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Complexification-averaging method via the averaged Lagrangian

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Abstract

In this paper, the complexification-averaging (CX-A) method for multi-DOF nonlinear vibratory systems is rederived in a new way based upon the averaged Lagrangian. The complex variables are introduced to represent the original displacements and velocities, and then the fast–slow decomposition of the complex variables is made. The time averaging of the Lagrangian over the fast variables is performed. Two different expressions for the kinetic energy are presented, and this results in two schemes for deriving the governing equations of the slow variables. For the scheme I, through the order analysis of the derivatives of the slow variables, it is shown that the second-order terms appeared in the averaged Lagrangian can be omitted, and thus a reduced averaged Lagrangian is obtained. Via the reduced averaged Lagrangian, the corresponding Lagrangian equations are derived. For the scheme II, through time averaging, the averaged Lagrangian is obtained, and then the corresponding equations for the slow variables can be obtained. Finally, two nonlinear vibratory systems with two-DOF and four-DOF, respectively, are given as examples to illustrate the new procedure for the CX-A method. The loci of nonlinear normal modes on the potential surface are studied in the first example, and the frequency-energy plot is investigated in the second example.

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References

  1. Nayfeh, A.H., Mook, T.D.: Nonlinear Oscillations. Wiley, New York (1979)

    MATH  Google Scholar 

  2. Mickens, R.E.: Truly Nonlinear Oscillations. World Scientific, Singapore (2010)

    Book  MATH  Google Scholar 

  3. Manevitch, L.I.: The description of localized normal modes in a chain of nonlinear coupled oscillators using complex variables. Nonlinear Dyn. 25, 95–109 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Manevitch, A.I., Manevitch, L.I.: The Mechanics of Nonlinear Systems with Internal Resonances. Imperial College Press, London (2005)

    Book  Google Scholar 

  5. Manevitch, L.I., Musienko, A.I., Lamarque, C.-H.: New analytical approach to energy pumping problem in strongly nonhomogeneous 2dof systems. Meccanica 42, 77–83 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S.: Nonlinear Targeted Energy Transfer Mechanical and Structural Systems. Springer, Berlin (2008)

    Google Scholar 

  7. Kerschen, G., Vakakis, A.F., Lee, Y.S., Mcfarland, D.M., Kowtko, J.J., Bergman, L.A.: Energy transfers in a system of two coupled oscillators with essential nonlinearity: 1:1 resonance manifold and transient bridging orbits. Nonlinear Dyn. 42, 283–303 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)

    MATH  Google Scholar 

  9. Natarajan, R., Brown, R.A.: Third-order resonance effects and the nonlinear stability of drop oscillations. J. Fluid Mech. 183, 95–121 (1987)

    Article  MATH  Google Scholar 

  10. Whitham, G.B.: A general approach to linear and non-linear dispersive waves using a Lagrangian. J. Fluid Mech. 22, 273–283 (1965)

    Article  MathSciNet  Google Scholar 

  11. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)

    MATH  Google Scholar 

  12. Lee, Y.S., Vakakis, A.F., Bergman, L.A., McFarland, D.M., Kerschen, G., Nucera, F., Tsakirtzis, S., Panagopoulos, P.N.: Passive non-linear targeted energy transfer and its applications to vibration absorption: A review. J. Multi-Body Dyn. 222, 77–134 (2008)

    Google Scholar 

  13. Fermi, E., Pasta, J., Ulam, S.: Studies of nonlinear problems. In: E. Fermi, Collected Papers, pp. 977–988. University of Chicago Press, Chicago (1965)

    Google Scholar 

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Authors and Affiliations

  1. Center of Mechanics/State Key Lab for Mechanical Structural Strength and Vibration, School of Aerospace, Xi’an Jiaotong University, Xi’an, 710049, P.R. China

    Xinhua Zhang

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  1. Xinhua Zhang
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Correspondence to Xinhua Zhang.

