Variable Elasticity Effects in Rotating Machinery
The effects of variable elasticity in rotating machinery occur with a large variety of mechanical, electrical, etc., systems, in the present case, geometrical and/or mechanical problems. Parameters affecting elastic behavior do not remain constant, but va
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Variable Elasticity Effects in Rotating Machinery
Abstract The effects of variable elasticity in rotating machinery occur with a large variety of mechanical, electrical, etc., systems, in the present case, geometrical and/or mechanical problems. Parameters affecting elastic behavior do not remain constant, but vary as functions of time. Systems with variable elasticity are governed by differential equations with periodic coefficients of the Mathieu-Hill type and exhibit important stability problems. In this chapter, analytical tools for the treatment of this kind of equations are given, including the classical Floquet theory, a matrix method of solution, solution by transition into an equivalent integral equation and the BWK procedure. The present analysis is useful for the solution of actual rotor problems, as, for example, in case of a transversely cracked rotor subjected to reciprocating axial forces. Axial forces can be used to control large-amplitude flexural vibrations. Flexural vibration problems can be encountered under similar formulation.
2.1 Introduction Variable elasticity effects occur with systems in which the parameters affecting elastic behavior do not remain constant, but vary as functions of time. Equations of motion pertaining to such systems remain linear, but they possess time-dependent coefficients. Similar phenomena appear in all fields of physics and are generally associated with wave propagation in periodic media, a problem encountered as early as 1887 by Lord Rayleigh, and subsequently by other prominent physicists [1]. It was recognized that such phenomena are described by means of Hill and Mathieu differential equations. A detailed and comprehensive review on the work on waves in periodic structures was given by Brillouin [2].
A. D. Dimarogonas et al., Analytical Methods in Rotor Dynamics, Mechanisms and Machine Science 9, DOI: 10.1007/978-94-007-5905-3_2, Springer Science+Business Media Dordrecht 2013
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2 Variable Elasticity Effects in Rotating Machinery
Omitting a large number of non-mechanical phenomena mentioned in the above references, it is interesting to concentrate attention on the following vibrating mechanical systems [3–10]: 1. 2. 3. 4. 5. 6.
A rotating shaft with non-circular cross-section, i.e. non-uniform flexibility. A mass suspended from a taut string with time-varying tension. A pendulum with time-varying length. An inverted pendulum attached to a vertically vibrating hinge. The side-rod system of electric locomotives, exhibiting torsional vibrations. The rotating parts of small motors, which are subject to the time-varying action of electromagnetic fields. 7. A rotating flywheel carrying radially moving masses. 8. A rotating flywheel eccentrically connected to reciprocating masses. The list is by no means complete, but the examples are characteristic of the problems encountered. If one of these structures is constrained to move at constant circular frequency, the respective parameters are periodic functions and under proper conditions large vibration ampl
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