Taylor Expansion In Terms Research Articles

Dynamic characteristics of composite drive shaft in complex environment based on Carrera unified formulation

This article uses Carrera unified formulation (CUF) to analyze the dynamics characteristics of composite shaft with consideration of thermal effects, axial loads, and rotational speeds. The shaft radial displacement is described by Taylor expansion terms, and the axial displacement is described by improved Fourier series. Hamilton’s principle is used to obtain the control equations. The calculation results indicate that: The increase of temperature can reduce composite shaft natural frequency, and the influence is not obvious when the layer angle is small; The effect of axial load on composite shaft natural frequency is roughly linear, but the influence varies under different boundaries, length-radius ratio, layer orders; Due to the gyroscopic effect, the natural frequency of the composite shaft exhibits two states under the action of rotation speed: forward and backward mode; In forward mode, the influence of temperature and axial load on the composite shaft still follows the previous conclusion; In the backward mode, when the rotational speed exceeds the critical speed, the influence of temperature and axial load on the composite shaft shows the opposite conclusion. These findings are of great significance for the practical application of composite shaft in engineering.

Nonlinear vibrational analysis and linearization error investigation in size-dependent resonant pressure microsensors

Dynamic and vibrational behavior of nonlinear resonant pressure micro sensors is studied in this paper using a proposed dynamic model. The basic part of the device is a microplate deflects due to external pressure in presence of electrostatic actuation. The frequency changes because of the microplate deformation is used to estimate the external pressure. For modeling of the micro sensor, the nonlinear governing equation of size dependent microplate in polar coordinate system and based on the modified couple stress theory is presented. Assuming specified number of Taylor expansion terms for electrostatic force, the linear and nonlinear ODE is obtained and nonlinear equation is solved both with multiple time scales method (MTS) and finite difference numerical method. Results indicate that considering finite terms of Taylor expansion (1 term for linear and three terms for MTS method) is the source of error in estimation of pressure or system stability. Using sufficient terms of Taylor Expansion, the error in linear and MTS solutions is vanished. Therefore, the limited values of microplate vibrational amplitude is extracted to increase the accuracy of the sensor in linear action. The effect of size dependency, applied voltage on nonlinear behavior and linearization error is also studied.

A general method for the development of diabatic spin-orbit models for multi-electron systems.

Spin-orbit (SO) coupling can have significant effects on the quantum dynamics of molecular systems, but it is still difficult to account for accurately. One promising way to do this is to devise a diabatic SO model combined with the molecular potential energy. Few such models have been developed utilizing spatial and time-reversal symmetry. These models are tedious to derive and are specific for the molecular symmetry and included spin states. Here, we present a relatively simple approach to construct such models for various spin states with S≠12 from a basic one-electron SO case with S=12. The multi-electron fine structure states are expressed in terms of Slater determinants of single-electron spin functions (spinors). The properties of all single-electron matrix elements over the SO operator are derived and expressed as Taylor expansions in terms of symmetry-adapted nuclear coordinates. The SO matrix elements for the multi-electron case are then obtained from these single-electron matrix elements using the Slater-Condon rules. This yields the full SO matrix and symmetry properties of the multi-electron matrix elements in a straightforward way. The matrix elements are expressed as symmetry-adapted polynomials up to arbitrary order. This approach is demonstrated first for an abstract model of two electrons in a set of p orbitals in a C3v symmetric environment and then applied to set up a diabatic model for the photodissociation of methyl iodide (CH3I). The high accuracy of this new approach is demonstrated in comparison to an available analytic SO model for CH3I.

Precise integration method for natural frequencies and mode shapes of ocean risers with elastic boundary conditions

• A relatively new precise integration method for modal analysis of ocean risers is developed. • It is suitable for ocean risers with variable tension and cross-section and various boundaries. • The results are proved to be very accurate by comparing the analytical solution and literature. • Computing time is quite short and the method exhibits good convergence and high efficiency. • Effects of elastic constraints simulating damaged and undamaged boundaries on natural frequencies and mode shapes are investigated. The effects of the elastic constraints simulating damaged and undamaged boundaries on the natural frequencies and mode shapes of ocean risers with a variable axial tension are investigated using the precise integration method (PIM). The classical high-order variable-coefficient partial-differential governing equation of the free vibration of ocean risers is reduced to a set of first-order ordinary differential equations and efficiently solved by the PIM. The main advantages of the PIM are that the numerical results can be calculated with high accuracy even when total element number n , number of iterations N and Taylor expansion terms r are small. Moreover, the computing time is quite short. Various boundary conditions are modeled as linear elastic constraints using a pair of translational and torsional springs, and four types of boundary damage coefficients are proposed to investigate the effects of a damaged boundary on the natural frequencies. The results for specific boundary conditions show agreement with those reported in the literature, and the calculation errors are very small in comparison with the analytical solution. Overall, the methodology of PIM is applicable for the investigation of the natural frequencies and mode shapes of ocean risers with a variable axial tension and cross-section and various boundary conditions.

