application of the simultaneous iteration method to undamped vibration problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 3, 13-24 (1971)

APPLICATION OF THE SIMULTANEOUS ITERATION METHOD TO UNDAMPED VIBRATION PROBLEMS

ALAN JENNINGS* AND D. R. L. O R R t

Queen's University, Berfast, Northern Ireland

SUMMARY

The simultaneous iteration method of obtaining eigenvalues and eigenvectors is employed for the solution of undamped vibration problems. This method is of significance when a few of the dominant eigenvalues and eigenvectors are required from a large matrix, and hence is particularly suitable for vibration problems involving a large number of degrees of freedom. It is shown that advantage may be taken of both the symmetry and the band form of the mass and stiffness matrices, thus making it feasible to process on a computer larger order vibration problems than can be processed using transformation methods. A method of allowing for body freedom is given and some numerical tests are discussed.

INTRODUCTION

For dynamic analysis displacement variables are almost universally adopted. The free undamped vibration of a structural system gives rise to matrix equations of the form1T2

w2Mq = Kq (1)

where M and K are symmetric non-negative definite n x n mass and stiffness matrices and q is an n x 1 displacement vector defining the vibration mode associated with the natural frequency W.

This can be reduced to a symmetric eigenvalue problem by introducing auxiliary variables3

p = LTq (2) where L is the lower triangular matrix obtained from the Choleski factorization of K according to K = LLT. Equation (1) is reduced to

where A = L-lML-T. With the analysis expressed in this form it is seen that the largest eigenvalues of the matrix A give the lowest natural frequencies of the system.

For computer solution of general structural problems it is normally most convenient to use as variables displacement functions which extend over only local regions of the structure. Examples of this are the Livesley displacement co-ordinates for structural frames and finite element methods for plates and shells. Coupling terms only arise in the equations of motion when two displacement functions overlap. Hence both M and K are sparse and, where all the self-weight of the structure is included in the analysis, then M and K will have similar patterns of non-zero elements. Representation of structures in this way can require large numbers of displacement variables.

* Senior Lecturer, Department of Civil Engineering. f Formerly Postgraduate Student, Department of Computer Science.

Received 25 April 1969 Revised 16 December 1969

@ 1971 by John Wiley & Sons, Ltd.

13

14 ALAN JENNINGS AND D. R. L. ORR

Approximations in the idealization of the structure will tend to make the highest computed frequencies meaningless and intermediate frequencies only partially accurate. Hence just a limited number of the lower vibration frequencies will normally be required.

Whereas Householder’s method4 for the eigenvalues of symmetric matrices is convenient for analyses involving a moderate number of variables, with very large values of n the determination and subsequent transformation of the full matrix A would be a prohibitive computational pro- position. Like the new method by Gupta? the simultaneous iteration procedure presented in this paper not only takes advantage of the fact that a partial eigenreduction is required but also makes use of the sparseness of the mass and stiffness matrices to reduce storage space and computing time, thus making significantly larger problems within the scope of present day computers.

SIMULTANEOUS ITERATION METHODS Simultaneous iteration methods are extensions of the power method in which iteration is carried out with two or more trial vectors simultaneously in such a way that they converge onto the dominant eigenvectors of the matrix. Wilkinson4 describes several techniques of which Bauer’s bi-iteration process is most nearly related to the method adopted in this paper. Considering a symmetric matrix A of order n x n and m trial vectors h, then applying Bauer’s method would first involve simultaneous pre-multiplication of all the vectors by A to give

v = AU (4)

where u = [ul, u,, .. ., u,] is of order n x m. The trial vectors are required to be an orthogonal set, i.e. satisfying the relationship

Hence the pre-multiplied vectors v have to be subjected to an orthogonalizing procedure in order to generate a new orthogonal set for repeating the iterative process. Recent methodsa,‘ include an interaction analysis thus giving a full iterative cycle which can be summarized as follows :

U T U = I ( 5 )

1. Pre-multiplication 2. Interaction analysis 3. Orthogonalizing sequence 4. Test for convergence The interaction analysis is a method of predicting the approximate eigenvectors by examining

the m x m ‘interaction matrix’ B = U ~ A U = U ~ V

It will be noted that when convergence is obtained the interaction matrix will be diagonal and hence the presence of an off-diagonal element bii in matrix B signifies an interaction between the vectors ui and uj which needs to be eliminated. For the first method the element bij is used to estimate the coupling between ui and ui and the derived vectors vi and vi are accordingly modified. For the second method an eigenreduction of the interaction matrix is performed giving a more accurate prediction of the coupling. In numerical tests the Jacobi method was used to perform the eigenreduction of the interaction matrix. The numerical tests indicated that the second method appeared to be slightly more efficient in computing time and was used for the vibration analysis. An alternative procedure was also tried in which three pre-multiplications per iteration were employed. This could sometimes lead to a saving in computer time.

