Model reduction for structures with damping and gyroscopic effects

Transcription

1 Model reduction for structures with damping and gyroscopic effects M.I. Friswell, J.E.. Penny and S.D. Garvey Department of Mechanical Engineering, University of Wales Swansea, Swansea SA2 8PP, UK School of Engineering and Applied Science, Aston University, Birmingham B4 7E, UK School of Mechanical, Materials, Manufacturing Engineering and Management, Nottingham University, University Park, Nottingham NG7 2RD, UK Abstract Model reduction of dynamic models of structures is well established for undamped, non-rotating structures. Methods such as Guyan (or static) reduction, dynamic reduction, the Improved Reduced System approach (including iterated and dynamic versions) or the System Equivalent Expansion Process may be used to good effect. However, the situation is more difficult for damped or rotating structures. Methods based on state space models, using concepts of controllability and observability via balanced realisations are well established in the control literature. Unfortunately obtaining a model in second order form from the reduced state space model is not always (or indeed not often) possible because the constraints between the displacement and velocity parts of the state vector are not enforced during the reduction. his paper reviews the state of the art in reduction of models of damped or rotating structures, and discusses the errors introduced by using methods based on the undamped model, and highlights the problems in using the balanced realisation approach. hese approaches are tested on models of both damped and rotating structures, and the errors introduced compared. 1. Introduction his paper is concerned with systems that may be modelled in the second order form [ ] Mq&& + D+ G q& + Kq= f (1) where M, D and K are the symmetric mass, viscous damping and stiffness matrices, and G is the skewsymmetric gyroscopic matrix. G may also include effects due to Coriolis forces. Model reduction is a topic on which much has been written and whilst it is necessary to touch on some aspects, the bulk of this is beyond the scope of this paper. In general, the expansion is encapsulated in a n m co-ordinate transformation matrix,, which transforms the n analytical degrees of freedom (DoF) to the m reduced DoF. hus if q m is a response or mode at the reduced (sometimes called master) DoF, and q is the equivalent at the analytical DoF, then, qm = q (2) In general n>>m. In this paper, it will be assumed that is independent of frequency, although some modal expansion methods allow for the natural frequency associated with a particular mode-shape to be taken into consideration. Since has more rows than columns, the expanded vector is constrained to lie in the subspace spanned by the columns of, that is [ ]. his subspace uniquely defines the reduction / expansion transformation, although equivalent transformations may seem different due to square transformations of the reduced order model. he full system matrices may be reduced using the same transformation. he condensed mass, stiffness, viscous damping and gyroscopic matrices will be denoted M r, K r, D r and G r and the relation between the full-size matrices and their reduced counterparts is, Mr = M Kr = K Dr = D Gr = G (3) Note that in most case the reduction transformation is derived from the mass and stiffness matrices, but also applied to the damping and gyroscopic matrices. If the reduction transformation was derived from asymmetric matrices, then the same transformation would not

