Journal of Sound and Vibration (1995) 183(5), 925–927

LETTER TO THE EDITOR

NUMERICAL EXPERIMENTS ON THE STODOLA–VIANELLO METHOD

P. A. A. L

Institute of Applied Mechanics (CONICET)and

Department of Engineering, Universidad Nacional del Sur, 8000 Bahı́a Blanca, Argentina

L. E

Mechanical Systems Analysis Group, Facultad Regional Bahı́a Blanca (U.T.N.), Argentina

G. S´ S

Department of Physics, School of Engineering, Universidad Nacional de Buenos Aires,Argentina

(Received 6 April 1994, and in final form 26 September 1994)

1.

As stated by Hildebrand in his classical textbook [1], ‘‘the method of Stodola and Vianellois a useful iterative procedure which allows for the approximate determination of thecharacteristic numbers and functions of a boundary value problem’’. Hildebrandillustrated the procedure by considering the problem of a vibrating string fixed at its endsand governed by the differential system

d2y/dx2 =−ly, y(0)= y(L)=0, (1a–c)

where l is the eigenvalue under investigation. Replacing y on the right side of equation(1a) by a first approximation y1(x) which satisfies equations (1b, c), one is able to integrateequation (1a) and obtain a second approximation which can be written in the form

y= lf1(x). (2)

Requiring now that y1(x) and lf1(x) agree as well as possible (in some sense), for instanceintegrating them over the interval (0, L) one is able to obtain an approximate value of l.One may improve further the result by repeating the procedure: e.g., substituting f1(x) inequation (1a) and obtaining y= lf2(x). As shown by Hildebrand convergence of theprocedure is achieved in some instances and upper bounds are obtained when polynomialswith integer powers of the variable are used.

The present note describes numerical experiments in which the approximate co-ordinatefunctions are simple polynomials which contain a minimization exponential parameter g.By judiciously selecting g one is able to improve the calculated eigenvalue determined inthe first cycle, although one cannot guarantee the existence of a lower or upper bound sincenon-integer powers of the independent variable are used.

Three numerical examples are presented in the following section.

2.

In the case of the differential system (1) one takes

y1(x)= x−L(x/L)g. (3)925

0022–460X/95/250925+03 $08.00/0 7 1995 Academic Press Limited

926

Substituting this into the right side of equation (1a), integrating twice and using theboundary conditions (1b, c) one obtains

y(x)= l$ 1Lg−1

xg+2

(g+1)(g+2)−

x3

6+L2016−

1(g+1)(g+2)1x%. (4)

Integrating equations (3) and (4) over the interval (0, L) and equating the results, oneobtains

l(1)1 L2 =$12−

1g+1%>$ 1

(g+1)(g+2)(g+3)−

12(g+1)(g+2)

+124%=

I1

I2. (5)

A second cycle of the interaction is performed by substituting equation (4) in equation (1a).An improved value of y(x) is obtained, and repeating the previous procedure one obtains

g(2)1 L2 = I3/I4, (6)

where

I3 =1S

−124

+016−1R1 1

2, I4 =−$1

U−

1720

+124 016−

1R1−

12

W%,R=(g+1)(g+2), S=R(g+3), T=S(g+4), U=T(g+5),

W=1T

−1

6R+

136

−1

120.

Minimizing expression (5) with respect to g, one obtains l(1)1 L2 =9·983, while the same

procedure applied to expression (6) yields g(2)2 L2 =9·88. The exact eigenvalue is p2 2 9·87,

and the approximate values obtained in reference [1] are g(1)1 L2 =10 and g(2)

1 L2 =9·882(which correspond to taking a fixed value of g=2). It is observed that a slightimprovement has been achieved.

Next, consider the buckling of a simply supported beam, the flexural rigidity of whichvaries according to the functional relation Cx/L. The differential system is defined by [1]

d2y/dx2 =−(m2/L)(y/x), y(0)= y(L)=0, (7a–c)

where m2 =PcrL2/C. Taking

y1(x)= x−L(x/L)g (8)

and following the previously explained procedure one obtains, as a first approximation,

m(1)1 =6$12−

1g+1%>$ 1

g(g+1)(g+2)+

12g(g+1)

−112%7

1/2

, (9)

This functional relation is a monotonically increasing function of g. Obviously, the valueg=1 is not valid, since equation (8) will be identically zero. On the other hand, valuesof g very close to unity, say g=1·01, will correspond to extremely flat deflection shapes.It seems reasonable to choose a value of g which differs considerably from g=2 (whichyields m(1)

1 =2 [1]) but which still maintains a reasonable shape for the deflected beamconfiguration. For instance, for g=1·10 one obtains l(1)

1 =1·90. On the other hand, forg=1·01 a value of g(1)

1 =1·898 results, and for g=1·40 one obtains g(1)1 =1·94.

Accordingly, a pragmatic selection of g (for instance, g=1·10) seems to be a reasonableone, but using other values will not introduce considerable differences, the exact valuebeing g1 =1·916 [1].

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Now consider the case of axisymmetric vibrations of a circular membrane of unit radius.The differential system is now

(1/r)(d/dr)(r du/dr)=−l2u, u(1)=0, (10a, b)

taking

u1 =1− rg (11)

and following the usual procedure one obtains

l(1)1 =6$12−

1g+2%>$ 1

(g+2)2(g+4)−

12(g+2)2 +

116%7

1/2

. (12)

With g=2 one obtains l(1)1 =2·45, while for g=1·01 the value l(1)

1 =2·39 is determined.Following the reasoning previously stated, one selects g=1=1·10 and determinesl(1)

1 =2·40, which is in excellent agreement with the exact eigenvalue: l1 =2·4048.

3.

Summarizing, one can conclude the following.The classical Stodola–Vianello iterative procedure converges in many situations of

mathematical physics of practical interest, and upper bounds of the exact eigenvalues aredetermined when co-ordinate functions with integer powers of the independent variableare used, as proved by Hildebrand in his well known treatise [1]. A large number ofiterations is possible nowadays by using a computer program, such as MACSYMA, toavoid the tedious algebra, as the complexities of the function increase from one iterationto the next.

The minimization procedure proposed in this note leads, obviously, to the use ofpolynomials with non-integer powers and does not yield upper bounds. However, froma practical viewpoint, and as shown in the three examples presented here, considerableimprovement is achieved in the determined eigenvalue when only the first cycle of theStodola–Vianello method is employed. It does certainly require judicious selection of thenon-integer optimization parameter. The technique seems convenient when used inconnection with more complex eigenvalue problems.

The present study has been sponsored by Secretarı́a de Ciencia y Technologı́a ofUniversidad Nacional del Sur (Project 1994–1995; Program Director Professor R. E.Rossi). The authors are indebted to Professor P. E. Doak and to the referee of the presentnote for their useful comments.

1. F. B. H 1962 Advanced Calculus for Applications. Englewood Cliffs, New Jersey,Prentice-Hall. See pp. 200–206.