properties of a semi-discrete approximation to the beam equation

8/10/2019 Properties of a Semi-discrete Approximation to the Beam Equation

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J. Inst. Maths Applies (1969) 5, 329-339

Properties of a Semi-discrete Approximationto the Beam Equation t

ROLA ND A. SW EET

Department of Computer Science,Cornell University, Ithaca, New York, U.S.A.

[Received 19 Jan ua ry 1968, and in revised form 29 Octo ber 1968]

The solution of the equationw(*)Hrt+[p(*)Jx = 0 , 0 < x < L, t>0,

where it is assumed that w and p are positive on the interval [0,L], is approximated byusing the method of straight lines. The resulting approximation is a linear system ofdifferential equations with coefficient matrix S. The matrix S is studied under very generalboundary conditions which result in a conservative system. In all cases the matrix S iseither an oscillation m atrix or possesses nearly all the properties of an oscillation matrix.

1. Introduction

THE STUDY of small transverse (or lateral) vibrations of a beam with non-uniformcross-sectional area and moment of inertia is of wide interest. Of particular importancein the design of structures which possess the characteristics of a beam, for instance,chimneys, are the natural frequencies and mode shapes. The equation describing them otio n is impo ssible to solve in most cases, hence informatio n is ob taine d by num ericalapproximating techniques. Linearly tapered cantilever beams have been studied byHou sner & Keightley (1963).

The general equation describing the free vibrations iswix)ult+[pix)ixx\n = 0, 0 0 and p(x) > 0, 0 < x < L.

These conditions are physically meaningful an d unrestrictive in vibration problem s.The primary goal of this paper is the study of the coefficient matrices which arise

through the use of the method of straight lines (a discretization of the space variableto produce a linear system of ordinary differential equations). We prove that undervery general boundary conditions the coefficient matrices possess all the propertiesof oscillation matrices (see Gantmakher & Krein, 1950).

2. Preliminaries

To solve equation (1) one must prescribe initial conditions

u(x,0) = 4>(x), ut(x,0) = Mx), (2)t This research was supported by an NDEA Fellowship at Purdue University, West Lafayette,

Indiana.

329

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330 ROLAND A. SWEET

for 0 < x < L, and four linearly independent boundary conditions. These boundaryconditions are usually given in the form of two conditions at the end x = 0 and twoat the end x = L, although this is not the only case which can occur (for example,

periodic boundary conditions can also occur). We will concern ourselves, however,only with the former type.We shall consider only those boundary conditions which correspond to conservative

problems. Multiplying (1) by ut, integrating over the rectangle 0 ^ x L, 0 t T,and using integration by parts twice we get

f r [ L

0 = > , + ( p O r ] dx dtJ o J o

= E(u,T)-E(u,0)+ \ T[B(u,L)-B(u,O)-]dt,J o

where 1 fL(".'i) = 2 [wuf+pull^dx

andB U,X) = (> KXI)* r-i'xx x


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SOLUTION OF THE BEAM EQUATION 33 1

Then, if w e denote the approximation of u x , , t ) by ufj), one hasC x t i k fxi + i

0 = w(x)u tt dx+\

= hwfit + [ (pOx]x ' -x t - i* (4)If we now replace all partial derivatives of x by central difference approximations,we obtain from (4)

0 = AW|fi(+A

(5)

TABLE 1

Boundary Conditions

Case Condition (at x = 0) Condition (at x = L)

3 xx(0,0 = C P"x*)x(0,')=04 ux(0 , t ) = 0 ) ^ ( 0 , 0 = 0 u x{L, t) = (pu^L,5t uJ0,t) = 0 u x(L,t) = 0

6t

7 t

8 t

9 t

au(0 , t ) = G M ^ C X X X O , /

a(0,r) = (pu^JP, )cujp,t) = u^O.t)

-bu^Tt^ujltt)(p$x(lojUf

bu L,t) = (p uxJx(L,t)-d ux(L,t) = Uxx(L,t)

t Note: we assume a > 0, b > 0, c > 0, and d > 0.

If an equation of the type of (5) is written for each unknown , one obtains as theapproximation, a linear system of coupled ordinary differential equations. The systemhas the form

H i ) + / t ^ S U = F, (6)where H is a diagonal matrix with positive elements, 5 is a pentadiagonal matrix,F is a vector whose components depend on the nature of the boundary conditions,and U is the vector whose ith component is the function ut).

