GENERAL CASE
2.8.3 Dynamic Analysis of General Systems .1 Equations of motion for general rotor-stator systems
2.8.3.2 Eigen-properties of general LTI system matrices
The previous subsections have revealed some trends that are to be expected in the matrices which are generated for the equations of motion of system with rotating components and it is appropriate here to illustrate the properties of such matrices in a more general form than has been shown
The essential features of interest are those which result in a loss of symmetry of the velocity-dependent or displacement-dependent matrices (referred to loosely a s the 'damping' and 'stiffness' matrices, respectively). I t may be readily seen that any matrix can be expressed a s the linear combination of a symmetric matrix and a skew-symmetric matrix and it is convenient to think of these dynamic system matrices in this way because most of the reasons for a loss of symmetry are, in fact, the introduction of a degree of skew-symmetry
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the gyroscopic forces in the first case and the internal damping effects in the second. Thus it is useful to examine the effect of a skew-symmetric damping matrix and then of a skew-symmetric stiffness matrix. These two effects will be described and illustrated by representative numerical examples based on the 2DOF system already studied.(a) Skew-symmetry in the damping matrix
As we have already seen, the effect of introducing a skew-symmetric damping matrix to otherwise symmetric system matrices is to generate complex eigenvectors (mode shapes) but to retain real eigenvalues undamped natural frequencies with no decay component). A set of
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numerical examples based on the
2DOF
system are shown in the Table below. For the case where the three system matrices are described by:we obtain the following results for the second mode a s (the extent of skew symmetry) is varied from to 1:
(b) Skew-symmetry in the stiffness matrix
We can conduct a similar exercise for the second example, in which the stiffness matrix is found to have a skew-symmetric element due to the effect of internal damping. Thus, for the same basic
2DOF
system, we can introduce the following variable elements in the stiffness matrix:which yields the results shown in the following table, again focusing on the second mode of vibration. In many ways, these results are more interesting than those from the previous set because they show a transition from a n undamped system to an unstable damped one. I n this example, the extent of the skew-symmetry could be adjusted by varying the speed of rotation of the rotor since this parameter appears directly in the off-diagonal terms that relate to the internal damping.
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Thus we arrive a t the conclusion that as the speed of rotation increases, the system becomes less and less stable until there comes a point when the boundary of stability is crossed so that both the eigenvalues and eigenvectors are complex and the system is unstable. In real structures, this boundary of stability will be affected by all the sources of damping which are present (only the effect of internal damping on the stiffness matrix has been included here) but this analysis demonstrates in a modal analysis context and, using a simple example, the well-known phenomenon that 'internal damping has a destabilising effect on a rotor that is running a t a super-critical speed'.
Before leaving this topic, it is worth noting that there is another common phenomenon which comes into the same category in that it gives rise to a non-symmetric stiffness matrix, and that is
induced rub between rotating and a stationary components. When forces created by rubs are included in the equations of motion, they will frequently have the effect of contributing a non-symmetric component to the displacement-dependent (stiffness) matrix. Such a feature will have the same possible consequences a s those we have seen above: in some cases, depending on the specific numerical values of all the relevant parameters, this effect can give rise to a n unstable mode of vibration.
Although it seems instinctively to be unlikely, friction forces developed by a rotor rubbing on the stator can cause instability, and this property has been witnessed in countless situations, often to the distress of the relevant components.
2.8.4 D y n a m i c Analysis of R o t a t i n g Flexible Disc-like S t r u c t u r e s
2.8.4.1 Classification and modal p r o p e r t i e s of flexible disc-like s t r u c t u r e s
It is found that many of the most critical vibration-prone) components in rotating machines fall into a class of which is
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generally described as 'quasi-axisymmetric': components which are 'disc-like' (wheels, gears, discs, impellers,...). The various types of structure which concern us here are:
(i) axisymmetric;
cyclically-periodic, or
slightly asymmetric, or aperiodic (which means slightly imperfect structures of the first and second types), referred to a s
periodic'.
