Relationship between the contact force strength and numerical inaccuracies in piecewise-smooth systems

Approximating piecewise nonlinearities in dynamic systems with sigmoid functions: advantages and limitations

“…To this end, the MATLAB built-in event function [21] is used: this function allows the switch of non-smooth properties during the integration procedure, identifying the exact location of the change. The procedure adopted in this paper is similar to the one adopted in [6,22,24]: for each event occurrence, the time integration stops and the final conditions are used as initial conditions for the next integration step. The numerically integrated ODE is modified accordingly with the position of the first mass, x 1 , as described by Eq.…”

“…Mechanical systems with free-play gaps and piecewise constraints have aroused the interest of the scientific community in the past decades. They find practical engineering applications in the study of impacting capsule systems [1], impact drilling systems [2], cracked systems [3], mechanical oscillators [4][5][6][7] and aeroelastic systems with free-play gaps [8,9], buildings subjected to earthquakes [10], and impact oscillators with rigid walls [11][12][13]. These systems, generally referred to as non-smooth dynam-ical systems, incorporate discrete-events which guide the associated mathematical solution through regions of the domain where different characteristics are applied.…”

“…Although analytical solutions [4,16,17] and mapping techniques [12,15,[18][19][20] exist, numerical methods have been widely adopted in engineering applications. Direct numerical integration has been particularly used in the literature [5,6,[8][9][10]21,22]. It can handle non-smooth properties in two different ways: either locating the exact discontinuity point 2 and restarting the integration process or directly integrating the function over its discontinuity.…”

“…Dai et al [8] studied an aeroelastic system with a free-play gap; they proved that the adoption of the exact event location with Henon's method allows identifying correctly chaos and transient chaos responses that could not have been found without the event location or with rational polynomial approximation. Recently, Saunders et al [6] performed similar studies on a single-degree-of-freedom (SDOF) Duffing system with a free-play gap [25]. They used the MATLAB built-in function and proved that the Basins of Attraction (BOAs), the phase plots, and the transition to the chaos of the system are not strongly affected by errors in the event location, however, poorly converged solutions and basins boundaries modifications have been encountered without the exact event location.…”

“…This clearance introduces a piecewise characteristic in the system which is modelled by a continuous function of class C 0 , as typically happens in engineering applications (e.g. see [6,8,9,26,42,43,45,50,60]). Systems with similar properties fall in the category of the piecewise-smooth continuous systems for which the Picard-Lindelöf Theorem guarantees the uniqueness and existence of the solution in the whole domain.…”