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We had already obtained the analytical solutions for the compression wave and steady-state oscillating flow in a pipe with a circular cross section [Sato Y, Kanki H. Formulas for compression wave and oscillating flow in circular pipe. To be published in Appl Acoust [accepted 11 Sept. 2006]]. This work contains three key-components. The first key is to simplify the formulas using the unique mathematic technique without losing the accuracy. Simplifying the formulas is the one of the most important factors for formulas used in engineering use. The results will enable us to greatly reduce the work and computation costs. The second is to verify the flow distribution calculated by our formulas. The third is to study the possibility of application of our method to the analysis in turbulent flow region. Kawata et al. have represented the validity of one-dimensional quasi-analysis in turbulent flow region, using the method of D’Souza et al. and the shear viscosity. Therefore, in this paper the validity of the analysis in a turbulence region was verified by proving theoretically that the methods of D’Souza et al. and ours are intrinsically equivalent. The proof of equivalence was accomplished using the formulas simplified in this paper.

The effectiveness of squeeze-film dampers in controlling vibrations in rotating machinery may be limited by the nonlinear interactions between large rotor imbalance forces with fluid-film forces induced by dampers operating in cavitated conditions. From a practical point of view, the occurrence of nonsynchronous and chaotic motion in rotating machinery is undesirable and should be avoided as they introduce cyclic stresses in the rotor, which in turn may rapidly induce fatigue failure. The bifurcations in the response of a rigid rotor supported by cavitated squeeze-film dampers resulting from such interactions are presented in this paper. The effects of design and operating parameters, namely the bearing parameter (B), gravity parameter (W), spring parameter (S) and unbalance parameter (U), on the bifurcations of the rotor response are investigated. Spring parameter (S) values between 0 and 1 are considered. A spring parameter value of S=0 represents the special case of dampers without centering springs. With the exception of the case S=1, jump phenomena appeared to be a common bifurcation that occurred at certain combinations of B, W and U irrespective of the value of S. Period-doubling and secondary Hopf bifurcations which occurred for low values of S (⩽0.3) were not observed for the higher values S⩾0.5. For very low stiffness values (S

Numerical investigation on the unbalance response of a rigid rotor supported by squeeze-film dampers without centering springs revealed some complex bifurcation features that have not been previously reported in the literature. With the variation of the unbalance parameter (U), the period-1 solution was found to undergo a sequence of period-doubling bifurcations that eventually resulted in chaotic motion. The existence of a period-3 solution, which formed a closed bifurcation curve consisting of a pair of saddle nodes, was for the first time observed in such a system. The chaotic attractor arising from the period-doubling cascade of the period-1 solution, which was observed to co-exist with the period-3 attractor in a narrow range of U values, was eventually annihilated in a collision with the unstable period-3 orbit in a boundary crisis. Similar to the bifurcations of the period-1 solution, the period-3 solution was also found to bifurcate into solutions of period-6 and period-12, which eventually led to chaotic motion. A chaotic attractor was also observed to co-exist with a period-4 orbit. The period-4 orbit was found to undergo a sequence of reverse period-doubling bifurcations resulting in a large amplitude period-1 orbit. The occurrence of non-synchronous and chaotic motion in rotating machinery is undesirable and should be avoided as they introduce cyclic stresses in the rotor, which in turn may rapidly induce fatigue failure. The magnitude of rotor unbalance where non-synchronous and chaotic motion were observed in this study, although higher than the permissible unbalance level for rigid rotating machinery, may nevertheless occur with in-service erosion of the rotor or in the event of a partial or an entire blade failure.

The continuation method is employed to investigate the non-linear response of a rigid rotor supported by squeeze-film dampers without centering springs. The Floquet transition matrix method, which was used to examine the stability of the periodic response, indicated that the rotor may lose stability via period-doubling and saddle-node bifurcations at certain parameter values. The continuation method, which allows for switching from one solution branch to another, enabled the solution branches of period-two, four, eight and sixteen to be mapped out. By simultaneously varying the unbalance and bearing parameters, the combination of these two parameters that avoids the occurrence of non-synchronous vibration or jump phenomena in this system was determined.