On the free vibrations of spinning paraboloids

Journal of Sound and Vibration (1986) 111(2), 279-296

ON THE FREE VIBRATIONS OF SPINNING PARABQLOIDSt W. L. SHOEMAKER

Department of Civil Engineering, Auburn University, Auburn, Alabama 36849, U.S.A. AND

S. UTKU

Department of Civil Engineering and Computer Science, Duke University, Durham, North Carolina 27706, U.S.A. (Received 23 July 1985, and in revised form 3 January 1986) The formulation of the discrete equations of dynamics for spinning, imperfect, paraboloids made of linearly elastic material is presented in a form suitable for computer generation and solution. The deformations may be non-axisymmetric and are approximated with admissible trial functions. These trial functions are composed of known yet unspecified co-ordinate functions in the meridional and circumferential directions of the paraboloid and of unknown functions of time. The strain-displacement and velocity-displacement relations which are needed to express the strain energy and kinetic energy in terms of the admissible trial functions are explicitly given for the linear formulation and reference is given to the excluded geometric non-linear contributions. The coefficient matrices of the discrete equations of dynamics are presented in terms of basic matrices which can be generated from the generic partitions that are provided. The partitions provided are specialized to the axisymmetdc case, but can be easily extended by including the nonaxisymmetric terms of the strain-displacement and velocity-displacement relations which are given. Numerical studies of the free vibration for the axisymmetric case are presented. Continuous (global) co-ordinate functions are used in the meridional direction in the form of a power series expansion. After examining the convergence of the natural frequencies and mode shapes as a function of the number of terms included in the power series representing the co-ordinate functions and the number of stations used in the volume integration, studies of the effect of bending rigidity and spin rate are presented. Results are also presented for the subset of the spinning disk in which the focal length of the paraboloid goes to infinity and compared to results for spir~ning disks found in the literature.

1. INTRODUCTION The free vibration p r o b l e m o f spinning structural systems has been primarily focused on spinning beams or cantilevers (e.g., references [1-3]), on spinning disks (e.g., references [4-6]), a n d on spinning plates (e.g., reference [7]) which have applications to helicopter blades, saw blades, and turbine blades, respectively. The free vibration study o f spinning structures for use in space is also o f recent interest (e.g., reference [8]). A knowledge o f the free vibration characteristics o f a structure is important to the u n d e r s t a n d i n g o f its behavior and to avoid d a m a g i n g resonance between its natural frequencies and excitations in the vicinity o f the structure. Free vibrations o c c u r in the absence o f external loads a n d therefore the m o t i o n o f a spinning structure can be represented as a set o f h o m o g e n e o u s second order quasi-linear "l-The work described herein was sponsored by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration (NASA). 279 0022-460X/86/230279+18 S03.00/0 9 1986 Academic Press Inc. (London) Limited

280

w.L.

S H O E M A K E R A N D S. U TK U

ordinary differential equations such as Mi~+C~+Ke=O,

(1)

where M, C, and K represent the mass, gyroscopic, and stiffness matrices respectively and c represents the generalized co-ordinates measured from the steaOy-state or equilibrium position. For the free vibration problem involving only axisymmetric deformations, the gyroscopic forces are absent so that the problem reduces to Mi~+ Kc = 0.

(2)

Observing that each of the normal modes of free vibration executes a single harmonic motion with an associated natural frequency to, one can write c = y e i~'',

j = x/'2"T.

(3)

Substituting expression (3) into equation (2) gives Ky = to2My

(4)

which is a general eigenvalue problem. The distinction between linear and non-linear axisymmetric vibrations is whether or not the coefficient matrices given in equation (2) include the geometric non-linear contributions associated with large deformations. The non-linear problem of axisymmetric vibrations can also be solved by using equation (4) except an iterative scheme must be used (e.g., reference [9]) since the coefficient matrices are a function of the eigenvector y. The method of obtaining the discrete equations of dynamics given by equation (1) is the Rayleigh-Ritz procedure in which a trial function, composed of undetermined functions of time (generalized co-ordinates c) and known yet unspecified co:ordinate functions of the two independent spatial variables, is introduced into the principal functional of dynamics and Hamilton's Principle is applied to yield Lagrange's equation of motion. The complete procedure and the resulting non-linear coefficient matrices are detailed in [I0] and [I 1] while only the linear terms needed for this numerical study will be presented here. The following special notation is utilized in this paper. All numerical vectors, including the description of free vectors in an appropriately selected co-ordinate system, are shown by lower case bold face type characters. Matrices with more than one column are shown by upper case bold face type characters. Light face type characters are used for scalars. The transpose of a matrix is denoted by superscript T. Repeated indices imply summation over the range of the index unless otherwise noted, and a comma in the subscript implies differentiation with respect to the quantity (or quantities) following the comma. A dot over a variable represents differentiation with respect to time. 2. GEOMETRY OF THE PARABOLOID In the formulation presented in references [10] and [11], the paraboloid could be offset translationally and rotationally from an inertial reference frame but in this application, the global reference system of the paraboloid is assumed coincidental with the inertial reference frame. In this case, four right-handed orthogonal curvilinear co-ordinate systems are needed as shown in Figure 1. The I system is the global reference system with its origin located at the apex of the paraboloid. The X and Y axes define the tangent plane at the apex of the paraboloid and Z (unit vector i3) corresponds to its axis of revolution. The paraboloid spins about its axis of revolution at a constant angular velocity/~. The J system is the local co-ordinate system with its origin located at a generic point on the

