I

V

VIBRATION ANALYSIS OF A POLE STRUCTURE

by

JOHN ROBERT LANKFORD, B.S. in Engr. Phys

A THESIS

IN

MECHANICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

Approved

Accepted

August, 1972

ACKNOWLEDGMENT

I would like to express my sincere thanks to Professor L. J.

Powers for his direction of this thesis and for his helpful criticism

and suggestions.

n

/

TABLE OF CONTENTS

Page

ACKNOWLEDGMENT ii

LIST OF TABLES v

LIST OF ILLUSTRATIONS - vi

LIST OF SYMBOLS USED vii

ABSTRACT viii

I. INTRODUCTION 1

II. SIMPLIFIED MODEL 3

Assumptions 3

Bending 5

Torsion 10

III. TRANSFER MATRIX MODEL 12

Assumptions 12

Bending 13

Simplified Model 17

Pole With Concentrated Masses 21

Torsion 23

Damping 26

Bending From Dynamic Loads 33

IV. RESULTS AND CONCLUSIONS 35

Computer Programs 35

Results of Programs 37

Conclusions 52

m

LIST OF REFERENCES 53

APPENDIX 54

A. Details of the Structure 55

B. Computer Programs 57

TV

LIST OF TABLES

Table Page

1. Bending Natural Frequencies of the Pole Structure 33

2. Torsional Natural Frequencies of the Pole Structure 39

LIST OF ILLUSTRATIONS

Figure Page

1. Simplified Model 4

2. Forces and Moments Acting on the Pole 5

3. Beam for Deflection Equations 8

4. Cantilever Beam 13

5. Beam Segment Connecting Concentrated Masses 14

6. Concentrated Mass 15

7. Pole Composed of 28 Discrete Masses 17

8. Profile View of Simplified Model 18

9. Mass less Torsional Beam Connecting Disks 24

10. Disk Representing Concentrated Mass 25

vl

LIST OF SYMBOLS USED

<{ Displacement

E Young's modulus

G Shear modulus

h Constant of structural damping

H Angular momentum

I Moment of iner t ia (area, unless otherwise specif ied)

j \/T

L Length of a section

m Mass

M Bending moment

0 Slope

e Torsional rotat ion about an axis

T Torsion

u Poisson's ra t io

V Shear

y Displacement in the y di rect ion

Oi Circular frequency of v ibrat ion

v n

ABSTRACT

A vibration analysis employing transfer matrices is developed

for a pole structure with an intermediate support and a varying cross-

section. The analysis is applied to an existing structure and the bend­

ing diagrams are plotted for different modes. A method is included for

accounting for structural damping.

vm

CHAPTER I

INTRODUCTION

The vibration response of oscillating structures is a major fac­

tor in their design. The prediction of the bending diagrams for differ­

ent modes is important for a thorough analysis and can result in the

design of more reliable structures.

This paper concerns the application of a transfer matrix method

of analysis to a pole structure. The method allows the successive

determination of the deflection, slope, moment, and shear diagrams for

any chosen frequency. Boundary conditions can be applied to the analy­

sis by a relatively straightforward means, and structural damping can

be included by introducing a complex impedance. The transfer matrix

method is particularly well adapted to computerized computation and

avoids the problem of inverting matrices.

The particular problem chosen for study is that of the pole

structures serving as light standards in the football stadium at Texas

Tech University. The structures exhibit vibratory amplitudes of several

feet during high wind velocities, and such oscillations can be expected

much of the time. The configuration of the structures presents many of

the features encountered in analyzing complicated structures. The diam­

eter of the pole varies from 12 inches to 32 inches and then to 8 1/2

inches. The boundary conditions must account for a support at the base,

a support 40 feet above the base, and a large distributed mass at the

top of the structure. The principal dimensions are shown in the

1

Appendix.

Structural problems with the standards have occurred since their

construction in 1959, and on two occasions poles on the east side of the

stadium have collapsed. The first failure involved the upper supports

failing in a 60 mile-an-hour wind, and the second failure was the result

of a tornado. Since then, the poles on the east side of the stadium

have been shortened and a larger diameter cross-section used for the

upper portion of the pole. The poles dealt with in this analysis, those

on the west side of the stadium, are of the original construction.

The analysis to be developed uses a simplified model of the

standards to develop the boundary conditions and then applies the con­

ditions to a more refined model. This refined model considers the pole

to consist of a series of concentrated masses linked together with beam

sections corresponding to appropriate sections of the pole. From the

refined model the deflection, slope, moment, and shear diagrams for

different frequencies can be plotted. The principal reference for this

paper is Matrix Methods in Elastomechanics by Eduard C. Pestel and

Frederick A. Leckie.

CHAPTER II

SIMPLIFIED MODEL

Assumptions

As a first approximation, the pole is considered to be a mass-

less spring with the bank of lights acting as a rigid, distributed mass

at the top. Bending is assumed to act only in the plane perpendicular

to the face of the lights. Neither axial loading nor buckling stability

are considered. The assumption that the bank of lights acts as a unit

is justified in part by the rigid, structural bracing to which the light­

ing fixtures are fastened. The pole in this model has a uniform cross-

section of 20 inches in diameter. Linear stress-strain relations and

other assumptions common to elementary mechanics of solids apply.

