NASA Contractor Report 181862 

ICASE Report No. 89-39 

ICASE 

AN INTEGRATING MATRIX FORMULATION FOR 

BUCKLING OF ROTATING BEAMS XNCLUDING THE 

EFFECTS OF CONCENTRATED MASSES 

William D. Lakin 

Raymond G. Kvaternik 

[NASA-CR-181862) AN 1ITEI;RATING HATRIX N89-263 42 POBEULATIOI POB BUCKLING Of ROTATING EEAMS 

INCLUDING THE EFFECTS OF COUCENTRATED HASSES 

Final Report (XICASE) 25 p CSCL 12A Unclas 

G3/64 0224120- .. _.. 

Contract Nos. NAS 1-1 8 107, NAS 1-1 8605 

May 1989 

Institute for Computer Applications in Science and Engineering 

NASA Langley Research Center 

Hampton, Virginia 23665-5225 

Operated by the Universities Space Research Association 

National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton. Virginia 23665-5225 
AN INTEGRATING MATRIX FORMULATION FOB BUCKLING OF ROTATING BW 

INCLUDING TRE EFFECTS OF CONCENTRATED MASSES 

William D. Lakin 

Old Dominion University 

and 

Institute for Computer Applications in Science and Engineering 

and 

Raymond G. Kvaternik 

NASA Langley Research Center 

ABSTRACT 

This paper extends the integrating matrix technique of computational 

mechanics to include the effects of concentrated masses. The stability of a 

flexible rotating beam with discrete masses is analyzed to determine the 

critical rotational speeds for buckling in the inplane and out-of-plane 

directions. In this problem, the beam is subjected to compressive centrifugal 

forces arising from steady rotation about an axis which does not pass through 

the clamped end of the beam. To determine the eigenvalues from which 

stability is assessed, the differential equations of motion are solved 

numerically by combining the extended integrating matrix method with an 

eigenanalysis. Stability boundaries for a discrete mass representation of a 

uniform beam are shown to asymptotically approach the stability boundaries for 

the corresponding continuous mass beam as the number of concentrated masses is 

increased. An error in the literature is also noted for the discrete mass 

problem concerning the behavior of the critical rotational speed for inplane 

buckling as the radius of rotation of the clamped end of the beam is reduced. 

Research for the first author was supported by the National Aeronautics and 

Space Administration under NASA Contract Nos. NASl-18107 and NAS1-18605 while 

he was in residence at the Institute for Computer Applications in Science and 

Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665. 
2 

NOTATION 

5 

D 

E 

L 

m 

M 

n 

Px 9 P y s P z 

R 

5 S 

T 

t 

5 T 

v ,w 

XYZ 

X 

5 A 

discontinuous function defined by Eq. 15 

diagonal matrix obtained from 

Young’s modulus of elasticity 

second moments of area about y and z axes, respectively 

integrating matrix 

length of beam 

mass distribution of beam (continuous plus concentrated) 

concentrated mass 

number of intervals into which the beam is divided 

distributed inertial loads in x, y, and z directions, respectively 

radius of rotation of clamped end of beam 

diagonal matrix defined by Eq. 24 

tensile force in beam 

time 

diagonal matrix defined by Eq. 25 

deformation of elastic axis in y and z directions, respectively 

coordinate system which rotates with beam such that X-axis lies 

along the initial or undeformed position of elastic axis and passes 

through the axis of rotation, which is parallel to the Z axis 

running coordinate along X-axis measured from clamped end of beam 

centroidal principal axes of beam cross section 

i n t e rpol a ti on mat ri x 

Dirac delta function 

variable of integration 

De (x) 
3 

eigenvalue 

continuous mass distribution of beam 

position of discrete mass measured from clamped end of beam 

rotational speed of beam 

time derivative, - 

spatial derivative, - 

ax 

square matrix 

diagonal matrix 

column matrix 

a 

at 

a 
4 

INTRODUCTION 

The dynamic behavior of flexible rotating beams continues to receive considerable attention in the literature as it constitutes a fundamental problem 

