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Flexural vibrations of nonlinearly elastic circular rings

  • Advances in Dynamics, Stability and Control of Mechanical Systems
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Abstract

This paper treats free nonlinear flexural vibrations of circular elastic rings in the context of a geometrically exact formulation accounting for the effects of nonlinear material behavior. A direct asymptotic approach based on the method of multiple scales is used to investigate such vibrations. It is shown that the flexural motions are softening for linearly elastic rings, in agreement with previous results in the literature, while there are nonlinearly elastic rings for which the motions are hardening. There are thresholds in the nonlinear constitutive laws separating softening from hardening behaviors.

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Acknowledgments

The financial support of Lacarbonara and Arena through a 2010-2011 MIUR (Italian Ministry of Education, University and Scientific Research) PRIN Grant is gratefully acknowledged. The work of Antman was partially supported by a grant from the National Science Foundation.

Author information

Authors and Affiliations

  1. Department of Structural and Geotechnical Engineering, Sapienza University of Rome, 00184 , Rome, Italy

    Walter Lacarbonara & Andrea Arena

  2. Department of Mathematics, Institute for Physical Science and Technology, and Institute for Systems Research, University of Maryland, College Park, MD , 20742-4015, USA

    Stuart S. Antman

Authors
  1. Walter Lacarbonara
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  2. Andrea Arena
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  3. Stuart S. Antman
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Corresponding author

Correspondence to Walter Lacarbonara.

Appendix

Appendix

The expressions of the integrand functions appearing in the effective nonlinearity coefficient (42) are given next.

