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Dualities in vibrating rods and beams: continuous and discrete models
Journal of Sound and Vibration (1995) 184(5), 759–766
DUALITIES IN VIBRATING RODS AND BEAMS: CONTINUOUS AND DISCRETE MODELS Y. M. R Department of Mechanical Engineering
S. E Department of Computer Science, University of Adelaide, Adelaide, South Australia 5005 (Received 28 March 1994) Consider an axially vibrating rod with constant material properties but with a varying cross-sectional area. There exist two rods, with generally different cross-sections, which have the same spectrum. The connection between the eigenfunctions of one rod and the other is given. It is shown that this duality carries across from the continuous to the analogous discrete model. Similar results are established for a certain model of the vibrating beam. 7 1995 Academic Press Limited
1. INTRODUCTION
The axial vibration of a rod with modulus of elasticity E, mass per unit length r and cross-sectional area a leads to the eigenvalue problem
0
1
d d Ea u + hrau = 0, dx dx
0 Q x Q 1,
u(0) = 0,
d u(1) = 0. dx
(1)
Our present interest is in the case, common in practice, where E and r are constant and only a = a(x) q 0 varies along the length x of the rod. It is thus sufficient to consider the normalized problem
0 1
d d a u + lau = 0, dx dx
0 Q x Q 1,
u(0) = 0,
d u(1) = 0, dx
(2)
with l = hr/E. Let
aˆ (x) = 1/a(1 − x).
(3)
By the dual problem we mean
0 1
d d aˆ v + maˆv = 0, dx dx
0 Q x Q 1,
v(0) = 0,
d v(1) = 0, dx
(4)
because systems (2) and (4) have common eigenvalues mi = li , i = 1, 2, 3, . . . . This, not widely known, property is mentioned in Fundamentals of Musical Acoustics [1, pp. 410]: Consider a pair of horns such that the product A1 A2 of their cross sectional areas is a constant from one end to the other . . . . If the small ends of both of these 759 0022–460X/95/300759 + 08 $12.00/0
7 1995 Academic Press Limited
. . .
760
horns are closed off, the two air columns turn out to have identical natural frequencies! We have observed [2] that the discrete mathematical model of the vibrating rod preserves this duality. In this paper we show that there also exist similar dualities for the continuous and discrete models of certain transversely vibrating beams. In addition, we give the explicit relation between the eigenfunctions (eigenvectors for the discrete case) of the primal and dual problems. Thus, if an eigenpair is known for the primal problem, then its counterpart can be simply obtained without solving the dual problem, and vice versa. Turning now to the finite difference models of systems (2) and (4), we define the n × n shift matrix S = [sij ], sij = di + 1,j , dij being the Kronecker delta, and denote F = I − ST. Let D = diag {d1 , d2 , . . . , dn }, where the di are the discretization values of a(x) at n equally spaced intervals. Then, system (2) may be modelled, with finite differences, as n 2FTDFU = DUV,
(5)
where V = diag {v1 , v2 , . . . , vn } and U = [u1 , u2 , . . . , un ] is the column partitioning of the eigenvector matrix U. Similarly, we denote D = ED−1E,
(6)
where E = [eij ] is the exchange matrix defined by eij = di,n − j + 1 . Then, the dual problem (4) has the finite difference model n 2FTD FV = D VG,
(7)
where G = diag {g1 , g2 , . . . , gn } and V = [v1 , v2 , . . . , vn ]. The eigenpairs (vi , ui ) and (gi , vi ) are the finite difference approximations to the first n eigenpairs of systems (2) and (4), respestively. We have shown [2] that vi = gi , i = 1, 2, . . . , n. In section 4 we give an explicit characterization of the eigenvector matrix V in terms of D and U. Thus, there is a direct analogy between the duality in the continuous system and its discrete model. Consider a Bernoulli–Euler beam, aligned along the x- axis of an x, y, z Cartesian space, with unit width in the z direction and cross-section y which varies only with x. Let u(x) be the transverse displacement of the beam in the z direction. With appropriate normalization, the associated eigenvalue problem is