in applied mechanics. Further, beams comprise parts of many rotating 

structures of engineering significance. One area which is receiving attention 

is the problem of the buckling of beams due to rotation-induced compressive 

centrifugal forces. Such forces arise, for example, in a rotating beam whose 

axis of rotation does not pass through the clamped end of the beam. As the 

beam rotates, its clamped end describes a circle of nonzero radius about the 

axis of rotation. This geometrical arrangement can be specified by a dimensionless parameter involving the radius of this circle and the length of the 

beam. If such a rotating beam has one of its cross-sectional principal axes 

parallel to the axis of rotation, transverse buckling can occur either in the 

plane of rotation or out of the plane of rotation, depending, among other 

factors, on the value of the geometrical parameter and the rotational speed. 

The problem of determining the vibrations and stability of rotating beams subjected to compressive centrifugal forces has been treated by several investigators [l-91. References 8 and 9 solved the differential equations of motion 

numerically using an integrating matrix (Refs. 10-12) in combination with an 

eigenanalysis to determine the eigenvalues from which stability is assessed. 

The method of solution was shown to be numerically exact and not to exhibit 

the convergence problems associated with the approximate methods of solution 

which have been applied by others to the study of rotation-induced buckling. 

The integrating matrix procedure has been extended to include both 

nonuniformly spaced grid points (Ref. 13) and concentrated masses (Ref. 14). 

The purpose of this paper is to employ the integrating matrix procedure 
I 

, 

extended to include concentrated masses in a more complete and rigorous 

examination of the buckling behavior of the discrete mass representation of 

the flexible rotating beam which has been treated in Ref. [6]. 

FORMUMTION 

A geometrical arrangement giving rise to rotation-induced radial centrifugal forces which can be compressive rather than tensile, and thus leading to 

the possibility of a buckling-type instability, is depicted in Fig. 1. A beam 

of length L is clamped to the inside of a ring of radius R which is rotating with constant angular velocity ii about an axis perpendicular to the 

plane of the ring and passing through its center. If the principal bending 

planes of the beam are oriented perpendicular and parallel to the plane of the 

ring, transverse buckling can occur either in the plane of rotation or perpendicular to the plane of rotation, depending on the value of such parameters as 

the geometric ratio R/L and the rotational speed. The use of an integrating 

matrix procedure in the numerical solution of the buckling behavior of the 

uniform rotating beam depicted in Fig. 1 for 0 < R/L - < 2.0 , and in the 

identification of the limiting values of R/L below which buckling cannot 

occur, has been treated in Ref. [9]. The integrating matrix formulation for 

the case of a beam with continuous properties described in Ref. [91 has been 

extended to include the case in which concentrated masses are also present 

[14]. The modifications required to account for discrete masses result in new 

matrices which simply have to be added to the matrices previously established 

in Ref. [9] for a beam with continuous massI Because the formulation for the 

beam with continuous mass remains unchanged, for brevity here reference will 

- 
6 

be made to Ref. [91 for pertinent details and explicit expressions. The 

notation of Ref. [91 is maintained as much as possible. 

Governing Equations 

The equations of motion which describe the uncoupled out-of-plane and inplane bending of the beam in Fig. 1 are, from Ref. [91, 

.. 

mw - (Tw’)’ + (E1 = 0 

YY 

2 .. 

m(v - 51 v) - (Tv’)’ + (EIZzvOe)” = 0 

where the tensile force T(x) is given by 

A positive value of T indicates tension. The boundary conditions corresponding to a clamped end at x = 0 and a free end at x = L are 

w(0,t) = w’(0,t) = v(0,t) = v’(0,t) = 0 

w”(L,t) = WPO‘(L,t) = v”(L,t) = V”&(L,t) = 0. 

A procedure using an integrating matrix as an operator to eliminate the spatial dependence from the partial differential equations of motion, and the 

reduction of the resulting equations to matrix eigenvalue form for 
7 

determination of the eigenvalues from which the stability of the beam can be 

assessed, is fully described in Ref. 191 for a beam with a continuous mass 

distribution. 

The extension of the integrating matrix technique to include concentrated 

masses proceeds initially along the same lines employed previously in Ref. 

