Vibrations of Shafts, Blades and Disks
This chapter examines the oscillation problems of shafts, turbine blades and the joint vibrations of blades and disks of a steam turbine’s rotor.
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Vibrations of Shafts, Blades and Disks
This chapter examines the oscillation problems of shafts, turbine blades and the joint vibrations of blades and disks of a steam turbine’s rotor. Shaft vibrations. Shafts are the basic carrier parts of the rotors of various types of turbomachines. The remaining parts of rotors are fastened to the shafts, and in calculations are considered as additional masses. Therefore, when discussing the vibrations of rotors, we are really talking about the vibrations of shafts. Let us consider three reasons which should be taken into account when studying these oscillations. (1) The action of unbalanced centrifugal forces generated during rotation of a rotor. (2) The variability of shaft parameters in two mutually perpendicular planes, such as twofold bending stiffness. (3) The presence of support anisotropy.
10.1
Bending Vibrations of a Rotating Shaft on Pivot Supports Under Unbalanced Centrifugal Forces
Let us consider vibrations of a shaft of circular cross-section on pivot bearings. Let s be the coordinate directed along the axis of the shaft; qðsÞ its mass per unit length; and bðsÞ is distributed pliability at bending. We consider a perfect geometry (i.e. assume that the shaft is an axisymmetric cylinder, whose line of cross-section geometric centers (elastic line) is straight in the absence of loads (shaft is considered to be weightless)). The shaft can be regarded as a thin rod. We introduce two coordinate systems: x; y; s is stationary and u; v; s is rotary with the shaft at speed x. The top of the coordinate system is compatible with the axis of a straight shaft. We suppose a uniform rotation such that the angle between two specified coordinate systems is considered equal to a ¼ xt: © Springer Nature Singapore Pte Ltd. 2018 V. Fridman, Theory of Elastic Oscillations, Foundations of Engineering Mechanics, DOI 10.1007/978-981-10-4786-2_10
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10 Vibrations of Shafts, Blades and Disks
Assume that the center of gravity of the shaft in an arbitrary cross-section in which the longitudinal coordinate s, generally speaking, does not coincide with the geometric center of this cross-section. In the coordinate system connected with the rotating shaft, we can represent the eccentricity of the cross-section gravity center of the shaft relative to its geometric center as a complex function eðsÞ ¼ eu ðsÞ þ iev ðsÞ;
ð10:1Þ
where eu ðsÞ and ev ðsÞ are projections of eccentricity on the axes of the rotating system u; v: In a fixed coordinate system the cross-section center of gravity can be determined by the vector eeixt . Rotation of the shaft induces bending vibrations because of centrifugal forces. As a result the geometrical center of a cross-section with coordinate s shifts relative to the axes x and y. Let xðs; tÞ; yðs; tÞ be the elastic line displacement relative to the axes x and y; and My ðs; tÞ; Mx ðs; tÞ the bending moments relative to the axes y and x. We introduce complex deflections and moments as zðs; tÞ ¼ xðs; tÞ þ iyðs; tÞ; Mðs; tÞ ¼ My ðs; tÞ þ iMx ðs; tÞ:
ð10:2Þ
Th
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