0
1
d2 d2 u − lau = 0, 2 a dx dx 2
0 Q x Q 1,
u(0) =
d d2 d3 u(0) = 2 u(1) = 3 u(1) = 0. dx dx dx
(8)
d d2 d3 v(0) = 2 v(1) = 3 v(1) = 0, dx dx dx
(9)
We show, in section 5, that its dual,
0
1
d2 d2 v − maˆv = 0, 2 aˆ dx dx 2
0 Q x Q 1,
v(0) =
where aˆ is given again by equation (3), has the same eigenvalues. With similar notation, the finite difference analogue of system (8) is n 4(F2 )TDF2U = DUV,
(10)
n 4(F2 )TD F2V = D VG,
(11)
and we show that its dual is
where D is given by equation (6). We also present, in section 5, the explicit relation between
761
the eigenfunctions u and v of the continuous models and the relation between the eigenvector matrices U and V of the discrete model. It is remarkable that, despite the well known inconsistency between the asymptotic behaviour of the eigenvalues of the continuous problem and those of it discrete model [3], this duality carries across fully from the continuous to the discrete case for both the rod and the beam. The duality itself leads to some counterintuitive phenomena. For example, using duality we may design two rods with quite different cross-sectional areas, say trapezoidal and hyperbolic, which have the same spectrum. This case is discussed in section 3. From the preceding discussion, it is clear that if we have one solution of the inverse eigenvalue problem of reconstructing a(x) (apart from a scale factor) from its spectrum, then a complementary solution aˆ (x) may be obtained immediately.
2. THE FINITE DIFFERENCE ROD MODEL
The duality result concerning the equivalence of the spectra of the finite difference models (5) and (7) was established in reference [2]. For completeness, we briefly review the relevant result here. Recall that the spectrum of a pencil A − lB, denoted by s(A, B), is invariant under left or right multiplication by a non-singular matrix. Then, using (I − S)−1D(I − D−1SD) = D, we have s(FTDF, D) = s((I − S)−1FTDF(I − D−1SD), D),
(12)
s(FTDF, D) = s(FTDF + C, D),
(13)
which simplifies to
where C = D2STD−1S − SDST = diag
6
7
d2i − di + 1 , di − 1
and d0 :a, dn + 1 = 0. Now, by equivalence transformation,
s(FTDF, D) = s(ED−1(I − S)D(I − ST)D−1E, ED−1E).
(14)
(15)
Replacing D in equations (13) and (14) with D gives s(FTD F, D ) = s(FTD F + C , D ),
(16)
C = ED−2SDES − SD ST.
(17)
ED−1(I − S)D(I − ST)D−1E = FTD F + C .
(18)
where
It is easily shown that
Hence, it follows from equations (15)–(18) that s(FTDF, D) = s(FTD F, D ),
(19)
or, equivalently, V = G in equations (5) and (7), establishing the duality result. We now present a simple example. All the numerical calculations in this paper were performed in IEEE double precision standard floating point arithmetic. We display five correctly rounded decimal digits.
. . .
762
2.1. Let D = diag {1, 2, 3, 4, 5}. Then D = diag {1/5, 1/4, 1/3, 1/2, 1}. The eigensolutions of the primal (5) and dual (7) problems are V = G = 25 diag {0·03703, 0·78626, 2·10475, 3·52101, 4·63428}, and the matrices U and V are
F G G U=G G G f
1·00000
1·00000
1·00000
1·00000
1·00000
1·48148
1·10687
0·44762
−0·26051
−0·81714
J G G 1·76590 0·59793 −0·54872 −0·48935 0·49600 G 1·93016 −0·13637 −0·42979 0·63127 −0·24310 G G 2·00439 −0·63804 0·38903 −0·25040 0·06689 j
F G G V=G G G f
1·00000 1·77037
and
J G G 2·29898 0·60871 −0·73008 0·15573 2·54165 G. 2·59463 −0·08521 −0·26984 0·57187 −2·34478 G G 2·69441 −0·39868 0·24425 −0·22684 0·64519 j 1·00000 1·17099
1·00000 0·11620
1·00000 1·00000 −0·01681 −1·90742
Although U and V are the eigenvector matrices for dual problems (with the same spectrum) the above analysis does not indicate how to find one from the other. We return to this point in section 4, after the duality for the continuous problem is discussed.
