Trigonometric Approximation Of Functions Research Articles

Pendulum vibration absorber with a nonlinear viscous damping mechanism

SummarySimple pendulum vibration absorbers, as one of the most common models of absorbers, experience nonlinear behaviors in the large amplitude of oscillations, and under such conditions, they are not effective enough and are more prone to lose their stability. To overcome some of these drawbacks, a complementary damping mechanism including two nonlinear elements is introduced. Alongside the ordinary linear viscous damper, a damping mechanism is also utilized to control the oscillations of the dynamic structure exposed to the excitation forces more intense than the ones considered in the optimization. In other words, they make this system more robust and stable in the face of excitations stronger than the presupposed conditions. The complementary elements of this mechanism are proposed to offer an appropriate approximation of various likely models which can be used as practical nonlinear damping mechanisms. The steady‐state solutions of the nonlinear governing equations are achieved with pinpoint precision with the help of the harmonic balance method and more precise approximations of trigonometric functions. To enhance the impacts of this mechanism, optimization is performed for a compound objective function in addition to the conventional one. This study explains how such mechanisms are capable of contributing to the higher robustness and stability of pendulum vibration absorbers under a wider range of excitation intensities. The performance of this absorber is also studied for an N‐story shear structure as a multidegree of freedom (MDOF) system.

Trigonometric approximation of functions in $$L_1$$-norm

Trigonometric approximation of functions in $$L_1$$-norm

Trigonometric approximation of functions in seminormed spaces

Trigonometric approximation of functions in seminormed spaces

Trigonometric approximation of functions f(x,y) of generalized Lipschitz class by double Hausdorff matrix summability method

In this paper, we establish a new estimate for the degree of approximation of functions f(x,y) belonging to the generalized Lipschitz class Lip ((xi _{1}, xi _{2} );r ), r geq 1, by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from Lip ((alpha ,beta );r ) and Lip(alpha ,beta ) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and (C, gamma , delta ) means.

On the trigonometric approximation of functions in a weighted Lipschitz class

Mishra et al. (Applied Mathematics and Computation 237:252–263, 2014) obtained the degree of approximation of $$\tilde{f}$$ , conjugate of a 2π periodic function $$f$$ belonging to the weighted Lipschitz class $$W(L_{r} ,\beta ,\xi )(r \ge 1)$$ , through the Cesaro-Norlund means of the conjugate Fourier series of f, which is an extension of the results of Lal (Applied Mathematics and Computation 209:346–350, 2009) and Singh et al. (International Journal of Mathematics and Mathematical Sciences 2012:1–12, 2012). Zhang (Applied Mathematics and Computation 247:1139–1140, 2014) has pointed out that the conclusions of the theorem of Mishra et al. (Applied Mathematics and Computation 237:252–263, 2014) hold only for constant functions, and thus the results are trivial. In this paper, we redefine the problems of Lal (Applied Mathematics and Computation 209:346–350, 2009) and Mishra et al. (Applied Mathematics and Computation 237:252–263, 2014) for $$f \in W(L_{r} ,\beta ,\xi )(r \ge 1)$$ and its conjugate $$\tilde{f}$$ . We obtain the degree of approximation through more general summability means under weighted norm which in turn resolve the issues raised by Zhang (Applied Mathematics and Computation 247:1139–1140, 2014). We also derive some corollaries from our results.

Trigonometric Approximation of Functions from L_{2pi }^{p(x)}

We generalize and improve the results of A. Guven, D. Israfilov, Xh. Z. Krasniqi and T. N. Shakh-Emirov. We consider the general methods of summability of Fourier series of functions from L_{2pi }^{p(x)} with pleft( xright) ge 1. For estimate of the error of approximation of functions by the matrix means we use a modulus of continuity constructed by the Steklov functions of the increments of considered functions without of absolute values.