[91. The governing equations are first formally integrated to isolate the deri vat i ves w’ e and VI‘. To this end, equations (1) and (2) are written in 

the form 

w”)’~ = p, + (Tw’)’ (6 1 (EIYY 

where 

2 = -mil (R - x) PX 

-- 2 = -m(v - Q v) pY 

.. = -mw. 

PZ 

Formally integrating equations (6) and (7) twice from x to L and applying 

the boundary conditions of zero shear at the free end of the beam leads to 
8 

As these equations are linear, it is sufficient to consider extension of the 

analysis for the case of a single concentrated mass, M, at an arbitrary position x = 6 (0 < 5 < L). The aggregate effect of multiple concentrated 

masses may then be obtained by superposition. 

Let the continuous mass distribution of the beam be denoted by ~(x). 

Then, the mass distribution m(x) of the beam (distributed plus concentrated) 

can be written as 

m(x) = ~(x) + M 6(x - 6) (11) 

where 6(x) is the usual Dirac delta function. Substituting (11) into (9) 

and (10) and assuming time solutions of the form 

- At v(x,t) = v(x)e 

equations (9) and (10) take the form 
9 

where, by the properties of the Dirac delta function, DS(x) 

tinuous function given by 

is the disconEquations (13) and (14) are valid for all values of x along the beam. 

In particular, they are valid at n + 1 discrete grid points (stations) with 

ordering 

0 = xo < x1 < < Xn-l < xn = L. 

The points need not have uniform spacing, and the location 5 of the concentrated mass need not coincide with a grid point. Writing equations (13) 

and (14) at n + 1 discrete points along the beam the resulting sets of 

equations may be cast into matrix form. The integrals appearing in these 

equations are associated with the continuous mass distribution and are 

evaluated by introducing an integrating matrix operator as described in Ref. 

[9]. The integrating matrix operator is also used to express the displacements v and w in terms of the curvatures 7’- and we* so that 

the curvatures appear as the dependent variables in the final matrix 

equations. Because the underlying differential equations are uncoupled, the 

- - - 
10 

resulting matrix equations are also uncoupled. However, following Ref. [91, 

for computational convenience the resulting equations are written in the 

combined form. 

or, in condensed notation, 

where 

A 

[ 111 = Ell] + [Gl1I 

A 

[ 221 = [G221 + [G221 

A 

[ 111 = [Hl1I + [Hl1I 

A 

[ 221 = [H221 + [H221' 

continuous properties of the beam, rotational speed, and integrating matrix 

operators. Explicit expressions for these submatrices are given in Ref. 

[91. The submatrices [Gll], [G22], [HI1] and [H22] are associated with 

the concentrated mass and are functions of the magnitude and location of the 

A A A A 
11 

mass, rotational speed, and integrating matrix operator. Explicit expressions 

for these submatrices are derived below. 

It should be noted that the inclusion of a concentrated mass leads to an 

eigenvalue problem of the same form as for a beam with only continuous mass. 

Further, the effects of a concentrated mass enter only through terms which are 

additive to terms obtained previously in Ref. [9] for a beam with no concentrated masses. Thus, the size of the eigenvalue problem given by equation 

(18) is not increased when concentrated mass effects are included. The 

eigenvalue problem given by equation (18) can be solved using standard 

eigensolution techniques to yield the eigenvalues and eigenvectors which 

characterize the dynamic behavior and stability of both the inplane and outof-plane motions of the beam. Since this is a kinetic approach [I51 for 

assessing stability, the solution of equation (18) will identify all the 

instabilities of the system, both dynamic and static. However, because of the 

absence of both Coriolis forces and damping forces in the present equations, 

only static (buckling) instabilities can occur. 

Explicit Form of Additive Matrices for Concentrated lhss 

Explicit expressions for the submatrices A A A A 

[Gll], [G221, [Hl11, and [Hz21 

which arise from the concentrated mass terms in equations (13) and (14) are 

rather straightforward to derive following the procedures outlined in Ref. 

[14]. However, special considerations are required for the case where the 

mass is not at one of the grid points established along the beam and in the 

treatment of the discontinuous function Dg(x). 

