# Global Residue Harmonic Balance Method for a System of Strongly Nonlinear Oscillator

Authors: Huaxiong Chen; Wei Liu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, the global residue harmonic balance method is applied to obtain the approximate periodic solution and frequency for a well-known system of strongly nonlinear oscillator in engineering. This method can improve accuracy by considering all the residual errors in deriving each order approximation. With this procedure, the expressions of the higher-order approximate solution and corresponding frequency for the considered system can be determined easily. The comparison of the obtained results with previously existing and corresponding exact solutions shows the high accuracy and efficiency of the method.


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## Body

## 1. Introduction

It is well known that the most models can be described by nonlinear differential equations. It is difficult to obtain exact solutions for these nonlinear problems. Thus, to find analytically approximate solutions to these nonlinear problems has been the desire of many researchers for a long time.The classical perturbation methods [1–3] for solving nonlinear equations are limited to the weakly nonlinear systems. To overcome this shortcoming, many improved or effective methods applicable to the strongly nonlinear systems have been developed. These methods include the homotopy perturbation method (HPM) [4–6], the energy balance method [7], the harmonic balance method [8], the variational iteration method [9], and the Lattice Boltzmann method [10–12]. Also, many researchers have applied these methods to study the models in many different fields. By applying the homotopy analysis method (HAM), Easif et al. [13] obtained the approximate solutions of the variable coefficient variant boussinesq system, Vakilzadeh et al. [14] investigated the dynamic pull-in instability of a suspended microchannel resonator, and Suharsono et al. [15] studied the solution of the reverse flow reactor model. Abed [16] studied the Kuramoto–Sivashinsky equation by the new iterative method. Talarposhti et al. [17] analysed the solution to micropolar nanofluid between two parallel plates in a rotating system by using AGM. Gerdroodbary [18] simulated the magnetohydrodynamic (MHD) flow of a nanofluid inside an inclined enclosure with a sinusoidal wall by the CVFEM method. Khalouta and Kadem [19] investigated the solution of a class of nonlinear fractional partial differential equations by using the reduced differential transform method. Ramos [20] solved Bratu’s and Troesch’s problems by the hybrid block method. Peiravi and Alinejad [21] studied a 3D Newtonian droplet impact by the volume of fluid (VOF) method.Among these strongly nonlinear methods, the global residue harmonic balance method (GRHBM) is an effective and analytical one. The GRHBM was the first presented by Ju and Xue [22] to obtain the periodic solutions of the jerk equation. By introducing a parameter p and viewing it as a small parameter, the GRHBM can be independent on the large parameter and can give the higher-order analytical approximation. And, to improve the accuracy, the residual errors are considered in the approximations of each order. Thus, the GRHBM has found applications in various nonlinear problems [23–26].In this paper, we study a system of strongly nonlinear oscillators in engineering, namely, the large amplitude free vibration of a restrained uniform beam carrying an intermediate lumped mass. This system was first presented by Hamdan and Shabaneh [27] and then has been analysed by Ganji et al. [28] by applying AFF and HPM. However, the higher-order analytical approximate solution for this system has not been obtained. To obtain high-order and more accurate results, this system is revisited in this paper by using the GRHBM. The higher-order approximate solution and corresponding frequency have been obtained. Comparison of the results obtained by using GRHBM with those obtained by exact solutions and other methods shows that the GRHBM can be very accurate, effective, and straightforward. It also can be extended to other nonlinear oscillators in engineering and physics.