3. THE CONTINUOUS MODEL FOR THE ROD
We now focus our attention on the continuous model for the rod. Changing the variable from x to 1 − x in the differential equation (2) gives d d a(1 − x) u(1 − x) = −la(1 − x)u(1 − x). dx dx
0
1
(20)
v(x) = a(1 − x)(d/dx)u(1 − x)
(21)
Now, substituting
into the left side of the first of equations (4) and using equation (20) we have
0 1
0 0
d d d d d a v + maˆv = aˆ aˆ− u(1 − x) dx dx dx dx 1 dx
11
+m
d u(1 − x) dx
d d (aˆ (laˆ−1u(1 − x))) + m u(1 − x) dx dx
=−
= (m − l)(d/dx)u(1 − x).
(22)
The expression on the right will vanish, and so the differential equation (4) will be satisfied, whenever l = m. Now consider the boundary conditons for the first of equations (4). Putting z = 0 into
763
equation (21) yields v(0) = a(1)(d/dx)u(1) = 0, in view of equations (2). Similarly, equation (21) at x = 1 gives, upon using equation (20) again,
0
d d d v(1) = a(1 − x) u(1 − x) dx dx dx
1b
= −la(0)u(0) = 0.
x=1
Thus the following duality holds: if l is an eigenvalue of system (2) and u(x) is its corresponding eigenfunction, then l is also an eigenvalue of system (4) and its corresponding eigenfunction is v(x) = a(1 − x)(d/dx) u(1 − x). 3.1. (a) Uniform cross-sectional area. Here a(x) = k, k constant and aˆ (x) = 1/k. Since the eigenpairs of system (2) are invariant under scalar multiplication of a, we see that systems (2) and (4) have the same eigenvalues and eigenvectors. However, the duality result does not in this case, lead to an essentially different rod configuration. (b) Exponential cross-sectional area. The case a(x) = ex leads to a dual problem with aˆ = ex − 1 , which can be written as aˆ (x) = (1/e)a(x). Here again, the primal and dual problems are effectively the same. (c) Linear cross-sectional area. Here, a(x) = 1 + x and aˆ = 1/(2 − x). The eigenvalues li of the primal and dual probelms here satisfy J1 (2zli )Y0 (zli ) − J0 (zli )Y1 (2zli ) = 0, where Jn and Yn are the nth order Bessel functions of the first and second kinds, respectively. The corresponding eigenfunctions of systems (2) and (4) are ui (x) = J0 ((1 + x)zli ) −
J0 (zli )
Y0 ((1 + x)zli ),
Y0 (zli )
0
vi (x) = (2 − x) J1 ((2 − x)zli ) −
J0 (zli )
1
Y1 ((2 − x)zli ) .
Y0 (zli )
Using these facts we may confirm the duality.
4. THE DISCRETE ROD MODEL REVISITED
Relation (21) may suggest the connection between the primal and dual eigenvector matrices U and V in the discrete case. Let us denote the discretization, at points separated by constant distance h, of a continuous function g(x), x $ [0, 1], by G = diag {g(x1 ), g(x2 ), . . . , g(xn )},
x1 = h,
xi + 1 = xi + h.
Then our finite difference approximation of its derivative, (d/dx)g(x), is (1/h)FTG. Furthermore, the discrete analogy of the transformation of co-ordinates from g(x) to g(1 − x) is the replacement of G by EGE for G a matrix and q by Eq, if q is a vector. Hence, the discrete analog of equation (21) for the matrices V and U is V = EDEFTEU. For any Toeplitz matrix A it is true that EATE = A.
(23)
Our hypothesis is that if U is the eigenvector matrix of equation (5) then, since F is Toeplitz, V = EDFU
(24)
. . .
764
is the eigenvector matrix of equation (7). We now prove this hypothesis. Substituting equation (24) into (7) gives n 2FTED−1EFEDFU = ED−1EEDFUG, or equivalently, n 2FTED−1FTDFU = EFUG. Now, using equation (5), this gives FTEUV = EFUG.
