- Email: [email protected]
Bounds to bending frequencies of a rotating beam
Journal of The Franklin Institute DEVOTEDTO
SCIENCE AND
THE
MECHANICARTS
October
Volume 294, .Number 4
1972
Bounds to Bending Frequencies of a Rotating Beami by
NATHAN
RUBINSTEIN
and
JAMES
T. STADTER
The Johns Hopkins University, Applied Physics Laboratory, Silver Spring, Maryland ABSTRACT vibration methods, provide virtues
: Two lower bound methods are applied to obtain bounds to frequencies of a rotating uniform in conjunction
rigorous
upper
beam simply
with the well-known and lower
bounds
supported
Rayleigh-Ritz to a wide
qf the methoda are discussed and numerical
of bending
at one end, free at the other. These procedure,
variety
can also be used to
of vibration
problems.
The
results are given.
I. Introduction
In structural design and development it is important to know precisely the vibration frequencies of elastic structures and structural elements. Only the most elementary structures can be analyzed exactly and even for these simple problems it is often impractical to compute the exact frequencies. Therefore it is necessary to use approximation methods to estimate the frequencies of most structures. The Rayleigh-Ritz procedure is one of the most popular approximation methods because it is easy to apply and it gives estimates which are upper bounds to the true values. Thus it is possible to use the Rayleigh-Ritz method in conjunction with a lower bound procedure and obtain approximate frequencies which bracket the true frequencies. Such a combination of upper and lower bounds yields not only frequency estimates but also a measure of the accuracy of these estimates. In this paper, we compute upper and lower bounds to bending frequencies of a rotating uniform beam simply supported at one end, free at the other. The upper bounds were obtained using the Rayleigh-Ritz method with two different sets of trial functions. The lower bounds were computed using two procedures developed by Bazley and Pox (1, t)-the method of intermediate problems with truncation and the method of sums of resolvable quadratic forms. One of these methods gave excellent lower bounds to the higher t This work was N 00017-72-C-4401.
supported
by
the
Department
217
of
the
Navy
under
Contract
Nathan Rubinstein and James T. Stadter eigenvalues while the other gave excellent bounds to the lower ones. Together with the Rayleigh-Ritz procedure, these methods yield bounds such that the ratio of their difference to their average is less than 3 per cent. One significant feature of the methods used here is that they give improvable bounds-as the size of the matrices used in the computations is increased the upper and lower bounds converge to the true frequencies. The purpose of the paper is to illustrate the use of these lower bound techniques ; however, the results are of interest in themselves. Although the methods are applicable to a wide variety of structural problems (see (%lO)) in this investigation we treat a relatively simple vibration problem, assuming the beam to be uniform and neglecting shear deformation and gyroscopic effects. II.
Formulation
of the Problem
The differential equation governing the free vibrations of a uniform rotating beam simply supported at one end and free at the other (see Fig. 1) is given by EIu”” _ ~,(~2_52)‘1’--*~2fz~c~ with boundary
= 0,
O
(1)
conditions U(0) = U”(0) = u”(L) = d’(L)
= 0,
(2)
where the prime denotes differentiation, u is the transverse displacement of the beam, I is the moment of inertia of the cross section about the principal axis in the plane of rotation, E is the modulus of elasticity, m is the mass per unit volume, L is the length of the beam, C is the cross-sectional area, Q is the angular velocity of rotation, and f is the natural frequency.
FIG. 1. Rotating
beam.
For convenience we introduce the nondimensional variable 5 = x/L and write the above differential equation as an eigenvalue problem, u ““-+az[(l-<2)U’]‘-k with boundary
= 0,
1,
O<<<
(3)
conditions U(0) = U”(0) = U”( 1) = urn(1) = 0.
(4)
Here the prime denotes differentiation with respect to 5, the parameter a2 is proportional to the angular velocity of rotation, a2 = mCL4Q2/EI,
218
(5)
Journal of The Franklin Institute
Bounds to Bending Prequencies of a Rotating Beam and the eigenvalue X is related to the natural frequency f by x = 49 mCL4f yEI.
(6)
We denote by A the differential operator of Eq. (3). This operator has for its domain of definition, IDA, the set of functions in class C4 for 0 < 5 < 1 which satisfy the boundary conditions Eq. (4). The inner product for two functions f and g defined over the range 0 4 5 6 1 is given by
(f,d = j-01f(5)s(5)dbAssociated expression
with the operator A is a quadratic form, JA(u), obtained from the
(Au,u) Integration
(7)
= /;&.ud&
(8)
by parts transforms this expression into the quadratic form
JA(u) =
1[(u”)2 +*a"(1-
s0
c2) (u’)2] dc.
The domain, DD,, of this quadratic form is the class of functions 0 < 5 < 1 satisfying the boundary condition, U(0) = 0.
C2 for (10)
We note that the quadratic form JA(u) as given by Eq. (9) is twice the potential energy expression for the rotating beam. It is well known that the eigenvalues and eigenfunctions of Eq. (3) can be characterized in terms of JA(u); in fact, the eigenvalues X, are the stationary values of the ratio Ja(u)/(u, u) for variations taken in DJA and the corresponding eigenfunctions are the functions which make this ratio stationary. III.
Lower
and Upper
Bound
Procedures
In this section we present a brief description of the methods used to compute lower and upper bounds. For a more detailed account of the theoretical procedures we refer to the original papers (2, 11-13). The first eigenvalue of this problem is known exactly, it is X, = as. The corresponding eigenfunction, normalized so that (ul,uJ = 1, is u1 = J(3)x. We therefore need not find bounds for this first eigenvalue, so we restrict our attention to functions which are orthogonal to the first eigenfunction. 1. Lower Bounds Using the Method of Intermediate Problems with Truncation
The method of intermediate problems requires that the quadratic associated with the problem be decomposable as follows: Ja(u) =
JAo(u) + (5% Tu),
form
(11)
where Jao(u) is a quadratic form associated with an operator A0 whose eigenvalues and eigenfunctions can be readily obtained, and T is a closed operator with domain of definition Br. It is further required that the eigenvalues of A0 be below the corresponding eigenvalues of the given
Vol.
294, No.
4, October
1972
219
Nathan Rub&stein and James T. Stadter operator A. We denote the eigenvalues and eigenfunctions of A0 by X,0and u: respectively. This operator A0 is referred to as a base operator and the corresponding eigenvalue problem as a base problem. For the rotating beam problem the quadratic form Ji(u) is given by JAo(u) = (AOu, u) =
1(uf’)2dc, s0
(12)
and its domain DJAOconsists of functions in the class C2 satisfying the boundary condition u(0) = 0. Through integration by parts Jao(u) can be put in the form Jao(u) = (A0 u, u) =
‘AOu.udc, s0
in which the base operator, A”, is shown explicitly
with boundary
(13)
to be
A’-Ju = a”“,
(14)
U(0) = u”(0) = uV(l) = u”(l) = 0.
