# Linear Barycentric Rational Collocation Method for Beam Force Vibration Equation

Authors: Jin Li; Yu Sang
Journal: Shock and Vibration
 (2021)

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## 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.


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## Body

## 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(1)ρA∂2u∂t2+EI∂4u∂x4=qx,t,0≤x≤l;t≥0.By takinga2=EI/ρA and fx,t=qx,t/ρA, we get the equation of EulerBernoulli beam as(2)∂2u∂t2+a2∂4u∂x4=fx,t,0≤x≤l;t≥0,with boundary conditions as follows:(3)ux,0=ϕ1x;utx,0=ϕ2x,0≤x≤1,u0,t=0,uxx0,t=0,t>0,u0,t=0,ux0,t=0,t>0,uxxx0,t=0,uxx0,t=0,t>0.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 intervala,b into 0=x0<x1<⋯<xm=l,h=l/m and 0,T into 0=t0<t1<⋯<tn=T,τ=T/n with Ω=0,l×0,T and. xi,tj,i=1,2,…,m;j=1,2,…,n.We set(4)uxi,t=uit,i=0,1,…,m,and its barycentric interpolation approximation is (5)ux,t=∑j=0mRjxujt,where (6)Rjx=wj/x−xj∑k=0nwk/x−xkis the basis function, and (7)wk=∑i∈Jk−1i∏j=i,j≠ki+d1xk−xjis the weight function. Taking equation (5) into equation (2), we have(8)∑j=0mRjxu¨jt+a2∑j=0mRj4xujt=fx,t,∑j=0mRjxiu¨jt+a2∑j=0mRj4xiujt=fxi,t,i=0,1,…,m,then, we change the form into the following equation:(9)∑j=1mδiju¨jt+a2∑j=1mCij4ujt=fit,i=0,1,…,m,where Rjxi=δij,Rj″xi=Cij2,Cijk,k=1,2fxi,t=fit,i=0,1,…,m. We get the matrix form as (10)u¨0t⋮u¨mt+a2C004⋯C0m4⋮⋮Cm04⋯Cmn4u0t⋮umt=f0t⋮fmt,where ut=u0t,u1t,…,umtT and ft=f0t,f1t,…,fmtT.By taking the notation, (11)uitj=uxi,tj=uij,i=0,1,…,m;j=0,1,…,n,uit=∑k=0nRktuik,i=0,1,…,m,we have (12)∑k=0nR¨ktju0k⋮∑k=0nR¨ktjumk+a2C004⋯C|m4⋮⋮Cm04⋯Cmn4∑k=0nRktju0k⋮∑k=0nRktjunk=f0tj⋮fmtj,where j=0,1,…,n.Its matrix form can be expressed as(13)In⊗D2+a2C4⊗InU=F.The matrix equation can also be written as(14)LU=F,where L=In⊗D2+a2C4⊗In and ⊗ is Kronecher product of matrix:(15)U=u00,u01,…,u0n,u10,u11,…,u1n,…,um0,um1,…,umnT,F=f00,f01,…,f0n,f10,f11,…,f1n,…,fm0,fm1,…,fmnT,fij=fxi,tj,i=0,1,…,m;j=0,1,…,n, and (16)Cij4=ri4xj,Cij2=ri″xj,Dij2=ri″tj,are the elements of the differentiation matrices with (17)ri′tj=wi/wjtj−ti,i≠j,ri′ti=−∑j≠iri′tj,i=j.Similarly, we have(18)rj′xi=wj/wixi−xk,j≠i,rj′xi=−∑j≠irj′xi,i=j,rj″xi=−2wj/wixi−xj∑k≠iwk/wixi−xk+1xi−xj,j≠i,ri″xi=−∑j≠irj″xi,i=j,for k≥2, according to mathematical induction, we obtain the recurrence formula of m-order differential matrix as (19)Cij4=4Cii3Cij1−Cij3xi−xj,i≠jCii4=−∑j=1,j≠inCij4,Dij2=2Dii1Dij1−Dij1ti−tj,i≠j,Dii2=−∑j=1,j≠inDij2.


