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DYNAMIC CONDENSATION OF MASS AND STIFFNESS MATRICES
Journal of Sound and Vibration (1995) 188(4), 601–615
DYNAMIC CONDENSATION OF MASS AND STIFFNESS MATRICES N. Z† Department of Mechanical Engineering, Monash University, Clayton, Victoria 3168, Australia (Received 17 September 1993, and in final form 27 January 1995) Details are given of a procedure for condensing the mass and stiffness matrices of a structure for dynamic analysis. The condensed model is based on choosing nc master co-ordinates Xc from the original n co-ordinates X and nc natural frequencies and the corresponding modes of the original model. The model is constructed so that (1) it has nc natural frequencies equal to those of the original model, (2) the modes fijc i, j=1, 2, . . . , nc are the same as those for the master co-ordinates in the corresponding modes of the original and (3) the responses of the condensed system at the co-ordinates Xc due to forces at these co-ordinates, at one particular chosen frequency, are the same as those of the original system. The natural frequencies, the corresponding modes and the dynamic responses used for the condensation can be obtained from finite element analysis of the original structure. The method has been applied to the modelling of two common structures to examine its applicability. Comparisons between the performance of the condensed models obtained by means of the dynamic condensation method and that of the models obtained by the Guyan method have been conducted. The results of the examples show that the condensed models determined by the dynamic condensation method retain the natural frequencies and modal shapes and perform better in describing the dynamic responses of the structures than do the corresponding models obtained by the Guyan method. 7 1995 Academic Press Limited
1. INTRODUCTION
It is often necessary to reduce the computation time of dynamic analysis of a complex structure. One of the effective ways of doing so is to reduce simultaneously the size of the non-diagonal mass and stiffness matrices for the eigenvalue analysis and response analysis. For the static analysis of a structure, Turner et al. [1] presented a procedure to eliminate the co-ordinates at which no force is applied and to obtain a small size stiffness matrix without losing accuracy. For the dynamic analysis, the most common procedure for reducing the size of the original model was presented by Guyan [2]. In the case of the reduced stiffness matrix, if the applied loads are static, none of the structural complexity applied is lost since all elements of the original stiffness contribute. However, to reduce the mass matrix, the purely static transformation between the eliminated co-ordinates and those retained co-ordinates is employed. The result is that the eigenvalue–eigenvector problem is closely but not exactly preserved [2]. Sotiropoulos [3] developed an improved procedure by taking the additional inertia effects into consideration, but he still retained the purely static transformation between the retained and eliminated.
†Present address: School of Mechanical Engineering, University of Technology, Sydney, 1 Broadway, Sydney, NSW 2007, Australia.
601 0022–460X/95/490601+15 $12.00/0
7 1995 Academic Press Limited
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Craggs [4] presented a component mode method for modelling the dynamic properties of turbo-generator sets. The method greatly improves the accuracy of the reduced model of structural systems by introducing a few of the lowest component modes and measured displacements. But for the higher modes of the reduced model, there still is some kind of deviation from the original modes if the number of component modes introduced is limited. For the vibration analysis of a complex structure excited by one or a number of harmonic forces in a wide range of frequencies, the reduced mass and stiffness matrices which have a good accuracy are required. Zhang and Hayama [5] developed a method for identifying the mass and stiffness matrices with a limited number of modes of a structure from experiment and the results showed that the identified mass and stiffness matrices are very accurate in the analysis of the forced vibration of structures. However, in practice, it is difficult to apply the method to large structures because of the large inertia. For dynamic analysis of rotor systems of rotating machinery, a condensed model of the rotor shaft is often needed and it is also required that the model has natural frequencies and corresponding modes which are as close as possible to those of the original model. The forces applied to a rotor system are usually due to the unbalance of the rotating parts and their frequencies are often identical to the operational frequencies. Hence, condensed models of rotor shafts are also required to perform accurately in describing their dynamic behavior at these operational frequencies. The common shortcoming of the available condensation procedures is that the resulting models do not retain precisely the natural frequencies and the corresponding modes from the original models. Besides, at the operating frequency, the resulting models can describe the dynamic behavior of the original system only approximately. In what follows, a procedure is presented for determining the condensed mass and stiffness matrices of a structure in which the dynamic contributions from all of the original co-ordinates of the structure are considered. The condensed model is based on choosing nc master co-ordinates Xc from the original n co-ordinates X and nc natural frequencies and corresponding modes of the original model. The model is constructed so that: (1) it has nc natural frequencies equal to those of the original model, (2) the modes fijc i, j=1, 2, . . . , nc are the same as those for the master co-ordinates in the corresponding modes of the original model and (3) the responses of the condensed system at the co-ordinates Xc due to forces at these co-ordinates, at one particular chosen frequency, are the same as those of the original system. The method has been applied to the modelling of two common structures, i.e., a cantilever pipe and a rotor shaft, to examine its applicability. Comparisons between the performance of the condensed models obtained by means of the dynamic condensation method and that of the models obtained by the Guyan method have been conducted. 2. PROCEDURE OF DYNAMIC CONDENSATION
2.1. A continuous structure can be represented with good accuracy by a discrete model of order n as long as the frequency range of interest is limited. For simplicity, it is assumed that the structure considered is free of damping. The forced motion of the structure is therefore described by the differential equation [M]{X (t)}+[K]{X(t)}={F(t)},
(1)
where [M] and [K] represent the mass and stiffness matrices of size n×n, respectively, and {X(t)}, {X (t)} and {F(t)} represent the displacement, the acceleration and the excitation
603
force vector of order n, respectively. Here it is emphasized that the response vector and the force vector are arranged in such a way that the upper nc elements of the vectors correspond to those master co-ordinates which one wants to keep in the reduced model. The lower ne (where ne=n−nc ) elements are the co-called slave co-ordinates which one wants to eliminate from the reduced model. If one chooses nc degrees of freedom of the original structural system to be retained in the condensed model, the motion of the structure at the chosen master co-ordinates can then be described by the equation [Mc ]{Xc (t)}+[Kc ]{Xc (t)}={Fc (t)},
(2)
where [Mc ] and [Kc ] are assumed to be, respectively, the condensed mass and stiffness matrices of size nc×nc , and {Xc (t)}, {Xc (t)} and {Fc (t)} represent the displacement, the acceleration and the excitation force vector of order nc , respectively, at the chosen master co-ordinates. Here the vectors {Xc (t)}, {Xc (t)} and {Fc (t)} are actually the same as the upper subvectors of the vectors {X(t)}, {X (t)} and {F(t)} in equation (1). If the excitation force is a harmonic force, the equation (2) can be re-written as [Ac ]{Xc }eivt=[Mc ]−1{Fc }eivt ,
i=z−1,
(3)
where [Ac ]=[−v 2Ic+M−1 c Kc ],
(4)
and {Xc } and {Fc } represent the amplitude vectors of response and force corresponding to the master co-ordinates, respectively. Here, one assumes that the structural system is excited by a harmonic force with unit amplitude from the first master co-ordinate to the last master co-ordinate. The response vectors of the structure can be obtained from nc equations [Ac ]{Xc1 }eivt=[Mc ]−1{Fc1 }eivt ,
[Ac ]{Xc2 }eivt=[Mc ]−1{Fc2 }eivt , . . . ,
[Ac ]{Xcnc }eivt=[Mc ]−1{Fcnc }eivt ,
(5)
where
89
1 {Fc1 }= 0 , * 0
89
0 {Fc2 }= 1 , . . . , * 0
89
0 {Fcnc }= 0 , * 1
(6)
and {Xc1 }, {Xc2 }, . . . , {Xcnc } are the amplitude vectors of the responses at the master co-ordinates with respect to the corresponding excitation forces. Equations (5) can be rewritten in matrix form as ivt −1 ivt [−v 2Ic+M−1 e , c Kc ][Vc ]e =[Mc ]
(7)
[Vc ]=[Xc1 , Xc2 , . . . , Xcmc ].
(8)
where
Since eivt is a non-zero factor, from equation (7) one has −1 [−v 2Ic+M−1 . c Kc ][Vc ]=[Mc ]
(9)
In order to make the condensed model best fit the original one, two essential requirements are introduced: (a) the condensed model retains nc natural frequencies and corresponding modes at the chosen master co-ordinates of interest from the original model;
.
604
(b) for the same harmonic forces applied at the master co-ordinates, the response matrix [Vc ] in equation (9) determined from the condensed model is the same as the one determined from the original model. 2.2. To meet the first requirement, the system matrix [M−1 c Kc ] of the condensed model in equation (9) is determined as −1 [Bc ]=[M−1 , c Kc ]=[F][L][F]
(10)
where
&
l12 0 0 2 [L]= 0 l2 0 * * * 0 0 0
' &
0 0 , * ln2c
f11 [F]= f21 * fnc 1
f12 · · · f22 · · · * * fnc 2 · · ·
'
f1nc f2nc . * fnc nc
(11, 12)
Here, li represents the ith natural frequency, while fji represents the modal coefficient at jth master co-ordinate. The sequence of the eigenvectors is the same as the sequence of the upper nc elements in the response vector {X} in equation (1). All these eigenvalues and eigenvectors of interest can be obtained by solving the eigenvalue problem from the original model (i.e., equation (1)) which is represented by the mass and stiffness matrices of the full system. Hence the condensed model with the system matrix [Bc ] given by equation (10) precisely retains nc eigenvalues and corresponding eigenvectors at the master co-ordinates of interest from the original model. To meet the second requirement, the response matrix [Vc ] must be determined from the original structural system by the original model, of a large number of degrees of freedom, and this is described in the next section. Based on equation (9), from these matrices [Bc ] and [Vc ], the mass matrix of the condensed model can then be determined: i.e. [Mc ]=[Vc ]−1 [−v 2I+Bc ]−1 .
(13)
Consequently, the stiffness matrix is determined as [Kc ]=[Mc ][Bc ].