(15)
conditions
The base problem is then given by U “.-Au u(0)
The eigenfunctions given by u,“(C) =
= =
u”(0)
0, =
u”(l) = u”(1) = 0,
of this base problem,
normalized
(16) I
so that (u, u) = 1, are
2 I__ (sinhol,sinol,~+sina,sinh~,<), sinh2 OI,, - sin2 OL,
and the corresponding
v = 1,2,3, . . . . (17)
eigenvalues are x,0= LX:, v=
1,2,3 ,,..,
(18)
where 0~”is the vth positive root of the transcendental tana-tanhar
equation
= 0.
(19)
A comparison of Eqs. (9) and (12) shows that, for a given function the quadratic forms Jao(u) and JA(u) satisfy
(20)
JAo(u)< JA(u). The eigenvalues inequalities
associated
with these quadratic A;<&,
v=
u( [),
forms satisfy the parallel
1,2,3 ,....
(21)
Thus, the base problem eigenvalues are lower bounds to the eigenvalues of the given problem, although, in general, they are crude lower bounds. To obtain improved lower bounds we construct solvable intermediate problems. The eigenvalues of these intermediate problems will be greater than the base problem eigenvalues yet still be lower bounds to the eigenvalues of the rotating beam.
220
Journal
of The Franklin
Institute
Bounds to Bending Frequencies of a Rotating Beam For the rotating beam problem the operator T expressed in the quadratic form decomposition Eq. (11) is given by Tu = ~~(l-~2)u’, with domain of definition given by functions condition U(0) = 0.
(22) in Cl satisfying the boundary (23)
The adjoint operator T* and its domain IDTI are determined from the relation (Tu, v) = (u, T* v), through integration
by parts. Explicitly, T*v
=.-fi
(24)
T* is given by
(25)
d2
and its domain consists of functions
satisfying the condition
limJ(l-
c2) w(t) = 0.
(26)
c-t1-
As shown in detail in (1) the procedure uses lc elements of a sequence {P,) of linearly independent functions from the domain of T* in conjunction with n base problem eigenvalues h,Oand eigenfunctions uf to generate intermediate quadratic forms which we denote by Jnpk and which satisfy J”,k(~) < Ja(u).
(27)
These intermediate quadratic forms are constructed in such a way that the corresponding eigenvalue problems are solvable. The eigenvalues of these intermediate problems are lower bounds to the eigenvalues of A and they are improvable in the sense that for a larger-order problem (larger value of n or k) the lower bounds increase, or at least do not decrease. That is, the eigenvalues of the intermediate problems satisfy the inequalities @,k < ‘?+r’” Y . hFk+l These eigenvalues equation
h,“,k are obtained det [A/(X+,
where the elements products A,
BiP = (T*p&)
Cij =
(Pi,Pj) =
Vol. 294, No. 4, October 1972
=
=
s
as the roots h of the determinantal
- A) + BDBTP + CJ = 0,
of the matrices
= (T*pi,T”pj)
lT*pi.u;d<, s0
(29)
A, B and C are given by the inner
lT”p,.T*pjd{, s0
01Pi*Pjd5,
(28)
"b
(1
i,j = 1,2 ,..., k,
i=
1,2 ,..., k,
p=
I,2 ,..., n,
;,j = I, 2 , ***, k,
221
Nathan Rubinstein and James T. Stadter and D is a diagonal matrix with elements
For the rotating beam problem domain of T* the set
we chose as our functions
pi(c) = ,/(1-~2)sini~~,
i = 1,2,3 ,... .
(pi) from the (30)
These functions, along with the base problem eigenvalues and eigenfunctions as given by Eqs. (17) and (18) were used to compute the elements of the matrices A, B, C and D. The resulting expressions are given in the appendix. 2. Lower Bounds Using the Method of Sums of Resolvable Quadratic Forms This method requires that the quadratic form associated with the given problem can be expressed as a sum of two or more quadratic forms whose associated eigenvalue problems are solvable. Thus, we express the quadratic form, JA, of the rotating beam problem as the sum of two quadratic forms
JA = JAI+JAe, where each of the corresponding
(31)
eigenvalue problems A,u-hu
= 0
(32)
A,u-hu
= 0
(33)
and
is solvable. We denote the eigenvalues and eigenfunctions of Eq. (32) by Xt and u$ respectively and those of Eq. (33) by X: and uz. All eigenfunctions are assumed to be normalized so that (%U) = I, where (u,u) denotes the inner product form Jai is given explicitly by
(34)
defined by Eq. (7). The quadratic
Ja,W = )u”)2d5, s
(35)
on functions in class C2 satisfying u(0) = 0, and Jda is given by
JA,W = ;j)-
12)b02 d5,
(36)
(37)
on functions in class C2 satisfying the same boundary condition Eq. (36). The quadratic form JAI is identical to JgO of Eq. (12) ; the corresponding eigenvalue problem is therefore identical to that given in Eq. (16). If we denote the normalized eigenfunctions and eigenvalues of the operator A, by u: and Xi, it follows that the ui’s are identical to the us’s of Eq. (17) and the h;‘s are identical to the hz’s of Eq. (18). The eigenvalue problem for the operator A, associated with JAais - +a2[( 1 - c2) u’]’ - hu = 0,
222
(33)
Journal of The Franklin Institute
Bounds to Bending Frequencies of a Rotating Beam with boundary
conditions U(0) = [(l - 52)U’]l_ = 0.
(39)
This is a solvable problem ; its normalized eigenfunctions u; = (-
v = 1,2,3,
l)“J(4v+3)P,,+,(~),
and eigenvalues are . ..)