The precise forms of H and S for various boundary conditions will be exhibitedin the next section. All the boundary conditions we consider yield an F which iszero, hence we shall solve (6) in its homogeneous form. To do this we seek a solutionof the form

U(f) = V(a sin VA /+ P cos JX t), (7)where V is a vector and a, /?, A, are constants to be determined. Substituting (7) into (6)

and abbreviating H ^S to just S we arrive at the condition(SV-AHV)(a sin Jk t+0 cos VA 0 = 0.

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3 3 2 ROLAND A. SWEET

F or this to be true for all values of /, it must be that

that is , k must be an eigenvalue of H-'S and V a corresponding eigenvector. If S and

H are of order m, then there are m eigenvalues X\,X2,...,X m and (we assume) m corre-spond ing eigenvectors V\, Vz,...,V m . Then for each Xk a nd Vk (7) is a solution of (6).Hence ,

U(0 = f ottsin W + ^ c o s W ) ^t= iis a solution of (6), the coefficients ak and pk (k = l,2, . . . ,m ) being determined by theinitial conditions (2), i.e.

U(0) = Xi> ... > km> 0.


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SO UT ON OF THE BEAM EQUATION 33 3

The eigenvector corresponding to the kth eigenvalue has exactly k\ sign changes andthe nodes of two successive eigenvectors alternate.THEOREM 2. A totally non-negative matrix A is oscillatory if and only if

(1) d(A) > 0, and(2) a u+i > Oandai+i, t > 0 (i = 1,2,...,m-1).

A generalization of totally non-negative and oscillation matrices has been madeby Gantmakher & Krein (1950). We state now one of their results.

Definition. A matrix A is called a fixed-sign matrix of class d if for any p ^ d, allthe non-zero minors of order p have the same sign e p . If for any p ^ d, all the pthorder minors are different from zero and have the same sign e p , than A is called strictlyfixed-sign. A fixed-sign matrix A of class d is called a matrix of class d+, if a positiveinteger k exists such that A* is strictly fixed-sign of class d.

THEOREM 3. Let the matrix A of class d+ have the sign pattern e k = 1 (k = 1,2, ...,d),and let its eigenvalues, X\, k 2 ,..., A m, be arranged in order of decreasing moduli. Then

(1) h > h > ... > h > IVil > IJrfd > - ^ I-U > 0,

(2) the kth eigenvector corresponding to X k has exactly k\ sign changes

(k = \,2,...,d). The nodes of two successive eigenvectors u k and u k+i (k = l,2,...,d 1)alternate.

3. The Structure of the Coefficient Matrix

Let us consider the boundary condition 9-9, i.e. condition 9 of Table 1 applyingat x = 0 and condition 9 applying at x = L. We can write these conditions in the form

Clux 0,t) = d lU J0,t)a 2u(L,t) = hipu^ULj)

W

-c2ux L,t) = d2uJ< L,i),where it can be assumed that the constants a h b,, c t, and d ( (i =1,2) are positive.Using central difference approximations we may replace (8) by

,*) = P 1U xx (X l,t)+-y-U 0

Let us definethe quantities

2 c 2h 2*> = a D d e * =

2ic lh+2d l)

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334 ROLAND A. SWEET

Then using equations (5) and conditions (9) we can write the equations for the un-knowns uo, ui, un and n+ i .

- i w o u o = h-*[(e2

1+4p oel+p1)uo-2(2poel+p1)u1+p lu2]

wj n =

Hence, the matrices of (6) areH = diag[iwo,H'i,...>M'11 ,iH'B+i]

ande21+4p 0e

22+p 1 -2(2poel+p1) Pi

-2i2poel+ Pl)

2(Pn+2pn+ ie2

3)un+ 1 ']

S = ooDefining the numbers

c ( = V>i. / = 0 ,1 ,2 , . . ., + 1 ,it is easy to see that S can be factored as the pr od uct

S = AAr

where A is the ( +2 ) x (n+4) matrix

2coe2 ci 00 -2coe2 2 ci - c 20 0 -ci 2c20 0 0 -c2A =

O

O c . 02c. - 2 c + i e 3 0We observe that A is a tridiagonal matrix J bordered by two columns (one on eachside). If we define the orthogonal matrices

andC =

then we haveS* = D S D = D A AT ) = ( D A Q ^ A7 ) ) = (DAC XDAC F =

wh ere F is the matrix obtained by replacing J by J* in A.The matrix J* has the following properties. It is non-negative and

d(J,*) = 2e2 'f l c, > 0, p = 1 ,2,. . . ,1 0

d(J*) = d ( J ?+ 2) = 0.

