Systematic generation of nonlinear discretized dynamic equilibrium equations of spinning cantilevers

Compulcrs & Slrucfurcs Vol. IS. No. 3, pp. 251L282, 1982 Printed in Great Britain.

@W-7949/82/030259-24$03.00/0 Pergamon Press Ltd.

SYSTEMATIC GENERATION OF NONLINEAR DISCRETIZED DYNAMIC EQUILIBRIUM EQUATIONS OF SPINNING CANTILEVERS? M. EL-ESSAWI$ and S. UTKU§ Civil Engineering Department, Duke University, Durham, NC 27706,U.S.A. and

M. SALAMA~ Division of Applied Mechanics, Jet Propulsion Laboratory, California Institute of Technology, Pasedena, CA 91103,U.S.A. (Received 23 May 1981;received for publication 24 June 1981) Abstract-The Rayleigh-Ritz procedure, in conjunction with any admissible trial solution in terms of undetermined functions of time and known yet unspecified coordinate functions of space, is systematized to obtain the coefficient matrices of the second order nonlinear ordinary .dZerential equations of dynamic equilibrium of a spinning cantilever with initial geometric imperfections. In the functional second order nonlinear strain-displacement and velocity-displacement relationships are used, and the material is assumed linearly elastic. Systematic forms for the discretized energy density expressions are provided. The fact that the coordinate functions are represented parametricallyserves to unify the discretizationapproaches.The choice of the coordinate functions includes both sets of continuous and piecewise continuous functions employed in the conventional Rayleigh-Ritz method and the finite element method, respectively. A computer program for the systematic generation of the coefficient matrices involved in the governing equations is described. The degree of the nonlinearity, the type and number of the coordinate functions, the ordering of the vetor representing the complete list of the undetermined functions of time, the imperfection functions, and the time independent forcing functions are taken as parameters of the program for the complete simulation.

1. INTRODUCTION

In a recent paper by the authors [ 11, the general nonlinear discretized equations of motion of spinning elastic solids and structures were given explicitly as a set of second order nonlinear ordinary differential equations for the case when the strain-displacement and the velocity-displacement relationships are nonlinear up to the second order. These equations are quasi-linear, and the coefficients of the vector of the undetermined functions of time and its derivatives involve up to second degree nonlinearities. In the present paper these governing equations are specialized for the case of a spinning cantilever with initial geometric imperfections. The coefficients involved in the discretization of the strain energy density, the kinetic energy density, and the density of loss of potential of prescribed forces are given explicitly in a suitable form for the systematic generation of the governing equations. 2. DJSCRETIZATION OF ENERGY DENSITY EXPRRSSIONS

As illustrated in Fig. 1, four coordinate systems are used to formulate the problem. The (XYZ) system is an inertial system. Tha constant spin Cl is about the Z-axis.

Point A with coordinates (R,, 0, 0) in the (XYZ) system is the fixed point of the cantilever. The (xyz), (at a2 al), and (/3, B2 p3) systems are related with the perfectunstressed, imperfect-unstressed, and imperfect-stressed states of the cantilever, respectively. Always the first coordinate axis is tangent to the longitudinal axis of the cantilever, the other two axes are preferred (or principal) axes of the cross-section. The (xyz) system is obtained from the (XYZ) system by two successive rotations, first through a constant cone angle c about Y-axis, and then through a constant pitch angle p about the rotated Xaxis. The longitudinal axis of the cantilever in the imperfect-unstressed state is defined by the known imperfection deformations u’(x), u’(x), w’(x), and 0’(x) along x,y,z, and about the x-axis, respectively. The longitudinal axis in the imperfect-stressed state is defined-by the total deformations C(x, t), 6(x, t), G(x, t), and 0(x, t) along x,y,z, and about the x-axis, respectively. Other points of the cross-section can be located by means of Bernou@Euler assumptions. The final deformations 4 = [ii, 17,KJ,i]’ consist of the initial geometric imperfections +’ = [u’, o’, w’, @‘IT plus the elastic deformations $ = [u, II, w, e]=, that is, ij=$‘t&

tThis paper presents one phase of research carried out at the Applied Mechanics and Technology Section, Jet Propulsion Laboratory, California Institute of Technology, under contract NAS 7-100 sponsored by the National Aeronautics and Space Administration. The effort was supported by Dr. S. Venneri, Materials and Structures Division, Office of Aeronautics and Space Technology, NASA. SDoctoral candidate. #Professor of Civil Engineering and Computer Science.

1[Memberof the TechnicalStaff.

(1)

Throughout the following, deformations shown with primes or bars correspond to the imperfect-unstressed state or the imperfect-stressed state, respectively, and the quantities without primes or bars represent elastic deformations. Also, partial differentiation with respect to time or with respect to a spatial variable(s) is indicated

by a dot above or a spatial variable(s) preceded by a comma in the subscript, respectively. In addition, the 259

M. EL-ESSAWI et al.

240 z

INERTIALLY FIXED COORDINATE SYSTEM

t STRESSED IMPERFECT CANTILEVER

VECTOR

UNSTRESSED IMPERFECT CANTILEVER

UNSTRESSED PERFECT CA?JTILEVER

Fig. 1. The geometryof the spinnjngcantileverand the coordinatesystems. notation of repeated indices indicates summation over the range. The discretization is introduced by assuming an admissible trial solution in terms of undetermined functions of time c,“(t), c,“(t), c,“(t), c,“(t), a = 1,. ..,m, and known yet unspecified coordinate functions of space u,(x), t),(x), we(x), t%(x), a = 1,. . . , m, that are sufficiently smooth in the solution domain and satisfy the essential boundary conditions. Assuming no support settlements, i.e. homogeneous essential boundary conditions, the coordinate functions should be selected such that U,(O)= V*(O)= w*(O)= t&(O)= u*,(O) = w,,(O) = 0, a=l,...,m.

involved in the discretization of the energy density expressions in a concise and systematic form, the entries of c will be ordered in a manner which is partitionable with respect to modes (or nodes like in the finite element method). Changing the ordering of the entries of c such that it would be partitionable with respect to degrees of freedom is implemented in the computer program, and will be discussed later in the paper. With the above in mind, the c vector can be displayed in the following form CT= 1c,=,QT.. . . , c,*, . . . , CmT]

(3)

where the subscripts denote the partition (or mode or node) number, and the cuth partition of c is shown by

(2a) C,

T-

-

[c,“,

c,“,

c,‘,

c,“],

a = 1,. . * , m.

(4)

Then one may write the trial solution as Or referring to the entries of c, by their sequential numbers in c, one may write

4x9 0

v(x,0 44x,0 =

ceT = ICKY+,c4a...2, c~~-I.c4J,

=

CN

w(x,t)

m, 6 , where again, repeated indices imply summation over the range m which is the number of coordinate functions for each degree of freedom. The total number of coordinate functions for all degrees of freedom is n =4m. Throughout the following the range of Greek indices (e.g. a and fi) is m, and the range of Latin indices (e.g. i and j) is n. Also, the complete list of the undetermined functions of time will be shown by the vector c = c(t). The entries of c will be referred to by either Latin or Greek indices, e.g. ci or c4a-3. The ordering of the entries of c is an impor~nt aspect of the discretixation, therefore it is taken as another parameter of the discretixation. However in order to obtain the coefficients

o!= 1,.. . , m.