FREE VIBRATIONS OF S P I N N I N G PARABOLOIDS

~

t

l

Z

la

t

281

[2

Figure 1. Geometry and co-ordinate systems.

middle surface of the perfect unstressed paraboloid. The K system represents the local system after being relocated by some known initial imperfection vector u' and the L system represents the local system in the final imperfect stressed position relocated by the unknown displacement vector u. The unit vectors in the J, K, and L local co-ordinate systems are defined as follows. The first unit vector (j~, k~, or l~) tangent to the surface at the corresponding generic point, is in the meridional plane, and is directed in the increasing direction of angle q5 as denoted on Figure 1. The second unit vector 02, k2, or 12) is tangent to the surface at the same point, is orthogonal to the first unit vector, and is directed in the increasing direction of angle 0. The third unit vector (j3, k3, or 13) is coincident with the normal direction of the surface at the same point and directed toward the center of curvature. 3. RAYLEIGH~

PROCEDURE

The method of replacing the variational problem defined by Hamiiton's Principle of dynamics for a conservative system with an ordinary extremum problem by using the Rayleigh-Ritz procedure was given in reference [I0] and will be summarized here. The objective is to find the time dependent elastic deflections w(qS, 0, t) caused by prescribed conservative force and moment magnitudes q acting at a unit area of unstressed imperfect middle surface related with the directions of the K system. Let q be denoted by qV= [ql, q2, q3, q4, qs],

(5)

where qt, q2, q3 are distributed prescribed force magnitudes acting in the kt, k2, and k3 directions respectively and q4, q5 are distributed prescribed moment magnitudes acting about k~ and k2, respectively. The unknown deflections associated with these prescribed forces will be denoted by WT = [H, 1), W,

w,o/r2,-w,~/R,],

(6)

where u, v, w are the components of u in the k~, k2, and k3 directions, respectively. Also note that R~ is the principal radius of curvature in the meridional direction and r~ is the parallel radius. 3.1. HAMILTON'S VARIATIONAL PRINCIPLE The equation of dynamics and the associated boundary conditions can be derived using Hamilton's Principle for a conservative system given by

8

(U+T-V)dvdt=O, to

(7)

282

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S H O E M A K E R A N D S. U T K U

where 8 is the first variation, to and tf define the time interval [to, ty] for a given initial and final configuration, ~, indicates volume, and U, V, T are the densities of the strain energy, loss of potential of prescribed forces, and the kinetic energy, respectively. 3.2. A D M I S S I B L E T R I A L F U N C T I O N Let the trial function be denoted by ~ ( f f , 0, t) which can be expressed in terms o f undetermined functions of time c(t) and known yet unspecified co-ordinate functions o f space ~(qS) and 0 ( 0 ) that are admissible in the solution domain. Admissibility is governed by sufficient smoothness and satisfaction of the essential boundary conditions which are the constraints involving the components listed in equation (6). The admissible co-ordinate functions may range from smooth continuous functions (conventional global l~ayleighRitz method) to piecewise continuous functions of class C i, i = 0, 1 , . . . (finite element form of Rayleigh-Ritz method). Let m,, and m/3 denote the number of co-ordinate functions per component or degree of freedom listed in equation (6) for ~(qS) and O(0) respectively. The total number o f unknown functions of time and the total number of co-ordinate function products for all degrees of freedom is n = 5m where m = m,~mD. One can now define the trial function as

co.(t)~o( ,~)o ~(o)

v(,b, 0, t u(4,, 0, t)t)

,

w= w,o(~,o,t )

t

,

2 ::)O ~( 2 O) c 2~ ( t)O,,( 3 3 3 catj(t)cI"~(~)OD(O)

w(fb, O, t)

=~(,/,,0, t)= c~(t)~'~(g,)o'~(o)

ct = 1 , . . . , m~,

/3=1,...,m~

(8)

r2 - w ~ (~ , 0, t)

5 ( Co

5

o)

where the superscripts will denote the degree of freedom in the order listed by equation (6) and the ranges of a a n d / 3 will henceforth be as defined above. Note that the trial function represents an approximation to the actual response w and the accuracy obtained is dependent on the type and number of co-ordinate functions chosen. The ordering of c can be taken as a parameter of the discretization but to obtain the coefficients needed in the discretization of the energy density expressions in a more concise form the entries of c will be ordered partitionable with respect to modes (conventional Rayleigh-Ritz terminology or nodes (finite element terminology). The complete list of the unknown functions of time can be written as CT = [ClTh t I T 2 , . .

T 9 9 9 C,,.,,~], T 9, %D,

(9)

where the subscripts indicate the sequential partition node number which is defined by 3/= (c~ - 1)m D+/3,

(10)

where T = 1. . . . , m and the a/3th partition of c is defined as T

!

2

3

4

5

e~, = [c~/3, c~,, C~D, c,~,, cat3].

(11)

With the entries of c,,, indicated by their sequential numbers in c, T

CoS =

['C5T--4, Csy--3, C5T--2, C5y--l, C5y].

(12)

Let F denote the complete listing of the co-ordinate function products (i.e., ~,~(qS) Ot3(O)) ordered in the same manner as e as r

T

T

T

T T ., r~o~,] r~,..

(13)

FREE VIBRATIONS OF SPINNING PAKABOLOIDS

283

and, similar to that in equation (11), the aflth partition is =[q)aO~,q).O~,

3 ~b.,O~, ,~ q).,Ot3], 5 5 q ~3 Or3,

(14)

and sequentially F,,~)T_--[F5~,-4, F5~,-3, F5~,-2, Fs~,-,, r5~,].

