Jacobian Matrix Method Research Articles

Periodic solutions preserved by nonstandard finite-difference schemes for the Lotka–Volterra system: a different approach

The Lotka–Volterra predator–prey system x′ = x − xy, y′ = − y+xy is a good differential equation system for testing numerical methods. This model gives rise to mutually periodic solutions surrounding the positive fixed point (1,1), provided the initial conditions are positive. Standard finite-difference methods produce solutions that spiral into or out of the positive fixed point. Previously, the author [Roeger, J. Diff. Equ. Appl. 12(9) (2006), pp. 937–948], generalized three different classes of nonstandard finite-difference methods that when applied to the predator–prey system produced periodic solutions. These methods preserve weighted area; they are symplectic with respect to a noncanonical structure and have the property that the computed points do not spiral. In this paper, we use a different approach. We apply the Jacobian matrix procedure to find a fourth class of nonstandard finite-difference methods. The Jacobian matrix method gives more general nonstandard methods that also produce periodic solutions for the predator–prey model. These methods also preserve the positivity property of the solutions.

Evolution of discrete local levels into an impurity band in solidified inert gas solution

The density of states g(ω) of disordered solutions of solidified inert gases are calculated using the Jacobian matrix method. The transformation of a discrete vibrational level into an impurity band at a growing concentration of light impurity atoms is investigated. It is shown that a 1-10% change in the impurity concentration leads to smearing the local discrete level into an impurity band. As this occurs, additional resonance levels appear which carry important information about the impurity-impurity and impurity-basic lattice force interactions in such solutions.

Permanence and stability of equilibrium for a two-prey one-predator discrete model

In this letter, a predator-prey system of two-prey one-predator discrete model is investigated. It is proved that the system is permanence under some appropriate conditions. By Jacobian matrix method, a sufficient and necessary condition is derived for the local asymptotic stability of a equilibrium of the system. Meanwhile, we give a suitable example for supporting our theoretical result.

Kinematic analysis and error modeling of TAU parallel robot

The TAU robot presents a new configuration of parallel robots with three degrees of freedom. This robotic configuration is well adapted to perform with a high precision and high stiffness within a large working range compared with a serial robot. It has the advantages of both parallel robots and serial robots. In this paper, the kinematic modeling and error modeling are established with all errors considered using Jacobian matrix method for the robot. Meanwhile, a very effective Jacobian approximation method is introduced to calculate the forward kinematic problem instead of Newton–Raphson method. It denotes that a closed form solution can be obtained instead of a numerical solution. A full size Jacobian matrix is used in carrying out error analysis, error budget, and model parameter estimation and identification. Simulation results indicate that both Jacobian matrix and Jacobian approximation method are correct and with a level of accuracy of micron meters. ADAMS's simulation results are used in verifying the established models.

DEVELOPMENT OF TESTING MACHINE USING PARALLEL MECHANISM

The bench test machine of single shaft is used to develop the motorcycle frame now. It is not expressible for the simple apparatus though combined load is added to the frame while the motorcycle is actually running. Parts of the motorcycle might break down some time. This is very dangerous and efficient test method is desired. But application of conventional test method of single or multiple actuators is difficult, because test object is relatively small and fragile. In this situation, we developed new method of load test using parallel mechanism. Advantage of this method is possibility to generate any assigned force or moment vector at desired small point. That is, combined skew force or three-axis moment test becomes possible by only software change. We actually designed practical parallel mechanism actuating system for motorcycle. End-effector force and moment vector are converted to cylinder pressure command by Jacobian matrix method.

Configuration Bifurcations Analysis of Six Degree-of-Freedom Symmetrical Stewart Parallel Mechanisms

The configuration bifurcations of Stewart parallel manipulators at singular positions induce the uncertainty of the moving trends of the manipulative platform. The Jacobian matrix method can determine the singular position of Stewart manipulators, but it cannot determine the configuration variation trend in the vicinity of the singular position. In order to investigate the concrete motion behaviors of the Stewart parallel manipulator at singular positions, we construct the algorithm for determining all the configuration branches and bifurcation points. Through detailed investigations of configuration bifurcation characteristics, we have found that with a decrease of the extensible legs’ length, the bifurcation points of configuration branches of the movable platform get together gradually and the bifurcation type changes from turning to dual-point bifurcation, and then, finally, it becomes multiple-point bifurcation.

