Stepwise Iterated Improved Reduction (SIIR)

If substitution of accelerations

{ }

x&& and _a x&& into equation (5.126) is o

repeated, for the subsequent iterations the constraint equation becomes

⎣

^oa( )ⁱ

⎦ ^{{ }}

^a

o G x

x = ⁺¹ , where

[

( )

] [ ⎣ ⎦ ( ⎣ ⎦ ⎣ ⎦

^{( )}

) ^{[ ] [ ]}

i i

]

i oa oo oa

oa oo i

oa K K M M G M K

G ⁺¹ =− ⁻¹ − + ⁻¹ .

The reduction to a-DOFs becomes

{ } [ ]

⎣ ⎦

^oaa( )ⁱ

^{{ }}

^a

^{[ ]{ }}

^{i a}

o

a x T x

G I x

x ⎥ =

⎦

⎢ ⎤

⎣

=⎡

⎭⎬

⎫

⎩⎨

⎧ ,

{ }

x =

[ ] [ ]{ }

P T_i x_a ,

where the subscript i denotes the ith iteration. After one reduction step, the SIIR homogeneous equation of motion of the damped gyroscopic system is

[ ] { } [ ] { } [ ] { } { }

M_i x&&_a ⁺ C_i x&_a ⁺ K_i x_a ^{= 0} , where

[ ] [ ] [ ] [ ][ ][ ]

M_i = T_i ^T P^T M P T_i ,

[ ] [ ] [ ] [ ] [ ][ ]

K_i = T_i ^T P^T K P T_i ,

[ ] [ ] [ ] [ ] [ ][ ]

C_i = T_i ^T P^T C P T_i . (5.132) At each reduction step, the effect of the removed o-DOF is redistributed to all the remaining a-DOFs, so that the next reduction will remove the o-DOF with the highest k_jj m_jj ratio in the reduced mass and stiffness matrices. The procedure is applied until the highest ratio k_jj m_jj is equal to or less than

ω

_c². At this point the a-DOFs represent the selected active DOFs of the reduced model.

Indeed, for sinusoidal excitation with frequency ω, the lower part of equation (5.126) yields

(

^{K M}

) ( ⎣

^K

⎦ ⎣

^M

⎦ ) ^{{ }}

^{x .}

K

x_o =− _oo⁻¹ 1−ω² _oo⁻¹ _oo ⁻¹ _oa −ω² _{oa a} (5.133) If the first parenthesis in (5.133) is approximated by the truncated binomial expansion

(

1−ω²Koo⁻¹Moo

) (

⁻¹≅ 1+ω²Koo⁻¹Moo

)

(5.134) and the terms in

ω

⁴ are ignored, then a constraint equation is obtained as

⎣ ⎦ ( ⎣ ⎦ ⎣ ⎦ )

[

^{K M M G}

] ^{{ }}

^{x .}

K

x_o =− _oo⁻¹ _oa −ω² _oa + _{oo oa}^ST _a (5.135) When

{ }

x_a is a mode shape of the reduced conservative problem, equation (5.135) becomes equation (5.130). But equation (5.134) is valid only for frequencies ω²<<K_oo/M_oo. This means that if a limit

ω

_c² is established, then the elimination of o-DOFs can be done until the ratio K_oo/M_oo is equal to or smaller than this value. In this way, the minimum number of a-DOFs is automatically determined, as well as their location.

For conservative non-gyroscopic systems, the SIIR method converges monotonically to a reduced model that preserves the lower eigenvalues and the corresponding reduced eigenvectors of the full system. For damped gyroscopic systems, the arbitrary reduction of damping and gyroscopic matrices in (5.132) may lead to better prediction accuracy of the SIR method for some eigenmodes.

After solving the reduced eigenvalue problem, equation (5.129) is used in the Inverse SIIR (ISIIR) method to expand the a-DOF vector to the size of the full problem, using the transformation matrix

[ ] [ ][ ]

T =

(

P T_i

)

₁

( [ ][ ]

P T_i

)

₂...

( [ ][ ]

P T_i

)

_n, where the subscript i is the number of iterations in the SIIR method.

A measure of the accuracy of the expanded mode shapes is given by the relative mode shape error

(

_FEM

)

abs

(

_expanded

)

_FEM 100

( )

%

absΦ − Φ Φ ⋅ .

Example 5.1

Figure 5.42 shows the single-disk multi-stepped shaft rotor system with two identical isotropic bearings of reference [5.8]. It was modeled with 76 DOFs, using 18 Timoshenko shaft elements including gyroscopic effects and neglecting internal damping [5.11]. The reduction procedure, applied with a cut-off frequency of 8000 rad/sec, selected 8 active DOFs: 3, 9, 25, 33, 41, 47, 63, 71.

The damped eigenfrequencies, computed at a spin speed n=30000rpm, are given in Table 5.3. The "true" values listed in the second column correspond to the full eigenvalue problem (76 DOFs). Columns three to five list natural frequencies computed using SGR, SIR and SIIR.. Note that the 8 translation DOFs are located at 4 nodes, the a-DOFs selected in one plane being selected in the other plane too.

Fig. 5.42

Columns six to eight list the relative error of expanded mode shapes by Inverse Stepwise Reduction. For some modes, the iterations in SIIR (and ISIIR) do not improve on the values in SIR (and ISIR).

