Solving Homogeneous Trees of Sturm-Liouville Equations using an Infinite Order Determinant Method

Transcription

1 Paper Civil-Comp Press, Proceedings of the Eleventh International Conference on Computational Structures Technology,.H.V. Topping, Editor), Civil-Comp Press, Stirlingshire, Scotland Solving Homogeneous Trees of Sturm-Liouville Equations using an Infinite Order Determinant Method. Watson, W.P. Howson, C. James and C. Williams Department of eronautical and utomotive Engineering Loughorough University, United ingdom Cardiff School of Engineering Cardiff University, United ingdom stract Consideration is given to determining the eigenvalues of a graph with tree topology on which the second order Sturm Liouville operator is acting. This is directly analogous to the free viration prolem of a networ of ars with identical topology. The eigenvalues are determined using the well-estalished Wittric-Williams algorithm in conjunction with a dynamic stiffness approach. plot of the determinant of the stiffness matrix versus trial eigenparameter has several poles and eigenvalues of the system can occur oth when the determinant of the stiffness matrix is zero or infinite. recent exact procedure has een estalished that transforms this plot into an equivalent one that has no poles and where all eigenvalues correspond to zero values of the determinant. The determinant of the transformed matrix is defined as the infinite order determinant. This paper uses such a procedure and it is shown that the infinite order determinant plot tends to zero for large eigenvalues. However a small modification is seen to produce a plot that repeats at regular intervals and corresponds to the and gap structure of the spectrum for an infinitely long tree. In other pulished wor it has een shown that the eigenvalues of a finite, n level tree are otainale from a family of n stepped ars. oth techniques can e used to estimate the edges of the spectrum for the equivalent infinite system and can e extended to higher order Sturm Liouville prolems. However in oth cases the stepped ar technique is shown to e much superior computationally. eywords: exact dynamic stiffness matrix, Wittric-Williams algorithm, infinite order determinant, graph theory, Sturm Liouville operator. Introduction Using the Wittric Williams algorithm [] applied to the Sturm Liouville prolem on trees [] it has een demonstrated that a numerical procedure can e used to

2 provide insight on the spectral ehaviour of the free Laplacian in one dimension acting on a metric tree [3]. The prolem is directly analogous to the viration analysis of a networ of ars or eams connected together to form a tree topology [3]. The eigenvalues of the tree correspond to the square of the natural frequency of free viration. In this paper the approach adopted is a mathematical one. The structural analogy is shown elow, however Ref [] discusses this analogy in more detail. In the case of the infinitely long tree, the eigenvalues form ands of continuous spectra with an eigenvalue of infinite multiplicity in the middle of the gaps. In this paper, as in previous wor [], only finite length trees are analysed. Eigenvalues are found using the well-estalished Wittric-Williams algorithm and a dynamic stiffness approach. The plot of the determinant of the stiffness matrix versus trial eigenparameter has several poles and eigenvalues of the system can occur oth when the determinant of the stiffness matrix is zero or infinite. The Wittric-Williams algorithm locates with certainty, and up to computer accuracy, all the eigenvalues of the system. If the eigenvalue count on either side of a pole, in the plot of determinant versus trial eigenparameter, is identical then no eigenvalues are present. Typically poles occur at individual memer eigenvalues that have clamped oundary conditions. In the case of the infinitely long tree the eigenvalue in the middle of the gap corresponds to the clamped/clamped memer eigenvalue. For the finite length tree, if the Wittric-Williams algorithm shows an eigenvalue is present as the determinant plot passes through a pole, the clamped/clamped memer eigenvalue is also an eigenvalue of the system. The structure of the spectrum, for the infinite tree, is a repeating and of continuous spectra with an eigenvalue in the middle of the gap. For the finite tree there are no ands of continuous spectra, however all eigenvalues fall within the lower and upper ounds of each and of continuous spectra. These ands of eigenvalues form repeating patterns. The determinant plot versus trial eigenparameter starts from a positive value at zero on the ascissa and after crossing the ascissa one or more times tends to infinity as the trial eigenparameter tends to the clamped/clamped memer eigenvalue. Williams et al [5] estalished a procedure to transform this plot so that it has no poles and all eigenvalues occur when the transformed stiffness matrix determinant ecomes zero. The determinant of the transformed matrix is nown as the infinite order determinant. This paper uses such a procedure and it is seen that the infinite order determinant plot tends to zero for large values of trial eigenparameter. However a small modification can produce a plot that repeats at regular intervals and corresponds to the and gap structure of the spectrum for an infinitely long tree. From the results of finite length trees it is then possile to estimate the edges of the spectrum for the infinite system. Williams et al [6] show that the eigenvalues of the n level tree are otainale from a family of n stepped ars. From the results of the stepped ar approach it is then possile to estimate the edges of the spectrum for the infinite system.