(15)

4. RESULTS OF RAYLEIGH-RITZ PROCEDURE The substitution of the admissible trial function into Hamilton's Principle (7) through the energy expressions yields Lagrange's equation of motion for a conservative system in terms of the unknown functions of time c (or generalized co-ordinates) as O,c+ 9,~- T,c+ (d/dt) T,,e= 0,

(16)

where a bar above a variable represents integration over the volume. This is a set of second order quasi-linear ordinary differential equations which may be restated as [M(c)]~+ [C(c, 0 ] ~ + [K(c)]c = p, (17) where the three terms on the left represent inertial, gyroscopic, and restoring forces respectively and p represents the loading of the system. In order to produce linear equations of dynamics with respect to the generalized co-ordinates, it can be seen from equation (16) that one needs to obtain expressions for the energy densities to the second degree prior to differentiation. 5. ENERGY DENSITY EXPRESSIONS 5.1. STRAIN ENERGY DENSITY ( U ) It is assumed that the paraboloidal material is linearly elastic such that the constitutive equation is tr =D'E, (18) where ~r is the list of independent stresses at a material point denoted by (19)

o ' T = [O'll, 0"22, 0"33, 0"12, 0"13, 0"23]

and e is the corresponding list of independent strains denoted by (20)

I~T----"JEll, e22, E33, ")/12, "Y13, "Y23].

Matrix D' is the material matrix which is real, symmetric, and positive definite. The 0nl)~ requirement of the material properties is that they be symmetric with respect to the tangent plane of the middle surface (i.e., orthotropic) at the material point which was suggested in reference [12] to remove the inconsistency of Love's postulate that both the transverse normal stress and normal strain vanish. Therefore, the matrix D' may be displayed for this general case as d~2 d~2 D'= Sym

d[3 d~3 dh

d~4

0

0

d,~4

0

0

d~4

0

0

d•

0

0

d~5

d~6 d~6

(21)

284

w . L . SHOEMAKER AND S. UTKU

The complete strain tensor is denoted as E and is defined by E! l E=

El2 E22

Sym.

El31

e23/ , E33.]

(22)

where e12-- 3/12/2, el3 = 3/13/2, and e23 = 3/23/2. Note that the indices of the components of the strain tensor and corresponding stresses (i.e., 1, 2, 3) refer to the directions of kt, k2, and k3, respectively. The following K_irchhoff assumptions for thin shells are now imposed. The transverse normal stress tr33 is negligible and the two transverse shear strains 3'~3, 3'23 are zero if one assumes that normals to the middle surface remain normal after deformati6n. The modified lists of stress and strain components are now given as o ' T = o"T~-'~ ['0"11, 0"22, 0, O"12, O"13, 0-23)]

and

e T = I~'T= JEll, E22, E33, 3"12, 0, 0"1.

(23, 24) The strain energy density U can be defined by U= 89

"',

U=89

or

(25,26)

Using equation (18) and a reduced material matrix D due to the assumption that 0-33=0 yields U = 89eTD e, (27) where E T = [ E I I , E22,3'I2]

and

D = D ' - (1/d33)d3d , , 3,T,

(28, 29)

where d~ represents the third column of D'. Note that although E33 is not zero, it does not contribute to the strain energy density since 033 which is to be multiplied by eaa is zero due to the negligible transverse normal stress assumption. The'strain energy density may be defined in terms of the elastic deflections w by means of the strain-displacement relations. 5.2. KINETIC ENERGY DENSITY ( T )

Let p denote the unit mass of the paraboloidal material; then the kinetic energy density T may be expressed as T=~pvTv, (30) where v is the description of the particle velocity in the I co-ordinate system. The kinetic energy density may be defined in terms of the elastic deflections w by means,of the velocity-displacement relations. : 5.3. DENSITY OF THE LOSS OF POTENTIAL ENERGY ( V ) OF CONSERVATIVE LOADS

Although not needed for the free vibration problem, the loss of potential energy of the conservative loads q may be defined as V = --(1/h)wTq,

(31)

where h is the paraboloidal thickness. 6. STRAIN-DISPLACEMENT RELATIONS General expressions for the strain-displacement relations including the effects of initial imperfections have been given in reference [10]. The explicit terms derived from these

F R E E VIBRATIONS OF S P I N N I N G P A R A B O L O I D S

285

general expressions have been presented in reference [11]. Note that, from equation (26), only the three strain components, en, e22 and Y12, are needed for the strain energy density U. Also, since no initial strains are present, the strain e will be of degree one and higher with respect to the elastic deflections. Therefore, only linear terms of e will b e needed here to yield U containing the required second degree terms. Since the terms also contain initial imperfections u' and the arclength ~" measured along J3, k3, or 13 (i.e., the distance of a material particle from the middle surface), an order o f magnitude assumption is needed to decide which terms to retain in the strain-displacement relations. It Will be assumed that the elastic deflections and initial imperfections are of the same order o f magnitude and both are up to one order o f magnitude greater than the paraboloidal thickness. Each time a term is multiplied by an additional u, u', or ~" it is reduced ifi magnitude because it is also divided by a radius of curvature. Since Kirchhoff's assumption has been imposed with respect to linearly varying strains across the thickness of the paraboloid, the strain can be written as e = e ~ et~".