(14)
In respect to the lack of symmetry of the [K], [M] matrices, it should be noted that this is not a matter of computational deficiency, but is fundamental. The condensed model is a mathematical, rather than a physical one; it is not a reciprocal system: at a general frequency v, the response at i due to a force at j is not equal to the response at j due to that force at i. However, it is reciprocal at the chosen frequency v0 , or in other words the matrix [Kc−v02 Mc ] is symmetric. One more point is that the nc chosen natural frequencies, the corresponding modes and the responses matrix [Vc ] can be obtained by solving the eigenvalue problem of the finite element model which is presented by the mass and stiffness matrices of the structure considered. It requires no more advanced knowledge than do the other reduction procedures but a little bit more computation. 2.3. - To determine the dynamic response matrix [Vc ], let the excitation force be a simple harmonic force. Equation (1) can then be re-written as [A]{X}eivt={F}eivt ,
i=z−1,
(15)
605
[A]=[−v 2M+K],
(16)
where
and {X} and {F} represent the amplitude vectors of response and force vector corresponding to the master co-ordinates and the slave co-ordinates, respectively. As in the procedures described in section 2.1, one assumes that the structural system is excited by a harmonic force with unit amplitude from the first master co-ordinate to the last master co-ordinate. The response vectors of the structure can be obtained from the nc equations [A]{X1 }eivt={F1 }eivt ,
[A]{X2 }eivt={F2 }eivt , . . . ,
[A]{Xnc }eivt={Fnc }eivt ,
(17)
where
89
1 {F1 }= 0 , * 0
89
0 {F2 }= 1 , . . . , * 0
89
0 {Fnc }= * , 1 0
(18)
and {X1 }, {X2 }, . . . , {Xnc } are the amplitude vectors of the responses at the master co-ordinates and slave co-ordinates due to the corresponding excitation forces. Similarly, equation (17) can be re-written in matrix form as [A][V]eivt=[F]eivt ,
(19)
where [V]=[X1 , X2 , . . . , Xnc ],
[F]=[F1 , F2 , . . . , Fnc ].
(20)
Furthermore, equation (19) can be partitioned into the form
$
A11 A12 A12 A22
%$ % $ %
V11 ivt I e = 11 eivt , V21 0
(21)
where [A]=
$
%
A11 A12 , A21 A22
[V]=
$ %
V11 , V21
[F]=
%$I11 . 0
(22, 23)
In equation (21) the submatrix [V11 ] of size nc×nc contains all the amplitude vectors of the master co-ordinates in the case that the excitation force is applied from the first co-ordinate to the nc th co-ordinate. Since eivt is a non-zero factor, the following equations are obtained from equation (21): [A11 ][V11 ]+[A12 ][V21 ]=[I11 ],
[A21 ][V11 ]+[A22 ][V21 ]=[0].
(24, 25)
The submatrix [V21 ] can be determined from equation (25) as [V21 ]=−[A22 ]−1 [A21 ][V11 ].
(26)
Substituting equation (26) into equation (24) yields the solution for the submatrix [V11 ], i.e., −1 , [V11 ]=([A11 ]−[A12 ][A−1 22 ][A21 ])
(27)
.
606 and the inverse of the submatrix
[V11 ]−1=[A11 ]−[A12 ][A−1 22 ][A21 ].
(28)
Here, as long as the frequency of applied force is not equal to zero, the matrix does not have a singularity even if the structure considered is under free–free boundary conditions. The submatrix [V11 ] contains the dynamic contributions from the original co-ordinates due to the specified excitation force. This submatrix [V11 ] is actually the dynamic response matrix [Vc ] which is used to determine the condensed mass matrix in section 2.2. 2.4. - After the condensed model has been obtained, the response at the master co-ordinates due to the applied forces can be computed from equation (9). By following the reverse procedure described in section 2.3, the dynamic responses at the slave co-ordinates can also be obtained in terms of the computed responses at the master co-ordinates. Since the actual applied loads at the slave co-ordinates are considered to be zero, the response amplitudes at the original co-ordinates can be determined from equation (21) as
$
A11 A12 A21 A22
%6 7 6 7
Xc F = c , Xs 0
(29)
where {Xc } and {Xs } represent the response amplitudes at the master co-ordinates and the slave co-ordinates, respectively, and {Fc } represents the vector of applied loads at the master co-ordinates. Hence, from equation (29), the dynamic transformation between the condensed model and the original model can be obtained: i.e.,
67
Xc =[T][Xc ], Xs
(30)
where [T]=
$
%
Ic . −A−1 22 A21
(31)
Note that the dynamic transformation [T] is a function of the frequency of the applied loads. If the applied loads are static, i.e., v=0, the transformation becomes the well-known one [T]=
$
%
Ic . −K−1 22 K21
(32)
It should be pointed out that in this particular case, although the procedure presented and the Guyan method produce the same transformation between the retained and eliminated co-ordinates and the same stiffness matrix, the two procedures still produce different mass matrices. The procedure presented has the desired advantage that the resulting model retains precisely the natural frequencies and modes of interest.
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3. APPLICATION WITH CONSIDERATION OF VARIOUS APPLIED FORCES
3.1. The mass and stiffness of the dynamically condensed model represented by equations (13) and (14) are functions of the frequency of the excitation forces. The accuracy of the model in describing the response of the structural system depends on how close the frequency of the excitation force used for the condensation is to the actually applied force. Some suggestions for selecting the frequencies to be used to produce the condensed models are provided for different types of known applied forces in this section. Case 3.1.1, {Ft }=constant. When the applied force is constant or the frequency is equal to zero, the problem becomes a static problem. All the elements of the submatrix [V11 ] represent the static deflections at the master coordinates when the force is applied to the structure. The stiffness of the reduced model is equal to the inverse of matrix [V11 ]. One more point is that, in this special case, the matrix [A] in equation (16) becomes the stiffness matrix of the original system. Therefore, the reduced stiffness matrix is determined as [Kc ]=[K11 ]−[K12 ][K22 ]−1 [K21 ].
(33)
This stiffness matrix is exactly the same as that obtained by the Guyan method. It should be pointed out that, however, the mass matrix determined according to equation (13) is different from that obtained by the Guyan method. Case 3.1.2, {F(t) }={F0 }eivt . Theoretically, the dynamically condensed model of a structural system can describe the system precisely as long as the same frequency v is used to produce the condensed model. This is the case for rotating structures of most rotating machines since these machine are operated at certain frequencies. Case 3.1.3, {F(t) }=amj=1 {Fj }eivj t . For each frequency of the excitation force, the corresponding reduced mass and stiffness matrices can be obtained in case 3.1.2; the results are [Mcj ],
[Kcj ],
j=1, 2, . . . , m.
(34)
Here one can compute the dynamic responses of the structure considered separately in terms of the above reduced models, due to the corresponding harmonic components of the applied forces. Since the structure is linear, accurate dynamic responses can then be obtained from the sum of those computed separately. On the other hand, for simplicity, the approximated condensed mass and stiffness matrices can be determined as, [Mcf ]=
1 m s F [M ], Ff j=1 j cj
[Kcf ]=
1 m s F [K ], Ff j=1 j cj
(35, 36)
where m
Ff= s Fj .
(37)
j=1
The condensed model with these mass and stiffness matrices probably is fairly accurate but may not be the best one. An alternative condensed model can be produced by choosing just one harmonic component as the applied load. The frequency of the chosen component should be in the middle or below the middle of the range of frequency of interest. 3.2. - In engineering practice, there are forced vibration problems with non-harmonic periodic or random applied forces. It is not possible to obtain such a condensed model from the
608
.
original one to describe the structural system precisely. However, based on the information about the applied forces, it is always possible to obtain a condensed model which is fairly accurate in describing the dynamic behaviour of the structural system. Case 3.2.1, Known non-harmonic applied forces. Whether or not the non-harmonic applied forces are periodic, non-periodic or random, spectral analysis of the forces can always provide valuable information about the harmonic components and the corresponding amplitudes of the applied forces. Then, by following the procedure described in case 3.1.3, a possible condensed model which is fairly accurate can be determined. Case 3.2.2, Unknown applied forces. The procedure for condensing the original model dynamically is the same as that described in Case 3.1.3. But the excitation forces have unit amplitude and the frequencies are chosen to be between the natural frequencies of interest of the original structural system. 3.3. , - For most industrial applications, it is recommended that the lowest few natural frequencies and the corresponding modes of the original structural system be kept in the reduced model. It is recommended that the co-ordinates of interest such as the co-ordinates at which the external loads are applied are retained in the reduced model. As described in section 2.3, theoretically the responses of all eliminated co-ordinates can be computed. 4. DISCUSSION
For constant applied loads, i.e., v=0, if the structure considered is not under free–free boundary conditions, the stiffness matrix obtained by the method presented is exactly the same as that obtained by the Guyan method. The mass matrix is determined from the relationship between the condensed stiffness matrix and the retained natural frequencies and the corresponding modes at the master co-ordinates. Therefore, the resulting mass matrix is different from the one obtained by the Guyan method. The fundamental difference between the Guyan method and the method presented is that, the Guyan method is based only on the pure static transformation and the resulting model does not retain the natural frequencies and corresponding modes from the original one, but the method presented is based on the dynamic transformation and the chosen nc natural frequencies and corresponding modes at the master co-ordinates from the original model and the resulting model retains precisely nc natural frequencies and corresponding modes. Besides, the condensed model produced by the method presented has great accuracy in describing the dynamic behavior of the original structural system at the frequency which was used to produce the model. Since these retained natural frequencies, modes and the dynamic behavior at the chosen frequencies of the structural systems are what one is mostly interested in, such a condensed model has great advantages in describing the dynamic response of structures such as the rotor shafts of most rotating machines. There are some similarities between the method presented and the component mode method [4]. Instead of using the purely static transformation between the eliminated co-ordinates and the master co-ordinates, the component mode method makes use of a few of the lowest component modes of interest and the measured displacements to produce a reduced model for the dynamic analysis. The method improves the accuracy of the resulting model provided that good quality measurement of the displacements is guaranteed. Compared with the method presented here, the model obtained by the component method does not accurately retain the higher modes as the number of
609
component modes used to produce the reduced model is smaller than the number of chosen to be preserved in the model. Comparing the presented method with the experimental identification method [5] shows that theoretically both methods produce the same model provided that there are no errors included in all the required measurements. The advantage of the experimental identification method is that the model obtained preserves the physical properties of the original structure such as the boundary conditions and the damping properties more accurately. The modal parameters identified experimentally are more accurate than those obtained analytically for complicated structures [6]. The disadvantages, however, are the large work loads and the difficulty of applying large excitation forces for the identification of large complex structures. Theoretically, there is no special restriction on choosing the natural frequencies and the corresponding modes to be kept in the condensed model. However, since most applied loads are in the range of the lower frequencies, it is the author’s suggestion to keep a chosen number of lowest modes of interest in the condensed model unless one has particular interests in other higher modes. For the same reason, it is suggested that mainly lower frequencies are used to produce the condensed model. Theoretically, there would be no limitation on the application of the procedure presented to the modelling of large structural systems. Even for a full global system, as long as it is a linear system, the full-size mass and stiffness matrices can always be obtained analytically and, therefore, the original model of the system can also be condensed. 5. APPLICATION EXAMPLES
In principle, the method presented can be applied to all the problems for which reduced models are required for the dynamic analysis. To examine the applicability of the dynamic condensation method, two examples are given in this section for the modelling of two common structures. A few of the condensed models of the structures corresponding to different frequencies of the applied forces were obtained by means of the dynamic condensation method. Comparisons between the performances of the condensed models obtained by means of the dynamic condensation method and that of the models obtained by the Guyan method are conducted. 5.1. 1, Figure 1 shows schematically a cantilever pipe. The pipe was originally modelled by 20 equally sized beam elements. The number of original co-ordinates is 40 (each node has two degrees of freedom). As the first application, the five lowest natural frequencies and the corresponding modes of the pipe were chosen to retain in the condensed model. The number of degrees of freedom of the model is five. The first, fifth, ninth, 13th and 17th original co-ordinates were chosen as the master co-ordinates. Seven condensed models of the pipe were obtained by
Figure 1. A cantilever pipe. SGP A15; L=1000 mm, D=21·7 mm, d=16·1 mm.