He nce, by a lemma of Gantm akhe r & Krein (1950) J* is totally non-negative.


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SOLUTION OF THE BEAM EQUATION 3 3 5

Therefore, we have that all minors of F not involving the first row and columnor the last row and column are non-negative. But those minors which do involve theabove-mentioned rows and columns can be reduced to a minor of J* multiplied by

a positive number {a, e*, or eie*). Hence, we have that F is totally non-negative.By the Cauchy-Binet formula for the determinant of a product of two matrices,we have that S* is also totally non-negative. By the same formula S* is non-singularsince

2 ... n+2\ r /l 2 ... n+2\2 ... n + 2)

F{l 2 ... l)

fl ~cd2 > 0.i-0

Hence, by Theorem 2 S* is an oscillation matrix.We are really interested in the structure of H~iS, but since H is a diagonal m atrix

of positive elements it is clear that H = H* and that H- 1S* is oscillatory if S* is.We have proved:THEOREM 4. The coefficient matrix of the linear system of equations generated by theboundary conditions of case 9-9 is similar to an oscillation matrix.

This most general set of boundary conditions was considered first for the simplereason that all other cases follow from it almost immediately. For choosing appropriatevalues of a t, bh c,, and d, (i = 1,2) we can obtain any set of the boundary conditionswhich we have listed in Table 1.

TABLE 2

E ffects of boundary conditions

Conditionat end

123456789

Col. 1/Col. H + 4

XXXX

XX

Col. 2/Col. n+3

XX

X

R o w l /R O W / J + 1

XX

X

Excess ofcols over rows

10

- 1020102

In Table 2 we have tabulated the effects on the matrix A of the various boundaryconditions considered. An "X" placed in the table means that the particular rowor column listed at the top of the column is removed from A. The meaning of the lastcolumn of the table will be explained later.

In each case A always has the form of a tridiagonal matrix, whose star-matrix istotally non-negative, possibly bordered by a column or row. The only non-zero elementin this column or row is always on the main diagonal, and hence, in the matrix Fit appears as a positive element. For this reason F is always totally non-negative.

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336 ROLAND A. SWEET

Therefore, to check the oscillation of the matrix S* it is sufficient to check that ineach case S* is non-singular.

There are cases when we remove from A a sufficient number of columns to make its

rank smaller than the order of S. In these cases S must be singular since it is the productof A and A r. Hence, it cannot be oscillatory. However, practically all of the structureof the eigenvalues and eigenvectors remains.

The last column of Table 2 expresses the excess of the number of columns over thenumber of rows (for one end condition). In considering case / - / we add the twonumbers in rows / and / of the last column, and if the number is negative then thecoefficient matrix must be singular.

Immediately we see that cases 2-3, 3-4, 3-6, and 3-8 give singular matrices. Sodoes the case 3-3, but the treatment of this case will be taken up later. Singular matricesarise from the cases 4-4, 4 -8, 6-8 , and 8-8 also. This is due to the fact that the deter-

minan t of the coefficient matrix contains (10) as a multiplicative factor.By the same type of argument used to establish the non-singularity of S* in case 9-9,it can be shown that all other cases are non-singular. Using these results we can showthat the singular matrices listed above retain most of the structure of oscillationmatrices. We use now the generalization of oscillation matrices developed byGantmakher & Krein (1950).

To illustrate let us examine case 2-3. H ere we have

F =

2c2

2c-1 cc- i 2c0 c

which is of order n + l)x. Therefore, the rank of S* must be no greater than n.Since the first n rows and columns of S is just the matrix obtained in case 2-2, wehave by the above work that

2 ::: ) > 0 - ( 1 1 )

Hence, the rank of S* is precisely n . Furthermore, the last n rows and columns of Sis the matrix which arises in case 1-3 which we have shown to be non-singular. So

> 0.

Conditions (11) and (12) and the nature of the elements of S* allow us to prove thefollowing theorem.

