2.4 Insights into Linear LRVs
2.4.1 Systems with Nonsymmetric Matrices
The decomposition of any n×n matrix [A]into its symmetric (“s”) and skew-symmetric (“ss”) parts is an elementary technique of Matrix Algebra,
expressed as follows:
[Aij] =12
Aij+Aji
+12[Aij−Aji] ≡ [Asij] + [Assij] where [Asij] ≡12[Aij+Aji] and [Assij] ≡12[Aij−Aji] giving [Asij] = [Asij]T and [Assij] = −[Assij]T
(2.71)
As shown in Equations 2.71, the symmetric part of [A]is equal to its own transpose (“T,” i.e., interchange of rows and columns) whereas its skew-symmetric part is equal to minus its own transpose. This matrix decomposition technique can therefore be applied to any of the square matrices associated with the equations of motion for LRV. Clearly, if an n×n matrix is symmetric to begin with, then its skew-symmetric part is zero and this matrix decomposition does not accomplish anything.
Although most linearized vibration models have symmetric[M],[C], and [K] matrices, LRV models typically have some nonsymmetries. There are compelling physical reasons to justify that the 2×2 interaction-force gradient coefficient matrices [kij] and [cij] defined in Equation 2.70 can be nonsymmetric, and conversely that the 2×2 array [mij] should be symmetric. Furthermore, as already shown for spinning rotors in Equa- tions 2.17, the gyroscopic moment effect manifests itself in the motion equations as a skew-symmetric additive to the[C] matrix, for example, Equations 2.18, 2.52, and 2.54. In a series of papers some years ago, listed in the Bibliography at the end of this chapter, the author related the somewhat unique nonsymmetric structure of rotor–bearing dynamics equation-of- motion matrices to certain physical characteristics of these systems. The main points of those papers are treated in the remainder of this section.
The complete linear LRV equations of motion can be compactly expressed in standard matrix form as follows:
[M]{¨q} + [C]{˙q} + [K]{q} = {f(t)} (2.72) First, the matrices in this equation are decomposed into their symmetric and skew-symmetric parts as follows:
[K] = [Ks] + [Kss], [C] = [Cs] + [Css], [M] = [Ms] + [Mss] (2.73) where the decompositions in Equations 2.73 are defined by Equations 2.71.
The fundamental demonstration is to show that these decompositions amount to a separation of dynamical effects into energy conservative and energy nonconservative parts. That[Ks]is conservative,[Cs]is nonconser- vative, and[Ms]is conservative can automatically be accepted, being the standard symmetric stiffness, damping, and mass matrices, respectively.
[Css]is handled here first since there is a similarity in the treatments of [Kss]and[Mss].
Attention is first on some 2×2 submatrix within the [Css] matrix that contains[cijss], the skew-symmetric part of [cij] for a radial bearing, seal, or other fluid-containing confine between the rotor and nonrotating member.
The incremental work dw (i.e., force times incremental displacement) done on the rotor by the[cssij]terms at any point on any orbital path (refer to journal center orbital trajectory shown in Figure 2.10) is expressible as follows:
dw= − cijss x˙
˙ y
{dx dy} (2.74)
where[cssij] =
0 cssxy
−cssxy 0
.
Performing the indicated multiplications in Equation 2.74 and substitut- ing dx= ˙x dt and dy= ˙y dt yields the following result:
dw= −cssxy(x˙y˙− ˙yx)˙ dt≡0 (2.75) This result simply reflects that the force vector here is always perpen- dicular to its associated velocity vector, and thus no work (or power) is transmitted. Similarly, focusing on some 2×2 submatrix within the [Css]matrix that contains a pair of gyroscopic moment terms, as provided in Equations 2.17, the identical proof applies to the gyroscopic moment effects, shown as follows:
dw= −
0 ωIP
−ωIP 0 θ˙x
θ˙y
+dθxdθy
,
= −
0 ωIP
−ωIP 0 θ˙x
θ˙y
+θ˙xθ˙y
,dt≡0 (2.76)
The gyroscopic moment vector is perpendicular to its associated angu- lar velocity vector, and thus no work (or power) is transmitted. The skew-symmetric part of the total system[C] matrix thus embodies only conservative force fields and is therefore not really damping in the energy dissipation or addition sense, in contrast to the symmetric part of[C]which embodies only nonconservative forces.