## 2. System Discussion

A schematic of the beam under research is given in Figure1. Here, we give a brief description of this beam. For more details, one can refer to [27].Figure 1
A sketch of the beam system under research.The beam is considered to be uniform of lengthl and unit mass m, hinged at the base to a rotational spring Kr and carried a lumped mass M at an arbitrary point s=d along the beam span. ζ=s/l is a dimensionless arc length.According to [27], the Lagrangian of this beam should be(1)L=ml2∫011+μδζ−ηx˙2+y˙2−EIλ42my″2+λy′y″2+λ4y′4y″2dζ,where the primes and dots denote the derivatives with respect to ξ and t, respectively, η=d/l is the relative position of M, μ=M/ml is a mass ratio parameter, δζ−η is Dirac’s function, λ=1/l, and EI is the modulus of flexural rigidity.Assumex and y satisfy the constraint relation as follows:(2)1+λx′2+λy′2=1.Equation (2) can be rewritten as(3)1+λx′=1−λy′21/2.Assumeλy′2≪1 and yζ,t have a form as follows:(4)yζ,t=ϕζut,where ϕζ is an eigenfunction.Expanding the right side of equation (3) to the fifth-order term, then equation (3) yields(5)x=−12∫0ζλy′2+14λ3y′4dχ.Combining with equations (5) and (4), equation (1) can become(6)L=ml2α1u˙2+α3λ2u2u˙2+α4λ2u2u˙2+α5λ4u4u˙2+α6λ4u4u˙2−EIλ4mα2u2+α7λ2u4+α8λ4u6,where α1,α2,…,α8 are functions of ζ, which describe the mode shapes.Applying the Euler–Lagrange equation(7)ddt∂L∂u˙−∂L∂u=0,to equation (6), one obtains the following problem:(8)q¨+q+ɛ1q2q¨+ɛ1qq˙2+ɛ2q4q¨+2ɛ2q3q˙2+ɛ3q3+ɛ4q5=0,with(9)q0=A,q˙0=0,which ɛ1,…,ɛ4 are functions of α1,…,α8.


## 3. Review of the Global Residue Harmonic Balance Method

Consider the systems as follows:(10)Φq¨,q˙,q=0,q0=A,q˙0=0,where q˙=dq/dt, A is the amplitude of the system, and Φq¨,q˙,q is a nonlinear function satisfying(11)Φ−q¨,−q˙,−q=−Φq¨,q˙,q.By introducingτ=ωt, equation (10) becomes(12)Φω2q″,ωq′,q=0,q0=A,q′0=0,where q′=dq/dτ and ω is an frequency.Let the periodic solution of equation (12) exist, and it can be expressed by such a set of functions as follows:(13)cos2m−1τ|m=1,2,3,….According to equation (13), the initial approximation solution satisfying the initial conditions should be(14)q0τ=Acosτ,τ=ω0t,where ω0 can be determined later.Substituting equations (14) into (12), the initial residual is(15)R0τ=Φω02q0″,ω0q0′,q0.IfR0τ=0, then q0τ turns into the exact solution of equation (10). Generally, such case will not appear in the nonlinear systems.Eliminating the secular terms in equation (15), through which, ω0 can be determined. Then, the zeroth-order approximation solution is obtained of the form as follows:(16)q0τ=Acosτ,τ=ω0t,with the initial residual as follows:(17)R0τ=Φω02q0″,ω0q0′,q0.Next, we introduce a parameterp∈0,1, denote qτ and ω as qτ,p and ωp, and expand qτ,p and ωp as follows:(18)qτ,p=qk−1τ,p+pkqkτ+⋯,ω2p=ωk−12p+pkωk+⋯,in which(19)qk−1τ,p=q0τ+pq1τ+⋯+pk−1qk−1τ,ωk−12p=ω02+pω1+⋯+pk−1ωk−1.Obviously,qt,0 is the zeroth-order approximation and qt,1 is the required approximate solution of equation (12). Generally, the higher-order approximation to the solution can be provided by equation (18). For instance, the first-order approximation can be shown as(20)qτ=q0τ+pq1τ,ω2=ω02+pω1.Inserting equations (20) into (12) and equating the coefficients of p, we can get(21)F1τ,ω1,q1τ=ddpΦω2pq″τ,p,ωpq′τ,p,qτ,p|p=0=ω1∂∂ω2+q1″∂∂q″+q1′∂∂q′+q1∂∂qΦ0,where ∂Φ0/∂q denotes that ∂Φ/∂q is to be evaluated at the zeroth-order approximation.By taking equation (13) into account, let(22)q1τ=a3,1cosτ−cos3τ,and substitute it into equation (21), and we consider the following equation:(23)R1τ=F1τ,ω1,q1τ+R0τ,where R0τ has been introduced into equation (23) to improve the accuracy.The right hand side of equation (23) should not contain the terms cosτ and cos3τ. Letting their coefficients be zeros, one could obtain two linear equations, then the two unknowns ω1 and a3,1 can be determined. Till now, the first-order approximation can be obtained as(24)q1τ=q0τ+q1τ,ω12=ω02+ω1,τ=ω1t,with the residual(25)R1τ=Φω12q1″,ω1q1′,q1.For thek-th-order approximation, we may assume(26)qkτ=∑i=1ka2i+1,kcosτ−cos2i+1τ,k=2,3,…,.To determinea2i+1,k,i=2,…,k and ωk, substituting equations (18) into (12) and collecting the coefficients of p, one should get(27)Fkτ,ωk,qkτ=1k!dkdpkΦω2pq″τ,p,ωpq′τ,p,qτ,p|p=0,Rk−1τ=Φωk−121qk−1″τ,1,ωk−11qk−1′τ,1,qk−1τ,1.Considering the equation(28)Rkτ=Fkτ,ωk,qkτ+Rk−1τ,eliminating the coefficients of cosτ,cos3τ,… and cos2k+1τ in equation (28), then the unknowns a3,k,a5,k,…,a2k+1,k and ωk can be obtained. Thus, the k-th-order approximation should be obtained as(29)qkτ=qk−1τ,1+qkτ,ωk2=ωk−121+ωk.


