Partial Differential Equations in Applied Mathematics

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Partial Differential Equations in Applied Mathematics 6 (2022) 100414

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journal homepage:www.elsevier.com/locate/padiff

Free vibration analysis of nonlinear axially loaded beams using a modified harmonic balance method

M. Wali Ullah

^a^,^b^,^∗

, M. Saifur Rahman

^c

, M. Alhaz Uddin

^a

aDepartment of Mathematics, Khulna University of Engineering & Technology, Khulna 9203, Bangladesh

bDepartment of Computer Science & Engineering, Northern University of Business and Technology Khulna, Khulna 9100, Bangladesh

cDepartment of Mathematics, Rajshahi University of Engineering & Technology, Rajshahi 6204, Bangladesh

A R T I C L E I N F O

Keywords:

Harmonic balance method Free vibration

Axially loaded beam Nonlinear beams

A B S T R A C T

The conventional harmonic balance method utilizes a numerical approach for solving a set of nonlinear algebraic equations to identify the unknown coefficients. In this research, a modified harmonic balance method has been represented to study a free vibration problem with an axial load. The success of this research is that it requires the solution of a single nonlinear algebraic equation and a system of linear algebraic equations to obtain the desired results. As a result, it takes less effort to compute than the traditional harmonic balance technique. To validate and justify the accuracy of the proposed approach, the acquired outcomes are compared to those obtained using the fourth-order Runge–Kutta technique. The results acquired from the proposed process nicely agree with the numerical results.

1. Introduction

Nonlinear vibration analysis of beam/plate is an interesting topic to the scientists, physicists, engineers and researchers due its various applications in science and engineering. In mathematical modeling, nonlinearity comes due to the significant amplitude vibration of the beam/plate. These nonlinear problems rarely have accurate or closed solutions. Therefore, many scientists and researchers have developed numerical or approximation methods to handle these nonlinear problems. Numerical techniques are relatively easy to program, but it requires harsh computational effort and appropriate initial values. Be- sides, the outcomes of these methods have no analytical expressions.

On the other hand, analytical approximation techniques are prefer- able to scientists because they are intrinsic in physical meanings and more worthy of the parametric study. A lot of analytical methods are used for solving nonlinear problems. For example, perturbation methods such as Lindstedt–Poincare (LP),¹Krylov–Bogoliubov–Mitropolskii (KBM),² and multiple time-scale methods³ are widely used. The lim- itation of the perturbation methods is that the approximate solutions must be expressed in a power series associated with small parameters. However, solutions to some nonlinear problems do not meet this requirement. Several analytical approximation techniques, like as Homotopy Analysis technique,^4,5 Homotopy Perturbation technique,⁶ and Variational Iteration technique,⁷ have been developed to solve differential equations both for weekly and strongly nonlinearity to overcome the limitations of the perturbation techniques. Recently, the harmonic balance method,^8–12 has attracted the attention of some

∗ Corresponding author at: Department of Computer Science & Engineering, Northern University of Business and Technology Khulna, Khulna 9100, Bangladesh.

E-mail address: [email protected](M.W. Ullah).

researchers. The solution to the harmonic balance method is expressed in terms of the truncated Fourier series, the coefficients of which are calculated by solving a set of linear equations with the help of nonlinear one. This method is valid to analyze the differential equations with both strongly and weekly nonlinearities respectively. The harmonic balance method has been widely utilized to solve several beam/plate problems.

Rahman et al.¹³ presented a residual multilevel harmonic balance method to analyze the free vibration of axial beams with third order and fifth order nonlinearity. Hasanet al.¹⁴introduced a novel harmonic balance technique named the multilevel residue harmonic balance approach for obtaining solutions for multimode nonlinear vibrating beams on an adjustable base. Motallebi et al.¹⁵studied the vibrations of a nonlinear beam subjected to axial force by using homotopy analysis technique. Arda and Aydogdu¹⁶ investigated the dynamic stability of harmonically stimulated nanobeams, considering the presence of axial inertia. Gao et al.¹⁷ used a two-step perturbation approach to perform free vibration analysis on nanotubes composed of functionally graded bi-semitubes. Ri et al.¹⁸investigated nonlinear forced vibration analysis using DQFEM and IHB on composite beams. Ullah et al.¹⁹ devised a modified harmonic balancing approach for handling second- order nonlinear ordinary forced vibration equations characterized by significant nonlinearity. Xu et al.²⁰demonstrated the dynamic stability of supported beams when parametric stimulation is multi-harmonic.