Modeling Phase-Locked Loop-Based Synchronization in Grid-Interfaced Converters

Grid converters need a synchronization which is often a phase-locked loop (PLL) to get connected to the grid. In the literature, impedance-based models using small-signal technique have been widely utilized to describe the behaviors of PLL-synchronized grid converter. However, most of them target at specific problems and are able to represent the phenomenon under certain scenarios. For the analysis during phase transient, like phase jump due to grid fault, an accurate model is still missing. To address this issue, a general impedance-based model using Maclaurin expansions for its trigonometric function approximation has been proposed in this paper, so that it can well represent the system behavior during phase transient. Moreover, different deepness of the general model can be scaled, leading to different models including phasorial model, current source model, impedance-based model with/without PLL effects. A guideline on how to choose a proper model based on the problems to be studied and grid conditions has been provided in this paper. Simulation and experimental results are given to validate the effectiveness of the model and the theoretical analysis.

A totally positive basis for circle approximations

We construct a six-dimensional space of polynomials containing approximations of trigonometric functions. In this way, an approximation of a circle can be represented as a parametric curve in this space. We show that the space has shape preserving representations and we construct the basis with optimal shape preserving properties.

A novel approach for computation of cosine function

Objective of trigonometric function approximation in digital systems are faster computation in less number of clock cycles, optimising hardware resources, accuracy in more number of bits, etc. This study proposes a novel method for cosine function computation and respective FPGA-based architecture. A triangle is presumably located in the first quadrant of a circle with unit radius whose one vertex is the centre, other two vertices touches the perimeter. Using the area of the triangle, it is observed that the y coordinate of the third vertex of the triangle is the sine value. The error is the difference between arch length and side length. Newton's interpolation method is used to formulate the error approximation function. This method is synthesised on Xilinx Spartan 3 xc3s200-5ft256 FPGA kit. The proposed method gives accuracy up to 14 bits or more in 96% cases in 11 clock cycles only at maximum speed of 89.977 MHz.

A novel approach for computation of cosine function

Objective of trigonometric function approximation in digital systems are faster computation in less number of clock cycles, optimising hardware resources, accuracy in more number of bits, etc. This study proposes a novel method for cosine function computation and respective FPGA-based architecture. A triangle is presumably located in the first quadrant of a circle with unit radius whose one vertex is the centre, other two vertices touches the perimeter. Using the area of the triangle, it is observed that the y coordinate of the third vertex of the triangle is the sine value. The error is the difference between arch length and side length. Newton's interpolation method is used to formulate the error approximation function. This method is synthesised on Xilinx Spartan 3 xc3s200-5ft256 FPGA kit. The proposed method gives accuracy up to 14 bits or more in 96% cases in 11 clock cycles only at maximum speed of 89.977 MHz.

Mixed modulus of smoothness with Muckenhoupt weights and approximation by angle

ABSTRACTMixed modulus of smoothness in weighted Lebesgue spaces with Muckenhoupt weights are investigated. Using mixed modulus of smoothness we obtain Potapov type direct and inverse estimates of angular trigonometric approximation of functions in these spaces. Also we obtain equivalences between mixed modulus of smoothness and K-functional and realization functional. Fractional order modulus of smoothness is considered as well.

Simultaneous approximation of functions in Orlicz spaces with Muckenhoupt weights

It is proved that theorems on simultaneous trigonometric approximation of functions given in the weighted Orlicz spaces , with a generating Young function that might be non-convex, are held. We mainly concentrate on weights satisfying the Muckenhoupt’s condition .

Trigonometric approximation of functions belonging to certain Lipschitz classes by C1⋅ T operator

The study of approximation properties of the periodic functions in Lp(p ≥ 1)-spaces, in general and in Lipschitz classes Lip α, Lip (α, p), Lip (ξ(t), p) and weighted Lipschitz class W(Lp, ω(t), β), in particular, through trigonometric Fourier series, although is an old problem and known as Fourier approximation in the existing literature, has been of a growing interests over the last four decades due to its application in filters and signals [E. Z. Psarakis and G. V. Moustakides, An L2-based method for the design of 1-D zero phase FIR digital filters, IEEE Trans. Circuits Systems I Fundam. Theory Appl., 44(7) (1997) 551–601]. The most common methods used for the determination of the degree of approximation of periodic functions are based on the minimization of the Lp-norm of f(x) - Tn(x), where Tn(x) is a trigonometric polynomial of degree n and called approximant of the function f. In this paper, we discuss the approximation properties of the periodic functions in the Lipschitz classes Lip α and W(Lp, ω(t), β), p ≥ 1 by a trigonometric polynomial generated by the product matrix means of the Fourier series associated with the function. The degree of approximation obtained in our theorems of this paper is free from p and sharper than earlier results.

