Chaos prediction in nonlinear viscoelastic plates subjected to subsonic flow and external load using extended Melnikov’s method

Abstract

Chaotic behavior of the viscoelastic plates subjected to subsonic flow and simultaneous external excitation is studied in this paper. The equation of motion of the plate is derived using the von-Kármán theory. Galerkin’s approach is adopted as the solution method. Corresponding extended Melnikov’s integral is obtained for the non-Hamiltonian system by numerical and analytical approaches. A parametric study is carried out, and effects of different parameters such as linear and nonlinear stiffness and structural damping on the chaotic behavior of the dynamical system are investigated. Chaos thresholds are obtained, and the correlation between the analytical and numerical results is evaluated.

Appendix

In order to use the Melnikov’s integral, one should analytically obtain the integrals of $I_{1}(\Omega )$, $I_{2}(\Omega )$, $I_{3}$ and $I_{4}$. These integrals are represented in the form of Eq. (18). One may show

$$\begin{aligned} \int \limits _{-\infty }^{+\infty } {\frac{e^{b t}}{(1+e^{a t})^{2}}\mathrm{d}t=\frac{(a-b)\pi }{a^{2}\sin \left( {\frac{\pi b}{a}} \right) }} =\frac{(b-a)\pi }{a^{2}\sin \left( {\frac{\pi (b-a)}{a}} \right) }\nonumber \\ \end{aligned}$$

(29)

So, $I_{1}(\Omega )$ can be represented by

$$\begin{aligned} I_1 (\Omega )= & {} \int \limits _{-\infty }^{+\infty } \sin \Omega t\;e^{\delta t}\dot{x}(t) \mathrm{d}t \nonumber \\= & {} \hbox {Im}\left[ {\int \limits _{-\infty }^{+\infty } {e^{i\Omega t}e^{\delta t}\dot{x}(t) \mathrm{d}t} } \right] \nonumber \\= & {} \hbox {Im}\left[ {\int \limits _{-\infty }^{+\infty } {e^{i\Omega t}e^{\delta t}\left\{ {-\tilde{c}(x_3^s -x_1^s )^{2}\frac{e^{\tilde{c}(x_3^s -x_1^s )t}}{(1+e^{\tilde{c}(x_3^s -x_1^s )t})^{2}}} \right\} \mathrm{d}t} } \right] \nonumber \\= & {} -\tilde{c}(x_3^s -x_1^s )^{2}\hbox {Im}\left[ {\int \limits _{-\infty }^{+\infty } {\frac{e^{\left( {i\Omega +\delta +\tilde{c}(x_3^s -x_1^s )} \right) t}}{(1+e^{\tilde{c}(x_3^s -x_1^s )t})^{2}}\mathrm{d}t} } \right] \nonumber \\= & {} -\hbox {Im}\left[ {\frac{\left( {i\Omega +\delta } \right) \pi }{\tilde{c}\sin \left( {\frac{\left( {i\Omega +\delta } \right) \pi }{\tilde{c}(x_3^s -x_1^s )}} \right) }} \right] \end{aligned}$$

(30)

Moreover, one can show

$$\begin{aligned} \frac{ia+b}{ic+d}=\left( {\frac{ac+bd}{c^{2}+d^{2}}} \right) +i\left( {\frac{ad-bc}{c^{2}+d^{2}}} \right) \end{aligned}$$

(31)

According to Eq. (31), if we define $a=\tilde{c}(x_3^s -x_1^s ), b=i\Omega +\delta +\tilde{c}(x_3^s -x_1^s )$, the Eq. (30) can be represented by

$$\begin{aligned} \hbox {Im}\left[ {\frac{\left( {i\Omega +\delta } \right) \pi }{\tilde{c}\sin \left( {\frac{\left( {i\Omega +\delta } \right) \pi }{\tilde{c}(x_3^s -x_1^s )}} \right) }} \right] =\frac{1}{\tilde{c}^{2}}\frac{\pi \Omega \sin \frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}\cosh \frac{\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}-\pi \delta \cos \frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}\sinh \frac{\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}}{\sin ^{2}\frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}\cosh ^{2}\frac{\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}+\cos ^{2}\frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}\sinh ^{2}\frac{\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}} \end{aligned}$$

(32)

and

$$\begin{aligned} \hbox {Re}\left[ {\frac{\left( {i\Omega +\delta } \right) \pi }{\tilde{c}\sin \left( {\frac{\left( {i\Omega +\delta } \right) \pi }{\tilde{c}(x_3^s -x_1^s )}} \right) }} \right] =\frac{1}{\tilde{c}^{2}}\frac{\pi \Omega \cos \frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}\sinh \frac{\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}+\pi \delta \sin \frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}\cosh \frac{\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}}{\sin ^{2}\frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}\cosh ^{2}\frac{\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}+\cos ^{2}\frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}\sinh ^{2}\frac{\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}} \end{aligned}$$

(33)