The most significant error terms are due to the (m + 1)th eigenvector as this error component is not eliminated by the interaction analysis. The component of the (m+ 1)th eigenvector in the trial vector converging on thej th eigenvector will be reduced by a factor APn+JAj at each pre- multiplication. In order to yield fast convergence more trial vectors should be used than the

(6)

SIMULTANEOUS ITERATION METHOD 15

number of eigenvectors required accurately. The highest eigenvectors tend to be produced to a higher accuracy than the lower eigenvectors and all the eigenvalues tend to be more accurate than their eigenvectors.

PROCEDURE FOR VIBRATION ANALYSlS

To determine the dominant eigenvalues and eigenvectors of A by any simultaneous iteration method the only operation involving A is the matrix multiplication of equation (3). With A in the form given by equation (2) this multiplication can be replaced by the following equivalent operations:

1. Solve LTx = u (7)

2. Form y = M x (8) 3. Solve Lv = y (9)

Equations (7) and (9) are simply solved on account of the triangular form of L amounting to a back substitution and forward substitution respectively. When M and K are sparse the sparsity of M is fully retained, while in the triangular decomposition of K to give L the sparsity is partially retained.

Any form of storage for M would be suitable, but for large systems of equations it would seem most beneficial to have a sparse storage scheme in which registers are only used for non-zero elements or submatrices on one side of the diagonal together with appropriate position counters. L can be formed by overwriting K as long as the storage scheme for K allows space for all elements of K which become non-zero during the triangular decomposition. The possible systems for store allocation will be similar to those suitable for Gaussian elimination methods of solving symmetric simultaneous equations. The conventional band storage scheme as described by Weave$ would be adequate for many structural analyses, although a storage scheme involving local variation in the bandwidth would be more versatile for general use.g Peters and WilkinsonlO give an algorithm for the triangular decomposition of band matrices.

Once the eigenvectors have been determined to a sufficient accuracy then the vibration modes can be determined by back substitution for q in equation (2).

UNCONSTRAINED STRUCTURES

In the case of a structure unconstrained in one or more possible body freedoms, the stiffness matrix is singular, and therefore the matrices L and A cannot be constructed. Associated with a displacement in a body freedom there must be a zero frequency of vibration due to the lack of restoring force. A method of overcoming this difficulty whilst at the same time retaining the diagonal bandform of the equations is to use a modificationll of a method given by Cox, which involves rewriting equation (1) in the form

(d+a)Mq = (K+aM)q (1 0) where 01 is a constant recommended to be positive. The analysis can proceed as before using a modified stiffness matrix = KS a M to determine L, the only difference being that the natural frequencies are obtained from the eigenvalues hi of the matrix A according to the formula

a; = l / x i -a (1 1) For every body freedom of the structure there will be an eigenvalue hi = 1/a with the corresponding eigenvector pi yielding the displacement form of the body freedom when substituted in equation (2) for qi. For positive a these eigenvalues will be the largest and hence will appear in the simultaneous iteration results.


Because the body freedoms are known in advance it would be possible to determine their associated eigenvectors pi by using equation (2). If this were done then, with extra programming, the trial vectors could be orthogonalized with respect to the body freedoms at each iteration so preventing the trial vectors converging onto any of the body freedoms and allowing a lower number of trial vectors to be used.

INCOMPLETE MASS MATRIX

The mass matrix will not be singular if all the contributions, including structure self-weight, are taken into account. However, some contributions may be ignored for parts of the structure having a negligible mass or when the deflections in the required vibration modes are small. In such cases the mass matrix could be described as incomplete and may be singular. The only situation in which the feasibility of the analysis is affected by a singular mass matrix is when there is zero mass associated with a body freedom for an unconstrained structure, a situation which should not occur in practice. However, in cases where a large number of the dynamic equations contain no mass terms then it niight be expedient to pre-eliminate these variables thus deriving a con- densed but full stiffness matrix.l

When using simultaneous iteration with an incomplete mass matrix some of the eigenvalues of the matrix A will be zero. It is advisable that no more trial vectors are used than the number of non-zero eigenvalues because the convergence criterion is not satisfied for zero eigenvalues and the inclusion of the associated vectors does not improve the rate of convergence.

