Constant Coefficients In Terms Research Articles

Stability of asymmetric rotors with asymmetric flexible disks

The presented work examines the dynamic behavior of an asymmetric rotor with asymmetric flexible disk, contrary to previous works on the subject, where researchers examined the effect of either rigid disk asymmetry or disk flexibility at a time. Account for the asymmetry of flexible disk in rotors constitutes the new contribution in this work. The suggested mathematical model combines Lagrangian approach with Rayleigh–Ritz method to derive the governing equations of motion of the rotor. Account for asymmetry of the flexible disk results in complicated and lengthy expressions for the potential and kinetic energies of the rotor, required in the adopted Lagrangian approach. Using symbolic computation simplified the derivation of the governing equations of motion with constant coefficients in terms of rotating coordinate system. Solution of the resulting eigenvalue problem provided numerical results for rotors with symmetric and asymmetric flexible disks, required to assess the effect of disk flexibility and asymmetry on the resulting frequencies and stability boundaries of the examined rotor system.

Note on some representations of general solutions to homogeneous linear difference equations

It is known that every solution to the second-order difference equation x_{n}=x_{n-1}+x_{n-2}=0, nge 2, can be written in the following form x_{n}=x_{0}f_{n-1}+x_{1}f_{n}, where f_{n} is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

Rubber accelerated ageing life prediction by Peck model considering initial hardness influence

Accelerated degradation test is an effective means for rapid and convenient prediction of rubber life. In the existing research, the influence of rubber hardness on performance degradation is neglected. Thus, the life prediction results are inaccurate. A rubber ageing model considering hardness influence was established on the basis of modified traditional dynamic degeneration trajectory equation, with which the effect of dispersion of the initial hardness of rubber on service life was described. Three types of natural rubber with different hardness were considered as research objects. On the basis of different high-temperature accelerated ageing tests, the deformation trajectory equation was used to model the compression set of each sample. Identification results show the relatively stable power exponential term coefficient and constant term coefficient, whereas there is large fluctuation in the reaction rate term coefficient. To simplify the research, the power exponent term coefficient and constant term coefficient were set as fixed values. Then, the reaction rate coefficient was estimated again. The rationality of parameter re-identification result was verified by the goodness of fit. The Arrhenius and Peck models were introduced in accordance with the obtained reaction rate coefficient identification results to simulate the reaction rate coefficients. Combined with Peck and dynamic models, a modified rubber ageing life prediction model was established, with which Monte Carlo method was employed to analyse the influence of initial hardness dispersion of rubber on service life under natural environment. The effectiveness of the model was verified by comparison with measured natural ageing data. The proposed ageing life prediction model can simultaneously consider the influences of initial hardness and temperature on rubber ageing with high accuracy. This study can provide a reference for the design and reliability evaluation of rubber parts.

Analytical solution of two-dimensional advection–dispersion equation with spatio-temporal coefficients for point sources in an infinite medium using Green’s function method

In the present study analytical solutions of a two-dimensional advection–dispersion equation (ADE) with spatially and temporally dependent longitudinal and lateral components of the dispersion coefficient and velocity are obtained using Green’s Function Method (GFM). These solutions describe solute transport in infinite horizontal groundwater flow, assimilating the spatio-temporal dependence of transport properties, dependence of dispersion coefficient on velocity, and the particulate heterogeneity of the aquifer. The solution is obtained in the general form of temporal dependence and the source term, from which solutions for instantaneous and continuous point sources are derived. The spatial dependence of groundwater velocity is considered non-homogeneous linear, whereas the dispersion coefficient is considered proportional to the square of spatial dependence of velocity. An asymptotically increasing temporal function is considered to illustrate the proposed solutions. The solutions are validated with the existing solutions derived from the proposed solutions in three special cases. The effect of spatially/temporally dependent heterogeneity on the solute transport is also demonstrated. To use the GFM, the ADE with spatio-temporally dependent coefficients is reduced to a dispersion equation with constant coefficients in terms of new position variables introduced through properly developed coordinate transformation equations. Also, a new time variable is introduced through a known transformation.

Fractional Laplace transform method in the framework of the CTIT transformation

We propose an adapted Laplace transform method that gives the solution of a linear fractional differential equation with constant coefficients in terms of exponential function. After we mention what the utilized transformation, the CTIT transformation, is based on, we explain how it can reduce the problem from fractional form to ordinary form when it is used with Laplace transformation, via some examples for 0<α<2 where α is the order of fractional derivative. Finally, we illustrate the applications of our results.