A strut connects each light pole to the stadium structure forty

feet above the base. The strut prevents deflection at the connection

but provides no restraining moment. Therefore, this support can be

approximated by a pinned connection. The support at the base supports

the axial load of the structure and provides restraint of horizontal

movement. Although initially constructed as a fixed support, it will be

considered as a pinned connection. The concrete foundation for the base

is capable of withstanding only small moments, and cracks have occurred

since installation making the base flexible rather than rigid. Evidence

of this is shown in photographs 3 and 4 in Appendix A, along with the

dimensions of the structure. Figure 1 represents this model.

z

r^ T 2c = 15

b = 119'

CI C3y-

a = 40'

Figure 1.--Simplified Model

Because the distributed mass of the pole is neglected, computa­

tions based on this model will have but limited accuracy. The simplifi­

cations will allow, however, a less obstructed development of the

equations governing the effect of the mass of the lights upon the pole.

It will also give an approximation of the vibration response of the

structure. Results of the computations for the simplified model will be

compared to those for a more precise transfer matrix analysis.

i^nn^tmammam^^'.-.'m

Bending

The force and moment equations of dynamics can now be written

for the model. Consider Figure 2.

^ y

Figure 2.--Forces and Moments Acting on the Pole

The displacement of the center of mass of the bank of lights can

be expressed as y + c sin i . Linearizing, this is approximated as

^= (y + c0x)

The velocity of the center of mass is then

V = (y + c^y)

The acceleration is

a = (y + ciy)

The angular momentum of the model is found from^

• • .

^x = Ixx ^ - ^xy'^y " xzl z • • .

^V "" "^yx^x " ^yy^y ' lyz^z • • •

^z = -^zx^x - zyJ y ^ zz z

or [H] = [I] [0] where [I] is the moment of inertia matrix. The moment

equations are

Mx = Hx - Hy0z + Hz^y

My = Hy - H^^x + Hx^z

Mz = Hz - Hx0y + Hy0x

For principal axes whose origin is located at the centroid of the bank

of lights,

^xy ^xz ^yx ~ ^yz ~ ^zx ~ • zy ~ ^

From Figure 2,

02 = 0y = 0

Substituting these values in the moment equation gives

\ = Hx = Ixxl»x

Numbers refer to similarly numbered references in List of References at the end of the report.

48 2 where I^^ is found from I^^ = .?-, m.jX where m- corresponds to the

mass of one of the lights and x- is its distance from the x axis which

has been translated to the center of the bank of lights.

Summation of forces about the junction of the bank of lights

with the pole is

V^ = -m (y + c0^)

where m represents the total mass of the lights

Summation of moments is

• • • • M^ = -m (y + c0^) c - I^^ 0^

Expressing the relat ions in matrix form.

f s

V.

M.

\ /

m

mc

mc

Ixx + nic

• >

2 4

• •

y • •

0 s t

Assuming periodic motion.

y = A sin wt,

0 = B sin (»>t.

y = -<Ai^y

2 0 = -ui 0

so that the matrix relations can be expressed as

/ V

\

\ k /

•

mar

mew <w

N

mcoj

( I + mc2)u)^ XX /

^ s

y

0 V. /

The slope and deflection of the pole in Figure 2 are assumed to

be continuous with the section of the pole supporting the bank of lights

aOa

8

To relate the equations governing the bank of lights to the structure as

a whole, the deflection and slope of the portion of the pole below the

lights is expressed in terms of V and M . For this purpose consider

Figure 3.

L Rl

iR'

V P X

Mr

Figure 3.--Beam for Deflection Equations

Summing moments about R ,

-R,a + V (a + b) + H = 0 , R2 = '^P^^ ") p

Summing moments about R2,

V b + M - R-,a + Vpb + M = 0 , Rl = V ° "P

The moment equation for the beam is

M = RT X - Rp <x - a>

where <x - a> = 0 for X less than a

< x - a > = x - a for x greater than a

then d^y ^ Mx = Rl^ - R2<x - a> dx2 ^ EI ' T T EI

dy R-|x2 R2 <x - a> 2 — = 0 = + dx 2EI 2EI

y = R-|x3 R <x - a>

6EI 6EI + C-j X + C2

From the boundary conditions.

a t x = o , y = o , C 2 = o

at X = a , y = 0 , C"! = f l FET

Substituting the expressions for R] and R2 into the equations

for y and 0 along with x = a + b.

^ = IT (1 k ' vp - IT ( I + I ) "P

In matrix form

y

0

ab2 ,1 . b X ab /I . 1 X

r r (3 ^ 3? U^l^Ti^ ab /I + b \ a_ f l + b_N

rr ^3 W EI 3 a

^P

"p

Combining these relat ions with those developed for M,_ and V^,

0 s y

abi ( 1 + ^ ) EI 4 3a^

ab / I . b > ET ^ I " " ^ ^

EI ^3 2a^

^ f - + - ) EI ^3 ^ a^

m<#> mc<v^

mcai 2 ( I + mc2)u»2 XX /

iiilrtafc"- •

10

After the matrices are multiplied together, the y and 0 terms

can be collected on one side of the equation. The determinant of the

coefficients of the homogeneous equations must equal zero for non-zero

values of y and 0. The eigenvalues of CJ are the natural frequencies of

the model. Using a diameter of 20 inches, solution of the determinant

gives natural frequencies of 0.25 cycles per second and 9.2 cycles per

second.