Performance Features of a Stationary Stochastic Novikov Engine.

In this article a Novikov engine with fluctuating hot heat bath temperature is presented. Based on this model, the performance measure maximum expected power as well as the corresponding efficiency and entropy production rate is investigated for four different stationary distributions: continuous uniform, normal, triangle, quadratic, and Pareto. It is found that the performance measures increase monotonously with increasing expectation value and increasing standard deviation of the distributions. Additionally, we show that the distribution has only little influence on the performance measures for small standard deviations. For larger values of the standard deviation, the performance measures in the case of the Pareto distribution are significantly different compared to the other distributions. These observations are explained by a comparison of the Taylor expansions in terms of the distributions’ standard deviations. For the considered symmetric distributions, an extension of the well known Curzon–Ahlborn efficiency to a stochastic Novikov engine is given.

Improved quadratic cuts for convex mixed-integer nonlinear programs

This paper presents scaled quadratic cuts based on scaling the second-order Taylor expansion terms for the decomposition methods Outer Approximation and Partial Surrogate Cuts for solving convex Mixed Integer Nonlinear Programing problems. The scaled quadratic cut is proved to be a stricter and tighter underestimation for convex nonlinear functions than classical supporting hyperplanes, which results in the improvement of Outer Approximation and Partial Surrogate Cuts based solution methods. We integrate the strategies of scaled quadratic cuts with multi-generation cuts for Outer Approximation and Partial Surrogate Cuts and develop six types of Mixed Integer Nonlinear Programming solution methods with scaled quadratic cuts. These cuts are incorporated in the master problem of the decomposition methods leading to a Mixed Integer Quadratically Constrained Programming problem. Numerical results of benchmark Mixed Integer Nonlinear Programming problems demonstrate the effectiveness of the proposed Mixed Integer Nonlinear Programming solution methods with scaled quadratic cuts.

General parametrization of axisymmetric black holes in metric theories of gravity

Following previous work of ours in spherical symmetry, we here propose a new parametric framework to describe the spacetime of axisymmetric black holes in generic metric theories of gravity. In this case, the metric components are functions of both the radial and the polar angular coordinates, forcing a double expansion to obtain a generic axisymmetric metric expression. In particular, we use a continued-fraction expansion in terms of a compactified radial coordinate to express the radial dependence, while we exploit a Taylor expansion in terms of the cosine of the polar angle for the polar dependence. These choices lead to a superior convergence in the radial direction and to an exact limit on the equatorial plane. As a validation of our approach, we build parametrized representations of Kerr, rotating dilaton, and Einstein-dilaton-Gauss-Bonnet black holes. The match is already very good at lowest order in the expansion and improves as new orders are added. We expect a similar behavior for any stationary and axisymmetric black-hole metric.

Curvature of the freeze-out line in heavy ion collisions

We calculate the mean and variance of net-baryon number and net-electric charge distributions from Quantum Chromodynamics (QCD) using a next-to-leading order Taylor expansion in terms of temperature and chemical potentials. We compare these expansions with experimental data from STAR and PHENIX, determine the freeze-out temperature in the limit of vanishing baryon chemical potential, and, for the first time, constrain the curvature of the freeze-out line through a direct comparison between experimental data on net-charge fluctuations and a QCD calculation. We obtain a bound on the curvature coefficient, kappa_2^f < 0.011, that is compatible with lattice QCD results on the curvature of the QCD transition line.

The Method to Calculate the Length of the Catenary for Anchored Vessel

The length of the anchor chain released directly influence the holding power of anchor and holding power of anchor decides the safety of the ship in the state of anchoring. In order to calculate the length of the catenary for anchored vessel in static hydraulic condition, Parabola and catenary mathematical model has been used in this paper to the data processing. These two kinds of thoughts are both based on the infinitesimal dividing model. The parabola method is using the Taylor series expansion to calculate the results of former 3~5 terms respectively, while catenary one uses the differential solving method. Through the example analysis to compare the accuracy of results obtained by these two methods, the conclusion can be obtained that parabola method is closely related to Taylor expansion terms. For a certain range of the anchored catenary length, the more Taylor series expansion, the more accurate results we can get; the gap of the results calculated by these two methods can be neglected. The catenary method is more accurate than that of parabolic one when the length of the catenary beyond this range.