$${{\varUpsilon }_{0m}^{\varphi }}^{(2)}= \gamma _{01m}' \left( 3 \psi _m''+\psi _m''''+2 \varphi _m'\right) +2 \gamma _{02m}' \left( 3 \psi _m''+\psi _m''''+2 \varphi _m'\right) +3 \gamma _{01m}'' \psi _m'+6 \gamma _{02m}'' \psi _m'+\gamma _{01m}'''' \psi _m'+2 \gamma _{02m}'''' \psi _m'+\psi _m''' \gamma _{01m}''+2 \psi _m''' \gamma _{02m}''+\gamma _{01m}''' \psi _m''+2 \gamma _{02m}''' \psi _m''+2 \gamma _{01m} \psi _m'''+4 \gamma _{02m} \psi _m'''+2 \gamma _{01m}''' \psi _m+4 \gamma _{02m}''' \psi _m+2 \phi _{01m}' \psi _m'-\psi _m''' \phi _{01m}'+4 \phi _{02m}' \psi _m' -2 \psi _m''' \phi _{02m}'+\phi _{01m} \psi _m''+2 \phi _{02m} \psi _m''+2 \psi _m \phi _{01m}''-\phi _{01m}'' \psi _m'' +4 \psi _m \phi _{02m}''-2 \phi _{02m}'' \psi _m''+\phi _{01m} \psi _m''''+2 \phi _{02m} \psi _m''''-\gamma _{01m}''' \varphi _m' -2 \gamma _{02m}''' \varphi _m'+\varphi _m \gamma _{01m}''+2 \varphi _m \gamma _{02m}''+2 \gamma _{01m} \varphi _m''+4 \gamma _{02m} \varphi _m''-\gamma _{01m}'' \varphi _m''-2 \gamma _{02m}'' \varphi _m''+\gamma _{01m}'''' \varphi _m+2 \gamma _{02m}'''' \varphi _m -3 \phi _{01m}' \varphi _m''-6 \phi _{02m}' \varphi _m''-3 \phi _{01m}'' \varphi _m'-6 \phi _{02m}'' \varphi _m' $$
(48)
$${{\varUpsilon }_{0m}^{\psi }}^{(2)}=-4 \psi _m''' \gamma _{01m}'-8 \psi _m''' \gamma _{02m}'-4 \gamma _{01m}''' \psi _m'-8 \gamma _{02m}''' \psi _m'-6 \gamma _{01m}'' \psi _m''-12 \gamma _{02m}'' \psi _m''-2 \gamma _{01m} \psi _m''''-4 \gamma _{02m} \psi _m''''-2 \gamma _{01m}'''' \psi _m-4 \gamma _{02m}'''' \psi _m;+\phi _{01m}' \left( -3 \psi _m''+2 \psi _m''''+5 \varphi _m'''\right) +\phi _{02m}' \left( -6 \psi _m''+4 \psi _m''''+10 \varphi _m'''\right) -3 \phi _{01m}'' \psi _m'-6 \phi _{02m}'' \psi _m'+\phi _{01m}'''' \psi _m'+2 \phi _{02m}'''' \psi _m'+4 \psi _m''' \phi _{01m}'';+8 \psi _m''' \phi _{02m}''+3 \phi _{01m}''' \psi _m''+6 \phi _{02m}''' \psi _m''-2 \phi _{01m}''' \psi _m-4 \phi _{02m}''' \psi _m;-3 \gamma _{01m}'' \varphi _m'-6 \gamma _{02m}'' \varphi _m'-3 \gamma _{01m}' \varphi _m''-6 \gamma _{02m}' \varphi _m'';+2 \gamma _{01m}'''' \varphi _m'+4 \gamma _{02m}'''' \varphi _m'+\varphi _m'''' \gamma _{01m}'+2 \varphi _m'''' \gamma _{02m}'+3 \varphi _m''' \gamma _{01m}''+4 \gamma _{01m}''' \varphi _m''+8 \gamma _{02m}''' \varphi _m''+6 \varphi _m''' \gamma _{02m}'';-2 \gamma _{01m} \varphi _m'''-4 \gamma _{02m} \varphi _m'''+5 \phi _{01m}''' \varphi _m'+10 \phi _{02m}''' \varphi _m'+\phi _{01m} \varphi _m'';+2 \phi _{02m} \varphi _m''+\varphi _m \phi _{01m}''+8 \phi _{01m}'' \varphi _m''+2 \varphi _m \phi _{02m}''+16 \phi _{02m}'' \varphi _m'';+\phi _{01m} \varphi _m''''+2 \phi _{02m} \varphi _m''''+\phi _{01m}'''' \varphi _m+2 \phi _{02m}'''' \varphi _m $$
(49)
$${{\varUpsilon }_{1m}^{\varphi }}^{(2)}=\gamma _{01m} \left( \psi _m'+\varphi _m\right) +2 \gamma _{02m} \left( \psi _m'+\varphi _m\right) +\gamma _{01m}' \psi _m''+2 \gamma _{02m}' \psi _m''+\gamma _{01m}'' \psi _m'+2 \gamma _{02m}'' \psi _m'+\psi _m \gamma _{01m}'+2 \psi _m \gamma _{02m}'+3 \gamma _{11m}' \psi _m''+6 \gamma _{12m}' \psi _m''+3 \gamma _{11m}'' \psi _m'+6 \gamma _{12m}'' \psi _m'+\psi _m'''' \gamma _{11m}'+2 \psi _m'''' \gamma _{12m}'+\gamma _{11m}'''' \psi _m'+2 \gamma _{12m}'''' \psi _m'+\psi _m''' \gamma _{11m}''+2 \psi _m''' \gamma _{12m}''+\gamma _{11m}''' \psi _m''+2 \gamma _{12m}''' \psi _m''+2 \gamma _{11m} \psi _m'''+4 \gamma _{12m} \psi _m'''+2 \gamma _{11m}''' \psi _m+4 \gamma _{12m}''' \psi _m+\phi _{01m} \psi _m''+2 \phi _{02m} \psi _m''+\phi _{01m} \psi _m+2 \phi _{02m} \psi _m+2 \phi _{11m}' \psi _m'+4 \phi _{12m}' \psi _m'-\psi _m''' \phi _{11m}'-2 \psi _m''' \phi _{12m}'+\phi _{11m} \psi _m''+2 \phi _{12m} \psi _m''+2 \psi _m \phi _{11m}''-\phi _{11m}'' \psi _m''+4 \psi _m \phi _{12m}''-2 \phi _{12m}'' \psi _m''+\phi _{11m} \psi _m''''+2 \phi _{12m} \psi _m''''+\varphi _m \gamma _{01m}''+2 \varphi _m \gamma _{02m}''+2 \gamma _{11m}' \varphi _m'+4 \gamma _{12m}' \varphi _m'-\gamma _{11m}''' \varphi _m'-2 \gamma _{12m}''' \varphi _m'+\varphi _m \gamma _{11m}''+2 \varphi _m \gamma _{12m}''+2 \gamma _{11m} \varphi _m''+4 \gamma _{12m} \varphi _m''-\gamma _{11m}'' \varphi _m''-2 \gamma _{12m}'' \varphi _m''+\gamma _{11m}'''' \varphi _m+2 \gamma _{12m}'''' \varphi _m-3 \phi _{11m}' \varphi _m''-6 \phi _{12m}' \varphi _m''-3 \phi _{11m}'' \varphi _m'-6 \phi _{12m}'' \varphi _m' $$
(50)
$${{\varUpsilon }_{1m}^{\psi }}^{(2)}=-\gamma _{01m} \varphi _m'-2 \gamma _{02m} \varphi _m'+2 \phi _{01m}' \varphi _m'+4 \phi _{02m}' \varphi _m'+\gamma _{01m}'' \varphi _m'+2 \gamma _{02m}'' \varphi _m'-3 \gamma _{11m}'' \varphi _m'-6 \gamma _{12m}'' \varphi _m'+5 \phi _{11m}''' \varphi _m'+10 \phi _{12m}''' \varphi _m'+2 \gamma _{11m}'''' \varphi _m'+4 \gamma _{12m}'''' \varphi _m'-\psi _m \phi _{01m}'-2 \psi _m \phi _{02m}'-\gamma _{01m} \psi _m''-2 \gamma _{02m} \psi _m''+\phi _{01m}' \psi _m''+2 \phi _{02m}' \psi _m''-3 \phi _{11m}' \psi _m''-6 \phi _{12m}' \psi _m''-\psi _m \gamma _{01m}''-2 \psi _m \gamma _{02m}''-6 \psi _m'' \gamma _{11m}''-12 \psi _m'' \gamma _{12m}''+\phi _{01m} \varphi _m''+2 \phi _{02m} \varphi _m''+\phi _{11m} \varphi _m''+2 \phi _{12m} \varphi _m''+\gamma _{01m}' \varphi _m''+2 \gamma _{02m}' \varphi _m''-3 \gamma _{11m}' \varphi _m''-6 \gamma _{12m}' \varphi _m''+8 \varphi _m'' \phi _{11m}''+16 \varphi _m'' \phi _{12m}''-4 \gamma _{11m}' \psi _m'''-8 \gamma _{12m}' \psi _m'''+4 \phi _{11m}'' \psi _m'''+8 \phi _{12m}'' \psi _m'''+4 \varphi _m'' \gamma _{11m}'''+8 \varphi _m'' \gamma _{12m}'''-2 \gamma _{11m} \varphi _m'''-4 \gamma _{12m} \varphi _m'''+5 \phi _{11m}' \varphi _m'''+10 \phi _{12m}' \varphi _m'''+3 \gamma _{11m}'' \varphi _m'''+6 \gamma _{12m}'' \varphi _m'''-2 \psi _m \phi _{11m}'''+3 \psi _m'' \phi _{11m}'''-4 \psi _m \phi _{12m}'''+6 \psi _m'' \phi _{12m}'''-2 \gamma _{11m} \psi _m''''-4 \gamma _{12m} \psi _m''''+2 \phi _{11m}' \psi _m''''+4 \phi _{12m}' \psi _m''''-2 \psi _m \gamma _{11m}''''-4 \psi _m \gamma _{12m}''''+\phi _{11m} \varphi _m''''+2 \phi _{12m} \varphi _m''''+\gamma _{11m}' \varphi _m''''+2 \gamma _{12m}' \varphi _m''''+\varphi _m \left( \phi _{01m}+2 \phi _{02m}+\phi _{01m}''+2 \phi _{02m}''\right. \left. +\phi _{11m}''+2 \phi _{12m}''+\phi _{11m}''''+2 \phi _{12m}''''\right) +\psi _m' \left( -\gamma _{01m}'-2 \gamma _{02m}'+\phi _{01m}''\right. \left. +2 \phi _{02m}''-3 \phi _{11m}''-6 \phi _{12m}''-4 \gamma _{11m}'''-8 \gamma _{12m}'''+\phi _{11m}''''+2 \phi _{12m}''''\right) $$
