GENERAL CASE
2.8.2.2 Modes of the undamped rotating system
If we now return to the basic system, with no damping and no external excitation, and allow the disc to spin at speed R,, then there will be some additional forces generated by the fact that, when vibrating in the y directions, the disc is rotating simultaneously about more t h a n one axis. These forces are due to the Coriolis accelerations set up by this complex motion and are usually referred to as 'gyroscopic' forces.
Essentially, simultaneous rotation about the z-axis (at angular speed and about the y-axis (with angular speed can only exist if there is a moment applied to the system about the third (x-) axis with a magnitude M, = L . When added to the equations of motion, this moment, and its counterpart for the other combination of rotations, has the effect of coupling the two equations, which now take the form:
This equation is more conveniently written as:
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Note that these equations include velocity-dependent terms (although these are not damping effects) but that they are not symmetric: indeed, the velocity-dependent matrix is skew-symmetric. The free vibration solution of these equations reveals that there are still two modes of vibration and that now they have different natural frequencies, even when the two bearing stiffnesses are identical. Furthermore, it can be seen t h a t these are still the natural frequencies of a n undamped system, even though there are velocity-dependent forces present. I t will also be noted that this type of equation has two different sets of eigenvectors - the so-called 'left-hand' and 'right-hand' sets, the latter representing the mode shapes themselves while the former are associated with preferred excitation patterns.
(a) Symmetric stator
The essential solution for the case where the vertical and horizontal stiffnesses are identical, and both equal to k, is a s follows. Assume, a s before, a simple harmonic solution of the type:
which, applying the symmetry of the stator so that = = k , leads to:
and this, in turn, to the following characteristic equation:
This equation can be solved to find the natural frequencies, and using the following notation, to give:
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where
Fig. shows the dependence of the two natural frequencies on the speed of rotation, , indicating the two relevant asymptotes, =
and = respectively. Also shown is the once-per-rev line,
(sometimes referred to a s the 'fist engine order' line), and the two critical speeds where this line intersects the two natural frequency lines, a t
respectively.
Completion of the free vibration solution reveals that the mode shapes corresponding to the two natural frequencies are complex - entirely complex, in fact, in that the two elements are exactly in quadrature with each other. The right-hand eigenvector for the lower- frequency of the two modes, = 1, can be shown to take the form:
while t h a t for the second mode, 2, is:
mode shapes is to the lower natural The interpretation of these two complex
straightforward: the one (which corresponds
frequency) represents a motion which constitutes a circular orbit of the disc centre which is backwards with respect to the spinning motion of the rotor, while the second mode shape, corresponding to the higher natural frequency, represents a forward circular orbiting motion, in the same direction a s the spinning of the disc itself. Fig. 2.2003) seeks to illustrate these mode shapes. and further explanation of the
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Fig. 2.20 Modal properties of symmetric 2DOF system.
(a) Natural frequencies; (b) Mode shapes
interpretation of complex modes in terms of stationary and rotating components is provided in later sections of the book.
These two mode shapes combine to form the matrix of the RH eigenvectors, which can be compared with its left-hand
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counterpart, and shown to observe the orthogonality conditions for this type of system, which are of the rather complicated form previously encountered in the case of general viscously-damped systems (see Section 2.7, equations (2.80)).
Non-symmetric stator
It is relatively straightforward to extend the above analysis to include the case where the two bearing (stator) stiffnesses are not identical, but have the relationship:
I n this case, the two system natural frequencies differ from those quoted previously, a s illustrated in Fig. but the main difference is seen i n the two mode shapes, which once again reflect backward and forward orbits but this time of a n elliptical rather than circular form, a s shown by the relevant right-hand eigenvectors:
and which are illustrated alongside the natural frequency plot in Fig.
F R F s o f t h e rotating s t r u c t u r e w i t h e x t e r n a l damping If we extend our analysis further, to include external damping and excitation forces, then we can
-
as before-
derive expressions for the FRF characteristics of this system. There are again four FRFs applicable a t the bearing support point, but this time they contain important differences from those developed earlier for the non-spinning case. Now, we find that the equations of motion for damped forced vibration of the symmetric system can be written as:and the corresponding expressions for the same four FRFs a s shown previdusly are found by setting and to be and , respectively, to yield:
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Fig. 2.21 Modal properties of non-symmetric 2DOF system.
(a) Natural frequencies; Mode shapes
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Plots of these are shown in Fig. The loss of reciprocity between and is visible in these plots and is evident from the formula, as is the fact that a force in the vertical direction now generates a response in the orthogonal horizontal (x-) direction, and vice versa, in direct contrast to the results which apply without any spinning of the rotor. Also clear from the plots is the high degree of complexity of the mode shapes, as witnessed by the significant imaginary part of the transfer FRF curves.
Similar expressions apply to the more general case in which the stator stiffnesses are not identical in the two planes, although the
are of detail only, and bring no additional features.
Of course, the response of the system to the simultaneous application of several forces a t the same frequency can be derived by appropriate superposition of the relevant components. Such a situation could be envisaged if there were simultaneous horizontal and vertical forces applied to the bearing in the current example. I t also applies in the important case of internally-generated out-of-balance forces, which will be dealt with in the next section.
2.8.2.4 Response of externally-damped rotating structure to