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Dynamic Response of an Inverted Pendulum System in Water Under Parametric Excitations for Energy Harvesting: a Conceptual Approach

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Dynamic Response of an Inverted Pendulum System in Water Under Parametric Excitations for Energy Harvesting: a Conceptual Approach energies Article Dynamic Response of an Inverted Pendulum System in Water under Parametric Excitations for Energy Harvesting: A Conceptual Approach Saqib Hasnain 1, Karam Dad Kallu 2, Muhammad Haq Nawaz 3, Naseem Abbas 4,5,* and Catalin Iulin Pruncu 6,7,* 1 School of Mechanical Engineering, Pusan National University, 30 Jangjeon-dong, Guemjeong-gu, Busan 46241, Korea; [email protected] 2 Robotics and Intelligent Machine Engineering (RIME), School of Mechanical and Manufacturing Engineering (SMME), National University of Science and Technology (NUST), H-12, Islamabad 44000, Pakistan; [email protected] 3 School of Engineering and Information Technology (SEIT), University of New South Wales at Australian Defense Force Academy, Canberra 7916, Australia; [email protected] 4 Department of Mechanical Engineering, University of Central Punjab, Lahore 54000, Pakistan 5 School of Mechanical Engineering, Chung-Ang University, Seoul 06974, Korea 6 Mechanical Engineering, Imperial College London, London SW7 2AZ, UK 7 Design, Manufacturing & Engineering Management, University of Strathclyde, Glasgow G1 1XJ, UK * Correspondence: [email protected] (N.A.); [email protected] (C.I.P.) Received: 19 August 2020; Accepted: 30 September 2020; Published: 7 October 2020 Abstract: In this paper, we have investigated the dynamic response, vibration control technique, and upright stability of an inverted pendulum system in an underwater environment in view point of a conceptual future wave energy harvesting system. The pendulum system is subjected to a parametrically excited input (used as a water wave) at its pivot point in the vertical direction for stabilization purposes. For the first time, a mathematical model for investigating the underwater dynamic response of an inverted pendulum system has been developed, considering the effect of hydrodynamic forces (like the drag force and the buoyancy force) acting on the system. The mathematical model of the system has been derived by applying the standard Lagrange equation. To obtain the approximate solution of the system, the averaging technique has been utilized. An open loop parametric excitation technique has been applied to stabilize the pendulum system at its upright unstable equilibrium position. Both (like the lower and the upper) stability borders have been shown for the responses of both pendulum systems in vacuum and water (viscously damped). Furthermore, stability regions for both cases are clearly drawn and analyzed. The results are illustrated through numerical simulations. Numerical simulation results concluded that: (i) The application of the parametric excitation control method in this article successfully stabilizes the newly developed system model in an underwater environment, (ii) there is a significant increase in the excitation amplitude in the stability region for the system in water versus in vacuum, and (iii) the stability region for the system in vacuum is wider than that in water. Keywords: underwater inverted pendulum; parametric excitations; stability border; hydrodynamics forces; energy harvesting 1. Introduction Global climate changes are rapidly transforming into more fatal and complex environmental disasters, happening all around the world. Abundant emission of hazardous gases form coal fired Energies 2020, 13, 5215; doi:10.3390/en13195215 www.mdpi.com/journal/energies Energies 2020, 13, 5215 2 of 15 energy plants is one such kind of severe climate change. That is why the world energy paradigm is Energies 2020, 13, x FOR PEER REVIEW 2 of 15 shifting from existing hazardous gas emission energy plant to finding new renewable and clean energy sourcesshifting (suchfrom asexisting photo-voltaic hazardous solar gas cells, emission nuclear energy energy, plant and to ocean finding energy, new etc.). renewable The basic and aim clean of thisenergy research sources article (such is as to photo-voltaic propose an initial solar conceptcells, nuclear related energy, to energy and harvestingocean energy, from etc.). water The waves basic devices.aim of this Moreover, research this article study is hasto propose potential an benefits initial concept in the field related of underwater to energy under-actuatedharvesting from robotic water armwaves system. devices. Moreover, this study has potential benefits in the field of underwater under-actuated roboticIn thisarm article, system. the dynamic response characteristics of an inverted pendulum system in viewpoint of a futureIn this wave article, energy the harvesting dynamic deviceresponse in an characte underwaterristics environment of an inverted are