(5)

2.1 Discretization of the strain energy density expression The description of the position vector F of a generic point on the impe~ect-stressed state in the (xyzf system is given in [I, 21with up to second degree nonlin~rities as

F=

where

Nonlinear discretized dynamic equilibrium equations

sections remain planes. In addition, if the torsional degree of freedom is absent, then y12 and y13 are zero, which indicates that plane sections remain normal to the longitudinal axis. Assuming that lll, y12 and y13 are small as compared to unity, yet the displacements and the rotations are large, the strain energy density is given in [l] as

(9)

In the above, i is the shortening of the longitudinal axis of the cantilever due to transverse displacements, A is the axial shortening due to warping, x is the warping function of the cross-section, and & is the Euler angle of rotation about the longitudinal axis of the cantilever. Similar expressions for the imperfect-unstressed state can be obtained by replacing bars with primes in (3t(6). Using P and r’ from (6), the strain-displacement relationships may be obtained from dtT df - dr’* dr’ = 2 daTE da

lJ=;EaTa

-(l-2uIX)(W.XX

(12)

where

’ 7’ 1 =

(10)

T

2% 22

a=[e,,

rl

(13)

1

x42(1+ VI)

(14)

and E is Young’s modulus. Using (1) and then (2) in (1l), it can be shown that (13) can be represented as the sum of parts which are linear and quadratic in the elastic deformations, i.e.

where E is the Lagrangina strain tensor, and dCrT= [da,, da*, da& The second order nonlinear strain-displacement relationships specialized for a uniaxial state of stress are given in [2]. In terms of engineering strains, they are El1=(l -&)u.x

261

a=a’ta4

(15)

where

1 +%w,XX)+- u: 2 -x(?.x - rIx)

a’ = (Ao),c, a4 = [c4~(A4~--3)~ + c.&A4~-2).

+ c48-k448-r)a (16)

The subscript a may be interpreted also as the partition number of subscripted coefficient matrices A,,, A,, . . . ,A,, i.e. A0 = KAo)l,. . . , aaL,. . . , &M Ai = [(A:),, t ee9(Ai),, . . . 9(Ai)m], i = 1,. . . , n.1 E22=E33=

(17)

-vet1

NotethatAi,i=l,..., n, matrices correspond to A4B_K, k = 3, 2, 1, 0: /.I = 1,. *. , m, [see (4) and (S)]. The a-th partitions of the subscripted coefficient matrices may be expressed as

(11)

y23=0

where u is Poisson’s ratio. In the absence of warping x, i is linear in a2 and a3 [see (6)] which indicates that plane - uIx)u,,

a2ulxxu,,,

(a2G

-

a3wIJu,+,

- a30’- a2a3wlxxt a20’-

[a2(1

a34,wmJ

-

2uIJ

-

[a30

-

24,) a2a3&

- a2 ~,xxl~~,xx - a32wIxx1w,w ---___________~ -________________________ 2,

-

(a2wL

-

a3uldL

t ( az2+ -

a32

,de:xe,,

(18) (Ad, = j 1 - [a2(1

-&)e,,

1 rl *

0

0

II ,a=l,...,m

x..3 I

cl

e:,e,,, dr

!l

262

hf.EL-ESSAWIetPI.

; ue.xu-,xa2k3JxVD.x _________.___,

0

: , a3uf3,,,~e,l

_-.~~~~_~_~__~___~~_:

.:,

_________________~

0 0 0 _____________ .----------________-: ______.____________, j

c4g-3L =

0

0

0

I

i

&P-2L =

0 --______________

(A+-,)* =

0

0

0

/

L

0

$(a:+ 0 i .-________-_ ;

a32

.---___N__________

;

xPJ,,&,

x

i

0

&Sf~ =

-

.~_~-~-__~_~~__~~~~_~--~~______________

0

1

ea,,~a,Tdr I 0 -..----_-_--____ : _-__________. 1_~~~*_*__“_________~~~~___-_____~_~_ X 0 I 0 $X.=3 I * b.,q7 d7 (19) :a,B=l

Noting that there are n = 4m entries of c, one may write (15) in the form a = A& + Ci Aic

(20)

where the ar-th partitions of Aa and AS,i = 1,. . . , n, are defined in (18) and (19).

g+2

,...,m

where c and p are the constant cone and pitch angles mentioned earlier. Using (1) and then (2) in (6), it can be shown that (6) can be written as the sum of parts which are constant, linear and quadratic in the elastic deformations, i.e.

2.2 Discretization of the kinetic energy density expression The kinetic energy density is given by

where p is the unit mass of the material, and v is the description of the velocity vector of a generic point on the imperfect-stressed state in the (wyz)system. Vector v is given in [l] in the following form

where

az+ VI-

x

u&w',,dr I0 > ~~_~~~~~_~~_~-~~~_~~-~~__~___-________-~~________~~~~ x o3 + w’t (y2et - -1 a,( w”.X+ e*Z) - a z v&w:, dr 2 i I0

and $I=cosCcosP;

**=coscsinp;

$4 = cos p;

$5 =

sin p

+ [email protected]

$3=sinC; (24)

+ ~4~(B4&]cn

(28)

and the a-th partitions of the subscripted coefficient

Nonlinear discretized dynamic equilibrium equations matrices appearing in (27) and (28) may be expressed

& +

((Y2uIx

+ a3wlx)u,,,

(-

a2

+

+

a3e’bLI

(Y2uIx

263

as

(-

a3

-

a297

+

a36

was

--______----_-_~ : -v,

-

- (a3 t a2e’)e,

(a2vIx x

(BoL=

0

+

a3

f0

vimwc,,

(29)

dr

1

+

a3

wI,u,._

f0

dr

.___________________

_______________. : ,_ x

“+a,,

ff2 f 0

-

(a2-

dr

a,ew,

0

J

,a=l,...,m

x

I

‘w,+,, ______________

0 (B4~-2)a

1

-

:

f

0

VD,JL,, dr

0 ;________-________-__-----~~~-.

---___----_______-_________ : ______________________

1 - a253Ju,,,

-

!I 0

a3vi3,b,x

I

I[

a3 QQ+L,, dr 0 : _________________----__-----______-_j --____---____________________ +

=

0

x

)

0

-

a2

f0

v~.nw.z,~ d7 j 0

(30)

(B4~--l)a =

____________________.

x

0 1 agOsh, _______________i ______---7s-----

@4&

=

-

a2epwa4

/

_-_----____-.

j - a2e,e, 0 j 0 _____________-.::.____________________________ .________________-__--------1

0

CAS Vol. IS. No%E

d7 I ep.,eu,,

-x 0

!

0

0 a,/?=1 ,...,

;

m.

