Structural analogies for hybrid discrete-continuum systems of deformable bodies coupled with non-linear layers

Abstract

The chapter is addressed at phenomenological mapping and mathematical analogies of oscillatory regimes in hybrid discrete-continuum systems of coupled deformable bodies. Systems consist of connected deformable bodies like plates, beams, belts, or membranes that are coupled through visco-elastic non-linear layer. The layer is modeled by continuously distributed elements of Kelvin–Voigt type with non-linearity of third order. Using the mathematical analogies, the similarities of structural models in systems of plates, beams, belts, or membranes are explained. The mathematical models consist by a set of two coupled non-homogenous partial non-linear differential equations. The proposed solution is divided into space and time domains by classical Bernoulli–Fourier method. In the time domains, the systems of coupled ordinary non-linear differential equations are completely analog for different systems of deformable bodies and are solved using the Krilov–Bogolyubov–Mitropolski asymptotic method. This paper presents the power of mathematical analytical calculus which is similar for physically different systems. The mathematical numerical experiments are a great and useful tool for making the final conclusions between many input and output values. The conclusions about non-linear phenomena in multi-body systems dynamics are revealed from the specific example of double plate’s system stationery and no stationary oscillatory regimes.

Appendices

Appendix A

$$\begin{aligned} h_{0i}=\frac{\int _0^r\int _0^{2\pi }\tilde{F}_{(0i)}\tilde{\tilde{F}}_{(i)}(r,\phi )W_{(i)nm}(r,\phi )r\mathrm{d}r\mathrm{d}\phi }{\int _0^r\int _0^{2\pi }[W_{(i)nm}(r,\phi )]^2r\mathrm{d}r\mathrm{d}\phi }. \end{aligned}$$