(40)
and ~,2=a2(v+1)(2v+1), where P2Y+1(?Jis the Legendre expressed by
1,2,3 ,...,
V=
polynomial
(41)
of degree 2v + 1 which can be
v+l
with (2a+2v)! CL+1=22y+1(v-(T+1)!(v+(T)!(20.-1)!~
(43)
As shown in (2) the method of sums of resolvable quadratic forms uses a finite number, say n,, of the eigenfunctions u3 and the eigenvalues Xz to construct a quadratic form JAI nl which is smaller than JAI, and a finite number, say n2, of the eigenfunctions u,” and the eigenvalues X,Zto construct a quadratic form Jy2 which is smaller than JA2. These quadratic forms are constructed in such a manner that they are monotonically increasing with respect to the indices n, and n2, that is, J”1 J-417 (44) Al’< J”l+‘< A1 and
Jnz < J”z+~< JAZ’ AZ’ AZ
(45)
These smaller quadratic forms are used to define a new quadratic form Jynz
=
JT~
+
Jm AZ’
(46)
It follows from Eqs. (31), (44) and (45) that J 2”” is monotonic in the indices n, and n2 and smaller than JA, the quadratic form of the rotating beam, (47)
Consequently, the eigenvalues X,, n1~n2 of the operator satisfy the parallel inequalities
G&O
v=
corresponding
1,2,3 ,...,
to J?y
(48)
and so are lower bounds to the eigenvalues X, of the rotating beam. These lower bounds are obtained in terms of the known eigenvalues and eigenfunctions hz, hz, u: and u;. In fact, the lower bounds are the solutions X of the matrix eingenvalue problem of order n, +n, det [(hX,+1+ Xi,,,) I -B
- hl] = 0,
(49)
where I is the identity matrix of order n, + n2,
Vol. 294, No. 4, Oct.ober 1972
223
Nathan Rubinstein and James T. Xtadter D is a block matrix (50) in which the elements of any block DC@’are given by
i= 1,2,...,n,, -xq)t(U~,~)(X~~+l-h~)l, a$ = (h~~+~
j = 1,2,...,
ng.
(51) The formulas for the inner products (u:, u$ shown in Eq. (51) are given in the appendix. 3. The Rayleigh-Ritz
Upper Bound Procedure
An Nth-order Rayleigh-Ritz procedure is based on a family of linearly independent trial functions from the domain of the quadratic form Ja. These trial functions are chosen to be orthonormal, that is,
The functions +@ are used in conjunction with the quadratic form, JA, of the given problem to generate a symmetric matrix R whose elements are given by R,, =
J,&o+y),
p-L,v = 1,2, . . ..N.
(53)
where J-4($p, A) is the bilinear form associated with the quadratic JA(+). This symmetric bilinear form is obtained from the expression
form
(54) by integration by parts. Upper bounds to the eigenvalues of the rotating beam are obtained as the solutions X of the Nth-order matrix eigenvalue problem det [R-XI]
= 0,
(55)
where I is the Nth-order identity matrix. Two different sets of trial functions were used for computing RayleighRitz upper bounds. For our first set we chose the eigenfunctions of the base problem, the uz as given by Eq. (17). The elements of the matrix R are then given by R,, = JA(u;,u,o) =
j
:(u;,,” ($?)“dC+;
j;(l-5”)
(u;)‘(u;)‘dL
/.L,v = 1,2 . ..) N. (56)
For comparison, we used the functions uf as given by Eq. (40) as a second set of trial functions in the Rayleigh-Ritz procedure. The matrix elements for this set of functions are given by R,, = J(u;,u;)
224
=
j
I($)”
(u;)” d5 +;
j;( 1 - 5”) (u;)’ (u;)‘d5,
p,v=
1,2 ,..., N. (57)
Journal of The Franklin Institute
Bounds to Bending Frequencies of a Rotating Beam The expressions for the matrix elements indicated by Eqs. (56) and (57) are given in the appendix. IV. Numerical
Results and Conclusions
Upper bounds to eigenvalues were computed using eleventh-order matrices for parameters a2 = 5, 50 and 500 with two sets of trial functions. The upper bounds using trial functions u: of Eq. (40) were obtained by diagonalizing the matrix whose elements are given by Eq. (57) ; the first six bounds are presented in Table I. The trial functions uz of Eq. (17) were used to construct TABLE I Bounds
to eigenvalues
of a simply supported
a2 = 5.0
Y 1 2 3 4 5 6
Lower bounds
rotating
a2 = 50.0 Lower bounds
Upper bounds
Upper bounds
beam a2 = 500.0
Lower bounds
Upper bounds
500~0000 5*000000 5~000000 50~00000 50~00000 500*0000 269.6703 269.6703 554.4686 554.4686 3316.361 3316.362 2585.934 2585.935 3384.199 3384.200 10958.27 10958.30 11046.94 11047.50 12655.42 12659.44 28158.83 28159.26 32050.80 32083.38 34496-l 1 34805.81 59447.73 61333.86 74291.20 74461.55 76907.85 78590.17 103385.1 119165.7
Upper bounds: Lower bounds
Rayleigh-Ritz
procedure using trial functions given by Eq. (40).
: Method of sums of resolvable quadratic forms. TABLE II
Bounds
to eigenvalues
a2 = 5.0
Y 1 2 3 4 5 6
Lower bounds
of a simply
supported
a2 = 50.0 Upper bounds
Lower bounds
Upper bounds
rotating
beam a2 = 500.0
Lower bounds
Upper bounds
5*000000 5~000000 50~00000 50*00000 500~0000 500~0000 268.2520 269.7044 540.7962 557.2948 3171.512 3425.917 2581.294 2585,994 3338.263 3390.091 10497.09 11353.97 11036.89 11047.56 12552~55 12665.52 27018.22 28703.15 32065.73 32083.44 34627~62 34811.71 59407.33 61924.26 74429.85 74460.46 78280.81 78595.19 115787.9 119763.5
Upper bounds: Rayleigh-Ritz procedure using trial functions given by Eq. (17). Lower bounds : Method of intermediate problems with truncation.
the matrix whose elements are given by Eq. (56) ; this matrix was diagonalized to give upper bounds of which the first six are presented in Table II. The lower bounds using the method of sums of resolvable quadratic forms were obtained by solving the matrix eigenvalue problem, Eq. (49), of order
Vol. 294. No. 4. October 1972
225
Nathan Rubinstein and James T. Stadter n, + n2 = 11 (nr = 7, nz = 4) for a2 = 5, 50 and 500. The results are given in Table I. Lower bounds were also obtained using the method of intermediate problems with truncation. An intermediate problem of order k = 11 and base problem of order n = 11 yielded a determinant Eq. (29) of order 11 whose roots are the desired bounds. The first six lower bounds from this procedure are given in Table II for a2 = 5, 50 and 500.”
IV.
Summary
and Conclusions
The method of sums of resolvable quadratic forms yields excellent bounds to the lower order eigenvalues, however, due to overflow in computing the coefficients PG+;+l,Eq. (43), of the Legendre polynomials, the order of the problem had to be limited to 11. Consequently these bounds cannot be improved due to the numerical difficulties. The method of intermediate problems with truncation yields good lower bounds to the higher-order eigenvalues and presents no numerical difficulties as those encountered above ; thus the bounds can be improved by increasing the order of the determinant, Eq. (29). The process of finding roots of largeorder determinants, Eq. (29), is substantially more involved than diagonalizing large-order matrices. In most problems the lower bound methods usually do not present numerical difficulties, consequently the bounds can be easily improved by increasing the size of the matrix computation. When such numerical difficulties do occur however, it is possible to avoid them either by choosing another decomposition of the quadratic form Jd or by applying a different lower bound procedure. In fact, this is precisely what we have done here in applying the method of intermediate problems after we experienced numerical difficulties with our decomposition of the quadratic form JA in the method of sums of resolvable quadratic forms.