## 4. Solution Procedure

In this section, the GRHBM is applied to solve system (8) and (9). With τ=ωt, system (8) and (9) becomes(30)ω2q″+ɛ1q2q″+ɛ1qq′2+ɛ2q4q″+2ɛ2q3q′2+q+ɛ3q3+ɛ4q5=0,and(31)q0=A,q′0=0,where q′=dq/dτ.


### 4.1. Zeroth-Order Approximation

Firstly, the initial approximation can be written as(32)q0τ=Acosτ,τ=ω0t.Substituting equation (32) into (30) yields(33)R0τ=F01cosτ+F03cos3τ+F05cos5τ,in which(34)F01=8A−8ω02A−4ω02ɛ1A3−3ω02ɛ2A5+6ɛ3A3+5ɛ4A58,F03=4ɛ3A3+5ɛ4A5−8ω02ɛ1A3−7ω02ɛ2A516,F05=ɛ4A5−3ω02ɛ2A516.SolvingF01=0 gives(35)ω0=8+6ɛ3A2+5ɛ4A48+4ɛ1A2+3ɛ2A4,which is the same as the result by Ganji et al. [28]. Then, we obtain the zeroth-order approximation (32) and the initial residual error as follows:(36)R0τ=F03cos3τ+F05cos5τ.