## 3. Convergence and Error Analysis

For the barycentric rational interpolants of functionfx with rx, its error convergence rate is(20)ex≔fx−rx=x−xi,…,x−xi+dfxi,xi+1,…,xi+d,x,(21)ex≔maxa≤x≤bex=∑i=0n−dλixfx−pix∑i=0n−dλix=AxBx=Ohd+1,where d is the degree of polynomial rx:(22)Ax≔∑i=0n−d−1ifxi,…,xi+d,x,Bx≔∑i=0n−dλix,where(23)λix=−1ix−xi,…,x−xi+d.For the barycentric rational interpolants of functionux,t with rm,nx,t, we can get the barycentric rational interpolants:(24)rm,nx,t=∑i=0m∑j=0nwi,j/x−xit−tjui,j∑i=0m∑j=0nwi,j/x−xit−tj,where(25)wi,j=−1i−d1+j−d2∑k1∈Ji∏h1=k1,h1≠jk1+d11xi−xh1∑k2∈Jj∏h2=k2,h2≠jk2+d21tj−th2,Ji=k1∈Im:i−d1≤k1≤i, Im=0,…,m−d1, Jj=k2∈In:j−d2≤k2≤j, In=0,…,n−d2, and d1andd2 are the degree of polynomial of rm,nx,t.(26)ex,t≔ux,t−rm,nx,t=x−xi,…,x−xi+d1uxi,xi+1,…,xi+d1,x+t−tj,…,t−tj+d2utj,tj+1,…,tj+d2,t.The following lemma has been proved by Jean-Paul Berrut in [10].Lemma 1.
(see reference [10]). For the ex defined in equation (20), we have(27)ex≤Chd+1,u∈Cd+2a,b,e′x≤Chd,u∈Cd+3a,b,e″x≤Chd−1,u∈Cd+4a,b,d≥1.Theorem 1.
For theex,t defined in equation (26) and ux,t∈Cd1+2a,b×Cd2+20,T, we have(28)ex,t≤Chd1+1+τd2+1.Proof.
Forx,t, the function wi,jx,t is well-defined, and the error functional can be expressed as(29)ux,t−rm,nx,t=∑i=0n1−d1∑j=0n2−d2λixλjtux,t−rm,nx,t∑i=0n1−d1∑j=0n2−d2λixλjt,
By the Newton error formula,(30)ux,t−rm,nx,t=ux,t−u1x,t+u1x,t−rm,nx,t=x−xi,…,x−xi+d1uxi,xi+1,…,xi+d1,x,t+t−tj,…,t−tj+d2utj,tj+1,…,tj+d2,x,t,we reach that(31)ux,t−rm,nx,t=∑i=0n1−d1−1iuxi,xi+1,…,xi+d1,x,t∑i=0n1−d1λix+∑j=0n2−d2−1jutj,tj+1,…,tj+d2,x,t∑j=0n2−d2λjt.
By the similarly analysis in Li and Cheng [15], we have(32)∑i=0n1−d1λix≥1d1!hd1+1,∑j=0n2−d2λjt≥1d2!τd2+1.
Combining equations (29)–(31) together, the proof of Theorem 1 is completed.Corollary 1.
For theex,t defined in equation (26), we have(33)ettx,t≤Chd1+1+τd2−1,ux,t∈Cd1+3a,b×Cd2+20,T,d2≥1,exxxxx,t≤Chd1−3+τd2+1,ux,t∈Cd1+4a,b×Cd2+20,T,d1≥3,
Taking the numerical scheme,(34)∑j=1nujRj″x,t+a2∑j=1nujRj4x,t=fx,t.
Combining equations (1) and (34), we have(35)Tex≔e″x,t+a2e4x,t=Rfx,t,where Rfx,t=fx,t−Rfxk,t,k=1,2,⋯,n.
Based on the above lemma, we get the following theorem.Theorem 2.
LetRfx,t=fx,t−Rfxk,t,k=1,2,…,n, we have(36)ux,t−uxm,tn≤Chd1−3+τd2−1.Proof.
As(37)uttx,t+a2uxxxxx,t−uttxm,tn+a2uxxxxxm,tn=uttx,t−uttxm,tn+a2uxxxxx,t−uxxxxxm,tn≔R1x,t+R2x,t,where(38)R1x,t=uttx,t−uttxm,tn,R2x,t=a2uxxxxx,t−uxxxxxm,tn.
As for theR1x,t, we have(39)R1x,t=uttx,t−uttxm,tn=uttx,t−uttxm,t+uttxm,t−uttxm,tn=s∑i=0m−d1−1iuttxi,xi+1,…,xi+d1,x,t∑i=0m−d1λix+∑j=0n−d2−1jutttj,tj+1,…,tj+d2,x,t∑j=0n−d2λjt=ettx,tn+ettxm,tn.
Similarly, forR2x,t, we have(40)R2x,t=uxxxxx,t−uxxxxxm,tn=exxxxx,tn−exxxxxm,tn.
Then, we have(41)ux,t−uxm,tn≤∑j=1MjxTex≤Chd1−3+τd2−1.
The proof is completed.