Furthermore, it is easy to show

$$\begin{aligned}&\sin ^{2}\frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}\cosh ^{2}\frac{\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}\nonumber \\&\qquad +\,\cos ^{2}\frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}\sinh ^{2}\frac{\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}\nonumber \\&\quad =\frac{1}{2}\left( {\cosh \frac{2\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}-\cos \frac{2\pi \delta }{\tilde{c}(x_3^s -x_1^s )}} \right) \end{aligned}$$

(34)

Thus, one can represent $I_{1}(\Omega )$ as

$$\begin{aligned}&I_1 (\Omega ) =\frac{-2\pi }{\tilde{c}^{2}}\nonumber \\&\quad \times \,\left( {\frac{\Omega \sin \frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}\cosh \frac{\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}-\delta \cos \frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}\sinh \frac{\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}}{\cosh \frac{2\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}-\cos \frac{2\pi \delta }{\tilde{c}(x_3^s -x_1^s )}}} \right) \nonumber \\ \end{aligned}$$

(35)

In a similar way, in order to obtain $I_{2}(\Omega )$ one can write

$$\begin{aligned} I_2 (\Omega )= & {} \int \limits _{-\infty }^{+\infty } {\cos \Omega t\;e^{\delta t}\dot{x}(t) \mathrm{d}t=\hbox {Re}\left[ {\int \limits _{-\infty }^{+\infty } {e^{i\Omega t}e^{\delta t}\dot{x}(t) \mathrm{d}t} } \right] }\nonumber \\= & {} \hbox {Re}\left[ {\int \limits _{-\infty }^{+\infty } {e^{i\Omega t}e^{\delta t}\left\{ {-\tilde{c}(x_3^s -x_1^s )^{2}\frac{e^{\tilde{c}(x_3^s -x_1^s )t}}{(1+e^{\tilde{c}(x_3^s -x_1^s )t})^{2}}} \right\} \mathrm{d}t} } \right] \nonumber \\= & {} -\tilde{c}(x_3^s -x_1^s )^{2}\hbox {Re}\left[ {\int \limits _{-\infty }^{+\infty } {\frac{e^{\left( {i\Omega +\delta +\tilde{c}(x_3^s -x_1^s )} \right) t}}{(1+e^{\tilde{c}(x_3^s -x_1^s )t})^{2}}\mathrm{d}t} } \right] \nonumber \\= & {} -\hbox {Re}\left[ {\frac{\left( {i\Omega +\delta } \right) \pi }{\tilde{c}\sin \left( {\frac{\left( {i\Omega +\delta } \right) \pi }{\tilde{c}(x_3^s -x_1^s )}} \right) }} \right] \end{aligned}$$

(36)

Using Eqs. (31), (33) and (34) gives

$$\begin{aligned}&I_2 (\Omega )=\frac{-2\pi }{\tilde{c}^{2}}\nonumber \\&\qquad \times \,\left( {\frac{\Omega \cos \frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}\sinh \frac{\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}+\delta \sin \frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}\cosh \frac{\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}}{\cosh \frac{2\pi \Omega }{\tilde{c}(x_3^s -x_1^s )}-\cos \frac{2\pi \delta }{\tilde{c}(x_3^s -x_1^s )}}} \right) \nonumber \\ \end{aligned}$$

(37)

In order to calculate $I_{3}$, one should write

$$\begin{aligned} I_3= & {} \int \limits _{-\infty }^{+\infty } {x^{2}\dot{x}^{2}e^{\delta t}} \mathrm{d}t=\int \limits _{-\infty }^{+\infty } \left\{ \left( {x_1^s +\frac{x_3^s -x_1^s }{1+e^{\tilde{c}(x_3^s -x_1^s )t}}} \right) \right. \nonumber \\&\left. \times \,\left( {-\tilde{c} e^{\tilde{c}(x_3^s -x_1^s )t}\frac{(x_3^s -x_1^s )^{2}}{\left( {1+e^{\tilde{c}(x_3^s -x_1^s )t}} \right) ^{2}}} \right) \right\} ^{2}e^{\delta t}\mathrm{d}t\nonumber \\ \end{aligned}$$

(38)

Calculating the expression $\bigg \{ \big ( x_1^s +\frac{x_3^s -x_1^s }{1+e^{\tilde{c}(x_3^s -x_1^s )t}} \big )\big ( {-\tilde{c} e^{\tilde{c}(x_3^s -x_1^s )t}\frac{(x_3^s -x_1^s )^{2}}{\big ( {1+e^{\tilde{c}(x_3^s -x_1^s )t}} \big )^{2}}} \big ) \bigg \}^{2}$ leads to rewrite $I_{3}$ in the form

$$\begin{aligned} I_3= & {} \tilde{c}^{2}e^{2\tilde{c}(x_3^s -x_1^s )t}(x_3^s -x_1^s )^{4}\left( (x_3^s )^{2}I_{3,1} +2x_1^s x_3^s I_{3,2} \right. \nonumber \\&\left. +\,(x_1^s )^{2}I_{3,3} \right) \end{aligned}$$