--_--__________________

-

a,e,e,

M. EL-ESSAWI et al.

264

Differentiating (25) with respect to time t, it can be shown that +=;I+;4 (31) where i’ = (B&C,

(32)

2.3 Discretization of the expression of the density of loss of potential of prescribed forces

For a system of prescribed forces q=[K(x), F*(X), K(x), F4(x)]'acting per unit length of the imperfectunstressed state of the cantilever along x, y, z and about the longitudinal axis, the density of loss of potential of prescribed forces is v= -d=q

+ = [c++&,+&,

+ c~~-&t~&,

+ c+-,(B+-,)a

+ cdB&JL

(38)

where (33) d = $[(ri - A)-(I/-A’),

Noting that there are n = 4m entries of c, one may write (22) with the help of (25) and (31) in the form

v, w, &- e$]=

(39)

and A is the cross-sectional area of the cantilever. Using V=Boi:tCiBiCtn(g,tGoctciGiC) (34) (1) (7), (9) and (2) in (391, after the algebraic manipulations, one obtains where the subscripted coefficients go, Go, Gi, i = 1,. . . , n, d=d’+d’ (40) are defined by 80 = W, rb.Ro, - 45Rol=+ TP’

(35)

Go=T&

(36)

where d’ = (Do),,c, d4 = ]++&-3)u

Gi=iTBi,

i=l,...,n

+ c4p-2(D4o-A

(41) +

c4~-1(D4,+1L

(37) +

The a-th partitions of B,, and Bi, i = 1,. . . , n. are defined in (29) and (30), respectively, and T is as in (23).

c4a(D4aLlccs

(42)

and the a-th partitions of the subscripted coefficient matrices appearing in (41) and (42) may be expressed as

r

a, ----:

r&w,,, d7 10 ____________________________ 1j.________________________.,._____

0 j % 0 /O -------I ---_-_____------___________. 1I.________________________,I______

(Do), = ;

0 ] 0 W, 10 -------I -----_____________________-. j.________________________.~______

;a=l,...,m

(D4,+3L

=

0

0

0 @4,9-21,

=

;

(0

________________________~, I ._______

0

(0

_________________________.‘,_______

x

-

I 0

uj+,w,.,d7 10

(43)

NonIineardiscretizeddynamicequ~lib~umequations

265

I-

O

-- 1 =

I

wi~w~,, dr 0 2 0 .__________________--~-~-~._______

0

I

I .--___-_____,._________________.

I

10

0

0

0

1, __________~ ,__________________.

._______

0 i 0 0 ____________i___________________

0 _______.

10

t 0

0

0

;a,@=1

,..I,

Noting that there are n = 4m entries of c, one may write (40) in the form

(45) where the a-th partitions of DOand Di, i = 1,. . . , n, are defined in (43) and (44), respectively. DECODED

DYNAMICEQUIP

~UA~ONS

The general nonlinear discretized equations of motion of spinning elastic solids and structures, for the case when the strain-displacement and the velocity-displacement relationships are nonlinear up to the second degree, are derived in [l] using Hamilton’s variational principle for elastodynamics in conjunction with the Rayleigh-Ritz discretization procedure. These equations may be expressed as [M(c)]?f [C(c, e)]i:+ [K(c)]c = p

.

m.

where the three terms on the left side of (46) represent the inertial, the gyroscopic, and the restoring forces, respectively. The term on the right side represents the loading of the system which, is assumed, time independent. When the coordinate functions used in the discretization are of the type of the generalized pyramid functions employed in the finite element method, the n-th order coelkient matrices M(c), C(c,c), K(c) and p represent the mass, the gyroscopic, the stiffness matrices, and the load vector of the structure, respectively. For other choices of the coordinate functions, these coefficients bear no immediate physical representation. However, independent of the choice of the coordinate functions, this terminology will be used. The governing matrix equation (46) is quasi-linear. Up to cubic nonlinearity is present in not only the restoring forces, but in the inertia1 and gyroscopic forces as well. Explicit expressions for the coefficients M, C, K, and p are given in [l]. Specializing these expressions for the spinning cantilever, one may write

d=Ds+ciDic

3. NONL~E~

(44

(46)

(47) C = p ((&%j [

dir) - (iQ$iiT)T)

+ i ((~~)T~~jiiT

+ cjijT)) -

((~)=~(Cjii~

+

CiijT))Tf

t (CiBiTBo+ a (dicj t c$f)Bi7Bj)]

+ &[((BoTC,) - (BXJ’)

+ (((m

t ((m(qqI

+ m)(CiI

f ((B,‘pQiT)-(BiTg&lT)T)

-+ ci:)) - ((&=Gi + BiTGo)(CiIt diT))T)

+ c(ciijT +&I-

fBiTGj(WjI

+ C(CiijT + CjbT)))T)I

(48)

266

M. EL-ESSAWI et al.

t(m)=)

K = E[&A%+ is((m) ti ((((m)

~ t (AoTAi)T)&T) + (((AoTAi) t (AoTAi)‘)&‘)‘)

.ti Cj((((Ai’Aj) t (&T&)T)diT) t (((AiTAj) t (AiTAj)‘)ciiT)‘)] ___ __ __ - fi2p[GoTGot ((igoTGi) t (iiTaTGi)T) + i cj ((a)+

t $((((a) ty!j

t (GoTGi)T)ciiT) t (((a)

cj((((GiTGj)t

(m)‘)CiiT)t(((m)t T

-

[(ia Di) t

t

(GoTGi)T)

(coTCi)T)CiiT)T)

(m)T)CiiT)T)l

(49)

@Di)‘l

p=~tn$zC.

In (47)-(50) a bar above indicates integration over the volume, I and ij (where j = 1,. . . , n) are the n-th order identity matrix and its j-th column, respectively. Note that M, C, and K all contain parts which are constant, linear, and quadratic in c. Terms with p represent contribution from the mass of the material, terms with fip represent contribution from coriolis acceleration, terms with R2p represent centrifugal acceleration contribution, terms with E represent contribution from the material stiffness, and terms with q represent contribution from the loading of the cantilever. It is seen from (47) to (49) that M and K are symmetric. C is skew symmetric excluding some terms involving p only. These exclusive terms are nongyroscopic and arise because of the inclusion of nonlinear terms in the velocity vector expression for a non-spinning cantilever (see (34) when R = 0). For the linear case, the terms in (49) which represent spin-stiffness of the cantilever are - G’p[a

•t ((iin)

t (ii=)],

where the first term represents softening effect and the second represents stiffening effect (see the expression for g, in (35)). From (47) to (50), it is seen that the computation of the following basic matrices are sufficient to define the coefficient matrices in terms of c and i:

G,@&,G,GiTgo2,DiTq,i=l,...,

n (51)

AorA,,_,GoTGi,BoTGi,Bi’co,i=l,...,

n (52)

A,rAi,m,a,fl,i,j=l,...,

n.