2 necessarily be used to transform the forces. hus a different transformation matrix would be used to premultiply the structural matrices, for example, r = L R. M M Furthermore the transformation is constrained to be real valued. When the model is reduced the analytical response is constrained to the subspace spanned by the transformation matrix. Note that if two transformations, 1 and 2, span the same subspace, then the reduced matrices obtained from equation (3) will yield exactly the same natural frequencies. In the rotordynamics community most researchers either reduce the undamped shaft models before coupling these shafts to the damped bearings [1, 2], or they reduce the state space form of the equations of motion. In state space form the popular methods of reduction are complex modal reduction [3] and balanced realisations [3, 4]. 2. Static and Dynamic In static or Guyan reduction [5, 6], the deflection and force vectors, q and f, and the mass and stiffness matrices, M and K, are re-ordered and partitioned into separate quantities relating to master (retained) and slave (discarded) degrees of freedom. Assuming that no force is applied to the slave degrees of freedom and the damping is negligible, the equation of motion of the structure becomes Mmm Mms q&& m + M sm Mss q && s Kmm Kms qm fm = Ksm K ss q s 0 (4) he subscripts m and s relate to the master and slave co-ordinates respectively. By neglecting the inertia terms in the second set of equations the slave degrees of freedom may be eliminated so that q m I = 1 q = q qs K ss Ksm m s m (5) where s denotes the static transformation between the full displacement vector and the master coordinates. he reduced mass and stiffness matrices are then given by equation (3). Note that any frequency response functions generated by the reduced matrices in equation (4) are exact only at zero frequency. As the excitation frequency increases the inertia terms neglected in equation (2) become more significant. Dynamic reduction may be performed by using the dynamic stiffness matrix of the system at a fixed frequency rather than the static stiffness matrix in equation (5). he question arises as to how the master coordinates are to be selected. Here the underlying assumption in Guyan reduction must be borne in mind, that at slave co-ordinates the inertia forces are negligible compared to the elastic forces. hus the slaves should be chosen where the inertia is low and the stiffness is high so that the mass is well connected to the structure. Conversely the master co-ordinates are chosen where the inertia is high and the stiffness is low. his process can be automated [7] by examining the ratio of the diagonal terms in the stiffness and mass matrices, k ii m ii, for the i th co-ordinate. If kii m ii is small then there are significant inertia effects associated with this coordinate and thus it should be retained as a master, if kii m ii is large then the i th co-ordinate should be chosen as a slave and removed. he slave co-ordinates are not chosen according to the above rule en bloc, but rather chosen and removed one at a time. here are two advantages to this procedure. Firstly at each stage the effect of each co-ordinate removed is redistributed to all the remaining co-ordinates so that the next reduction will remove the co-ordinates with the highest kii m ii ratio in the reduced mass and stiffness matrices. Secondly there is a very simple algorithm for performing this sequential process of co-ordinate selection and removal. 3. Improved Reduced System O Callahan [8] improved the static reduction method by introducing a technique known as the Improved Reduced System (IRS) method. he method perturbs the transformation from the static case by including the inertia terms as pseudo-static forces. Obviously it is impossible to emulate the behaviour of a full system with a reduced model and every reduction transformation sacrifices accuracy for speed in some way. O Callahan s technique results in a reduced system which matches the low frequency resonances of the full system better than static reduction. However, the IRS reduced stiffness matrix will be stiffer than the Guyan reduced matrix and the reduced mass matrix is less suitable for orthogonality checks than the reduced mass matrix from Guyan reduction [9]. he IRS transformation, IRS, may be conveniently written as, [8],

3 1 IRS = s + SMs MR K R (6) 0 0 where S = 1 and 0 K s is the transformation ss from static reduction. he reduced mass and stiffness matrices in equation (6) are obtained using static reduction, but the matrices obtained by using IRS reduction may be used in an iterative scheme [10, 11]. Although equation (6) is a convenient form for expressing the IRS transformation, in practice it is inefficient to compute the transformation in this way. he IRS method may be extended by using dynamic rather than static reduction [10]. he transformation is then exact at a non-zero frequency chosen by the analyst, rather than the zero frequency in the standard IRS method. 4. System Equivalent Expansion Process he System Equivalent Expansion Process (SEREP) [12, 13] uses the modes of the undamped analytical model to generate the transformation. he subset of the analytical eigenvectors, Φ, are partitioned into master and slave DoF, Φ m and Φ s, and the transformation is, Φm + SEREP = Φm, (7) Φs where ( ) 1 + m m m m Φ = Φ Φ Φ is the pseudo inverse. he subspace spanned by the transformation in the SEREP approach is obviously the space obtained from the analytical eigenvectors used. hus any mode or response that is contained in this subspace is likely to be reproduced in the reduced model or expanded accurately. 5. via s hus far the reduction methods have been based on the undamped equation of motion. he usual approach to the analysis of damped systems is to write the equations of motion in state space. A popular method of model reduction in state space is the balanced realisation approach [14, 15]. However the result is a reduced model in state space, and this is not guaranteed to be realisable as a model in second order form [16]. his section will introduce balanced model reduction, before deriving an approximate real transformation for the physical DoF. Equation (1) may be written in state space form as where & (8) x= Ax+ Bfm, y = qm = Cx, q x =, q& 0 I A = 1 1 M K M ( D+ G ), 0 B = 1 M H, C= H 0 and the matrix H selects the master DoFs, and f m is the force at the master DoF. here are other, equivalent, state space forms, but this one is sufficient for our purpose. A system is controllable if an input exists so that the states of the system may be driven to any arbitrary configuration. he controllability grammian, W c, for the system described by equation (8) is the solution of the equation, AWc + Wc A + BB = 0 (9) he system is controllable if the grammian is full rank, and the condition number of the grammian is a measure of the controllability of the system. Similarly the observability grammian, W o, may be obtained from the solution of A Wo + Wo A+ C C= 0 (10) If the system is observable, then the state vector may be reconstructed from knowledge of the current and past values of the output vector. he degree of controllability and observability cannot be determined in isolation [15]. A convenient approach is to transform the state so that the observability and controllability grammians are equal. his is a balanced realisation. It is also convenient to make these grammians diagonal, so that the controllability and observability of individual states may be determined immediately. he least controllable and observable transformed states may then be removed. Skelton [15] outlined the method and gives a convenient algorithm. On the assumption that the derivative of the slave states is zero, then a reduced state space model based solely on the retained states may be formed. Calculating balanced realisations is computationally intensive. Furthermore, methods that generate the transformation using a Cholesky decomposition often fail for systems with a large number of states, because the grammians are rank deficient to the numerical precision used in typical software packages. Skelton [15] suggests removing uncontrollable states before calculating the