Moving State Marine SINS Initial Alignment Based on High Degree CKF

A new moving state marine initial alignment method of strap-down inertial navigation system (SINS) is proposed based on high-degree cubature Kalman filter (CKF), which can capture higher order Taylor expansion terms of nonlinear alignment model than the existing third-degree CKF, unscented Kalman filter and central difference Kalman filter, and improve the accuracy of initial alignment under large heading misalignment angle condition. Simulation results show the efficiency and advantage of the proposed initial alignment method as compared with existing initial alignment methods for the moving state SINS initial alignment with large heading misalignment angle.

A derivative-free distributed filtering approach for sensorless control of nonlinear systems

This article examines the problem of sensorless control for nonlinear dynamical systems with the use of derivative-free Extended Information Filtering (EIF). The system is first subject to a linearisation transformation and next state estimation is performed by applying the standard Kalman Filter to the linearised model. At a second level, the standard Information Filter is used to fuse the state estimates obtained from local derivative-free Kalman filters running at the local information processing nodes. This approach has significant advantages because unlike the EIF (i) is not based on local linearisation of the nonlinear dynamics (ii) does not assume truncation of higher order Taylor expansion terms thus preserving the accuracy and robustness of the performed estimation and (iii) does not require the computation of Jacobian matrices. As a case study a robotic manipulator is considered and a cameras network consisting of multiple vision nodes is assumed to provide the visual information to be used in the control loop. A derivative-free implementation of the EIF is used to produce the aggregate state vector of the robot by processing local state estimates coming from the distributed vision nodes. The performance of the considered sensorless control scheme is evaluated through simulation experiments.

Galactic parameters from masers with trigonometric parallaxes

Spatial velocities of all currently known 28 masers having trigonometric parallaxes, proper motion and line-of-site velocities are reanalyzed using Bottlinger's equations. These masers are associated with 25 active star-forming regions and are located in the range of galactocentric distances 3<R<14 kpc. To determine the Galactic rotation parameters, we used the first three Taylor expansion terms of angular rotation velocity {\Omega} at the galactocentric distance of the Sun R0=8 kpc. We obtained the following solutions: {\Omega}o=-31.0 +/- 1.2 km/s/kpc, {\Omega}o'=4.46 +/- 0.21 km/s/kpc^2, {\Omega}o"=-0.876 +/- 0.067 km/s/kpc^3, Oort constants: A=17.8 +/- 0.8 km/s/kpc, B=-13.2 +/- 1.5 km/s/kpc and circular velocity of the Solar neighborhood rotation Vo=248 +/- 14 km/s. Fourier analysis of galactocentric radial velocities of masers VR allowed us to estimate the wavelength {\lambda}=2.0 +/- 0.2 kpc and peak velocity f_R=6.5 +/- 2 km/s of periodic perturbations from the density wave and velocity of the perturbations 4 +/- 1 km/s near the location of the Sun. Phase of the Sun in the density wave is estimated as {\chi}o ~ -130^o +/- 10^o. Taking into account perturbations evoked by spiral density wave we obtained the following non-perturbed components of the peculiar Solar velocity with respect to the local standard of rest (LSR) (Uo,Vo,Wo)LSR=(5.5,11,8.5) +/- (2.2,1.7,1.2) km/s.

Bayesian model-independent evaluation of expansion rates of the universe

Marginal likelihoods for the cosmic expansion rates are evaluated using the `Constitution' data of 397 supernovas, thereby updating the results in some previous works. Even when beginning with a very strong prior probability that favors an accelerated expansion, we obtain a marginal likelihood for the deceleration parameter $q_0$ peaked around zero in the spatially flat case. It is also found that the new data significantly constrains the cosmographic expansion rates, when compared to the previous analyses. These results may strongly depend on the Gaussian prior probability distribution chosen for the Hubble parameter represented by $h$, with $h=0.68\pm 0.06$. This and similar priors for other expansion rates were deduced from previous data. Here again we perform the Bayesian model-independent analysis in which the scale factor is expanded into a Taylor series in time about the present epoch. Unlike such Taylor expansions in terms of redshift, this approach has no convergence problem.

New perturbation–iteration solutions for Bratu-type equations

Perturbation–iteration theory is systematically generated for both linear and nonlinear second-order differential equations and applied to Bratu-type equations. Different perturbation–iteration algorithms depending upon the number of Taylor expansion terms are proposed. Using the iteration formulas derived using different perturbation–iteration algorithms, new solutions of Bratu-type equations are obtained. Solutions constructed using different perturbation–iteration algorithms are contrasted with each other as well as with numerical solutions. It is found that algorithms with more Taylor series expansion terms yield more accurate results.