### 4.2. First-Order Approximation

Substituting equations (20) and (22) into equation (30) and considering the coefficient of p, we have(37)F1τ,ω1=F11cosτ+F13cos3τ+F15cos5τ+F17cos7τ16,in which(38)F11=16a3,1−16a3,1ω02−16Aω1−8A3ω1ɛ1+5A4a3,1ω02ɛ2−6A5ω1ɛ2+24A2a3,1ɛ3+25A4a3,1ɛ4,F13=144a3,1ω02−16a3,1+56A2a3,1ω02ɛ1−8A3ω1ɛ1+31A4a3,1ω02ɛ2−7A5ω1ɛ2−12A2a3,1ɛ3−5A4a3,1ɛ4,F15=72A2a3,1ω02ɛ1+61A4a3,1ω02ɛ2−3A5ω1ɛ2−12A2a3,1ɛ3−15A4a3,1ɛ4,F17=31A4a3,1ω02ɛ2−5A4a3,1ɛ4.According to equation (23), we have(39)R1τ=F1τ,ω1+R0τ.Eliminating thecosτ and cos3τ in equation (39) yields(40)0=a3,1−a3,1ω02−Aω1+ɛ1ω1A32+5a3,1ɛ2ω02A416−3ɛ2ω1A58+3a3,1ɛ3A22+25a3,1ɛ4A416,(41)0=−a3,1+9a3,1ω02−ɛ1ω02A32+7a3,1ɛ1ω02A22−ɛ1ω1A32−7ɛ2ω02A516+31a3,1ɛ2ω02A416−7ɛ2ω1A516+ɛ3A34−3a3,1ɛ3A24+5ɛ4A516−5a3,1ɛ4A416.Solving equations (40) and (41) gives(42)a3,1=2A38+4ɛ1A2+3ɛ2A4H2304ω02−256A2ɛ1+2176A2ω02ɛ1−208A4ɛ2−256+F+G,(43)ω1=A216−16ω02+5ɛ2ω02A4+24ɛ3A2+25ɛ4A4H2304ω02−256A2ɛ1+2176A2ω02ɛ1−208A4ɛ2−256+F+G,in which(44)H=8ɛ1ω02+7ɛ2ω02A2−4ɛ3−5ɛ4A2,G=448A4ω02ɛ12−240A6ɛ2ɛ3−192A2ɛ3−240A6ɛ1ɛ4−205A8ɛ2ɛ4,F=1472A4ω02ɛ2+544A6ω02ɛ1ɛ2+151A8ω02ɛ22−80A4ɛ4−288A4ɛ1ɛ3.Thus, we obtain the first-order approximation as follows:(45)ω1=ω02+ω1,q1t=A+a3,1cosω1t−a3,1cos3ω1t,where ω0, ω1, and a3,1 are given in equations (35), (42), and (43).With the procedure going on, the higher-order approximation can be obtained similarly.