(51)
$${{\varUpsilon }_{2m}^{\varphi }}^{(2)}=\gamma _{11m} \left( \psi _m'+\varphi _m\right) +2 \gamma _{12m} \left( \psi _m'+\varphi _m\right) +\gamma _{11m}' \psi _m''+2 \gamma _{12m}' \psi _m''+\gamma _{11m}'' \psi _m'+2 \gamma _{12m}'' \psi _m'+\psi _m \gamma _{11m}'+2 \psi _m \gamma _{12m}'+\phi _{11m} \psi _m''+2 \phi _{12m} \psi _m''+\phi _{11m} \psi _m+2 \phi _{12m} \psi _m+\varphi _m \gamma _{11m}''+2 \varphi _m \gamma _{12m}'' $$
(52)
$${{\varUpsilon }_{2m}^{\psi }}^{(2)}=\psi _m' \left( -\gamma _{11m}'-2 \gamma _{12m}'+\phi _{11m}''+2 \phi _{12m}''\right) -\gamma _{11m} \psi _m''-2 \gamma _{12m} \psi _m''-\psi _m \gamma _{11m}''-2 \psi _m \gamma _{12m}''+\phi _{11m}' \psi _m''+2 \phi _{12m}' \psi _m''-\psi _m \phi _{11m}'-2 \psi _m \phi _{12m}'+\gamma _{11m}'' \varphi _m'+2 \gamma _{12m}'' \varphi _m'+\gamma _{11m}' \varphi _m''+2 \gamma _{12m}' \varphi _m''-\gamma _{11m} \varphi _m'-2 \gamma _{12m} \varphi _m'+2 \phi _{11m}' \varphi _m'+4 \phi _{12m}' \varphi _m'+\phi _{11m} \varphi _m''+2 \phi _{12m} \varphi _m''+\varphi _m \left( \phi _{11m}''+2 \phi _{12m}''+\phi _{11m}+2 \phi _{12m}\right) $$
(53)
$${{\varUpsilon }_{3m}^{\varphi }}^{(2)}=2 \psi _m''' \gamma _{11m}''+4 \psi _m''' \gamma _{12m}''+2 \gamma _{11m}''' \psi _m''+4 \gamma _{12m}''' \psi _m''+\gamma _{21m}' \psi _m''+2 \gamma _{22m}' \psi _m''+\gamma _{21m}'' \psi _m'+2 \gamma _{22m}'' \psi _m'+\gamma _{21m} \psi _m'+2 \gamma _{22m} \psi _m'+\psi _m \gamma _{21m}'+2 \psi _m \gamma _{22m}'+2 \psi _m''' \phi _{11m}'+4 \psi _m''' \phi _{12m}'+2 \phi _{11m}'' \psi _m''+4 \phi _{12m}'' \psi _m''+\phi _{21m} \psi _m''+2 \phi _{22m} \psi _m''+\phi _{21m} \psi _m+2 \phi _{22m} \psi _m+2 \gamma _{11m}''' \varphi _m'+4 \gamma _{12m}''' \varphi _m'+2 \gamma _{11m}'' \varphi _m''+4 \gamma _{12m}'' \varphi _m''+\varphi _m \gamma _{21m}''+2 \varphi _m \gamma _{22m}''+\gamma _{21m} \varphi _m+2 \gamma _{22m} \varphi _m+2 \phi _{11m}' \varphi _m''+4 \phi _{12m}' \varphi _m''+2 \phi _{11m}'' \varphi _m'+4 \phi _{12m}'' \varphi _m' $$
(54)
$${{\varUpsilon }_{3m}^{\psi }}^{(2)}=-2 \psi _m'''' \gamma _{11m}''-4 \psi _m'''' \gamma _{12m}''-2 \gamma _{11m}'''' \psi _m''-4 \gamma _{12m}'''' \psi _m''-4 \gamma _{11m}''' \psi _m'''-8 \gamma _{12m}''' \psi _m'''-\gamma _{21m}' \psi _m'-2 \gamma _{22m}' \psi _m'-\gamma _{21m} \psi _m''-2 \gamma _{22m} \psi _m''-\psi _m \gamma _{21m}''-2 \psi _m \gamma _{22m}''-2 \psi _m'''' \phi _{11m}'-4 \psi _m'''' \phi _{12m}'-4 \psi _m''' \phi _{11m}''-8 \psi _m''' \phi _{12m}''-2 \phi _{11m}''' \psi _m''-4 \phi _{12m}''' \psi _m''+\phi _{21m}' \psi _m''+2 \phi _{22m}' \psi _m''+\phi _{21m}'' \psi _m'+2 \phi _{22m}'' \psi _m'-\psi _m \phi _{21m}'-2 \psi _m \phi _{22m}'-2 \gamma _{11m}'''' \varphi _m'-4 \gamma _{12m}'''' \varphi _m'-4 \gamma _{11m}''' \varphi _m''-8 \gamma _{12m}''' \varphi _m''-2 \varphi _m''' \gamma _{11m}''-4 \varphi _m''' \gamma _{12m}''+\gamma _{21m}'' \varphi _m'+2 \gamma _{22m}'' \varphi _m'+\gamma _{21m}' \varphi _m''+2 \gamma _{22m}' \varphi _m''-\gamma _{21m} \varphi _m'-2 \gamma _{22m} \varphi _m'-2 \varphi _m''' \phi _{11m}'-4 \varphi _m''' \phi _{12m}'-2 \phi _{11m}''' \varphi _m'-4 \phi _{12m}''' \varphi _m'-4 \phi _{11m}'' \varphi _m''-8 \phi _{12m}'' \varphi _m''+2 \phi _{21m}' \varphi _m'+4 \phi _{22m}' \varphi _m'+\phi _{21m} \varphi _m''+2 \phi _{22m} \varphi _m''+\varphi _m \phi _{21m}''+2 \varphi _m \phi _{22m}''+\phi _{21m} \varphi _m+2 \phi _{22m} \varphi _m $$
