Abstract
Applications of a novel time-integration technique to the non-linear and linear dynamics of mechanical structures are presented, using an extended Picard-type iteration. Explicit discrete-mechanics approximations are taken as starting guess for the iteration. Iteration and necessary symbolic operations need to be performed only before time-stepping procedure starts. In a previous investigation, we demonstrated computational advantages for free vibrations of a hanging pendulum. In the present paper, we first study forced non-linear vibrations of a tower-like mechanical structure, modeled by a standing pendulum with a non-linear restoring moment, due to harmonic excitation in primary parametric vertical resonance, and due to excitation recordings from a real earthquake. Our technique is realized in the symbolic computer languages Mathematica and Maple, and outcomes are successfully compared against the numerical time-integration tool NDSolve of Mathematica. For out method, substantially smaller computation times, smaller also than the real observation time, are found on a standard computer. We finally present the application to free vibrations of a hanging double pendulum. Excellent accuracy with respect to the exact solution is found for comparatively large observation periods.
Highlights
The need for novel computationally efficient time-integration schemes is increasingly invoked by online and real-time applications of non-linear dynamics of mechanical structures
A novel efficient time-integration scheme recently derived by the present authors, see [11], is applied for simulating non-linear forced and free vibrations of mechanical structures, and its advantages against standard numerical time-integration methods, such as NDSolve, available in Wolfram Mathematica [12], are exemplarily demonstrated
In our previous investigation [11], computational advantages have been demonstrated for large free vibrations of a hanging rigid pendulum, for which exact solutions do exist, and which is used in the literature as a benchmark example for comparison with numerical time integration methods
Summary
The need for novel computationally efficient time-integration schemes is increasingly invoked by online and real-time applications of non-linear dynamics of mechanical structures. In our previous investigation [11], computational advantages have been demonstrated for large free vibrations of a hanging rigid pendulum, for which exact solutions do exist, and which is used in the literature as a benchmark example for comparison with numerical time integration methods. In the last part of the paper, as a first step towards applications to non-linear multi-body dynamic systems, we present the application to free vibrations of a hanging double pendulum, for which exact solutions do exist In this example, two explicit discrete-mechanics type solutions originally suggested by Greenspan [15] are used and compared as starting guesses in the first. Sci. 2021, 11, 3742 a comparatively large observation period, which proves that the method can be applied with high advantage for linear vibrations
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