(25)
Using equation (23) again proves statement (24). Furthermore, G = V. This is an alternative proof of the duality result in the discrete case. 4.1. Following on from Example 2.1 we have
F G G EDFU = G G G f
0·37114
−2·50830
4·09411 −4·40836
1·54994
0·65705 −2·93721 0·47573 4·48247 −2·95640 0·85324 −1·52683 −2·98903 −0·68652 3·93942 0·96297 0·21374 −1·10475 −2·52101 −3·63428 1·00000 1·00000 1·00000 1·00000 1·00000
J G G G, G G j
which, apart from a diagonal scaling of its columns, contains the same eigenvectors as V.
5. CONTINUOUS AND DISCRETE MODELS OF THE TRANSVERSELY VIBRATING BEAM
An analysis, similar to that of the previous sections, can be applied to the continuous and discrete beam models (8)–(11) of section 1. For the continuous model, the eigenfunctions of systems (8) and (9) which correspond to equal eigenvalues satisfy v(x) = a(1 − x)(d2/dx 2 )u(1 − x).
(26)
Similarly, for the discrete model, the eigenvector matrices U and V of equations (10) and (11) satisfy V = EDF2U.
(27)
The proofs closely follow the corresponding argument for the rod. Changing the variable from x to 1 − x in the differential equation (8) gives
0
1
d2 d2 u(1 − x) = la(1 − x)u(1 − x). 2 a(1 − x) dx dx 2
(28)
765
Substituting equation (26) into the left side of the first of equations (9) and using equation (28) we have
0
1
0 0
d2 d2 d2 d2 d2 aˆ v − maˆv = 2 aˆ aˆ−1 2 u(1 − x) dx 2 dx 2 dx dx 2 dx =
11
−m
d2 u(1 − x) dx 2
d2 d2 −1 u(1 − x))) − m 2 u(1 − x) 2 (aˆ (laˆ dx dx
= (l − m)(d2/dx 2 )u(1 − x).
(29)
So, the first of equations (9) is satisfied if l = m. Furthermore, the boundary conditions of system (9) give v(0) = a(1)(d2/dx 2 )u(1) = 0,
0
d d d2 v(0) = a(1 − x) 2 u(1 − x) dx dx dx
0
1b
= −a(1) x=0
d2 d2 d2 u(1 − x) 2 v(1) = 2 a(1 − x) dx dx dx 2
1b
d3 d d2 a(1) 2 u(1) = 0, 3 u(1) − dx dx dx = la(0)u(0) = 0, x=1
d3 d d d v(1) = (la(1 − x)u(1 − x))= x = 1 = −u(0) a(0) − a(0) u(0) = 0, dx 3 dx dx dx in view of equation (28) and the boundary conditions associated with system (8). This completes the proof of duality for the continuous model of the beam. For the discrete beam model, we substitute equations (27) and (6) into equation (11), and, recalling that F is Toeplitz and E2 = I, we obtain n 4(F2 )TED−1(F2 )TDF2U = EF2UG. Then, using equation (10) simplifies this to (F2 )TEUV = EF2UG. However, simce F2 is also a Toeplitz matrix we conclude that V = G, and this establishes equation (27).
6. CONCLUSIONS
Two axially vibrating rods, the first fixed at one end and free to oscillate at the other and the second configured oppositely, which have the property that the product of their cross-sectional areas is constant along the rods, have a common spectrum. In the present paper we have shown that this duality is reflected in both the continuous to the discrete models of the rod. We have also explicitly given the relation between (a) the eigenfunctions of the primal and dual in the continuous case and (b) the eigenvector matrices of the primal and dual in the discrete case. Thus, if the eigenfunctions (eigenvectors) of one of the rods are known, we may determine those of the other without solving the other problem. Furthermore, we have shown that there exists an analogous duality for the vibrating beam, that it carries over from the continuous to the discrete model, and we have given the relations between the eigensolutions of the primal and dual for that case. In the discrete model of the rod there are at most (n − 1)! different solutions [2]. The important problem of expressing other possible cross-sectional area rods with the same spectrum (for the discrete or continuous case) remains open. A similar solution pertains for the beam.
766
. . . REFERENCES
1. A. H. B 1976 Fundamentals of Musical Acoustics. Oxford: Oxford University Press. 2. Y. M. R and S. E 1993 University of Adelaide, Department of Computer Science, Technical Report 93–15. An inverse eigenvalue problem arising in the vibration of rods. 3. J. W. P, F. W. D H and R. S. A 1981 Computing 26, 123–139. On the correction of finite difference eigenvalue approximations for Sturm–Liouville problems.