### 4.3. Results and Discussion

To illustrate and verify the efficiency and accuracy of the presented method, the comparison of the analytical approximation obtained by the presented method with the exact solutions and other methods is discussed in this section. Here, the numerical results from fourth-order Runge–Kutta method are viewed as the exact solutions.For this system, the parametersɛii=1,2,3,4 describe the mode shapes and can be calculated by the values of base stiffness S, mass ratio μ, and position η. Here, the zeroth- and first-order approximate frequencies and their relative errors RE by the different values of ɛ1∼ɛ4 and A are shown in Table 1. From Table 1, it can be seen that the zeroth-order approximations of frequencies the same as the first-order results by HPM and AFF have a good adjustment with the exact ones with the maximal relative error 0.79887, and the relative errors of the first-order approximate frequencies are much less than the zero-order. It indicates that the first-order approximate frequencies can be more closer to the exact frequencies than the zero-order ones, and the GRHBM is more efficient and more accurate for solving strongly nonlinear problems.Table 1
Comparison of frequency corresponding to various parameters of this system.
ModeAɛ1ɛ2ɛ3ɛ4ω0=ω1[28] REω1REωex11.00.3268450.1295790.2325980.0875841.00705 (0.30664)1.010180 (0.00297)1.0101520.51.6420330.9130550.3135610.2042970.93255 (0.40947)0.935268 (0.11982)0.9363930.24.0514861.6652320.2814180.1496770.96546 (0.12144)0.965852 (0.08152)0.9666440.38.2055783.1453680.2723130.1337080.85970 (0.52752)0.864408 (0.01712)0.8642611.20.3038440.1150760.2325980.0875841.02135 (0.67990)1.027492 (0.08246)1.0283420.80.8939810.4673990.3135610.2042970.94117 (0.68750)0.947891 (0.02226)0.9476830.45.0746361.9719620.2814180.1496770.85275 (0.65326)0.858566 (0.02516)0.8583540.35.3716261.9958200.2723130.1337080.90371 (0.32569)0.906268 (0.04324)0.9066611.10.3338610.1319140.3331050.1299231.05601 (0.50254)1.060826 (0.04843)1.0613420.61.5129300.8027000.3794170.2502830.92528 (0.59649)0.930113 (0.07810)0.9308430.44.2789911.7154240.3182010.1733130.87526 (0.53749)0.879977 (0.00147)0.8799940.310.0150073.8204950.2971090.1492310.83552 (0.79887)0.841654 (0.06958)0.8422410.90.3272310.1270220.3331050.1299231.03871 (0.31394)1.040568 (0.13551)1.0419820.71.3569640.6960430.3794170.2502830.91850 (0.74083)0.925756 (0.04388)0.9253530.44.2333891.6574640.3182010.1733130.87662 (0.52277)0.881255 (0.00284)0.8812340.111.1785634.2002500.2971090.1492310.97418 (0.14969)0.974368 (0.13038)0.97564To further illustrate and verify the accuracy of the presenting method, we select some parameter values as follows:(46)A=0.3,ɛ1=8.205578,ɛ2=3.145368,ɛ3=0.272313,ɛ4=0.133708,(47)A=0.4,ɛ1=5.074636,ɛ2=1.971962,ɛ3=0.281418,ɛ4=0.149677,(48)A=0.4,ɛ1=4.233389,ɛ2=1.657464,ɛ3=0.318201,ɛ4=0.173313,(49)A=0.5,ɛ1=1.642033,ɛ2=0.913055,ɛ3=0.313561,ɛ4=0.204297.Figures2–5 provide the comparisons of the current analytical solutions to the exact solutions. From Figures 2–5, it is noteworthy that the first-order approximate solutions are almost similar to the exact solutions. This shows that the accuracy of the results obtained by this method is very high.Figure 2
Comparison between approximate and exact solutions under the parameter values (46);  exact solution;  zeroth-order approximation;   first-order approximation.Figure 3
Comparison between approximate and exact solutions under the parameter values (47);  exact solution;  zeroth-order approximation;  first-order approximation.Figure 4
Comparison between approximate and exact solutions under the parameter values (48);  exact solution;  zeroth-order approximation;  first-order approximation.Figure 5
Comparison between approximate and exact solutions under the parameter values (49);  exact solution;  zeroth-order approximation;  first-order approximation.


## 4.1. Zeroth-Order Approximation

Firstly, the initial approximation can be written as(32)q0τ=Acosτ,τ=ω0t.Substituting equation (32) into (30) yields(33)R0τ=F01cosτ+F03cos3τ+F05cos5τ,in which(34)F01=8A−8ω02A−4ω02ɛ1A3−3ω02ɛ2A5+6ɛ3A3+5ɛ4A58,F03=4ɛ3A3+5ɛ4A5−8ω02ɛ1A3−7ω02ɛ2A516,F05=ɛ4A5−3ω02ɛ2A516.SolvingF01=0 gives(35)ω0=8+6ɛ3A2+5ɛ4A48+4ɛ1A2+3ɛ2A4,which is the same as the result by Ganji et al. [28]. Then, we obtain the zeroth-order approximation (32) and the initial residual error as follows:(36)R0τ=F03cos3τ+F05cos5τ.