Appendix The 01:s which appear below are the positive roots of the transcendental A. Matrix
Elements
Used in the Method
(1) The elements of {(T*pi,
Problema
I
&&
l)i+i+l
16a2ij(i”+ 4i2j2+j4) I
(iz_j2)4~2
(16i4&+40iZ+-
15),
* All cctlculations were done on the IBM 7094 computer Laboratory.
226
with Truncation
T*p,)}.
(_
(T*pi, T*z~r) =
of Intermediate
Eq. (19).
foriZj, for i = j.
at the Applied
Physics
Journal of The Franklin
Institute
Bounds to Bending Frequencies
of a Rotating Beam
(2) The elements of {(T* p,, u:)}. 1601: ai3m3
sinh CY~ (TIPS*%)
=
a,-SirGar,)
J(sinh2
+
I[ (a$-G7P)2
201, aiTr(301;+ i2 79) (c$ - ia 7rZ)S
L
1
1
(-l)Ssiuor,
(-l)icosa,
-(a~~~~~~a)3[~:+2(3~~+i2rrz)-i2~2(2u:-iarr2)] P cash ati
2a, a&( 3oL;- i2 79)
+ $(sinh2 aP - sin2 CYJ[
(CL;+ i2 7P)3
+J(si&2
a,lsiuS
orll)[(a,JFz*)i .
I
1(-
l)i sin &@
I [at- 2(3aE-i2rr2)
+ i2 7r2(2a2Ir+ i2 59)] sin OL @’ (3) The elements of {(pi,pj)}.
.. ((PiYPj)
=
l)i+J+l (ia_$)2n2.
4i2T2+3
for i = j.
t 12i2p2’ B. Matrix
Elements
Used in the Method
The elements of {(u:_,,
(Ukl’ uf-l)
u:+)},
for izj,
of Sums of Resolvable
II = 2, 3, . . . . v = 2, 3, . . .
Quadratic
&S/L - 2) sinb CL-~ 5l)w:(20-l)! = j(sinhe (Y,_~- sin2 a,_,) ,,=1 r1+
( - l)y
cos (Y,_~- (2K + 1) sin c&l [%__1 (2K+l)!or::7=
(Cg
Forms
.
.
is defined by Eq. (43).)
C. Elements
RNV Used in the
Upper Bounds
In this formulation we use the ~4’s of the base problem, Eq. (1) The elements of (R,,}, p #v.
(17, as trial functions.
2a2 R,,
Vol.
=
294, No. 4. October
J(sinh2 (Y~- sin2 LQ)I/(sinh2 01,-sin% 01,)
1972
227
Nathan Rub&stein and James T. Xtadter
(2) The elements of {RfiY}, p = Y.
R,, =
(
(sin (Yecos a0 sinhz afi - sin2 0~~sinh a,, cash 01~) 2 +
D. Elements
R,,
5
(sin2 (Y~+ sinh2 a&)+ f sin2 01~sinh2 01~ + c$. I
used in the
In this formulation
Upper Bounds
we use the normalized,
odd-ordered Legendre polynomials,
Eq.
(40), as trial functions. The elements of {R,,}.
E,
=
~[3(~+1)(2~+1)-((~+1)(2~+1)+31,
forp#fv,
~~(~+1)(2~+1)+~(~)(~+1)(2~+1)(2~+3)(4~+3)[2(~+1)(2~+1)+31, for p = v.
References (1) N. W. BAZLEY and D. W. Fox, “Lower Bounds to Eigenvalues Using Operator Decompositions of the Form B*B”, Archive for Rat. Mech. and Anal., Vol. 10, pp. 352-360, 1962. (2) N. W. BAZLEY and D. W. Fox, “Methods for Lower Bounds to Frequencies of ContinuousElastic Systems”, Zeit. Angew. Math. Phys., Vol. 17, pp. l-37, 1966. (3) N. RUBINSTEIN, “Frequencies of Beams on Partial Elastic Foundations”, Applied Physics Lab., Johns Hopkins Univ., Tech. Rep. TG-804, March 1966. (4) N. RUBINSTEIN, V. G. SICILLITO and J. T. STADTER, “Torsional Frequencies of Uniform Shafts with Elastic Restraints and Elasticically Attached Masses”, Applied Physics Lab., Johns Hopkins Univ., Tech. Rep. TG-833, June, 1966. (5) N. RUBINSTEIN, V. G. SIGILLITOand J. T. STADTER, “Upper and Lower Bounds to Torsional Frequencies of Nonuniform Shafts and Application to Missiles”, Shock and Vibration BulEetin, Bull. 40, Part 4, pp. 163-170, Dec. 1969. (6) N. RUBINSTEIN, V. G. SI~ILLITO and J. T. STADTER, “Upper and Lower Bounds to Bending Frequencies of Nonuniform Shafts and Applications to Missiles”, Shock and Vibration BuZZetin, Bull. 38, Part 2, pp. 169-176, Aug. 1968. (7) N. RUBINSTEIN and J. T. STADTER, “Bounds to Frequencies of Uniform Beams with Elastically Attached Masses”, Applied Physics Lab., Johns Hopkins Univ., Tech. Rep. TG-1126, Aug. 1970. (8) N. W. BAZLEY, D. W. Fox and J. T. STADTER, “Upper and Lower Bounds for the Frequencies of Rectangular Cantilever Plates”, ZAMM, Vol. 47, pp. 251-260, 1967. (9) N. W. BAZLEY, D. W. Fox and J. T. STADTER, “Upper and Lower Bounds for the Frequencies of Rectangular Clamped Plates”, ZAMM, Vol. 47, pp. 191-198, 1967. (10) N. W. BAZLEY, D. W. Fox and J. T. STADTER, “Upper and Lower Bounds for the Frequencies of Rectangular Free Plates”, ZAMP, Vol. 18, pp. 445460, 1967.
228
Journal of The Franklin Institute
Bounds to Bending Prequencies of a Rotating Beam Methods for Eigenvalues of Completely Con(11) N. ARONSZAJN, “Approximation tinuous Symmetric Operators”, Proc. Symp. on Spectral Theory and Differential Problems, Stillwater, Okla., pp. 179-202, 1951. (12) S. H. GOULD, “Variational Methods for Eigenvalue Problems”, Toronto, Univ. of Toronto Press, 2nd edit., 1966. (13) A. WEINSTEIN, “Etudes des Spectres des Equations aux D&i&es Partielles de la Theorie des Plaques Elastiques”, Mdmorial des Sciences Mathkmatiques, Fascicule No. 88, Gauthier-Villars, Paris, 1937.