Turning attention to the skew-symmetric part of[K], consider some 2×2 submatrix within the [Kss] matrix that contains[kijss], the skew-symmetric part of [kij] for a radial bearing, seal, or other fluid-containing confine between rotor and nonrotating member. The incremental work done by the
[kijss]terms on any point on any orbital trajectory is expressible as follows:
dw= −
' 0 kssxy
−kxyss 0
* x y
{dx dy} = −kxyssy dx+kssxyx dy≡fxdx+fydy (2.77)
∴ ∂fx
∂y = −kssxy and ∂fy
∂x =kxyss
Obviously,(∂fx/∂y)=(∂fy/∂x), that is, dw here is not an exact differential;
hence, the[kijss]energy transferred over any portion of a trajectory between two points “A” and “B” is path dependent, and thus the force field is non- conservative. The skew-symmetric part of the total system[K]matrix thus embodies only nonconservative force fields and is therefore not really stiff- ness in the energy conservative sense, in contrast to the symmetric part of[K]which embodies only conservative forces. An additional interesting insight is obtained here by formulating the net energy-per-cycle exchange from the[kijss]terms (see Figure 2.11).
Ecyc =
dw= −kxyss
(y dx−x dy) (2.78) Splitting the integral in Equation 2.78 into two line integrals between points “A” and “B,” and integrating the dy terms “by parts” yield the following result:
Ecyc=2kssxy
xB
xA
(y2−y1)dx (2.79)
The integral in Equation 2.79 is clearly the orbit area. Typically, kxyss ≥0 for journal bearings, seals, and other rotor–stator fluid annuli, even com- plete centrifugal pump stages. Thus, the kssxy effect represents negative damping for forward (corotational) orbits and positive damping for back- ward (counter-rotational) orbits. Only for orbits where the integral in
x A
xA B
y2(x)
y1(x) xB
FIGURE 2.11 Any periodic orbit of rotor relative to nonrotating member.
Equation 2.79 is zero will the net exchange of energy per cycle be zero. One such example is a straight-line cyclic orbit. Another example is a “figure 8”
orbit comprised of a positive area and a negative area of equal magnitudes.
The complete nonconservative radial interaction force vector{P}on the rotor at a journal bearing, for example, is thus embodied only in the sym- metric part [csij] and the skew-symmetric part [kijss], and expressible as follows (actually, csxx=cxxand csyy=cyy):
Px
Py
= −
csxx csxy csxy csxx
˙ x˙ y
−
0 kxyss
−kxyss 0 x y
(2.80) The parametric equations, x=X sin(Ωt+φx)with y=Y sin(Ωt+φy), are used here to specify a harmonic rotor orbit for the purpose of formu- lating the energy imparted to the rotor per cycle of harmonic motion, as follows:
Ecyc=
(Pxdx+Pydy)=
2π/Ω 0
(Pxx dt˙ +Pyy dt)˙
= −π Ω
csxxX2+2csxyXY cos(φx−φy)+csyyY2
−2kssxyXY sin(φx−φy)
(2.81) By casting in the x–y orientation of the principal coordinates of[csij], the csxyterm in Equation 2.81 disappears, yielding the following result, which is optimum for an explanation of rotor dynamical instability self-excited vibration:
Ecyc= −π Ω
cpxxX2+cpyyY2
−2kxyssXY sin(φx−φy)
(2.82) Since[kijss]is an isotropic tensor, its coefficients are invariant to orthogonal transformation, that is, do not change in transformation to the principal coordinates of[cijs]. Furthermore,Ω, cpxx, cpyy, kxyss, X and Y are all posi- tive in the normal circumstance. For corotational orbits the difference in phase angles satisfies sin(φx−φy) >0, and conversely for counterrota- tional orbits sin(φx−φy) <0. For a straight-line orbit, which is neither forward nor backward whirl,φx=φy so sin(φx−φy)=0, yielding zero destabilizing energy input to the rotor from the kssxy effect. From Equa- tion 2.82, one thus sees the presence of positive and negative damping effects for any forward whirling motion. Typically, as rotor speed increases, the kxyss effect becomes progressively stronger in comparison with the csij (squeeze-film damping) effect. At the instability threshold speed, the two effects exactly balance on an energy-per-cycle basis, andΩis the natural
