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DINAME 2017
XVII International Symposium on Dynamic Problems of Mechanics
CHAOS CONTROL OF AN SMA-PENDULUM SYSTEM
Submission Author:
Dimitri Danulussi Alves Costa , SP
Co-Authors:
Dimitri Danulussi Alves Costa, Marcelo Savi
Presenter: Dimitri Danulussi Alves Costa
doi://10.26678/ABCM.DINAME2017.DIN17-0077
Abstract
XVII International Symposium on Dynamic Problems of Mechanics - DINAME 2017 March 5th to 10th, 2017 Beach Hotel Cambury & Sunset, São Sebastião, SP, Brazil Dimitri D. A. Costa 1, Marcelo A. Savi 2 Universidade Federal do Rio de Janeiro, COPPE - Departament of Mechanical Engineering, 21.945.972, Rio de Janeiro - RJ, Brazil 1dda.costa@mecanica.coppe.ufrj.br 2savi@mecanica.ufrj.br CHAOS CONTROL OF AN SMA-PENDULUM SYSTEM Preferred area/sub-area of the symposium: Nonlinear dynamics ABSTRACT Pendulum systems are extensively studied due to a variety of reasons, being a source of inspiration for several discoveries in nonlinear dynamics and engineering. Studies show several complex responses that include bifurcations, chaos and transient chaos on those systems. Pendulum-like systems are also employed to model biological systems motors systems, cranes and their swings among others. Chaos is present on a variety of systems including non-smooth mechanical systems, smart material systems, machining, electronics, lasers, among others. To get advantages of these behaviours and the embedded unstable periodic orbits (OPIs) present on this kind of responses chaos control methods where proposed. These controllers greatly reduce the energy cost to stabilise these systems on periodic responses. Initially a discrete control was proposed by Ott et al. (1990) which was followed by a continuous controller proposal using time delayed feedback (Pyragas, 1992). After these two main works numerous modifications were proposed blooming the field. Recently, Pyragas (2001) uses a unstable time delayed feedback controller to stabilise OPIs with odd number of positive Floquet multipliers, eliminating the restrictions, noted by Ushio (1996) and further investigated by Hooton and Amann (2012), of previous controllers. Various applications of chaos control were analysed in electronics, energy harvesting, mechanical systems, satellite orbits and others. As new controllers are prosed there is a need to investigate its applications and adaptability to control restrictions as asymmetries on the controller’s rate of change. Shape memory alloys (SMAs) belong to the smart materials family presenting interesting properties for applied dynamics being applied in vibration control, origami structures, robotics and energy harvesting, showing a broad use and potential for a variety of applications. Their unique behaviour is due to its thermomechanical coupling in phase transformations that imply some of their traditional effects as shape memory and pseudoelasticity. These remarkable properties can be used for control purposes in robotics to create rotorless motions and mimic biological movements. This work deals with the nonlinear dynamics of an SMA-system composed of a nonlinear pendulum coupled with SMA springs. This system presents chaotic responses and adaptive behaviour, presenting a temperature dependent response. Dynamical investigation is carried out and the system is used to analyse chaos control methods, showing a potential application of SMA in this type of control. The use of thermal loads on the SMA-springs to control the system is proposed considering asymmetric heat-cooling speeds on a zero dimensional model and verification of this control is numerically performed. Figure 1 shows a preliminary result for control purposes. It illustrates a situation where temperature variations are employed to change the global system response. It is built by initializing the system at x_0=(-6 rad,0 rad/s), with initial temperature T_0=286.65 K. In these conditions the system presents a chaotic response (attractor Poincaré section on Figure 1b). The temperature is maintained for 600 cycles and afterwards increased at a constant rate of 0.1 K/s until it reaches T=288.15 K, pushing the solution to a periodic orbit. The temperature is maintained for 400 cycles and them is decreased at the same rate to its initial value returning the system to a chaotic response. Figure 2 also has a preliminary result displaying the stabilization of a 6-period orbit utilizing the time delay feedback controller (Pyragas, 1992). The actuator in this case can produce any signal value. Figure 1: Control strategy based on temperature change. (a) Chaotic-periodic-chaotic cycle. (b) Poincaré section of chaotic and periodic responses. Figure 2: Time delay feedback control on a chaotic region. a) System response against time. b) Stabilized orbit. Keywords: Nonlinear Dynamics analysis, Chaos Control, Shape Memory Alloys. References: Hooton, E.W., Amann, A., 2012. Analytical Limitation for Time-Delayed Feedback Control in Autonomous Systems. Phys. Rev. Lett. 109. doi:10.1103/PhysRevLett.109.154101 Ott, E., Grebogi, C., Yorke, J.A., 1990. Controlling chaos. Phys. Rev. Lett. 64, 1196–1199. Pyragas, K., 2001. Control of Chaos via an Unstable Delayed Feedback Controller. Phys. Rev. Lett. 86, 2265–2268. doi:10.1103/PhysRevLett.86.2265 Pyragas, K., 1992. Continuous Control of Chaos by Self-Controlling Feedback. Phys. Lett. A 170, 421–428. Ushio, T., 1996. Limitation of de layed feedback control in nonlinear discrete-time systems. IEEE Trans. Circuits Syst. - Fundam. Theory Appl. 43, 815–816.
Keywords
nonlinear dynamics, Chaos Control, Shape Memory Alloys