Torsion

Because of the large projected area of the lights, non-uniform

wind loading of the structure can be expected. For such loading, rota­

tional motion about the vertical (z) axis of the pole will result

(Figure 1). If the frequencies of oscillation for the rotation are

close to the frequencies for bending, coupling between the modes can be

expected. The effect of this coupling is that the bending mode oscilla­

tion and the torsional mode oscillation can occur simultaneously with

one mode, then the other, dominating.

For rotational motion, the support at the base should prevent

rotational deflection since the bottom of the pole is bolted to a con­

crete foundation. The upper support provides but little restraint for

torsional oscillation and will be neglected for this model.

For the mass less beam model being considered as a first approxi­

mation, the angular momentum of the lights is found from

[H] = [I] [e]

For principal axes at the centroid of the mass and rotation considered

11

about the z axis only

Mz = Hz = Izz'e

where I^^ is found from I^^ = |^ m^-z^2 ^^g^g ^^ corresponds to the

mass of one of the lights and z is its distance from the z axis.

The torsional resistance of the pole is and is denoted as k.

Summing moments about the pole at its junction with the lights.

- k e^ = Izz^z

The solut ion to th is equation is

The natural frequency i s , therefore,

ou = J ^ or f = 1 . T^ ^' ' " 27r \ /I

zz zz

Substituting the values of Figure 1 into this expression results in

f = 3.64 cycles per second. This frequency is of the order of those

computed for bending of this model, and some coupling between bending

and torsion is to be expected.

CHAPTER III

TRANSFER MATRIX MODEL

Assumptions

This method of analysis considers the pole to consist of a

series of concentrated masses connected by tubular beam sections. The

relations which will link adjacent masses are formulated using the

assumptions made for linear elastic materials. Each mass corresponds

to the weight of one particular section of the pole. The beam section

which connects it to adjacent masses has a cross-section corresponding

to the average dimensions of the tapered section of the pole represented.

The relations can be mathematically combined with similar expressions

for adjoining sections so that the beam equations can be represented

as a series of matrices. The boundary conditions developed for the

previous model are applied to the matrices corresponding to the sections

at which these conditions occur. After computing the matrix resulting

from the series of matrices, the determinant of the coefficients is set

equal to zero to determine the eigenvalues or natural frequencies of

the model. With these frequencies, the deflection, slope, moment, and

shear distributions for the loads can be plotted from the matrix analysis

To illustrate the application of the theory to the pole, the

previous model is analyzed using the method of transfer matrices. The

equivalence of the results confirms the accuracy of the transfer matrices

as a method of analysis. The model is then applied to a pole consisting

of 28 discrete masses and elastic sections. A general outline of the

12

13

application follows, and a computer program is given in Appendix B.

While consideration of damping is not essential to this application, a

method of applying it is outlined.

Bending

The following formulas are used from elementary mechanics of

solids: shear V = EI -^ , moment M = EI ^ , slope 0 = -^ , and

deflection y = y. As a basis for the transfer beam relations, consider

a cantilever beam loaded as shown in Figure 4.

M

V V / > X

1'

y

Figure 4.--Cantilever Beam

V(x) = V

M(x)

EI 0(x) = -MX -

EI y(x)

M + Vx + Cl

Vx2 + C 2'

_ M X 2 _ Vx3 ^ MLx + ^ + C T

C = 0

C2

Co

= ML + VL'

ML2 VL3

For X = 0 V(o)

M(o)

0

V

M

ML + ML2

rr 2EI ML2 _VL^

2EI 3EI

14

Applying these relations to a generalized segment between masses

n and n-1 of Figure 5,

Figure 5.--Beam Segment Connecting Concentrated Masses

>n

y'n

,R M'- L VL L2 0[;_-| + "n "-n , 'n '-n

(EI)^ 2(EI)n

" ^ "- " 2(EI). 3(EI) (EI)n 3(EI)n

^n ^n-1

ML = M^ 1 + Vn L n n-1 ^ n

Subst i tut ing the las t two equations into the f i r s t two,

,L .R + ^n-1 ^ ^ C l ^ ^ ' - V l ^ (EI)n 2 (E I ) ,

MR , I 2 v^ , I 3 wL = wR , . MR , L - ^^n-1 Ln . ^n-l l-n ^n J^n-1 V l n 2(EI)n 6(EI)n

15

These relations can be expressed in matrix form as

-y

0

M

V ^ ^n

0

0

0

L

1

0

0

_Li 2EI

L EI

0

L 3 ^

6EI

L2

2EI

L

1

-y

0

M

V

R

n-1

where the negative deflection is used in the column matrices in order to

have a positive square matrix symmetric about one diagonal. This rep­

resentation is sometimes referred to as a field transfer matrix and can

be represented as

zL = F ZR n n n-1

The relations governing the mass elements are developed in the

same manner as were those for the beam sections. The relations for

Figure 6 are

n

"n r t O M "n VR n

" n n

Figure 6.--Concentrated Mass

'n J'n

^'n - ^n^

"n = "n

Vn = V^ - m^yn

Assuming harmonic motion, m^y^ = -aj2m^y^

16

Expressing these relations in matrix form.

