on the dynamic properties of reduced systems in

ON THE DYNAMIC PROPERTIES OF REDUCED SYSTEMS IN STRUCTURAL COUPLING

Urgueira, A.P.V.(*),

Maia, N.M.M.(**),

Almeida, R.A.B.(*)

(*) Faculdade de Ciencias e Tecnologia, Dep. Eng.Mec., 2825 Monte de Caparica, Portugal (**) Instituto Superior Tecnico, Dep. Eng.Mec., Av. Rovisco Pais, 1096 Lisboa Codex, Portugal

ABSTRACT

INTRODUCTION

Reduced systems are of major importance in the study of the dynamic behaviour of coupled structures, since they provide an efficient way of handling models which require a large number of coordinates for their spatial description. However, the reduction operated on the number of coordinates not always preserve the dynamic properties of the component models. In this paper several techniques are used to condense the dynamic information of spatial and response models. The practical implementation of these methods is then discussed and demonstrated via case studies.

It is generally acknowledged that an analysis of a complex structure with Finite Element method (FE) requires an high computational effort because there is a need to manage high level matrices. In practical situations the computational limitations lead to a reduction of the order of the system, resulting in a better efficiency of the algorithms. This is reached by reducing the order of the system, or, in different words, limiting the study to a restrict number of coordinates and/or to certain modes, with recognised importance to solve the problem.

However, the reduction techniques usually employed, such as Guy an reduction [I], are not exact and introduce some errors into the calculation of the reduced mass and stiffness matrices. In the last years, new techniques to reduce systems have emerged, trying to eliminate or minimize these errors.

NOMENCLATURE

[K]: system stiffness matrix [M ]: system mass matrix

[K 11 l:

One of these reduction processes is the Improved Reduced System (IRS) [3], which makes use of the Guyan reduction to obtain an estimation of the reduced system matrices and then makes some adjustments to compensate the inertia effect which are ignored in a Guyan process

reduced stiffness matrix

["-1 11 : reduced mass matrix [T]: transformation matrix

[ct>]:

modal matrix

[z]: dynamic stiffness matrix

Another different process, referred to as System Equivalent Reduction Expansion Process (SEREP) [4], was developed to reduce a finite element model down to an exact model in a reduced space which correctly preserves the eigenvalues and eigenvectores at the reduced set of test coordinates for a specific set of modes. The transformation matrix uses the eigenvectores of the global system, and therefore, the reduced matrices then obtained, preserves exactly the dynamic characteristics of global system for the set of frequencies selected.

[H]: accelerance matrix [s]: flexibility matrix [j

1

:

standard inverse of a matrix

[ ].
=

[ l· [

I'}~

[z 1J. =[r Jr[[~.l.'l'][~".,]l·[r J

( 12)

11

.l n

nrn

In most practical cases, the number of primary coordinates is greater than the number of desired modes. For this case the generalised inverse can be calculated as,

[·

z,"

] . [

z"

]

/)_rn

(17)

This reduced matrix is presented although in a different context.

(13)

Zhang work [5],

111

However when the number of primary coordinates is equal to the number of modes, the transformation matrix takes a similar form to both methods previously presented, this is

As it was previously mentioned, in an experimental point of view, the available matrix from the measurements is the accelerance matrix [H ].

[ jl[J "k ] [T ]-[[I]][t] - [J J'

[H]=

S/ol
-

By using stiffness equations present in

(14)

the transformation matrix [Y:,F/1/ol'] the reduced and mass matrices can be calculated using (2) and (3 ). More details about this method are reference [4].

[Hl.' l': ffl.''] 11," : [H""]

Thus, if the partitioned matrix 1s selected, it is important to notice that the inversion of this matrix provides the correspondent reduced dynamic matrix

[z 11 L

111

as in

(17),

[zil] - [H

Dynamic Reduction

/Jl'/1 -

lf a spatial model is used to represent the dynamic characteristics of a component, the correspondent spatial properties of the reduced model can be obtained via the Guyan or IRS reduction techniques. The reduced dynamic stiffness is then obtained by using equation ( 4). In this case the accuracy of the reduced model is dependent on the selected primary coordinates, as it will be shown on the applications. ln practical situations, the stiffness and mass matrices of the global system sometimes are not known. Honever, in the experimental via it is easy to measure the accelerance matrix [H] being the dynamic stiffness matrix of global system [z] calculated, as [z]= [H]- 1. Once this matrix has been