(53)

Computations of the basic matrices listed in (51) define the constant parts of the coefficient matrices, whereas computations of the basic matrices appearing in (52) and (53) define the parts of the coefficient matrices which are linear and quadratic in c and I?,respectively. In (51), in order to facilitate the computations of the basic matrices, vectors go, and go are introduced such

(50)

that go=go,+go,

(54)

with go as defined in (35) and g,n = 10,Uo,

-

(bjRo1’

&, = f T(2P”).

(55) (56)

Note that there are 4 individual n X n and 3 t (9/2)n number of n x 1 basic matrices in (51) (see the definition Of Di, i=l,... , n, in (44)). In (52), 5n number of n x n basic matrices are listed, and in (53) (5/2)n2 t (3/2)n number of n x n basic matrices appear. Therefore, the total number of n x n basic matrices is (5/2)n* t (13/2)n +4, and the total number of n x 1 basic matrices is (9/2)n t 3. The total number of entries of these basic matrices is (5/2)n4t (13/2)n’ + (17/2)n2 t 3n, which shows that the cost of generation of all the basic matrices is proportional to the fourth power of the total number of degrees of freedom. Once the coordinate functions, the imperfection functions, and the forcing functions are specified, the basic matrices appearing in (51)-53) can be computed. The problem can now be posed in two different ways. If one is interested in the quantitative response, i.e. the response trajectories in the state space as time progresses, the problem can be analyzed as an initial value problem with appropriate initial conditions. On the other hand, if one is interested in the qualitative response, i.e. the stability of the motion in the neighborhood of an equilibrium configuration, the problem can be analyzed as a stability problem. In the first approach, the governing equations (46) may formally be transformed in to a state variable form as follows 8 = f(5)

(57)

jT = [cT,CT]

(58)

f(&=At+h

(59)

where

Nonlinear discretized dynamic equilibrium equations and

In the second approach (the stability problem), the reduction to the state variable form (57) may not be appropriate, instead other forms such as the ones described in [3,4] may be used. 4.THRCOMPUTERPROGRAM

A computer program called GOMOSC (written in FORTRAN IV language, in double precision, and about 1100 statements long) to systematically generate the basic matrices appearing in (51)-(53) is briefly described below. In the subsequent study, this computer program will be augmented with two additional links. The first will be used to solve the governing nonlinear equations (57) as an initial value problem with appropriate initial conditions using numerical integration techniques. The second will be devoted to the local nonlinear dynamic stability investigation within a specified domain of interest. As input, the GOMOSC program requires the definitions of the geometry, the material, the discretization parameters, and some other procedure-control parameters. In addition, the coordinate functions, the imperfection functions, and the time independent forcing functions are left to the user to specify through a user supplied subprogram function. As output, the program provides a list of the input data (the amount of which depends on a specified input parameter), a list of some intermediate results (if requested), and the basic matrices (if requested) related to the degree of the nonlinearity specified in the input (i.e. linear problem, second degree nonlinear problem, or third degree nonlinear problem). The program uses seven labelled common blocks to store and transmit data from one program element to another. The length of each common block is directly related to the discretization parameters used in the simulation. These common blocks have pseudo object time dimensions, therefore none of the program elements has to be recompiled. When a user needs larger common block sizes, all he has to do is to recompile the main program which is thirteen statements long. 4.1 Computation strategy In order to evaluate the volume integrals appearing in (51)-(53), and as another discretization step, the cantilever is divided into equally spaced NS number of stations. The first station is coincident with the fixed end of the cantilever (x = 0), and the last one is coincident with the free end of the cantilever (x = L). The spatial functions specified by the user (i.e. coordinate functions, imperfection functions, and forcing functions) are then discretized (i.e. computed at these NS stations), normalized, and stored in an array for future use. In the computations of the basic matrices, no attempt has been made to group the repeated entities in the subscripted coefficient matrices such that similar entities in different basic matrices may be computed just once. Instead, each entity of a basic matrix is computed separately. Such a policy yielded a simpler program to debug and modify.

267

The maximum number of coordinate functions that can be used in the program depends on whether the basic matrices are to be stored in the primary memory or in a secondary storage space. Attention in the present is focused on using the primary memory to store the basic matrices. However, a built-in capability is provided in the program such that a future extension of the program, so that the basic matrices may be stored in a secondary storage space as they are generated, can easily be handled. When the basic matrices are to be stored in the primary memory, the maximum number of coordinate functions that can be used in the program is directly related to the degree of the nonlinearity considered in the simulation, and is limited by the maximum size of primary memory allowed. As was shown before, if the problem is simulated as a third degree nonlinear, second degree nonlinear, or linear, the total number of entries in the associated basic matrices is proportional to the fourth, third, or second power of the total number of degrees of freedom, respectively. Therefore, in these cases, one can use up to 3,7 or 28 coordinate functions for each degree of freedom, respectively. 4.2 Short description of the program The program consists of an input part and a generation part, each being accessed by a main driver subroutine called BOSS. The listing of the main program is given in Table 1, and a general-flow chart for subroutine BOSS is shown in Fie. 2. A brief descrintion of the source program elemeits and the labelled’ common blocks is given in Tables 2 and 3. In the input phase, subroutine BOSS (by means of its slave routines INPUT, ARNUM, IJ) first reads, stores, and prints out (if requested) the input data describing the procedure-control and the discretization parameters, the geometry, the material, the constant spin rate, the area integral constants, and two constants related to each imperfection function (amplitudes and frequencies if they are simulated by trigonometric functions). It also computes the dynamic memory allocation constants related to the labelled common blocks. Secondly, (by means of its slave routines SETMAT, POINTR, LKIJ) subroutine BOSS reads, stores, and prints out (if requested) a coding form defining the a-th partitions of the subscripted coefficient matrices used in the discretization of the energy density expressions [e.g. (A&, (Ai),]. Next, subroutine BOSS (by means of its slave routine SETMEM) checks for sufficient memory. The program is terminated if one (or more) of the common block sizes is less than the minimum size computed internally. An output message is printed out to indicate which common block size should be increased, and by how much. In the generation phase, subroutine BOSS (by means of its slave routines PHIOFX, PNORM, FUNC, Uwhich is a user supplied subprogram function defining the spatial functions) hrst computes, orders, normalizes, and stores the spatial functions at NS number of stations along the cantilever. The normalizing factor of each discretized spatial function (taken as the infinity norm of the vector listing the spatial function values at NS number of stations) is also stored (see Fig. 3). Secondly, (by means of its slave routines CONST, LINEAR, QUADRC, WHICH, ATB, IDNTFY, ARINT, PWCF, GRAND, PNORM, QUAD) subroutine BOSS generates, stores, and prints out (if requested) the basic matrices related to the degree of the nonlinearity specified in the input.