DUALITIES IN VIBRATING RODS AND BEAMS: CONTINUOUS AND DISCRETE MODELS Y. M. R Department of Mechanical Engineering
S. E Department of Computer Science, University of Adelaide, Adelaide, South Australia 5005 (Received 28 March 1994) Consider an axially vibrating rod with constant material properties but with a varying cross-sectional area. There exist two rods, with generally different cross-sections, which have the same spectrum. The connection between the eigenfunctions of one rod and the other is given. It is shown that this duality carries across from the continuous to the analogous discrete model. Similar results are established for a certain model of the vibrating beam. 7 1995 Academic Press Limited
1. INTRODUCTION
The axial vibration of a rod with modulus of elasticity E, mass per unit length r and cross-sectional area a leads to the eigenvalue problem
0
1
d d Ea u + hrau = 0, dx dx
0 Q x Q 1,
u(0) = 0,
d u(1) = 0. dx
(1)
Our present interest is in the case, common in practice, where E and r are constant and only a = a(x) q 0 varies along the length x of the rod. It is thus sufficient to consider the normalized problem
0 1
d d a u + lau = 0, dx dx
0 Q x Q 1,
u(0) = 0,
d u(1) = 0, dx
(2)
with l = hr/E. Let
aˆ (x) = 1/a(1 − x).
(3)
By the dual problem we mean
0 1
d d aˆ v + maˆv = 0, dx dx
0 Q x Q 1,
v(0) = 0,
d v(1) = 0, dx
(4)
because systems (2) and (4) have common eigenvalues mi = li , i = 1, 2, 3, . . . . This, not widely known, property is mentioned in Fundamentals of Musical Acoustics [1, pp. 410]: Consider a pair of horns such that the product A1 A2 of their cross sectional areas is a constant from one end to the other . . . . If the small ends of both of these 759 0022–460X/95/300759 + 08 $12.00/0
7 1995 Academic Press Limited
. . .
760
horns are closed off, the two air columns turn out to have identical natural frequencies! We have observed [2] that the discrete mathematical model of the vibrating rod preserves this duality. In this paper we show that there also exist similar dualities for the continuous and discrete models of certain transversely vibrating beams. In addition, we give the explicit relation between the eigenfunctions (eigenvectors for the discrete case) of the primal and dual problems. Thus, if an eigenpair is known for the primal problem, then its counterpart can be simply obtained without solving the dual problem, and vice versa. Turning now to the finite difference models of systems (2) and (4), we define the n × n shift matrix S = [sij ], sij = di + 1,j , dij being the Kronecker delta, and denote F = I − ST. Let D = diag {d1 , d2 , . . . , dn }, where the di are the discretization values of a(x) at n equally spaced intervals. Then, system (2) may be modelled, with finite differences, as n 2FTDFU = DUV,
(5)
where V = diag {v1 , v2 , . . . , vn } and U = [u1 , u2 , . . . , un ] is the column partitioning of the eigenvector matrix U. Similarly, we denote D = ED−1E,
(6)
where E = [eij ] is the exchange matrix defined by eij = di,n − j + 1 . Then, the dual problem (4) has the finite difference model n 2FTD FV = D VG,
(7)
where G = diag {g1 , g2 , . . . , gn } and V = [v1 , v2 , . . . , vn ]. The eigenpairs (vi , ui ) and (gi , vi ) are the finite difference approximations to the first n eigenpairs of systems (2) and (4), respestively. We have shown [2] that vi = gi , i = 1, 2, . . . , n. In section 4 we give an explicit characterization of the eigenvector matrix V in terms of D and U. Thus, there is a direct analogy between the duality in the continuous system and its discrete model. Consider a Bernoulli–Euler beam, aligned along the x- axis of an x, y, z Cartesian space, with unit width in the z direction and cross-section y which varies only with x. Let u(x) be the transverse displacement of the beam in the z direction. With appropriate normalization, the associated eigenvalue problem is
