Model condensation

⎜ ⎞

⎝ +⎛

=

=

&= . (5.93)

The variation of θ& as a function of tω is plotted in Fig. 5.36. It is seen that θ& varies between ω′=

( )

b a ω and ω′′=

( )

a b ω so that ω= ω′ω′′.

Fig. 5.36

In conclusion, the angular speed of the precession motion is not θ& - that of the point B along the elliptic orbit, but the speed ω of the points M and P along the generating circles of radii a and b, respectively.

5.8 Model order reduction

The initial finite element discretization of a rotor system needs a relatively large number of degrees of freedom (DOFs) for satisfactory accuracy. A reduction of the number of DOFs is sometimes necessary because, in actual applications, only a few lower modes are of concern, giving little justification for solving the complete equation of motion in the dynamic analysis.

5.8.1 Model condensation

Static and dynamic condensation techniques can be used to produce reduced models possessing eigensystems that approximate that of the original full system model.

5.8.1.1 Formalism of coordinate reduction Consider equation (5.50) in the form

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

A rectangular transformation matrix

[ ]

T is seeked, which relates the n

The transformation is time independent, so that

{ }

x =

[ ]{ }

T u ,

{ }

x& =

[ ]{ }

T u& ,

{ }

x&& =

[ ]{ }

T u&& . (5.96) Substituting equations (5.96) in (5.94) and premultiplying by

[ ]

T ^T, the energy equivalence yields the reduced equations of motion

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

A proper choice of

[ ]

T will drastically reduce the number of DOFs without altering the lower eigenfrequencies and the mode shapes of interest.

5.8.1.2 Guyan/Irons condensation

The basis for the Guyan/Irons reduction is to follow a standard procedure used in static structural analysis, namely, elimination of DOFs at which no forces are applied, whence the name of static condensation.

The coordinates (DOFs) are partitioned into two groups: a) the active (“master”, retained) coordinates, and b) the omitted (“slave”, discarded) coordinates, denoted by “a” and “o”, respectively.

Partitioning the equation (5.94) accordingly

[ ] [ ]

Assuming

{ } { }

f_o = 0 , the static force-deflection relationship reduces to

[ ] [ ]

The lower partition provides a static constraint equation

[

^K_oa

]{ } [

^x_a ^{+ K}_oo

]{ } { }

^x_o ^{= 0 (5.100)}

which can be written

{ }

x_o =−

[

K_oo

] [

⁻¹ K_oa

]{ }

x_a . (5.101) The original set of coordinates

{ }

x can be related to the subset of active coordinates by the equation

{ } { } { } [ ]

Equation (5.102) may be referred to as a Ritz transformation. The Ritz basis vectors, which are columns of the Ritz transformation matrix

[ ]

T , are the displacement patterns associated with unit-displacement of the respective a-coordinates while the o-a-coordinates are released

{ } [ ]{ } { } { } [ { } ] ∑ { }

The reduced set of equations has the form (5.97), where

{ } { }

u = x_a ,

Replacing the dynamic relationship between active and omitted DOFs by a static relationship, the Guyan/Irons reduction is an incomplete extension of the Static condensation, with inherent loss of accuracy.

One exception is worth mentioning: a lumped-mass model, consisting of point masses at the nodes where translational displacements are defined (mass moments of inertia neglected). With all the rotational DOFs as o-DOFs and all the translational DOFs as a-DOFs,

[

M_oo

] [ ]

= ⁰ ,

[

M_oa

] [ ]

= ⁰ ,

[

M_ao

] [ ]

= ⁰ ,

[

Mred

]

=

^[

Maa

^]

and the Guyan reduction is accurate.

Drawbacks: a) misapplication can lead to serious modeling errors; b) destroys the banded form of matrices; and c) requires insight, experience and skill to partition the DOFs, though automatic procedures exist for the selection of active DOFs. In conclusion, the accuracy of the refined finite element model one tries very hard to achieve may be lost by the Guyan/Irons reduction.

5.8.1.3 Use of macroelements

Rotating shafts have variable cross sections and usually it is necessary to use a large number of finite elements to obtain a good model of the rotor. The number of elements can be reduced by introducing macro-elements [5.9]. Several short cylindrical elements can be treated as one element. Formally, this is done by a static condensation, treating the interior coordinates at the steps of the cross section as o-DOFs and the boundary coordinates as a-DOFs. This increases numerical economy without loss of accuracy in the results.

a b Fig. 5.37

For the stepped shaft from Fig. 5.37, a, a macro-element is shown in Fig.