- - 

Equations (13) and (14) involve v and w evaluated at the concen- 
12 

trated mass position 5. If the mass M is not at a grid point, an interpolation polynomial (e.g., Lagrange interpolation) must be employed to express v(5) and w(5) in terms of the values of v and w at the 

grid points. Because the integrating matrix has elements which depend only on 

the grid point locations xo, ,xn, and not on the values of the function 

at the grid points, the interpolation polynomial employed for the concentrated 

mass must have coefficients which depend only on 5 and the grid point locations. Consistent with this requirement, consider an interpolation of the 

form 

- - - - 

If equation (20) is an mth order interpolant (m n) and 5 is not a 

grid point, then in general a (5) E 0 for k < y and/or k > y + m . Here, y is an appropriate reference integer and 

By choosing m sufficiently large (e.g., Lagrange interpolation with m = 7) 

the resulting interpolating polynomial gives a high degree of accuracy without 

the need to cluster grid points about 5. To use the interpolation polynomial given by equation (20) in the present context, let [As] be the 

square matrix of order 

k 

y+m 5 xn* <x <s<x xo- y 

n + 1 with the ijth element 

= a (51, j-1 j = 1,2, ,n+l 

i.e., [A5] 

coefficients in the interpolation polynomial. Premultiplying {u) by [A,] 

and using equation (20) leads to the relation 

has identical rows and each row has as elements the successive 
t 

[A, I(U1 = 1) (22) 

The matrix [A,] may be regarded as an interpolation matrix. If the mass 

is at a grid point, equation (20) shows that is a matrix with a single 

nonzero column consisting of ones. 

[A,] 

If the mass is not at a grid point, some care must also be taken in defining a diagonal matrix [D,] that gives the effect at the grid points of 

5 

multiplying by the discontinuous function 

to zero across x = 5. Suppose that 6 

x < 6 - < x j’ j = 1,2, , n. Then if 

order n + 1 with the diagonal elements 

j-1 

1 

[D,] (f) 

grid points. 

gives the vector of values of 

D,(x) which changes from unity 

lies in the subinterval 

[D5] is the diagonal matrix of 

i<j - 

i>j 

the function D,(x)f(x) at the 

Using the matrices [A,] and [DE] introduced above and introducing 

the diagonal matrices [S,] and [T,] defined by 

tS,I = Nx - 51 

the concentrated mass terms in equations (13) and (14) lead to the explicit 

matrix expressions 
14 

n “ 

where [I] is the identity matrix and 

[91 and [14]. To arrive at equation 

[I1] is the integrating matrix of 

26) use has been made of the relat 

Refs. 

on s 

- - 

to express the displacements v and w in terms of the curvatures VI@ 

and wee as was done in Ref. [91. The effect of several concentrated masses 

is obtained by superposition. 

- 

If the concentrated mass, M, is placed at the free end of the rotating 

- - 

beam, (i.e., 6 = L), the boundary conditions v”’(L) = 0 and w”’(L) = 0 

must be replaced by the differential equations for the inplane and out-ofplane motion of the concentrated mass. However, it is shown in Ref. [14] that 
15 

the application of an appropriate limiting procedure to the governing equations for v and w given by equations (13) and (14) for an interior concentrated mass as 5 tends to L produces the proper boundary conditions 

for a concentrated mass at 5 = L. Hence, equations (13) and (14), which 

have been derived assuming that the concentrated mass is not at the free end 

of the beam, remain valid for the case in which the mass is at the free end of 

the beam. Thus, for computation there is no need for any different treatment 

if the mass is at the free end. 

