Ing. Krištof Pučejdl

In this paper, we present a laboratory mechatronic platform for experimental demonstration and verification of various dynamical and control system phenomena exhibited by the Frenkel-Kontorova (FK) model – a spatially discretized version of the sine-Gordon equation. The platform consists of an array of spring-coupled pendulums pivoting around a single axis with the first and the last pendulums controlled by motors and all pendulums’ angles measured electronically. We first introduce and describe the platform, providing details of its mechatronic design and software architecture. All the design files are freely shared with the research community under an open-source license through a public repository. The platform can be used as a testbed for various control algorithms, e.g., for distributed control or control of flexible structures. In the second part of the paper, we showcase the platform using two control problem formulations tailored to the FK model. We discuss the practical motivation for studying these problems, propose methods for solving them, and experimentally demonstrate their functionality. In particular, the first control formulation deals with non-collocated stabilization of a single pendulum through one boundary of the array in the presence of a disturbance sent from the other boundary; the second control problem consists in synchronizing pendulums’ angular speeds. Other problems can certainly be formulated, solved, and demonstrated using the proposed platform whether for research or education purposes.

Rijke tube is a popular physical experiment demonstrating a spontaneous generation of sound in an open vertical pipe with a heat source. This laboratory instance of thermoacoustic instability has been researched due to its significance in practical thermoacoustic systems. We revisit the experiment, focusing on the possible use of the Rijke tube as a musical instrument. The novelty presented in this paper consists in shifting the goal from sound suppression to active sound generation. This called for modifying the previously investigated methods for stabilization of thermoacoustic oscillations into their excitation and control of their amplitude. On our way towards this goal, we developed a time-domain mathematical model that considers the nonlinear and time-varying aspects of the Rijke tube. The model extends the existing modeling and analysis approaches, which are mainly based in frequency domain. We also present an extension of the basic laboratory setup in the form of an array of Rijke tubes equipped with a single speaker used to control multiple Rijke tubes with different natural frequencies simultaneously.