-y ^

M

V n

0 0

0 0

0 0

mo»2 0 0

0

0

0

1

-y

0

M

k

R

This representation is sometimes referred to as a point transfer matrix

and can be represented as

zL = p zR n n n

With these relations a beam of discrete elements such as Figure

7 can be represented as

^28 " ^28''27^27^26 •*• ^1^1 o

28 27

17

J£ 2 1

Figure 7.--Pole Composed of 28 Discrete Masses

The boundary conditions can be applied by modifying the appropriate

matrices to account for the conditions acting on a particular section.

Simplified Model

The model used in Chapter II can be analyzed using transfer

matrices and the results from the different methods compared. The mass-

less beam analysis can then be expanded to an analysis for the model for

the pole consisting of concentrated masses. Figure 8 represents the

model analyzed in Chapter II.

18

^ >

- * hi

t 2c

b

a

1

EI = Constant

Outside Diameter = 20 Inches

Inside Diameter = 19 1/2 Inches

0

O ?//^ Figure 8.—Prof i le View of Simplif ied Model

The boundary conditions are

X = 0 ,

X = 8,

X = 28,

y = 0

y

V

M

mu|2y + mcu)20

mcu2y + ( I ^ ^ + mc2)aj2|z5

The governing expression is

A -' FzPlFlZg

where

zR ^0

0

0

\A rL _

0

mui y + mcu»20

^

mcw2 + (1^2 + mc^)cx)20

19

The point matrix P has zero for i t s def lect ion

for the reaction at R .

component and accounts

Then

l^ = F 1^ -1 "^0^0

zk =

0

0 0

0 0

1

0

a3 6EI

0

2EI

a EI

2EI

0

0

V

6EI

a2 2EI

0

1

< s

0

0

0

V Jo

At P.

' \ -

0 0 0 V

S . r>>

SO that a0^ + ^ V = 0 0 6EI 0

or V = 0

0 6EI

Subst i tut ing th is value into the previous 4 by 2 matr ix.

zL = F 7 ^ = 1 o'-o

0

-2

-6EI a

-6EI . ^

20

I P In order to go from Zir to Z ' the reaction R-] must be accounted for so

that

0 0

P,F Z^ 1 0 0

-2

•6EI a

•6EI

0

0

/

R l

Then

l\- 1 Pi^0^0

0

0

0

From the boundary conditions

0

0

2EI

b EI

1

0

_b3 6EI

_ ^

2EI

b

1

0

-2

.6EI a

6EI

0,

p 2 mm y„ + mco* 02

mew 2y2 + (Izz " mc2)oo20.

0

0

0

1

t,

R.

Equating the two expressions for Z^ results in four homogeneous equa­

tions and four unknowns. The four unknowns are 0o, R-], "i^, and 02- After

collecting these unknowns on one side of the equation, the determinant

of the coefficient matrix can be set equal to zero to determine the

21

eigenvalues ofoo. To simplify this operation, the expressions for Y2

and 02 from the first two equations can be substituted into the last

two equations which can then be solved for 00. When the values used in

the analysis of Chapter II were applied as indicated above, the computed

natural frequencies were the same as previously found.

Pole With Concentrated Masses

The method of solution applied to the previous model can be

expanded to solve the problem of a tapered pole consisting of a series

of masses by computing a set of transfer matrices for each mass. The

values of I, moment of inertia, in each field transfer matrix correspond

to the particular cross-section represented by the field transfer matrix

The boundary conditions are the same as those of the previous problem.

If the pole is considered to consist of 28 masses with pinned

supports at 0 and 8, the matrix equation is

^ 8 " ^28P27''27''26 •*• ^g^B^B '•• 1 0^0

where

0

0

0

y

and rL _ '28

0 p 2

mw y + mc<*> 0

mctt;2y + (I + mc2)oj20

The point matrix PQ has zero deflection and must account for the reaction o

RQ. As in the simplified model, this matrix for the upper support will o

have the form

2?

9 l l 912

§21 922

931 932

[941 942^

f N

0

V 0

- y

0

M

V

where y = 0.

8

Then g^^ 0^ + g V = 0, V = ' 9n0o nl2

Substituting this condition back into the matrix and accounting for the

reaction Rg,

rR _ -8

1 (912 92' gi2 ^^1^ 321

1 (912 93 1

911 922)

911 932) 912

1 , g^2 ^^12 941 - 911 942)

The matrix multiplication then continues for Z^g = F23P27F27

N

0

0

0

1 /

rR FgZg. This expression will be of the form

^28^27^27 ••• FgZg

Since Z 28

J l l

J21

J31

^41

J12

J22

J 32

J 42

f •

K h

\ ^

7 9 m«^y2 + mcu>^02

mcw2y2 + (Izz + mc^)u>20.

23

there are four equations and four unknowns for the analysis. Because

the equations are homogeneous, the unknowns can be collected on one side

of the equation and the determinant of their coefficients set equal to

zero to determine eigenvalues of6o.