5.37, b. Considering only the motion in the Y-X plane, the 8× macro-element 8 matrix has a banded form (Fig. 5.38). Reordering the nodal displacements, moving up the external DOFs selected as a-DOFs and moving down the internal DOFs selected as o-DOFs, destroys the banded form.

Fig. 5.38

Elimination of internal DOFs using the transformation (5.85) yields a condensed 44× matrix. This allows to maintain the banded structure of the system matrix (Fig. 5.39).

Fig. 5.39

5.8.1.4 Modal condensation

Consider the homogeneous part of the equation (5.50) of a damped anisotropic rotor system, written as

[ ]{ }

^{M x}&& ⁺

( [ ]

^{Cb +}Ω

^{[ ]}

^G

) ^{{ }}

^x& ⁺

( [ ] [ ]

^{Ks + Kb}

) ^{{ } { }}

^{x = 0} , (5.106) where

[ ]

^{Kb and}

[ ]

^Cb are the stiffness and damping matrices for bearings, respectively,

[ ]

^Ks is the shaft stiffness matrix,

[ ]

G is the (shaft+disks) gyroscopic matrix,

[ ]

M is the mass matrix, and

{ }

x is the 4n×1 state vector.

The corresponding complex eigenvalue problem yields the damped eigenfrequencies and the complex eigenvectors. For rotor systems with a large

number of DOFs, the complex eigenvalue procedure may run into numerical difficulties and may be time consuming.

One approach to avoid some of these problems is the modal condensation method. One variant is based on the analysis of the isotropic undamped non-gyroscopic part of equation (5.106) in the Y-X plane:

[ ]

^m

{ }

^Y&& ⁺

[

^k+kyy

] ^{{ } { }}

^{Y = 0} . (5.107) The shaft is assumed symmetrical, rotor gyroscopic effects are neglected, bearing damping is neglected, and only an average bearing principal stiffness term is considered, usually the symmetric component defined in equation (5.37). This (neighboring) associate conservative system has planar undamped modes entering as columns in the modal matrix

[ ]

Φ =

[ { } { }

Φ1 Φ2 _L

{ }

Φ2n

]

. (5.108) A truncated modal matrix is used as the transformation matrix, using only the L lower modes of the system described by (5.107).

Retaining the first L columns of the matrix (5.108)

[ ]

Φ^∗ =

[ ^{{ } { }}

Φ₁ Φ_{2 L}

^{{ }}

ΦL

]

. (5.109) The coordinate transformation is

{ } { } { } [ ] _{{ }} ^{{ }} [ ] ^{[ ]}

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

^u

u u u

u Z

x Y

z y z

y ∗

∗

∗ ∗ =

⎭⎬

⎫

⎩⎨

⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

=⎡

⎭⎬

⎫

⎩⎨

= ⎧

⎭⎬

⎫

⎩⎨

=⎧ Φ

Φ Φ Φ

0

0 (5.110)

where

{ }

u is the reduced state vector.

Substitution of (5.110) into (5.106) and pre-multiplication by

[ ]

^{Φ∗ T yields}

the reduced set of equations of motion (5.97)

where

[

^Mred

] [ ]

^{= Φ∗ T}

^{[ ]}

^M

[ ]

^{Φ∗ ,}

{ }

^{fred =}

[ ]

^{Φ∗ T}

^{{ }}

^{f ,}

[ ] [ ] [ ]

^{Cred = Φ∗ T}

(

^{Cb +}

^{[ ]}

^G

) [ ]

^{Φ∗ , (5.111)}

[ ] [ ] [ ] [ ]

^{Kred = Φ∗ T}

(

^{Ks + Kb}

) [ ]

^{Φ∗ .}

After determining the modal coordinates

{ }

u , the physical coordinates are calculated from equation (5.110).

One can say that up to 10-12 modal vectors

{ }

Φj have to be used in order to accurately determine the first 2-3 eigenvalues whose imaginary parts fall within the operating speed range. It is possible to introduce diagonal modal damping values accounting for the external and internal/structural damping.

The modal method does not require any intuition on mass lumping, component mode selection, and iterative procedures to improve the transformation matrix. The only basic assumption is that the linear combination of the Ritz basis vectors obtained by considering the undamped isotropic rotor system constitutes a good approximation to the complex eigenvectors of the heavily damped anisotropic rotor system.

In document M Rades - Dynamics of Machinery 2 (Page 64-70)