Chaotic dynamics of flexible Euler-Bernoulli beams.

Area of Science:

• Mechanical Engineering
• Nonlinear Dynamics
• Applied Mathematics

Background:

• Euler-Bernoulli beam theory is fundamental for structural analysis.
• Understanding chaotic dynamics in flexible structures is crucial for predicting complex behaviors.
• Geometric nonlinearities, like Kármán-type, significantly influence beam dynamics.

Purpose of the Study:

• To mathematically model and analyze spatio-temporal chaotic dynamics in flexible simple and curved Euler-Bernoulli beams.
• To investigate novel transition scenarios from periodic to chaotic and hyper-chaotic states.
• To explain the transition from symmetric to asymmetric vibrations and the development of temporal-space chaos.

Main Methods:

• Reduction of partial differential equations to Cauchy problems using Finite Difference Method (O(c(2)) approximation) and Finite Element Method.
• Numerical solution of the Cauchy problem using fourth and sixth-order Runge-Kutta methods.
• Analysis of chaotic dynamics using qualitative theory of differential equations, including phase portraits, Lyapunov exponents, and Poincaré maps.

Main Results:

• A novel scenario for the transition from periodicity to chaos was identified.
• A transition from chaos to hyper-chaos was demonstrated.
• The phenomenon of transition from symmetric to asymmetric vibrations was studied and explained.
• Vibration-type charts were generated based on excitation amplitude and frequency.
• The development of temporal-space chaos was detected and illustrated.

Conclusions:

• The study provides a comprehensive analysis of chaotic dynamics in flexible Euler-Bernoulli beams under nonlinear conditions.
• New insights into transition mechanisms and the emergence of complex spatio-temporal chaos were obtained.
• The findings contribute to a deeper understanding of nonlinear structural dynamics and chaos theory.