Stability of rocking, impacting, and slipping systems
The aim of the project is to analyze the dynamics of mechanical systems modelled as rigid components in contact with one another. These problems include the earthquake design of rocking structures or the locomotion design of walking robots. The research focuses on stability analysis and active stabilization of equilibria or motion patterns on various engineering applications Contact-induced interactions (friction, impacts) give rise to complex dynamical behavior, which makes traditional methods of local stability analysis inapplicable. Stability against large perturbations is an even tougher challenge, and thumb rules or crude approximations are in common use in engineering practice. The task of the student is to develop new stability conditions, and design methodologies, and to apply them to engineering problems.
- Posa, Michael, Mark Tobenkin, and Russ Tedrake. "Stability analysis and control of rigid-body systems with impacts and friction." IEEE Transactions on Automatic Control 61.6 (2016): 1423-1437.
- Di Egidio, Angelo, Daniele Zulli, and Alessandro Contento. "Comparison between the seismic response of 2D and 3D models of rigid blocks." Earthquake Engineering and Engineering Vibration 13.1 (2014): 151-162.
- Dimitrakopoulos, Elias G., and Matthew J. DeJong. "Revisiting the rocking block: closed-form solutions and similarity laws." Proc. R. Soc. A. Vol. 468. No. 2144. The Royal Society, 2012.
- Leine, R. I., and N. Van De Wouw. "Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact." Nonlinear Dynamics 51.4 (2008): 551-583.
- Pang, J‐S., and J. Trinkle. "Stability characterizations of rigid body contact problems with coulomb friction." ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 80.10 (2000): 643-663.
- Nonlinear Dynamics
- J. Nonlinear Science
- International J. Solids and Structures
- International J. Robotics Research
- IEEE Transactions in Robotics
- IEEE Transctions in Automation Science and Engineering
The aim of the research program is to examine mechanical systems, in which friction and impacts play a significant role. The two most important questions concerning such systems are: how to model contact induced phenomena, and how to describe the dynamic behavior induced by these phenomena. This proposal focuses on the latter question.
Simple models of friction and impacts have been known for centuries. The concept of the "coefficient of restitution" was introduced by Newton and even earlier by Leonardo da Vinci. The most popular model of dry friction was worked out by physicists of 17th-18th centuries (C-A. de Coulomb and G. Amontons) [5]. Despite their simplicity, such models induce complex dynamic behavior. Dry friction is inherently non-smooth, and impacts induce hybrid dynamics with continuous and instantaneous components.
Typical phenomena include non-smooth transitions (stick-slip and slip-stick), instantaneous jumps (impact), infinitely many switches within finite time intervals (Zeno behavior [17]), singularities, such as contact forces diverging to infinity[4], and non-uniqueness or apparent non-existence of a forward solution (Painlevé paradoxes [1]). these special properties call for special methods of analysis, and may pose limitations to the application of physical models based on rigid body theory [13]. At the same time, such problems have numerous applications in mechanical engineering, robotics, and structural dynamics.
We pose two principal research questions. We seek conditions of Lyapunov stability of equilibrium configurations, and we examine the behavior of systems under large perturbations. The latter is applied to earthquake design of structures.
Local stability
"Lyapunov stability" is a property that involves appropriate behavior of a system in the presence of sufficiently small perturbations. This concept is widely used in robotics -, [15], -. At the same time, there is no general strategy for testing Lypaunov stability in the presence of contacts. Engineers tend to use a priori conditions instead, which may lead to unexpected destabilization -.
This fact motivated me to analyze a model system, and to develop conditions of Lyapunov stability [10] [16] [15]. The task of a PhD student will be to make these conditions applicable to a wider class of systems. He/she could use an automated stability test recently developed by a research group at MIT -, which is theoretically applicable to any system, but in its present form does not give a definite answer in many important cases.
Non-local stability and earthquake resistance
Rocking structures have been known for long to be highly earthquake-resistant [6]. In contrast to traditional structures, they may lift off the ground and impact again, which absorbs significant amount of kinetic energy - [7] [14]. Earthquakes represent large perturbations for these structures, therefore the question of earthquake resistance goes beyond that of Lyapunov stability.
The behavior of rocking structures can be characterized by "rocking spectra", which show how the response of the structure depends on characteristic properties of an earthquake [9]. The important difference between classical 2D models of rocking, and real 3D structures was not realized until recently -. The goal of the research is to analyze simple models, that can explain the observed rocking spectra of three dimensional structures, and can predict their behavior. We also plan to examine unconventional methods to improve earthquake resistance.
[1] Champneys, Alan R., and Péter L. Várkonyi. "The Painlevé paradox in contact mechanics." IMA Journal of Applied Mathematics 81.3 (2016): 538-588.
[2] Di Egidio, Angelo, Daniele Zulli, and Alessandro Contento. "Comparison between the seismic response of 2D and 3D models of rigid blocks." Earthquake Engineering and Engineering Vibration 13.1 (2014): 151-162.
[3] Dimitrakopoulos, Elias G., and Matthew J. DeJong. "Revisiting the rocking block: closed-form solutions and similarity laws." Proc. R. Soc. A. Vol. 468. No. 2144. The Royal Society, 2012.
[4] Génot, Franck, and Bernard Brogliato (1999). "New results on Painlevé paradoxes". European Journal of Mechanics A. 18 (4): 653–678
[5] Halliday, David, Jearl Walker, and Robert Resnick. Fundamentals of Physics, Chapters 33-37. John Wiley & Sons, 2010.
[6] Housner, George W. "The behavior of inverted pendulum structures during earthquakes." Bulletin of the seismological society of America 53.2 (1963): 403-417.
[7] Koh, Aik-Siong, and Ghulani Mustafa. "Free rocking of cylindrical structures." Journal of engineering mechanics 116.1 (1990): 35-54.
[8] Leine, R. I., and N. Van De Wouw. "Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact." Nonlinear Dynamics 51.4 (2008): 551-583.
[9] Makris, Nicos, and Dimitrios Konstantinidis. "The rocking spectrum and the limitations of practical design methodologies." Earthquake engineering & structural dynamics 32.2 (2003): 265-289.
[10] Or, Yizhar, and Elon Rimon. "On the hybrid dynamics of planar mechanisms supported by frictional contacts. II: Stability of two-contact rigid body postures." Robotics and Automation, 2008. ICRA 2008. IEEE International Conference on. IEEE, 2008.
[11] Pang, J‐S., and J. Trinkle. "Stability characterizations of rigid body contact problems with coulomb friction." ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 80.10 (2000): 643-663.
[12] Posa, Michael, Mark Tobenkin, and Russ Tedrake. "Stability analysis and control of rigid-body systems with impacts and friction." IEEE Transactions on Automatic Control 61.6 (2016): 1423-1437.
[13] Stewart, David E. "Rigid-body dynamics with friction and impact." SIAM review 42.1 (2000): 3-39.
[14] Ther, Tamás, and László P. Kollár. "Refinement of Housner's model on rocking blocks." Bulletin of Earthquake Engineering: 1-15.
[15] Várkonyi, Péter L., and Yizhar Or. "Lyapunov stability of a rigid body with two frictional contacts." Nonlinear dynamics, in press.
[16] Várkonyi, Péter L., David Gontier, and Joel W. Burdick. "On the Lyapunov stability of quasistatic planar biped robots." Robotics and Automation (ICRA), 2012 IEEE International Conference on. IEEE, 2012.
[17] Zhang, Jun, et al. "Zeno hybrid systems." International journal of robust and nonlinear control 11.5 (2001): 435-451.