.
610
T 1 Comparison between the natural modes of model M10 and the true ones of model M01 of the cantilever pipe Freq. (Hz) Model Mode f1i f5i f9i f13i f17i
f1
f2
M01 M10 19·5 19·5 1·00 1·00 0·726 0·726 0·462 0·462 0·231 0·231 0·064 0·064
M01 121·0 1·00 0·073 −0·590 −0·689 −0·308
f3
f4
f5
M10 M01 M10 M01 M10 M01 M10 121·0 334·0 338·0 642·0 668·0 1036·0 1079·0 1·00 1·00 1·00 1·00 1·00 1·00 1·00 0·073 −0·392 −0·384 −0·651 −0·638 −0·628 −0·704 −0·589 −0·485 −0·481 0·314 0·312 0·728 0·890 −0·688 0·522 0·514 0·345 0·338 −0·709 −0·882 −0·308 0·623 0·616 −0·775 −0·754 0·657 0·760
using the two different methods or the same method but different frequencies of the applied forces. Model M10 was obtained by means of the Guyan method. Models M11, M12, M13 and M14 were obtained by means of the dynamic condensation method for the frequencies of 0 Hz, 50 Hz, 200 Hz and 500 Hz, respectively. Model M15 was obtained by taking the average of the three models of M11, M12 and M13. Model M16 was obtained by the dynamic condensation method with the harmonic forces,
0
19
1 i100kpt e k k=1
f(t)= s
1>
3·5477,
0
1
19 1 s =3·5477 . k k=1
The eigenvalue–eigenvector analyses of the models M11, M12, M13, M14, M15 and M16 clearly show that these models have retained the five lowest natural frequencies and the corresponding modes at the master co-ordinates of the pipe, which are exactly the same as those of the original finite element model. The natural frequencies and the corresponding modes of model M10 and the original model M01 are given in Table 1. It is noted that the results for M01 are also true for M11, M12, M13, M14, M15 and M16 From the table, it is seen that model M10 has preserved the first three lowest natural modes very well. But for the two higher modes, the model is not very accurate. The modal shapes of the highest mode of the model have relative errors of up to 20% when compared with the corresponding true values. The condensed mass and stiffness matrices of models M10, M11 and M15 are listed in Table 2. From this table, it is seen that the condensed stiffness matrices of model M10 and M11 are exactly the same but their condensed mass matrices are different. To compare the performances of these models with each other, dynamic analysis based on these models and on the original finite element model have been carried out by applying a few harmonic forces at the fifth original co-ordinate. The frequencies of the applied forces were 0 Hz, 50 Hz, 200 Hz, 500 Hz, 700 Hz and 1200 Hz. The relative errors, which are represented by percentages, between the response amplitudes obtained from the condensed models and those obtained from the original model were computed. The results are given in Table 3. In Table 3, from the left, the first column lists the frequencies of the applied loads, the second column lists the numbers of the master co-ordinates, the third column lists the response amplitudes, which were normalized according to the maximum computed from the original model, and the rest of the columns in the table are the relative errors. The results given in Table 3 clearly show that the accuracy of the model obtained by the dynamic condensation method depends on the frequencies which are used to produce
−0·0143 0·0244 0·2254 0·0295 −0·0162 −0·0173 0·0203 0·2347 0·0236 −0·0190 −0·0193 0·0197 0·2407 0·0236 −0·0205
K 0·0742 0·0407 G 0·0407 0·2387 G −0·0143 0·0244 G 0·0044 −0·0133 k −0·0013 0·0051
K 0·0749 0·0406 0·2431 G 0·0391 G −0·0131 0·0188 G 0·0048 −0·0125 k −0·0027 0·0090
K 0·0783 0·0431 G 0·0418 0·2495 G −0·0148 0·0182 G 0·0057 −0·0133 k −0·0032 0·0100
Mass (kg)
0·0106 −0·0135 0·0225 0·2378 0·0323
0·0093 −0·0129 0·0227 0·2324 0·0314
0·0044 −0·0133 0·0295 0·2225 0·0346
−0·0062 L K 0·3052 0·0077 G G −0·6761 −0·0157 G G 0·4580 0·0271 G G −0·0994 0·2232 l k 0·0147
−0·6707 1·8122 −1·7046 0·6839 −0·1354
−0·6675 1·7862 −1·6725 0·6808 −0·1620
Model M11 −0·0054 L K 0·2997 0·0070 G G −0·6675 −0·0146 G G 0·4512 0·0266 G G −0·1026 0·2187 l k 0·0244 Model M15
−0·6675 1·7862 −1·6725 0·6808 −0·1620
Model M10 −0·0013 L K 0·2997 0·0051 G G −0·6675 −0·0162 G G 0·4512 0·0346 G G −0·1026 0·2092 l k 0·0244
Stiffness (N/m)
0·4316 −1·6983 2·5223 −1·8684 0·7308
0·4512 −1·6725 2·4670 −1·8344 0·7507
0·4512 −1·6725 2·4670 −1·8344 0·7507
−0·0684 0·6858 −1·8766 2·5949 −2·0129
−0·1026 0·6808 −1·8344 2·5369 −1·9964
−0·1026 0·6808 −1·8344 2·5369 −1·9964
T 2 The condensed mass and stiffness matrices of the cantilever pipe (5 d.o.f. models)
−0·0054 L −0·1523 G 0·7622 G ×E06 −2·0443 G 3·2704 l
0·0244 L −0·1620 G 0·7507 G ×E06 −1·9964 G 3·2177 l
0·0244 L −0·1620 G 0·7507 G ×E06 −1·9964 G 3·2177 l
611
.
612
T 3 Effect of condensation procedures on dynamic responses of the cantilever pipe (5 d.o.f. models) Freq. (Hz)
No.
Amp. M01
M10
0
1 5 9 13 17
1·00 0·728 0·461 0·228 0·063
0·00 0·00 0·00 0·00 0·00
50
1 5 9 13 17
1·00 0·717 0·470 0·249 0·074
200
1 5 9 13 17
500
Relative errors of response amplitudes (%) M11 M12 M13 M14 M15 0·0 0·0 0·0 0·0 0·0
−0·05 −0·05 −0·05 −0·05 −0·05
−0·07 −0·07 −0·07 −0·07 −0·07
−0·05 −0·05 −0·05 −0·06 −0·06
1·00 0·340 0·180 0·183 0·093
−0·90 −1·59 −0·57 1·28 1·89
−0·90 −1·05 −0·31 0·38 0·78
1 5 9 13 17
1·00 0·009 0·854 −0·028 −0·841
5·39 −786·00 −1·68 129·00 −1·15
−7·28 −7·20 −6·14 −10·2 −10·5 −6·81 −2·54 −2·51 −2·15 −22·6 −22·4 −19·3 3·71 3·67 3·14
700
1 5 9 13 17
1·00 −0·914 0·103 0·587 −0·977
143·00 −101·00 419·00 81·2 −105·0
8·74 8·70 −10·4 −10·3 26·8 26·7 8·08 8·03 −9·53 −9·48
1200
1 5 9 13 17
1·00 −0·882 0·532 −0·510 0·492
44·9 −55·1 83·2 −65·0 34·2
M16
−0·82 −0·81 −0·80 −0·79 −0·79
−5·14 −5·10 −5·05 −5·00 −4·99
−0·29 −0·29 −0·29 −0·28 −0·28
−2·68 −2·65 −2·62 −2·60 −2·60
0·0 0·0 0·0 0·0 0·0
0·71 0·76 0·81 0·85 0·86
4·75 5·06 5·39 5·67 5·73
0·22 0·24 0·25 0·27 0·27
2·44 2·61 2·79 2·94 2·96
−0·85 −0·99 −0·29 0·35 0·73
0·0 0·0 0·0 0·0 0·0
4·78 5·54 1·71 −1·90 −4·16
−0·58 −0·68 −0·20 0·24 0·50
2·08 2·39 0·68 −1·01 −2·18
8·15 −9·64 25·0 7·49 −8·85
0·0 0·0 0·0 0·0 0·0
−6·87 −3·22 −9·16 −20·3 −2·40 −1·21 −21·4 −8·22 3·50 1·66
4·76 8·53 −5·44 −10·1 14·4 26·2 4·25 7·86 −5·06 −9·29
6·43 −7·63 19·4 5·92 −6·97
19·3 19·3 19·2 18·4 19·3 18·8 −27·6 −27·6 −27·1 −24·3 −27·4 −25·8 18·5 18·5 18·3 16·8 18·4 17·5 −2·61 −2·59 −2·54 −2·28 −2·58 −2·46 −9·33 −9·35 −9·28 −8·86 −9·32 −8·97
the model. The model has greatest accuracy in describing the dynamic response of a structure if the frequency of the actual applied loads is the same or close enough to the one used to produce the model. The results for model M15 also show that a model which is very accurate within a wide range of frequencies can be obtained as long as a few optimum frequencies are chosen to produce the model. The model obtained by the Guyan method is very accurate only in the range of the lower frequencies. In the range of higher frequencies (e500 Hz), however, the accuracy of the model is unacceptable. Compared with the model obtained by the Guyan method, the other models perform better within a wide range of frequencies which are of interest. Furthermore, the results obtained for the highest frequency (1200 Hz) of the applied load show that, for the applied loads of which the frequency is larger than the highest natural frequency retained in the models, none of the models are of good accuracy. Hence, the condensed models can be used only for the dynamic analysis of a structure of which the frequencies of the actual applied loads are lower than the highest retained natural frequency.
