Nonlinear dynamic behavior and bifurcation analysis of a rotating viscoelastic size-dependent beam based on non-classical theories


In this paper, the free and harmonically forced vibrations of a viscoelastic rotating microbeam have been analyzed by exploiting non-classical theories, such as the Modified Couple Stress Theory and Modified Strain Gradient Theory (MSGT). A continuous model is derived and reduced to a well-known Duffing–Rayleigh system via a Galerkin projection; then, the Multiple Scales Method is used to obtain amplitude equations. Free and forced nonlinear oscillations are studied via classical and non-classical theories, and their corresponding results are compared with each other. An expression for natural frequency is obtained in which the effects of the internal damping and small-scale parameters are obvious. It is observed that the natural frequency obtained using the MSGT has the highest value, while the Classical Theory (CT) predicts the lowest natural frequency value. Also, it is seen that by increasing the hub radius, the influence of rotational speed on natural frequency is increased. According to obtained results, the responses corresponding to the CT become completely stable for the lowest value of the excitation force and the highest value of the external damping coefficient. For MSGT, responses are stable when the values of the excitation force and external damping coefficients are highest and lowest, respectively. It is shown that for the MSGT, the system is stable when the internal damping coefficient is small, while for the CT, the beam is stable for the highest value of the internal damping coefficient. The system parameters—such as the detuning parameter, damping coefficients, and excitation force—affect the solution, such a way that the system may possess one or two stable spirals depending on the values of those parameters, as shown in phase plane portraits.

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$$ K_{1} = \frac{{\lambda_{2} \left( {\mu_{i} + \theta_{3} } \right) - \lambda_{6} \theta_{5} + \lambda_{1} \mu_{e} }}{{\lambda_{1} }} $$
$$ K_{2} = \frac{{\lambda_{2} \left( {1 + \theta_{2} } \right) - \lambda_{3} \varOmega^{2} - \lambda_{6} \theta_{4} }}{{\lambda_{1} }} $$
$$ K_{3} = \frac{{\lambda_{4} \left( {1 + \theta_{2} } \right) - \lambda_{7} \theta_{4} }}{{\lambda_{1} }} $$
$$ K_{4} = \frac{{\lambda_{5} \left( {\mu_{i} + \theta_{3} } \right) - \lambda_{8} \theta_{5} }}{{\lambda_{1} }} $$
$$ K_{5} = \frac{{\lambda_{9} }}{{2\lambda_{1} }} $$
$$ K_{6} = \frac{F}{{2\lambda_{1} }} $$
$$ \lambda_{1} = \mathop \int \limits_{0}^{1} \left[ {\phi \left( x \right)} \right]^{2} {\text{d}}x $$
$$ \lambda_{2} = \mathop \int \limits_{0}^{1} \phi \left( x \right)\phi^{\left( 4 \right)} \left( x \right){\text{d}}x $$
$$ \lambda_{3} = \mathop \int \limits_{0}^{1} \left[ {\beta \left( {1 - x} \right) + \frac{1}{2}\left( {1 - x^{2} } \right)} \right]\phi \left( x \right)\phi^{\prime\prime}\left( x \right){\text{d}}x - \mathop \int \limits_{0}^{1} \left( {d + x} \right)\phi \left( x \right)\phi^{\prime}\left( x \right){\text{d}}x $$
$$ \lambda_{4} = \mathop \int \limits_{0}^{1} \phi \left( x \right)\left[ {\phi^{\prime}\left( x \right)\left( {\phi^{\prime}\left( x \right)\phi^{\prime\prime}\left( x \right)} \right)^{'} } \right]^{'} {\text{d}}x $$
$$ \lambda_{5} = \mathop \int \limits_{0}^{1} \phi \left( x \right)\left[ {3\phi^{\prime}\left( x \right)\phi^{\prime\prime}\left( x \right)^{2} + 2\phi^{\prime}\left( x \right)^{2} \phi^{\prime\prime\prime}\left( x \right)} \right]^{'} {\text{d}}x $$
$$ \lambda_{6} = \mathop \int \limits_{0}^{1} \phi \left( x \right)\phi^{\left( 6 \right)} \left( x \right){\text{d}}x $$
$$ \lambda_{7} = \mathop \int \limits_{0}^{1} \phi \left( x \right)\left[ {6\phi^{\prime\prime\prime}\left( x \right)\phi^{\prime\prime}\left( x \right)^{2} + 3\phi^{\prime\prime\prime}\left( x \right)^{2} \phi^{\prime}\left( x \right) + 4\phi^{\prime}\left( x \right)\phi^{\prime\prime}\left( x \right)\phi^{\left( 4 \right)} \left( x \right)} \right]^{'} {\text{d}}x $$
$$ \lambda_{8} = \mathop \int \limits_{0}^{1} \phi \left( x \right)\left[ {18\phi^{\prime\prime\prime}\left( x \right)\phi^{\prime\prime}\left( x \right)^{2} + 9\phi^{\prime\prime\prime}\left( x \right)^{2} \phi^{\prime}\left( x \right) + 12\phi^{\prime}\left( x \right)\phi^{\prime\prime}\left( x \right)\phi^{\left( 4 \right)} \left( x \right) + 2\phi^{\prime}\left( x \right)^{2} \phi^{\left( 5 \right)} \left( x \right)} \right]^{'} {\text{d}}x $$
$$ \lambda_{9} = \mathop \int \limits_{0}^{1} \phi \left( x \right)\left( {\phi^{\prime}\left( x \right)\mathop \int \limits_{1}^{x} \mathop \int \limits_{0}^{x} \phi^{\prime}\left( x \right)^{2} {\text{d}}x{\text{d}}x} \right)^{'} {\text{d}}x $$

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Shahgholi, M., Ghasabi, S.A. Nonlinear dynamic behavior and bifurcation analysis of a rotating viscoelastic size-dependent beam based on non-classical theories. Eur. Phys. J. Plus 135, 944 (2020).

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