0
1
d2 d2 u − lau = 0, 2 a dx dx 2
0 Q x Q 1,
u(0) =
d d2 d3 u(0) = 2 u(1) = 3 u(1) = 0. dx dx dx
(8)
d d2 d3 v(0) = 2 v(1) = 3 v(1) = 0, dx dx dx
(9)
We show, in section 5, that its dual,
0
1
d2 d2 v − maˆv = 0, 2 aˆ dx dx 2
0 Q x Q 1,
v(0) =
where aˆ is given again by equation (3), has the same eigenvalues. With similar notation, the finite difference analogue of system (8) is n 4(F2 )TDF2U = DUV,
(10)
n 4(F2 )TD F2V = D VG,
(11)
and we show that its dual is
where D is given by equation (6). We also present, in section 5, the explicit relation between
761
the eigenfunctions u and v of the continuous models and the relation between the eigenvector matrices U and V of the discrete model. It is remarkable that, despite the well known inconsistency between the asymptotic behaviour of the eigenvalues of the continuous problem and those of it discrete model [3], this duality carries across fully from the continuous to the discrete case for both the rod and the beam. The duality itself leads to some counterintuitive phenomena. For example, using duality we may design two rods with quite different cross-sectional areas, say trapezoidal and hyperbolic, which have the same spectrum. This case is discussed in section 3. From the preceding discussion, it is clear that if we have one solution of the inverse eigenvalue problem of reconstructing a(x) (apart from a scale factor) from its spectrum, then a complementary solution aˆ (x) may be obtained immediately.
2. THE FINITE DIFFERENCE ROD MODEL
The duality result concerning the equivalence of the spectra of the finite difference models (5) and (7) was established in reference [2]. For completeness, we briefly review the relevant result here. Recall that the spectrum of a pencil A − lB, denoted by s(A, B), is invariant under left or right multiplication by a non-singular matrix. Then, using (I − S)−1D(I − D−1SD) = D, we have s(FTDF, D) = s((I − S)−1FTDF(I − D−1SD), D),
(12)
s(FTDF, D) = s(FTDF + C, D),
(13)
which simplifies to
where C = D2STD−1S − SDST = diag
6
7
d2i − di + 1 , di − 1
and d0 :a, dn + 1 = 0. Now, by equivalence transformation,
s(FTDF, D) = s(ED−1(I − S)D(I − ST)D−1E, ED−1E).
(14)
(15)
Replacing D in equations (13) and (14) with D gives s(FTD F, D ) = s(FTD F + C , D ),
(16)
C = ED−2SDES − SD ST.
(17)
ED−1(I − S)D(I − ST)D−1E = FTD F + C .
(18)
where
It is easily shown that
Hence, it follows from equations (15)–(18) that s(FTDF, D) = s(FTD F, D ),
(19)
or, equivalently, V = G in equations (5) and (7), establishing the duality result. We now present a simple example. All the numerical calculations in this paper were performed in IEEE double precision standard floating point arithmetic. We display five correctly rounded decimal digits.
. . .
762
2.1. Let D = diag {1, 2, 3, 4, 5}. Then D = diag {1/5, 1/4, 1/3, 1/2, 1}. The eigensolutions of the primal (5) and dual (7) problems are V = G = 25 diag {0·03703, 0·78626, 2·10475, 3·52101, 4·63428}, and the matrices U and V are
F G G U=G G G f
1·00000
1·00000
1·00000
1·00000
1·00000
1·48148
1·10687
0·44762
−0·26051
−0·81714
J G G 1·76590 0·59793 −0·54872 −0·48935 0·49600 G 1·93016 −0·13637 −0·42979 0·63127 −0·24310 G G 2·00439 −0·63804 0·38903 −0·25040 0·06689 j
F G G V=G G G f
1·00000 1·77037
and
J G G 2·29898 0·60871 −0·73008 0·15573 2·54165 G. 2·59463 −0·08521 −0·26984 0·57187 −2·34478 G G 2·69441 −0·39868 0·24425 −0·22684 0·64519 j 1·00000 1·17099
1·00000 0·11620
1·00000 1·00000 −0·01681 −1·90742
Although U and V are the eigenvector matrices for dual problems (with the same spectrum) the above analysis does not indicate how to find one from the other. We return to this point in section 4, after the duality for the continuous problem is discussed.