613
Figure 2. A free—free flexible rotor shaft.
5.2. 2, – The second structure analyzed is a free–free flexible rotor shaft, shown in Figure 2. The rotor was used in an experimental installation which consists of the rotor, four journal bearings and a steel–concrete foundation. Originally, the rotor was modelled by the finite element method with 55 elements in total. Based on this original FEM model, denoted as M02, more applications of the dynamic condensation procedure have been carried out. As the second application, a nine-degrees-of-freedom (9 d.o.f.) model of the rotor shaft is considered. The nine lowest natural frequencies and the corresponding modes, of which two are the free–free modes (i.e., f1=f2=0), and nine master co-ordinates were chosen for retention in the model. Among the chosen master co-ordinates, the second, fourth, sixth and eighth co-ordinates correspond to the four supporting bearing stations and the other five correspond to the unsupported stations; see Figure 2. To investigate the influence of the frequencies used in the condensation on the performance of the resulting model, four condensed models of the rotor were obtained by means of the dynamic condensation method and one model was obtained by the Guyan method. Model M20 was obtained by means of the Guyan method. Models M21, M22 and M23 were obtained by means of the dynamic condensation method with the frequencies 35 Hz, 75 Hz and 125 Hz respectively. Model M24 was obtained by using the applied force f(t)=(ei20pt+0·2ei70pt+0·04ei150pt )/1·24. The eigenvalue–eigenvector analyses of the models M21, M22, M23 and M24 clearly show that these models have retained the nine lowest natural frequencies and the corresponding modes of the original model M02. Table 4 gives the nine lowest natural frequencies and the corresponding modes of model M02 and those of model M20. It is noted that the results for M02 are also true for M21, M22, M23 and M24. Since theoretically the first two natural frequencies should be zero (corresponding to free–free motion), they can be set to zero in the condensation procedure. The results in the table show that model M20 does not preserve the higher modes accurately. T 4 Comparison between the natural frequencies of model M20 and the true ones of model M02 of the rotor shaft Model Freq. (Hz)
M02
M20
f1 f2 f3 f4 f5 f6 f7 f8 f9
0·001 0·941 21·275 49·398 105·791 151·636 234·115 490·338 546·990
0·849 1·526 21·281 49·464 106·584 158·014 250·778 508·066 677·594
.
614
The response amplitudes at the master co-ordinates corresponding to a few harmonic applied forces were computed based on the obtained models. A few harmonic forces were applied at the third master co-ordinate and the frequencies of the forces were 35 Hz, 75 Hz, 125 Hz and 175 Hz. The results are given in Table 5. The structure of Table 5 is the same as that of Table 3. The results listed in Table 5 show that the models M21, M22, M23 and M24 perform better than model M20 does in the range of frequency up to 175 Hz. If the actual applied forces have a few harmonic components in the frequency range (0–175 Hz), model M24 probably has the best performance in describing the dynamic response of the rotor shaft.
T 5 Effect of condensation procedures on dynamic responses of the rotor shaft (9 d.o.f. models) Freq. (Hz)
No.
Amp. M02
35
1 2 3 4 5 6 7 8 9
1·000 0·570 0·296 0·249 0·327 0·358 0·158 −0·225 −0·665
0·96 0·96 0·12 1·59 1·44 1·09 0·81 1·58 1·66
0·0 0·0 0·0 0·0 0·0 0·0 0·0 0·0 0·0
0·26 0·21 0·00 −0·13 0·09 0·24 0·32 −0·15 −0·13
0·88 0·70 0·00 −0·43 0·29 0·80 1·08 −0·50 −0·44
−0·04 −0·04 0·00 0·02 −0·02 −0·04 −0·06 0·03 0·02
75
1 2 3 4 5 6 7 8 9
0·167 0·249 0·568 0·842 0·587 0·094 −0·248 0·096 1·0000
7·04 0·48 0·88 0·15 0·31 3·53 0·28 2·01 0·07
1·13 −0·09 −0·18 −0·15 −0·25 −0·94 0·48 −0·30 −0·78
0·0 0·0 0·0 0·0 0·0 0·0 0·0 0·0 0·0
−2·65 0·22 0·43 0·36 0·59 2·22 −1·13 0·70 1·83
1·33 −0·10 −0·21 −0·18 −0·29 −1·10 0·56 −0·35 −0·91
125
1 2 3 4 5 6 7 8 9
0·388 −0·203 −0·462 −0·166 0·346 0·502 −0·105 −0·199 1·000
34.3 16·6 6·57 41·0 20·3 2·47 12·6 4·00 0·92
9·75 −4·89 −1·95 10·4 4·75 2·28 −2·34 1·37 −0·37
6·84 −3·44 −1·37 7·30 3·33 0·20 −1·64 0·96 −0·26
0·0 0·0 0·0 0·0 0·0 0·0 0·0 0·0 0·0
10·2 −5·13 −2·04 10·9 4·98 0·30 −2·45 1·43 −0·39
175
1 2 3 4 5 6 7 8 9
1·000 −0·560 −0·533 0·213 0·243 −0·072 −0·026 0·120 −0·285
23·0 15·7 4·99 62·6 7·15 164·3 20·0 64·5 78·1
−7·22 5·68 1·29 −23·1 −2·28 61·5 −7·82 −23·7 29·0
−6·22 4·89 1·12 −19·9 −1·98 52·9 −6·71 −20·4 25·0
M20
Relative errors of amplitudes (%) M21 M22 M23
−3·85 3·02 0·69 −12·3 −1·25 32·6 −4·10 −12·6 15·4
M24
−7·39 5·81 1·32 −23·6 −2·34 62·9 −7·98 −24·2 29·7
615
6. CONCLUSIONS
A dynamic condensation procedure has been presented for reducing the size of mass and stiffness matrices for a structure for the dynamic analysis. The condensed model is formulated in terms of the chosen natural frequencies, the corresponding modes and the dynamic responses at the retained master co-ordinates, which are determined from the original structural system. As a result, the resulting model satisfies the two essential requirements introduced into the dynamic condensation procedure: (a) it remains the chosen natural frequencies and the corresponding modes at the master co-ordinates from the original model; (b) the responses of the condensed system at the master co-ordinates due to forces at these co-ordinates, at one particular chosen frequency which is often identical to the operational one of a rotating machine, are the same as those of the original system. For constant loads, if the structure considered is not under free–free boundary conditions, the condensed stiffness matrix is exactly the same as the one obtained by the Guyan method but the two mass matrices are different. Unlike the Guyan method, that cannot accurately preserve the higher modes of interest in the condensed model, the dynamic condensation method can accurately retain both the lower and higher modes of interest in the resulting model. The results obtained from two examples show that the condensed model produced directly by means of the dynamic condensation method from the finite element model of the original structure retains the chosen number of lowest natural frequencies and the corresponding modes precisely. For known applied loads, the condensed model obtained by the dynamic condensation method can describe the dynamic responses of the structure with very good accuracy. For unknown applied loads, but within a limited range of frequency, the models produced by using optimal frequencies of the approximated applied loads have better performance than do the condensed models obtained by the Guyan method. The frequencies used to produce the condensed model of a structure should be chosen within the range of interest. The closer the frequencies are to those of the actual applied loads, the better the performance of the resulted model is in describing the dynamic responses of a structure. ACKNOWLEDGMENTS
The author would like to thank Dr J. M. Krodkiewski at the University of Melbourne and Professor S. Hayama at the University of Tokyo for valuable discussions and encouragement. REFERENCES 1. M. J. T, R. W. C, H. C. M and L. J. T 1956 Journal of Aeronautics Science 23, 805–823. Stiffness and deflection analysis of complex structures. 2. R. J. G 1965 American Institute of Aeronautics and Astronautics Journal 3, 380. Reduction of stiffness and mass matrices. 3. G. H. S 1984 Journal of Sound and Vibration 94, 150–153. Comment on the substructure synthesis methods. 4. A. C 1987 Journal of Sound and Vibration 117, 277–288. A component mode method for modelling the dynamics of turbo-generator sets. 5. N. Z and S. H 1991 International Journal of the Japan Society of Mechanical Engineers, Series 3, 34(1), 64–71. Identification of structural system parameters from time domain data (method for the identification of physical parameters of structural system from experimental data). 6. N. Z and S. H 1990 International Journal of the Japan Society of Mechanical Engineers, Series 3, 33(2), 168–175. Identification of structural system parameters from time domain data (identification of global modal parameters of structural system by improved state variable method).