Appendix B

$$\begin{aligned} A= & {} \delta _{1} +\delta _{2} -\varepsilon \left( \frac{P_{1} \cos \phi _{1s} }{\left( {\Omega _{1s} +{\hat{p}}_{1} } \right) a_{1s} }+\frac{P_{2} \cos \phi _{2s} }{\left( {\Omega _{2s} +{\hat{p}}_{2} } \right) a_{2s} } \right) ,\\ B= & {} \frac{3}{4}\frac{\alpha _{1} }{{\hat{p}}_{1} }a_{1s} \frac{\varepsilon P_{1} \sin \phi _{1s} }{\left( {\Omega _{1s} +{\hat{p}}_{1} } \right) }\nonumber \\&+\frac{\varepsilon ^{2}P_{1}^{2}\sin ^{2}\phi _{1s} }{\left( {\Omega _{1s} +{\hat{p}}_{1} } \right) ^{2}a_{1s}^{2} } -\frac{\varepsilon P_{1} \cos \phi _{1s} \left( {\delta _{2} +\delta _{1} } \right) }{\left( {\Omega _{1s} +{\hat{p}}_{1} } \right) a_{1s} }\nonumber \\&+\frac{3}{4}\frac{\alpha _{2} }{{\hat{p}}_{2} }a_{2s} \frac{\varepsilon P_{2} \sin \phi _{2s} }{\left( {\Omega _{2s} +{\hat{p}}_{2} } \right) } +\frac{\varepsilon ^{2}P_{2}^{2}\sin ^{2}\phi _{2s} }{\left( {\Omega _{2s} +{\hat{p}}_{2} } \right) ^{2}a_{2s}^{2} }-\\&-\frac{\varepsilon P_{2} \cos \phi _{2s} \left( {\delta _{2} +\delta _{1} } \right) }{\left( {\Omega _{2s} +{\hat{p}}_{2} } \right) a_{2s} } +\frac{\varepsilon P_{2} \cos \phi _{2s} }{\left( {\Omega _{2s} +{\hat{p}}_{2} } \right) a_{2s} }\frac{\varepsilon P_{1} \cos \phi _{1s} }{\left( {\Omega _{1s} +{\hat{p}}_{1} } \right) a_{1s} }\\&+\delta _{2} \delta _{1}, \\ C= & {} \left( {\delta _{2} -\frac{\varepsilon P_{2} \cos \phi _{2s} }{\left( {\Omega _{2s} +{\hat{p}}_{2} } \right) a_{2s} }} \right) \left( \frac{3}{4}\frac{\alpha _{1} }{{\hat{p}}_{1} }a_{1s}\frac{\varepsilon P_{1} \sin \phi _{1s} }{\left( {\Omega _{1s} +{\hat{p}}_{1} } \right) } \nonumber \right. \\&+\frac{\varepsilon ^{2}P_{1}^{2}\sin ^{2}\phi _{1s} }{\left( {\Omega _{1s} +{\hat{p}}_{1} } \right) ^{2}a_{1s}^{2} }\nonumber \\&\left. -\frac{\varepsilon P_{1} \cos \phi _{1s} \delta _{1} }{\left( {\Omega _{1s} +{\hat{p}}_{1} } \right) a_{1s} } \right) \\&+\left( {\delta _{1} -\frac{\varepsilon P_{1} \cos \phi _{1s} }{\left( {\Omega _{1s} +{\hat{p}}_{1} } \right) a_{1s} }} \right) \left( \frac{3}{4}\frac{\alpha _{2} }{{\hat{p}}_{2} }a_{2s} \frac{\varepsilon P_{2} \sin \phi _{2s} }{\left( {\Omega _{2s} +{\hat{p}}_{2} } \right) }\nonumber \right. \\&+\frac{\varepsilon ^{2}P_{2}^{2}\sin ^{2}\phi _{2s} }{\left( {\Omega _{2s} +{\hat{p}}_{2} } \right) ^{2}a_{2s}^{2} }\nonumber \\&\left. -\frac{\varepsilon P_{2} \cos \phi _{2s} \delta _{2} }{\left( {\Omega _{2s} +{\hat{p}}_{2} } \right) a_{2s} } \right) \hbox {and}\nonumber \\&D=\frac{1}{4}\frac{\beta _{1} \beta _{2} }{{\hat{p}}_{1} {\hat{p}}_{2} }a_{2s} a_{1s} \frac{\varepsilon ^{2}P_{2} P_{1} \sin \phi _{2s} \sin \phi _{1s} }{\left( {\Omega _{2s} +{\hat{p}}_{2}} \right) \left( {\Omega _{1s} +{\hat{p}}_{1} } \right) }+ \\&+\left( \frac{3}{4}\frac{\alpha _{1} }{{\hat{p}}_{1} }a_{1s} \frac{\varepsilon P_{1} \sin \phi _{1s} }{\left( {\Omega _{1s} +{\hat{p}}_{1} } \right) }+\frac{\varepsilon ^{2}P_{1}^{2}\sin ^{2}\phi _{1s} }{\left( {\Omega _{1s} +{\hat{p}}_{1} } \right) ^{2}a_{1s}^{2} }\right. \\&\left. -\frac{\varepsilon P_{1} \cos \phi _{1s} \delta _{1} }{\left( {\Omega _{1s} +{\hat{p}}_{1} } \right) a_{1s} } \right) \nonumber \\&\quad \left( \frac{3}{4}\frac{\alpha _{2} }{{\hat{p}}_{2} }a_{2s} \frac{\varepsilon P_{2} \sin \phi _{2s} }{\left( {\Omega _{2s} +{\hat{p}}_{2} } \right) }+\frac{\varepsilon ^{2}P_{2}^{2}\sin ^{2}\phi _{2s} }{\left( {\Omega _{2s} +{\hat{p}}_{2} } \right) ^{2}a_{2s}^{2} }\nonumber \right. \\&\left. \quad -\frac{\varepsilon P_{2} \cos \phi _{2s} \delta _{2} }{\left( {\Omega _{2s} +{\hat{p}}_{2} } \right) a_{2s} } \right) . \end{aligned}$$