Abd-el-Malek et al.²¹ developed a modified perturbation solution to the strongly nonlinear initial value problem of the KdV–KP equation.

Sobamowo and Yinusa²² investigated nonlinear finite element analysis of vibration of multi-walled carbon nanotubes with geometric

https://doi.org/10.1016/j.padiff.2022.100414

Received 14 May 2022; Received in revised form 24 June 2022; Accepted 25 June 2022

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effort than the existing classical harmonic balance technique. The attained results are associated with the corresponding numerical results in figures that show good agreement.

2. Theoretical model

The nonlinear differential equation of a beam can be written as follows if the mid-plane stretching is not ignored¹⁵

𝐸𝐼 𝑊^′′′′+𝑚̈ 𝑊+

(

−𝑁−𝐸𝐴 2𝐿∫

𝐿 0

( 𝑊^′)2

𝑑𝑥+𝐸𝐴 8𝐿 ∫

𝐿 0

( 𝑊^′)4

𝑑𝑥 )

𝑊^′′

= 0 (2.1)

where 𝑊 denotes the transverse displacement,𝑥is the physical coordinate of the beam, 𝑊^′, 𝑊^′′ and𝑊^′′′′ are the first, second, and fourth-order derivatives with respect to𝑥respectively, while ̈

𝑊 is the second-order derivative with respect to time𝑡,𝑁is the axial force, the length of the beam is 𝐿, 𝐴 = 𝐵×ℎ =is the beam’s cross-sectional area, the beam width is 𝐵, while the beam thickness is ℎ, 𝐼 is the moment of inertia of a beam,𝐸is the beam’s Young’s modulus,𝑚 is the mass per unit length and𝜌is the material density. The following non-dimensional parameters are used for this study

𝑥= 𝑥 𝐿, 𝑤= 𝑤

𝑟, 𝑡=𝑡

√𝐸𝐼

𝑚𝐿⁴, 𝐹 =𝑁 𝐿²

𝐸𝐼 (2.2)

where𝑟= (

𝐼 𝐴

)¹₂

is the radius of gyration of the cross-section, 𝑥,𝑤, and𝑡are the normalized longitudinal beam coordinate, the normalized transverse displacement, and the normalized time respectively, while 𝐹 denotes the non-dimensional axial force. Non-dimensional form of Eq.(2.1)is written as

𝑊^′′′′+𝑊̈ + (

−𝐹−1 2∫

1 0

(𝑊^′)2

𝑑𝑥+ 𝑟² 8𝐿²∫

1 0

(𝑊^′)4

𝑑𝑥 )

𝑊^′′= 0 (2.3) The boundary condition associated with the Simply Supported (SS) beam is written as follows:

𝑊(𝑥, 𝑡) =𝑊^′′(𝑥, 𝑡) = 0 at𝑥= 0,1 (2.4) Assuming 𝑊(𝑥, 𝑡) = 𝑞(𝑡)𝜑(𝑥), where 𝑞 is the amplitude,𝜑 is the dimensionless structural mode shape of the beam,𝜑(𝑥) = sin (𝑛𝜋𝑥)for simply supported beams. Applying the Galerkin method,¹³Eq.(2.3)is written as

̈

𝑞+𝜔²₁𝑞+𝛼𝑞³+𝛽𝑞⁵= 0 (2.5)

where 𝜔²

1=∫₀¹𝜑𝜑^𝑖𝑣𝑑𝑥−𝐹∫₀¹𝜑𝜑^′′𝑑𝑥

∫₀¹𝜑²𝑑𝑥

, 𝛼= −

1

2∫₀¹𝜑𝜑^′′𝑑𝑥.∫₀¹( 𝜑^′)2

𝑑𝑥

∫₀¹𝜑²𝑑𝑥 ,

𝛽=

𝑟²

8𝐿∫₀¹𝜑𝜑^′′𝑑𝑥.∫₀¹( 𝜑^′)4

𝑑𝑥

∫₀¹𝜑²𝑑𝑥 .