## 4. Numerical Examples

Example 1.
Consider the beam force vibration equation:(42)∂2u∂t2+1x+x4120∂4u∂x4=0,12≤x≤1;t≥0,with the following conditions:(43)ux,0=0,0<x<1;utx,0=1+x5120,12≤x≤1,u12,t=1+0.55120sint,uxx12,t=148sint,u1,t=121120sint,uxx1,t=16sint.
Its analysis solutions is(44)ux,t=1+x5120sint.
In this example, we test the linear barycentric rational for the equidistant nodes. Table1 shows the convergence rate is Ohd1−1 with d2=7 firstly given for the space area for t=π. In Table 2, for the space area partition d1=7 firstly given, the convergence rate of times is Oτd2 which agrees with our theorem analysis.
In Figures1 and 2, the errors of LBRCM by equidistant nodes and quasi-equidistant nodes with m=n=12 and d1=d2=9 are presented. From the figure, we know that the accuracy of equidistant node is higher than quasi-equidistant node.
In Figures3 and 4, the errors of equidistant nodes and quasi-equidistant nodes with m=n=12 are presented. From the figure, we know that the accuracy of equidistant node is higher than the quasi-equidistant node.
In Tables3 and 4, we test the linear barycentric rational for the quasi-equidistant nodes; Table 3 shows the convergence rate is Ohd1−1 with d2=7 firstly given for the space area for t=π. In Table 4, for the space area partition d1=7 firstly given, the convergence rate of times is Oτd2.Table 1
Convergence rate of equidistant nodes with differentd1d2=7 values.
nd1=2d1=3d1=4d1=58×83.5062e − 01.7148e − 042.7309e − 052.7309e − 05416×162.1219e − 00.75324.1154e − 052.0873.6634e − 089.5703.6630e − 089.5704677832×321.1641e − 00.87219.9209e − 062.0586.9359e − 119.0509.5628e − 118.5875496964×646.0977e − 00.93432.4447e − 062.0225.7157e − 07—3.7940e − 07—522Table 2
Convergence rate of equidistant nodes with differentd2d1=7 values.
nd1=2d1=3d1=4d1=58×87.5819e − 01.3995e − 034.6760e − 041.7882e − 04316×167.9829e − 03.27634.7529e − 054.9081.3026e − 055.1941.1934e − 067.256047532×329.0685e − 03.14402.0817e − 064.5194.3378e − 074.9141.3851e − 073.113151464×641.6500e − 02.45981.5216e − 06—1.0236e − 06—6.4063e − 07—5Figure 1
Errors of LBRCM by equidistant nodes withm=n=12andd1=d2=9.Figure 2
Errors of LBRCM by quasi-equidistant nodes withm=n=12andd1=d2=9.Figure 3
Errors of equidistant nodes withm=n=12.Figure 4
Errors of quasi-equidistant nodes withm=n=12.Table 3
Convergence rate of quasi-equidistant nodes with differentd1d2=7 values.
nd1=2d1=3d1=4d1=58×81.4433e − 01.7229e − 049.1628e − 069.1628e − 06416×161.6054e − 0—7.4842e − 077.9178.0479e − 1013.541.6691e − 0912.49342532×327.6781e − 01.07916.9197e − 06—1.4486e − 06—1.5017e − 05—564×641.4474e − 0—3.4807e + 0—8.0897e − 01—1.1143e − 01—20Table 4
