For rotors coupled by gears, the appropriate model for TRV coupling could be flexible or rigid, depending on the particulars of a given appli- cation. When the gear teeth contact and gear wheel combined equivalent torsional stiffness is much greater than other torsional stiffnesses in the system, it is best to model the geared connection as rigid so as to avoid any computational inaccuracies stemming from large disparities in con- necting stiffnesses. The configuration shown in Figure 3.3 contains both a geared connection and a pulley–belt connection of a three-shaft assembly.
Although the shown shafts are mutually parallel, this is not a restriction for the TRV models developed here.
The configuration shown in Figure 3.3 is categorized as an unbranched system. The full impact of this designation is fully clarified in the next sub- section that treats branched systems. Whether a TRV system is branched or unbranched made a lot of difference in computer programming complexity when older solution algorithms (e.g., transfer matrix method) were used.
With the modern finite-element-based matrix approaches used exclusively in this book, additional programming complexities with branched systems are not nearly so significant as with the older algorithmic methods that were better matched to the relatively limited memory of early generation com- puters. A coupled-rotor TRV system is defined here as unbranched when each of the coupled rotors in the drive train is connected to the next or previous rotor only at its two end stations (i.e., first or last), is not connected to more than one rotor at either end station, and is not connected to the same rotor at both end stations. When this is the case, the stiffness matrix for the coupled system is tridiagonal, just like the individual rotor stiffness matrices. The simplest example of this is the stiffness matrix for coaxial same-speed coupled rotors, Equation 3.12, which is tridiagonal. The Figure 3.3 system also clearly fits the definition of an unbranched TRV system, and is
Gear set
Pulley–belt set Rotor-1
Rotor-2
Rotor-3 qi,1
qi,2
qi,3
FIGURE 3.3 Unbranched three-rotor system with a gear set and a pulley–belt set.
here used to show the formulation for TRV equation-of-motion matrices of arbitrary-speed-ratio rotors with rigid and flexible unbranched connections.
3.4.2.1 Rigid Connections
The gear set of the system in Figure 3.3 will be assumed to be torsionally much stiffer than other torsional flexibilities of the system, and thus taken as perfectly rigid. The TRV angular displacements of the two gears are then constrained to have the same ratio as the nominal speed ratio of the two-gear set. Thus, one equation of motion must be eliminated either from rotor-1 (last station) or rotor-2 (first station). Here the equation of motion for the first station of rotor-2 is absorbed into the equation of motion for last station of rotor-1. The concentrated inertia of the rotor-2 gear is thus transferred to the rotor-1 station with the mating gear. Defining n21as the speed ratio of rotor-2 to rotor-1, andθi,jas ith angular coordinate of the jth rotor, the TRV angular coordinate of the rotor-2 gear is expressed in terms of the rotor-1 gear’s coordinate, as follows. Note the opposite positive sense forθi,2, Figure 3.3.
θ1,2=n21θN1,1 (3.13) where N1=number of stations on rotor-1=station number rotor-1’s last station.
The TRV kinetic energy of the two rigidly coupled gears is thus expressible as follows:
T12gears=12I(Nd1),1θ˙2N1,1+12I(1,2d)θ˙21,2=12
IN(d1),1+n221I1,2(d)
θ˙2N1,1 (3.14)
where Ii,j(d)≡nonstructural concentrated inertia for the ith station of the jth rotor.
The combined TRV nonstructural inertia of the two gears is thus lumped in the motion equation for station N1of rotor-1 as follows:
d dt
(∂T12gears
∂θ˙N1,1
)
= IN(d)
1,1+n221I1,2(d)
θ¨N1,1 (3.15)
As previously detailed in Section 3.3, shaft element structural mass is included by using either the lumped mass or the distributed mass approach.
