11.11 Stream Whirl

Steam-excited whirl of high pressure turbine rotors is caused by shroud and shaft labyrinth seal (see Fig. 3.32) forces and by steam flow forces on the blades. The seal whirl force is destabilizing for some conditions and stabilizing for others (Wright, 1978). The problem of steam whirl is one of the technological limits that now prohibit the development of power-generating turbo-machinery substantially above 1 GW. Due to steam flow, self-excited vibrations develop at high loads in the form of stable limit cycles that, at even higher loads, deteriorate to chaotic vibration of high amplitude ( Dimarogonas and Gomez-Mancilla, 1994).

In the 1940s, two non-condensing turbines, having very flexible rotors, built by the General Electric Company experienced very violent whirl at high loads (Dimarogonas & Paipetis, 1983). This whirl could not be corrected with balancing and was present only at high loads. Thomas (1956, 1958) reported similar problems with AEG ( Allgemeine Elektricitäts-Gesellschaft, literally General Electricity Company ) turbines and was the first to give a rational explanation supported by a simple analytical model and experiments. Thomas identified the source of the excitation and developed a stability criterion based on a combination of analytical results and ex perimental validation. He concluded that the excitation is due to the steam flow through the seal clearances and the stabilizing effects come from the damping forces. He reported also that the problem was corrected, mainly, by decreasing the span and thus raising the critical speed and, sometimes, by bearing changes. Landzberg ( 1960) used a transfer matrix technique to evaluate the stability of tur bine rotors to steam excitation. Alford ( 1965) reported that modification to the steam path largely eliminated the prob lem of the General Electric turbines that experienced this type of instability. Both Thomas and Alford agreed that the vibration occurred at frequencies equal to the critical speed of the rotor. Alford, however, stated that variation of the whirl speed in jet engines varied within a wide range, depending on the amplitude and the test conditions. In the late 1960s the problem came back into atten tion, and several papers published in Germany dealt with the problem along the same lines with Thomas (Gash, 1965; Kraemer, 1968; Vogel, 1970, 1971). At the same time, large turbines of General Electric (1 GW size) had steam whirl problems and Dimarogonas ( 1971 a, 1971 b, 1972) reported on related analytical and experimental investigations. Black (1974) investigated flow induced vibration in high-speed centrifugal pump rotors and instabilities due to local reversed flows at very low flow rates. Shapiro and Colsher ( 1977) examined the influence of bearings on steam whirl, and Pollman et al. ( 1977) and Wright (1977) investigated the excitation mechanisms. Now through simple analytical treatment the steam whirl phenomenon will be explained.

Equations of motion of a Jeffcott rotor for free vibrations with the steam whirl can be written as

(11.190)

and

(11.191)

where m is the rotor mass, c is the damping parameter, k is the shaft stiffness and is the stiffness coefficient of the steam whirl. It should be noted that the steam whirl force gives the cross-coupled restoring force. In matrix form of above equations, the stiffness matrix is a skew-symmetric. Defining the complex displacement as

Then equations of motion can be combined as

The general solution of the following form can be assumed

where S is the complex amplitude in stationary coordinate system and is the eigen value. On substituting equation (11.194) into equation (11.193), we get

which is the frequency equation and can be written as

(11.196)

with

the solution of it takes the following form

which has the following form