Mathematical modelling of nonlinear oscillations of a biodynamical system in the form of a complex cantilever

Abstract

To ensure the stability of a young seedling of a tree and proper rooting, tree stems are usually staked with one, two or three stakes. In this paper, we present a mathematical model and its approximations of a nonlinear biodynamical system in the form of a complex discrete structure on a cantilever. The nonlinear biodynamical system corresponds to a staked tree with one stake. The geometric nonlinearity of the system is introduced by a spring with cubic nonlinear properties that oscillates in the vertical plane. The influence coefficients of deflection of the cantilever were used for describing single-frequency forced oscillations of a complex biodynamical system under an external single-frequency force. This external single-frequency force has a circular frequency that is close to frequencies from a set of eigen circular frequencies of a corresponding linearized system. The system oscillates in two orthogonal planes - the horizontal and the vertical plane. Forced oscillations, in the vertical direction, of this discrete complex system, are described by subsystems of nonlinear differential equations that are solved in the first approximation by using two methods: the method of variation of constants in combination with the method of averaging and the other extended asymptotic method of nonlinear mechanics Krylov- Bogoliubov- Mitropolsky. The results of the qualitative analysis of the solution of the system of nonlinear differential equations by amplitudes and phases of four nonlinear modes are presented and discussed. In both cases, the frequencies of external force are within the range of resonant frequencies, but constant in time in the case when the generalized method of variation of constants with the method of averaging was used. In the case when the Krylov-Bogoliubov- Mitropolsky asymptotic method of nonlinear mechanics of approximation was used, the frequencies of external force have changeable values within time- frequency changes with different speeds. This system is simpler than a system of governing differential equations along amplitudes and phases of generalized coordinates, and permits a qualitative analysis of nonlinear phenomena in systems with nonlinear dynamics. The method described in this paper is suitable for studying the stability and instability of amplitudes and of phases of nonlinear modes in the first approximation. The asymptotic method of nonlinear mechanics Krylov-Bogoliubov- Mitropolsky allows the analysis of stationary discrete modes of nonlinear small oscillations in the resonant range as well as of non-stationary nonlinear modes. This method can be applied to studying nonlinear dynamics of more complex systems based on cantilevers with complex structure. The most valuable and important elements of the work are emphasized.