Lyapunov spectrum determination from the FEM simulation of a chaotic advecting flow

The problem of the determination of the Lyapunov spectrum in chaotic advection using approximated velocity fields resulting from a standard FEM method is investigated. A fourth order Runge–Kutta scheme for trajectory integration is combined with a third order Jacobian matrix method with QR-factorization. After checking the algorithm on the standard Lorenz and coupled quartic oscillator systems, the method is applied to a model 3-D steady flow for which an analytical expression is known. Both linear and quadratic approximated velocity fields succeed in predicting the Lyapunov exponents as well as describing the chaotic or regular regions inside the flow with satisfactory accuracy. A more realistic flow is then studied in order to delineate the possible limitations of the approach. Copyright © 2005 John Wiley & Sons, Ltd.

Simulation of a Multi-Frequency Continuous-Wave Reconstruction Technique for Subsurface Conductivity and Dielectric-Constant Profiles

In this paper, an efficient inversion method to reconstruct conductivity and dielectric-constant profiles by launching a continuous sinusoidal wave from the surface of a layered medium is presented. The reconstruction code consists of two main partsthe forward model and the inversion algorithm. Two forward models are developed for applications in shallow subsurface sensing and nondestructive evaluation of disbonding in a layered dielectric medium. In the inversion process, the Jacobian Matrix method is used to modify the inverted profile iteratively toward the real formation profile. A constrained optimization method is applied in order to obtain maximum stability when noise is present. The choice of the discrete frequencies depends on the required spatial resolution, investigation depth, and the parameters to be inverted. At the present stage, only synthetic models with signal-to-noise ratio around 40 dB are used to test the inversion method. The conductivity and the dielectric constant profiles for different kinds of synthetic models are satisfactorily reconstructed by using this method.

Effect of external Gaussian random field on phonon spectrum of linear atomic chains

We present the theoretical study of the effect of external random field characterized by a Gaussian probability distribution function on the continuous phonon spectrum of one-dimensional (1D) chain, based on the Jacobian matrix method. The cumulative effect of the random field and simple isotopic defect is studied analytically and numerically. The Gaussian random field removes a square-root divergence appearing in the phonon spectrum of ideal 1D chain. The impurity phonon DOS shows strong dependence on the variance and the mean of the random field and exhibits very different behavior from the non-random case: the continuous spectrum is expanded and the δ-peak, describing discrete impurity vibrations in the non-random chain with the impurity, falls into a continuous zone.

Force Transmissibility Performance of Parallel Manipulators

AbstractIn this paper, a new force transmission index called the mean force transmission index (MFTI) is proposed, and the force transmissibility analysis procedure is established for parallel manipulators. The MFTI is an extended definition of the force transmission index (FTI) introduced by the authors previously. It is shown that the FTI is a function of the input velocity ratio (IVR) for a multi‐DOF mechanism of the same configuration. To represent the force transmissibility by a definite value, the MFTI is defined as the mean value of the normalized FTIs function over the whole range of the IVR. The force transmissibility analysis of two planar parallel manipulators is illustrated using the MFTI method. The result is compared with that of the Jacobian matrix method and the joint force index (JFI) method. It shows that, especially for symmetric parallel manipulators, an approximate inverse‐proportionality relationship exists between the JFI and MFTI, and between the maximum input torque/force and MFTI. It is concluded that the MFTI can be used as a quantitative measure of the force transmissibility performance for parallel manipulators. In the end, a design optimization problem is studied by taking the global force transmission index as the objective function. © 2003 Wiley Periodicals, Inc.

The force transmissivity index of planar linkage mechanisms

In this paper, a new force transmissivity index (FTI) of planar linkage mechanisms is proposed. The index is used to quantitatively measure the force transmission quality from the input link to an output link. Traditionally, the transmission angle is used to measure the ability to transmit motion for planar four-bar linkages. However, the transmission angle is usually limited to be used at four-bar linkages due to the structural simplicity. For complex multi-loop mechanisms, it is often difficult to define the force transmissivity. Here we have established a procedure of force transmissivity analysis for planar linkage mechanisms. The method is based on the static force analysis and the concept of power flow path. It is found that the force transmissivity of a mechanism depends not only on the configurations of the mechanism, but also on the selection of the output link and the forms of the loading. We have compared the results based on the FTI with the results from the Jacobian matrix method and the joint force index (JFI) method for four-bar mechanisms. It shows that the proposed FTI can describe the force transmission performance more accurately than other methods do. It is concluded that the index can be used as a better measure of force transmissivity analysis for planar linkage mechanisms.