DYNAMIC CONDENSATION OF MASS AND STIFFNESS MATRICES N. Z† Department of Mechanical Engineering, Monash University, Clayton, Victoria 3168, Australia (Received 17 September 1993, and in final form 27 January 1995) Details are given of a procedure for condensing the mass and stiffness matrices of a structure for dynamic analysis. The condensed model is based on choosing nc master co-ordinates Xc from the original n co-ordinates X and nc natural frequencies and the corresponding modes of the original model. The model is constructed so that (1) it has nc natural frequencies equal to those of the original model, (2) the modes fijc i, j=1, 2, . . . , nc are the same as those for the master co-ordinates in the corresponding modes of the original and (3) the responses of the condensed system at the co-ordinates Xc due to forces at these co-ordinates, at one particular chosen frequency, are the same as those of the original system. The natural frequencies, the corresponding modes and the dynamic responses used for the condensation can be obtained from finite element analysis of the original structure. The method has been applied to the modelling of two common structures to examine its applicability. Comparisons between the performance of the condensed models obtained by means of the dynamic condensation method and that of the models obtained by the Guyan method have been conducted. The results of the examples show that the condensed models determined by the dynamic condensation method retain the natural frequencies and modal shapes and perform better in describing the dynamic responses of the structures than do the corresponding models obtained by the Guyan method. 7 1995 Academic Press Limited
1. INTRODUCTION
It is often necessary to reduce the computation time of dynamic analysis of a complex structure. One of the effective ways of doing so is to reduce simultaneously the size of the non-diagonal mass and stiffness matrices for the eigenvalue analysis and response analysis. For the static analysis of a structure, Turner et al. [1] presented a procedure to eliminate the co-ordinates at which no force is applied and to obtain a small size stiffness matrix without losing accuracy. For the dynamic analysis, the most common procedure for reducing the size of the original model was presented by Guyan [2]. In the case of the reduced stiffness matrix, if the applied loads are static, none of the structural complexity applied is lost since all elements of the original stiffness contribute. However, to reduce the mass matrix, the purely static transformation between the eliminated co-ordinates and those retained co-ordinates is employed. The result is that the eigenvalue–eigenvector problem is closely but not exactly preserved [2]. Sotiropoulos [3] developed an improved procedure by taking the additional inertia effects into consideration, but he still retained the purely static transformation between the retained and eliminated.
†Present address: School of Mechanical Engineering, University of Technology, Sydney, 1 Broadway, Sydney, NSW 2007, Australia.
601 0022–460X/95/490601+15 $12.00/0
7 1995 Academic Press Limited
602
.
Craggs [4] presented a component mode method for modelling the dynamic properties of turbo-generator sets. The method greatly improves the accuracy of the reduced model of structural systems by introducing a few of the lowest component modes and measured displacements. But for the higher modes of the reduced model, there still is some kind of deviation from the original modes if the number of component modes introduced is limited. For the vibration analysis of a complex structure excited by one or a number of harmonic forces in a wide range of frequencies, the reduced mass and stiffness matrices which have a good accuracy are required. Zhang and Hayama [5] developed a method for identifying the mass and stiffness matrices with a limited number of modes of a structure from experiment and the results showed that the identified mass and stiffness matrices are very accurate in the analysis of the forced vibration of structures. However, in practice, it is difficult to apply the method to large structures because of the large inertia. For dynamic analysis of rotor systems of rotating machinery, a condensed model of the rotor shaft is often needed and it is also required that the model has natural frequencies and corresponding modes which are as close as possible to those of the original model. The forces applied to a rotor system are usually due to the unbalance of the rotating parts and their frequencies are often identical to the operational frequencies. Hence, condensed models of rotor shafts are also required to perform accurately in describing their dynamic behavior at these operational frequencies. The common shortcoming of the available condensation procedures is that the resulting models do not retain precisely the natural frequencies and the corresponding modes from the original models. Besides, at the operating frequency, the resulting models can describe the dynamic behavior of the original system only approximately. In what follows, a procedure is presented for determining the condensed mass and stiffness matrices of a structure in which the dynamic contributions from all of the original co-ordinates of the structure are considered. The condensed model is based on choosing nc master co-ordinates Xc from the original n co-ordinates X and nc natural frequencies and corresponding modes of the original model. The model is constructed so that: (1) it has nc natural frequencies equal to those of the original model, (2) the modes fijc i, j=1, 2, . . . , nc are the same as those for the master co-ordinates in the corresponding modes of the original model and (3) the responses of the condensed system at the co-ordinates Xc due to forces at these co-ordinates, at one particular chosen frequency, are the same as those of the original system. The method has been applied to the modelling of two common structures, i.e., a cantilever pipe and a rotor shaft, to examine its applicability. Comparisons between the performance of the condensed models obtained by means of the dynamic condensation method and that of the models obtained by the Guyan method have been conducted. 2. PROCEDURE OF DYNAMIC CONDENSATION
2.1. A continuous structure can be represented with good accuracy by a discrete model of order n as long as the frequency range of interest is limited. For simplicity, it is assumed that the structure considered is free of damping. The forced motion of the structure is therefore described by the differential equation [M]{X (t)}+[K]{X(t)}={F(t)},
(1)
where [M] and [K] represent the mass and stiffness matrices of size n×n, respectively, and {X(t)}, {X (t)} and {F(t)} represent the displacement, the acceleration and the excitation
603
force vector of order n, respectively. Here it is emphasized that the response vector and the force vector are arranged in such a way that the upper nc elements of the vectors correspond to those master co-ordinates which one wants to keep in the reduced model. The lower ne (where ne=n−nc ) elements are the co-called slave co-ordinates which one wants to eliminate from the reduced model. If one chooses nc degrees of freedom of the original structural system to be retained in the condensed model, the motion of the structure at the chosen master co-ordinates can then be described by the equation [Mc ]{Xc (t)}+[Kc ]{Xc (t)}={Fc (t)},
(2)
where [Mc ] and [Kc ] are assumed to be, respectively, the condensed mass and stiffness matrices of size nc×nc , and {Xc (t)}, {Xc (t)} and {Fc (t)} represent the displacement, the acceleration and the excitation force vector of order nc , respectively, at the chosen master co-ordinates. Here the vectors {Xc (t)}, {Xc (t)} and {Fc (t)} are actually the same as the upper subvectors of the vectors {X(t)}, {X (t)} and {F(t)} in equation (1). If the excitation force is a harmonic force, the equation (2) can be re-written as [Ac ]{Xc }eivt=[Mc ]−1{Fc }eivt ,
i=z−1,
(3)
where [Ac ]=[−v 2Ic+M−1 c Kc ],
(4)
and {Xc } and {Fc } represent the amplitude vectors of response and force corresponding to the master co-ordinates, respectively. Here, one assumes that the structural system is excited by a harmonic force with unit amplitude from the first master co-ordinate to the last master co-ordinate. The response vectors of the structure can be obtained from nc equations [Ac ]{Xc1 }eivt=[Mc ]−1{Fc1 }eivt ,
[Ac ]{Xc2 }eivt=[Mc ]−1{Fc2 }eivt , . . . ,
[Ac ]{Xcnc }eivt=[Mc ]−1{Fcnc }eivt ,
(5)
where
89
1 {Fc1 }= 0 , * 0
89
0 {Fc2 }= 1 , . . . , * 0
89
0 {Fcnc }= 0 , * 1
(6)
and {Xc1 }, {Xc2 }, . . . , {Xcnc } are the amplitude vectors of the responses at the master co-ordinates with respect to the corresponding excitation forces. Equations (5) can be rewritten in matrix form as ivt −1 ivt [−v 2Ic+M−1 e , c Kc ][Vc ]e =[Mc ]
(7)
[Vc ]=[Xc1 , Xc2 , . . . , Xcmc ].
(8)
where
Since eivt is a non-zero factor, from equation (7) one has −1 [−v 2Ic+M−1 . c Kc ][Vc ]=[Mc ]
(9)
In order to make the condensed model best fit the original one, two essential requirements are introduced: (a) the condensed model retains nc natural frequencies and corresponding modes at the chosen master co-ordinates of interest from the original model;
.
604
(b) for the same harmonic forces applied at the master co-ordinates, the response matrix [Vc ] in equation (9) determined from the condensed model is the same as the one determined from the original model. 2.2. To meet the first requirement, the system matrix [M−1 c Kc ] of the condensed model in equation (9) is determined as −1 [Bc ]=[M−1 , c Kc ]=[F][L][F]
(10)
where
&
l12 0 0 2 [L]= 0 l2 0 * * * 0 0 0
' &
0 0 , * ln2c
f11 [F]= f21 * fnc 1
f12 · · · f22 · · · * * fnc 2 · · ·
'
f1nc f2nc . * fnc nc
(11, 12)
Here, li represents the ith natural frequency, while fji represents the modal coefficient at jth master co-ordinate. The sequence of the eigenvectors is the same as the sequence of the upper nc elements in the response vector {X} in equation (1). All these eigenvalues and eigenvectors of interest can be obtained by solving the eigenvalue problem from the original model (i.e., equation (1)) which is represented by the mass and stiffness matrices of the full system. Hence the condensed model with the system matrix [Bc ] given by equation (10) precisely retains nc eigenvalues and corresponding eigenvectors at the master co-ordinates of interest from the original model. To meet the second requirement, the response matrix [Vc ] must be determined from the original structural system by the original model, of a large number of degrees of freedom, and this is described in the next section. Based on equation (9), from these matrices [Bc ] and [Vc ], the mass matrix of the condensed model can then be determined: i.e. [Mc ]=[Vc ]−1 [−v 2I+Bc ]−1 .
(13)
Consequently, the stiffness matrix is determined as [Kc ]=[Mc ][Bc ].