(55)
$${{\varUpsilon }_{4m}^{\varphi }}^{(2)}=2 \gamma _{21m}' \varphi _m'+4 \gamma _{22m}' \varphi _m'+2 \phi _{01m}'' \varphi _m'+4 \phi _{02m}'' \varphi _m'-3 \phi _{21m}'' \varphi _m'-6 \phi _{22m}'' \varphi _m'+2 \gamma _{01m}''' \varphi _m'+4 \gamma _{02m}''' \varphi _m'-\gamma _{21m}''' \varphi _m'-2 \gamma _{22m}''' \varphi _m'+2 \psi _m' \phi _{21m}'+4 \psi _m' \phi _{22m}'+\phi _{21m} \psi _m''+2 \phi _{22m} \psi _m''+3 \gamma _{21m}' \psi _m''+6 \gamma _{22m}' \psi _m''+\varphi _m \gamma _{21m}''+3 \psi _m' \gamma _{21m}''+2 \varphi _m \gamma _{22m}''+6 \psi _m' \gamma _{22m}''+2 \gamma _{21m} \varphi _m''+4 \gamma _{22m} \varphi _m''+2 \phi _{01m}' \varphi _m''+4 \phi _{02m}' \varphi _m''-3 \phi _{21m}' \varphi _m''-6 \phi _{22m}' \varphi _m''+2 \gamma _{01m}'' \varphi _m''+4 \gamma _{02m}'' \varphi _m''-\gamma _{21m}'' \varphi _m''-2 \gamma _{22m}'' \varphi _m''+2 \psi _m'' \phi _{01m}''+4 \psi _m'' \phi _{02m}''+2 \psi _m \phi _{21m}''-\psi _m'' \phi _{21m}''+4 \psi _m \phi _{22m}''-2 \psi _m'' \phi _{22m}''+2 \gamma _{21m} \psi _m'''+4 \gamma _{22m} \psi _m'''+2 \phi _{01m}' \psi _m'''+4 \phi _{02m}' \psi _m'''-\phi _{21m}' \psi _m'''-2 \phi _{22m}' \psi _m'''+2 \gamma _{01m}'' \psi _m'''+4 \gamma _{02m}'' \psi _m'''+\gamma _{21m}'' \psi _m'''+2 \gamma _{22m}'' \psi _m'''+2 \psi _m'' \gamma _{01m}'''+4 \psi _m'' \gamma _{02m}'''+2 \psi _m \gamma _{21m}'''+\psi _m'' \gamma _{21m}'''+4 \psi _m \gamma _{22m}'''+2 \psi _m'' \gamma _{22m}'''+\phi _{21m} \psi _m''''+2 \phi _{22m} \psi _m''''+\gamma _{21m}' \psi _m''''+2 \gamma _{22m}' \psi _m''''+\varphi _m \gamma _{21m}''''+\psi _m' \gamma _{21m}''''+2 \varphi _m \gamma _{22m}''''+2 \psi _m' \gamma _{22m}'''' $$
(56)
$${{\varUpsilon }_{4m}^{\psi }}^{(2)}=-3 \phi _{21m}' \psi _m''-6 \phi _{22m}' \psi _m''-6 \gamma _{21m}'' \psi _m''-12 \gamma _{22m}'' \psi _m''-2 \phi _{01m}''' \psi _m''-4 \phi _{02m}''' \psi _m''+3 \phi _{21m}''' \psi _m''+6 \phi _{22m}''' \psi _m''-2 \gamma _{01m}'''' \psi _m''-4 \gamma _{02m}'''' \psi _m''-3 \varphi _m' \gamma _{21m}''-6 \varphi _m' \gamma _{22m}''+\phi _{21m} \varphi _m''+2 \phi _{22m} \varphi _m''-3 \gamma _{21m}' \varphi _m''-6 \gamma _{22m}' \varphi _m''-4 \varphi _m'' \phi _{01m}''-8 \varphi _m'' \phi _{02m}''+\varphi _m \phi _{21m}''-3 \psi _m' \phi _{21m}''+8 \varphi _m'' \phi _{21m}''+2 \varphi _m \phi _{22m}''-6 \psi _m' \phi _{22m}''+16 \varphi _m'' \phi _{22m}''-4 \gamma _{21m}' \psi _m'''-8 \gamma _{22m}' \psi _m'''-4 \phi _{01m}'' \psi _m'''-8 \phi _{02m}'' \psi _m'''+4 \phi _{21m}'' \psi _m'''+8 \phi _{22m}'' \psi _m'''-4 \varphi _m'' \gamma _{01m}'''-4 \psi _m''' \gamma _{01m}'''-8 \varphi _m'' \gamma _{02m}'''-8 \psi _m''' \gamma _{02m}'''-4 \psi _m' \gamma _{21m}'''+4 \varphi _m'' \gamma _{21m}'''-8 \psi _m' \gamma _{22m}'''+8 \varphi _m'' \gamma _{22m}'''-2 \gamma _{21m} \varphi _m'''-4 \gamma _{22m} \varphi _m'''-2 \phi _{01m}' \varphi _m'''-4 \phi _{02m}' \varphi _m'''+5 \phi _{21m}' \varphi _m'''+10 \phi _{22m}' \varphi _m'''-2 \gamma _{01m}'' \varphi _m'''-4 \gamma _{02m}'' \varphi _m'''+3 \gamma _{21m}'' \varphi _m'''+6 \gamma _{22m}'' \varphi _m'''-2 \varphi _m' \phi _{01m}'''-4 \varphi _m' \phi _{02m}'''-2 \psi _m \phi _{21m}'''+5 \varphi _m' \phi _{21m}'''-4 \psi _m \phi _{22m}'''+10 \varphi _m' \phi _{22m}'''-2 \gamma _{21m} \psi _m''''-4 \gamma _{22m} \psi _m''''-2 \phi _{01m}' \psi _m''''-4 \phi _{02m}' \psi _m''''+2 \phi _{21m}' \psi _m''''+4 \phi _{22m}' \psi _m''''-2 \gamma _{01m}'' \psi _m''''-4 \gamma _{02m}'' \psi _m''''-2 \varphi _m' \gamma _{01m}''''-4 \varphi _m' \gamma _{02m}''''-2 \psi _m \gamma _{21m}''''+2 \varphi _m' \gamma _{21m}''''-4 \psi _m \gamma _{22m}''''+4 \varphi _m' \gamma _{22m}''''+\phi _{21m} \varphi _m''''+2 \phi _{22m} \varphi _m''''+\gamma _{21m}' \varphi _m''''+2 \gamma _{22m}' \varphi _m''''+\varphi _m \phi _{21m}''''+\psi _m' \phi _{21m}''''+2 \varphi _m \phi _{22m}''''+2 \psi _m' \phi _{22m}'''' $$
(57)
$${{\varUpsilon }_{5m}^{\varphi }}^{(2)}=2 \left[ \psi _m''' \gamma _{21m}''+2 \psi _m''' \gamma _{22m}''+\gamma _{21m}''' \psi _m''+2 \gamma _{22m}''' \psi _m''\right. +\phi _{21m}' \left( \psi _m'''+\varphi _m''\right) +2 \phi _{22m}' \left( \psi _m'''+\varphi _m''\right) +\phi _{21m}'' \psi _m''+2 \phi _{22m}'' \psi _m''+\gamma _{21m}''' \varphi _m'+2 \gamma _{22m}''' \varphi _m'+\gamma _{21m}'' \varphi _m''\left. +2 \gamma _{22m}'' \varphi _m''+\phi _{21m}'' \varphi _m'+2 \phi _{22m}'' \varphi _m'\right] $$
(58)
$${{\varUpsilon }_{5m}^{\psi }}^{(2)}=-2 \left[ \psi _m'''' \gamma _{21m}''+2 \psi _m'''' \gamma _{22m}''+\gamma _{21m}'''' \psi _m''+2 \gamma _{22m}'''' \psi _m''+2 \gamma _{21m}''' \psi _m'''\right. +4 \gamma _{22m}''' \psi _m'''+\psi _m'''' \phi _{21m}'+2 \psi _m'''' \phi _{22m}'+2 \psi _m''' \phi _{21m}''+4 \psi _m''' \phi _{22m}''+\phi _{21m}''' \psi _m''+2 \phi _{22m}''' \psi _m''+2 \varphi _m'' \left( \gamma _{21m}'''+2 \gamma _{22m}'''+\phi _{21m}''+2 \phi _{22m}''\right) +\gamma _{21m}'''' \varphi _m'+2 \gamma _{22m}'''' \varphi _m'+\varphi _m''' \gamma _{21m}''+2 \varphi _m''' \gamma _{22m}''\left. +\varphi _m''' \phi _{21m}'+2 \varphi _m''' \phi _{22m}'+\phi _{21m}''' \varphi _m'+2 \phi _{22m}''' \varphi _m'\right]$$
(59)
$${{\varUpsilon }_{0m}^{\varphi }}^{(3)}=-3 \psi _m'{}^2 \varphi _m''-9 \psi _m' \varphi _m''{}^2-15 \psi _m' \psi _m'' \varphi _m'-9 \varphi _m \psi _m' \varphi _m''-15 \psi _m''' \psi _m' \varphi _m''-12 \varphi _m''' \psi _m' \psi _m''-9 \varphi _m \psi _m'' \varphi _m'-27 \psi _m \varphi _m' \varphi _m''-3 \psi _m'' \varphi _m' \varphi _m''-9 \psi _m''' \psi _m'' \varphi _m'+6 \varphi _m \psi _m'{}^2-3 \varphi _m'''' \psi _m'{}^2-18 \psi _m' \varphi _m'{}^2+18 \psi _m \psi _m' \varphi _m'+9 \psi _m''' \varphi _m \psi _m'-12 \varphi _m''' \psi _m' \varphi _m'-9 \psi _m'''' \psi _m' \varphi _m'-6 \varphi _m \varphi _m'''' \psi _m'-9 \psi _m \psi _m''' \varphi _m'-9 \psi _m'''' \varphi _m \varphi _m'+6 \varphi _m \psi _m''{}^2+9 \psi _m \varphi _m \psi _m''+9 \psi _m{}^2 \varphi _m''-6 \psi _m''{}^2 \varphi _m''-9 \psi _m \psi _m'' \varphi _m''-15 \psi _m''' \varphi _m \varphi _m''-12 \varphi _m \varphi _m''' \psi _m''-3 \psi _m''' \varphi _m{}^2+9 \psi _m \psi _m'''' \varphi _m+12 \psi _m' \psi _m''{}^2+27 \psi _m \psi _m' \psi _m''+6 \psi _m'{}^3+12 \psi _m''' \psi _m'{}^2+9 \psi _m \psi _m'''' \psi _m'+9 \psi _m \psi _m''' \psi _m''+9 \psi _m{}^2 \psi _m'''+12 \varphi _m'{}^2 \varphi _m''-6 \varphi _m \varphi _m'{}^2-12 \varphi _m \varphi _m''' \varphi _m'-9 \varphi _m \varphi _m''{}^2-6 \varphi _m{}^2 \varphi _m''-3 \varphi _m{}^2 \varphi _m'''' $$