## 4.2. First-Order Approximation

Substituting equations (20) and (22) into equation (30) and considering the coefficient of p, we have(37)F1τ,ω1=F11cosτ+F13cos3τ+F15cos5τ+F17cos7τ16,in which(38)F11=16a3,1−16a3,1ω02−16Aω1−8A3ω1ɛ1+5A4a3,1ω02ɛ2−6A5ω1ɛ2+24A2a3,1ɛ3+25A4a3,1ɛ4,F13=144a3,1ω02−16a3,1+56A2a3,1ω02ɛ1−8A3ω1ɛ1+31A4a3,1ω02ɛ2−7A5ω1ɛ2−12A2a3,1ɛ3−5A4a3,1ɛ4,F15=72A2a3,1ω02ɛ1+61A4a3,1ω02ɛ2−3A5ω1ɛ2−12A2a3,1ɛ3−15A4a3,1ɛ4,F17=31A4a3,1ω02ɛ2−5A4a3,1ɛ4.According to equation (23), we have(39)R1τ=F1τ,ω1+R0τ.Eliminating thecosτ and cos3τ in equation (39) yields(40)0=a3,1−a3,1ω02−Aω1+ɛ1ω1A32+5a3,1ɛ2ω02A416−3ɛ2ω1A58+3a3,1ɛ3A22+25a3,1ɛ4A416,(41)0=−a3,1+9a3,1ω02−ɛ1ω02A32+7a3,1ɛ1ω02A22−ɛ1ω1A32−7ɛ2ω02A516+31a3,1ɛ2ω02A416−7ɛ2ω1A516+ɛ3A34−3a3,1ɛ3A24+5ɛ4A516−5a3,1ɛ4A416.Solving equations (40) and (41) gives(42)a3,1=2A38+4ɛ1A2+3ɛ2A4H2304ω02−256A2ɛ1+2176A2ω02ɛ1−208A4ɛ2−256+F+G,(43)ω1=A216−16ω02+5ɛ2ω02A4+24ɛ3A2+25ɛ4A4H2304ω02−256A2ɛ1+2176A2ω02ɛ1−208A4ɛ2−256+F+G,in which(44)H=8ɛ1ω02+7ɛ2ω02A2−4ɛ3−5ɛ4A2,G=448A4ω02ɛ12−240A6ɛ2ɛ3−192A2ɛ3−240A6ɛ1ɛ4−205A8ɛ2ɛ4,F=1472A4ω02ɛ2+544A6ω02ɛ1ɛ2+151A8ω02ɛ22−80A4ɛ4−288A4ɛ1ɛ3.Thus, we obtain the first-order approximation as follows:(45)ω1=ω02+ω1,q1t=A+a3,1cosω1t−a3,1cos3ω1t,where ω0, ω1, and a3,1 are given in equations (35), (42), and (43).With the procedure going on, the higher-order approximation can be obtained similarly.