Kinematic Modeling and Hardware Development of 5-DoF Robot Manipulator

This paper presents a 5-DoF articulated robot manipulator and proposes a strategy for solving its inverse kinematics. The Denavit – Hartenberg (D-H) parameterization has been used to model the kinematics of the manipulator. As degree of freedom of manipulator increases, the geometrical solution for inverse kinematics becomes difficult; hence an analytical method for the same is presented. Novelty in the method presented is that no approximations of trigonometric functions are used resulting in a theoretical positional accuracy of 10-10mm of the end-effector. The articulated robotic manipulator developed makes use of integrated actuators and rapid prototyping technology enabling easy replication for educational purposes. The robot arm has been used for manipulation tasks in its workspace successfully.

Trigonometric approximation of functions belonging to Lipschitz class by matrix ( C 1 ⋅ N p ) operator of conjugate series of Fourier series

In the present paper, a new theorem on the degree of approximation of a function f ˜ , conjugate to a 2π periodic function f belonging to the Lipα (0<α≤1) class without the monotonicity condition on the generating sequence { p n } has been established, which in turn generalizes the results of Lal (Appl. Math. Comput. 209: 346-350, 2009) on a Fourier series.MSC:40G05, 41A10, 42B05, 42B08.

Calculation and approximation of the cnoidal function in cnoidal wave theory

A concise method for calculating the cnoidal function in cnoidal wave theory is presented. The method is based on the “precise integration method” and an improvement on the data preliminary treatment is employed to ensure the precision. For practical usage, a trigonometric function approximation expression for the cnoidal function is provided with brief verification. When the modulus of the wave is less than 0.95, the maximum error and absolute average error are acceptable for engineering.

Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent

We investigate the approximation properties of the trigonometric system in $ L_{2\pi }^{p\left( \cdot \right)} $ . We consider the moduli of smoothness of fractional order and obtain direct and inverse approximation theorems together with a constructive characterization of a Lipschitz-type class.

Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces with nonstandard growth

Abstract This work deals with basic approximation problems such as direct, inverse and simultaneous theorems of trigonometric approximation of functions of weighted Lebesgue spaces with a variable exponent on weights satisfying a variable Muckenhoupt A p(·) type condition. Several applications of these results help us transfer the approximation results for weighted variable Smirnov spaces of functions defined on sufficiently smooth finite domains of complex plane ℂ.

Approximation of trigonometric functions by trigonometric polynomials with interpolation

The paper studies the approximation order of periodic functions by trigonometric polynomials with interpolation in arbitrary set of nodes. A method of construction of Hermite interpolation polynomials is pointed out.

Using infinite matrices to approximate functions of class [formula omitted] using trigonometric polynomials

Given a function f in the class Lip ( α , p ) ( 0 < α ⩽ 1 , p ⩾ 1 ) , Chandra [P. Chandra, Trigonometric approximation of functions in L p -norm, J. Math. Anal. Appl. 275 (2002) 13–26] approximated such an f by using trigonometric polynomials, which are the nth terms of either certain weighted mean or Nörlund mean transforms of the Fourier series representation for f. He showed that the degree of its approximation is O ( n − α ) . In this paper we obtain the same degree of approximation for a more general class of lower triangular matrices, and deduce some of the results of [P. Chandra, Trigonometric approximation of functions in L p -norm, J. Math. Anal. Appl. 275 (2002) 13–26] as corollaries.