M. EL-ESSAWIet al.

268

Table 1. Main program of GOMOSC C

MAIN

PROGRAM

IMPLICIT

PZAL

lS

(A-H.0.W

COMMON

/ICONST/

IN(150)

COMMON

/RCONST/

m(150)

COMHON

/MATS/

IM(758)

CoplpylN /XFUNC/

PHI(3737)

CaMMoN

/AUXIL/

A(404)

COMMON

/COEFF/

CO ( 15000)

COM”ON /INFORM/ IN(l)

= 1

IN(Z)

= 3

CW

BOSS

INF (1000)

(150,150,758,3737,404,15000,1000)

STOP END

C~mncm locations 1 and 2 of IN are for the lOgiCa Unit numbers of formatted input and output files of the operating system When changing camnon si.zcst,note that each cmon size appears twice. Coon sizes me functions of the discretization parameters used in the simulation.

t Cmm dimensions user would DATA which

blocks ICONST end RCONST.have fixed are used for them in all subroutines. like to change their sizes, he should initializes some variables associated

sizes. If for

Pseudo variable any season, the

do that also in the BLOCK with these twd camcm

blocks.

The ordering of the entries of a typical basic matrix, consequently the ordering of the entries of the c vector, is being taken care of in subroutine ATB. From the definitions of the basic matrices in (51t_(53) one may observe that up to four spatial functions are involved in the computations of a typical entry of a typical basic matrix. Whenever an entry is computed it is directly placed in the appropriate (j, j) position consistent with the ordering indicated by an input parameter. In the computation of a typical entry of a typical basic matrix, the volume integral is handled first as an area integral then as an integral along the cantilever. The area integral type number implied by the quantities involved is first identified. The integration along the cantilever (the x-integral) is skipped whenever the area integral type number corresponds to a zero area integral value (user input or user-generated through a dummy routine), and the entry is assigned the value zero. Also, if the product of the normalizing factors (infinity norms) of the corresponding discretized spatial functions is found to be less than a user-specified threshold input value, the integration is skipped. An output message is printed out if requested. Otherwise, the numerical computation of the x-integral is initiated. The i-th entry (where i = 1,. . . , AS) of the integrand vector is the product of the i-th entries of the corresponding discretized and normalized spatial functions.

The value of the x-integral is equal to the x-integral of the integrand vector multiplied by the normalizing factors. The absolute value of the x-integral of the integrand vector is therefore equal to EL, 0 s EI 1. The x-integral of the integrand vector is assigned the value zero if its absolute value is found to be less than a second userspecified threshold input value. For illustration of the above, consider the (S, Jo) partition of mO, S, Jo= 1,..., m. In the absence of the imperfections and the warping function, and assuming a symmetrical crosssection, it is given by (see eqns 17 and 18)

(62) where

Now consider the following two integrals:

I

L

0

v:,_ dx and

I

L

0

v,~,v~,, dx.

269

Nonlinear discretized dynamic eq~ib~um equations about the source pro~m elements (excludi~ common blocks)

Table 2. Short isolation lapy

Description Performs the volume

ATE

T A TB

- _-

(e.g..

f;Go

_

*

integral of a typical be&c

*,

$‘Bo

matrix

of

the

type

I being ths n-th order

identify matrix)

Identifiesthe area integral type nmaber, hence the area intagral, for a typical contributionto a typical entry of a typical basic matrix,

ARXNT

area integral type number - 1, .... 39 (see Table 8). AIINLNI

Duaulysubroutinefor the nlmerica1 evaluation of the a?x(tintegrais (by whatever qoadrature rule the user would like to use).

Boss

Driver subroutine (sea Fig. 21.

CCNST

Generates the basic matrices related to the constant coefficientscase. Slave subroutineto PHIOFX routine. indicateswhat type of spatial function (coordinate, imperfection,forcing) is indicated in the current computationof a typical entry of the vector containingthe discretfzedspatial functions.

FONC

GRAND

Canputes the final integrandvector (representingthe integrand function) and its spatial integrdl (usingsubroutineQUAD) related to a typical entry of a typical basic matrix.

IDNTEY

Inrerpretsthe descriptorsof a typical term of a typical entry of the a-th partition of a typical subscriptedcoefficientmat&x.

IJ

Integer function,canputes the one dtinsional array subscriptof UI entry of a two dimensionalarray.

INPZPT

Reads

and

prints-out input data, generates soaw other Constants.

Generates the basic coefficients case.

matrices related to the linearlyvarying

LKIJ

Integer function,computes the one dimensionalarray subscriptof an entry of a four dimensionalarray.

MAIN

Main program (see Table 1)

PHIOFX

Generates the nonniLli%ed spatial functionvalues at NS stations the cantilever (see Fig. 3)

PNORM

Finds the infinity norm

POINTR

Slave subroutineof SETMAT subroutine

PWCF

Slave subroutineof ATS subroutine,cauputes the differencebetween the highest and lowest node mnnbess involved in the c-at&ion of a typical entry of a typical basic matrix, and detewines whether the x-integralshould be initiated. It is used if only ICRDFN - 1 (see Table 5)

of

along

a vector

Performs the quadratureSimpson's rule? to the integrandvector

find

the

x-integralof

Generates the basic matrices related with the quadraticallyvarying coefficientscase. SNTMAT

Reads, stores, and prints out (if requested)the coding forms defining the wth partition of the subscriptedcoefficientmatrices.

SETMEM

Checks for sufficientmemory

U

User supplied subprogramfunction (having14 entries: UX, UXX, V, VX, VXX, W. WX, WXX. ST, STX, PRIM. PRIM, PRIMXX. F) to define the spatial functionsof the independentvariable x.

WSICR

Identifieswhich subscriptedcoefficientmatrices (the a-th pertitione are to be involved in the casputationof e typical basic matrix

only)

t

The argumentsof QUAD routine are the integrandvector and its length. Therefore,using a note sophisticatedquadraturerule can easily be handled, if the results obtained using Simpson's hlle are not satisfactory.

M. EL-ESSAW et al.

270

I

Call INPUT to read the problem parameters

1

Call SETMAT to read the a-th partitions of the subscripted coefficient matrices J I

Call SETMEM to check for sufficient memory

Call CONST to generate basic matrices related to the constant coefficients case

Call LINEAR to generate basic matrices related to the linearly varying coefficients case

Call QUADRC to generate basic matrices related to the quadratically varying coefficients case

23

RETURN

Fig. 2. General flowchart for subroutineBOSS.

