We study a mathematical model for unforced and undamped, initially straight beams. This system is governed by an integro-partial differential equation, and its energy is conserved: It is an infinite-degree-of-freedom Hamiltonian system. We can derive “exact” finite-degree-of-freedom mode truncations for it. Using the differential Galois theory for Hamiltonian systems, we prove that any two or more modal truncations for the model are nonintegrable in the following sense: The Hamiltonian systems do not have the same number of “meromorphic” first complex integrals which are independent and in involution, as the number of their degrees of freedom, when they are regarded as Hamiltonian systems with complex time and coordinates. This also means the nonintegrability of the infinite-degree-of-freedom model for the beams. We present numerical simulation results and observe that chaotic motions occur as in typical nonintegrable Hamiltonian systems.

Issue Section:
Technical Papers
Keywords:
vibrations, chaos, modelling, partial differential equations, integro-differential equations, simulation, eigenvalues and eigenfunctions, Galois fields
Topics:
Algebra, Chaos, Computer simulation, Degrees of freedom, Differential equations, Eigenfunctions, Eigenvalues, Vibration, Trajectories (Physics), Modeling, Partial differential equations, Simulation, Computation, Dynamics (Mechanics)
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