COMPARISON WITH OTHER METHODS

A comparison may be obtained from the approximate numerical efficiency of various methods, and for this Householder’s transformation method and Gupta’s Sturm sequence method have been compared with simultaneous iteration. Consider the analysis for the r lowest natural frequencies of vibration of a structure with n displacement variables for which the equations have a bandwidth 2b + 1 , where n is large. It will be assumed that all sparse matrices including M and K will be stored in bandform using approximately nb registers. For matrix multiplications involving band matrices M or K then computing time may be salved by taking account of non-zero elements within the band and a factor of f will be applied to the number of multiplications to make allowance for this possible saving.

Table I(a). Comparison of computational efficiency-general formulae

Approximate maximum Approximate total number storage requirement of multiplications

Householder :n2 + 2nb 36 n 2 + ) n 3 + n 2 r Gupta 5nb 25nbz r . S.I. 2nb + 3nm + +m2 (3nbm + 2)nm2 + +m3) c + &nb2

Table I shows the estimated values for storage requirement and number of multiplications, the latter being an approximate guide to computing time. For Householder’s method the reduction of A to tridiagonal form takes up most of the computing time although the formation of A will tend to require more storage space than the eigenreduction of it. Gupta’s ’method requires storage for the sparse matrices M and K and also the reduction of M- AK. The values given in Table I for storage space could be reduced for cases where there are sufficient non-zero elements within


the bandwidth of M and K to warrant only storing the non-zero elements and position counters. Most of the computing time will be spent examining different A values. The number of multiplications given in Table I is based on the figures given in Gupta’s paper of up to 2nb2 multiplications for examining one h value, assuming that about twelve h values have to be examined for every frequency predicted. The formulae for simultaneous iteration are based on c iteration cycles being required with m trial vectors using the bandform procedure given by equations (7), (8) and (9). The most significant terms for both storage space and number of multiplications are the first two terms given in Table I(a). The arithmetic involved in the interaction analysis has been ignored, but will normally tend to be a function of m2.

From Table I(b) it is seen how Householder’s method soon becomes prohibitive to use with large values of n. Gupta’s method and simultaneous iteration have fairly similar storage require- ments, but the simultaneous iteration method should be significantly more efficient particularly for large order matrices. Gupta’s method is most sensitive to the bandwidth of the matrices, while for simultaneous iteration methods the optimum efficiency depends on choosing a reasonable value of m such that the specified accuracy for all the required vibration modes will be obtained

Table I(b). Comparison of computational efficiency-specific numerical values

Store ( K = lo3) Number of multiplications ( M = lo6) n b Householder Gupta S.1. Householder Gupta s.1.

50 5 1.8K 1.3K 2.OK 0.12M 0.22M 0.13M 200 10 24K 10K 10K 6.2M 3.5M 0.7M

1000 20 540K IOOK 70 K 704M 70M 5.3M

200 20 28K 20K 14K 6.8M 14M 1.1M 1000 40 580K 200K 1 IOK 740M 280M 9.5M

50 10 2.3K 2.5K 2.6K 0.14M 0.88M 0.17M

With r (No. of eigenvectors required) = 7 rn (No. of trial vectors for S.I.) = 10 c (No. of iteration cycles for S.I.) = 6

in only a few iterations. The choice of m is best undertaken with some knowledge of the spectrum of eigenvalues but by making m - r % 3 slow convergence is very unlikely. In the numerical example given in Table I(b) it has been assumed that six iterations will be required to obtain sufficient accuracy for the first seven eigenvalues when iterating with ten trial vectors. If w7 z w1,/2 then A7 z 44, and about six iterations should produce the seventh eigenvector to about three figure accuracy, at which stage the corresponding eigenvalue should be accurate to about six figures while the more dominant eigenvalues and eigenvectors should have a higher accuracy than the seventh. It is probable that such an accuracy would be beyond the meaningfbl accuracy of the basic equations and any further iteration would be of no consequence.

In the case of an incomplete mass matrix it may be possible to pre-eliminate some of the displacements and use a transformation method of eigenreduction. This may give an efficient form of solution, but if the number of displacements after pre-elimination is more than about 100, then it is unlikely to be more efficient than applying simultaneous iteration to the basic banded equations in which all the displacements are retained.