Appendix: The equations of the harmonic coefficients for the 4-DOF FPU model

Appendix: The equations of the harmonic coefficients for the 4-DOF FPU model

$$\begin{aligned} H_1 =&18 {A_{{1}}}^{3}-9 {A_{{2}}}^{3}+4 A_{{2}}A_{{5}}A_{{6}}-4 A_{{6}} A_{{5}}A_{{1}} \\ &{}+4 {A_{{5}}}^{2}A_{{1}}+2 A_{{1}}{A_{{6}}}^{2}+27 {A_ {{2}}}^{2}A_{{1}}+6 A_{{2}}A_{{5}}A_{{1}}\\ &{}-6 A_{{1}}A_{{2}}A_{{6}}-2 {A_{{5}}}^{2}A_{{2}}-24 A_{{2}}{ \omega}^{2} \\ &{}-24 A_{{1}}{\omega}^{4} +48 A_{{1}}{\omega}^{2}-2 A_{{2}}{A_{{6}}}^{2}-27 A_{{2}}{A_{{1}}}^ {2} \\ &{}-6 A_{{5}}{A_{{1}}}^{2}+3 {A_{{1}}}^{2}A_{{6}}+3 A_{{6}}{A_{{2}} }^{2}-3 {A_{{2}}}^{2}A_{{5}}, \\ H_2 =&3 {A_{{2}}}^{2}A_{{5}}-3 {A_{{1}}}^{2}A_{{6}}-6 A_{{6}}{A_{{2}}}^{2 }+27 A_{{2}}{A_{{1}}}^{2} \\ &{}+3 A_{{5}}{A_{{1}}}^{2}-24 A_{{1}}{\omega} ^{2}+4 A_{{2}}{A_{{6}}}^{2}+2 {A_{{5}}}^{2}A_{{2}} \\ &{}+48 A_{{2}}{ \omega}^{2}-27 {A_{{2}}}^{2}A_{{1}}-2 {A_{{5}}}^{2}A_{{1}}-2 A_{{1} }{A_{{6}}}^{2} \\ &{}-27 A_{{3}}{A_{{2}}}^{2}-2 A_{{3}}{A_{{7}}}^{2}-2 {A_{{6}}}^{2}A_{{3}}+2 {A_{{7}}}^{2}A_{{2}}\\ &{}+27 {A_{{3}}}^{2}A_{{2}}+3 {A_{{3}}}^{2}A_{{7}}-3 A_{{6}}{A_{{3}}}^{2}+3 {A_{{2}}}^{2}A_{{7}} \\ &{}-24 A_{{3}}{\omega}^{2}-24 A_{{2}}{ \omega}^{4}-9 {A_{{1}}}^{3}+18 { A_{{2}}}^{3}-9 {A_{{3}}}^{3} \\ &{}+6 A_{{1}}A_{{2}}A_{{6}}-6 A_{{2}}A_{{5 }}A_{{1}}-4 A_{{2}}A_{{5}}A_{{6}} \\ &{}+4 A_{{6}}A_{{5}}A_{{1}}-4 A_{{6}}A_{{2}}A_{{7}}-6 A_{{3}}A_{{7}}A_{{2}}\\ &{}+6 A_{{6}}A_{{2}}A_{{3}}+4 A_ {{6}}A_{{3}}A_{{7}}, \\ H_3 =&3 A_{{6}}{A_{{2}}}^{2}-2 A_{{2}}{A_{{6}}}^{2}-24 A_{{2}}{ \omega}^{2 }+27 A_{{3}}{A_{{2}}}^{2} \\ &{}+4 A_{{3}}{A_{{7}}}^{2}+2 {A_{{6}}}^{2}A_{ {3}}-2 {A_{{7}}}^{2}A_{{2}}-27 {A_{{3}}}^{2}A_{{2}} \\ &{}-6 {A_{{3}}}^{2} A_{{7}}+3 A_{{6}}{A_{{3}}}^{2}-3 {A_{{2}}}^{2}A_{{7}}+48 A_{{3}}{ \omega}^{2}\\ &{}-9 {A_{{2}}}^{3}+18 {A_{{3}}}^{3}+4 A_{{6}}A_{{2}}A_{{7} }+6 A_{{3}}A_{{7}}A_{{2}} \\ &{}-6 A_{{6}}A_{{2}}A_{{3}}-4 A_{{6}}A_{{3}}A _{{7}}-6 A_{{8}}A_{{4}}A_{{3}} \\ &{}+6 A_{{7}}A_{{4}}A_{{3}}+4 A_{{4}}A_{ {8}}A_{{7}}-9 {A_{{4}}}^{3}+27 A_{{3}}{A_{{4}}}^{2} \\ &{}+2 A_{{3}}{A_{{8 }}}^{2}-2 {A_{{8}}}^{2}A_{{4}}-27 A_{{4}}{A_{{3}}}^{2}-2 {A_{{7}}}^ {2}A_{{4}} \\ &{}+3 {A_{{3}}}^{2}A_{{8}}+3 {A_{{4}}}^{2}A_{{8}}-3 {A_{{4}} }^{2}A_{{7}}-24 A_{{3}}{ \omega}^{4}\\ &{}-24 A_{{4}}{\omega}^{2}-4 A_{{7} }A_{{3}}A_{{8}}, \\ H_4 =&3 {A_{{3}}}^{2}A_{{7}}-24 A_{{3}}{\omega}^{2}+48 A_{{4}}{ \omega}^{2 }-3 {A_{{3}}}^{2}A_{{8}} \\ &{}-6 {A_{{4}}}^{2}A_{{8}}+3 {A_{{4}}}^{2}A_{{ 7}}-24 A_{{4}}{ \omega}^{4}+4 A_{{7}}A_{{3}}A_{{8}} \\ &{}-2 A_{{3}}{A_{{7} }}^{2}-4 A_{{4}}A_{{8}}A_{{7}}+2 {A_{{7}}}^{2}A_{{4}}+4 {A_{{8}}}^{ 2}A_{{4}}\\ &{}+27 A_{{4}}{A_{{3}}}^{2}-9 {A_{{3}}}^{3}-2 A_{{3}}{A_{{8}} }^{2}-6 A_{{7}}A_{{4}}A_{{3}} \\ &{}-27 A_{{3}}{A_{{4}}}^{2}+6 A_{{8}}A_{{ 4}}A_{{3}}+18 {A_{{4}}}^{3}, \\ H_5 =&-216 A_{{5}}{ \omega}^{4}+48 A_{{5}}{\omega}^{2}-24 A_{{6}}{\omega}^ {2}-18 {A_{{1}}}^{3} \\ &{}+9 {A_{{2}}}^{3}-{A_{{6}}}^{3}+2 {A_{{5}}}^{3}- 3 A_{{6}}{A_{{5}}}^{2}-27 {A_{{2}}}^{2}A_{{1}} \\ &{} -36 A_{{2}}A_{{5}}A_{{1}}+36 A_{{1}}A_{{2}}A_{{6}}+27 A_{{2}}{A_{{1}}}^{2}\\ &{}+36 A_{{5}}{A_ {{1}}}^{2}-18 {A_{{1}}}^{2}A_{{6}}+3 A_{{5}}{A_{{6}}}^{2} \\ &{}-18 A_{{6} }{A_{{2}}}^{2}+18 {A_{{2}}}^{2}A_{{5}}, \\ H_6 =&-24 A_{{7}}{\omega}^{2}-216 A_{{6}}{\omega}^{4}+3 A_{{6}}{A_{{7}}}^ {2}-3 {A_{{6}}}^{2}A_{{7}} \\ &{}-{A_{{7}}}^{3}+36 A_{{6}}{A_{{2}}}^{2}+27 A_{{3}}{A_{{2}}}^{2}-27 {A_{{3}}}^{2}A_{{2}} \\ &{} -18 {A_{{3}}}^{2}A_{{7}}+18 A_{{6}}{A_{{3}}}^{2}-18 {A_{{2}}}^{2}A_{{7}}-18 {A_{{2}}}^{3} \\ &{}+9 {A_{{3}}}^{3}+36 A_{{3}}A_{{7}}A_{{2}}-36 