(60)
$${{\varUpsilon }_{0m}^{\psi }}^{(3)}=6 \varphi _m'{}^3+45 \psi _m'' \varphi _m'{}^2-36 \varphi _m''' \varphi _m'{}^2-9 \psi _m'''' \varphi _m'{}^2-18 \psi _m'{}^2 \varphi _m'+45 \psi _m''{}^2 \varphi _m'-60 \varphi _m''{}^2 \varphi _m'+6 \varphi _m \psi _m' \varphi _m'-27 \psi _m \psi _m'' \varphi _m'+9 \varphi _m \varphi _m'' \varphi _m'+93 \psi _m' \varphi _m'' \varphi _m'+24 \varphi _m \psi _m''' \varphi _m'+60 \psi _m' \psi _m''' \varphi _m'-36 \varphi _m'' \psi _m''' \varphi _m'+45 \psi _m \varphi _m''' \varphi _m'-27 \psi _m'' \varphi _m''' \varphi _m'+18 \psi _m \psi _m'''' \varphi _m'-9 \varphi _m \varphi _m'''' \varphi _m'-9 \psi _m' \varphi _m'''' \varphi _m'+6 \psi _m''{}^3-27 \psi _m \psi _m''{}^2+36 \psi _m \varphi _m''{}^2-24 \psi _m'' \varphi _m''{}^2+3 \varphi _m{}^2 \psi _m''-36 \psi _m'{}^2 \psi _m''-9 \varphi _m \psi _m' \psi _m''+9 \psi _m \varphi _m \varphi _m''-27 \psi _m \psi _m' \varphi _m''+39 \varphi _m \psi _m'' \varphi _m''+87 \psi _m' \psi _m'' \varphi _m''-36 \psi _m \psi _m' \psi _m'''+24 \varphi _m \psi _m'' \psi _m'''+24 \psi _m' \psi _m'' \psi _m'''+36 \psi _m \varphi _m'' \psi _m'''-9 \psi _m{}^2 \varphi _m'''+3 \varphi _m{}^2 \varphi _m'''+24 \psi _m'{}^2 \varphi _m'''+27 \varphi _m \psi _m' \varphi _m'''+27 \psi _m \psi _m'' \varphi _m'''-21 \varphi _m \varphi _m'' \varphi _m'''-21 \psi _m' \varphi _m'' \varphi _m'''-9 \psi _m{}^2 \psi _m''''+6 \varphi _m{}^2 \psi _m''''+6 \psi _m'{}^2 \psi _m''''+12 \varphi _m \psi _m' \psi _m''''+9 \psi _m \varphi _m \varphi _m''''+9 \psi _m \psi _m' \varphi _m'''' $$
(61)
$${{\varUpsilon }_{1m}^{\varphi }}^{(3)}=-6 \varphi _m \psi _m'' \varphi _m'-6 \varphi _m \psi _m' \varphi _m''-6 \psi _m' \psi _m'' \varphi _m'-3 \psi _m'{}^2 \varphi _m''-\frac{3}{2} \varphi _m{}^2 \psi _m'+\frac{3}{2} \varphi _m \psi _m'{}^2-3 \psi _m' \varphi _m'{}^2+6 \psi _m \varphi _m \psi _m''+3 \psi _m{}^2 \varphi _m+6 \psi _m \psi _m' \psi _m''+\frac{3}{2} \psi _m'{}^3+3 \psi _m{}^2 \psi _m'-3 \varphi _m \varphi _m'{}^2-3 \varphi _m{}^2 \varphi _m''-\frac{1}{2} 3 \varphi _m{}^3 $$
(62)
$${{\varUpsilon }_{1m}^{\psi }}^{(3)}=-3 \psi _m'' \varphi _m'{}^2+6 \psi _m \psi _m'' \varphi _m'-6 \psi _m' \varphi _m' \varphi _m''+9 \varphi _m \psi _m' \psi _m''+6 \psi _m \psi _m' \varphi _m''+6 \psi _m \varphi _m'{}^2-3 \psi _m{}^2 \varphi _m'+\frac{15}{2} \psi _m'{}^2 \varphi _m'+9 \varphi _m \psi _m' \varphi _m'+\frac{9}{2} \varphi _m{}^2 \psi _m''+6 \psi _m \varphi _m \varphi _m''+3 \psi _m \varphi _m{}^2+\frac{9}{2} \psi _m'{}^2 \psi _m''-3 \psi _m \psi _m'{}^2-3 \psi _m{}^2 \psi _m''-6 \varphi _m \varphi _m' \varphi _m''-3 \varphi _m'{}^3+\frac{3}{2} \varphi _m{}^2 \varphi _m' $$
(63)