If one selects u,(x) and uZ(x) as the first two bending modes of a cantilever (note that in the present formulation, there are not any generalized coordinates associated with slopes), then the first integral is greater than zero, whereas the second one is identically zero because of the orthogonality property (Appendix). Let “1, and v+ be the discretized and normalized vectors correspondmg to u,,(x) and v2?(x), respectively. Let I(v,,~~](~ and /v2,]lm be their normahzing factors. One may observe then that)]v,,(): and I(v,,(]~. (Iv~~~(/~ are of the same order [O(L- )]. Thus, the second threshold value is important to judge on the merit of a typical x-integral of a typical integrand vector. Such an elimination of terms like Jk uI_u2, dx in the above example is important. Due to the discretization error associated with the quadrature rule used to perform the x-integral, and due to round-off errors, such a term may not be computed as identically zero. It would in turn spoil the character of the corresponding basic matrix, especially in cases where the area integral values are very small. For example, one

such case is encountered in the Heliogyro sail concept. Typical data for the length, width, and highest thickness of a cantilever is 7000 meters, 8 meters, and 25 x 10m6 meters respectively. Another important aspect arises when dealing with piecewise continuous coordinate functions, i.e. the type of coordinate functions used in the finite element method, and which are referred to as shape functions. Figure 4 shows a possible choice for the a-th piecewise continuous coordinate function of class C’, (Y= 1,..., m, for the transverse displacement u(x, t), (also shown with solid lines are the corresponding element shape functions). Note that when such piecewise continuous coordinate functions are used, the essential boundary conditions shown in (2a) affect the first one only, since all others and their first derivatives vanish at x =O. The use of the class C’ coordinate functions shown in Fig. 4 makes all basic matrices banded as explained in the next paragraph. It is clear that a typical integral of the type

271

Nonlinear discretized dynamic equilibrium equations Table 3. Short information about the labelled common blocks Minimlrm

Description

length*

AUXIL

Variable:

4 x N.Sl

COEFE-

Variable; NDIM : if basic matrices are stored in the primaty -ry

scratch space to generate the final integrand vector related to a typical term of a typical entry of a typical basic matrix. Storage space for the generated basic matrices if they are stored in the ptimery memory. Otherwise it is a scratch space to temporarily store are typical basic matrix currently being genetated.

N x N: otherwise storage space the problem.

for the integer canstents of

ICONST

Fixed

INFORM

Variable;

MmS

Fixed

Storage space for tie coding fczm data defining the wkh partition of the subscripted coefficient matrices, and other pertinent infwations.

RCONST

Fixed

Storag! space

2xNTmAL

Storege space for coded infomations and pointers associated with the gmerated basic matrices.

forthe real constants of

the problem. Variablei (11 x Ml + 4) x NSl

XFUNC

t Nmencleture

Storage spacefor the discretized and nomalized spatial functions (see Fig.

used in this column:

N

: ~lrmberof coordinate functions for each degree : Total number of degrees of freedom = 4 x M

NDIM

:

NS1

: Number of stations + 1, i.e., NS+l.

Ml

NTCTAL:

3).

Total number of entries be generated.