Convergence rate of quasi-equidistant nodes with differentd2d1=7 values.
nd1=2d1=3d1=4d1=58×81.2079e − 01.9645e − 041.1008e − 044.7330e − 05316×162.6034e − 05.60646.9740e − 078.2082.0174e − 079.1621.0751e − 0812.175154232×321.2548e − 0—8.6849e − 07—7.2689e − 06—5.8886e − 06—364×645.1823e − 04.60111.0338e − 03—7.2953e − 03—5.1897e − 03—Example 2.
Consider the beam force vibration equation:(45)EI∂4w∂x4+ρ∂2w∂t2=Qsinπxlsinpt,0≤x≤l;t≥0,with the boundary conditions as follows:(46)ux,0=0,utx,0=0,0<x<l,(47)u0,t=0,uxx0,t=0,ul,t=0,uxxl,t=0,t≥0.
Its analysis solutions is(48)wx,t=Qρsinπxlsinpt−p/wsinwtw2−p2,w=π2EIρ4,Mx,t=EI∂2w∂x2=QEIπ2ρl2sinπxlsinpt−p/wsinwtw2−p2,w=π2EIρ4,where EI=4.7726×107,ρ=420,Q=107,l=10,andp=2π/0.28335.
In Figures5 and 6, the errors of deflection with quasi-equidistant nodes m=n=28, d1=d2=25, and m=n=28 are presented. From the figure, we know that the accuracy of quasi-equidistant node with m=n=28 and d1=d2=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; Table5 shows the convergence rate is Ohd2 with d1=7 firstly given for the space area for t=1. In Table 6, for the space area partition d1=7firstly given, the convergence rate of times is Oτd2 which agrees with our theorem analysis.
In this example, we test the linear barycentric rational for deflection and bending moment with the equidistant nodes; Table7 shows the convergence rate is Ohd2−1 with d1=7 firstly given for the space area for t=1. In Table 8, for the space area partition d1=7 firstly given, the convergence rate of times is Oτd2 which agree with our theorem analysis.
In Tables9 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 Ohd1 with d2=7 firstly given for the space area for t=1. In Table 10, for the space area partition d2=7 firstly given, the convergence rate of times is Oτd1 which agrees with our theorem analysis.
In Tables11 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 Ohd1 with d2=7 firstly given for the space area for t = 1. In Table 12, for the space area partition d2=7 firstly given, the convergence rate of times is Oτd1 which agrees with our theorem analysis.Figure 5
Errors of deflection with quasi-equidistant nodesm=n=28.Figure 6
Errors of deflection with quasi-equidistant nodesm=n=28andd1=d2=25.Table 5
Convergence rate of deflection with equidistant nodesd1d2=7.
nd1=2d1=3d1=4d1=58×82.3676e + 03.7721e + 04.9328e + 04.7610e + 0222216×164.1793e + 02.50211.3027e + 0—6.6585e + 0—7.5280e + 02—13232×325.2574e + 0—3.3896e + 08.5861.5076e + 08.7861.2155e + 009.27461020864×643.7355e + 0—7.0233e − 012.2701.0927e − 013.7864.0207e − 024.9179193Table 6
Convergence rate of bending moment with equidistant nodesd1d2=7.