Vol. 294, No. 4, October1972
229
SCIENCE AND
THE
MECHANICARTS
October
Volume 294, .Number 4
1972
Bounds to Bending Frequencies of a Rotating Beami by
NATHAN
RUBINSTEIN
and
JAMES
T. STADTER
The Johns Hopkins University, Applied Physics Laboratory, Silver Spring, Maryland ABSTRACT vibration methods, provide virtues
: Two lower bound methods are applied to obtain bounds to frequencies of a rotating uniform in conjunction
rigorous
upper
beam simply
with the well-known and lower
bounds
supported
Rayleigh-Ritz to a wide
qf the methoda are discussed and numerical
of bending
at one end, free at the other. These procedure,
variety
can also be used to
of vibration
problems.
The
results are given.
I. Introduction
In structural design and development it is important to know precisely the vibration frequencies of elastic structures and structural elements. Only the most elementary structures can be analyzed exactly and even for these simple problems it is often impractical to compute the exact frequencies. Therefore it is necessary to use approximation methods to estimate the frequencies of most structures. The Rayleigh-Ritz procedure is one of the most popular approximation methods because it is easy to apply and it gives estimates which are upper bounds to the true values. Thus it is possible to use the Rayleigh-Ritz method in conjunction with a lower bound procedure and obtain approximate frequencies which bracket the true frequencies. Such a combination of upper and lower bounds yields not only frequency estimates but also a measure of the accuracy of these estimates. In this paper, we compute upper and lower bounds to bending frequencies of a rotating uniform beam simply supported at one end, free at the other. The upper bounds were obtained using the Rayleigh-Ritz method with two different sets of trial functions. The lower bounds were computed using two procedures developed by Bazley and Pox (1, t)-the method of intermediate problems with truncation and the method of sums of resolvable quadratic forms. One of these methods gave excellent lower bounds to the higher t This work was N 00017-72-C-4401.
supported
by
the
Department
217
of
the
Navy
under
Contract
Nathan Rubinstein and James T. Stadter eigenvalues while the other gave excellent bounds to the lower ones. Together with the Rayleigh-Ritz procedure, these methods yield bounds such that the ratio of their difference to their average is less than 3 per cent. One significant feature of the methods used here is that they give improvable bounds-as the size of the matrices used in the computations is increased the upper and lower bounds converge to the true frequencies. The purpose of the paper is to illustrate the use of these lower bound techniques ; however, the results are of interest in themselves. Although the methods are applicable to a wide variety of structural problems (see (%lO)) in this investigation we treat a relatively simple vibration problem, assuming the beam to be uniform and neglecting shear deformation and gyroscopic effects. II.
Formulation
of the Problem
The differential equation governing the free vibrations of a uniform rotating beam simply supported at one end and free at the other (see Fig. 1) is given by EIu”” _ ~,(~2_52)‘1’--*~2fz~c~ with boundary
= 0,
O
(1)
conditions U(0) = U”(0) = u”(L) = d’(L)
= 0,
(2)
where the prime denotes differentiation, u is the transverse displacement of the beam, I is the moment of inertia of the cross section about the principal axis in the plane of rotation, E is the modulus of elasticity, m is the mass per unit volume, L is the length of the beam, C is the cross-sectional area, Q is the angular velocity of rotation, and f is the natural frequency.
FIG. 1. Rotating
beam.
For convenience we introduce the nondimensional variable 5 = x/L and write the above differential equation as an eigenvalue problem, u ““-+az[(l-<2)U’]‘-k with boundary
= 0,
1,
O<<<
(3)
conditions U(0) = U”(0) = U”( 1) = urn(1) = 0.
(4)
Here the prime denotes differentiation with respect to 5, the parameter a2 is proportional to the angular velocity of rotation, a2 = mCL4Q2/EI,
218
(5)
Journal of The Franklin Institute
Bounds to Bending Prequencies of a Rotating Beam and the eigenvalue X is related to the natural frequency f by x = 49 mCL4f yEI.
(6)
We denote by A the differential operator of Eq. (3). This operator has for its domain of definition, IDA, the set of functions in class C4 for 0 < 5 < 1 which satisfy the boundary conditions Eq. (4). The inner product for two functions f and g defined over the range 0 4 5 6 1 is given by
(f,d = j-01f(5)s(5)dbAssociated expression
with the operator A is a quadratic form, JA(u), obtained from the
(Au,u) Integration
(7)
= /;&.ud&
(8)
by parts transforms this expression into the quadratic form
JA(u) =
1[(u”)2 +*a"(1-
s0
c2) (u’)2] dc.
The domain, DD,, of this quadratic form is the class of functions 0 < 5 < 1 satisfying the boundary condition, U(0) = 0.
C2 for (10)
We note that the quadratic form JA(u) as given by Eq. (9) is twice the potential energy expression for the rotating beam. It is well known that the eigenvalues and eigenfunctions of Eq. (3) can be characterized in terms of JA(u); in fact, the eigenvalues X, are the stationary values of the ratio Ja(u)/(u, u) for variations taken in DJA and the corresponding eigenfunctions are the functions which make this ratio stationary. III.
Lower
and Upper
Bound
Procedures
In this section we present a brief description of the methods used to compute lower and upper bounds. For a more detailed account of the theoretical procedures we refer to the original papers (2, 11-13). The first eigenvalue of this problem is known exactly, it is X, = as. The corresponding eigenfunction, normalized so that (ul,uJ = 1, is u1 = J(3)x. We therefore need not find bounds for this first eigenvalue, so we restrict our attention to functions which are orthogonal to the first eigenfunction. 1. Lower Bounds Using the Method of Intermediate Problems with Truncation
The method of intermediate problems requires that the quadratic associated with the problem be decomposable as follows: Ja(u) =
JAo(u) + (5% Tu),
form
(11)
where Jao(u) is a quadratic form associated with an operator A0 whose eigenvalues and eigenfunctions can be readily obtained, and T is a closed operator with domain of definition Br. It is further required that the eigenvalues of A0 be below the corresponding eigenvalues of the given
Vol.
294, No.
4, October
1972
219
Nathan Rub&stein and James T. Stadter operator A. We denote the eigenvalues and eigenfunctions of A0 by X,0and u: respectively. This operator A0 is referred to as a base operator and the corresponding eigenvalue problem as a base problem. For the rotating beam problem the quadratic form Ji(u) is given by JAo(u) = (AOu, u) =
1(uf’)2dc, s0
(12)
and its domain DJAOconsists of functions in the class C2 satisfying the boundary condition u(0) = 0. Through integration by parts Jao(u) can be put in the form Jao(u) = (A0 u, u) =
‘AOu.udc, s0
in which the base operator, A”, is shown explicitly
with boundary
(13)
to be
A’-Ju = a”“,
(14)
U(0) = u”(0) = uV(l) = u”(l) = 0.