(32)

Defining the arclength sr as zero at the middle surface of the paraboloid yields a volume integration across the thickness from - h / 2 to + h / 2 . Therefore, energy terms with odd powers of ~" will vanish and only zero and even powers of ~" in the strain energy density will contribute to the total strain energy. These terms will emerge from the products U~ = 89176176162176 U' = 89

(33)

Since the latter of the two equations (33) contains one "order of smallness" due to ~.2, this implies that the largest contributing term of U ~ would be comparable to a third degree term. Therefore, with the order of magnitude assumptions introduced, the second degree terms of e ~ are o f the same importance as the first degree terms of e 1. However, in this study only the first degree terms of e ~ along with the first degree terms o f e I will be included. This gives rise to the following strain-displacement relations: e~ = ki(u,+ - w),

e~t = k21(3u tan 4 - 2 u , + + w ) + kl(w~.+),

e ~ = k2(u cos 4 + V,o - w sin 4),

(34-35) (36)

e~2 = k 2 [ - u ( k l cos 4 + k2 sin 4 cos 4 ) -2V, o(k2 sin 4 ) + w(k2 sin 2 4 ) - wo.o + w+(cos 4)], (37)

T~ = k,(v,e,)+ k2(u,o - v cos 4), 712=2k2[-k,(u,o)+v(k2sin

4 cos 4 ) - v , + ( k l

sin 4 ) + w o cos 4 + w+.e].

(38) (39)

Note that k, = 1 / R , , wo = w,o/r2,

k2 = 1/r2,

(40)

',,re, = - w , ~ / R l .

(41)

7. VELOCITY-DISPLACEMENT RELATIONS The non-linear velocity displacement relations which are needed to express the kinetic energy in terms of the elastic deflections were derived in reference [10] and explicitly presented in reference [11] up to and including fourth degree non-linearities. To summarize here, if one defines a position vector b in the J co-ordinate system, then the velocity

286

W.L.

SHOEMAKER

AND

S. U T K U

vector v in the I system is given by v =S(b+.Oi~ • b),

(42)

where J is the co-ordinate transformation matrix from J to the I s_y.stem and i~ is the description of i3 in the J system. The position vector b can be expressed in a similar manner as the list of strains with respect to the arclength ~" as b=b~

(43)

In this case, b ~ and b 1 contain zero degree (constant) terms and higher. Due t,a the presence of these constant terms, one needs to include second degree terms in thgposition vector to recover all of the second degree terms in the kinetic energy expression (30). This means that second degree terms relating to the meridional shortening due to bending must be included in the descriptions of the imperfection vector u' and the elastic deflections vector u. A similar discussion of foreshortening effects in rotating beams was presented in reference [13]. When the same order of magnitude assumptions are used for the strain displacement relations, b~ needs to contain second degree terms and b ~ need only contain first degree terms. Denoting the components of b as b,, b2, and b3 in the it, J2, and J3 directions, respectively, one has b~ = 2f(sin 4~)(1 + 89

-~!

2 q~)+ u ' - ~

[(k,w~)2]R, dq~+u

fo'

i [(klw, e)2]R, d~ - v(k,v~,)w[kl(u'+ w~)],

bl=k~(-u'-w' -u+w,~),

b~=v'+u(k~v',)+v-wk2(w'o+v'sinr~),

(44) (45-46)

b~ = -k2(v' sin ~ + w~o+v sin 4~+ w,o), b~=-f(sin4tan4)+w'+uk~(u'+w~,)+V[kE(V' sinr~+w~o)]+w,

(47)

b~=l, (48, 49)

where f is the focal length of the paraboloid and u', v', w' are the components of the imperfection vector u' defined in the J co-ordinate system. 8. EXPRESSIONS FOR GENERALIZED STRAIN AND POSITION VECTOR 8.1. G E N E R A L I Z E D STRAIN Using the admissible trial solution (8) in the strain displacement relations, one can define the strain e given by equations (34)-(39) in terms of the generalized co-ordinates C as

e = ~ e22~ = Aoc+ O(2),

(50)

I "Yi2J where 0(2) represents terms of second degree and higher which are not included in this study. The Ao matrix is 3 • n (3 rows by n columns) and contains m partitions as does e in equation (9) such that Ao = [(Ao)a,, (Ao),2,..., (Ao),,~,..., (Ao)r,:~].

(51)

FREE

VIBRATIONS

OF SPINNING

PARABOLOIDS

287

The general partition (Ao)~a can be separated into two components as AGe = [(Ao~176 + (A~)~'11%~.

(52)

In the notation of equation (8), the aflth partitions of the coefficient matrices for the axisymmetric case given by equation (52) are given in Figure 2.

[_ k,'r

1

2 1 3k1~,,6) aI t a n

q~

I

-klk2tlb~e~ COS q~

=

II

0

l] 0 ,

r~

[

(ADo

I

_ [ . 2 d,i I /'~1

0

II 0

_/,.'~1"~" t.h3/~3 a v/3

r--r--

/.,.2r1~3 /'~3a n. I "ur a ,,.,,

0 [

01

.,.-k2~ae~slnqs,P_t 0/

1

T----

I I I

2"*-,~,-,~ sin 2 ~b

I I

5

I

~ 0

5

0 I k2~atgt3 cos I

I I

551

kl~,,., O~

1----

,~2~a,-'~ sin q~ cos ~b I 0

II

--r-~'-'

0 ~

0

F i g u r e 2. a f l t h p a r t i t i o n s o f Ao c o e f f i c i e n t m a t r i c e s .