4 observability grammian. his can be taken further, and many of the least controllable states could be removed at this stage. Sufficient states should be kept to ensure that there are enough observable states to generate the reduced model. For the large scale systems in structural dynamics the initial model will also have to be reduced to an intermediate size model using eigensystem truncation, described above, or one of the standard methods for undamped structures, for example static reduction, IRS or SEREP. Meyer and Srinivasan [17] considered the balanced realisation based on the second order form of the equations of motion, equations (1), and gave a method of calculating the balancing transformation in second order form. Assumptions have to be made concerning the initial velocity and Meyer and Srinivasan assume that either the initial velocity is zero (the zero velocity grammian) or the initial velocity is available as a variable during the optimisation (the free velocity grammian). If the state space grammians are partitioned as Gc11 Gc12 Gc = G c12 G c22 Go11 Go12 Go = G o12 G o22 then the second order free velocity grammians are (11) Gˆ cfv = G ˆ c11, GoFV = G o11 (12) the second order zero velocity grammians are -1 Gˆ czv = Gc11 Gc12Gc22Gc12 ˆ -1 GoZV = Go11 Go12Go22Go12 (13) he balancing transformation may then be obtained in a similar way to the state space formulation. he approach adopted below works well for low order systems, and variations similar to those above are required for large systems. he Cholesky decomposition of the controllability grammian (either free or zero velocity) is obtained as Gˆ c = E E (14) which is then used to form EGˆ oe. his symmetric positive definitive matrix is then decomposed as 2 NΣ N = EGˆ oe (15) where N N= I and Σ is diagonal. he balancing transformation is then 1 2 b = Σ E N (16) If the decomposition in equation (15) orders the diagonal elements of Σ then the most controllable and observable DoFs may be easily identified and the remaining DoF are reduced out. For a free-free structure a transformation based on the undamped modes must be undertaken, so that the rigid body modes are decoupled from the structural modes, to ensure that the model is asymptotically stable and thus meets the requirements for the application of the balancing approach. 6. Clifford he final reduction approach considered is called Clifford reduction, because the approach arises naturally as the equivalent of static reduction when the second order equations of motion are cast in their Clifford algebra form [18]. his approach is not adopted here since space does not permit the introduction of the mathematical concepts required for the development. However this reduction produces mass, damping and stiffness matrices that ensure that the expansion of the receptance as a power series in frequency is correct to the terms of degree 2. o aid the derivation, assume that D is a general matrix that may have a skew-symmetric part to account for gyroscopic effects. Assume also that this matrix is partitioned in the same way as the mass and stiffness matrices in equation (4). hen the receptance, in terms of the master degrees of freedom, is 2 ( ) αmm ω = Mmmω + Dmm jω+ K mm 2 Mmsω + Dms jω+ K ms 1 2 Mssω + Dss jω+ K ss 2 Msmω + Dsm jω+ K sm (17) he inverse may be expanded as a series in ω, and the terms in ascending powers of ω collected together, and compared to 2 3 ( ) j ( ) αmm ω = r + r ω rω + O ω K D M. (18) he resulting reduced stiffness and damping matrices (including the gyroscopic effects) are