This procedure to solve foroi is carried out on a digital computer

by substituting successive values ofau into the transfer matrices and

computing successive values of the determinant until there is a change

in the sign. When this happens, a value ofoo causing a zero value for

the determinant has been passed. The computer is programed to then use

smaller intervals of w until the value of the natural frequency is deter­

mined to within the desired accuracy. The program will then continue its

search for higher natural frequencies.

Because the matrix relation written for the structure consists of

equations for the deflection, slope, moment, and shear of each concen­

trated mass of the structure, the four unknowns in the four equations can

be solved for if a forcing term is applied to section 28 and the value

of cw is specified. These values will comprise the Z^ column matrix. By

successively multiplying this matrix through the 28 sections of the pole,

the deflection, slope, moment and shear for each mass can be computed.

This procedure is explained in the section for forced vibrations.

Torsion

The equations for torsional vibrations for a beam consisting of

a series of discrete masses can be developed along the same lines as

those for the theory applying to bending vibrations. An expression con­

sisting of a series of transfer matrices can be developed and the

24

boundary conditions applied to it to determine the eigenvalues of 6J.

Figure 9 represents a general segment of the beam connecting the

masses n and n-1.

T ^ ^-'n-1 <-MR Vl

n

>

T

^ e

n

n

Figure 9.--Massless Torsional Beam Connecting Disks

The relations governing the torsional deflection and torque of the beam

section are

T L = TL n T^i- 1

T? , L. e n - "n-1

e? . ^ n-1 "n (JG)

n

Expressed in matrix form

e

T

or

R

n

L 1 JG

0

/• N

e

T

R

n-1

yL - F 7R

The relations for the general mass n of Figure 10 are

25

TL ^ n "^

R = ^ T n _x Ie

n

Figure 10.--Disk Representing Concentrated Mass

Assuming harmonic motion.

n n

n = n ^ '^n

e = -to ^e

The mass matr ix, Zf| = P^Z[:J , is then

e

T

R > ^ N

n " -w2i

0

0

e

J . n

The boundary conditions applied to this problem are the same as

those used in the massless beam model. For a pole divided into 28 masses

where

rR _ •28 "

rR _

^28^27^27 I 0 0

0 T

and Z R _ e 28 - 0

and P20 has a value of I which accounts for the rotational moment of

inertia of the light bank about the pole. For free vibrations, no

26

external torque is applied to the structure. Values of u> resulting in

element 2, 1 of Z2g being zero represent eigenvalues for torsional

oscillation of the structure.

This formulation was programed for a digital computer solution

similar to that of the bending analysis. A listing of the program is

given in Appendix B and the results are included with those for bending

oscillation in Chapter IV.

Damping

The analysis thus far has neglected the effect of damping in the

system. Because mechanical dampers are not used, the amount of damping

in the structure is expected to be small. A comparison of a damped and

undamped analysis of a cantilever beam having only structural damping

has shown that while damping is present, it does not significantly affect

the natural frequencies.^ However, an explanation of a method for

including structural damping is made and an outline of its application

is given.

Damping is usually accounted for as a force proportional to the

velocity of mass under consideration. For structural damping, it has

been shown to be useful to have the coefficient of the damping term to

be proportional to the frequency of oscillation.^ This can be expressed

as

cH = k • , 00

where k is the spring constant and h is dependent upon the material

(0.005 and 0.01 for a welded structure).

27

The impedance of a damped-spring-mass system is R(t) = kx + ex.

For structural damping, the impedance is R(t) = kx + x. Assuming

oscillatory motion, x = A cos oi»t or in complex notation, x = Re(AeJ'*' )

where Re refers to the real part of eJ^t. For structural damping, then,

R(t) = Re [(k + jkh) Ae>t], jhg complex impedance I, therefore, is

k(l + jh) and k is proportional to 11 in the undamped beam formulation. L3

To account for damping then, k(l + jh) is proportional to ^ ^ 3 ^^^ .

Since I and L are dependent upon the dimensions of the structure, a com­

plex modulus of elasticity is the more desirable way of including

damping.

Then T = E(l + jh)

and ^ = G(l + jh)

where G = pd + y) cind u is assumed to be real.

This form of E replaces the purely real form used in the previous

formulations. Because this causes the transfer matrices to have real

and imaginary parts, a method of handling the two parts separately is

needed. To accomplish this, the complex impedance can be expressed as

Z = Z"" + Zi

U = U"" + jui

where l„ = Ti„ l^_-^

Then zr + jzi = (U*" + juM (Z"" + jzi)^ .

-K^'n-^ -"nCl) ^m^-^ u zi;.i)

28

In matrix form th is is

z'

V

-

n

r

- u ^ ^ -1

ui s.

•

z-"

z ' n-1

The f i e l d t ransfer matrix then becomes

• >

-yr

r

H>"

V"

-yi

i'

Mi

vi . /

R 1 L

0 1

L2_ L L 2EI 6EI

EI

0 0 1

0 0 0

n

0 0

0 0

0 0

0 0

-hL EI

0

0

and the point matrix is

-y

0

M

V

- y

0

M

V

R '

=

n *

0

L 2 _ 2EI

L

1

-hL^ ihJLl 2EI 6EI

-hL2

0

0

1 0 0 0

1 0 0

0 0 1 0

w^m 0 0 1

0

0 0

0 0

0

0

0

0

hL/ 2EI

hL, EI

0

0

L'

0

0

1 0

1

0

0

1—1

_J U

J

1

0

0

hL/ 6EI

hii 2EI

0

0

2EI 6EI

L2 WT

L

1

-y

0 ^

Mr

V^

0"

M""

V n-1

•

1 0 0 0

0 1 0 0

0 0 1 0

I't^^m 0 0 1

-y

0

M

V

- y

0

M

V ^n

29

For a complex formulation, the boundary conditions for the base

of the pole structure being considered are

.yr

^r

W

V

-yi

0i

Ml

. v i .