discussed. pendulum The system problems in associatedviewpoint withof a future the dynamic wave energy response harvesting and parametric device excitationin an underwater control ofenvironment the pendulum are systemdiscussed. are alsoThe addressed.problems associated Simple water with waves the dynamic are simulated respon throughse and theparametric concept ofexcitation parametric control excitations. of the Parametricpendulum excitationsystem are phenomena also addressed. are widely Simple used water in thewaves control are designsimulated of variousthrough mechanical the concept and of electricalparametric dynamical excitations. systems. Parametric Moreover, excitation nowadays, phenomena parametrically are widely excited used in pendulum the control systems design are of gettingvarious a mecha huge attentionnical and in electrical the development dynamical of systems. wave energy Moreover, converters nowadays, and in variousparametrically kind of excited energy harvestingpendulum devices.systems Itare is getting believed a thathuge wave attention energy in canthe development be successfully of harvested wave energy if pivot converters point of and an invertedin variou pendulums kind of en systemergy oscillatesharvestin atg d aev constantices. It speed.is believ Thised pivotthat w point,ave e oscillatingnergy can withbe s aucc constantessfully speedharvested in the if pivot horizontal point directionof an inverted will be pendulum attached tosystem a generator. oscillates The at attacheda constant generator speed. This will pivot then producepoint, oscillating electrical energy.with a Figureconsta1 ntshows speed the in proposed the hori conceptualzontal direction schematic will diagram be attached of the wave to a energygenerator. harvesting The attached system. generator will then produce electrical energ FigureFigure 1.1. Proposed conceptual schematic diagramdiagram ofof thethe wavewave energyenergy harvestingharvesting system.system. ItIt isis alreadyalready known known that that the the dynamic dynamic response response behavior behavior and and stability stability control control of a system of a system in water in iswater different is different than in vacuumthan in [vacuum1]. The mathematical [1]. The mathematical analysis and analysis simulation and ofsimulation underwater of systemsunderwater can besystems veryuseful can be for very understanding useful for understanding their physical their characteristics. physical characteristics. All underwater All underwater systems experience systems hydrodynamicexperience hydrodynamic forces during forces their during operations, their andoper theseations, forces and these cause forces problems cause in problems their control in their and stabilization.control and stabilization. That is why That parametric is why parametric excitations excitations cannot be cannot directly be applied directly successfully applied successfully in water in water without the development of a new model of an inverted pendulum system for stabilization purposes. Therefore, it is necessary first to obtain the dynamic model of a pendulum system while keeping in view the impact of the hydrodynamic forces on its surface. A pendulum-based system model has been used to show the response of water on a submerged body [2]. In addition, the Energies 2020, 13, 5215 3 of 15 without the development of a new model of an inverted pendulum system for stabilization purposes. Therefore, it is necessary first to obtain the dynamic model of a pendulum system while keeping in view the impact of the hydrodynamic forces on its surface. A pendulum-based system model has been used to show the response of water on a submerged body [2]. In addition, the dynamic response of a viscously damped and non-rotating single degree of freedom (DOF) pendulum system is investigated theoretically and experimentally. Several other works related to the development of dynamic models of different systems in water can be referred to in previous studies [3–6]. For at least one century, an inverted pendulum system has been considered as a benchmark system in designing, controlling, stabilizing, and testing of numerous nonlinear mechanical control applications, like boxingbots [4], segway [7], and an inverted pendulum cart system [8]. There are a number of standard control techniques that exist in the control engineering area, which have been tested on an inverted pendulum model prior to their implementation on the real systems [4,7–11]. For example, the design of an active stabilizing system for a single-track vehicle system was studied [12], and an intelligent control and balancing technique for a robotics system has been formulated [13]. A fuzzy controller has been suggested to solve the trajectory tracking problem of the inverted pendulum attached to a cart system [14], while a particle swarm optimization-based neural network controller has been designed for solving
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