(15)
conditions
The base problem is then given by U “.-Au u(0)
The eigenfunctions given by u,“(C) =
= =
u”(0)
0, =
u”(l) = u”(1) = 0,
of this base problem,
normalized
(16) I
so that (u, u) = 1, are
2 I__ (sinhol,sinol,~+sina,sinh~,<), sinh2 OI,, - sin2 OL,
and the corresponding
v = 1,2,3, . . . . (17)
eigenvalues are x,0= LX:, v=
1,2,3 ,,..,
(18)
where 0~”is the vth positive root of the transcendental tana-tanhar
equation
= 0.
(19)
A comparison of Eqs. (9) and (12) shows that, for a given function the quadratic forms Jao(u) and JA(u) satisfy
(20)
JAo(u)< JA(u). The eigenvalues inequalities
associated
with these quadratic A;<&,
v=
u( [),
forms satisfy the parallel
1,2,3 ,....
(21)
Thus, the base problem eigenvalues are lower bounds to the eigenvalues of the given problem, although, in general, they are crude lower bounds. To obtain improved lower bounds we construct solvable intermediate problems. The eigenvalues of these intermediate problems will be greater than the base problem eigenvalues yet still be lower bounds to the eigenvalues of the rotating beam.
220
Journal
of The Franklin
Institute
Bounds to Bending Frequencies of a Rotating Beam For the rotating beam problem the operator T expressed in the quadratic form decomposition Eq. (11) is given by Tu = ~~(l-~2)u’, with domain of definition given by functions condition U(0) = 0.
(22) in Cl satisfying the boundary (23)
The adjoint operator T* and its domain IDTI are determined from the relation (Tu, v) = (u, T* v), through integration
by parts. Explicitly, T*v
=.-fi
(24)
T* is given by
(25)
d2
and its domain consists of functions
satisfying the condition
limJ(l-
c2) w(t) = 0.
(26)
c-t1-
As shown in detail in (1) the procedure uses lc elements of a sequence {P,) of linearly independent functions from the domain of T* in conjunction with n base problem eigenvalues h,Oand eigenfunctions uf to generate intermediate quadratic forms which we denote by Jnpk and which satisfy J”,k(~) < Ja(u).
(27)
These intermediate quadratic forms are constructed in such a way that the corresponding eigenvalue problems are solvable. The eigenvalues of these intermediate problems are lower bounds to the eigenvalues of A and they are improvable in the sense that for a larger-order problem (larger value of n or k) the lower bounds increase, or at least do not decrease. That is, the eigenvalues of the intermediate problems satisfy the inequalities @,k < ‘?+r’” Y . hFk+l These eigenvalues equation
h,“,k are obtained det [A/(X+,
where the elements products A,
BiP = (T*p&)
Cij =
(Pi,Pj) =
Vol. 294, No. 4, October 1972
=
=
s
as the roots h of the determinantal
- A) + BDBTP + CJ = 0,
of the matrices
= (T*pi,T”pj)
lT*pi.u;d<, s0
(29)
A, B and C are given by the inner
lT”p,.T*pjd{, s0
01Pi*Pjd5,
(28)
"b
(1
i,j = 1,2 ,..., k,
i=
1,2 ,..., k,
p=
I,2 ,..., n,
;,j = I, 2 , ***, k,
221
Nathan Rubinstein and James T. Stadter and D is a diagonal matrix with elements
For the rotating beam problem domain of T* the set
we chose as our functions
pi(c) = ,/(1-~2)sini~~,
i = 1,2,3 ,... .
(pi) from the (30)
These functions, along with the base problem eigenvalues and eigenfunctions as given by Eqs. (17) and (18) were used to compute the elements of the matrices A, B, C and D. The resulting expressions are given in the appendix. 2. Lower Bounds Using the Method of Sums of Resolvable Quadratic Forms This method requires that the quadratic form associated with the given problem can be expressed as a sum of two or more quadratic forms whose associated eigenvalue problems are solvable. Thus, we express the quadratic form, JA, of the rotating beam problem as the sum of two quadratic forms
JA = JAI+JAe, where each of the corresponding
(31)
eigenvalue problems A,u-hu
= 0
(32)
A,u-hu
= 0
(33)
and
is solvable. We denote the eigenvalues and eigenfunctions of Eq. (32) by Xt and u$ respectively and those of Eq. (33) by X: and uz. All eigenfunctions are assumed to be normalized so that (%U) = I, where (u,u) denotes the inner product form Jai is given explicitly by
(34)
defined by Eq. (7). The quadratic
Ja,W = )u”)2d5, s
(35)
on functions in class C2 satisfying u(0) = 0, and Jda is given by
JA,W = ;j)-
12)b02 d5,
(36)
(37)
on functions in class C2 satisfying the same boundary condition Eq. (36). The quadratic form JAI is identical to JgO of Eq. (12) ; the corresponding eigenvalue problem is therefore identical to that given in Eq. (16). If we denote the normalized eigenfunctions and eigenvalues of the operator A, by u: and Xi, it follows that the ui’s are identical to the us’s of Eq. (17) and the h;‘s are identical to the hz’s of Eq. (18). The eigenvalue problem for the operator A, associated with JAais - +a2[( 1 - c2) u’]’ - hu = 0,
222
(33)
Journal of The Franklin Institute
Bounds to Bending Frequencies of a Rotating Beam with boundary
conditions U(0) = [(l - 52)U’]l_ = 0.
(39)
This is a solvable problem ; its normalized eigenfunctions u; = (-
v = 1,2,3,
l)“J(4v+3)P,,+,(~),
and eigenvalues are . ..)
(40)
and ~,2=a2(v+1)(2v+1), where P2Y+1(?Jis the Legendre expressed by
1,2,3 ,...,
V=
polynomial
(41)
of degree 2v + 1 which can be
v+l
with (2a+2v)! CL+1=22y+1(v-(T+1)!(v+(T)!(20.-1)!~
(43)
As shown in (2) the method of sums of resolvable quadratic forms uses a finite number, say n,, of the eigenfunctions u3 and the eigenvalues Xz to construct a quadratic form JAI nl which is smaller than JAI, and a finite number, say n2, of the eigenfunctions u,” and the eigenvalues X,Zto construct a quadratic form Jy2 which is smaller than JA2. These quadratic forms are constructed in such a manner that they are monotonically increasing with respect to the indices n, and n2, that is, J”1 J-417 (44) Al’< J”l+‘< A1 and
Jnz < J”z+~< JAZ’ AZ’ AZ
(45)
These smaller quadratic forms are used to define a new quadratic form Jynz
=
JT~
+
Jm AZ’
(46)
It follows from Eqs. (31), (44) and (45) that J 2”” is monotonic in the indices n, and n2 and smaller than JA, the quadratic form of the rotating beam, (47)
Consequently, the eigenvalues X,, n1~n2 of the operator satisfy the parallel inequalities
G&O
v=
corresponding
1,2,3 ,...,
to J?y
(48)
and so are lower bounds to the eigenvalues X, of the rotating beam. These lower bounds are obtained in terms of the known eigenvalues and eigenfunctions hz, hz, u: and u;. In fact, the lower bounds are the solutions X of the matrix eingenvalue problem of order n, +n, det [(hX,+1+ Xi,,,) I -B
- hl] = 0,
(49)
where I is the identity matrix of order n, + n2,
Vol. 294, No. 4, Oct.ober 1972
223
Nathan Rubinstein and James T. Xtadter D is a block matrix (50) in which the elements of any block DC@’are given by
i= 1,2,...,n,, -xq)t(U~,~)(X~~+l-h~)l, a$ = (h~~+~
j = 1,2,...,
ng.