(39)

in which

$$\begin{aligned} I_{3,1}= & {} \int \limits _{-\infty }^{+\infty } {\frac{e^{\left( {\delta +2\tilde{c}(x_3^s -x_1^s )} \right) t}}{(1+e^{\tilde{c}(x_3^s -x_1^s )t})^{6}}\mathrm{d}t} \nonumber \\= & {} \frac{\pi \delta }{5!\tilde{c}^{6}(x_3^s -x_1^s )^{6}\sin \frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}}\left( {\delta ^{2}-\tilde{c}^{2}(x_3^s -x_1^s )^{2}} \right) \nonumber \\&\times \,\left( {\delta -2\tilde{c}^{2}(x_3^s -x_1^s )} \right) \left( {\delta -3\tilde{c}^{2}(x_3^s -x_1^s )} \right) \end{aligned}$$

(40)

$$\begin{aligned} I_{3,2}= & {} \int \limits _{-\infty }^{+\infty } {\frac{e^{\left( {\delta +3\tilde{c}(x_3^s -x_1^s )} \right) t}}{(1+e^{\tilde{c}(x_3^s -x_1^s )t})^{6}}\mathrm{d}t}\nonumber \\= & {} \frac{\pi \delta }{5!\tilde{c}^{6}(x_3^s -x_1^s )^{6}\sin \frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}}\left( {\delta ^{2}-\tilde{c}^{2}(x_3^s -x_1^s )^{2}} \right) \nonumber \\&\times \left( {\delta -2\tilde{c}^{2}(x_3^s -x_1^s )} \right) \left( {\delta +2\tilde{c}^{2}(x_3^s -x_1^s )} \right) \end{aligned}$$

(41)

$$\begin{aligned} I_{3,3}= & {} \int \limits _{-\infty }^{+\infty } {\frac{e^{\left( {\delta +4\tilde{c}(x_3^s -x_1^s )} \right) t}}{(1+e^{\tilde{c}(x_3^s -x_1^s )t})^{6}}\mathrm{d}t} \nonumber \\= & {} \frac{\pi \delta }{5!\tilde{c}^{6}(x_3^s -x_1^s )^{6}\sin \frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}}\left( {\delta ^{2}-\tilde{c}^{2}(x_3^s -x_1^s )^{2}} \right) \nonumber \\&\times \,\left( {\delta +2\tilde{c}^{2}(x_3^s -x_1^s )} \right) \left( {\delta +3\tilde{c}^{2}(x_3^s -x_1^s )} \right) \end{aligned}$$

(42)

Substituting Eqs. (40–42) in Eq. (39) results in

$$\begin{aligned} I_3= & {} \frac{\pi \delta }{5!\tilde{c}^{4}(x_3^s -x_1^s )^{2}\sin \frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}}\left( {\delta ^{2}-\tilde{c}^{2}(x_3^s -x_1^s )^{2}} \right) \nonumber \\&\times \,\left\{ {(x_3^s )^{2}\left( {\delta -2\tilde{c}(x_3^s -x_1^s )} \right) \left( {\delta -3\tilde{c}(x_3^s -x_1^s )} \right) } \right. \nonumber \\&\left. +\,2x_1^s x_3^s \left( {\delta -2\tilde{c}(x_3^s -x_1^s )} \right) \left( {\delta +2\tilde{c}(x_3^s -x_1^s )} \right) \right. \nonumber \\&\left. +\,(x_1^s )^{2}\left( {\delta +2\tilde{c}(x_3^s -x_1^s )} \right) \left( {\delta +3\tilde{c}(x_3^s -x_1^s )} \right) \right\} \nonumber \\ \end{aligned}$$

(43)

In order to find $I_{4}$, one should calculate the below integral

$$\begin{aligned} I_4= & {} \int \limits _{-\infty }^{+\infty } {\dot{x}(t)e^{\delta t}} \mathrm{d}t\nonumber \\ \!= & {} \!\int \limits _{-\infty }^{+\infty }\! {\left( {-\tilde{c}(x_3^s \!-\!x_1^s )^{2}\frac{e^{\tilde{c}(x_3^s -x_1^s )t}}{(1+e^{\tilde{c}(x_3^s -x_1^s )t})^{2}}} \right) e^{\delta t}} \mathrm{d}t \end{aligned}$$

(44)

Multiplying the term $e^{\delta t}$ in the parenthesis and setting $a=\tilde{c}(x_3^s -x_1^s ),\;b=\tilde{c}(x_3^s -x_1^s )+\delta$, gives

$$\begin{aligned} I_4= & {} -\tilde{c}(x_3^s -x_1^s )^{2}\left( {\frac{\pi \delta }{\tilde{c}^{2}(x_3^s -x_1^s )^{2}\sin \frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}}} \right) \nonumber \\= & {} \frac{-\pi \delta }{\tilde{c}\sin \frac{\pi \delta }{\tilde{c}(x_3^s -x_1^s )}} \end{aligned}$$

(45)