3. THE CONTINUOUS MODEL FOR THE ROD
We now focus our attention on the continuous model for the rod. Changing the variable from x to 1 − x in the differential equation (2) gives d d a(1 − x) u(1 − x) = −la(1 − x)u(1 − x). dx dx
0
1
(20)
v(x) = a(1 − x)(d/dx)u(1 − x)
(21)
Now, substituting
into the left side of the first of equations (4) and using equation (20) we have
0 1
0 0
d d d d d a v + maˆv = aˆ aˆ− u(1 − x) dx dx dx dx 1 dx
11
+m
d u(1 − x) dx
d d (aˆ (laˆ−1u(1 − x))) + m u(1 − x) dx dx
=−
= (m − l)(d/dx)u(1 − x).
(22)
The expression on the right will vanish, and so the differential equation (4) will be satisfied, whenever l = m. Now consider the boundary conditons for the first of equations (4). Putting z = 0 into
763
equation (21) yields v(0) = a(1)(d/dx)u(1) = 0, in view of equations (2). Similarly, equation (21) at x = 1 gives, upon using equation (20) again,
0
d d d v(1) = a(1 − x) u(1 − x) dx dx dx
1b
= −la(0)u(0) = 0.
x=1
Thus the following duality holds: if l is an eigenvalue of system (2) and u(x) is its corresponding eigenfunction, then l is also an eigenvalue of system (4) and its corresponding eigenfunction is v(x) = a(1 − x)(d/dx) u(1 − x). 3.1. (a) Uniform cross-sectional area. Here a(x) = k, k constant and aˆ (x) = 1/k. Since the eigenpairs of system (2) are invariant under scalar multiplication of a, we see that systems (2) and (4) have the same eigenvalues and eigenvectors. However, the duality result does not in this case, lead to an essentially different rod configuration. (b) Exponential cross-sectional area. The case a(x) = ex leads to a dual problem with aˆ = ex − 1 , which can be written as aˆ (x) = (1/e)a(x). Here again, the primal and dual problems are effectively the same. (c) Linear cross-sectional area. Here, a(x) = 1 + x and aˆ = 1/(2 − x). The eigenvalues li of the primal and dual probelms here satisfy J1 (2zli )Y0 (zli ) − J0 (zli )Y1 (2zli ) = 0, where Jn and Yn are the nth order Bessel functions of the first and second kinds, respectively. The corresponding eigenfunctions of systems (2) and (4) are ui (x) = J0 ((1 + x)zli ) −
J0 (zli )
Y0 ((1 + x)zli ),
Y0 (zli )
0
vi (x) = (2 − x) J1 ((2 − x)zli ) −
J0 (zli )
1
Y1 ((2 − x)zli ) .
Y0 (zli )
Using these facts we may confirm the duality.
4. THE DISCRETE ROD MODEL REVISITED
Relation (21) may suggest the connection between the primal and dual eigenvector matrices U and V in the discrete case. Let us denote the discretization, at points separated by constant distance h, of a continuous function g(x), x $ [0, 1], by G = diag {g(x1 ), g(x2 ), . . . , g(xn )},
x1 = h,
xi + 1 = xi + h.
Then our finite difference approximation of its derivative, (d/dx)g(x), is (1/h)FTG. Furthermore, the discrete analogy of the transformation of co-ordinates from g(x) to g(1 − x) is the replacement of G by EGE for G a matrix and q by Eq, if q is a vector. Hence, the discrete analog of equation (21) for the matrices V and U is V = EDEFTEU. For any Toeplitz matrix A it is true that EATE = A.
(23)
Our hypothesis is that if U is the eigenvector matrix of equation (5) then, since F is Toeplitz, V = EDFU
(24)
. . .
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is the eigenvector matrix of equation (7). We now prove this hypothesis. Substituting equation (24) into (7) gives n 2FTED−1EFEDFU = ED−1EEDFUG, or equivalently, n 2FTED−1FTDFU = EFUG. Now, using equation (5), this gives FTEUV = EFUG.
(25)
Using equation (23) again proves statement (24). Furthermore, G = V. This is an alternative proof of the duality result in the discrete case. 4.1. Following on from Example 2.1 we have
F G G EDFU = G G G f
0·37114
−2·50830
4·09411 −4·40836
1·54994
0·65705 −2·93721 0·47573 4·48247 −2·95640 0·85324 −1·52683 −2·98903 −0·68652 3·93942 0·96297 0·21374 −1·10475 −2·52101 −3·63428 1·00000 1·00000 1·00000 1·00000 1·00000
J G G G, G G j
which, apart from a diagonal scaling of its columns, contains the same eigenvectors as V.