( 18)

If"

1-J

(19)

Pt>J

This impor1ant property is very useful in an experimentally derived model, since it is only necessary to undertake measurements on the selected primary coordinates. The reduced stiffness matrices obtained via the other techniques do not preserve the dynamic properties of the correspondent systems and are related as

obtained, one can select the primary and secondary coordinates of the system as

[zl?] * !Jyll

i'

[zil] ~

)'/oR/of'

(20)

APPLICATIONS To test the advantages and disadvantages of each reduction methods, a discrete system defined by Mottershead [6] with

1447

six discrete masses considered:

IS

used. Two different situations are

preserve the natural frequencies and correspondent modes of the original system. The first method, based on the static and dynamic characteristics of system, preserves all natural frequencies and corresponding modes of original system.

I o Case We define as primary coordinates the first five coordinates corresponding to the position of the first five masses. The secondary coordinate corresponds only to the coordinate of the sixth mass (figure /).

6.

We can also conclude that the SEREP method and dynamic condensation method must be used when the mass and stiffness matrices are unknown.

REFERENCES In this case the primary coordinates were defined as the first four coordinates corresponding to the first four masses, while the secondary coordinates corresponds to the others masses (figure 2).

[I]

Guyan, R.J., "Reduction of Stiffness and Mass Matrices", AlAA Journal, Vol. 3, February 1965.

[2]

Irons, B.M., " Structural Eigenvalue Problems: Elimination of Unwanted Variables", AIAA Journal,

In both cases the masses of coordinates to be eliminated will be increased, aiming to test the influence of mass values of the eliminated coordinates on each reducing methods.

Vol. 3, May 1965.

[3] CONCLUSIONS

0' Callahan, J.C., "A Procedure for an Improved Reduce System (IRS) Model", Seventh International Modal Analysis Conference, Las Vegas, Nevada, pp. 17-21, February, 1989,

From the graphs presented in the annex we can conclude:

[4] I.

2.

The dynamic reduction method has proved to be the only one, which preserves the dynamic characteristics of all the systems tested. Additionally, it was shown that the inversion of the partitioned accelerance matrix corresponding to the primary coordinates leads to the same reduced stiffness matrix as obtained via the dynamic reduction. Either in Guyan or IRS reduction techniques the validity of the models depends on the values of the masses of the secondary coordinates. Better results are obtained when the coordinates to be eliminated have small masses. The second method preserves the information about the mass of the secondary coordinates onto the transformation matrix, thus the results obtained are more accurate.

3.

By the nature of its formulation the SEREP reduction method preserves all of the dynamics of the full model in the reduced state for the selected modes, and the response does not depend on the mass values of the secondary coordinates.

4.

For a zero frequency, the transformation matrix, obtained by dynamic reduction, is the same as obtained with Guyan's method.

5.

The main difference between the dynamic reduction and Guy an's method was that the last is based only on transformations using static characteristics of system. This is the reason why this method sometimes does not

0' Callahan, J.C., Avitable, P.,Rirmer, R., "System Equivalent Reduction Expansion Process (SEREP) ", Seventh International Modal Analysis Conference, Las Vegas. Nevada, February, pp.29-37, 1989.

[5]

Zhang, N., "Dynamic Condensation of Mass and Stiffness Matrices", Journal of Sound and Vibration, pp 601-615, 1995.

[6]

1448

Mottershead, John.E., Friswell, Michael I, Ahmadian, Hamid., "Cross Validation and 'L 'Curves for the Regularisation of ill-Conditioned Equations in Model Updating", Sixteen International Modal Analysis Conference, Santa Barbara, California, 1998.

K 1 =K, =K6 =K, =K10 =IOOON/m

Cas-e 1

K 2 =1250Nim K, =1500N/m

K4 = !OOOON lm K 1 =5000Nim K, = 7000N/m

M 1 =M4 =0.9kg

M, =0.2 kg M, =M, =0.1 kg M 6 = O.lkg

Figure I 0.0

0.0

iii' -25.0

iii' -25.0 ~

~

" -50.0

~

]

C'ooplete S;Mem

-75.0

~

-50.0

~

-75.0

§

Gr1.J.W1 Reduction

or

l.Okg -~--1

l

· ...