(51) The formulas for the inner products (u:, u$ shown in Eq. (51) are given in the appendix. 3. The Rayleigh-Ritz
Upper Bound Procedure
An Nth-order Rayleigh-Ritz procedure is based on a family of linearly independent trial functions from the domain of the quadratic form Ja. These trial functions are chosen to be orthonormal, that is,
The functions +@ are used in conjunction with the quadratic form, JA, of the given problem to generate a symmetric matrix R whose elements are given by R,, =
J,&o+y),
p-L,v = 1,2, . . ..N.
(53)
where J-4($p, A) is the bilinear form associated with the quadratic JA(+). This symmetric bilinear form is obtained from the expression
form
(54) by integration by parts. Upper bounds to the eigenvalues of the rotating beam are obtained as the solutions X of the Nth-order matrix eigenvalue problem det [R-XI]
= 0,
(55)
where I is the Nth-order identity matrix. Two different sets of trial functions were used for computing RayleighRitz upper bounds. For our first set we chose the eigenfunctions of the base problem, the uz as given by Eq. (17). The elements of the matrix R are then given by R,, = JA(u;,u,o) =
j
:(u;,,” ($?)“dC+;
j;(l-5”)
(u;)‘(u;)‘dL
/.L,v = 1,2 . ..) N. (56)
For comparison, we used the functions uf as given by Eq. (40) as a second set of trial functions in the Rayleigh-Ritz procedure. The matrix elements for this set of functions are given by R,, = J(u;,u;)
224
=
j
I($)”
(u;)” d5 +;
j;( 1 - 5”) (u;)’ (u;)‘d5,
p,v=
1,2 ,..., N. (57)
Journal of The Franklin Institute
Bounds to Bending Frequencies of a Rotating Beam The expressions for the matrix elements indicated by Eqs. (56) and (57) are given in the appendix. IV. Numerical
Results and Conclusions
Upper bounds to eigenvalues were computed using eleventh-order matrices for parameters a2 = 5, 50 and 500 with two sets of trial functions. The upper bounds using trial functions u: of Eq. (40) were obtained by diagonalizing the matrix whose elements are given by Eq. (57) ; the first six bounds are presented in Table I. The trial functions uz of Eq. (17) were used to construct TABLE I Bounds
to eigenvalues
of a simply supported
a2 = 5.0
Y 1 2 3 4 5 6
Lower bounds
rotating
a2 = 50.0 Lower bounds
Upper bounds
Upper bounds
beam a2 = 500.0
Lower bounds
Upper bounds
500~0000 5*000000 5~000000 50~00000 50~00000 500*0000 269.6703 269.6703 554.4686 554.4686 3316.361 3316.362 2585.934 2585.935 3384.199 3384.200 10958.27 10958.30 11046.94 11047.50 12655.42 12659.44 28158.83 28159.26 32050.80 32083.38 34496-l 1 34805.81 59447.73 61333.86 74291.20 74461.55 76907.85 78590.17 103385.1 119165.7
Upper bounds: Lower bounds
Rayleigh-Ritz
procedure using trial functions given by Eq. (40).
: Method of sums of resolvable quadratic forms. TABLE II
Bounds
to eigenvalues
a2 = 5.0
Y 1 2 3 4 5 6
Lower bounds
of a simply
supported
a2 = 50.0 Upper bounds
Lower bounds
Upper bounds
rotating
beam a2 = 500.0
Lower bounds
Upper bounds
5*000000 5~000000 50~00000 50*00000 500~0000 500~0000 268.2520 269.7044 540.7962 557.2948 3171.512 3425.917 2581.294 2585,994 3338.263 3390.091 10497.09 11353.97 11036.89 11047.56 12552~55 12665.52 27018.22 28703.15 32065.73 32083.44 34627~62 34811.71 59407.33 61924.26 74429.85 74460.46 78280.81 78595.19 115787.9 119763.5
Upper bounds: Rayleigh-Ritz procedure using trial functions given by Eq. (17). Lower bounds : Method of intermediate problems with truncation.
the matrix whose elements are given by Eq. (56) ; this matrix was diagonalized to give upper bounds of which the first six are presented in Table II. The lower bounds using the method of sums of resolvable quadratic forms were obtained by solving the matrix eigenvalue problem, Eq. (49), of order
Vol. 294. No. 4. October 1972
225
Nathan Rubinstein and James T. Stadter n, + n2 = 11 (nr = 7, nz = 4) for a2 = 5, 50 and 500. The results are given in Table I. Lower bounds were also obtained using the method of intermediate problems with truncation. An intermediate problem of order k = 11 and base problem of order n = 11 yielded a determinant Eq. (29) of order 11 whose roots are the desired bounds. The first six lower bounds from this procedure are given in Table II for a2 = 5, 50 and 500.”
IV.
Summary
and Conclusions
The method of sums of resolvable quadratic forms yields excellent bounds to the lower order eigenvalues, however, due to overflow in computing the coefficients PG+;+l,Eq. (43), of the Legendre polynomials, the order of the problem had to be limited to 11. Consequently these bounds cannot be improved due to the numerical difficulties. The method of intermediate problems with truncation yields good lower bounds to the higher-order eigenvalues and presents no numerical difficulties as those encountered above ; thus the bounds can be improved by increasing the order of the determinant, Eq. (29). The process of finding roots of largeorder determinants, Eq. (29), is substantially more involved than diagonalizing large-order matrices. In most problems the lower bound methods usually do not present numerical difficulties, consequently the bounds can be easily improved by increasing the size of the matrix computation. When such numerical difficulties do occur however, it is possible to avoid them either by choosing another decomposition of the quadratic form Jd or by applying a different lower bound procedure. In fact, this is precisely what we have done here in applying the method of intermediate problems after we experienced numerical difficulties with our decomposition of the quadratic form JA in the method of sums of resolvable quadratic forms.
