ISSN: 2475-4811 (print)
ISSN: 2475-482X (online)
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland

Email: pawel.olejnik@p.lodz.pl

C. Steve Suh (editor)

Texas A&M University, USA

Email: ssuh@tamu.edu

Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China

Email: tuoxianguo@suse.edu.cn

Smoothing Transforms for Dynamical Systems with Non-smooth Input

Journal of Vibration Testing and System Dynamics 5(2) (2021) 113--120 | DOI:10.5890/JVTSD.2021.06.001

Carmen Chicone, Z. C. Feng

Department of Mathematics, University of Missouri, Columbia, MO 65211, USA Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA

Abstract

Many control systems incorporate input from actuators. The actuators can be represented as force or displacement input. When an actuator is modeled as a displacement input, there are many practical applications for which the displacement is not smooth and its time derivative, i.e. the velocity, is not continuous everywhere. Numerical study for such displacement input becomes tedious since the state variables experience a finite jump at the time of acceleration singularity. Although the acceleration singularity can be avoided if the actuator dynamics is modeled by a force input, such an approach increases the order of the overall system. To preserve the simplicity of the displacement input of the actuators, we propose transforms of state variables so that only the velocity of the displacement input appears explicitly in the equations of the dynamical system. In short, we introduce transforms that reduce the order of time derivatives of the input. This approach simplifies the numerical solutions for non-smooth displacement input. Moreover, when the transform is applied to a parametrically excited pendulum, the stabilizing effect of the parametric excitation is shown explicitly.

References

1.  [1] Luo, A.C.J. (2011), Regularity and Complexity in Dynamical Systems, Springer: New York.
2.  [2] Singhose, W., Singer, N., Seering, W. (1995), Comparison of command shaping methods for reducing residual vibration, Proceedings of 3rd European Control Conference, Rome, Italy, September 1995. 1126-1131.
3.  [3] Conker, C., Yavuz, H. and Bilgic, H.H. (2016), A review of command shaping techniques for elimination of residual vibrations in flexible-joint manipulators, Journal of Vibroengineering, textbf{18} (5), 2947-2958.
4.  [4] Nayfeh, A.H. and Mook, D.T. (1979), Nonlinear Oscillation, John Wiley: New York.
5.  [5] Acheson, D.J. (1993), A pendulum theorem, Proceedings of Royal Society of London A, 443, 239-245
6.  [6] Acheson, D.J. (2009) Surprises in Maths: https://www.youtube.com/watch?v=BJ7\_fFABc9s.
7.  [7] Chicone, C . (2006)Ordinary Differential Equations with Applications, 2nd Ed. Springer-Verlag; New York.
8.  [8] Depetri, G.I., Pereira, F.A.C., Marin, B., Baptista, M.S., and Sartorelli, J.C. (2018), Dynamics of a parametrically excited simple pendulum, Chaos: An Interdisciplinary Journal of Nonlinear Science, 23(3), 033103.
9.  [9] Kapitza P.L. (1951), Dynamic stability of a pendulum when its point of suspension vibrates, Soviet Phys. JETP. 21, 588--597.
10.  [10] Kapitza P.L. (1951), Pendulum with a vibrating suspension, Usp. Fiz. Nauk. 44 7--15.
11.  [11] Levi, M. (1988) Stability of the inverted pendulum-a topological explanation, SIAM Review 30 (4) 639--644.
12.  [12] Goharoodi, S.K, Dekemele, K., Dupre, L., Loccufier, M., and Crevecoeur, G. (2020), Sparse identification of nonlinear Duffing oscillator from measurement data, https://arxiv.org/pdf/1805.03499.pdf