- 

NUMERICAL RESULTS 

The stability of a beam which is rotating in a plane perpendicular to the 

axis of rotation and clamped off this axis as indicated in Fig. 1 has been 

analyzed to determine the critical rotational speeds for buckling in the 

inplane and out-of-plane directions. An application of the integrating matrix 

technique to treat the case of a beam with a continuous mass distribution is 

described in Ref. [91. In the present paper, attention is directed to the 

case in which the mass distribution is approximated by concentrated masses at 

discrete points along the length of the beam. There are several wellestablished discrete mass approximations which can be used in dynamic analyses 

of beams. However, the less usual concentrated mass representation employed 

in Ref. [61 is adopted here to allow a rigorous comparison of the present 

numerical results with the analytical results given in Ref. [6]. It should be 

emphasized that this choice is not restrictive as all the observations made 

and the conclusions reached based on the numerical studies using the mass 

model of Ref. [61 are equally valid for other discrete mass representations. 
16 

In the present studies, three concentrated mass approximations to a 

uniform beam are considered, as indicated in Fig. 2. In each case, the sum of 

the concentrated masses is maintained equal to the total mass of the uniform 

beam. As in Ref. [91, numerical results here are obtained by dividing the 

beam into ten equal segments and using an integrating matrix [Ill which 

expresses the integrand as a seventh-degree polynomial in the form of Newton’s 

forward-difference interpolation formula. The computer program which is 

employed to solve the eigenvalue problem given in equation (17) uses the 

double shift QR algorithm (Ref. [16]) and is written to take advantage of any 

uncoupling which may be present in the combined matrix equation. Stability 

analyses were made for each of the three concentrated mass representations depicted in Fig. 2 over a range of values of the ratio R/L for a fixed value of 

the beam length. The resulting stability boundaries are shown in Fig. 3, 

where the nondimensional critical rotational speeds for buckling in the inplane and out-of-plane directions are presented as a function of R/L. Also 

shown in Fig. 3 for comparison with the present results are the corresponding 

boundaries for the beam with a uniform mass distribution obtained in Ref. 

[9]. The nondimensional rotational speed in Fig. 3 involves the inplane bending stiffness EIZz. The numerical results are for the particular case in 

which the inplane and out-of-plane bending stiffnesses are equal, that is, 

EI,, = EIyy* 

For the continuous mass case, it is seen that instability is first encountered in the plane of rotation for all values of R/L. Although the entire 

beam is in tension for R/L < 0.5, the term dl v which appears in the inplane equation of motion represents a component of centrifugal force normal to 

the beam which decreases the inplane natural frequency with increasing rota2 
17 

tional speed. Thus, for sufficiently large rotational speeds, the possibility 

of buckling in the plane of rotation can not be precluded even when the beam 

is entirely under tension. The results (which are numerically exact) indicate 

that the limiting value of the ratio R/L below which buckling cannot occur 

is R/L = 0.0. The results also show that the stability boundary is 

asymptotic to the vertical line given by R/L 0.0, so that the beam is 

stable for R/L = 0.0 for all finite values of the rotational speed. The 

numerical results for the case of buckling out of the plane of rotation 

indicate that the limiting value of the ratio R/L below which buckling cannot 

occur is R/L = 0.50. As in the case of inplane buckling, the stability 

boundary is asymptotic to the vertical line through the limiting value of R/L, 

so that the beam is stable for R/L = 0.50 for all finite values of the 

rotational speed. This limiting value is consistent with the fact that for 

R/L < 0.50 the beam is entirely in tension and no other destabilizing 

centrifugal forces exist as in the case of the inplane direction. The reader 

is referred to Ref. [9] for further discussion of this case and comparison 

with the results of similar work by others which has appeared in the 

literature. 