5. CONTINUOUS AND DISCRETE MODELS OF THE TRANSVERSELY VIBRATING BEAM
An analysis, similar to that of the previous sections, can be applied to the continuous and discrete beam models (8)–(11) of section 1. For the continuous model, the eigenfunctions of systems (8) and (9) which correspond to equal eigenvalues satisfy v(x) = a(1 − x)(d2/dx 2 )u(1 − x).
(26)
Similarly, for the discrete model, the eigenvector matrices U and V of equations (10) and (11) satisfy V = EDF2U.
(27)
The proofs closely follow the corresponding argument for the rod. Changing the variable from x to 1 − x in the differential equation (8) gives
0
1
d2 d2 u(1 − x) = la(1 − x)u(1 − x). 2 a(1 − x) dx dx 2
(28)
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Substituting equation (26) into the left side of the first of equations (9) and using equation (28) we have
0
1
0 0
d2 d2 d2 d2 d2 aˆ v − maˆv = 2 aˆ aˆ−1 2 u(1 − x) dx 2 dx 2 dx dx 2 dx =
11
−m
d2 u(1 − x) dx 2
d2 d2 −1 u(1 − x))) − m 2 u(1 − x) 2 (aˆ (laˆ dx dx
= (l − m)(d2/dx 2 )u(1 − x).
(29)
So, the first of equations (9) is satisfied if l = m. Furthermore, the boundary conditions of system (9) give v(0) = a(1)(d2/dx 2 )u(1) = 0,
0
d d d2 v(0) = a(1 − x) 2 u(1 − x) dx dx dx
0
1b
= −a(1) x=0
d2 d2 d2 u(1 − x) 2 v(1) = 2 a(1 − x) dx dx dx 2
1b
d3 d d2 a(1) 2 u(1) = 0, 3 u(1) − dx dx dx = la(0)u(0) = 0, x=1
d3 d d d v(1) = (la(1 − x)u(1 − x))= x = 1 = −u(0) a(0) − a(0) u(0) = 0, dx 3 dx dx dx in view of equation (28) and the boundary conditions associated with system (8). This completes the proof of duality for the continuous model of the beam. For the discrete beam model, we substitute equations (27) and (6) into equation (11), and, recalling that F is Toeplitz and E2 = I, we obtain n 4(F2 )TED−1(F2 )TDF2U = EF2UG. Then, using equation (10) simplifies this to (F2 )TEUV = EF2UG. However, simce F2 is also a Toeplitz matrix we conclude that V = G, and this establishes equation (27).
6. CONCLUSIONS
Two axially vibrating rods, the first fixed at one end and free to oscillate at the other and the second configured oppositely, which have the property that the product of their cross-sectional areas is constant along the rods, have a common spectrum. In the present paper we have shown that this duality is reflected in both the continuous to the discrete models of the rod. We have also explicitly given the relation between (a) the eigenfunctions of the primal and dual in the continuous case and (b) the eigenvector matrices of the primal and dual in the discrete case. Thus, if the eigenfunctions (eigenvectors) of one of the rods are known, we may determine those of the other without solving the other problem. Furthermore, we have shown that there exists an analogous duality for the vibrating beam, that it carries over from the continuous to the discrete model, and we have given the relations between the eigensolutions of the primal and dual for that case. In the discrete model of the rod there are at most (n − 1)! different solutions [2]. The important problem of expressing other possible cross-sectional area rods with the same spectrum (for the discrete or continuous case) remains open. A similar solution pertains for the beam.
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. . . REFERENCES
1. A. H. B 1976 Fundamentals of Musical Acoustics. Oxford: Oxford University Press. 2. Y. M. R and S. E 1993 University of Adelaide, Department of Computer Science, Technical Report 93–15. An inverse eigenvalue problem arising in the vibration of rods. 3. J. W. P, F. W. D H and R. S. A 1981 Computing 26, 123–139. On the correction of finite difference eigenvalue approximations for Sturm–Liouville problems.