Linear Barycentric Rational Collocation Method for Beam Force Vibration Equation

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.


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.
ere 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.
ere are lots of numerical methods [3][4][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. e barycentric formula is studied in [6][7][8] and has been used to solve Volterra equation and Volterra integro-differential equation [9,10]. Cirillo et al. [11][12][13][14] have proposed a rational interpolation scheme which has high numerical stability and interpolation accuracy on both equidistant and special distributed nodes. In [15][16][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][19][20][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. e convergence rate of linear barycentric rational collocation methods has been proved. At last, two examples are presented to illustrate our theorem analysis.
with boundary conditions as follows: e free vibration frequency of the beam is only related to the geometric and material parameters of the beam. e forced vibration of beam under external load is the result of superposition of free vibration and external excitation. We 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 have then, we change the form into the following equation: where We get the matrix form as where . . , f m (t)] T .By taking the notation, we have where j � 0, 1, . . . , n. Its matrix form can be expressed as e matrix equation can also be written as where L � (I n ⊗ D (2) ) + a 2 (C (4) ⊗ I n ) and ⊗ is Kronecher product of matrix: U � u 00 , u 01 , . . . , u 0n , u 10 , u 11 , . . . , u 1n , . . . , u m0 , u m1 , . . . , u mn T , f ij � f(x i , t j ), i � 0, 1, . . . , m; j � 0, 1, . . . , n, and 2 Shock and Vibration are the elements of the differentiation matrices with Similarly, we have for k ≥ 2, according to mathematical induction, we obtain the recurrence formula of m-order differential matrix as

Convergence and Error Analysis
For the barycentric rational interpolants of function f(x) with r(x), its error convergence rate is where d is the degree of polynomial r(x): where For the barycentric rational interpolants of function u(x, t) with r m,n (x, t), we can get the barycentric rational interpolants: . . , n − d 2 , and d 1 and d 2 are the degree of polynomial of r m,n (x, t).
Lemma 1 (see reference [10]). For the e(x) defined in equation (20), we have For the e(x, t) defined in equation (26) and Proof. For (x, t), the function w i,j (x, t) is well-defined, and the error functional can be expressed as By the Newton error formula, we reach that By the similarly analysis in Li and Cheng [15], we have Combining equations (29)-(31) together, the proof of eorem 1 is completed.

Shock and Vibration
Based on the above lemma, we get the following theorem.
Proof. As where As for the R 1 (x, t), we have � e tt x, t n + e tt x m , t n .

(39)
Similarly, for R 2 (x, t), we have en, we have e proof is completed.

Numerical Examples
Example 1. Consider the beam force vibration equation: with the following conditions: Its analysis solutions is Shock and Vibration              Figures 5 and 6, the errors of deflection with quasiequidistant nodes m � n � 28, d 1 � d 2 � 25, and m � n � 28 are presented. From the figure, we know that the accuracy of quasi-equidistant node with m � n � 28 and d 1 � d 2 � 25 is higher than m � n � 28.
In this example, we test the linear barycentric rational for deflection and bending moment with the equidistant nodes; Table 5 shows the convergence rate is O(h d 2 ) with d 1 � 7 firstly given for the space area for t � 1. In Table 6, for the space area partition d 1 � 7firstly given, the convergence rate of times is O(τ d 2 ) which agrees with our theorem analysis.
In this example, we test the linear barycentric rational for deflection and bending moment with the equidistant nodes; Table 7 shows the convergence rate is O(h d 2 − 1 ) with d 1 � 7 firstly given for the space area for t � 1. In Table 8, for the space area partition d 1 � 7 firstly given, the convergence rate of times is O(τ d 2 ) which agree with our theorem analysis.
In Tables 9 and 10, we test the linear barycentric rational for deflection and bending moments with the quasi-equidistant nodes; Table 9 shows the convergence rate is O(h d 1 ) with d 2 � 7 firstly given for the space area for t � 1. In Table 10, for the space area partition d 2 � 7 firstly given, the  In Tables 11 and 12, we test the linear barycentric rational collocation methods for deflection and bending moment with the quasi-equidistant nodes; Table 11 shows the convergence rate is O(h d 1 ) with d 2 � 7 firstly given for the space area for t � 1. In Table 12, for the space area partition d 2 � 7 firstly given, the convergence rate of times is O(τ d 1 ) which agrees with our theorem analysis.

Conclusion
In this paper, linear barycentric rational collocation methods have been presented to solve the beam force vibration equation. With the help of matrix equation of discrete beam force vibration equation, the time and space variable can be solved at the same time. As the coefficient matrix is full for the collocation methods, there are certain properties such as circularity and symmetry that can be studied in the near future. e 2 + 1 dimensional beam force vibration equation can also be solved easily by barycentric rational collocation methods.
Data Availability e data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.