nd1=2d1=3d1=4d1=58×88.63191.2547e + 01.5671e + 01.5214e + 011116×166.7548e − 013.67572.1839e + 0—1.1063e + 00.5021.2536e + 01—11432×328.5357e − 01-5.4495e − 028.6462.4465e − 028.8201.9834e − 029.30396864×646.0317e − 010.50091.1256e − 022.2751.7640e − 033.7936.5673e − 044.916558Table 7
Convergence rate of deflection with equidistant nodesd2d1=7.
nd1=2d1=3d1=4d1=58×81.2258e + 01.7906e + 02.5725e + 03.5441e + 0222216×162.5305e + 02.27622.2997e + 02.9606.5860e + 01.9655.9839e + 012.56621191732×323.2139e + 02.97704.7977e + 02.2611.2807e + 05.6846.7902e + 003.13960000464×649.0199e − 011.83313.2947e − 013.8642.3183e − 012.4651.4751e − 015.524518Table 8
Convergence rate of bending moment with equidistant nodesd2d1=7.
nd1=2d1=3d1=4d1=58×83.9561e + 00 5.7800e + 08.3046e + 01.1442e + 0100016×164.2155e − 013.23033.8310e − 013.9151.0971e + 02.9209.9683e − 013.520830232×325.2460e − 023.00647.8314e − 022.2902.0905e − 025.7131.1084e − 013.16894764×641.4697e − 021.83575.3686e − 033.8663.7775e − 032.4682.4037e − 035.527163Table 9
Convergence rate of deflection with quasi-equidistant nodesd1d2=7.
nd1=2d1=3d1=4d1=58×84.1355e + 04.0577e + 04.3916e + 04.3806e + 0222216×162.7542e + 03.90832.0053e + 04.3382.0135e + 04.4472.0022e + 014.45151181032×324.4083e + 0—1.9063e − 016.7162.1971e − 029.8391.8656e − 0210.06819964×641.9575e + 01.17122.1823e − 023.1263.5110e − 045.9673.7910e − 058.9428195Table 10
Convergence rate of bending moment with quasi-equidistant nodesd1d2=7.
nd1=2d1=3d1=4d1=58×87.5367e + 07.9725e + 08.6829e + 08.6637e + 0000016×165.5240e − 013.77014.1382e − 014.2674.1551e − 014.3854.1284e − 014.39139232×327.3745e − 01—3.1740e − 037.0263.7026e − 0410.133.1421e − 0410.3606264×643.1548e − 011.22501.7330e − 030.8732.2647e − 03—5.4457e − 03—0Table 11
Convergence rate of deflection with quasi-equidistant nodesd2d1=7.
nd1=2d1=3d1=4d1=58×82.7216e + 03.1196e + 03.5186e + 03.9417e + 0222216×162.5198e + 0—1.6983e + 00.8778.6747e + 05.3427.7764e + 005.66362230032×323.0502e − 019.69027.9164e − 0211.066.0185e − 027.1713.4646e − 027.81027364×648.5111e − 035.16348.7507e − 046.4994.7864e − 046.9741.7242e − 047.650633Table 12
Convergence rate of bending moment with quasi-equidistant nodesd2d1=7.
nd1=2d1=3d1=4d1=58×85.4001e + 06.1897e + 06.9811e + 07.8208e + 0000016×165.1958e + 0—3.5019e + 00.8211.7887e − 015.2861.6035e − 015.6080007532×325.1369e − 039.98221.3332e − 0311.351.0136e − 037.4635.8348e − 048.10239364×641.7818e − 02—4.4772e − 02—3.3747e − 02—2.6845e − 02—


## 5. 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. The2+1 dimensional beam force vibration equation can also be solved easily by barycentric rational collocation methods.



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