Three-to-one internal resonance in a two-beam structure connected with nonlinear joints

Abstract

A two-degree-of-freedom model for a two-beam structure with nonlinear joints is presented and used to investigate the nonlinear responses of the system to a primary resonance of its first two modes in the presence of three-to-one internal resonance. By using the equilibrium conditions between the beams and the joints, the joint dynamic characteristics included linear and nonlinear torsional stiffness are introduced into the model of the system. The first two global mode functions of the two-beam structure, which can be obtained by the global mode method, are used to formulate the nonlinear dynamic model of the system and the specific linear torsional stiffness that result in three-to-one internal resonance is mainly considered. The method of multiple scales is employed to obtain the governing equations of the amplitudes and phases for the two-degree-of-freedom nonlinear dynamical system under the three-to-one internal resonance condition. Based on the modulation equations obtained by the method of multiple scales, the approximation solutions are derived and compared with those obtained by the numerical integration method. Through the cases of primary resonance of the first two modes with the three-to-one internal resonance condition, the frequency–response and force-response curves are plotted for investigating the nonlinear dynamic behavior in the two-beam structure connected with nonlinear joints.

Appendices

Appendix 1

The relevant terms in Eq. (5) are given as

$$ \begin{aligned} M_{s} & = \int_{-L_{1}}^{{L_{1} }} {\rho_{1} A_{1} \left[ {\varphi_{1s} (x)} \right]}^{2} {\mathrm{d}} x + \int_{-L_{2}}^{{L_{2} }} {\rho_{2} A_{2} \left[ {\varphi_{2s} (x)} \right]}^{2} {\mathrm{d}} x,\quad c_{s} = \left( {aM_{s} + bK_{s} } \right), \\ K_{s} & = \int_{-L_{1}}^{{L_{1} }} {E_{1} I_{1} \left[ {\varphi^{\prime\prime}_{1s} (x)} \right]^{2} } {\mathrm{d}} x{ + }\int_{-L_{2}}^{{L_{2} }} {E_{2} I_{2} \left[ {\varphi^{\prime\prime}_{2s} (x)} \right]^{2} } {\mathrm{d}} x + k_{1}^{L} \Theta_{1s}^{{2}} + k_{2}^{L} \Theta_{2s}^{{2}} , \\ d_{s}^{jkr} & = k_{1}^{N} \Theta_{1j} \Theta_{1k} \Theta_{1r} \Theta_{1s} + k_{2}^{N} \Theta_{2j} \Theta_{2k} \Theta_{2r} \Theta_{2s} , \\ F_{s} (t) & = \int_{-L_{1}}^{{L_{1} }} {\rho_{1} A_{1} F\varphi_{is} (x) {\mathrm{d}}x + \int_{-L_{2}}^{{L_{2} }} {\rho_{2} A_{2} F\varphi_{2s} (x){\mathrm{d}}x} } . \\ \end{aligned} $$

Appendix 2

The coefficients listed in Eqs. (7) and (8) are

$$ \begin{aligned} \hat{\mu }_{s} & = \frac{1}{2}\left( {a + b\omega_{s}^{{2}} } \right),\quad \hat{\alpha }_{s1} = \frac{1}{{M_{s} }}d_{s}^{111} ,\quad \hat{\alpha }_{s2} = \frac{1}{{M_{s} }}\left( {d_{s}^{112} + d_{s}^{121} + d_{s}^{211} } \right), \\ \hat{\alpha }_{s3} & = \frac{1}{{M_{s} }}\left( {d_{s}^{122} + d_{s}^{212} + d_{s}^{221} } \right),\quad \hat{\alpha }_{s4} = \frac{1}{{M_{s} }}d_{s}^{222} ,\quad \hat{f}_{s} (t) = \frac{1}{{M_{s} }}F_{s} (t),\quad s = 1,2. \\ \end{aligned} $$