Eq.(2.5)is known as the nonlinear governing differential equation of axially loaded beams with the initial conditions

𝑞(0) =𝑏_𝑜, ̇𝑞(0) = 0 (2.6)

where𝑏_𝑜is the maximum non-dimensional amplitude of oscillation.

frequency,𝑓_𝑖(𝑞)are given nonlinear functions of𝑞such that𝑓(−𝑞) =

−𝑓(𝑞)and𝛼_𝑖(𝑖= 3,5,7,…, 𝑛) are constants. According to the proposed method, the approximate analytical solution of Eq.(3.1) is assumed as

𝑞=𝑎cos (𝜔𝑡) +𝑎₃cos (3𝜔𝑡) +⋯ (3.2)

where 𝑎, 𝑎₃,… are the unknown coefficients. Now differentiating Eq.(3.2)twice with respect to time𝑡and then putting it into Eq.(3.1) and expanding𝑓_𝑖(𝑞)as a truncated Fourier series expansion and taking the coefficients of equal harmonics from both sides, we obtain the following system of algebraic equations

𝑎(

−𝜔²+𝜔²₁) +𝐶₁(

𝑎, 𝑎₃,…)

= 0 (3.3)

𝑎₃(

−9𝜔²+𝜔²₁) +𝐶₃(

𝑎, 𝑎₃,…)

= 0 (3.4)

Applying initial conditions, we get

𝑎=𝑏₀−𝑎₃ (3.5)

Eliminating𝜔²from the Eq.(3.4)with the help of Eq.(3.3), we get

−8𝜔²₁𝑎₃− 9𝑎₃𝑆₁( 𝑏₀, 𝑎₃,…)

+𝑆₃( 𝑏₀, 𝑎₃,…)

= 0 (3.6)

Now expanding𝑎₃in a power series of the small parameter𝜇(𝛼, 𝛽) in the form

𝑎₃=𝜉₀+𝜉₁𝜇+𝜉₂𝜇²+⋯ (3.7)

After substituting Eq.(3.7)into Eq.(3.6)and equating the coeffi- cient of like power of small parameter𝜇(𝛼, 𝛽)we get,𝜉₀, 𝜉₁,…which are the functions of𝑏₀. Finally, substituting the value of𝑏₀, the value of𝑎₃ is determined. Consequently, the desired values of𝑎and𝜔are calculated.

4. Application of the proposed method

According to the proposed method, the solution of Eq. (2.5) is considered as

𝑞=𝑎cos (𝜔𝑡) +𝑎₃cos (3𝜔𝑡) (4.1)

Here, two harmonic terms are considered in the trial solution since it agrees well with the numerical solution for the different values of the system parameters and it gives the acceptable accuracy. On the other hand, if we add more harmonic terms in the trial solution, the obtained results do not change significantly. Inserting Eq.(4.1)with their derivatives in Eq.(2.5)and hence equating the coefficients of like harmonics on both sides, we get

𝑎( 𝜔²₁−𝜔²)

+3 4𝑎³𝛼+5

8𝑎⁵𝛽+3

4𝑎²𝛼𝑎₃+25

16𝑎⁴𝛽𝑎₃+3 2𝑎𝛼𝑎²₃ + 15

4𝑎³𝛽𝑎²₃+15

8𝑎²𝛽𝑎³₃+15

8𝑎𝛽𝑎⁴₃= 0 (4.2)

𝑎₃( 𝜔²

1− 9𝜔²) +𝑎³𝛼

4 +5𝑎⁵𝛽 16 +3

2𝑎²𝛼𝑎₃+15 8𝑎⁴𝛽𝑎₃ + 15

8𝑎³𝛽𝑎²₃+3𝛼𝑎³

3

4 +15

4𝑎²𝛽𝑎³₃+5𝛽𝑎⁵

3

8 = 0 (4.3)

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Fig. 1(a). Time response curve of a nonlinear axially loaded beam. Comparison of the analytical and numerical solutions for the initial value𝑞(0) = 0.1when𝜔= 9.91255, 𝑎= 0.09999, 𝑎₃= 0.000007745.