THEOREM 5. Let A = (o y )7 be a totally non-negative matrix of order m and rankm-l.If

(1) a/,+i > 0 andai+i,t > 0 (i = l , 2 , . . . , m - l ) , and2 ... m - l \ .(2 3 ... m\

2 ... m-l)A

{2 3 ... m j> 0

'then A is of class (m l) + .


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3 3 8 ROLAND A SWEET

Let B be the matrix compo sed of rows /i,ii + l , . . . , ip and columns ki,ki + l,...,k pof A. Then by a lemma proved by Gantmakher & Krein (1950), B must have rankp l. Let h = max (/i,A:i). Then

h,ki < h,h+p-l < i p ,k p .Therefore, the minor

(h h + l ... h+p-l\_ (h h + l ... h + p-l\a \h h + l ... h+p-l) ~ \ h h + l ... h + p-lj

is of order p, and so must be zero.We have h > l . I f A = l . t h e n

h+p-l < l+m-l-l = m-l.

Usin g (13) we get

(I 2 ... m-l\ (h h + l ... h + p-l\ (h+p ... m-l\_[l 2 ... m-l) A \h h + l ... h + p-l) \h + p ... m-l}~ '

which contradicts assumption (2). If h > 1, then

(2 3 ... m\ (2 ... h-l\ (h h + l ... h+p-l\ (h + P ... m\_A \2 3 ... m)** A \2 ... h-l) X {h h + l ... h + p-l) A \h + p ... m)- '

which also contradicts assumption (2).Hence (14) is not t rue. In other words, all quasi-principal minors of order p are

positive as was to be show n.The matrix S* of order n + 1 and rank n arising from case 2-3 satisfies cond ition (1)

and (2) of Theorem 1, so it is of class n+ . Then S* satisfies Theorem 3.In the same man ner it can be shown that all other singu lar m atrices belong t o class

r+ , where r is one less tha n the order of the ma trix, except the ma trix of case 3-3which belongs to class (r1)+. Again, it should be noted that S* in the class r+implies (H- 1S )* is also in the class r+ since H is a diagon al m atrix with positivediagonal elements.

4. Summary

For each possible set of boundary conditions we have investigated the nature ofthe solutions of the linear system of order m

SV = ( ; j^)HV, 15)

where S an d H have been defined by (6). Let p be the num ber of zero eigenvalues a ndord er the eigenvalues as

0 = Ai = h = ... = k p < ;. p + i < .. . < l m .

Let V t be the eigenvector associated with Xk.

Table 3

summarizes the relevant prop erties of

the eigenvalues for

each set of

boun dary conditions. The num ber of sign changes in an eigenvector V t correspondingto a positive eigenvector is equal to k p l.

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SOLUTION OF THE BEAM EQUATION 339

TABLE 3

Properties of Matrix S

Number ofOrder Number of positive distinct

Cases of S zero eigenvalues eigenvalues

1-1,-2,-7;-)2 - 2 , - 7 ; \1-1 J

1-3, -4, -5 , -6, -8 , -9 ;2-4, -5, -6. -8, -9;3-7; 4-7; 5-7; 6-7;7-8, -9

2-3

3-5,-9:4-5,-6,-9;-)5-5, -6, - 8 , -9 ; }6-6, 6-9; 8-9; 9-9 J

3-4, -6 , - 8 ;)4-4,-8; [6-8; 8-8 J

3-3

n

n+1

n+1

n+2

n+2

n+2

0

0

1

0

1

2

n+1

n+2

n+1

The author would like to thank Professor John S. Maybee for suggesting thisproblem and to the referees who offered many helpful criticisms.

REFERENCES

BEREZIN, I. S. & ZHTDKOV, N. P. 1965 Computing Methods, IL Oxford: Pergamon Press.GANTMAKHER, F. R. & KRETN, M. G. 1950 Oscillation Matrices and Kernels and Small

Vibrations of Mechanical Systems. Moscow: State Publishing House for Technical

Theoretical Literature. (An English translation available from Office of Tech. Serv.,Dept. of Commerce.)GENIN, J. & MAYBEE, J. S. 1966 / . Inst. Maths Applies, 2 (4), 343-357.HOUSNER, G. W. & KOGHTLEY, W. O . 1963 Trans. Am. Soc. Civ. Engrs 128 (1), 1020-1054.

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properties of a semi-discrete approximation to the beam equation

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