Insights on the continuous representations of piecewise-smooth nonlinear systems: limits of applicability and effectiveness

• Published in: Nonlinear dynamics Vol. 107; no. 2; pp. 1479 - 1494
• Main Authors: Saunders, B. E., Vasconcellos, R., Kuether, R. J., Abdelkefi, A.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 2022; Springer Nature B.V
• Subjects: Aerospace engineering; Algorithms; Automotive Engineering; Behavior; Classical Mechanics; Contact force; Continuity (mathematics); Control; Costs; Duffing oscillators; Dynamical Systems; Engineering; Functions (mathematics); Mechanical Engineering; Nonlinear systems; Nonlinearity; Numerical analysis; Original Paper; Polynomials; Representations; System effectiveness; Time integration; Vibration; Nonlinear dynamics; Freeplay nonlinearity; Chaotic responses; Piecewise-smooth representation
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-021-06436-w

Dynamical systems subject to intermittent contact are often modeled with piecewise-smooth contact forces. However, the discontinuous nature of the contact can cause inaccuracies in numerical results or failure in numerical solvers. Representing the piecewise contact force with a continuous and smooth function can mitigate these problems, but not all continuous representations may be appropriate for this use. In this work, five representations used by previous researchers (polynomial, rational polynomial, hyperbolic tangent, arctangent, and logarithm-arctangent functions) are studied to determine which ones most accurately capture nonlinear behaviors including super- and subharmonic resonances, multiple solutions, and chaos. The test case is a single-DOF forced Duffing oscillator with freeplay nonlinearity, solved using direct time integration. This work intends to expand on past studies by determining the limits of applicability for each representation and what numerical problems may occur.