(14)
In respect to the lack of symmetry of the [K], [M] matrices, it should be noted that this is not a matter of computational deficiency, but is fundamental. The condensed model is a mathematical, rather than a physical one; it is not a reciprocal system: at a general frequency v, the response at i due to a force at j is not equal to the response at j due to that force at i. However, it is reciprocal at the chosen frequency v0 , or in other words the matrix [Kc−v02 Mc ] is symmetric. One more point is that the nc chosen natural frequencies, the corresponding modes and the responses matrix [Vc ] can be obtained by solving the eigenvalue problem of the finite element model which is presented by the mass and stiffness matrices of the structure considered. It requires no more advanced knowledge than do the other reduction procedures but a little bit more computation. 2.3. - To determine the dynamic response matrix [Vc ], let the excitation force be a simple harmonic force. Equation (1) can then be re-written as [A]{X}eivt={F}eivt ,
i=z−1,
(15)
605
[A]=[−v 2M+K],
(16)
where
and {X} and {F} represent the amplitude vectors of response and force vector corresponding to the master co-ordinates and the slave co-ordinates, respectively. As in the procedures described in section 2.1, one assumes that the structural system is excited by a harmonic force with unit amplitude from the first master co-ordinate to the last master co-ordinate. The response vectors of the structure can be obtained from the nc equations [A]{X1 }eivt={F1 }eivt ,
[A]{X2 }eivt={F2 }eivt , . . . ,
[A]{Xnc }eivt={Fnc }eivt ,
(17)
where
89
1 {F1 }= 0 , * 0
89
0 {F2 }= 1 , . . . , * 0
89
0 {Fnc }= * , 1 0
(18)
and {X1 }, {X2 }, . . . , {Xnc } are the amplitude vectors of the responses at the master co-ordinates and slave co-ordinates due to the corresponding excitation forces. Similarly, equation (17) can be re-written in matrix form as [A][V]eivt=[F]eivt ,
(19)
where [V]=[X1 , X2 , . . . , Xnc ],
[F]=[F1 , F2 , . . . , Fnc ].
(20)
Furthermore, equation (19) can be partitioned into the form
$
A11 A12 A12 A22
%$ % $ %
V11 ivt I e = 11 eivt , V21 0
(21)
where [A]=
$
%
A11 A12 , A21 A22
[V]=
$ %
V11 , V21
[F]=
%$I11 . 0
(22, 23)
In equation (21) the submatrix [V11 ] of size nc×nc contains all the amplitude vectors of the master co-ordinates in the case that the excitation force is applied from the first co-ordinate to the nc th co-ordinate. Since eivt is a non-zero factor, the following equations are obtained from equation (21): [A11 ][V11 ]+[A12 ][V21 ]=[I11 ],
[A21 ][V11 ]+[A22 ][V21 ]=[0].
(24, 25)
The submatrix [V21 ] can be determined from equation (25) as [V21 ]=−[A22 ]−1 [A21 ][V11 ].
(26)
Substituting equation (26) into equation (24) yields the solution for the submatrix [V11 ], i.e., −1 , [V11 ]=([A11 ]−[A12 ][A−1 22 ][A21 ])
(27)
.
606 and the inverse of the submatrix
[V11 ]−1=[A11 ]−[A12 ][A−1 22 ][A21 ].
(28)
Here, as long as the frequency of applied force is not equal to zero, the matrix does not have a singularity even if the structure considered is under free–free boundary conditions. The submatrix [V11 ] contains the dynamic contributions from the original co-ordinates due to the specified excitation force. This submatrix [V11 ] is actually the dynamic response matrix [Vc ] which is used to determine the condensed mass matrix in section 2.2. 2.4. - After the condensed model has been obtained, the response at the master co-ordinates due to the applied forces can be computed from equation (9). By following the reverse procedure described in section 2.3, the dynamic responses at the slave co-ordinates can also be obtained in terms of the computed responses at the master co-ordinates. Since the actual applied loads at the slave co-ordinates are considered to be zero, the response amplitudes at the original co-ordinates can be determined from equation (21) as
$
A11 A12 A21 A22
%6 7 6 7
Xc F = c , Xs 0
(29)
where {Xc } and {Xs } represent the response amplitudes at the master co-ordinates and the slave co-ordinates, respectively, and {Fc } represents the vector of applied loads at the master co-ordinates. Hence, from equation (29), the dynamic transformation between the condensed model and the original model can be obtained: i.e.,
67
Xc =[T][Xc ], Xs
(30)
where [T]=
$
%
Ic . −A−1 22 A21
(31)
Note that the dynamic transformation [T] is a function of the frequency of the applied loads. If the applied loads are static, i.e., v=0, the transformation becomes the well-known one [T]=
$
%
Ic . −K−1 22 K21
(32)
It should be pointed out that in this particular case, although the procedure presented and the Guyan method produce the same transformation between the retained and eliminated co-ordinates and the same stiffness matrix, the two procedures still produce different mass matrices. The procedure presented has the desired advantage that the resulting model retains precisely the natural frequencies and modes of interest.
607
3. APPLICATION WITH CONSIDERATION OF VARIOUS APPLIED FORCES
3.1. The mass and stiffness of the dynamically condensed model represented by equations (13) and (14) are functions of the frequency of the excitation forces. The accuracy of the model in describing the response of the structural system depends on how close the frequency of the excitation force used for the condensation is to the actually applied force. Some suggestions for selecting the frequencies to be used to produce the condensed models are provided for different types of known applied forces in this section. Case 3.1.1, {Ft }=constant. When the applied force is constant or the frequency is equal to zero, the problem becomes a static problem. All the elements of the submatrix [V11 ] represent the static deflections at the master coordinates when the force is applied to the structure. The stiffness of the reduced model is equal to the inverse of matrix [V11 ]. One more point is that, in this special case, the matrix [A] in equation (16) becomes the stiffness matrix of the original system. Therefore, the reduced stiffness matrix is determined as [Kc ]=[K11 ]−[K12 ][K22 ]−1 [K21 ].
(33)
This stiffness matrix is exactly the same as that obtained by the Guyan method. It should be pointed out that, however, the mass matrix determined according to equation (13) is different from that obtained by the Guyan method. Case 3.1.2, {F(t) }={F0 }eivt . Theoretically, the dynamically condensed model of a structural system can describe the system precisely as long as the same frequency v is used to produce the condensed model. This is the case for rotating structures of most rotating machines since these machine are operated at certain frequencies. Case 3.1.3, {F(t) }=amj=1 {Fj }eivj t . For each frequency of the excitation force, the corresponding reduced mass and stiffness matrices can be obtained in case 3.1.2; the results are [Mcj ],
[Kcj ],
j=1, 2, . . . , m.
(34)
Here one can compute the dynamic responses of the structure considered separately in terms of the above reduced models, due to the corresponding harmonic components of the applied forces. Since the structure is linear, accurate dynamic responses can then be obtained from the sum of those computed separately. On the other hand, for simplicity, the approximated condensed mass and stiffness matrices can be determined as, [Mcf ]=
1 m s F [M ], Ff j=1 j cj
[Kcf ]=
1 m s F [K ], Ff j=1 j cj
(35, 36)
where m
Ff= s Fj .
(37)
j=1
The condensed model with these mass and stiffness matrices probably is fairly accurate but may not be the best one. An alternative condensed model can be produced by choosing just one harmonic component as the applied load. The frequency of the chosen component should be in the middle or below the middle of the range of frequency of interest. 3.2. - In engineering practice, there are forced vibration problems with non-harmonic periodic or random applied forces. It is not possible to obtain such a condensed model from the
608
.
original one to describe the structural system precisely. However, based on the information about the applied forces, it is always possible to obtain a condensed model which is fairly accurate in describing the dynamic behaviour of the structural system. Case 3.2.1, Known non-harmonic applied forces. Whether or not the non-harmonic applied forces are periodic, non-periodic or random, spectral analysis of the forces can always provide valuable information about the harmonic components and the corresponding amplitudes of the applied forces. Then, by following the procedure described in case 3.1.3, a possible condensed model which is fairly accurate can be determined. Case 3.2.2, Unknown applied forces. The procedure for condensing the original model dynamically is the same as that described in Case 3.1.3. But the excitation forces have unit amplitude and the frequencies are chosen to be between the natural frequencies of interest of the original structural system. 3.3. , - For most industrial applications, it is recommended that the lowest few natural frequencies and the corresponding modes of the original structural system be kept in the reduced model. It is recommended that the co-ordinates of interest such as the co-ordinates at which the external loads are applied are retained in the reduced model. As described in section 2.3, theoretically the responses of all eliminated co-ordinates can be computed. 4. DISCUSSION
For constant applied loads, i.e., v=0, if the structure considered is not under free–free boundary conditions, the stiffness matrix obtained by the method presented is exactly the same as that obtained by the Guyan method. The mass matrix is determined from the relationship between the condensed stiffness matrix and the retained natural frequencies and the corresponding modes at the master co-ordinates. Therefore, the resulting mass matrix is different from the one obtained by the Guyan method. The fundamental difference between the Guyan method and the method presented is that, the Guyan method is based only on the pure static transformation and the resulting model does not retain the natural frequencies and corresponding modes from the original one, but the method presented is based on the dynamic transformation and the chosen nc natural frequencies and corresponding modes at the master co-ordinates from the original model and the resulting model retains precisely nc natural frequencies and corresponding modes. Besides, the condensed model produced by the method presented has great accuracy in describing the dynamic behavior of the original structural system at the frequency which was used to produce the model. Since these retained natural frequencies, modes and the dynamic behavior at the chosen frequencies of the structural systems are what one is mostly interested in, such a condensed model has great advantages in describing the dynamic response of structures such as the rotor shafts of most rotating machines. There are some similarities between the method presented and the component mode method [4]. Instead of using the purely static transformation between the eliminated co-ordinates and the master co-ordinates, the component mode method makes use of a few of the lowest component modes of interest and the measured displacements to produce a reduced model for the dynamic analysis. The method improves the accuracy of the resulting model provided that good quality measurement of the displacements is guaranteed. Compared with the method presented here, the model obtained by the component method does not accurately retain the higher modes as the number of
609
component modes used to produce the reduced model is smaller than the number of chosen to be preserved in the model. Comparing the presented method with the experimental identification method [5] shows that theoretically both methods produce the same model provided that there are no errors included in all the required measurements. The advantage of the experimental identification method is that the model obtained preserves the physical properties of the original structure such as the boundary conditions and the damping properties more accurately. The modal parameters identified experimentally are more accurate than those obtained analytically for complicated structures [6]. The disadvantages, however, are the large work loads and the difficulty of applying large excitation forces for the identification of large complex structures. Theoretically, there is no special restriction on choosing the natural frequencies and the corresponding modes to be kept in the condensed model. However, since most applied loads are in the range of the lower frequencies, it is the author’s suggestion to keep a chosen number of lowest modes of interest in the condensed model unless one has particular interests in other higher modes. For the same reason, it is suggested that mainly lower frequencies are used to produce the condensed model. Theoretically, there would be no limitation on the application of the procedure presented to the modelling of large structural systems. Even for a full global system, as long as it is a linear system, the full-size mass and stiffness matrices can always be obtained analytically and, therefore, the original model of the system can also be condensed. 5. APPLICATION EXAMPLES
In principle, the method presented can be applied to all the problems for which reduced models are required for the dynamic analysis. To examine the applicability of the dynamic condensation method, two examples are given in this section for the modelling of two common structures. A few of the condensed models of the structures corresponding to different frequencies of the applied forces were obtained by means of the dynamic condensation method. Comparisons between the performances of the condensed models obtained by means of the dynamic condensation method and that of the models obtained by the Guyan method are conducted. 5.1. 1, Figure 1 shows schematically a cantilever pipe. The pipe was originally modelled by 20 equally sized beam elements. The number of original co-ordinates is 40 (each node has two degrees of freedom). As the first application, the five lowest natural frequencies and the corresponding modes of the pipe were chosen to retain in the condensed model. The number of degrees of freedom of the model is five. The first, fifth, ninth, 13th and 17th original co-ordinates were chosen as the master co-ordinates. Seven condensed models of the pipe were obtained by
Figure 1. A cantilever pipe. SGP A15; L=1000 mm, D=21·7 mm, d=16·1 mm.