5 identical to those obtained using static reduction. he reduced mass matrix is M r = M S ms + S ss ss sm + D t D K D D ss t S (19) 1 [ ] where M S is the reduced mass matrix obtained -1 using static reduction, and ts = KssK sm is the transformation between the slave and master degrees of freedom in static reduction. 7. Numerical Examples We now illustrate the application of the model reduction to a discrete model of a continuous structure with a discrete damper. he structure considered is a 1 mm thick, steel plate with a slot on one side and clamped along two other sides, as shown in Figure 1. he full model has 90 degrees of freedom and is reduced to the 6 degrees of freedom shown in Figure 1. All masters are translations out of the plane of the plate. his choice of co-ordinates represents the best selection of co-ordinates for Guyan reduction on the basis of the relative importance of the diagonal stiffness and inertia terms [7, 19]. many masters as the number of required eigenvalues [20]. he results for the natural frequencies of the damped structure are less satisfactory. A 50 Ns/m damper is placed at the node shown in Figure 1, which gives the natural frequencies and damping ratios shown in ables 2 and 3. Also shown in these tables are the natural frequencies and damping ratios when the second order model is reduced using the methods outlined in this paper. All the methods have severe difficulty and produce large errors, even for the lower modes. Note that static, IRS and SEREP reduction have changed the mode ordering. he Clifford reduction performs best at low frequency, but becomes inaccurate quite quickly. Note that the SEREP method is accurate for the lightly damped modes, but has difficulty with heavy non-proportional damping. he results for the balanced realisation approach are very interesting. Because the model is not frequency limited, the very lightly damped modes are chosen, rather than the low frequency modes. It may be that these modes are outside the frequency range of interest, and care must be exercised in applying this approach. Figure 2 shows a typical receptance plot. he plot is a point receptance at the uppermost master degree of freedom in Figure 1. Although it is difficult to distinguish individual reduction schemes, it is clear that all the methods, apart from balanced realisation, are quite accurate up to 30 Hz..3m Figure 3: Overhung rotor example. 0.25m Figure 1: he plate example. he solid dots represent the optimum set of master co-ordinates. he circle gives the position of the discrete damper. able 1 gives the first 6 natural frequencies of the undamped structure, and also the natural frequencies of the reduced model obtained by using static reduction, IRS and SEREP. he results are reasonably good and as expected the lower modes converge more quickly than the higher modes and in practice one would retain approximately twice as he second example is the overhung rotor shown in Figure 3. he solid shaft has diameter 25 mm, and the disk has diameter 250 mm and is 40 mm thick. Both bearings are short and the left bearing has stiffness 50 kn/m horizontally and 100 kn/m vertically, but no damping. he right bearing has the same stiffness but also includes a damper of magnitude 2 kns/m in both directions. Natural frequencies and damping ratios are calculated for this system at a running speed of 3000 rpm, and the results are shown in ables 4 and 5. he full model is obtained using 6 finite elements of equal length, resulting in 28 DoFs. he 8 reduced DoF consist of the deflections and rotations in both directions at the right bearing and at the disk. It is clear that all

