded by the use of influence coefficient matrices that simplify re-analysis and recalculation of fields to the product and sum of matrices.

Charge transport in viscous vortex flows

The physics of charge transport in a bent channel is analyzed using a vorticity transport model for fluid flow to better represent recirculation conditions at corners. The electrostatics is governed by the coupled nonlinear set comprised of the Poisson and current continuity equations. Electrostatic body forces are assumed to have a negligible effect on the flow field. The boundary element method (BEM) is used to solve the vorticity-stream function integral equation formulation for the flow field. Two advantages of this method are no requirement for discretization of the interior, and no need for specification of vorticity boundary conditions. The Poisson equation is also solved using the BEM. Influence coefficient matrices are used to facilitate both reanalysis and rapid calculations of field parameters on a fixed grid. This technique allows desired quantities to be computed by matrix multiplication rather than time-consuming repetitive integrations. The current continuity equation is solved using the method of characteristics. The solution algorithm alternates between the methods for field and source solution. Sample results for full ON and full OFF of the ion current are discussed for both inviscid potential and viscous flows.