Table. 5.3 Damped eigenfrequency, Hz Mode shape error, % Mode “True”

FEM SGR SIR SIIR

5 iter ISGR ISIR ISIIR 5 iter 1 246.01 246.78 246.27 246.21 2.04 1.46 1.32 2 296.49 296.91 296.81 296.74 1.54 1.82 1.66 3 774.00 777.26 773.00 772.79 3.59 1.07 1.23 4 808.31 808.38 807.32 807.20 2,16 0.91 1.12 5 1165.7 1287.3 1174.9 1166.8 24.96 7.57 3.44 6 1367.4 1425.4 1366.1 1369.7 14.71 0.69 3.35 7 1959.9 1997.6 1960.0 1958.7 3.59 1.26 0.57 8 2020.6 2054.1 2020.0 2020.7 3.29 0.61 0.53

Example 5.2

Figure 5.43 shows the four-disk three-bearing rotor from reference [5.12]

with variable inner diameter. The rotor was modeled with 13 nodes (52 DOFs), using Timoshenko shaft elements with consistent mass and gyroscopic matrices.

Fig. 5.43

The reduction procedures applied with a cut-off frequency of sec

rad

3000 selected 24 a-DOFs. Further reduction applied to select only 8 active DOFs, using the K M criterion, selected the four DOFs of node 1, and the translational DOFs of nodes 10 and 12 [5.11].

Table. 5.4 Damped eigenfrequency, Hz Mode shape error, % Mode “True”

FEM SGR SIR SIIR ISGR ISIR ISIIR

1 46.391 46.419 46.395 46.394 0.28 0.10 0.07 2 59.573 59.611 59.580 59.578 0.50 0.13 0.09 3 178.44 179.77 178.42 178.40 4.71 0.17 0.10 4 179.41 180.76 179.39 179.38 4.83 0.19 0.07 5 380.29 399.07 380.75 380.40 18.5 3.33 1.31 6 441.37 469.27 443.79 441.66 34.2 9.59 3.11 7 461.39 507.28 463.35 459.26 43.5 13.8 1.21 8 463.96 508.54 465.16 461.70 46.7 14.1 1.70

Table 5.4 lists the damped eigenfrequencies computed at a spin speed of 3000 rpm for the first eight modes of precession. Again, the accuracy is very good for the SIIR method with only 5 iterations. Mode shape expansion by the ISIIR method gives excellent results.

Example 5.3

The reduction procedures have been applied to the rotating shaft of a vertical Kaplan hydraulic unit (Fig. 5.44) from [5.13]. The shaft was modeled with

14 Timoshenko elements (60 DOFs) and simplified equivalent constant properties for the three horizontal bearings [5.11].

Fig. 5.44

Using a cut-off frequency of 500 rad/sec, the selected a-DOFs were: 1, 9, 21, 29, 31, 39, 51, 59, six of them for the three large masses: turbine, generator and auxiliary rotor. The eight lowest eigenfrequencies, computed for an angular speed of 3000 rpm, are listed in Table 5.5 together with the relative error of the expanded mode shapes.

Table. 5.5 Damped eigenfrequency, Hz Mode shape error, % Mode “True”

FEM SGR SIR SIIR ISGR ISIR ISIIR

1 25.337 25.357 25.368 25.371 1.21 1.49 1.61 2 28.762 28.884 28.799 28.796 2.91 1.61 1.58 3 38.092 38.681 38.084 38.084 9.54 0.38 0.71 4 38.386 38.946 38.389 38.379 10.3 1.50 0.75 5 45.026 45.779 45.040 45.045 7.94 1.37 1.68 6 47.195 48.015 47.245 47.226 8.49 2.59 2.02 7 65.724 66.776 65.774 65.705 7.43 2.38 0.90 8 68.203 69.263 68.203 68.198 6.88 1.20 0.95

References

5.1 Ehrich, F. F. (ed.), Handbook of Rotordynamics, McGraw Hill, New York, 1992.

5.2 Lalanne, M., Ferraris, G., Tran, D. M., Quéau, J. P., and Berthier, P., Comportement dynamique des rotors de turbomachines, I.N.S.A. Lyon, 1982.

5.3 Marguerre, K. and Wölfel, H., Mechanics of Vibration, Sijthoff & Noordhoff, Alphen aan den Rijn, 1979.

5.4 Genta, G. and Gugliotta, A., A conical element for finite element rotor dynamics, J. Sound Vib., vol.120, no. 1, p. 175-182, 1988.

5.5 Lee, C.-W., Vibration Analysis of Rotors, Kluwer Academic Publ., Dordrecht, 1993.

5.6 Someya, T. (ed), Journal-Bearing Databook, Springer, Berlin, 1988.

5.7 Childs, D., Turbomachinery Rotordynamics: Phenomena, Modeling and Analysis, Wiley, 1993.

5.8 Wang, W., and Kirkhope, J., New eigensolutions and modal analysis for gyroscopic/rotor systems, Part I: Undamped sytems, J. Sound Vib., vol.175, no.2, pp 159-170, 1994.

5.9 Gasch, R., and Knothe, K., Strukturdynamik, Band 2, Kontinua und ihre Diskretisierung, Springer, Berlin, 1989.

5.10 Shah, V. N., and Raymund, M., Analytical Selection of Masters for the Reduced Eigenvalue Problem, Int. J. Num. Methods in Engineering, vol.18, pp 89-98, 1982.

5.11 Radeş, M., Rotor-bearing model order reduction, Proc. 5th Int. IFToMM Conference on Rotor Dynamics, Vieweg, pp 148-159, 1998.

5.12 Rajan, M., Rajan, S. D., Nelson, H. D., and Chen, W. J., Optimal Placement of Critical Speeds in Rotor-Bearing Systems, ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, vol.109, pp 152-157, 1987.

5.13 Gmür, T. C., and Rodrigues, J. D., Shaft Finite Elements for Rotor Dynamics Analysis, ASME Journal of Vibration and Acoustics, vol.113, pp 482-493, 1991.

6 .

In document M Rades - Dynamics of Machinery 2 (Page 79-85)