Fig. 1(b). Time response curve of a nonlinear axially loaded beam. Comparison of the analytical and numerical solutions for the initial value𝑞(0) = 0.5when𝜔= 10.1308, 𝑎= 0.499073, 𝑎₃= 0.000926821.

Fig. 1(c). Time response curve of a nonlinear axially loaded beam. Comparison of the analytical and numerical solutions for the initial value𝑞(0) = 2when𝜔= 13.0335, 𝑎= 1.96422, 𝑎₃= 0.0357805.

Fig. 1(d). Time response curve of a nonlinear axially loaded beam. Comparison of the analytical and numerical solutions for the initial value𝑞(0) = 3when𝜔= 16.0727, 𝑎= 2.92072, 𝑎₃= 0.0792812.

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Fig. 1(e). Time response curve of a nonlinear axially loaded beam. Comparison of the analytical and numerical solutions for the initial value𝑞(0) = 5when𝜔= 23.2325, 𝑎= 4.82464, 𝑎3= 0.175364.

Fig. 2(a). Comparison of the 1st resonant backbone curve of simply supported beam acquired from the numerical method and proposed method when𝐹= 10.

Fig. 2(b). Comparison of the 3rd resonant backbone curve of simply supported beam acquired from the numerical method and proposed method when𝐹= 10.

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Fig. 2(c). Comparison of the 1st resonant backbone curve of a simply supported beam acquired from the numerical method and the proposed method when𝐹= −8.

Fig. 2(d). Comparison of the 3rd resonant backbone curve of a simply supported beam acquired from the numerical method and the proposed method when𝐹= −8.

Table 1

Comparison between the analytical and numerical solutions.

Time,𝑡 𝑏_𝑜= 0.1,𝜔= 9.912 𝑏_𝑜= 5,𝜔= 25.297

Numerical solution,𝑞_𝑛𝑢 Analytical solution,𝑞_𝑎𝑝𝑝 Numerical solution,𝑞_𝑛𝑢 Analytical solution,𝑞_𝑎𝑝𝑝

0 0.1 0.1 5 5

0.5 0.061 0.062 4.979 4.979

1 −0.023 −0.023 4.915 4.916

1.5 −0.090 −0.090 4.811 4.814

2 −0.088 −0.089 4.667 4.673

2.5 −0.020 −0.020 4.489 4.496

3 0.063 0.064 4.277 4.286

3.5 0.099 0.099 4.036 4.047

4 0.060 0.059 3.770 3.781

4.5 −0.025 −0.025 3.481 3.493

5 −0.091 −0.091 3.175 3.186

Eliminating 𝜔² from Eq. (4.3) with the help of Eq. (4.2), and ignoring the terms whose responses are insignificant and then we obtain

4𝑎³𝛼+ 5𝑎⁵𝛽− 84𝑎²𝛼𝑎₃− 60𝑎⁴𝛽𝑎₃− 108𝑎𝛼𝑎²₃− 195𝑎³𝛽𝑎²₃− 204𝛼𝑎³₃

− 480𝑎²𝛽𝑎³

3− 270𝑎𝛽𝑎⁴₃− 260𝛽𝑎⁵₃− 128𝑎₃𝜔²

1= 0 (4.4)

Applying the initial condition𝑎=𝑏₀−𝑎₃ in Eq.(4.4)and ignoring the terms whose responses are insignificant and then we obtain 72𝛼𝑎²₃𝑏₀+4𝛼𝑏³₀+95𝛽𝑎²₃𝑏³

0+5𝛽𝑏⁵₀+(−128𝜔²₁−96𝛼𝑏²₀−85𝛽𝑏⁴₀)𝑎₃= 0 (4.5)

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Fig. 3(a). 1st resonant backbone curve for various non-dimensional axial forces of a simply supported beam.

Fig. 3(b).3rd resonant backbone curve for various non-dimensional axial forces of a simply supported beam.