.
610
T 1 Comparison between the natural modes of model M10 and the true ones of model M01 of the cantilever pipe Freq. (Hz) Model Mode f1i f5i f9i f13i f17i
f1
f2
M01 M10 19·5 19·5 1·00 1·00 0·726 0·726 0·462 0·462 0·231 0·231 0·064 0·064
M01 121·0 1·00 0·073 −0·590 −0·689 −0·308
f3
f4
f5
M10 M01 M10 M01 M10 M01 M10 121·0 334·0 338·0 642·0 668·0 1036·0 1079·0 1·00 1·00 1·00 1·00 1·00 1·00 1·00 0·073 −0·392 −0·384 −0·651 −0·638 −0·628 −0·704 −0·589 −0·485 −0·481 0·314 0·312 0·728 0·890 −0·688 0·522 0·514 0·345 0·338 −0·709 −0·882 −0·308 0·623 0·616 −0·775 −0·754 0·657 0·760
using the two different methods or the same method but different frequencies of the applied forces. Model M10 was obtained by means of the Guyan method. Models M11, M12, M13 and M14 were obtained by means of the dynamic condensation method for the frequencies of 0 Hz, 50 Hz, 200 Hz and 500 Hz, respectively. Model M15 was obtained by taking the average of the three models of M11, M12 and M13. Model M16 was obtained by the dynamic condensation method with the harmonic forces,
0
19
1 i100kpt e k k=1
f(t)= s
1>
3·5477,
0
1
19 1 s =3·5477 . k k=1
The eigenvalue–eigenvector analyses of the models M11, M12, M13, M14, M15 and M16 clearly show that these models have retained the five lowest natural frequencies and the corresponding modes at the master co-ordinates of the pipe, which are exactly the same as those of the original finite element model. The natural frequencies and the corresponding modes of model M10 and the original model M01 are given in Table 1. It is noted that the results for M01 are also true for M11, M12, M13, M14, M15 and M16 From the table, it is seen that model M10 has preserved the first three lowest natural modes very well. But for the two higher modes, the model is not very accurate. The modal shapes of the highest mode of the model have relative errors of up to 20% when compared with the corresponding true values. The condensed mass and stiffness matrices of models M10, M11 and M15 are listed in Table 2. From this table, it is seen that the condensed stiffness matrices of model M10 and M11 are exactly the same but their condensed mass matrices are different. To compare the performances of these models with each other, dynamic analysis based on these models and on the original finite element model have been carried out by applying a few harmonic forces at the fifth original co-ordinate. The frequencies of the applied forces were 0 Hz, 50 Hz, 200 Hz, 500 Hz, 700 Hz and 1200 Hz. The relative errors, which are represented by percentages, between the response amplitudes obtained from the condensed models and those obtained from the original model were computed. The results are given in Table 3. In Table 3, from the left, the first column lists the frequencies of the applied loads, the second column lists the numbers of the master co-ordinates, the third column lists the response amplitudes, which were normalized according to the maximum computed from the original model, and the rest of the columns in the table are the relative errors. The results given in Table 3 clearly show that the accuracy of the model obtained by the dynamic condensation method depends on the frequencies which are used to produce
−0·0143 0·0244 0·2254 0·0295 −0·0162 −0·0173 0·0203 0·2347 0·0236 −0·0190 −0·0193 0·0197 0·2407 0·0236 −0·0205
K 0·0742 0·0407 G 0·0407 0·2387 G −0·0143 0·0244 G 0·0044 −0·0133 k −0·0013 0·0051
K 0·0749 0·0406 0·2431 G 0·0391 G −0·0131 0·0188 G 0·0048 −0·0125 k −0·0027 0·0090
K 0·0783 0·0431 G 0·0418 0·2495 G −0·0148 0·0182 G 0·0057 −0·0133 k −0·0032 0·0100
Mass (kg)
0·0106 −0·0135 0·0225 0·2378 0·0323
0·0093 −0·0129 0·0227 0·2324 0·0314
0·0044 −0·0133 0·0295 0·2225 0·0346
−0·0062 L K 0·3052 0·0077 G G −0·6761 −0·0157 G G 0·4580 0·0271 G G −0·0994 0·2232 l k 0·0147
−0·6707 1·8122 −1·7046 0·6839 −0·1354
−0·6675 1·7862 −1·6725 0·6808 −0·1620
Model M11 −0·0054 L K 0·2997 0·0070 G G −0·6675 −0·0146 G G 0·4512 0·0266 G G −0·1026 0·2187 l k 0·0244 Model M15
−0·6675 1·7862 −1·6725 0·6808 −0·1620
Model M10 −0·0013 L K 0·2997 0·0051 G G −0·6675 −0·0162 G G 0·4512 0·0346 G G −0·1026 0·2092 l k 0·0244
Stiffness (N/m)
0·4316 −1·6983 2·5223 −1·8684 0·7308
0·4512 −1·6725 2·4670 −1·8344 0·7507
0·4512 −1·6725 2·4670 −1·8344 0·7507
−0·0684 0·6858 −1·8766 2·5949 −2·0129
−0·1026 0·6808 −1·8344 2·5369 −1·9964
−0·1026 0·6808 −1·8344 2·5369 −1·9964
T 2 The condensed mass and stiffness matrices of the cantilever pipe (5 d.o.f. models)
−0·0054 L −0·1523 G 0·7622 G ×E06 −2·0443 G 3·2704 l
0·0244 L −0·1620 G 0·7507 G ×E06 −1·9964 G 3·2177 l
0·0244 L −0·1620 G 0·7507 G ×E06 −1·9964 G 3·2177 l
611
.
612
T 3 Effect of condensation procedures on dynamic responses of the cantilever pipe (5 d.o.f. models) Freq. (Hz)
No.
Amp. M01
M10
0
1 5 9 13 17
1·00 0·728 0·461 0·228 0·063
0·00 0·00 0·00 0·00 0·00
50
1 5 9 13 17
1·00 0·717 0·470 0·249 0·074
200
1 5 9 13 17
500
Relative errors of response amplitudes (%) M11 M12 M13 M14 M15 0·0 0·0 0·0 0·0 0·0
−0·05 −0·05 −0·05 −0·05 −0·05
−0·07 −0·07 −0·07 −0·07 −0·07
−0·05 −0·05 −0·05 −0·06 −0·06
1·00 0·340 0·180 0·183 0·093
−0·90 −1·59 −0·57 1·28 1·89
−0·90 −1·05 −0·31 0·38 0·78
1 5 9 13 17
1·00 0·009 0·854 −0·028 −0·841
5·39 −786·00 −1·68 129·00 −1·15
−7·28 −7·20 −6·14 −10·2 −10·5 −6·81 −2·54 −2·51 −2·15 −22·6 −22·4 −19·3 3·71 3·67 3·14
700
1 5 9 13 17
1·00 −0·914 0·103 0·587 −0·977
143·00 −101·00 419·00 81·2 −105·0
8·74 8·70 −10·4 −10·3 26·8 26·7 8·08 8·03 −9·53 −9·48
1200
1 5 9 13 17
1·00 −0·882 0·532 −0·510 0·492
44·9 −55·1 83·2 −65·0 34·2
M16
−0·82 −0·81 −0·80 −0·79 −0·79
−5·14 −5·10 −5·05 −5·00 −4·99
−0·29 −0·29 −0·29 −0·28 −0·28
−2·68 −2·65 −2·62 −2·60 −2·60
0·0 0·0 0·0 0·0 0·0
0·71 0·76 0·81 0·85 0·86
4·75 5·06 5·39 5·67 5·73
0·22 0·24 0·25 0·27 0·27
2·44 2·61 2·79 2·94 2·96
−0·85 −0·99 −0·29 0·35 0·73
0·0 0·0 0·0 0·0 0·0
4·78 5·54 1·71 −1·90 −4·16
−0·58 −0·68 −0·20 0·24 0·50
2·08 2·39 0·68 −1·01 −2·18
8·15 −9·64 25·0 7·49 −8·85
0·0 0·0 0·0 0·0 0·0
−6·87 −3·22 −9·16 −20·3 −2·40 −1·21 −21·4 −8·22 3·50 1·66
4·76 8·53 −5·44 −10·1 14·4 26·2 4·25 7·86 −5·06 −9·29
6·43 −7·63 19·4 5·92 −6·97
19·3 19·3 19·2 18·4 19·3 18·8 −27·6 −27·6 −27·1 −24·3 −27·4 −25·8 18·5 18·5 18·3 16·8 18·4 17·5 −2·61 −2·59 −2·54 −2·28 −2·58 −2·46 −9·33 −9·35 −9·28 −8·86 −9·32 −8·97
the model. The model has greatest accuracy in describing the dynamic response of a structure if the frequency of the actual applied loads is the same or close enough to the one used to produce the model. The results for model M15 also show that a model which is very accurate within a wide range of frequencies can be obtained as long as a few optimum frequencies are chosen to produce the model. The model obtained by the Guyan method is very accurate only in the range of the lower frequencies. In the range of higher frequencies (e500 Hz), however, the accuracy of the model is unacceptable. Compared with the model obtained by the Guyan method, the other models perform better within a wide range of frequencies which are of interest. Furthermore, the results obtained for the highest frequency (1200 Hz) of the applied load show that, for the applied loads of which the frequency is larger than the highest natural frequency retained in the models, none of the models are of good accuracy. Hence, the condensed models can be used only for the dynamic analysis of a structure of which the frequencies of the actual applied loads are lower than the highest retained natural frequency.