The stability boundaries for the beam with concentrated masses are seen 

to exhibit a behavior which is similar to that obtained for the beam with a 

continuous mass distribution. Again, because of the destabilizing term mil v 

in the inplane equation of motion, instability is first encountered in the 

plane of rotation for all values of R/L. For values of R/L greater than about 

1.2, both the inplane and out-of-plane stability boundaries approach the 

corresponding continuous mass boundaries from below as the number of 

concentrated masses is increased. As R/L is reduced, the limiting values of 

2 
18 

R/L beloy which buckling cannot occur approach the limiting values for the 

beam with continuous mass as the number of discrete masses is increased. For 

the inplane case, it should be noted that all the boundaries are asymptotic to 

the same vertical line given by R/L = 0.0, and that they approach the 

boundary for the continuous mass beam from the left as the number of 

concentrated masses is increased. For out-of-plane buckling the limiting 

values of R/L below which buckling cannot occur for the one, two and five-mass 

approximations are 1.0, 0.75, and 0.60, respectively. These limiting values 

of R/L correspond to values of R/L below which the beam is entirely in 

tension. The corresponding stability boundaries are asymptotic to the 

vertical lines through the limiting values of R/L and approach the boundary 

for the continuous mass beam from the right as the number of concentrated 

masses is increased. This is in contrast to the behavior for the inplane 

direction. The difference in behavior of the inplane and out-of-plane 

stability boundaries as R/L is reduced is associated with the term m!d v 

which appears in the inplane equation of motion. As mentioned earlier, this 

term represents a component of centrifugal force acting normal to the beam and 

decreases the inplane natural frequency with increasing rotational speed. As 

the number of concentrated masses representing the continuous mass beam 

increases, there are more and more masses being "shifted" toward the clamped 

end of the beam (where the displacements in the buckling mode are smaller) 

thereby reducing the destabilizing effect associated with the term m!d V. 

/ 

2 

v 

2 

The inplane and out-of-plane buckling of a rotating beam having both one 

and two concentrated masses has been treated in Ref. [61, which presents results from an exact solution of the differential equations obtained by applying Newton's law to the mass hypothetically cut from the beam and from equi- 
19 

librium considerations of the massless flexible beam. The present numerical 

results for the out-of-plane buckling behavior are in excellent agreement with 

the results presented in Ref. [61. However, the present results for the inplane buckling behavior are not in agreement with the results presented in 

Ref. [6]. In particular, the inplane stability boundaries in Ref. [61 do not 

exhibit the correct limiting behavior as R/L is reduced to zero. It appears 

that the development in Ref. [61 has incorrectly accounted for the mS2 v 

term. 

2 

CONCLUDING 

The stability of a flexible beam with discrete masses which is rotating 

in a plane perpendicular to the axis of rotation and clamped off this axis was 

analyzed to determine the critical rotational speeds for buckling in the inplane and out-of-plane directions. The differential equations of motion were 

solved numerically using an extended integrating matrix method which includes 

the effect of concentrated masses in combination with an eigenanalysis to 

determine the eigenvalues from which stability was assessed. Extension of the 

integrating matrix method to include concentrated masses and its application 

appear to be new. It was shown that the extended integrating matrix procedure, when applied to the governing differential equations of motion for a 

beam which includes discrete masses, leads to a matrix eigenvalue problem in 

which the matrices associated with the discrete matrices are sinply additive 

to the matrices previously derived for a beam with continuous mass. The stability boundaries obtained using the present method of solution were shown to 

asymptotically approach the corresponding boundaries for a beam with a 
20 

continuous mass distribution as the number of concentrated masses is 

increased. As the ratio of the radius of the circle traced out by the clamped 

end of the rotating beam to the length of the beam is reduced, the inplane and 

out-of-plane behavior was shown to be different due to the presence of a 

destabilizing inertial term which appears in the inplane equation of motion. 

These results have identified what appears to be an error in the literature 

with respect to the limiting behavior of the critical rotational speed for 

inplane buckling as the radius of rotation of the clamped end of the beam is 

reduced. 
REFERENCES 

1. Mostaghel, N. and I. Tadjbakhsh, Buckling of rotating rods and plates, 

Int. J. Mech. Sci., Vol. 15, 1973, pp. 429-434. 

2. Rammerstorfer, F. G., Comment on the buckling of rotating rods and 

plates, Int. J. Mech. Sci., Vol. 16, 1974, pp. 515-517. 

3. Nachman, A., The buckling of rotating rods, J. Apple Mech., Vole 42, 

1975, pp. 222-224. 

4. Wang, J. T. S., On the buckling of rotating rods, Int. J. Mech. Sci., 

Vol. 18, 1976, pp. 407-411. 

5. Lakin, W. D., Vibrations of a rotating flexible rod clamped off the axis 

of rotation, J. Engrg. Math., Vol. 10, 1976, pp. 313-321. 

6. Weber, H. I., A note on the stability of a rod subjected to compression 

by centrifugal force, J. Sound and Vibration, Vol. 46, 1976, pp. 105-111. 