Optimization of higher order differential inclusions with initial value problem

This paper concerns the sufficient conditions of optimality for initial value problem with higher order differential inclusions (HODIs) and free endpoint constraints. Formulation of the transversality conditions plays a substantial role in the next investigations without which hardly any necessary or sufficient conditions would be obtained. In terms of Euler–Lagrange and Hamiltonian forms the sufficient conditions of optimality both for convex and “non-convex” HODIs are based on the apparatus of locally adjoint mappings. Moreover, by applying the main result to a Bolza problem described by a polynomial differential operator with constant coefficients in terms of the adjoint differential operator the sufficient condition of optimality is obtained.

Analysis of Orbital Lifetime Prediction Parameters in Preparation for Post-Mission Disposal

Atmospheric drag force is an important source of perturbation of Low Earth Orbit (LEO) orbit satellites, and solar activity is a major factor for changes in atmospheric density. In particular, the orbital lifetime of a satellite varies with changes in solar activity, so care must be taken in predicting the remaining orbital lifetime during preparation for post-mission disposal. In this paper, the System Tool Kit (STK) Long-term Orbit Propagator is used to analyze the changes in orbital lifetime predictions with respect to solar activity. In addition, the STK Lifetime tool is used to analyze the change in orbital lifetime with respect to solar flux data generation, which is needed for the orbital lifetime calculation, and its control on the drag coefficient control. Analysis showed that the application of the most recent solar flux file within the Lifetime tool gives a predicted trend that is closest to the actual orbit. We also examine the effect of the drag coefficient, by performing a comparative analysis between varying and constant coefficients in terms of solar activity intensities.

A Formula for the General Solution of a Constant-coefficient Difference Equation

We give a formula for the general solution of a d th-order linear difference equation with constant coefficients in terms of one of the solutions of its associated homogeneous equation. The formula neither uses the roots of the characteristic equation nor their multiplicities. It can be readily generalized to the case where the domain of the difference equation is the real numbers, and the initial values are given by a function defined on the interval [0,d ). In both cases, we express the general solution of the difference equation in terms of a single solution of its associated homogeneous equation at integer arguments.

Similar Operators and a Functional Calculus for the First-Order Linear Differential Operator

The general problem of finding solutions of linear functional equations is undoubtedly one of the main problems of applied mathematics. Among such equations the classes of linear differential equations and linear difference equations are very important and numerous theories and meth- ods for their solution have been developed. The methods based on integral transforms and the operational methods are certainly the ones used most often. In the present paper we present some general linear algebra results that may be considered as a unified approach to the transform and operational methods. We use the linear algebra foundations of the Laplace transform and the concept of similarity to generalize a description of the general solution of linear differential equations with constant coefficients in terms of certain basis for the space of exponential polynomials, and a convolution product defined in an algebraic way using the basis. The generalization gives an explicit construction of the general solution of any * Research partially supported by a fellowship from CONACYT-Mexico. ´ † Research partially supported by a grant from CONACYT-Mexico. E-mail: verde@xan- ´

Propagation of Gevrey singularities for solutions of linear differential operators with constant coefficients

In [4]L. Hormander has given sufficient conditions for propagation of C∞singularities for solutions of linear differential operators P with constant coefficients in terms of limit operators called «localization of P at infinity». In this paper a result (Theorem 1.2) of the same type concerning the propagation of Gevrey singularities is given.

A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov

ABSTRACTThe second method of Liapunov is a useful technique for investigating the stability of linear and non-linear ordinary differential equations. It is well known that the second method of Liapunov, when applied to linear differential equations with real constant coefficients, gives rise to sets of necessary and sufficient stability conditions which are alternatives to the well-known Routh-Hurwitz conditions. In this paper a direct proof of the Routh-Hurwitz conditions themselves is given using Liapunov's second method. The new proof is ‘elementary’ in that it depends on the fundamental concept of stability associated with Liapunov's second method, and not on theorems in the complex integral calculus which are required in the usual proofs. A useful by-product of this new proof is a method of determining the coefficients of a linear differential equation with real constant coefficients in terms of its Hurwitz determinants.