We shall not describe the vibration properties of these structures in detail here, but simply note that the essential modal properties are a s follows:
.
axisymmetric structures (plain discs, wheels, etc.) have modes whose shapes are described in the circumferential direction, , by variations of the form:= cos
+
a,)and are therefore known a s 'n-nodal diameter' or modes;
cyclically-periodic structures (such a s bladed discs, impellers, gear wheels) have modes whose shapes are described in the circumferential direction, , by expressions of the form:
where N is the number of blades, vanes, teeth etc. These are also referred to a s 'n-nodal diameter modes' although this description is less precise in this case than for the preceding one because the mode shape clearly includes more components than just the
one. It is possible to find modes in which the largest component is cos , and such a mode is well described a s one with
but, equally, modes exist in which the most significant component in the mode shape is the second one ((N - or a n even higher one, and in these cases, the description is not the most appropriate. should be noted that if such mode shapes are defined by determining the amplitude ratios only a t the N discrete points around the rim which carry the
then the discrete Fourier description which results will be incapable of discriminating above the first term in series
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( N - n) would be indistinguishable from n hence the classification simply a s 'n-nodal diameter' modes.);
quasi-periodic structures, in which the loss of axisymmetry is small, usually due to manufacturing tolerances, have mode shapes which are described (in the circumferential direction) by:
= A,
+
a,)where 1, 2, 3, but is generally
-
although not always-
dominated by n, N - n , N + n , etc. When these modes are so dominated by a single, or few, terms of this type, the nodal diameter label is still used, although it is rather less precise than for the other cases. However, it'should be noted that there are situations where small deviations from true or cyclic symmetry, can lead to mode shapes which contain many significant diametral components and which cannot then be realistically described a t all by the label.
A second feature of the modes of these disc-like structures concerns their natural frequencies. In the case of type (i) and structures, most of the modes exist in pairs of 'double' modes: two modes with identical natural frequencies and mode shapes which differ only in the angular orientation of the nodal lines, in As is the case generally, when there are two or more modes with identical natural frequencies, any combination of the individual two mode shapes is also a mode shape.
This can lead to some unexpected features in the case of these axisymmetric structures where, for example, a valid mode shape can be produced by a combination of
+
a,) plus xsin+
a,),the result of which is a cos mode shape rotating around the structure in a travelling wave motion. Type (iii) structures also possess these double modes but in this case the natural frequencies of each pair of modes are slightly separated, or 'split', resulting in two distinct modes not repeated roots, in the mathematical sense) but which may be very difficult to distinguish from each other in observed response characteristics because of the inevitable 'coupling' effects of the structure's damping properties.
One consequence of these features is that the structures which display such modal properties can be much more difficult to test than are structures with single modes. As a result, measured modal properties of axisymmetric, or quasi-axisymmetric, structures are often in error, sometimes through ignorance on the part of the analyst and sometimes because of the inherent difficulties in making the measurements.
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2.8.4.2 R e s p o n s e properties of s t a t i o n a r y disc-like s t r u c t u r e s As intimated above, the response properties of the family of structures included in this 'disc' category can be very complicated as a direct result of the special 'symmetry' they possess, even when they are stationary.
(Note t h a t this 'symmetry' is unconnected with the symmetry of the system matrices which has been discussed a t length earlier in this section.) While it is not appropriate to provide a full exposition of these vibration properties here, it is useful to illustrate the basic phenomena using a simple example. The simplest illustration of the effects of interest can be provided by considering a plain uniform disc which is excited a t a single point on its rim with a n harmonic force tuned to the exact natural frequency of the fundamental mode with two nodal diameters. The resulting response can be summarised by the sketch in Fig. which shows the two nodal diameter lines symmetrically disposed about the excitation point. If we were to relocate the point of application of the excitation force around the disc rim by 45 degrees, we might then expect to obtain no resonant response, because excitation a t a nodal point has that particular result. However, what we find in practice is t h a t the nodal lines 'move' around to follow the excitation point so t h a t the response pattern obtained in the second case is a s shown in Fig. This result, which is both intuitive and correct, can only be explained if there is more than one mode with the two nodal diameter shape: indeed - two, each with the same natural frequency but with mode shapes that are orthogonal to each other
-
one a s sin 20 and the other a s cos 20 - and this is exactly what happens.Fig. 2.26 Forced response of disc in 2ND resonance
Each double mode of a 'disc' structure can be described a s two modes, with identical or very close (depending upon the exact degree of
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symmetry which obtains) natural frequencies and mode shapes which are essentially of the form:
= sin and =
(in which we have dispensed with the arbitrary offset, for simplicity). We shall now suppose that there is a point harmonic excitation applied at circumferential location = 9 , and note t h a t this excitation condition can also be described more completely as:
It can be seen that such a n excitation can also be described as:
and that the first term will excite the first of the pair of modes while the second term will excite the second one, so that the response will be of the form:
= (cos
+
(sin (at)where A , , depend on the proximity of the excitation frequency, to the natural frequency of each mode, and the damping, but which will be proportional to when the two modes have identical natural frequencies, in which case:
t) = (cosn - (a t )
This brief analysis serves to explain the above example of the 2ND modes on the simple disc by showing that in the case of a perfectly tuned axisymmetric) disc the nodal lines will align themselves to be symmetric with respect to the excitation, no matter where that is applied.