DEVELOPMENT OF TESTING DEVICE USING PARALLEL MECHANISM

The bench test machine of single shaft is used to develop the motorcycle frame now. It is not expressible for the simple apparatus though combined load is added to the frame while the motorcycle is actually running. Parts of the motorcycle might break down some time. This is very dangerous and efficient test method is desired. But application of conventional test method of single or multiple actuators is difficult, because test object is relatively small and fragile. In this situation, we developed new method of load test using parallel mechanism. Advantage of this method is possibility to generate any assigned force or moment vector at desired small point. That is, combined skew force or three-axis moment test becomes possible by only software change. We actually designed practical parallel mechanism actuating system for motorcycle. End-effector force and moment vector are converted to cylinder pressure command by Jacobian matrix method.

Low Cost Numerical Calculation Method for On-line Inverse Kinematics of Redundant Manipulators

A large calculation cost is required to solve the inverse kinematics problem by the numerical iterative method ( i.e. Newton-Raphson method) if the dimension of the Jacobian matrix becomes large. In this paper a new high speed numerical calculation method (SMJM method: Simply Modified Jacobian Matrix method) to solve the inverse kinematics of non-redundant manipulators and redundant manipulators are proposed. The calculation time is decreased, by replacing some elements of the Jacobian matrix with zeroes. The Monte Carlo method had been used to verify the advantage of the proposed SMJM method for the six links non-redundant manipulator and the four links redundant manipulator.

Oscillatory Spectra of Surface Atoms in Layered Crystal (Quasi-1D Behavior)

Vibrations localized near the surface have been analyzed by the Jacobian matrix method taking into account discreteness of the lattice. It has been shown that localized surface vibrations in layered crystals with the complex lattice have quasi-one-dimensional character and their properties are described by exact solutions obtained in the framework of the one-dimensional model.

NORMAL MULTI-MODES OF NON-LINEAR EULER BEAMS

For a strongly non-linear multi-degree-of-freedom system, in general, one cannot consider one mode at a time as in linear modal analysis. In the absence of external excitation, the natural vibration often involves more than one mode at a time resulting in quasi-periodic or multi-periodic (toroidal) vibration. The normal multi-mode in free vibration have been formulated by means of the action-angle transformation and the resulting ordinary differential equations embedded in partial differential equations. Final multi-periodic solutions have been obtained by extending the newly developed Toeplitz Jacobian matrix method with multi-periodic fast Fourier transforms.

Toeplitz Jacobian Method for Nonlinear Double-Periodic Excitations

The Toeplitz Jacobian matrix method is an efficient algorithm for computing the steady state solutions of nonlinear periodic vibration. In this paper, the method is generalized by using multiple time scales to double-periodic solutions in a multi-frequency excited system. The method is combined with a standard multi-dimensional FFT algorithm to accurately simulate the nonlinear oscillators with widely separated frequencies. The continuation technique can also be incorporated with the Newton–Raphson iteration to further increase its efficiency, and to achieve the complete frequency response characteristics.

Toeplitz Jacobian Method for Nonlinear Double-Periodic Excitations

The Toeplitz Jacobian matrix method is an efficient algorithm for computing the steady state solutions of nonlinear periodic vibration. In this paper, the method is generalized by using multiple time scales to double-periodic solutions in a multi-frequency excited system. The method is combined with a standard multi-dimensional FFT algorithm to accurately simulate the nonlinear oscillators with widely separated frequencies. The continuation technique can also be incorporated with the Newton–Raphson iteration to further increase its efficiency, and to achieve the complete frequency response characteristics.

The effective nuclear charge model—IX. On the transferability of effective nuclear charges defined from homonuclear diatomic molecules

Effective nuclear charges of many triatomic molecules have been calculated inversely by the least squares method (Jacobian matrix method) from the experimental force constants which are determined by normal coordinate analyses using the observed vibrational data. The values of effective nuclear charges thus obtained are compared with those defined from homonuclear diatomic molecules. The results show that the transferability of effective nuclear charges from homonuclear diatomic molecules to triatomic molecules is moderately good. This gives support to the utility of the effective nuclear charge model.