For the lumped mass approach, the shaft element connecting the rotor-2 first and second stations has half its inertia lumped at the last station of rotor-1 (with the n221 multiplier) and half its inertia lumped at the rotor-2 second station. For the distributed mass approach (in TRV, same as the consistent mass approach), the shaft element kinetic energy is integrated along the first shaft element of rotor-2, as is similarly shown in Equations 3.5 and 3.6.
That is, postulating a linear variation of angular velocity along the shaft element, and substituting from Equation 3.13 forθ1,2, yields the following equation for the TRV kinetic energy of shaft element-1 of rotor-2.
T1,2(s)= I1,2(s) 6
n221θ˙2N1,1+n21θ˙N1,1θ˙2,2+ ˙θ22,2
(3.16) The following equation-of-motion distributed mass inertia contributions of this shaft element to the stations that bound it are accordingly obtained:
d dt
(∂T1,2(s)
∂θ˙N1,1
)
= 13n221I1,2(s)θ¨N1,1+16n21I1,2(s)θ¨2,2
d dt
(∂T1,2(s)
∂θ˙2,2
)
= 16n21I1,2(s)θ¨N1,1+13I1,2(s)θ¨2,2
(3.17)
where Ii,j(s)≡structural inertia for the ith shaft element of the jth rotor.
Postulating a rigid connection between the two gears in Figure 3.3 elimi- nates one DOF (i.e., the first station of rotor-2). The corresponding detailed formulations needed to merge the rotor-1 and rotor-2 mass matrices are contained in Equations 3.13 through 3.17. Merging the rotor-1 and rotor-2 stiffness matrices must also incorporate the same elimination of one DOF.
Specifically, shaft element-1 of rotor-2 becomes a direct torsional stiffness between the last station of rotor-1 and the second station of rotor-2. This stiffness connection is almost as though these two stations were adjacent to each other on the same rotor, except for the speed-ratio effect. The eas- iest way to formulate the details for merging rotor-1 and rotor-2 stiffness matrices is to use the potential energy term of the Lagrange formulation for the equations of motion, as follows (see Equation 2.50):
V1,2=12K1,2(θ1,2−θ2,2)2 (3.18) whereVi,j≡TRV potential energy stored in ith shaft element of the jth rotor and Ki,j≡Torsional stiffness of the ith shaft element of the jth rotor.
Substituting from Equation 3.13 forθ1,2into Equation 3.18 thus leads to the following terms for merging rotor-1 and rotor-2 stiffness matrices:
∂V1,2
∂θN1,1 =K1,2(n221θN1,1−n21θ2,2)
∂V1,2
∂θ2,2 =K1,2(−n21θN1,1+θ2,2)
(3.19)
Before implementing the terms for connecting rotor-1 to rotor-2, the detailed formulations for connecting rotor-2 to rotor-3 are first developed so that the mass and stiffness matrices for the complete Figure 3.3 system can be assembled.
3.4.2.2 Flexible Connections
The pulley–belt set in Figure 3.3 connecting rotor-2 to rotor-3 is assumed to be a flexible connection and thus no DOF is eliminated, contrary to the rigid connection case. A flexible connection does not entail modifications to the mass matrix of either of the two flexibly connected rotors. Only the stiffness of the belt must be added to the formulation to model the flexible connection. It is assumed that both straight spans of the belt connecting the two pulleys are in tension, and thus both spans are assumed to have the same tensile stiffness, kb, and their TRV stiffness effects are additive like two springs in parallel. The easiest way to formulate the merging rotor-2 and rotor-3 stiffness matrices is to use the potential energy term of the Lagrange formulation, as shown in Equation 2.50. To model gear-set flexibility, replace 2kb with pitch-line kg and define Rj as jth pitch radius, not jth pulley radius.