3 Theory The theory presented in this section relates to the general form of the classical second-order Sturm-Liouville equation, which may e written as d dy p qy λwy ) dx dx where p, q and w are all real valued, positive constants and λ is the eigenparameter. y imposing the following oundary conditions y y at x and y y at x L ) leading to y cot L y csc L csc L y cot L y 3) where. This is the well nown edge equation with suscripts denoting the left and right end of each edge that is used to form the tree. Figure typifies the tree topology, shows Dirichlet oundary conditions and defines the edge and vertex levels, together with n and, the numer of vertex levels and ranching numer, respectively. ll trees are classified as repetitive or non-repetitive, depending upon whether or not the edge properties at all levels are identical, and such trees are su-divided into uniform or non-uniform, depending upon whether or not L, p, q and w are all constant. Hence, a repetitive uniform tree is a homogeneous one, whereas a repetitive nonuniform tree is not. This paper deals only with homogeneous trees with pwl and q. For the analysis of an n level tree the required eigenvalues are otainale from n stepped ars each containing r memers, where r,,,n. The stepped ar representations for a tree with n3 and 3 are shown in Figure. ll edges at the same level are part of the same memer in this representation. It should now e noted that Equation ) is exactly analogous to the axial viration equation of a non-uniform ar on a uniform elastic foundation, as follows d dy E κ y ω my ) dx dx where E is the extensional rigidity of the ar, κ is the stiffness per unit length of the elastic foundation, m is the mass per unit length of the ar and ω is the natural radian frequency. 3

4 y y Levels: Edge 3 n-) Vertex 3 5 n) a) five level, two ranching tree i.e. n 5, with Dirichlet oundary conditions ) n 3, 3 c) n 3, Figure : The topology of typical trees showing edge and vertex levels together with their ranching numer Single memers a) 3 level, stepped ar ) level stepped ar c) level stepped ar Figure : The stepped ar representations for a tree with n3 and 3. ll edges at the same level are part of the same memer in this representation so that the three level stepped ar has three memers not thirteen.

5 5. Dynamic stiffness matrix The necessary eigenvalue relationship for a single edge, evaluated at a trial eigenparameter,, is given y Equation 3) as y y 5) where L cot and L csc. The edge equation of equation 3) is used to assemle the gloal stiffness matrix, ), for the tree. For a tree with 3, n3 and Dirichlet oundary conditions, as shown Figure ), the gloal stiffness matrix is given as ) 6) Using Gauss elimination this matrix can e converted to upper triangular form to give Δ 3 6 ) 7) The determinant of ) is then ) 3csc 6 cot cot 6 ) ) 8) It can e seen that the value of the determinant ecomes large as the trial eigenparameter ecomes large. t an eigenvalue the determinant will e zero). If the value of / ) versus is plotted instead, then this leads to a repeating plot that still has poles.. Infinite Order Determinant Method The procedure using the infinite order determinant method is descried in detail elsewhere [5]. The essence of using this approach is to e ale to plot a determinant

