On the Reynolds Equation and the Load Problem in Lubrication: Literature Review and Mathematical Modelling
In this chapter, we provide a literature review concerning the theory of hydrodynamic lubrication, especially applied to journal bearings. The device consists of an external cylinder surrounding a rotating shaft, both separated by a lubricant to prevent c
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Abstract In this chapter, we provide a literature review concerning the theory of hydrodynamic lubrication, especially applied to journal bearings. The device consists of an external cylinder surrounding a rotating shaft, both separated by a lubricant to prevent contact. In particular, we derive the fluid film thickness model for journal bearings, considering both the parallel and the misaligned case. The hydrodynamic Reynolds equation with cavitation phenomenon, through both Reynolds and ElrodAdams models are fully derived in this chapter. Subsequently, we pose two suitable variational formulations for the hydrodynamic problem considering both cavitation models. In addition, we present the admissible range of misalignment angle projections for prescribed values of the shaft eccentricity and angular coordinate. Finally, we properly state the problem of a loaded misaligned journal bearing for stationary regime, considering the balance of force and torque components involved. Keywords Reynolds equation · Hydrodynamic lubrication · Journal bearing · Misalignment · Cavitation · Inverse problem
1 On the Beginning of the Theory of Hydrodynamic Lubrication Hydrodynamic lubrication is a phenomenon characterized by a lubricant flowing in the narrow gap between two closely spaced surfaces in relative motion. Important and well-known scientists, engineers and tribologists investigated in the past H. Lombera Rodríguez Centro de Informática Industrial, Universidad de las Ciencias Informáticas, 19370 La Habana, Cuba e-mail: [email protected] J. I. Tello (B) Depto Matemática Aplicada a las TIC. ETSIS Sistemas Informáticos, UPM, 28037 Madrid, Spain e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_1
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H. Lombera Rodríguez and J. I. Tello
the relationship between friction, wear and lubrication, especially applied to journal bearings. Gustave Adolph Hirn (1815–1890), accomplished in 1847 the first experiments on hydrodynamic lubrication and rediscovered the laws of Amontons and of Coulomb. In 1879, Robert Henry Thurston (1839–1903), published the results of his study on friction and lubrication. He showed that, with increasing speed, the friction coefficient of a lubricated bearing diminishes below its static value, passes through a minimum and then increases. He also specified that the speed corresponding to the minimum of friction depends on the load applied to the bearing [48]. In 1883, Nikolai Pavlovich Petrov (1836–1920) introduced the results of his studies and tests on lubricated bearings. He proved that, among the physical characteristics of an oil, the viscosity has a preponderant role in bearing friction. He stipulated that a fluid film totally separates the surfaces of both shaft and bearing, and that a constant pressure should be produced in this film. Petrov, also looked through the work of Hirn and reused the term of mediate f
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