ALTERNATIVE TIME DOMAIN BOUNDARY INTEGRAL EQUATIONS FOR THE SCALAR WAVE EQUATION USING DIVERGENCE-FREE REGULARIZATION TERMS

In this short note alternative time domain boundary integral equations (TDBIE) for the scalar wave equation are formulated on a surface enclosing a volume. The technique used follows the traditional approach of subtracting and adding back relevant Taylor expansion terms of the field variable, but does not restrict this to the surface patches that contain the singularity only. From the divergence-free property of the added-back integrands, together with an application of Stokes' theorem, it follows that the added-back terms can be evaluated using line integrals defined on a cut between the surface and a sphere whose radius increases with time. Moreover, after a certain time, the line integrals may be evaluated directly. The results provide additional insight into the theoretical formulations, and might be used to improve numerical implementations in terms of stability and accuracy.

Isentropic thermodynamics in the Polyakov–Nambu–Jona-Lasinio model

We discuss the isentropic trajectories on the QCD phase diagram in the temperature and the quark chemical potential plane using the Nambu--Jona-Lasinio model with the Polyakov loop coupling (PNJL model). We impose a constraint on the strange quark chemical potential so that the strange quark density is zero, which is the case in the ultra relativistic heavy-ion collisions. We compare our numerical results with the truncated estimates by the Taylor expansion in terms of the chemical potential to quantify the reliability of the expansion used in the lattice QCD simulation. We finally discuss the strange quark chemical potential induced by the strangeness neutrality condition and relate it to the ratio of the Polyakov loop and the anti-Polyakov loop.

Perturbative Derivation and Comparisons of Root-Finding Algorithms with Fourth Order Derivatives

Perturbation theory is systematically used to generate root finding algorithms with fourth order derivatives. Depending on the number of correction terms in the perturbation expansion and the number of Taylor expansion terms, different root finding formulas can be generated. Expanding Taylor series up to fourth order derivatives and taking two, three and four correction terms in the perturbation expansions, three different root finding algorithms are derived. The algorithms are contrasted numerically with each other as well as with the Newton-Raphson algorithm. It is found that the quadruple-correction-term algorithm performs better than the others.

Generation of root finding algorithms via perturbation theory and some formulas

Perturbation theory is systematically used to generate root finding algorithms. Depending on the number of correction terms in the perturbation expansion and the number of Taylor expansion terms, different root finding formulas can be generated. The way of separating the resulting equations after the perturbation expansion alters the root-finding formulas also. Well known cases such as Newton–Raphson and its second correction, namely the Householder’s iteration, are derived as examples. Moreover, higher order algorithms which may or may not be the corrections of well known formulas are derived. The formulas are contrasted with each other as well as with some new algorithms obtained by modified Adomian Decomposition Method proposed in Ref. [S. Abbasbandy, Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method, Applied Mathematics and Computation 145 (2003) 887–893].

Large eddy simulation and spectrum analysis of flow around two square cylinders arranged side by side

Large eddy simulation cooperated with the second order full extension ETG (Euler-Taylor-Galerkin) finite element method was applied to simulate the flow around two square cylinders arranged side by side at a spacing ratio of 1.5. The second order full extension ETG finite element method was developed by Wang and He. By means of Taylor expansion of terms containing time derivative, time derivative is replaced by space derivative. The function of it is equal to introducing an artificial viscosity term. The streamlines of the flow at different moments were obtained. The time history of drag coefficient, lift coefficient and the streamwise velocity on the symmetrical points were presented. Furthermore, the symmetrical problem of the frequency spectrum of flow around two square cylinders arranged side by side were studied by using the spectral analysis technology. The data obtained at the initial stage are excluded in order to avoid the influence of initial condition on the results. The power spectrums of drag coefficient, lift coefficient, the streamwise velocity on the symmetrical points were analyzed respectively. The results show that although the time domain process of dynamic parameters is nonsymmetrical, the frequency domain process of them is symmetrical under the symmetrical boundary conditions.

Thermodynamics at non-zero baryon number density: a comparison of lattice and hadron resonance gas model calculations

We compare recent lattice studies of QCD thermodynamics at non-zero quark chemical potential with the thermodynamics of a hadron resonance gas. We argue that for T⩽Tc the equation of state derived from Monte Carlo simulations of two flavour QCD at non-zero chemical potential can be well described by a hadron resonance gas when using the same set of approximations as used in current lattice calculations. We estimate the importance of truncation errors arising from the use of a Taylor expansion in terms of the quark chemical potential and examine the influence of unphysically large quark masses on the equation of state and the critical conditions for deconfinement.