Some further information on the effectiveness of simultaneous iteration methods for the analysis of undamped vibration problems may be obtained from a survey of possible methods by Bronlund.12


NUMERICAL RESULTS

Three series of numerical tests were carried out in order to confirm the viability of the procedure in which the diagonal bandform of the original equations is maintained and also to demonstrate convergence of the iterative procedure. In each case the second simultaneous iteration method' was employed and a segment of the unit matrix was used for trial vectors. It is likely that such trial vectors will be poor for vibration problems because the leading vibration modes will tend to contain displacements in all or most of the degrees of freedom, the transformation of equation (2) being unlikely to have a strong effect on the form of the eigenvectors.

Vibration of a three storey shear building

In a shear building the floors are assumed to be so stiff that only sway deformations take place. Rogers13 has derived the following stiffness and mass matrices for a three storey shear building:

19,000 -9,000 0 K = [ -9,000 17,000 -8,000 ] k.p.i.

0 -8,000 8,000

M = 0 15 0 lbsec2/in. [loS 1 PSI The following eigenvalues of A and associated natural frequencies of vibration have been obtained by simultaneous iteration

A, = 0.0080780172, fi = 1.771 C.P.S.

A2 = 0.001 1053375, f2 = 4.787 C.P.S.

A, = 0-0005249786, f3 = 6.946 C.P.S.

The natural frequencies and mode shapes agree with Rogers' results. Using one trial vector only the simultaneous iteration method was the same as the power

method using the Rayleigh quotient to predict the eigenvalue. Using three trial vectors then the correct solution was obtained from the Jacobi routine in the first iteration, but a further round of iteration was required before the tolerance criterion was satisfied. In this example there was no advantage gained from using simultaneous iteration over the power method for the first natural frequency or direct methods for all the frequencies because of the small number of degrees of freedom. Large shear buildings can yield many degrees of freedom for which the equations have a narrow bandwidth.14 Inclusion of joint rotations and vertical displacements into the vibration of building frames may be necessary to obtain reliable frequencie~.~ This will involve a much larger number of displacement variables per storey with the ratio of bandwidth to total number of variables depending on the number of storeys.

Transverse vibration of a .fuseIage-wing combination

A uniform wing of length 2L and mass 2Mw distributed uniformly over the span has been represented by a uniform beam of flexural stiffness EI with a fuselage mass of 2MF located at the centre. The rotary inertia of the fuselage has been neglected. Using a four element idealization of the wing, the ten possible degrees of freedom for the half wing are shown in Figure 1. Neglecting rotary inertia effects on the wing and also shear deformations, Przemienieckil has derived stiffness


Figure 1 . Four element idealization of fuselage-wing combination

and mass matrices. With displacement variables {xl 18, x3 lo4 x5 18, x, 18, x, 101,,] corresponding to the directions shown in Figure 1 then

1 2 3 4

EI 5 i3 6

7

8 9

10

K = -

7 8 9

10

where R = MF/M,,,.

12 6 4

-12 - 6 24 6 2 0 8 (Symmetric)

-12 -6 24 2 0 8 0 -12 -6 24

0 - 12

6

8 - 6 2

12 -6 4

156 + 1680R 22 4 54 13 312 - 13 - 3 0 8 (Symmetric)

- 1 3 -3 8 13 312

-13 - 3 0 8 0 54 13 156

0 0 -13 -3 -22 4

Due to symmetry the modes of vibration will separate into a symmetric set and an antisymmetric set, the symmetric set deriving from the given equations with row and column two omitted, and the antisymmetric set by omitting row and column one. Both symmetric and antisymmetric analyses involve nine degrees of freedom with one body freedom and therefore will

20 ALAN JENNlNGS AND D. R. L. ORR

require 01 to be non-zero. The results given by Przemieniecki were confirmed by simultaneous iteration.

Table I1 shows typical figures for numbers of iterations taken from the case where R = 0 in the symmetric analysis, two eigenvectors being demanded to a tolerance of in each test. Vectors converging onto the third and higher eigenvectors being correct to a lower accuracy. For this problem all the natural frequencies are reasonably well separated and as long as the number of

Table 11. Symmetric wing analysis-number of iterations for convergence using different values of a

Number of trial vectors 3 6

I 3 3

10,000 21 8 .( 100 4 3

trial vectors is at least one more than the number of eigenvectors required accurately then convergence is fairly rapid. Increasing cy does tend to bring the eigenvalues closer together and this is seen to slow down the convergence rate when 01 is much greater than co: (313 in this example). An 01 somewhere in the range 0 3 2 to w:/10 would seem to be suitable.

