Abstract
In the previous analysis, the dynamic behaviour of a nonlinear plate embedded into a fractional derivative viscoelastic medium has been studied by the method of multiple time scales under the conditions of the internal resonances two-to-one and one-to-one, as well as the internal combinational resonances for the case when the linear parts of nonlinear equations of motion occur to be coupled. A new approach proposed in this paper allows one to uncouple the linear parts of equations of motion of the plate, while the same method, the method of multiple time scales, has been utilized for solving nonlinear equations. The influence of viscosity on the energy exchange mechanism between interacting nonlinear modes has been analyzed. It has been shown that for some internal resonances there exist such particular cases when it is possible to obtain two first integrals, namely, the energy integral and the stream function, which allows one to reduce the problem to the calculation of elliptic integrals. The new approach enables one to solve the problems of vibrations of thin bodies more efficiently.
Highlights
It is well known that the nonlinear vibrations of plates are an important area of applied mechanics, since plates are used as structural elements in many fields of industry and technology
Nonlinear vibrations could be accompanied by such a phenomenon as the internal resonance, resulting in multimode response with a strong interaction of the modes involved [14] accompanied by the energy exchange phenomenon
Let us consider the dynamic behavior of a free supported nonlinear thin rectangular plate (Figure 1), vibrations of which in a viscoelastic medium are described in the Cartesian system of coordinates by the following three differential equations written in the dimensionless form [2, 3, 21]: uxx β22 12
Summary
It is well known that the nonlinear vibrations of plates are an important area of applied mechanics, since plates are used as structural elements in many fields of industry and technology. For example, the experimental data obtained by Abdel-Ghaffar and Housner [17] and Abdel-Ghaffar and Scanlan [18] during ambient vibration studies of the Vincent-Thomas Suspension Bridge and the Golden Gate Bridge, respectively, show that different vibrational modes feature different amplitude damping factors, and the order of smallness of these coefficients highlights the low damping capacity of suspension combined systems, resulting in prolonged energy transfer from one partial subsystem to another. To lead the theoretical investigations in line with the experiment, fractional derivatives were introduced by Rossikhin and Shitikova [19] for describing the processes of internal friction proceeding in suspension combined systems at nonlinear free vibrations, which allowed the authors to obtain the damping coefficients dependent on the natural frequency of vibrations. The good agreement between the theoretical results and the experimental data has been found through the appropriate choice of the fractional parameter (the order of the fractional derivative) and the viscosity coefficient [20]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