0

0'

0

v^

0

0i

0

vi

This results in P IFIZQ =

r 1 1 0 0 0

0 1 0 0

0 0 1 0

m<4»2 0 0 1

0

1

1 0

0 1

0 0

mu)2 0

0

0

0

1

0

0

0

0

1 >

L

1

0

0

0

0

0

0

L3

6EI(l+h2)

L2

2EI(l+h2)

L

1

-hL^ 6EI(l+h2)

-hL2 2EI(l+h2)

0

0

O

OO

O

L

1

0

0

hL^ 6EI(l+h2)

hL2 2EI(l+h2)

0

0

L3

6EI(l+h2)

L2

2EI(l+h2)

L

1

«s vs

For a pinned support at section 8, the matrix will have the general form

8 " Vz'^B ••• l 'l o

30

^ 0 ^ ^

0

M

V

0

0

M

V 8

911

921

931

941

951

961

971

981

912

922

932

942

952

962

972

982

9l3

923

933

943

953

963

973

983

914

924

934

944

954

964

974

984

r t\

vj

At section 8 the reaction Pg must be accounted for. In order to main­

tain the same number of unknowns in the formulation, two of the unknowns

at section 8 must be expressed in terms of the remaining two. From the

zero deflection of the pinned connection,

9ll ^^ + 9i2 V' + 9i3 ' + 9i4 V = 0

951 ^^ + 952 V'' + 953 ^^ + 954 V* = 0

From the equations above.

r _

(952

i _

(954

-1

- 954 gi2) 914

-1

- 952 q n . 9i4

(gsi - M g i 2 ) 0^+ (g53 " |5i gi3) 0 914

(951 -f^^ll' ^' " ^ 53

914

(g,, - ! ^ g i 3 ) 0 ' 912

V"" and V" are now expressed in terms of 0"" and 0"' alone and can be sub­

stituted back into the expression for FgP^Fg ... Pi^-jZ^ to have a matrix

in terms of 0" and 0"' alone. Designating the expressions for V^ and V" as

31

V» = X0'" + Y0T

vi = U0'' + Z0i

and introducing the reaction Rg, the matrix expression for Zg is

0

921 + 922^ " 924U

931 + 932X + g34U

941 + 942^ " 944U

0

0

0

0

1

0

961 + 962^ + 964^ 0

971 + 972^ + 974^ 0

981 - 982^ + 984^ ^

0

923 "*• 922^ "*• 924Z

933 - 932Y + 934z

943 + 942Y + 944Z

0

963 •" 962^ + 964^

973 "" 972Y + 974Z

983 + 982^ + 984Z

0

0

0

1

0

0

1

ac

y

«ii

pi

—

-y^

0 -

Mr

V

-y^'

0i

Mi

vi

After section 8 the process of matrix multiplication is continued

until Z28 is reached. At section 28 the boundary conditions developed

to describe the effect of the bank of lights are incorporated into the

point matrix Ppp. The relations acting at the point are

-y L = 28 ^28

0 28 0 R 28

"^^^^^28^ (^xx^^^^)^^ ^28 = ^28

mu>2 y28 "•" nicu) 02g = V28

32

The point transfer matrix at section 28 is then.

-y^

0 ^

Mr

V

-y'

0i

Ml

vi

R f

28^

1

0

-mcu)2

m 2

0

1

(I^x+mc^

2

0

V

0

0

•0

0

0

0

0

0

1

1

0

-mco)

-mar

0

0

XX

mcuj*

J

0 0

0 0

0 0

0 0 >

Ml"

V

-yi

Hi

Mi

. V i j 28

For free vibration

R

'28

-y^

0r

0

0

0i

0

s ^ .

Rows 3, 4, 7, and 8 represent four homogeneous equations with four

unknowns. The eigenvalues of co can be found by setting the determinant

of the coefficients equal to zero.

Dynamic forces can be applied to the formulation using the method

explained in the next section. When a forcing term and a> are specified.

33

the four unknowns in rows 3, 4, 7, and 8 can be solved for. These can

then be used to find the real and imaginary components of the deflection,

slope, moment, and shear for each of the 28 sections of the pole. For

an oscillating force with a frequency close to the natural frequency of

the pole, the real components represent bending in phase with the force,

while the imaginary components describe the bending 90° out of phase.

Bending From Dynamic Loads

The structure being considered has a low fundamental frequency,

as is shown in the results of Chapter IV. The period of the first mode

is approximately 2 1/2 seconds. Because wind gusts with this period are

possible, the effect of forced vibrations near resonance is an important

factor in the design of the pole.