of freedan + 1

of all the basic matrices required to

total number of basic matrices,to be generated,

~~~~(x)u~(x)~(x)dx, where h(x) could be an imperfection function or a forcing function, should vanish whenever 16- ~12 4 for the type of coordinate functions shown in Fig. 4. Here 6 and p represent the node numbers. In general, such an integral shoutd vanish whenever 18-pi 2 IBAND, where now 18- ~1 represent the maximum difference between the node numbers involved in the computation of a typical entry of a typical basic matrix, and IBAND is an input parameters. When such coordinate functions are used, the x-integral is initiated whenever (8 - PI< IBAND, otherwise it is automatically skipped. In this respect, another input parameter (ICRDFN) is needed to indicate the type of coordinate functions used in the simulation (i.e. continuous or piecewise continuous functions). 4.3 Preparation of the input The arrangement of the input cards of a given run is as shown in Table 4 where every card group is referred to as an input item. ascriptions of the contents of the input items are given in Tables 5-9. A typical term in a typical entry of the a-th partition of a typical subscripted coefficient matrix is inputed using a coding strategy involving two descriptors as shown in Table 10. For example, the two descriptors of terms such as - J~uj,vp,_ dT and - a3ws v, would be (0101112, 02012121) and (0101313, 020~1~), respectively. Note that a typical entry of the a-th partition of a

typical subscripted coefficient matrix has, in general, more than one term. As shown in Table 9, the maximum of number of nonzero terms in any entry in the matrix, and the total number of nonzero terms in all the entries of the matrix are required as input for the automatic allocation in MATS common block. These quantities can be obtained from the definitions of the cr-th petitions of the subscripted coefficient matrices given earlier. The two descriptors of a typical term in an entry is preceded in the input by (K, Z, J), where K is the term number in the entry, and Z and J are the row and column numbers, respectively. The above mentioned coding strategy is thought to be better read from cards rather than initialized through a block data. This has the advantage of achieving a simulation capability which is applicable not only to spinning cantilevers, but also to other spinning elastic systems involving the same degrees of freedom. Another advantage of such a policy is the easy implementations of special cases. Suppose the axial impe~ection u’(x) is considered to be zero in the simulation. This can be accomplished using one of the following two alternatives. In the first one, any terms involving u’(x) or its derivatives should be disregarded from the two-descriptor input, and dummy values for the amplitude and frequency of u’(x) as well as dummy definitions in the user supplied subprogram function should be provided. The second alternative is to keep all terms associated

272

M. EL-ESSAWI et al.

NP=2 ‘I1 NP=4

I

I

NP=5

NP=6

Nomenclature and symbols: a :

mode (or node) number.

m :

number of coordinate functions for each degree of freedom.

NS:

total number of stations.

NP:

partition number, = 3 x D.O.P.R+ Derivative%-2 = 12 otherwise.

IP:

, if function is not a forcing function.

address of a typical entry of a typical discretized and normalized spatial function. = (NP-1) (m+l) (NS+l) + (MN-11 (NS+l) + K, MN is the mode number, MN = 1, .., m+l ; K = 1, .., NS+l , if the function is not a forcing function. MN = M+l designates imperfection functions, K = NS+l designates the normalizing factor (m - norm). = ll(NS+l) (ntcl)+ (D.O.F. n-1) (NS+l) + K , K = l,.., NS + 1 otherwise.

0:

address of first entry in camnon block = 1.

0:

continuation.

0:

address of last entry in common block = ll(NS+l) (l'n+l) + 4(NS+l).

..:

designates a discretized and normalized spatial function.

+:

vector of length NS+l. The first NS entries represent the normalized values of a typical spatial function at NS number of stations. The NS+l entry represent the normalizing factor (- - norm).

Fii. 3. Arrangement of the discretized and normalized spatial functions in XFUNC common block.

in the two-descriptor input, to provide dummy definitions in the user supplied subprogram function, and to input a zero and a dummy value for the amplitude and frequency of u’(x), respectively. Obviously, the lirst approach is better because it minimizes the computation cost. Automatic elimination of a degree of freedom is not provided in the program. However, it can be handled by deleting all terms associated with this degree of freedom from the two- descriptor input, or by specifying zero coordinate functions corresponding to this degree of freedom. In this respect a subprogram function is needed for the condensation of the generated basic matrices. The handling of concentrated loads can easily be implemented through the implicit use of the Dirac Delta function. A concentrated load p. acting in the positive y direction at x =x0 may be expressed as F*(x) = poS(x, x0), where 6(x, x0) is the Dirac Delta function with singularity at x0. Such concentrated loads may be with u’(x) or its derivatives

properly scaled according to the quadrature rule used to compute the x-integral (see footnote in Table 2). For illustration, suppose that Simpson’s rule is used to compute the x-integral. Suppose also that a concentrated load p. is acting in the positive y direction at the free end of the cantilever (x = xNS), where NS is the last station number. The forcing function F*(x) may be expressed as F*(x) = po8(x, xNs). Let Ax = L/(NS - 1) be the width of the integration interval. Noting that Jk h(x)&(x) dx = poh(xNs), where h(x) could be a coordinate function or an imperfection function, etc. this concentrated load can be scaled such that all entries of the vector representing F*(x) values at NS number of stations are zero except the last entry, which should be defined as p; = 3p,/Ax. 4.4 Description of output All possible output items (except error messages) are shown chronologically in Table 11, together with the condition determining their production. The output is

I

0

c; (t) vg+J

X
.Y a-2

2 0 or x>xc(+2 t XM2

max (0, Xa_2) -'x f min (L,x~+~)
Fig. 4. Possible choice for the a-th piecewise continuous coordinate function u,(x - x,), a = 1,. . . , m,of class C’, and the corresponding (a, g + 1) element shape functions.

a+2 c B-a-1

-(1/16h6) ~.-x~_~~~x-x,_~(X-X,+~)(X-Xa+2)2

"(a,a+l)(x,t) -

va(x-xu) =

M. Et-ESSAWI ef al.

Table 4. Input items of input data Input item nwber

Brief

Table n&e1 for detailed description

description

~~c&ure-control

card

5

2

Material, geometry, spin, BIUIthreshold values.

6

3

tmperfectio" constants

7

4

area integrals

S

5

coding

9

1

form data for the a-th partitions of the subscripted coefficient matrices.

Table 5. Pracedure-controlcard of input data (input item 1)

rTYPE

1

type: Q-linear: l-Zad degree nonlinear; 2-3rd degree nonlinear.

I?.

O-2

Problem

M

2-4

I3

O-249?

Number of coordinate functions for each degree of freedom.

NS

5-8

I4

o-9999

Number of equally spaced stations alwg the cantilever (odd "wber).

fau?ER

9

If

O-l

Ordering of the entries of f ; O-partitioneble with respect to mcrles (or nodes); 1-partitionable with respect to degrees of freedom.

rm

10

11

U-I

~rea integrals indicator: O-read from cards; l-generate "umerifally using the duaay subroutine ARNUM

11

11

O-l

Basic matrices St*raye i"dicatoz : O-store in psimary memory i l-store in secondary menoxy.

12

I1

o-2

Print out level indicator: 0-minti output; 1-intermediatei Z-detailed.

I3

fT

a-9

14

11

O-1

coordinate functions type "amber: O-continwus functions; I-piecewise continuous functions.

15

I1

o-9

Difference in node labels at which the x-integral of a typical term Of a typical entry of a typical basic matrix vanishes fd%.muny value is required if ICFDFN = 0).

IwiND

'When using the proysam as a primary-memory program, M is limited to 21) if ITYPE = 0, 7 if ITYPE = 1, and 3 if ITYPE = 2.

minimum, intermediate, or detailed, according to whether the value of INP is 0, 1, or 2, respectively. The output items are self explanatory. Samples of output items 1, 2, 3, 4 and 5 are provided in Figs. 5 and 6. These samples do not belong to the same job, and they contain only portions of the long output items.

In output item 3, the subscripted coefficient matrices Bo>I&X, B4P--2,Bae--r and Bdp are displayed as BO, Bl, B2, B3 and B4, respectively. The same applies for al1 other subscripted coefficient matrices. In output item 5, a basic matrix is referred to by a type number and an index pair. The type numbers are sefected

Nonlinear discretized dynamic equilibrium equations Table 6. Description of input item 2 NlllaQ

of field

card columna of field

RN0

F-t

l-8

E 8.5

9-16

Description

E 8.5

Young's mcduius of the material..

17-24

E 8.5

Poisson's ratio of the material.

RO

25-32

I 8.5

Distance between origin of the inertial system and the fixed point of the cantiieYer k4ee Fig. If.

SPN

33-40

E 8.5

Length of the cantilever.

PANGIS

41-48

E 8.5

Pitch angle (in radians).

CANGIS