6 methods have some difficulty, but that IRS and SEREP are the most accurate. 8. Conclusions his paper has considered the application of model reduction to damped and gyroscopic systems. transformations generated from the undamped model can introduce large errors. he balanced realisation approach can also have problems if lightly damped high frequency modes are present. None of the methods are totally satisfactory, and the search is continuing for more suitable second order reduction approaches. Acknowledgements Dr. Friswell gratefully acknowledges the support of the EPSRC through the award of an Advanced Fellowship. References 1. R. Subbiah, R.B. Bhat and.s. Sankar, Dynamic response of rotors using modal reduction techniques, J. of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 111, pp , (1989). 2. G. Genta and C. Delprete, Acceleration through critical speeds of an anisotropic, non-linear, torsionally stiff rotor with many degrees of freedom, J. of Sound and Vibration, Vol. 180, No. 3, pp , (1995). 3. G.W. Fan, H.D. Nelson, P.E. Crouch and M.P. Mignolet, LQR-based least-squares output feedback control of rotor vibrations using the complex mode and balanced realization methods, J. of Engineering for Gas urbines and Power, Vol. 115, pp , (1993). 4. J.. Sawicki and W.K. Gawronski, model reduction and control of rotor-bearing systems, J. of Engineering for Gas urbines and Power, Vol. 119, pp , (1997). 5. R.J. Guyan, of stiffness and mass matrices, AIAA Journal, Vol. 3, No. 2, p. 380, (1965). 6. B. Irons, Structural eigenvalue problems: elimination of unwanted variables, AIAA Journal, Vol. 3, pp , (1965). 7. R.D. Henshell and J.H. Ong, Automatic masters for eigenvalue economisation, Earthquake Engineering and Structural Dynamics, Vol. 3, pp , (1975). 8. J.C. O Callahan, A procedure for an improved reduced system (IRS) model, Proceedings of the 7th International Modal Analysis Conference, Las Vegas, pp , (1989). 9. J.H. Gordis, An analysis of the improved reduced system (IRS) model reduction procedure, Modal Analysis: he International Journal of Analytical and Experimental Modal Analysis, Vol. 9, No. 4, pp , (1994). 10. M.I. Friswell, S.D. Garvey and J.E.. Penny, Model reduction using dynamic and iterated IRS techniques, J. of Sound and Vibration, Vol. 186, No. 2, pp , (1995). 11. M.I. Friswell, S.D. Garvey and J.E.. Penny, he convergence of the iterated IRS method, J. of Sound and Vibration, Vol. 211, No. 1, pp , (1998). 12. D.C. Kammer, est-analysis-model development using an exact modal reduction, he International Journal of Analytical and Experimental Modal Analysis, Vol. 2, No. 4, pp , (1987). 13. J.C. O Callahan, P. Avitabile and R. Riemer, System equivalent reduction expansion process (SEREP), Proceedings of the 7th International Modal Analysis Conference, Las Vegas, pp , (1989). 14. D.J. Inman, Vibration: with Control, Measurement and Stability, Prentice-Hall International, (1989). 15. R.E. Skelton, Dynamic Systems Control: Linear Systems Analysis and Synthesis, John Wiley & Sons, (1988). 16. M.I. Friswell, S.D. Garvey and J.E.. Penny, Extracting second order systems from state space representations, AIAA Journal, Vol. 37, No. 1, pp , (1999). 17. D.G. Meyer and S. Srinivasan, Balancing and model reduction for second-order form linear systems, IEEE ransactions on Automatic Control, Vol. 41, No. 11, pp , (1996). 18. S.D. Garvey, M.I. Friswell and J.E.. Penny, A Clifford Algebraic Approach to Second-Order Systems, AIAA Journal of Guidance, Control and Dynamics, to appear, (2000). 19. J.E.. Penny, M.I. Friswell and S.D. Garvey, Automatic choice of measurement locations for dynamic testing, AIAA Journal, Vol. 32, No. 2, pp , (1992).

7 20. K-J. Bathe and E.L. Wilson, Solution methods for eigenvalue problems in structural mechanics, International Journal of Numerical Methods in Engineering, Vol. 6, pp , (1972) FRF Magnitude (m/n) Full Model Static Clifford IRS SEREP SEREP Frequency (Hz) Figure 2: A typical point receptance. Mode Full Model Static (0.24%) (3.36%) (4.27%) (8.00%) (25.18%) (24.65%) IRS (0.02%) (0.20%) (3.99%) (4.01%) SEREP able 1. Natural frequencies (Hz) for the undamped plate (percentage error in brackets).

8 Static Mode Full Model (-1.23%) (3.98%) (-9.72%) (8.00%) (21.32%) (24.35%) IRS (-0.85%) (1.12%) (-13.80%) (0.19%) (0.43%) (3.43%) SEREP (-1.04%) (1.45%) (-15.67%) (-1.98%) Clifford (0.21%) (1.83%) (11.96%) (8.00%) (25.72%) (24.43%) able 2. Natural frequencies (Hz) for the damped plate (percentage error in brackets). Mode Full Model Static IRS SEREP Clifford able 3. Damping ratios (percent) for the damped plate. Mode Full Model Static IRS SEREP Clifford Real Eigen values able 4. Natural frequencies (Hz) at 3000 rpm for the overhung rotor. Mode Full Model Static IRS SEREP Clifford able 5. Damping ratios (percent) for the overhung rotor.

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