6 that does not have poles at any trial value of the eigenparameter. The procedure involves a numer of steps. The first of these steps is to evaluate the determinant at zero eigenparameter i.e.. Using the matrix derived aove for the 3 and n3 tree we otain the following ) 6 6 3) 8 9) The second step is to examine each individual memer to estalish the normalised memer stiffness determinant. For the repetitive tree all memers are identical hence only one memer needs to e analysed. From Ref. [5] this determinant is given as For all memers The infinite order stiffness determinant is then given as sin Δ m ) Δ Δ m ) m ) ) Δ ) ) ) For the tree considered 3, n3 and m3 numer of memers). Hence ) cos 3 sin 6 9 sin 6 cos 9 3 cot 6 cot 8 3) 3csc ) 3) The infinite order determinant tends to a small numer for large values of if the determinant is non-zero. If this determinant is multiplied y 9 then a repeating plot is produced and is shown in Figure 3. Figure 3 shows three plots which are: i) ) v ; ii) ) v ; and 9 iii) ) v. This plot is amplified y a factor of ). There are some points to note aout Figure 3. The plot of ) v has a non-zero value at. s seen aove ) 8. ecause this plot has an amplification 6

7 factor of. the ordinate value is.8. The plot of ) tends to zero very. rapidly. This is due to the presence of 9 in the denominator. This plot then does 9 not appear to give any useful information. The plot of ) v only has transcendental terms and is therefore cyclical. Herein this determinant is referred to as the normalised infinite order determinant ecause it repeats at fixed intervals and does not have any poles. This plot has een amplified y a factor of SSL SSU.5 Figure 3: Determinant plots for the n3, 3 tree. The three plots are the ordinary determinant; infinite order determinant and normalised infinite order determinant plots. The theoretical upper and lower ounds given y Soolev and Solomya [] are shown as solid vertical lines and laelled as SSL and SSU for the lower and upper ound, respectively. 3 Higher order trees The stiffness matrix for an n, ranching tree is shown elow. It shows how the stiffness matrix is constructed for higher order trees. The stiffness matrix is square and the order of the matrix is equal to the numer of vertices in the tree. The numer of vertices, N, can e equated as follows: ) 7

8 For each new additional level in tree length the order of the matrix increases y a factor of. Performing analytical methods on large trees gets difficult very quicly due to this exponential growth in numer of vertices and edges. ) ) ) 5) When this matrix is converted to upper triangular form the leading diagonal is made up of: i) diagonal elements containing ) ii) diagonal elements containing ) ; and ) ii) element containing ) ; ) ) s trees get larger it can e see that for each new level the aove elements grow y a factor of and the new eigenvalues are given y the last diagonal element of the matrix. Note that the locations of the individual terms, in the matrix, are as a consequence of the numering technique employed. Stepped ar model The analysis of the stepped ar representation of the tree is different to that of the tree. The principal difference lies in the construction of the stiffness matrix. The 8

9 9 memers at each level are not identical. However the terms p and w in Equation ) vary down the length of the stepped ar. This needs to e accounted for as follows. y y i 6) where i is the factor for each memer at level i as defined in the tree model of Figure. The construction of the stiffness matrix then ecomes for a three memer stepped ar with two internal vertices as follows: S ) ) 7) Performing Gauss elimination we otain Δ S ) ) ) 8) Hence ) ) ) ) ) ) ) ) S 9) Sustituting for and ) ) ) S csc cot csc cot ) Using the same techniques as used for the tree to otain the infinite order determinant plot we otain the following: ) ) S ) It can e shown that for all memers m sin Δ )

10 For all memers The Infinite Order Stiffness Determinant is then given as Δ Δ m 3) S m S ) ) Δ ) ) ) For the stepped with ar 3 and n3, there are 3 memers hence 3 S ) cot csc ) sin 6 cos 3) sin S ) 5) 39 3 The normalised plots for n; n; n8 and n6 are shown in Figure. Figure : Normalised infinite order determinant plots for n, n, n8 and n6 stepped ar. The theoretical upper and lower ounds given y Soolev and Solomya are shown as solid vertical lines and laelled as SSL and SSU for the lower and upper ound respectively. Tale shows the first eigenvalue for the four stepped ars of Figure along with the lower ound of Soolev and Solomya SSL). numerical convergence