Aircraft, missile and ship structural vibration problems can often be treated as beams involving transverse and torsional displacement degrees of freedom with possible body freedoms. The number of elements to be adopted in the analysis will depend on the number of natural frequencies required and the nature of the mass distribution and stiffness variation along the beam.

Vibration of rectangular grillages

Rectangular beam grillage configurations have been considered with various numbers of beams spanning in each direction. The beams were rigidly supported at the edges so that the only degrees of freedom were the displacement z and rotations 8 and 4 at each joint. With s beams spanning in the x direction at a spacing of p, and t beams spanning in the y direction at a spacing

Figure 2. A 3 x 5 beam grillage


of q then n = 3st. Ordering the joints by rows as shown in Figure 2 then the stiffness matrix for s = 3, t = 5 can be written in submatrix form as

1 ' 2 3 4 5 6 7 8 9

10 11 12 13 14 15

' D M D O M D N O O D

N O M D N O O D (Symmetric)

N O M D N O O D

N O M D N O M D

N O M D . o O N O M D

where D, M and N depend on the second moments of area II and I , and the torsional stiffnesses Kl and K, in the x and y directions as follows

1 D = e [ o 8EllP + 2EK2Iq 0

M = e [ o -EK2Iq 0

z 24E(1,lp3 + 12/q3) 0 0

+ 0 0 8EI,l9 + 2EKlIP z - 12E12/q3 0 - 6E12/q2

5b 6E12/q2 0

z - 12EIJp3 -6EIJp' N = 0 [ 6El11p2 2ElJp + 0 0

Equal masses and small rotary inertias were located at each of the joints. Parameters for the three grillages analysed are shown in Table 111. The 3 x 5 grillage was found to have its first fifteen

Table 111. Parameters for rectangular grillages analysed

h S X t n Maximum Average

3 x 3 27 10 7 3 x 5 45 10 8 5 x 8 120 16 14

eigenvalues 6045, 3970, 1902, 946, 879, 811, 639, 614,487, 382, 323, 313, 288, 250 and 220. The largest natural frequency agreed approximately with the frequency obtained by Rayleigh's

22 ALAN JENNINGS AND D. R . L. ORR

method using an assumed vibration mode of

z = (1 - c o s q ) (1 - C O S T ) sin wt

The tendency for the lower eigenvalues to be closer together than the higher eigenvalues was apparent in all problems analysed. Table IV shows a comparison of the convergence rates using different numbers of trial vectors.

Table IV. Number of iterations for a 3 x 5 grillage

Number of trial vectors

5 7 9

1 1 13 15

Eigenvectors required to three figure accuracy

3 5 7 9 1 1 1 3

6 6 10 6 7 8 5 6 7 1 1 4 5 5 7 1 0 4 4 4 4 4 4

_____

The 3 x 3 grillage was doubly symmetric and had two pairs of equal natural frequencies. The eigenvalues were predicted accurately but the eigenvectors associated with equal eigenvalues did not satisfy the tolerance criterion although all the other eigenvectors were seen to converge satisfactorily. The first nine eigenvalues are 3258.256,779.223, 779.223,438.224, 308.21 3, 307.593, 235.717,235.717 and 161-928. Table V shows the predicted eigenvalues obtained from the vectors

Table V. Predicted eigenvalues for a 3 x 3 grillage using 7 trial vectors

1 2 3 4 ~~

1 424-749 0.024 3 12.095 0.024 2 3257.930 775.798 774.038 405.757 3 3258.256 779.210 779.162 439.099 4 3258.256 779.223 779.222 437.955 5 3258.256 779-223 779.223 438-201 6 3258.256 779.223 779,223 438.222 7 3258.256 779,223 779.223 438.224

which converged onto the four dominant eigenvectors when iterating with m = 7. Due to the fact that eigenvalues beyond the ninth were much smaller then, when iterating with m = 9, all nine eigenvalues were obtained to six significant figures after only three iterations. Equal eigenvalues could have been avoided by making use of symmetry to solve only one half or one quarter of the grillage.

A set of tests were also performed with the rotary inertias of the 3 x 3 grillage set to zero, thus giving an incomplete mass matrix and only nine natural frequencies. It was found that the iteration could break down in the first cycle due to more than one predicted vector being null if a segment of the unit matrix was used for starting vectors. It is therefore safer to use trial vectors containing no zero elements.


THREE MATRIX MULTIPLICATIONS PER ITERATION

Tests were also carried out with the basic simultaneous iteration method modified to perform three matrix multiplications per iteration instead of one. This was found to be beneficial in computer time where the eigenvalues were closely spaced and is not recommended in other cases, particularly when the ratio &/A, is very large.