49.56

E 8.5

Cone angle (in radiana).

OMEGA

57-64

E 8.5

Spin rate, n

NPSNRH

65-72

E 8.5

Thresholdvalue for skipping integration (to be compared with the product of the non~lisinq factors of the discretizedspatial functionainvolved in the computationof e typical term of a typical entry of a typiCaX basic matrix).

73-80

E a.5

Thresholdvalue for ignoring an x-integral W&le. EPSINT is multipliedby SPN (internally)to establishthe x-integral cwarison quantity.

Table 7. Descriptionof input item 3 of field

Nanm

AMP(X)

card coltmrns of field

FOrmat

l-32

4 E 8.5

Amplitudesof the impexfectiondeformations u', Y', w'* and 8'; respectively

33-64

4 E 8.5

Frequencies of tba imperfection

x51,.-,4 FRQW Ial,.+,

Description

deformationsu', vi, w', and B'i respectively

+This Table provides two constants fat each imperfectiondeformation. Xt ie thought that en imperfectiondeformationcan be sfmulated as a trigonaoletrlcfunction,e.g., u' = uo sin mu y , where L is the length of the cantilever,and uo, mu ere referred to as the amplitude end frepuena';reSP,pe;CtiV%lF.

as 1,2, 3,4,$,and 7~~es~ondi~ basic mat&@ of the form ATA, B-, GT$, BT’i$ G’g, F@: and DTq, respectively. The first index m the mdex pair is the index of the first subsc~pted c~~cient matrix ~~ea~ng in the expression of the basic matrix, and the second index is the index of the second subscripted coe~cient matrix, For example, r& and w4 are referred to as matrix

type 1 with (0,4) index pair, and matrix type 4 with (3,4) index pair, respectiveIy. Note that for an (f, J) index pair, the ranges of I and J = 0,. . . , n. In addition, the sequential number of a typical basic matrix dnring generation, as wet1 as a code num~r are provided. The code number of a basic matrix of type L and with (I, J) index pair is lo” x L + lo-’X Ii-J,

276

M. EL-ESSAWI et al. Table 8. Description of input item 4t Name of field

Card c01wans of field

Format

AR(I),

3(1-801,

3(10 E S.S/)

I=1,..,39

(l-72)

,9 E 8.5

Description

Area integrals, where the area integrands are ordered in the following sequence: 1

r ap f a3 s x e x,, I x,a ,

2 a;? , a203 r a2x

3

2

2

I a2x,, I alx o * ajta3x* 2

'3

t

Tbrs input item should not be provided if the variable XARFZAin the control card has the value 1. the area integrals can then be computed using the dummy subroutine ARINT where any pertinent data can be read-in in place of this input item.

Table 9. Description of input item 5 List of input statements that read the associated input item cards

~ITA~I~,I=1.5~,~ITB~Il,I-1.5~. ~ITD~I~,I-1,5~,~ITGS~I~rI=lr2~.ITQS

Format (outside parantheses indicate the possibility of multiple cards) 18 12

Nc~~enclature:

ITA(I),I=l,S

: maximm of number of noneero terms in any (Jt18_3)a. (AsB_21a , ($sB_l)a , ($+B)cr,

ITBt~),I-1,s

:

ITDa),I=1,5

:

r6cdxmm

entry of (!oo'a, respectively.

of number of nonzero terms in any entry of (Foja,

(DlrB_S)a,

(D48_i)a, (elrB_,JaI (fI*B)a , +espectively.

tITD(Z) and ITD(5) shotid be zero, see (44)1. 1lYzi(1).1=1,2

:

maxtium

of number

of nonzero terms in any entry of yaol ,

r= + respectively.

ITPS

maXimum of numbzr of zWnzar0 terms in any entry of q.

mn

total YlURlber Of n0nZefb tarIDsin all the entries of the a-* partition of a typical subscripted coefficient matrix.

K

term number in an entry of the a-th partition of a subscripted coefficient matrix.

I

row number of the enin the a-tb partition of a typical subscripted coefficient matrix.

3

column number of the entry in the a-th partition of a typical subscripted coefficient matrix.

rn(L),L-1,2

t See

:

First and second descriptors+ of a typical term of e typical entry of the a-th partition of a typical subscripted coefficient matrix.

section 4.3 and Table 10.

Nonlinear discretized dynamic equilibrium equations Table 9(Contd.)

Table 10. The two descriptors of a typical term in a typical entry of the a-th partition of a typical subscripted coefficient matrix

Descriptor Of

12 format

1 format COnStant

co-

efficient number in the array cc show

econd descriptor

(17 format)

First descriptor

variable coefficient number in the array vc shovn below

1' 2 3 4 5 6 7 8 9 LO 11 12 13 14 15 16 17 LB 19

(16 format)

-

f,i fx)

2

format

2

format

,I

format

Lformat ?greeof reedom lmber

DE?fC3

onstant oefficient umber in he array 'C

riva-

ariable oefficiw umber in he array c

ve nber

m ,. q .. * .. m .. 3 .. N ._ , .. ,+

,

Table 11. Chronological listing of output items and condition for their production Item number

Desctiption of output item

Of

format

INP

1

Degree of the nonlinearity and procedure-control parameters.

2

Problem description: material constants, geometrical constants, spin rate, imperfection characteristics. area integrals, and threshold values for skipping integration and ignoring the value of an integral, respectively.

3

Two-descriptor inputs for the a-th partition of the subscripted coefficient matrices.

21

4

Discretized and normalized spatial functions.

2

5

Generated basic matrices

2

6

Cont(?ntsof INFORM ccumnonblock (listing of coding forms and pointer associated with the generated basic matrices).

21

7

N,aw&r of evaluations of the following routines: WHICH, ATB. IDNTPY, ARINT, GRAND, PNORM, QUAL3,PWCF.

M. EL-ESSAWI et ai,

0 N

Q

a: .

Nonlinear discretized dynamic equilibrium equations

CAS Vol. IS. No. G-F

219

M. EL-ESSAW et al.

280

0

m

--.--_I

w....

‘

.

.

.

.

.

*..*.**..*..1*

: : t

i

O-3

Fig. 6. Sample of output items 3,4 and 5 of GOMOSC program.

I

282

M. EL-EssAWI

In Fimmple of output item 5 is shown for the basic matrix &,=A,, where the (S, CL)partition, S, p = 1,. . . , m, is as shown in (62). The parameters used in the computations are those of items 1 and 2 shown in Fig. 5. The coordinate functions are selected as the first two free vibrational axial, bending, and torsional modes of a nonspinning cantilever, i.e.

u,(x) = 19,(x) = sin L E

G(x)= W,(X)=~[coshp~ t-COSP. i- ja(sinhp, f -sinp,f

)I

,oSxIL,cr=1,2

(63)

where 1,=1;1*=3; P, = 1.875104068712;Pz = 4.694091132974;

etd.

gram for the systematic generation of the governing quasi-linear dynamic equations is presented. The degree of the nonlinearity, the type and number of the coordinate functions, the ordering of the entries of the vector of the undetermined functions of time, the imperfection functions, and the forcing functions are left to the user to specify. Thus the problem can be simulated using a wide variety of models for both quantitative and qualitative analyses. REFERENCES 1.S. Utku, M. Salama and M. El-Essawi, On numerical nonlinear analysis of highlyflexiblespinningcantilevers.Compuf. Sfructures Z. 13(l-2). 349-355(1981). 2. M. Salama, M.‘Trubert, M. l%Essawi and S. Utku, Second order nonlinear equations of motion for spinning highly tlexible line-elements. To appear, .Z.Sound Vibrat. Vol. 80, No. 4, (Feb. 1982). Also presented in 20th Structures, Structural Dynamics, and Materials Conference, St. Louis, MO. (April 1979). 3. L. Meirovitch, A new method for solution of the Eigen value problem for gyroscopic systems. AZAA J. 12, 1337-1342 (1974). 4. K. K. Gupta, Free vibration analysis of spinning structural systems. Znt. J. Num. Meth. Engng 15, 395-415(1973). APPENDIX

f, = 0.7340955137589;j2 = 1.018467318759. 5.SUMMARYANDCONCLUSIONS

The nonlinear discretized equations of motion for a spinning cantilever with initial geometric imperfections, considering second degree nonlinearities in both the strain-displacement and the velocity-displacement relationships, are given explicitly. Up to and including third degree nonlinearities may be present in not only the restoring forces, but also in the inertial and gyroscopic forces. The governing discretized equations are quasilinear, and they may encompass a wide variety of geometrically nonlinear problems. In order to unify the discretization approaches associated with the stationary principles, the type and number of the coordinate functions used in the admissible trial solution are considered as parameters of the discretization. The choice of the coordinate functions includes both sets of continuous and piecewise continuous functions employed in the conventional Rayleigh-Ritz method and the finite element method, respectively. The subscripted coefficient matrices involved in the discretization of the energy density expressions are provided explicitly in concise and systematic forms adaptable to computer programming. A computer pro-

Orthogonality of the second derivatives of bending modes of a non-spinning cantilever

The bending modes of a non-spinning cantilever are given by o,(x) = cash P. : - cosp, i - f.(sinh p, i - sin pa ,osx~L,cr=l,...

i >

(Al)

wherep. and f., (I = 1,. . . , are constants, and L is the length of the cantilever. Noting that v,(x) in (Al) satisfy the following conditions v.(O)= v~, (0) = 0, v,,_(L) = II,,_ (L) = 0, and Jo%& dx = b,,&, (no sum on (I and p), where See is the Kronecker delta and b,, is the constant, one has L

L

0

0

I vqx~, dx= [~.,v~,loL- I v~,,vs_

dx

=[-V” a a,loL + lLU.Ufim’dx

= (P&)~ lL u.vs dx = (pB/L)4b.86.8 (no sum on (2 and B) where as can be observed from (Al),

v,+,_

= (p$Q4t+

(A2)