Comment on “Nonlinear Response of Strong Nonlinear System Arisen in Polymer Cushion”

1. Introduction

In [1], the following nonlinear oscillator was studied: 𝑋̈+tanh𝑋+𝜆₂tan𝜆₁𝑋+𝛽₅tan³𝜆₁𝑋(0)𝑋̇(0)𝑇𝑋=0,=0,=𝑉=⁰𝐿2𝑔ℎ=2𝛽₁𝑔ℎ𝛽₃. (1) Similar nonlinear oscillators arising in packing systems and textile engineering are available in [2–7].

In [1], (1) is approximated by Taylor’s series for the nonlinear terms: 𝑋̈+𝜔²₀₁𝑋+−13+𝜆₃𝜆³₁+13𝜆₂𝜆³₁𝑋³+2+215𝜆15²𝜆⁵₁+𝜆₃𝜆⁵₁𝑋⁵+11𝜆15³𝜆⁷₁𝑋⁷. (2) By using Taylor’s expansion function of MATLAB’s symbolic function, the nonlinear terms tanh(𝑋), tan(𝜆₁𝑋), and tan³(𝜆₁𝑋) are expanded to 8th order. After careful calculation, (2) should be corrected as 𝑋̈+𝜔²₀₁𝑋+−13+𝜆₃𝜆³₁+13𝜆₂𝜆³₁𝑋³+2+215𝜆15²𝜆⁵₁+𝜆₃𝜆⁵₁𝑋⁵+−17+31517𝜆315²𝜆⁷₁+11𝜆15³𝜆⁷₁𝑋⁷, (3) where 𝜔₀₁=1+𝜆₁𝜆₂.

2. Variational Iteration Method

In [1], the following variational iteration algorithm was constructed: 𝑋₁=𝑋₀+1𝜔₀₁^𝑡₀sin𝜔₀₁()×𝑋̈𝑠−𝑡₀+𝜔²₀₁𝑋₀+−13+𝜆₃𝜆³₁+13𝜆₂𝜆³₁𝑋³₀+2+215𝜆15²𝜆⁵₁+𝜆₃𝜆⁵₁𝑋⁵₀+11𝜆15³𝜆⁷₁𝑋⁷₀𝑑𝑠, (4) which should be corrected as 𝑋₁=𝑋₀+1𝜔₀₁^𝑡₀sin𝜔₀₁()×𝑋̈𝑠−𝑡₀+𝜔²₀₁𝑋₀+−13+𝜆₃𝜆³₁+13𝜆₂𝜆³₁𝑋³₀+2+215𝜆15²𝜆⁵₁+𝜆₃𝜆⁵₁𝑋⁵₀+−17+31517𝜆315²𝜆⁷₁+11𝜆15³𝜆⁷₁𝑋⁷₀𝑑𝑠. (5) In order to simplify the solution process, we denote 𝑎=𝜔²₀₁1𝑏=−3+𝜆₃𝜆³₁+13𝜆₂𝜆³₁2𝑐=+215𝜆15²𝜆⁵₁+𝜆₃𝜆⁵₁𝑑=−17+31517𝜆315²𝜆⁷₁+11𝜆15³𝜆⁷₁. (6) Equation (5) becomes 𝑋₁=𝑋₀+1𝜔₀₁^𝑡₀sin𝜔₀₁()×𝑋̈𝑠−𝑡₀(𝑠)+𝑎𝑋₀(𝑠)+𝑏𝑋³₀(𝑠)+𝑐𝑋⁵₀(𝑠)+𝑑𝑋⁷₀(𝑠)𝑑𝑠. (7) We begin with 𝑋₀=𝐴sin(Ω𝑡) (8) and use the following relationships: 𝑋³₀=𝐴³sin³()𝐴Ω𝑡=−³4sin(3Ω𝑡)+3𝐴³4()𝑋sinΩ𝑡₅₀=𝐴⁵sin⁵()=𝐴Ω𝑡⁵()−16sin5Ω𝑡5𝐴⁵()+16sin3Ω𝑡10𝐴⁵()𝑋16sinΩ𝑡₇₀=𝐴⁷sin⁷()𝐴Ω𝑡=−⁷()+64sin7Ω𝑡7𝐴⁷()−64sin5Ω𝑡21𝐴⁷()+64sin3Ω𝑡35𝐴⁷().64sinΩ𝑡 (9) By (7), we have 𝑋₁=𝑋₀+1𝜔₀₁^𝑡₀sin𝜔₀₁()×𝐴𝑠−𝑡₁sinΩ𝑠+𝐴₂sin(3Ω𝑠)+𝐴₃sin(5Ω𝑠)−𝐴₄sin(7Ω𝑠)𝑑𝑠, (10) where 𝐴₁=−𝐴Ω²+𝑎𝐴+3𝑏𝐴³4+10𝑐𝐴⁵+1635𝐴⁷𝑑𝐴64₂=−𝑏𝐴³4−5𝑐𝐴⁵−1621𝐴⁷𝑑𝐴64₃=𝑐𝐴⁵+167𝐴⁷𝑑𝐴64₄𝐴=−⁷𝑑.64 (11) Consider the following identity: sin𝜔^𝑡₀₀₁(𝑠−𝑡)sin𝑘Ω𝑠𝑑𝑠=𝑘Ωsin𝜔₀₁𝑡−𝜔₀₁sin𝑘Ω𝑡𝜔²₀₁−𝑘²Ω²,𝑘=1,3,5,7. (12) From (10), we obtain 𝑋₁1=𝐴sinΩ𝑡−𝜔₀₁Ω²−𝜔²₀₁𝐴₁Ωsin𝜔₀₁𝑡−𝜔₀₁−1sinΩ𝑡𝜔₀₁9Ω²−𝜔²₀₁𝐴₂3Ωsin𝜔₀₁𝑡−𝜔₀₁−1sin3Ω𝑡𝜔₀₁25Ω²−𝜔²₀₁𝐴₃5Ωsin𝜔₀₁𝑡−𝜔₀₁−1sin5Ω𝑡𝜔₀₁49Ω²−𝜔²₀₁𝐴₄7Ωsin𝜔₀₁𝑡−𝜔₀₁sin7Ω𝑡, (13) where 𝐴_𝑗 (𝑗 = 1~4) are defined above.

3. Discussion and Conclusion

The result of (7) in [1] should be corrected as that given in (13). In [1], actually the following variational iteration algorithm is used: 𝑋_𝑛+1=𝑋_𝑛+1𝜔₀₁^𝑡₀sin𝜔₀₁()×𝑋̈𝑠−𝑡_𝑛(𝑠)+𝑎𝑋_𝑛(𝑠)+𝑏𝑋³_𝑛(𝑠)+𝑐𝑋⁵_𝑛(𝑠)+𝑑𝑋⁷_𝑛(𝑠)𝑑𝑠, (14) which is called the variational iteration algorithm I in [8, 9]; for the present problem, the variational iteration algorithm II in [10] is much more effective, which reads 𝑋_𝑛+1=𝑋₀+1𝜔₀₁^𝑡₀sin𝜔₀₁()×𝑠−𝑡𝑏𝑋³_𝑛(𝑠)+𝑐𝑋⁵_𝑛(𝑠)+𝑑𝑋⁷_𝑛(𝑠)𝑑𝑠. (15) The variational iteration method is especially effective in solving nonlinear oscillators.