Implicit in the above analysis is the fact that a mode with a cos mode shape will only be excited by a n excitation force pattern which has
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a corresponding spatial distribution in it: the excitation force must contain a component of in it. Clearly, a single point excitation force can be seen to contain components of all orders, from = to infinity. From this result there follows the property that response in certain modes can be suppressed by arranging for a n excitation pattern that eliminates certain orders from its circumferential decomposition.
Thus, if we apply two single-point excitation forces which are spaced 90 degrees apart around the disc rim, then we shall that this excitation has a zero component of excitation and, a s a result, modes with will not be excited.
The above comments relate primarily to the reference case where the structure is tuned, or perfectly or cyclically symmetric. This is not always the case, and it is necessary to extend the foregoing analysis to the more general case where the symmetry is imperfect, in which case the two modes of each pair (or double mode) have slightly different natural frequencies. I n this condition, the two modes will resonate a t different frequencies and, strictly, the nodal lines will be fixed in the disc and will not move around to follow the excitation. Instead, two resonances will be observed, one at the first natural frequency with nodal lines in their orientation, independent of the location of the excitation, and a second resonance a t the second natural frequency with its nodal lines likewise fixed in the disc. The one caveat to relatively straightforward description concerns the transition from the former, tuned case, to the latter, mistuned situation. How much natural frequency separation or 'split' is necessary before we can observe the above two-resonance phenomenon? The answer depends upon the level of damping which prevails in the structure because if the frequency separation or split between the two modes is very small, and the damping is moderate, then the two resonances will not be individually discernable and the structure will still appear to be tuned. As a rough We shall now extend the above discussion of excitation forces applied to the disc-like structures of interest in many rotating machines to the case where these structures are rotating. We shall start by considering again a single point harmonic force, which is applied a t a fixed point in space, = and which can be defined by:
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We next consider the effect of this force pattern a s 'seen' in the rotating axes frame by setting =
+
Rt (where is the speed of rotation), so t h a t we obtain:This. expression shows that the single-harmonic fixed excitation, Foezot, will generate a series of excitations applied to the rotating component with the particular feature that all modes (in the rotating disc) which have a or form modes with n nodal diameters) will be excited a t two frequencies, Thus, a single harmonic excitation fixed in space will vibrations in the rotating components a t a large number of different frequencies, two for every n, although the strength of each of the different response components will vary according to the proximity of its frequency to a natural frequency of one of the corresponding modes of the disc. This result means that although it is possible to understand and to explain the reason for these complicated characteristics, it will nevertheless be very difficult to derive FRF data in the usual format if excitation is to be applied and measured i n the stationary axes set (for example) and the response is to be measured in the rotating axes set.
A special case of widespread interest can be noted here: when the excitation force is a static force when 0), there will still be a n effective dynamic excitation experienced by the rotating disc and this will be experienced a t a frequency of for a mode with
It can now be noted that the previously-reported example of this effect with the rigid rotor (in equation (2.120)) constitutes a special case of this more general analysis. Such a rotor only has the possibility of vibrating in modes for which (axial and torsional motion) or 1 (lateral motion) and in the example quoted only the second of these two groups was active, hence the two response frequencies of a s described in the earlier sections.
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2.9 COMPLEX MODES
2.9.1 Real and Complex Modes, Stationary and Travelling