11 technique is used to estimate the lower ound ased upon the finite values of n and compares reasonaly well with the analytical value. n Estimate of SSL SSL % difference Tale : First eigenvalue,, for stepped ar of length n and estimate of Soolev Solomya lower ound SSL. The % difference compares Estimate of SSL to SSL. Figure has several interesting features. The n6 stepped ar has the same eigenvalues as all the smaller stepped ars plus additional ones. This is a consequence of choosing the stepped ar lengths. ll the zeros of the four determinants plotted are eigenvalues except for the origin where all four plots start. The eigenvalues shown in the figure are not the complete set for the n6 tree. To otain a complete set one would need to include the same plots for all stepped ars in the range to 6 that are not already shown. s n tends to get larger it can e seen the amplitude of the plots is reducing inside the SSL and SSU range. In fact the plots have een amplified y factors 3, 5, 5 and for each of n,, 8 and 6 respectively. s n tends to infinity the determinant plot would e a flat line from the value SSL to SSU representing a set of solutions which form a continuous and of spectra []. The infinitely long stepped ar would thus have all the eigenvalues of the tree although their multiplicity would not e correct. lthough not shown, all the four plots repeat every interval of π and it can e seen that the numer of eigenvalues given y an n level stepped ar is exactly n- in the interval of SSL to SSU. 6 Summary and Conclusions The analysis presented shows that large trees can e analysed very easily with use of the infinite order determinant method and normalising the numerical data to produce plots that repeat at intervals of π. To achieve this, the stepped ar representation was used, which reduced the order of the prolem sustantially. The length of the stepped ar and order of its stiffness matrix varies linearly, so reasonaly long stepped ars e.g. n6 can e modelled. The order of the tree stiffness matrix is a function of the ranching numer and length of the tree, hence the order of the stiffness matrix increases exponentially with each new level. For the stepped ar, the order of the stiffness matrix is a linear function of the length of the ar only and is independent of the ranching numer.

12 Each stepped ar does not contain the information of the eigenvalues of the whole tree and so a series of stepped ars must e considered in order to otain this information. However the first eigenvalue of the stepped ar of length n has the same first eigenvalue of the tree and so only one shorter stepped ar is required to carry out a numerical convergence analysis to converge on the value for the edge of the spectrum. It was shown that for n8 and n6, the eigenvalues otained provided a very good estimate on the edge of the spectrum. This was validated y comparing with the theoretical result. References [] W.H. Wittric & F.W. Williams, F. W. general algorithm for computing natural frequencies of elastic structures, Q. J. Mech. ppl. Math,, 63 8, 97. [].V. Soolev, M. Solomya, Schrödinger operators on homogeneous metric trees spectrum in gaps, Reviews in Mathematical Physics 5), - 67,. [3] F.W. Williams, W.P. Howson,. Watson, pplication of the Wittric- Williams algorithm to the Sturm-Liouville prolem on homogeneous trees: structural mechanics analogy, Proceedings of the Royal Society series : Mathematical, Physical and Engineering Sciences, 65), 3-68,. [] W.P. Howson &. Watson, Homogeneous Trees of Second Order Sturm- Liouville Equations: General Theory and Program, Computers & Structures, -5C, 3-, [5] F.W. Williams, D. ennedy, M.S. Djoudi, The memer stiffness determinant and its uses for the transcendental eigenprolems of structural engineering and other disciplines, Proceedings of The Royal Society: Mathematical, Physical and Engineering Sciences, 595), -9, 3. [6] F.W. Williams,. Watson, W.P. Howson,.J. Jones. Exact solutions for Sturm-Liouville prolems on trees via novel sustitute systems and the Wittric-Williams algorithm, Proceedings of the Royal Society series : Mathematical, Physical and Engineering Sciences, 6388),