Vb= 12(2kb)(θN2,2R2−θ1,3R3)2
=kb(θ2N2,2R22−2θN2,2θ1,3R2R3+θ21,3R23) (3.20)
∂Vb
∂θN2,2 =2kb(θN2,2R22−θ1,3R2R3)
∂Vb
∂θ1,3 =2kb(−θN2,2R2R3+θ1,3R22)
(3.21)
where Rj ≡pulley radius for the jth rotor, Vb≡TRV potential energy in belt and N2=Number of stations on rotor-2=Station number rotor-2’s last station.
At this point, all components needed to write the equations of motion for the TRV system in Figure 3.3 are ready for implementation.
3.4.2.3 Complete Equations of Motion
For the individual rotors, the distributed mass approach is used here simply because it is better than the lumped mass approach, as discussed earlier.
Thus, Equation 3.7 is applied for construction of the three single-rotor mass matrices, [M1],[M2], and [M3]. Equation 3.9 is used to construct the three single-rotor stiffness matrices[K1],[K2], and [K3], adding any to-ground flexible connections to the free–free TRV stiffness matrices from
Equation 3.8. At this point, constructing the total system mass and stiffness matrices only entails catenating the single-rotor matrices and implement- ing the already developed modifications to the matrices dictated by the rigid and flexible connections. Employing modifications extracted from Equations 3.15 and 3.17,[M1]is augmented as follows. Superscript “rc”
refers to rigid connection.
(d) (s)
n212I1,2 13n212I1,2
N1×N1
+ [M1*] [M1] [M1rc], where [M1rc]
All elements in [M1rc]are zero except element (N N1, 1).
= +
(3.22)
Eliminating its first row and first column,[M2] is reduced to [M∗2]. The complete system mass matrix can be assembled at this point, catenating [M∗1],[M∗2], and[M∗3], and adding the cross-coupling terms contained in Equation 3.17, as follows:
[M] =
⎡
⎣ M1∗
Mcc
Mcc M∗2
[M3]
⎤
⎦
N×N
(3.23)
Subscript “cc” refers to cross-coupling.
MN1,N1+1=MN1+1,N1 = 16n21I1,2(s)≡Mcc, N=N1+(N2−1)+N3 The complete system stiffness matrix [K] is similarly constructed. Employ- ing modifications extracted from Equation 3.19, [K1] is augmented as follows:
[K1*] [K1] [K1rc], where [K ]
n K
1rc
212
1,2 N1×N1 All elements in [K1rc] are
zero except element (N N1, 1).
= + =
(3.24)
Eliminating its first row and first column, [K2] is reduced to[K2#], which is augmented to form[K2∗] as follows. Superscript “fc” refers to flexible connection.
[K2*] [K#2] [Kfc2]
fc 2
fc
, where [K2]
k Rb 22 2
N N2*× 2*
All elements in [K ]are zero except element(N N2*, 2*).
= + =
(3.25)
[K3]is augmented to form [K3*]as follows.
[K3*] [K3] [K3fc], where[K ] k R
fc b 3
32 2
N N3× 3
All elements in [Kfc3]are zero except element (1,1).
N2*=N2-1
= + =
(3.26)
The complete system stiffness matrix can be assembled at this point, cate- nating[K1∗],[K∗2], and[K3∗], and adding the cross-coupling terms contained in Equations 3.19 and 3.21, as follows:
[K] =
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎣
⎡
⎣ K∗1
⎤
⎦
Kcc1,2
Kcc1,2⎡
⎣ K2∗
⎤
⎦
K2,3cc K2,3cc⎡
⎣ K∗3
⎤
⎦
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎦
N×N
(3.27)
Kcc1,2≡ −n21K1,2; extracted from Equation 3.19 Kcc2,3≡ −2kbR2R3; extracted from Equation 3.21
The complete TRV equations of motion for the Figure 3.3 system are thus expressible in the same matrix format as Equation 3.10, that is,[M]{¨θ} + [K]{θ} = {m(t)}. The [M] and [K] matrices here are tridiagonal, which is consistent with the designation of unbranched. The formulations devel- oped here are readily applicable to any unbranched TRV system of coupled rotors.