FURTHER CONSIDERATIONS

Subsequent to these numerical tests being performed Rutishauser15 has published a simultaneous iteration method in which an eigenreduction of the matrix vTv is used to predict new trial vectors. This is equivalent to the interaction analysis of the methods previously described, but has the advantage that the orthogonalizing process becomes unnecessary. Rutishauser also gives a procedure for performing more than one pre-multiplication per iteration, the number of pre- multiplications being chosen dynamically from the spread of the predicted eigenvalues.

Further investigation of the simultaneous iteration method for undamped vibration analysis is to be undertaken in order to obtain more information on the alternative simultaneous iteration procedures and the effect of the choice in number and form of the initial trial vectors for various types of problem. There is a likelihood of possible improvements in both Gupta’s Sturm sequence method and simultaneous iteration methods through cross fertilization of ideas. For close eigenvalues then Gupta’s method may benefit from a simultaneous inverse iteration procedure, and the simultaneous iteration method may benefit in some cases by information about the grouping of eigenvalues which could be obtained from the Sturni sequence analysis for one or two h values,

CONCLUSION

The simultaneous iteration method is a feasible means of obtaining natural frequencies and modes of vibration. Where the stiffness and mass matrices are symmetric and banded it is possible to retain the bandform of the equations during computer analysis, even when body freedoms are present. From a rough assessment of computational efficiency it has been shown that this should be particularly beneficial where the number of displacement co-ordinates is very large.

Some numerical tests have been carried out starting with a segment of the unit matrix to define the trial vectors. Although this is easy to generate automatically it can be demonstrated to be a poor choice of trial vectors. Despite this, convergence has been obtained in a moderate number of iterations. Better choice of trial vectors should reduce the number of iterations in specific applications. The lowest natural frequencies and vibration modes tend to be produced to a higher accuracy than the higher ones. This is likely to be very suitable in many vibration applications as the lower frequencies are least affected by errors arising from the basic assumptions and tend to have the greatest influence in most dynamical response situations.

ACKNOWLEDGEMENTS

The authors would like to thank Dr. D. W. Martin of N.P.L. for helpful criticism of the original script and B. R. Corr of the Civil Engineering Department, Queen’s University, for assisting with some numerical tests.

REFERENCES

1. J. S. Przemieniecki, Theory of Matrix Structura/ Analysis, McGraw-Hill, New York and London, 1968. 2. R. E. D. Bishop, G. M. L. Gladwell and S. Michaelson, The Matrix Analysis of Vibration, Cambridge

3. R. S. Martin and J. H. Wilkinson, ‘Reduction of a symmetric eigenproblem Ax = ABx and related problems University Press, 1964.

to standard form’, Num. Math., 11, 99-110 (1968).


4. J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. 5. K. K. Gupta, ‘Vibration of frames and other structures with banded stiffness matrix’, Int. J. num. Meth.

6. A. Jennings, ‘A direct iteration method of obtaining latent roots and vectors of a symmetric matrix’, Proc.

7 . A. Jennings and M. Clint, “The evaluation of eigenvalues and eigenvectors of real symmetric matrices by

8. W. Weaver Jr. . Computer Programs .for Structural Analysis, Van Nostrand, Princeton, New Jersey, 1967. 9. A. Jennings, ‘A compact storage scheme for the solution of symmetric linear simultaneous equations’,

10. R. S. Martin and J. H. Wilkinson, ‘Symmetric decomposition of positive definite band matrices’, Nurn.

11. A. Jennings, ‘Natural vibrations of a free structure’, Air. Engng, 34, 81-83 (1962). 12. 0. E. Bronlund, ‘Eigenvalues of large matrices’, Symp. Finite Elem. techn., Institut fur Statik und Dynamik

der Luft- und Raumfahrtkonstruktionen, University of Stuttgart, 1969. 13. G. L. Rogers, Dynamics of Framed Structures, Wiley, New York, 1959. 14. W. Weaver, M. F. Nelson and T. A. Manning, ‘Dynamics of tier buildings’, Am. Soc. civ. Engrs Jnl, 94,

15. H. Rutishauser, ‘Computational aspects of F. L. Bauer’s simultaneous iteration method’, Nurn. Math.,

Engng, 2, 221-228 (1970).

Camb. phil. Soc., 63, 755-765 (1967).

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Comput. J., 9, 281-285 (1966).

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application of the simultaneous iteration method to undamped vibration problems

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