A_{{6}}A_{{2}}A_{{3}}- 24 A_{{5}}{ \omega}^{2}\\ &{}+48 A_{{6}}{\omega}^{2}+3 A_{{6}}{A_{{5}}}^{2 }+27 {A_{{2}}}^{2}A_{{1}}-27 A_{{2}}{A_{{1}}}^{2} \\ &{}-18 A_{{5}}{A_{{1} }}^{2}+18 {A_{{1}}}^{2}A_{{6}}-3 A_{{5}}{A_{{6}}}^{2}-18 {A_{{2}}}^ {2}A_{{5}} \\ &{}+9 {A_{{1}}}^{3}-{A_{{5}}}^{3}+2 {A_{{6}}}^{3}+36 A_{{2}}A_{{5}}A_{{1}} \\ &{}-36 A_{{1}}A_{{2}}A_{{6}}, \\ H_7 =&48 A_{{7}}{ \omega}^{2}-3 A_{{6}}{A_{{7}}}^{2}+3 {A_{{6}}}^{2}A_{{7} }+2 {A_{{7}}}^{3} \\ &{}-18 A_{{6}}{A_{{2}}}^{2}-27 A_{{3}}{A_{{2}}}^{2}+ 27 {A_{{3}}}^{2}A_{{2}} \\ &{}+36 {A_{{3}}}^{2}A_{{7}}-18 A_{{6}}{A_{{3}}} ^{2}+18 {A_{{2}}}^{2}A_{{7}}\\ &{}+9 {A_{{2}}}^{3}-18 {A_{{3}}}^{3}-36 A _{{3}}A_{{7}}A_{{2}}+36 A_{{6}}A_{{2}}A_{{3}} \\ &{}-24 A_{{6}}{\omega}^{2} -{A_{{6}}}^{3}+36 A_{{8}}A_{{4}}A_{{3}}-36 A_{{7}}A_{{4}}A_{{3}} \\ &{}+9 {A_{{4}}}^{3}-27 A_{{3}}{A_{{4}}}^{2}+27 A_{{4}}{A_{{3}}}^{2}-18 {A _{{3}}}^{2}A_{{8}} \\ &{} -18 {A_{{4}}}^{2}A_{{8}}-{A_{{8}}}^{3}+18 {A_{{4}}}^{2}A_{{7}}-24 A_{{8}}{ \omega}^{2}\\ &{}-216 A_{{7}}{\omega }^{4}+3 {A_{{ 8}}}^{2}A_{{7}}-3 {A_{{7}}}^{2}A_{{8}}, \\ H_8 =&-18 {A_{{3}}}^{2}A_{{7}}+18 {A_{{3}}}^{2}A_{{8}}+36 {A_{{4}}}^{2}A_ {{8}} \\ &{}-18 {A_{{4}}}^{2}A_{{7}}-24 A_{{7}}{\omega}^{2}-3 {A_{{8}}}^{2 }A_{{7}}+3 {A_{{7}}}^{2}A_{{8}} \\ &{}-216 A_{{8}}{ \omega}^{4}+48 A_{{8}}{ \omega}^{2}-27 A_{{4}}{A_{{3}}}^{2}+9 {A_{{3}}}^{3}\\ &{}+2 {A_{{8}}}^{3} -{A_{{7}}}^{3}+36 A_{{7}}A_{{4}}A_{{3}}+27 A_{{3}}{A_{{4}}}^{2} \\ &{}-36 A_{{8}}A_{{4}}A_{{3}}-18 {A_{{4}}}^{3}. \end{aligned}$$

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Zhang, X. Complexification-averaging method via the averaged Lagrangian. Nonlinear Dyn 75, 429–437 (2014). https://doi.org/10.1007/s11071-013-1075-5

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