8.2.

GENERALIZED

POSITION

VECTOR

Similarly, using equation (8) in the description of the position vector b given by equations (44)-(49), one can define a generalized position vector as b=

b2 =bo+Boe+ciB~r b3

(53)

where bo, Bo, Bi represent the constant, linear and quadratic terms with respect to the co-ordinate functions, respectively. As with the strains, the linear term can be written as Boco = [(Bo~176

(Bol)a~'t]c.~

(54)

and the quadratic term can be written as ciBi c =

o o + (Bs~-k),,S3r l 1]%~(no CS~-R[(Bs~-k)~

sum on y, k),

(55)

where y = 1, 2 , . . . , m, and k = 4 , 3, 2, 1, 0 corresponds to i = 1, 2 , . . . , n. Note that B~ is also 3 x n and contains m partitions such that B, = [(B,),,, (B,),2, 9 9 9 ( B , ) ~ , . . . , (B,)m.m~].

(56)

Vector bo is given by 2f(sin q~)(1 +89tan 2 40 + u'- 89

b~176

v'

Io'

[(klw~,)2]R, d~

- f s i n ~b tan q~+w'

f The aflth

1

[' J

1

partitions of Bo and B~ are given in Figures 3 and 4.

288

W. L. SHOEMAKER AND S. UTKU

(B~

=

O,,Ot3'' II 0 Ii _kl~b30~(u,+ w:,) , , , kl 1 i , __t O__t - k 2 cl) ,~ O . o ( w . o + v ' sin ~b) 1 I t I "3 3 k , ~ o O ~ ( u 9 +w~,) I 0 i O~Op

f

o_L 0_] oi-61 ,,--o-j

- k ~ O ~ O ~t l

I I "n tI vN tI "n tI ' ~d~St'~5"] ,,'"~l " ___.1___1 r - - - r - - 70- - - .r.l.0. .. . . . . . . 0. | I " j Io I 0 It 0 II 0 ] Figure 3. aflth partitions of Bo coefficientmatrices.

(B~)~=

0"

0 ', 0 ', 0 ', 0 ___l___t.__ __L (B~

=

0

I 0

I

0

Ii 0

o,,olo:o

-!2

f~

[(Fs.l(q,S.OS~)]Rtdr 0

o

1

Figure 4. aflth partitions of Bi coefficientmatrices. 9. THE DISCRETIZED EQUATIONS OF DYNAMICS Substituting the expression for the strain energy (27) into the strain energy term of equation (16) gives O.,= ~TDE. (59) The derivative of the generalized strain (50) with respect to c is e c = Ao.

(60)

Substituting equations (60) and (50) into equation (59) produces the strain energy term in terms of the generalized co-ordinates as IJ x = (AoTDAo)c.

(61)

The kinetic energy given by expression (30) can be expressed by using the velocitydisplacement relation (42) as T = 89p (l~+ .OTb)T(I~+ .QTb ).

(62)

Note that in equation (62) the cross-product has been replaced by a skew-symmetric matrix T where T = cos ~b 0

0

- s i n 4,

s i n 4'

0

(63)

and that since J is an orthogonal matrix, j x j is an identity matrix. Substituting the generalized position vector b given by equation (53) into the kinetic energy terms of equation (16) gives ( d / d / ) T~- T,, = p(BTBo)e+ ap[BoTTBo- (i3TTBo)T]~ - n2p [BTTTTBo + (iibTTTTB~ + (i~bTTrTB,)T)]e. (64) A complete derivation of these terms including up to cubic contributions can be found in reference [11]. Note that index i varies from 1 to n and ii refers to the ith column of an n order identity matrix.

F R E E VIBRATIONS OF S P I N N I N G P A R A B O L O I D S

289

It can be shown that the second term of the fight-hand side of equation (64) will vanish when only axisymmetric deformations are considered. The remaining terms can be grouped such as in equation (2) as follows: M = pBoTBo,

K = AoXDAo- f22p [BTTTTBo+ (i,bTTTTBi + (i,bTTTTBF)T)].

(65)

The coefficient matrix M of the inertial forces is due to mass of the material and emerges from the translational kinetic energy of a material point. It is real, symmetric and an nth order matrix. The coefficient matrix K of the restoring forces is composed of the first term which is associated with the material strain energy and other terms associated with the centrifugal component o f the kinetic energy. All of these stiffness terms are real, symmetric and are nth order matrices. 10. NUMERICAL STUDIES The basic matrices that need to be generated for the linear, axisymmetric free vibration study are given by equation (65). The active degrees of freedom for the axisymmetric case are u, w, and -w, d R ~o f those listed in equation (6). The axisymmetric deformations considered are due to transverse and meridional motion. The torsional modes whose deflections are of the form u =0, w = 0 and v = v ( ~ ) are excluded in this study. Continuous co-ordinate functions were chosen for this study because the boundary conditions at the fixed apex could be easily satisfied, and because one would expect fewer modes are needed for convergence than finite element nodes needed to define sufficiently the surface of the paraboloid. Also, if one uses continuous functions, the.transverse displacement w and the rotation about the k2 direction -w,~/R~ can be readily combined into one degree of freedom. The reason for separating the rotations in the formulation is done with piecewise co-ordinate functions in mind. The co-ordinate functions in the meridional direction were chosen as g'~ = \