The wind acts on the entire structure, but due to the large area

presented by the lights, the dominant force on the pole acts at the

section supporting the lights. For the discrete mass model, this is a

shear force acting at section 28. To account for such an oscillating

force, its frequency is used for60 in the transfer matrices and its mag­

nitude is used as a shear force applied to Z28. For an undamped bending

analysis with a force of P cos wt representing wind gusts acting on the

lights.

•28 2 2 mtr + mcoj mcu)2 + (I^z + mc^)(A)2 +P

34

Recalling that Z^F^ . . F^P^F^ . . . F2g is a four-by-two matrix

expression, the moment and shear equations at Z28 represent two equations

with two unknowns, 0^ and Rg. At section 8, V^ was determined from the

boundary condition to be V^ = - g r [ 0^ . With these three quant i t ies ,

913 the de f lec t ion , slope, moment and shear for sections 1 to 8 can be found

from

-y

M

n

11

'21

•31

•41

N

21

22

32

42 >

• N

"0

bo\

and for sections 8 through 28 from

-y

0

M

V n

N

u •2]

''31

L^41

'12

•22

•32

-42

0 0

R 8

rL _ where the e's represent the elements of the matrix ZJ: = F^Pn-i . - . FIZQ

fo r the r\^^ sect ion.

CHAPTER IV

RESULTS AND CONCLUSIONS

Computer Programs

To determine the natural frequencies and the bending diagrams for

these frequencies, Fortran programs were written for the undamped bend­

ing and torsional analyses of the discrete mass model. Appendix B con­

tains listings of the programs, but to aid in the understanding of them,

a brief description of the programs will be given.

In the undamped bending analysis of Chapter III, a four-by-two

matrix represented FIZQ. The bending program begins by generating this

matrix and the point transfer matrix Pi- The product of P-j x F-] is then

computed and stored as PF. The field transfer matrix for section 2 is

next generated and its product with PF is stored as Fl. The process is

then repeated for point mass 2 and section 3. Each time the process is

repeated, a new diameter and corresponding mass are generated. At section

8 the matrix Fl is respecified according to the relations given in the

bending analysis. At section 28 the elements for the boundary conditions

are computed. Because ZQFIPI ... P27E28 ^ ^28 represents four homogeneous

equations and four unknowns, the equations are valid only if the determ­

inant of the coefficients equals zero. Rather than evaluate a 4^^ order

determinant, the first two equations which represent expressions for

y28 and 028 in terms of the other variables alone can be substituted in

the remaining two equations to reduce the problem to that of evaluating

a second order determinant. As was mentioned in Chapter III, the entire

35

36

process just described is carried out for a single value of o) and the

value of the resulting determinant recorded. The process is then

repeated for a value of oj one cycle per second larger. If the value of

the determinant differs in sign from the previous determinant, a) returns

to its previous value and is increased by increments of yoo" °^ ^ cycle

per second until there is a sign change between two successive values

of the determinant. The value ofoi before the last sign change is then

recorded as a natural frequency. The increasing ofoj by increments of

y ^ of a cycle per second then continues until the next whole cycle

per second is reached. The increments then again become one cycle per

second. With double precision, the IBM Model 360 computer used has 16

decimal place accuracy. This allows accurate determination of natural

frequencies up to about 10 cycles per second. This frequency corresponds

to a value of w greater than 60. Wh-en values of this magnitude are used

in the 28 separate point transfer matrices and their product is computed,

the elements of the resulting matrix become too large for an accurate

determination of the value of its determinant.

Because the values of co being computed are to within ^r^^ of a

cycle per second of the value of to causing a zero value of the determi­

nant of ZQFI ... F28, such a frequency will result in unrealistic values

of the deflection, slope, moment, and shear for the structure. However,

for frequencies accurate only to within |g- of a cycle per second of the

natural frequency, representative values describing the bending mode can

be computed. When the program described is reduced in accuracy to j ^

of a cycle per second, a shear force can be applied to section 28 and

the unknowns of the matrix formulation solved for in the manner described

37

in the section on dynamic loads. After solving for the unknowns, the

deflection, slope, moment, and shear properties of each section can be

computed by multiplying the values of the unknowns times the F matrix

generated for each section. These properties compose the 28 x 4 matrix

Z which is transferred to a sub-routine and expressed in the output in

graphical form. By subscripting the diameter, area moment of inertia,

and cross-sectional area of each of the sections as they are computed,

they can be used to compute the maximum bending and shear stresses

for each of the sections once the respective moment and shear forces are

determined.

The magnitudes of the deflections and bending forces in the

graphical results are relative to the size of the oscillating force

applied. A 500 pound force which roughly corresponds to a 50 mile-an-

hour wind was used for the results shown.

The program for the rotational model is essentially the same as

the program for determining the natural frequencies of bending to within

yl^ of a cycle per second. All dimensions used in both programs are

expressed in units of feet, pounds, force,and seconds.

Results of Programs

The programs just described resulted in the tables and graphs

which follow.

