Open Access
Smith, Robert William
Graduate Program:
Doctor of Philosophy
Document Type:
Date of Defense:
April 22, 2008
Committee Members:
  • Steven Lurie Garrett, Committee Chair
  • Russell Lee Warley, Committee Member
  • Thomas B Gabrielson, Committee Member
  • Anthony A Atchley, Committee Member
  • Bellows
  • Thermoacoustic
  • Buckling
  • Parametric Stability
  • Rubber Composites.
Many electrically driven thermoacoustic refrigerators have employed corrugated metal bellows to couple work from an electro-mechanical transducer to the working fluid, typically, helium gas. As the power density in thermoacoustic refrigerators has increased, the stresses such bellows must endure have increased, leading to costly materials and few practical avenues toward performance improvement. In this study, an alternative bellows structure to mediate this power transfer is proposed: a laminated hollow cylinder comprised of alternating layers of rubber and metal. Axial compliance, needed to produce significant gas compression required of such a structure is conveyed by the rubber material properties, rather than structurally, as in a metal bellows. Two important considerations drive the proposed design: the capacity of the rubber to endure high cycle fatigue, and the desire to minimize viscoelastic power dissipation in the bellows; strain energy density plays a role in both. Assuming a rectangular cross-section in the rubber portions of the proposed bellows, equations are developed to estimate the bellows’ axial stiffness, peak local strain and section strain energy, in response to the combination of axial strain and oscillatory pressure. Optimal aspect ratios in the rubber section are also discussed. Consideration given to elastomeric material selection is presented, in terms of fatigue and loss characteristics, leading to the choice of unfilled natural rubber for prototypes that were constructed for this study. Comparisons of tearing energies estimated from known load cases and those obtained by finite element analysis for candidate dimensions are presented. It is observed that the phase speed of extensional waves in such a bellows is sufficiently low, that for the frequencies of interest, wave effects must be considered as these modify both the stresses and the power dissipation in the material. The metal layers, which experience negligible stress as a consequence of bellows axial strain, provide reinforcement to the elastomeric layers against pressure and are subject to out-of-plane buckling when the bellows is subject to external pressure loading. A bucking failure of this sort was observed in a prototype bellows and is described. The proposed structure also exhibits column instability when subject to internal pressure, as iv do metal bellows; for pressurized bellows and cylindrical sections, Haringx showed that internal pressures act in a manner analogous to axial compression of a beam. In the proposed bellows however, shear deflection cannot be ignored and this leads to column instability for both internal and external pressures, the latter being analogous to the case of tension buckling of a beam. Static column buckling with shear for internal and external pressure conditions is empirically demonstrated. In prototype bellows that were tested, transverse modes of vibration are believed to have been excited parametrically as a consequence of the oscillatory pressure the bellows are intended to produce. Unfortunately, the operating frequencies of interest in this study lie above the cut-on frequency at which Timoshenko beam theory (TBT) predicts multiple phase speeds. It is shown that TBT fails to accurately predict both mode shapes and resonance frequencies in this regime. TBT is also shown to predict multiple phase speeds in the presence of axial tension, or external pressures, at magnitudes of interest in this study, over the entire frequency spectrum. If one selects a mode which lies below cut-on in the absence of pressure or axial load, and is thus accurately represented by TBT, such modes are shown to converge to the static buckling results as the bellows is subjected internal or external pressure. TBT thus predicts decreasing resonance frequencies not only with axial compression, but also, at sufficient loads, in axial tension as well, and converges on solutions known to be valid. The implications of oscillatory pressure on a structure which exhibits such tuning behavior, in terms of parametric instability, are discussed; periodic solutions to the Whittaker-Hill equation are pursued to illustrate the shape of the parametric instability regions, and contrasted with the regions predicted by the more well-known Mathieu equation.