~

4,0 /

--- 0,

~

~5 = -kl

= 0, 3 ~,,.~,

\

4o/

'

a = I , . . . , nl~,,

(66)

where ~ba is the minimum angle of ~b and ~bo is the maximum angle of ~b. The use of q~a allows for the possibility of a truncated paraboloid or one with a rigid hub at the apex. The boundary conditions of a fixed inner boundary (i.e., u = w = 0, w,~ = 0) at ~b = ~b, is satisfied by expressions (66). For the circumferential direction, it can be considered that a Fourier cosine series is used such that

O~=cos(fl-1)O,

f l = l . . . . . ms,

r / = l , 2 . . . . . 5,

(67)

where ms is taken as one for the axisymmetric case. Each entry of matrices K and M from each contributing basic matrix must be integrated over the volume o f the antenna which for this case involves only an integration in the meridional direction. The transverse direction integration is handled through the separation of basic matrices into the powers of g" involved. A standard Simpson's rule algorithm was used to evaluate the meridional integration and a study o f the effect of the number o f volume integration stations used was first undertaken. For this case, m,~ = 5 and the spin rate was taken as zero. The results of this volume study are reported in Table 1. For this and all subsequent cases, it was found that the paraboloid exhibits a similarity to the natural frequencies o f a spherical shell with free edges (e.g., reference [14]) in which the frequencies arrange themselves into two groupings commonly referred to as the upper

290

W. L. SHOEMAKER AND S. LITKU TABLE 1 Effect of volume integration accuracy on natural frequencies ~ t

Number of volume stations 10 15 20 25 30

Mode ^ B1 0"05339241 0"05301253 0"05293423 0"05291236 0"05290457

B2

B3

M1

M2

M3

0"45219843 0"45274152 0"45282094 0"45284128 0"45284831

0"46879241 0"46933610 0"46942574 0"46944931 0"46945754

2"1732151 2"1733064 2"1733280 2"1733342 2"1733365

5"4547997 5"4537856 5"4536479 5"4536140 5"4536025

8"7004163 8"6808671 8"6781303 8"67744~9 8"6712157

t ~ = tor,~a~,/--op/E,and, for this study, m,, = 5, h~ rm~ = 0'00l,/-/= 0, v = 0"3 (Poisson's ratio), and rr~Jf = 1. b r a n c h and lower branch. The lower b r a n c h is m a d e up o f an infinite n u m b e r o f natural frequencies whose m o d e s are o f a bending character. The u p p e r b r a n c h is made up o f an infinite n u m b e r o f natural frequencies whose m o d e s are o f a m e m b r a n e character. These characteristics will be discussed further and are alluded to n o w for the identification o f the m o d e s in the studies. The bending modes will be labeled B1, B 2 , . . . a n d the m e m b r a n e m o d e s will be labeled M1, M 2 , . . . , as referred to in Table 1. Note that the n u m b e r o f v o l u m e stations refers to the n u m b e r o f breakpoints and that the functions are also evaluated at the midpoints in the classic Simpson style. The frequencies in Table 1 are n o n - d i m e n s i o n a l i z e d quantities as noted. For the remainder o f this study, it is felt that, based on these results, 20 volume stations is an adequate representation and correspondingly, only four significant digits are carried throughout. It is :interesting to note that the first b e n d i n g m o d e frequency B1 was a p p r o a c h e d from aboye,'with increased volume integration accuracy and the s e c o n d and third b e n d i n g f r e q u e n c i e s were a p p r o a c h e d from below, while the converse is true for the m e m b r a n e frequencies. The convergence o f the natural frequencies versus the n u m b e r o f m o d e s taken in the p o w e r series co-ordinate functions was examined and is summarized in Table 2 and Figure 5. T h e first b e n d i n g m o d e B1 or lowest natural frequency was the slowest to converge. Results with more than 10 modes required m o r e accurate v o l u m e integration to retain the linear i n d e p e n d e n c e o f the additional modes. W h e n the modes b e c a m e linearly d e p e n d e n t within the c o m p u t e r precision, the mass matrix M was no longer TABLE 2 Convergence o f natural frequencies ~ when power series co-ordinate functions were used Number of modes m,~ 1 2 3 5 7 9 10

Mode ^

r B1

B2

B3

M1

M2

M3

0"3540 0-1658 0.1016 0.0529 0.0338 0.0248 0.0220

-0.6099 0.4591 0.4528 0-4521 0.4517 0.4517

--0.6752 0.4694 0-4663 0.4641 0.4637

2"276 2.147 2.164 2.173 2.176 2.177 2-177

-6-391 5.569 5.454 5.455 5.455 5.456

--11.05 8.678 8.652 8.652 8.652

h~ rmax = 0'001, .O = 0, V= 0"3 (Poisson's ratio), rmaJf = 1.