Now expanding𝑎₃in a power series of the small parameter𝜇in the form

𝑎₃=𝜉₀+𝜉₁𝜇+𝜉₂𝜇² (4.6)

where,

𝜉₀= 4𝛼𝑏³₀+ 5𝛽𝑏⁵₀

(128𝜔²₁+ 96𝛼𝑏²₀+ 85𝛽𝑏⁴₀), 𝜉₁= 𝑏⁶

0(4𝛼+ 5𝛽𝑏²₀)² (96𝛼𝑏²₀+ 85𝛽𝑏⁴₀+ 128𝜔²₁)²,

𝜉₂= 2𝑏⁹₀(4𝛼+ 5𝛽𝑏²₀)³

(96𝛼𝑏²₀+ 85𝛽𝑏⁴₀+ 128𝜔²₁)³, 𝜇= 72𝛼𝑏₀+ 95𝛽𝑏³₀ (128𝜔²₁+ 96𝛼𝑏²₀+ 85𝛽𝑏⁴₀)

Now, inserting the value of𝑏₀, the value of𝑎₃ is obtained. Conse- quently after substituting the values of𝑏₀and𝑎₃in Eq.(3.5), the value

of𝑎is determined. Finally, putting𝑎and𝑎₃into Eq.(4.2), the value of 𝜔is obtained. The overall amplitude is𝑞(0) =𝑎+𝑎₃.

5. Results and discussion

The dimensions and material characteristics of the beam are pre- dicted as: beam dimensions=0.5 m × 0.2 m × 5 mm, Young’s modulus 𝐸= 71 × 10⁹N∕m², Mass density𝜌= 2700kg/m³, Poisson’s ratio=0.3, Radius of gyration𝑟=(_𝐼

𝐴

)¹₂

,𝐼is the inertial beam moment and𝐴is the cross-sectional area.

Figs. 1(a)–1(e)indicate system response over the time range. Back- bone curves for both 1st resonant frequency and 3rd resonant frequency of simply supported beams acquired from the numerical method and

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Fig. 4(a). 1st resonant backbone curve for various negative non-dimensional axial forces of a simply supported beam.

Fig. 4(b). 3rd resonant backbone curve for various negative non-dimensional axial forces of a simply supported beam.

proposed method are shown inFigs. 2(a)–2(d)for different values of the non-dimensional axial force𝐹. It is evident from these figures, the results attained from the modified harmonic balance method display good harmony with those numerical results.Figs. 3(a)and3(b)represent the 1st resonant and the 3rd resonant backbone curve of a simply supported beam for different axial forces respectively. From these figures, it is seen that when the amplitude and axial forces are zero, the frequency ratios are one, and, for positive (tension) and negative (compression) axial forces, the backbone curves are on the right side and left side of the zero axial force backbone curve, respectively. In other words,

the compressive axial force lowers the frequency of a beam while the tensile axial force raises it. Besides, the 3rd resonant backbone curves of nonzero non-dimensional axial forces are closer to that of zero axial force than the 1st resonant backbone curve. InFigs. 4(a)–4(b), the 1st resonant and the 3rd resonant backbone curves of simply supported beam for various negative axial forces are presented respectively.Fig. 5 compares the frequencies of the simply supported beam with and without fifth-order nonlinearity for various non-dimensional axial forces. It is shown that the frequency difference in each case is monotonically and exponentially increasing with overall amplitude. A comparison

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Fig. 5.Comparison between the frequencies of the simply supported beam with and without fifth-order nonlinearity for various non-dimensional axial forces.

between the analytical and numerical solutions is shown inTable 1and it shows good harmony between them.

6. Conclusion

The modified harmonic balance method has been established to obtain the free vibration response of axially loaded beams with third and fifth-order nonlinearity. The suitability of the present technique is that only a single nonlinear algebraic equation is needed to solve.

As a result, the computational effort is reduced and requires less effort than the existing harmonic balance method. The analytical solutions have been compared with the numerical solutions to justify the correctness of the method. The analytical solutions achieved by the modified harmonic balance method show an excellent harmony with the numerical solutions without any complexity. It is assumed that the present method is very significant and convenient for handling the free vibration problems of the axially loaded beam in the presence of third and fifth-order nonlinearity.

Declaration of competing interest

The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.

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