From Beale Number to Pole Placement Design of a Free Piston Stirling Engine

Abstract

In this paper, pole placement-based design and analysis of a free piston Stirling engine (FPSE) is presented and compared to the well-defined Beale number design technique. First, dynamic and thermodynamic equations governing the engine system are extracted. Then, linear dynamics of the free piston Stirling engine are studied using dynamic systems theory tools such as root locus. Accordingly, the effects of variations of design parameters such as mass of pistons, stiffness of springs, and frictional damping on the locations of dominant closed-loop poles are investigated. The design procedure is thus conducted to place the dominant poles of the dynamic system at desired locations on the s-plane so that the unstable dynamics, which is the required criterion for energy generation, is achieved. Next, the closed-loop poles are selected based on a desired frequency so that a periodical system is found. Consequently, the design parameters, including mass and spring stiffness for both power and displacer pistons, are obtained. Finally, the engine power is calculated through the proposed control-based analysis and the result is compared to those of the experimental work and the Beale number approach. The outcomes of this work clearly reveal the effectiveness of the control-based design technique of FPSEs compared to the well-known approaches such as Beale number.

Design of a traveling wave thermo‐acoustic engine based on genetic algorithm

“…Applying Laplace transform to Equation (13) results in the final formula describing the engine dynamics as follows:…”

“…Although the thermo-acoustic phenomenon is an old concept, the thermo-acoustic converters are known as new and advanced technologies ever presented. 12,13 Other categories of dynamic Stirling engines are known as Fluidyne and free piston Stirling engines, which are not within the scope of this article. The traveling wave thermo-acoustic Stirling engine (TASE) is a famous model of such converters.…”

Higher order modeling of a free-piston Stirling engine: analysis and experiment

“…8 Flowchart of the analytical procedure specification of SUTECH-SR-1. Readers are directed to the cited papers [12] for more details regarding specification of modern FPSEs.…”

“…Truthfully, in these methods, the dynamic analysis of the engine is based on the places of dominant poles of the closed-loop system, and working condition for starting the engine is that the location of closed-loop poles are either on the imaginary axis of the s-plane or on the right side of imaginary axis. The first analysis was carried out to set up and analyze the FPSE employing the wellknown linear method known as the fourth-order model [11,12], which tries to examine the effects of unknown parameters on the engine behavior. The basis of the fourth-order model is devised by Schmidt et al (also called Schmidt's theory [12], in which the pressure variations in the entire space of the engine compartment are supposed to be, and the heat transfer coefficient in the chamber are also assumed to be infinite.…”

“…The first analysis was carried out to set up and analyze the FPSE employing the wellknown linear method known as the fourth-order model [11,12], which tries to examine the effects of unknown parameters on the engine behavior. The basis of the fourth-order model is devised by Schmidt et al (also called Schmidt's theory [12], in which the pressure variations in the entire space of the engine compartment are supposed to be, and the heat transfer coefficient in the chamber are also assumed to be infinite. However, the other linear theory presented is known as the fifth-order model, in which the finite heat transfer coefficient is assumed to make it feasible to analyze the behavior of FPSE more precisely [13].…”