291

F R E E VIBRATIONS O F S P I N N I N G P A R A B O L O I D S 0.7

i(a),

i

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i

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3 4 5 6 7 8 Number of modes, m a

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1

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3

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Figure 5. Convergence of frequencies o3 when power series co-ordinate functions were used with h/rma~ = 0.001, .O = 0 , ~,=0.3 (Poisson's ratio), r m ~ J f = 1. (a) Bending B~; - - - , bending B2; . . . . , bending B3; (b) m e m b r a n e Ml; - - - , m e m b r a n e M2; - - - - - , m e m b r a n e M 3.

positive definite. Therefore, there is an upper bound on the number of modes that can be used in this method based on the computer precision available. This same type of loss of linear independence was reported in reference [15] when path derivatives were used as basis vectors in a Rayleigh-Ritz method; the linear dependence was continuous monitored to terminate the generation of basis vectors when necessary. Plots of the displacement patterns for the first three bending and membrane mode sfiapes are shown in Figure 6 for m,~ = 9. Clearly, the motion associated with the bending modes is predominantly transverse, while the motion associated with the membrane modes is predominantly meridional. It is also seen that the number of nodes (points of zero displacement) of it and w increase by one for each successive bending mode as would be expected. This displacement pattern is also repeated for the membrane modes. The addition of spin .O alters the stiffness due to the centrifugal effect. Of the two terms inside the bracket of equation (65), for the axisymmetric case, the first represents a softening effect related to membrane and bending stiffness respectively. The second represents a stiffening effect. It was found in reference [11] that it is essential to include the second degree meridional shortening effect in the description of the position vector to recover this stiffening effect. A study of the effect o f the bending rigidity of the shell and o f the effect o f the spin rate is presented in Table 3. When h/rm~x is taken as zero, the bending resistance is neglected and results would be comparable to a membrane theory solution. The spin rate is represented by a non-dimensional q u a n t i t y / 7 which is defined as J'~ = ~('2rmaxp / E.

(68)

The effect o f the bending rigidity on the bending and membrane modes is shown in Figure 7. It is seen that the bending modes are appreciably affected by the thickness of the shell with the variation increasing with each successive mode. However, the membrane modes rapidly lose variation after the first mode and are unaffected by changes in thickness.

292

W. L. S H O E M A K E R

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Figure 6. Mode shapes ( - - - , u; , w) for non-rotating paraboloid with m,, =9, h/rma~=O'O01, 1-1=0, 1'=0"3 (Poisson's ratio), rm~/f= 1. (a) Bending Bt; (b) bending B2; (c) bending B3; (d) membrane MI; (e) membrane M2; (f) membrane M3. This p h e n o m e n o n is characteristic of membrane modes as reported in reference [14] for spherical shells. The effect o f the spin rate on the bending and membrane modes is s h o w n in Figure 8. The increase o f spin rate shows the same trend as the increase o f thickness on the bending and membrane modes. N o results were available in the literature for the axisymmetric free vibrations o f a spinning paraboloid fixed at the apex. In reference [16], the equations o f motion for a paraboloid were presented in partial ditterential form and a method of solution using finite differences was suggested although no results were included. However, since results are available for spinning disks, it was found that by letting the curvature tend to zero as the focal length o f the paraboloid goes to infinity, a comparison could be made. For the non-rotating disk fixed at the center, exact results were published by Lamb and Southwell [4] and excellent agreement with the axisymmetric bending mode frequencies

293

FREE VIBRATIONS OF SPINNING PARABOLOIDS TABLE 3

Effect of bending rigidity and spin rate on natural frequencies (J Mode

h~ r,~x

B1

B2

B3

B4

B5

M1

M-Z

M3

0"00

0"000 0"001 0"005 0"010

0"0270 0"0285 0"0438 0"0608

0"4490 0"4519 0"4563 0"4601

0"4569 0"4650 0"4799 0"5018

0"4683 0"4792 0"5074 0"5825

0"4815 0"4901 0"5566 0"7370

2"176 2"176 2"714 5"256

5"455 5"455 5"455 5"456

8"652 8"652 8"652 8"652

0"01

0"000 0"001 0"005 0"010

0"0358 0"0367 0"0479 0"0630

0"4506 0"4529 0"4570 0"4608

0"4594 0"4671 0"4821 0"5044

0"4721 0"4830 0"5134 0"5883

0"4873 0"4990 0"5690 0"7465

2"176 2"176 2"717 5"257

5"455 5"455 5"455 5"456

8"652 8"652 8"652 8"652

0"05

0"000 0"001 0"005 0"010

0"0904 0"0905 0"0922 0"0966

0 " 4 6 9 8 0"5054 0"5484 0 " 4 7 0 1 0 " 5 0 9 3 0"5639 0"4726 0"5287 0"6166 0 " 4 7 6 7 0 " 5 5 7 1 0"6967

0"6140 0"6399 0"7543 0"9263

2"178 2"178 2-801 5"296

5"455 5"455 5"455 5"457

8"652 8"652 8"652 8"652

0"10

0"000 0"001 0"005 0"010

0"1427 0" 1428 0"1433 0"1448

0"5167 0"6211 0"7423 0 " 5 1 6 8 0 " 6 2 3 8 0"7593 0 " 5 1 8 9 0 " 6 4 6 1 0"8283 0 " 5 2 3 2 0"6809 0"9190

0"8934 0"9274 1"068 1"247

2"683 2"693 3"203 5"423

5"455 5"455 5"455 5"466

8'652 8"652 8"652 8"652

m, = 8, ~,= 0-3 (Poisson's ratio), rm~/f= 1. ()a . . . .