7. Lakin, W. D. and A. Nachman, Unstable vibrations and buckling of rotating 

rods, Q. Appl. Math., 1978, pp. 479-493. 

8. Kvaternik, R. G., W. F. White, Jr. and K. R. V. Kaza, Nonlinear flap-lagaxial equations of a rotating beam with arbitrary precone angle. Presented at the AIAA/ASME 19th Structures, Structural Dynamics and Materials Conference, Bethesda, Maryland, (AIAA Paper 78-4911, April 1978. 

9. White, W. F., Jr., R. G. Kvaternik, and K. R. V. Kaza, Buckling of rotating beams, Int. J. Mech. Sci., Vol. 21, 1979, pp. 739-745. 

10. Vakhitov, M. B., Integrating matrices as a means of numerical solution of 

differential equations in structural mechanics, Isvestiya VUZ. 

Aviatsionnaya Teknika, Vol. 3, 1966, pp. 50-61. 

11. Hunter, W. F., The integrating matrix method for determining the natural 

vibration characteristics of propeller blades, NASA TN D-6064, December 

1970. 
22 

12. White, W. F., Jr. and R. E. Malatino, A numerical method for determining 

the natural vibration characteristics of rotating non-uniform cantilever 

blades, NASA TM X-72751, October 1975. 

13. Lakin, W. D., Integrating matrices for arbitrarily spaced grid points, 

NASA CR-159172, November 1979. 

14. Lakin, W. D., Integrating matrix formulation for vibrations of rotating 

beams including the effects of concentrated masses, NASA CR-165954, June 

1982. 

15. Ziegler, H., Principles of Structural Stability, Blaisdell, Waltham, 

Mass. (1968). 

16. Wilkinson, J. H., The Algebraic Eigenvalue Problem, Clarendon Press, 

Oxford (1965). 
Report Documentation Page 

1. Repon No. 2. Government Accession No. 

NASA CR-181862 

ICASE Report No. 89-39 

4. Title and Subtitle 

3. Recipient's Catalog No. 

5. Report Date 

7. Authorid 

William D. Lakin 

Raymond G. Kvaternik 

8. Performing Organization Report No. 

89-39 

10. Work Unit No. 

9. Performing Organization Name and Address 

Institute for Computer Applications in Science and 

Engineering 

Mail Stop 132C, NASA Langley Research Center 

Hampton, VA 23665-5225 

12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Langley Research Center 

Hampton, VA 23665-5225 

505-90-21-01 

11. Contract or Grant No. 

NAS 1- 18 107 

NAS 1- 18605 

13. Type of Repon and Period Covered 

Contractor Report 

14. Sponsoring 4gency Code 

I 

15. Suppbmentary Notes 

17. Key Words (Suggested by Authoris)) 

Langley Technical Monitor: 

Richard W. Barnwell 

18. Diribution Statement 

International Journal of 

Mechanical Sciences 

19. Security Classif. (of this rewrt) 20. Security clad. (of thia page) 21. No. of pages 

Unclassified Unclassified 23 

Final Report 

IS. Abstract 

This paper extends the integrating matrix technique of computational 

mechanics to include the effects of concentrated masses. The stability of a 

flexible rotating beam with discrete masses is analyzed to determine the critical 

rotational speeds for buckling in the inplane and out-of-plane directions. In 

this problem, the beam is subjected to compressive centrifugal forces arising 

from steady rotation about an axis which does not pass through the clamped end 

of the beam. To determine the eigenvalues from which stability is assessed, the 

differential equations of motion are solvdd numerically by combining the extended 

integrating matrix method with an eigenanalysis. Stability boundaries for a 

discrete mass representation of a uniform beam are shown to asymptotically approacl 

the stability boundaries for the corresponding continuous mass beam as the 

number of concentrated masses is increased. An error in the literature is also 

noted for the discrete mass problem concerning the behavior of the critical 

rotational speed for inplane buckling as the radius of rotation of the clamped 

end of the beam is reduced. 

22. Price 

A0 3 

integrating matrices, rotating beam, 

buckling, concentrated masses 

64 - Numerical Analysis 

39 - Structural Mechanics 

NASA-Langley, 1989