## 4.3. Results and Discussion

To illustrate and verify the efficiency and accuracy of the presented method, the comparison of the analytical approximation obtained by the presented method with the exact solutions and other methods is discussed in this section. Here, the numerical results from fourth-order Runge–Kutta method are viewed as the exact solutions.For this system, the parametersɛii=1,2,3,4 describe the mode shapes and can be calculated by the values of base stiffness S, mass ratio μ, and position η. Here, the zeroth- and first-order approximate frequencies and their relative errors RE by the different values of ɛ1∼ɛ4 and A are shown in Table 1. From Table 1, it can be seen that the zeroth-order approximations of frequencies the same as the first-order results by HPM and AFF have a good adjustment with the exact ones with the maximal relative error 0.79887, and the relative errors of the first-order approximate frequencies are much less than the zero-order. It indicates that the first-order approximate frequencies can be more closer to the exact frequencies than the zero-order ones, and the GRHBM is more efficient and more accurate for solving strongly nonlinear problems.Table 1
Comparison of frequency corresponding to various parameters of this system.
ModeAɛ1ɛ2ɛ3ɛ4ω0=ω1[28] REω1REωex11.00.3268450.1295790.2325980.0875841.00705 (0.30664)1.010180 (0.00297)1.0101520.51.6420330.9130550.3135610.2042970.93255 (0.40947)0.935268 (0.11982)0.9363930.24.0514861.6652320.2814180.1496770.96546 (0.12144)0.965852 (0.08152)0.9666440.38.2055783.1453680.2723130.1337080.85970 (0.52752)0.864408 (0.01712)0.8642611.20.3038440.1150760.2325980.0875841.02135 (0.67990)1.027492 (0.08246)1.0283420.80.8939810.4673990.3135610.2042970.94117 (0.68750)0.947891 (0.02226)0.9476830.45.0746361.9719620.2814180.1496770.85275 (0.65326)0.858566 (0.02516)0.8583540.35.3716261.9958200.2723130.1337080.90371 (0.32569)0.906268 (0.04324)0.9066611.10.3338610.1319140.3331050.1299231.05601 (0.50254)1.060826 (0.04843)1.0613420.61.5129300.8027000.3794170.2502830.92528 (0.59649)0.930113 (0.07810)0.9308430.44.2789911.7154240.3182010.1733130.87526 (0.53749)0.879977 (0.00147)0.8799940.310.0150073.8204950.2971090.1492310.83552 (0.79887)0.841654 (0.06958)0.8422410.90.3272310.1270220.3331050.1299231.03871 (0.31394)1.040568 (0.13551)1.0419820.71.3569640.6960430.3794170.2502830.91850 (0.74083)0.925756 (0.04388)0.9253530.44.2333891.6574640.3182010.1733130.87662 (0.52277)0.881255 (0.00284)0.8812340.111.1785634.2002500.2971090.1492310.97418 (0.14969)0.974368 (0.13038)0.97564To further illustrate and verify the accuracy of the presenting method, we select some parameter values as follows:(46)A=0.3,ɛ1=8.205578,ɛ2=3.145368,ɛ3=0.272313,ɛ4=0.133708,(47)A=0.4,ɛ1=5.074636,ɛ2=1.971962,ɛ3=0.281418,ɛ4=0.149677,(48)A=0.4,ɛ1=4.233389,ɛ2=1.657464,ɛ3=0.318201,ɛ4=0.173313,(49)A=0.5,ɛ1=1.642033,ɛ2=0.913055,ɛ3=0.313561,ɛ4=0.204297.Figures2–5 provide the comparisons of the current analytical solutions to the exact solutions. From Figures 2–5, it is noteworthy that the first-order approximate solutions are almost similar to the exact solutions. This shows that the accuracy of the results obtained by this method is very high.Figure 2
Comparison between approximate and exact solutions under the parameter values (46);  exact solution;  zeroth-order approximation;   first-order approximation.Figure 3
Comparison between approximate and exact solutions under the parameter values (47);  exact solution;  zeroth-order approximation;  first-order approximation.Figure 4
Comparison between approximate and exact solutions under the parameter values (48);  exact solution;  zeroth-order approximation;  first-order approximation.Figure 5
Comparison between approximate and exact solutions under the parameter values (49);  exact solution;  zeroth-order approximation;  first-order approximation.


## 5. Conclusions

The global residue harmonic balance method has been employed to obtain the approximate periodic solution and frequency for a well-known system of strongly nonlinear oscillators in engineering, namely, the large amplitude free vibration of a restrained uniform beam carrying an intermediate lumped mass.(i)
The brief descriptions of the beam and the basic idea of the GRHBM are introduced.(ii)
The GRHBM can provide a high accuracy by considering the residual errors in the process of each order approximation. The solution procedure of the GRHBM is simple and clear. The higher-order approximate solution and frequency for the considered problem by GRHBM can be obtained easily.(iii)
By comparing with the previously existing results and the exact solutions, it is shown that the GRHBM is an accurate, convenient, and efficient method.(iv)
The GRHBM can be easily extended to other nonlinear oscillators in engineering and physics.



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*Source: 6641742-2021-09-09.xml*