Synchronization of temperature oscillations in heated plates with hysteretic oneoff control Miguel A. Barron a, * , Mihir Sen b a Departamento de Materiales, Universidad Autonoma Metropolitana-Azcapotzalco, Av. San Pablo 180, Mexico D.F., 02200, Mexico b Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA highlights  We model the thermal synchronization of two coupled plates with oneoff control.  We determine the self-oscillation frequency of an uncoupled controlled plate.  Frequency and phase locking arise when coupled plates share the deadband width.  Differences in the deadband width between plates cause frequency detuning.  A detuning window for locked synchronization is found for small values of detuning. article info Article history: Received 20 September 2013 Accepted 17 January 2014 Available online 28 January 2014 Keywords: Coupled behavior Coupled plates Frequency detuning Heated plate Self-oscillation frequency Oneoff control Temperature oscillations abstract The synchronized behavior of two coupled, top heated, square plates with oneoff control based on plate temperature is analyzed in this work. Each plate is represented by a two dimensional heat equation, and thermal communication between the plates is modeled by a thermal resistance; each plate also has an internal point at which the temperature is monitored for control. The problem is nonlinear, and nu- merical simulations are used to determine the long-time dynamic response of the system. As a first step the self-oscillation frequency of a single uncoupled controlled plate is determined as function of the deadband width of the controller. Then the dynamics of the coupled plates is analyzed, and the effect of the thermal resistance and the deadband width on synchronization of temperature oscillations is studied. Like in other complex systems with synchronization, a detuning window is found over which there is synchronization and beyond which the plates have different oscillation frequencies. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction Among the large variety of available control schemes, oneoff control with hysteresis (also called thermostatic, bangebang, or two-position control) is the simplest tool to regulate the dynamic performance of domestic and industrial thermal-fluid systems such as air-conditioners, water heaters, furnaces, and level controls [1e 3]. In open-loop operation, i.e. in the absence of control, these systems behave in a non-oscillatory manner, but exhibit oscillatory dynamics under control. The amplitude and frequency of closed- loop temperature oscillations depend on the width of the dead- band. When these systems are coupled, closed-loop oscillations interact and propagate through the ensemble. There have been few publications on the coupled behavior of interconnected systems with self-oscillations induced by control- lers. Among them can be cited the synchronization of oscillations in a thermal-hydraulic network [4]. Frequency locking, phase syn- chronization as well as phase slips are observed to occur due to thermal-hydraulic coupling between the controllers. In a later pa- per, the collective dynamics of a number of individually heated thermostatically controlled rooms arranged in the form of a ring was analyzed [5]. It was reported that the coupled system shows the presence of a rich array of synchronization dynamics in fre- quency and phase as well as clustering and coupling-induced amplitude death. The effect of walls on synchronization of ther- mostatic room-temperature oscillations has also been looked at [6]. Interconnections in the form of simple geometrical configurations, such as rings, can be studied if the self-oscillations are governed by ordinary differential equations [7]. The study of synchronization in coupled self-oscillation of systems governed by partial differential * Corresponding author. E-mail address: bmma@correo.azc.uam.mx (M.A. Barron). Contents lists available at ScienceDirect Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng 1359-4311/$ e see front matter Ó 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2014.01.026 Applied Thermal Engineering 65 (2014) 337e342