$${{\varUpsilon }_{4m}^{\varphi }}^{(3)}=6 \psi _m'{}^2 \varphi _m''+30 \psi _m' \psi _m'' \varphi _m'+6 \varphi _m \psi _m' \varphi _m''+6 \psi _m''' \psi _m' \varphi _m''+6 \varphi _m \psi _m'' \varphi _m'+18 \psi _m \varphi _m' \varphi _m''-30 \psi _m'' \varphi _m' \varphi _m''-6 \psi _m''' \psi _m'' \varphi _m'+12 \psi _m' \varphi _m'{}^2+6 \psi _m''' \varphi _m \psi _m'+6 \psi _m'''' \psi _m' \varphi _m'-12 \psi _m''' \varphi _m'{}^2+18 \psi _m \psi _m''' \varphi _m'+6 \psi _m'''' \varphi _m \varphi _m'+6 \varphi _m \psi _m''{}^2-6 \psi _m''{}^2 \varphi _m''+18 \psi _m \psi _m'' \varphi _m''+6 \psi _m''' \varphi _m \varphi _m''+6 \psi _m'''' \varphi _m \psi _m''+6 \psi _m'''{}^2 \varphi _m+18 \psi _m' \psi _m''{}^2+6 \psi _m'''' \psi _m' \psi _m''+6 \psi _m''' \psi _m'{}^2+6 \psi _m'''{}^2 \psi _m'+6 \psi _m''' \psi _m''{}^2+18 \psi _m \psi _m''' \psi _m''-24 \varphi _m'{}^2 \varphi _m'' $$
(64)
$${{\varUpsilon }_{4m}^{\psi }}^{(3)}=-36 \psi _m''{}^2 \varphi _m'-18 \psi _m'' \varphi _m'{}^2-54 \psi _m' \psi _m'' \varphi _m''+54 \varphi _m''' \psi _m'' \varphi _m'+18 \psi _m'''' \psi _m'' \varphi _m'+6 \varphi _m'''' \psi _m' \psi _m''-42 \psi _m' \varphi _m' \varphi _m''+84 \psi _m''' \varphi _m' \varphi _m''+18 \varphi _m''' \psi _m' \varphi _m''+6 \psi _m'''' \psi _m' \varphi _m''+18 \psi _m'''{}^2 \varphi _m'-48 \psi _m''' \psi _m' \varphi _m'-6 \varphi _m''' \psi _m'{}^2-6 \varphi _m \varphi _m''' \psi _m'-18 \psi _m \varphi _m''' \varphi _m'+12 \psi _m''' \varphi _m''' \psi _m'+18 \psi _m'''' \varphi _m'{}^2-6 \psi _m'''' \varphi _m \psi _m'-18 \psi _m \psi _m'''' \varphi _m'+6 \varphi _m'''' \psi _m' \varphi _m'+18 \varphi _m''' \psi _m''{}^2+48 \psi _m'' \varphi _m''{}^2-6 \varphi _m \psi _m'' \varphi _m''-12 \psi _m''' \varphi _m \psi _m''+48 \psi _m''' \psi _m'' \varphi _m''-18 \psi _m \varphi _m''' \psi _m''+6 \varphi _m \varphi _m'''' \psi _m''-18 \psi _m \varphi _m''{}^2-36 \psi _m \psi _m''' \varphi _m''+6 \psi _m'''' \varphi _m \varphi _m''+12 \psi _m''' \varphi _m \varphi _m'''-60 \psi _m''' \psi _m' \psi _m''-6 \psi _m'''' \psi _m'{}^2-18 \psi _m''{}^3-18 \psi _m \psi _m'''' \psi _m''-18 \psi _m \psi _m'''{}^2+66 \varphi _m' \varphi _m''{}^2+6 \varphi _m \varphi _m' \varphi _m''+36 \varphi _m''' \varphi _m'{}^2+6 \varphi _m \varphi _m'''' \varphi _m'+18 \varphi _m \varphi _m''' \varphi _m'' $$
(65)
$${{\varUpsilon }_{6m}^{\varphi }}^{(3)}=18 \psi _m'' \varphi _m' \varphi _m''+18 \psi _m''' \psi _m'' \varphi _m'+9 \psi _m''' \varphi _m'{}^2+9 \psi _m''{}^2 \varphi _m''+9 \psi _m''' \psi _m''{}^2+9 \varphi _m'{}^2 \varphi _m'' $$
(66)
$${{\varUpsilon }_{6m}^{\psi }}^{(3)}=-36 \psi _m''' \varphi _m' \varphi _m''-18 \varphi _m''' \psi _m'' \varphi _m'-18 \psi _m'''' \psi _m'' \varphi _m'-9 \psi _m'''' \varphi _m'{}^2-18 \psi _m'''{}^2 \varphi _m'-18 \psi _m'' \varphi _m''{}^2-36 \psi _m''' \psi _m'' \varphi _m''-9 \varphi _m''' \psi _m''{}^2-18 \psi _m'''{}^2 \psi _m''-9 \psi _m'''' \psi _m''{}^2-18 \varphi _m' \varphi _m''{}^2-9 \varphi _m''' \varphi _m'{}^2$$
(67)

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Lacarbonara, W., Arena, A. & Antman, S.S. Flexural vibrations of nonlinearly elastic circular rings. Meccanica 50, 689–705 (2015). https://doi.org/10.1007/s11012-014-0038-3

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