Applying Particle Swarm Optimization to Study the Effect of Dominant Poles Places on Performance of a Free Piston Stirling Engine

  • Sh. Zare
  • A. R. Tavakolpour-SalehEmail author
Research Article - Mechanical Engineering


In this paper, the effects of dominant closed-loop poles places of a free piston Stirling engine (FPSE) on the engine performance are investigated. First, linear and nonlinear formulations of an FPSE are presented. Then, based on the presented formulations, dynamic behavior of a prototype free piston Stirling engine, namely SUTECH-SR-1, is studied. Afterward, a part of the simulation results is compared with those of an experimental work to validate the mathematical models. Consequently, the obtained reasonable agreement between the experimental and simulation results affirms the validity of the proposed nonlinear theorem. Next, according to linear and nonlinear formulations, the necessary condition for startup state of the engine is investigated. In addition, it is demonstrated that existence of a stable limit cycle is the necessary condition for engine startup based on theory of nonlinear systems. Subsequently, the effects of two important factors including real and imaginary parts of the dominant closed-loop poles of the FPSE on the engine performance are investigated using particle swarm optimization. According to the obtained simulation outcomes, the free piston Stirling converter is more sensitive to variation of the real component of the dominant closed-loop poles than that of the imaginary part.


Free piston Stirling engine Particle swarm optimization Dominant poles 

List of Symbols


Crosscut area of the piston and displacer \(({\mathrm{m}^{2}})\)

\(A_r \)

Crosscut area of the displacer rod \(({\mathrm{m}^{2}})\)


Damping coefficient \(( {\hbox {N}\,\hbox {s}\,\hbox {m}^{-1}} )\)


Position of particle (m)


Spring stiffness \(( {\hbox {N}\,\hbox {m}^{-1}})\)


Piston mass (kg)


Gas mass (kg)


Pressure (Pa)


Velocity of particle \(( {\hbox {m}\,\hbox {s}^{-1}})\)


Gas constant for dry air \(( {\hbox {J}\,\hbox {kg}^{-1}\,\hbox {K}^{-1}})\)


Random vector


Temperature (K)


Time (s)


Random vector


Volume \(( {\hbox {m}^{3}})\)


Position of displacer piston (m)


Position of power piston (m)

Subscript and superscript


Cold gas


Displacer piston


Hot gas










Initial value



Greek symbols

\(\alpha \)


\(\beta \)


\(\theta \)

Phase difference (\(^{\circ }\))

\(\omega \)

Frequency \(( {\mathrm{rad}\, \mathrm{s}^{-1}})\)


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Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringShiraz University of TechnologyShirazIran

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