Appendix The 01:s which appear below are the positive roots of the transcendental A. Matrix
Elements
Used in the Method
(1) The elements of {(T*pi,
Problema
I
&&
l)i+i+l
16a2ij(i”+ 4i2j2+j4) I
(iz_j2)4~2
(16i4&+40iZ+-
15),
* All cctlculations were done on the IBM 7094 computer Laboratory.
226
with Truncation
T*p,)}.
(_
(T*pi, T*z~r) =
of Intermediate
Eq. (19).
foriZj, for i = j.
at the Applied
Physics
Journal of The Franklin
Institute
Bounds to Bending Frequencies
of a Rotating Beam
(2) The elements of {(T* p,, u:)}. 1601: ai3m3
sinh CY~ (TIPS*%)
=
a,-SirGar,)
J(sinh2
+
I[ (a$-G7P)2
201, aiTr(301;+ i2 79) (c$ - ia 7rZ)S
L
1
1
(-l)Ssiuor,
(-l)icosa,
-(a~~~~~~a)3[~:+2(3~~+i2rrz)-i2~2(2u:-iarr2)] P cash ati
2a, a&( 3oL;- i2 79)
+ $(sinh2 aP - sin2 CYJ[
(CL;+ i2 7P)3
+J(si&2
a,lsiuS
orll)[(a,JFz*)i .
I
1(-
l)i sin &@
I [at- 2(3aE-i2rr2)
+ i2 7r2(2a2Ir+ i2 59)] sin OL @’ (3) The elements of {(pi,pj)}.
.. ((PiYPj)
=
l)i+J+l (ia_$)2n2.
4i2T2+3
for i = j.
t 12i2p2’ B. Matrix
Elements
Used in the Method
The elements of {(u:_,,
(Ukl’ uf-l)
u:+)},
for izj,
of Sums of Resolvable
II = 2, 3, . . . . v = 2, 3, . . .
Quadratic
&S/L - 2) sinb CL-~ 5l)w:(20-l)! = j(sinhe (Y,_~- sin2 a,_,) ,,=1 r1+
( - l)y
cos (Y,_~- (2K + 1) sin c&l [%__1 (2K+l)!or::7=
(Cg
Forms
.
.
is defined by Eq. (43).)
C. Elements
RNV Used in the
Upper Bounds
In this formulation we use the ~4’s of the base problem, Eq. (1) The elements of (R,,}, p #v.
(17, as trial functions.
2a2 R,,
Vol.
=
294, No. 4. October
J(sinh2 (Y~- sin2 LQ)I/(sinh2 01,-sin% 01,)
1972
227
Nathan Rub&stein and James T. Xtadter
(2) The elements of {RfiY}, p = Y.
R,, =
(
(sin (Yecos a0 sinhz afi - sin2 0~~sinh a,, cash 01~) 2 +
D. Elements
R,,
5
(sin2 (Y~+ sinh2 a&)+ f sin2 01~sinh2 01~ + c$. I
used in the
In this formulation
Upper Bounds
we use the normalized,
odd-ordered Legendre polynomials,
Eq.
(40), as trial functions. The elements of {R,,}.
E,
=
~[3(~+1)(2~+1)-((~+1)(2~+1)+31,
forp#fv,
~~(~+1)(2~+1)+~(~)(~+1)(2~+1)(2~+3)(4~+3)[2(~+1)(2~+1)+31, for p = v.
References (1) N. W. BAZLEY and D. W. Fox, “Lower Bounds to Eigenvalues Using Operator Decompositions of the Form B*B”, Archive for Rat. Mech. and Anal., Vol. 10, pp. 352-360, 1962. (2) N. W. BAZLEY and D. W. Fox, “Methods for Lower Bounds to Frequencies of ContinuousElastic Systems”, Zeit. Angew. Math. Phys., Vol. 17, pp. l-37, 1966. (3) N. RUBINSTEIN, “Frequencies of Beams on Partial Elastic Foundations”, Applied Physics Lab., Johns Hopkins Univ., Tech. Rep. TG-804, March 1966. (4) N. RUBINSTEIN, V. G. SICILLITO and J. T. STADTER, “Torsional Frequencies of Uniform Shafts with Elastic Restraints and Elasticically Attached Masses”, Applied Physics Lab., Johns Hopkins Univ., Tech. Rep. TG-833, June, 1966. (5) N. RUBINSTEIN, V. G. SIGILLITOand J. T. STADTER, “Upper and Lower Bounds to Torsional Frequencies of Nonuniform Shafts and Application to Missiles”, Shock and Vibration BulEetin, Bull. 40, Part 4, pp. 163-170, Dec. 1969. (6) N. RUBINSTEIN, V. G. SI~ILLITO and J. T. STADTER, “Upper and Lower Bounds to Bending Frequencies of Nonuniform Shafts and Applications to Missiles”, Shock and Vibration BuZZetin, Bull. 38, Part 2, pp. 169-176, Aug. 1968. (7) N. RUBINSTEIN and J. T. STADTER, “Bounds to Frequencies of Uniform Beams with Elastically Attached Masses”, Applied Physics Lab., Johns Hopkins Univ., Tech. Rep. TG-1126, Aug. 1970. (8) N. W. BAZLEY, D. W. Fox and J. T. STADTER, “Upper and Lower Bounds for the Frequencies of Rectangular Cantilever Plates”, ZAMM, Vol. 47, pp. 251-260, 1967. (9) N. W. BAZLEY, D. W. Fox and J. T. STADTER, “Upper and Lower Bounds for the Frequencies of Rectangular Clamped Plates”, ZAMM, Vol. 47, pp. 191-198, 1967. (10) N. W. BAZLEY, D. W. Fox and J. T. STADTER, “Upper and Lower Bounds for the Frequencies of Rectangular Free Plates”, ZAMP, Vol. 18, pp. 445460, 1967.
228
Journal of The Franklin Institute
Bounds to Bending Prequencies of a Rotating Beam Methods for Eigenvalues of Completely Con(11) N. ARONSZAJN, “Approximation tinuous Symmetric Operators”, Proc. Symp. on Spectral Theory and Differential Problems, Stillwater, Okla., pp. 179-202, 1951. (12) S. H. GOULD, “Variational Methods for Eigenvalue Problems”, Toronto, Univ. of Toronto Press, 2nd edit., 1966. (13) A. WEINSTEIN, “Etudes des Spectres des Equations aux D&i&es Partielles de la Theorie des Plaques Elastiques”, Mdmorial des Sciences Mathkmatiques, Fascicule No. 88, Gauthier-Villars, Paris, 1937.
Vol. 294, No. 4, October1972
229