frequency of the rotor–bearing resonant mode which is on the threshold of “self-excitation.” From Equation 2.82 it therefore becomes clear as to why this type of instability always produces a self-excited orbital vibra- tion with forward whirl (corotational orbit), since the kxyssterm actually adds positive damping to a backward whirl. It also becomes clear as to why the instability mechanism usually excites the lowest-frequency forward-whirl mode, because the energy dissipated per cycle by the velocity-proportional drag force is also proportional toΩ, but the energy input per cycle from the kxyss destabilizing effect is not proportional toΩ. In other words, the faster an orbit is traversed, the greater the energy dissipation per cycle by the drag force. However, the energy input per cycle from the kxyss destabilizing effect is only proportional to the orbit area, not to how fast the orbit is traversed. Consequently, as rotor speed is increased, the first mode to be “attacked” by instability is usually the lowest-frequency forward-whirl mode.
Harmonic motion is also employed to investigate the mssxyeffect. The net energy per cycle imparted to the rotor by such a skew-symmetric additive to the mass matrix is accordingly formulated similar to Equation 2.78, as follows:
Ecyc = −mssxy
(y dx¨ − ¨x dy)=Ω2mssxy
(y dx−x dy) (2.83) The factor(−Ω2)comes from twice differentiating the sinusoidal func- tions for x and y to obtainx and¨ y, respectively. With reference to Figure 2.11,¨ utilizing the same steps in Equation 2.83 as in going from Equation 2.78 to Equation 2.79, the following result is obtained:
Ecyc = −2Ω2mssxy
xB
xA
(y2−y1)dx (2.84)
It is clear from Equation 2.84 that an mssxy effect would be nonconser- vative, similar to the kxyss effect, but differing by the multiplier (−Ω2).
For mssxy >0, such a skew-symmetric additive to an otherwise symmet- ric mass matrix would therefore “attack” one of the highest frequency backward-whirl modes of a rotor–bearing system and drive it into a self- excited vibration. Even if mssxy were very small (positive or negative), theΩ2multiplier would seek a high-enough-frequency natural mode in the actual continuous-media rotor system spectrum to overpower any velocity-proportional drag-force damping effect, which has only an Ω multiplier. No such very high-frequency backward-whirl (mssxy >0) or forward-whirl(mssxy<0)instability has ever been documented for any type of machinery. Thus, it must be concluded that mssxy =0 is consistent with physical reality. In other words, the mass matrix should be symmetric to be
consistent with real machinery. An important directive of this conclusion is the following: For laboratory experimental results from bearings, seals, or other fluid-containing confines between rotor and nonrotating mem- ber, schemes for fitting measured data to linear models like Equation 2.70 should constrain[mij]to symmetry.
Even with symmetry imposed on[mij], the model in Equation 2.70 still has 11 coefficients (instead of 12), which must be obtained either from quite involved computational fluid mechanics analyses or from quite specialized and expensive experimental efforts, as more fully described in Chapter 5.
Thus, any justifiable simplification to Equation 2.70 model that reduces the number of its coefficients is highly desirable. For conventional oil-film journal bearings, the justified simplification is to discount the lubricant’s fluid inertia effects, which automatically reduces the number of coefficients to eight. For seals and other rotor–stator fluid confines that behave more like rotationally symmetric flows than do bearings, the isotropic model is employed as described in Section 2.4.3.