Abstract

The linear barycentric rational collocation method for beam force vibration equation is considered. The discrete beam force vibration equation is changed into the matrix forms. With the help of convergence rate of barycentric rational interpolation, both the convergence rates of space and time can be obtained at the same time. At last, some numerical examples are given to validate our theorem.

1. Introduction

Beam vibration is the amount and direction of movement that a beam exhibits away from the point of applied force or the area of attachment. There are lots of application including the material used for the construction, length of the beam, construction of bridges, buildings, towers and the amount of force applied, and so on. Recently, applications of nanobeams in engineering structures [1, 2] like nonvolatile random access memory, nanotweezers, tunable oscillator, rotational motors, nanorelays, feedback-controlled nanocantilevers have also been developed.

There are lots of numerical methods [3–5] to solve the beam force vibration equation such as the finite difference method, finite element method, differential quadrature method, multiscale method, and spectral methods. The barycentric formula is studied in [6–8] and has been used to solve Volterra equation and Volterra integro-differential equation [9, 10]. Cirillo et al. [11–14] have proposed a rational interpolation scheme which has high numerical stability and interpolation accuracy on both equidistant and special distributed nodes. In [15–17], integro-differential equation, heat conduction equation, and biharnormic equation are solved by linear barycentric rational collocation method and the convergence rate is proved. In recent papers, Wang et al. [18–21] successfully applied the collocation method to solve initial value problems, plane elasticity problems, incompressible plane problems, and nonlinear problems which have expanded the application fields of the collocation method.

In this paper, we focus on the beam force vibration equation by barycentric rational interpolation methods. With the help of barycentric rational polynomial, the collocation scheme for beam force vibration equation and its matrix equation have been presented. The convergence rate of linear barycentric rational collocation methods has been proved. At last, two examples are presented to illustrate our theorem analysis.

2. Collocation Scheme for Beam Force Vibration Equation

In this article, we pay our attention to the numerical solution of beam force vibration as

By taking and , we get the equation of EulerBernoulli beam aswith boundary conditions as follows:

The free vibration frequency of the beam is only related to the geometric and material parameters of the beam. The forced vibration of beam under external load is the result of superposition of free vibration and external excitation.

We partition the interval into and into with and.

We set and its barycentric interpolation approximation is where is the basis function, and is the weight function. Taking equation (5) into equation (2), we havethen, we change the form into the following equation:where . We get the matrix form as where and .By taking the notation, we have where

Its matrix form can be expressed as

The matrix equation can also be written as where and is Kronecher product of matrix:, and are the elements of the differentiation matrices with

Similarly, we have for , according to mathematical induction, we obtain the recurrence formula of m-order differential matrix as

3. Convergence and Error Analysis

For the barycentric rational interpolants of function with , its error convergence rate iswhere is the degree of polynomial :where

For the barycentric rational interpolants of function with , we can get the barycentric rational interpolants:where, , , , and are the degree of polynomial of .

The following lemma has been proved by Jean-Paul Berrut in [10].

Lemma 1. (see reference [10]). For the defined in equation (20), we have

Theorem 1. For the defined in equation (26) and , we have

Proof. For , the function is well-defined, and the error functional can be expressed asBy the Newton error formula,we reach thatBy the similarly analysis in Li and Cheng [15], we haveCombining equations (29)–(31) together, the proof of Theorem 1 is completed.

Corollary 1. For the defined in equation (26), we haveTaking the numerical scheme,Combining equations (1) and (34), we havewhere
Based on the above lemma, we get the following theorem.

Theorem 2. Let , we have

Proof. AswhereAs for the , we haveSimilarly, for , we haveThen, we haveThe proof is completed.