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Figure 7. Effect of bending rigidity with m,~= 8, D = 0, v = 0.3 (Poisson's ratio), rmaJf = 1. (a) Bending Bl; - -, bending B2; - - - --, bending B3; . . . . , bending B4; , bending Bs; (b) membrane M~, - - -, membrane Mz; . . . . , membrane M3. -

o b t a i n e d b y u s i n g the present f o r m u l a t i o n is s h o w n in the first row of T a b l e 4. T h e m e m b r a n e m o d e s were n o t calculated i n reference [4]. F o r a s p i n n i n g disk fixed at the c-enter, L a m b a n d Southwell e m p l o y e d a n a p p r o x i m a t e t e c h n i q u e to b o u n d ' t h e lowest frequency. A n exact a p p r o a c h was used in reference [6] to o b t a i n transverse n a t u r a l frequencies for a s p i n n i n g disk with an infinitely stiff h u b o f v a r y i n g radius d e n o t e d here by rap~,. Very good a g r e e m e n t is s h o w n in T a b l e 4 for all h u b r a d i u s / m a x i m u m radius

294

w . L. S H O E M A K E R

AND

S. 1ATKU

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66666

66

66

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0

66666

66

66

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66

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VIBRATIONS

OF

SPINNING

0'7

295

PARABOLOIDS

9

i "O

(b) 8

/

O6 13

~6 ~0.5

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~

gO'4 ~O2' g

~4

*O

O

G~

Z

2 0.1 1 i

i

i

i

I

i

i

i

0.05 Non-dimensionalspinrote,.t~

i

I

0.10

1

o

,

,

,

I

J

,

005

I

0-~0

Non-dimension~ spinrote,

Figure 8. Effect of spin rate with m~ =8, h/rm~=O'O05, v=0,3 (Poisson's ratio), r~,a~/f= 1. (a) Bending Bl; - - - , bending B2; - - - - - , bending B3; (b) membrane MI; - - - , membrane M2; - - - - - , membrane M3. ratios and spin rates used. The results from reference [6], although obtained by an exact method, were read from curves; therefore they are intended to show trends in the results in Table 4 rather than to show a comparison of the approximate results obtained in this study to " e x a c t " results obtained in reference [6]. The non-dimensional spin rates used correspond to curves provided in reference [6]. 11. CONCLUSIONS The discrete linear equations of dynamics for spinning paraboloids in a form suitable for computer implementation have been presented. The equations of dynamics were given as a set o f second order quasi-linear ordinary differential equations. The independent variable is a vector o f unknown functions o f time and the coefficients are thus given as matrix partitions. The formulation can be extended to include non-linear effects by including non-linear terms in the strain-displacement and velocity-displacement relations [11]. The Rayleigh-Ritz method was used to discretize the equation of dynamics whereby an admissible trial solution consisting of unknown functions o f time and known yet unspecified co-ordinate functions was introduced into the principal functional o f dynamics and Hamilton's Principle was applied yielding Lagrange's equation of motion. A special purpose computer program was developed to demonstrate applicability of the formulation to the linear axisymmetric free vibration problem o f a spinning paraboloid fixed at its base. Natural frequencies and m o d e shapes are provided including the effect of spin rate and bending rigidity. Results for a spinning disk were also obtained and compare favorably to those found in the literature. REFERENCES 1. M. EL-ESSAWI 1982 Doctoral Dissertation, Duke University, Durham, North Carolina. A discrete model for nonlinear structural dynamics of rotating cantilevers. 2, G. L. ANDERSON 1975 International Journal of Non-linear Mechanics 10, 223-236. On the extensional and flexed vibrations of rotating bars.

296

w. L. SHOEMAKER AND S. UTKU

3. M. J. SCHILHANSL 1958 American Society of Mechanical Engineers. Bending frequency of a rotating cantilever beam. Journal of Applied Mechanics, 25, 28-30. 4. H. LAMB and R. V. SOUTHWELL 1921 Proceedings of the Royal Society of London, Series A 99, 272-280. The vibrations of spinning disk. 5. J. PRESCOTT 1946 Applied Elasticity. New York: Dover Publications. 6. W. EVERSMAN and R. DODSON, JR. 1969 American Institute of Aeronautics and Astronautics Journal 7, 2010-2012. Free vibration of a centrally clamped spinning circular disk. 7. M. A. DOKAINISH and S. RAWTAN! 1971 International Journal for Numerical Methods in Engineering 3, 233-248. Vibration analysis of rotating cantilever plates. 8. K. K. GUPTA 1973 International Journal for Numerical Methods in Engineering 5, 395-418. Free vibration analysis of spinning structural systems. 9. K. KANAKA RAJU and (3. VENKATESWARA RAO 1976 Journal of Sound and Vibration 44~ 327-333. Large amplitude asymmetric vibrations of some thin shells of revolution. 10. S. UTKU, W. L. SHOEMAKER and M. SALAMA 1983 Computers and Structures 16, 361-370. Nonlinear equations of dynamics for spinning paraboloidai antennas. 11. W. L. SHOEMAKER 1983 Doctoral Dissertation, Duke University, Durham, North Carolina. The nonlinear dynamics of spinning paraboloidal antennas. 12. F. B. HILDEBRAND, E. REISSNER and G. B. THOMAS 1949 NACA TN-1833. Notes on the foundations of the theory of small displacements of orthotropic shells. 13. K. R. KAZA and R. G. KVATERNIK 1977 American Institute of Aeronautics and Astronautics Journal 15, 871-874. Nonlinear flap-lag axial equations of a rotating beam. 14. A. KALNINS 1964 Journal of the Acoustical Society of America 36, 74-81. Effect of bending on vibration of spherical shells. 15. A. K. NOOR and J. M. PETERS 1980 American Institute of Aeronautics and Astronautics Journal 8, 455-462. Reduced basis technique for nonlinear analysis of structures. 16. J. T. WANG and C. LIN 1967 NASA CR-932. On the differential equation of the axisymmetric vibration of paraboloidal shells of revolution.