613
Figure 2. A free—free flexible rotor shaft.
5.2. 2, – The second structure analyzed is a free–free flexible rotor shaft, shown in Figure 2. The rotor was used in an experimental installation which consists of the rotor, four journal bearings and a steel–concrete foundation. Originally, the rotor was modelled by the finite element method with 55 elements in total. Based on this original FEM model, denoted as M02, more applications of the dynamic condensation procedure have been carried out. As the second application, a nine-degrees-of-freedom (9 d.o.f.) model of the rotor shaft is considered. The nine lowest natural frequencies and the corresponding modes, of which two are the free–free modes (i.e., f1=f2=0), and nine master co-ordinates were chosen for retention in the model. Among the chosen master co-ordinates, the second, fourth, sixth and eighth co-ordinates correspond to the four supporting bearing stations and the other five correspond to the unsupported stations; see Figure 2. To investigate the influence of the frequencies used in the condensation on the performance of the resulting model, four condensed models of the rotor were obtained by means of the dynamic condensation method and one model was obtained by the Guyan method. Model M20 was obtained by means of the Guyan method. Models M21, M22 and M23 were obtained by means of the dynamic condensation method with the frequencies 35 Hz, 75 Hz and 125 Hz respectively. Model M24 was obtained by using the applied force f(t)=(ei20pt+0·2ei70pt+0·04ei150pt )/1·24. The eigenvalue–eigenvector analyses of the models M21, M22, M23 and M24 clearly show that these models have retained the nine lowest natural frequencies and the corresponding modes of the original model M02. Table 4 gives the nine lowest natural frequencies and the corresponding modes of model M02 and those of model M20. It is noted that the results for M02 are also true for M21, M22, M23 and M24. Since theoretically the first two natural frequencies should be zero (corresponding to free–free motion), they can be set to zero in the condensation procedure. The results in the table show that model M20 does not preserve the higher modes accurately. T 4 Comparison between the natural frequencies of model M20 and the true ones of model M02 of the rotor shaft Model Freq. (Hz)
M02
M20
f1 f2 f3 f4 f5 f6 f7 f8 f9
0·001 0·941 21·275 49·398 105·791 151·636 234·115 490·338 546·990
0·849 1·526 21·281 49·464 106·584 158·014 250·778 508·066 677·594
.
614
The response amplitudes at the master co-ordinates corresponding to a few harmonic applied forces were computed based on the obtained models. A few harmonic forces were applied at the third master co-ordinate and the frequencies of the forces were 35 Hz, 75 Hz, 125 Hz and 175 Hz. The results are given in Table 5. The structure of Table 5 is the same as that of Table 3. The results listed in Table 5 show that the models M21, M22, M23 and M24 perform better than model M20 does in the range of frequency up to 175 Hz. If the actual applied forces have a few harmonic components in the frequency range (0–175 Hz), model M24 probably has the best performance in describing the dynamic response of the rotor shaft.
T 5 Effect of condensation procedures on dynamic responses of the rotor shaft (9 d.o.f. models) Freq. (Hz)
No.
Amp. M02
35
1 2 3 4 5 6 7 8 9
1·000 0·570 0·296 0·249 0·327 0·358 0·158 −0·225 −0·665
0·96 0·96 0·12 1·59 1·44 1·09 0·81 1·58 1·66
0·0 0·0 0·0 0·0 0·0 0·0 0·0 0·0 0·0
0·26 0·21 0·00 −0·13 0·09 0·24 0·32 −0·15 −0·13
0·88 0·70 0·00 −0·43 0·29 0·80 1·08 −0·50 −0·44
−0·04 −0·04 0·00 0·02 −0·02 −0·04 −0·06 0·03 0·02
75
1 2 3 4 5 6 7 8 9
0·167 0·249 0·568 0·842 0·587 0·094 −0·248 0·096 1·0000
7·04 0·48 0·88 0·15 0·31 3·53 0·28 2·01 0·07
1·13 −0·09 −0·18 −0·15 −0·25 −0·94 0·48 −0·30 −0·78
0·0 0·0 0·0 0·0 0·0 0·0 0·0 0·0 0·0
−2·65 0·22 0·43 0·36 0·59 2·22 −1·13 0·70 1·83
1·33 −0·10 −0·21 −0·18 −0·29 −1·10 0·56 −0·35 −0·91
125
1 2 3 4 5 6 7 8 9
0·388 −0·203 −0·462 −0·166 0·346 0·502 −0·105 −0·199 1·000
34.3 16·6 6·57 41·0 20·3 2·47 12·6 4·00 0·92
9·75 −4·89 −1·95 10·4 4·75 2·28 −2·34 1·37 −0·37
6·84 −3·44 −1·37 7·30 3·33 0·20 −1·64 0·96 −0·26
0·0 0·0 0·0 0·0 0·0 0·0 0·0 0·0 0·0
10·2 −5·13 −2·04 10·9 4·98 0·30 −2·45 1·43 −0·39
175
1 2 3 4 5 6 7 8 9
1·000 −0·560 −0·533 0·213 0·243 −0·072 −0·026 0·120 −0·285
23·0 15·7 4·99 62·6 7·15 164·3 20·0 64·5 78·1
−7·22 5·68 1·29 −23·1 −2·28 61·5 −7·82 −23·7 29·0
−6·22 4·89 1·12 −19·9 −1·98 52·9 −6·71 −20·4 25·0
M20
Relative errors of amplitudes (%) M21 M22 M23
−3·85 3·02 0·69 −12·3 −1·25 32·6 −4·10 −12·6 15·4
M24
−7·39 5·81 1·32 −23·6 −2·34 62·9 −7·98 −24·2 29·7
615
6. CONCLUSIONS
A dynamic condensation procedure has been presented for reducing the size of mass and stiffness matrices for a structure for the dynamic analysis. The condensed model is formulated in terms of the chosen natural frequencies, the corresponding modes and the dynamic responses at the retained master co-ordinates, which are determined from the original structural system. As a result, the resulting model satisfies the two essential requirements introduced into the dynamic condensation procedure: (a) it remains the chosen natural frequencies and the corresponding modes at the master co-ordinates from the original model; (b) the responses of the condensed system at the master co-ordinates due to forces at these co-ordinates, at one particular chosen frequency which is often identical to the operational one of a rotating machine, are the same as those of the original system. For constant loads, if the structure considered is not under free–free boundary conditions, the condensed stiffness matrix is exactly the same as the one obtained by the Guyan method but the two mass matrices are different. Unlike the Guyan method, that cannot accurately preserve the higher modes of interest in the condensed model, the dynamic condensation method can accurately retain both the lower and higher modes of interest in the resulting model. The results obtained from two examples show that the condensed model produced directly by means of the dynamic condensation method from the finite element model of the original structure retains the chosen number of lowest natural frequencies and the corresponding modes precisely. For known applied loads, the condensed model obtained by the dynamic condensation method can describe the dynamic responses of the structure with very good accuracy. For unknown applied loads, but within a limited range of frequency, the models produced by using optimal frequencies of the approximated applied loads have better performance than do the condensed models obtained by the Guyan method. The frequencies used to produce the condensed model of a structure should be chosen within the range of interest. The closer the frequencies are to those of the actual applied loads, the better the performance of the resulted model is in describing the dynamic responses of a structure. ACKNOWLEDGMENTS
The author would like to thank Dr J. M. Krodkiewski at the University of Melbourne and Professor S. Hayama at the University of Tokyo for valuable discussions and encouragement. REFERENCES 1. M. J. T, R. W. C, H. C. M and L. J. T 1956 Journal of Aeronautics Science 23, 805–823. Stiffness and deflection analysis of complex structures. 2. R. J. G 1965 American Institute of Aeronautics and Astronautics Journal 3, 380. Reduction of stiffness and mass matrices. 3. G. H. S 1984 Journal of Sound and Vibration 94, 150–153. Comment on the substructure synthesis methods. 4. A. C 1987 Journal of Sound and Vibration 117, 277–288. A component mode method for modelling the dynamics of turbo-generator sets. 5. N. Z and S. H 1991 International Journal of the Japan Society of Mechanical Engineers, Series 3, 34(1), 64–71. Identification of structural system parameters from time domain data (method for the identification of physical parameters of structural system from experimental data). 6. N. Z and S. H 1990 International Journal of the Japan Society of Mechanical Engineers, Series 3, 33(2), 168–175. Identification of structural system parameters from time domain data (identification of global modal parameters of structural system by improved state variable method).