TABLE

BENDING NATURAL FREQUENT IFS

OF THE POLE STRUCTURE

38

MODE CYCLES PEO SFCHNO -

1

2

3

4

5

6

0.3A

1.96

3 .60

6 .15

1 0 . 9 1

1A,59

39

TABLE

TOR SIGNAL NATURAL FRtCUENCIES

GF THE PfLE STRUCTURE

MCDE CYCLES PER SECOND -

I

2

1 .62

1A.9G

^PDF 1

CYCLES PEP SFCHND = 0 .^^0

40

SFCT ION

28 ?7 7H 25 2^ 23 22 21 2 ) 19 18 17 16 15 14 13 12 11 10 9 a 7 6 5 4 3 2 1 0

GFFLFfT ION

X X

X X

X X

X . X

X X

. X X

X ,x X X

X X X X X X X X X

X x

SLOPE

X X X X

X X X

X X

X X X

X X X

X X X X

X X X

X X

X X

X.

MAXIMUM DEFLECTION = 2 . 5 2 E 00 FEET

MAXIMUM SLOPE = 3 . 5 1 E - 0 2 FOOT/FOOT

41

MOOE

CYCLES PER SECONn = 0 . 4 0

SFCTI ON

23 27 ?6 25 24 23 22 21 2 0 19 13 17 16 15 14 13 12 11 10 q

8 7 6 5 4 3 2 1 0

MO' FNT

X. X X X .X

• . 4

X .X . X X

, X . X

X X X X X X X X X X X

X X

X X X

. X • • •

SHEAR

X . X X X ,

X X X . X X X , X X X ,

X X X X X X X X

• . 1 1 . •

X X X X X X X

MAXIMUM MOMENT ^ 1 . 3 4 E 05 FT-LBS

MAXIMUM SHEAR = 3 .37E 03 L^ S

42

^ODF 1

CYCLES PER SFCONO = 0 . 4 0

SECTION BENDING STRESS SHEAR STRESS

28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

X X

X X X

X X X X X X X X X X X X

X X

X X

X X X

X X X

X X X X X X X X X X X X X X X X

X X

X X

MAXIMUM RENDING STRESS ^ 1 . 5 5 E 0 ^ PS I

MAXIMUM SHEAR STRESS = 6 . 0 8 F ^l P S I

MODE

CYCIES PER SECOND = 2 . 0 0

43

SEC T I O N

28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10

9 a 7 6 5 4 3 2 1 0

X X X X X

X

>

X

X X

D E F L E C T I C N

X . X

X . X

<

X X

X X •

X. X .

• X .X .X . X . X .X

• . . « •

S L O P E

X X

X X X X X X X X X X X

X X

X X .X

X X

MAXIMUM DEFLECTION = 6 . 7 7 F - 0 1 FEET

MAXIMUM SLOPE = 4 . 6 8 E - 0 2 FOOT/FOOT

MODE

CYCLES PER SECOND = 2 .U0

44

SrC TIOM MOMENT SHEAR

28 27 26 ?5 24 23 22 21 20 19 18 17 16 15 14 13 12 11 1 0

9 3 7 6 5 4 3 7 \ 0

X .

X

X X

X

X X X

X X

X X X X X

X X

X X X X

X X X

• • • •

X X

X X X

MAXIMUM MOMENT = 1.51E 05 F T - L B S

MAXIMUM SHEAR = 6 . 7 5 E 03 LBS

45

MCDE

CYCLES PER SrCPND = 2 . 0 0

SECTION

28 27 26 25 24 23 ?7 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

t^FNDING STRESS

X

X , X,

X , X

X X X X X X X

• • «

•

X X

X X X X X

X X X

X X

X . X X

• .

X X X X

X X

SHEAR

X X X X , X,

X . X . X

X X

. • *

STRES

X . X . X

X X X X X X X

X

. .

MAXIMUM BENDING STRESS = 2 . 2 0 6 04 PSI

MAXIMUM SHEAR STRESS = 9 .14E 01 PSI

MODE

CYCIES PER SECOND = 3 . 6 0

46

S FCT ION

28 27 76 25 24 23 22 21 20 19 13 17 16 15 14 13 12 11 10 9 B 7 6 5 4 3 2 1 0

OEFL ECT ION

.

X .X X X X X .X .X . X . X . X . X . X . X . X .X .X .X .X X X X X

X . X. X. X

SL OPF

. X .X X X X

X. X X X X X X X X X X X X X X X X X X X X

MAXIMUM DEFLECTION = 6 . 8 9 E 00 FEET

MAXIMUM SLOPE = 1.03E 00 FOOT/FOOT

47

MODE

CYCLES PEW SECOND = 3 . 6 0

S E C T I O N

28 27 7 6 25 24 23 12 21 70 19 18 17 16 15 14 13 12 11 10

9 8 7 6 5 4 3 2 1 0

M O M E N T

X X

X X

X . X . X X

X • X .

X . X X X X

X . X •

X . X.

• X . X

X . X . X . X . X . X .X

• • • « •

X X

X X

X X

SHEAR

X, X

« . «

X X X

X X

X X

X . X .X

,x • X

. X , X

X . X

X > . .

MAXIMUM MOMENT = 1 . 1 7 E 06 FT-LRS

MAXIMUM SHEAR = 3 . 9 6 E 04 L3S

48

MODE

CYCLES PER SECOND = "^.6")

SFCT ION BENDING STRESS SHEAR STRESS

28 27 26 25 24 23 22 21 2 1 19 13 17 16 15 14 13 12 11 10

9 8 7 6 5 4 3 2 1 0

. X . X .X .X X X X X X

X . X.

X X X X X X X X X X X X X .X • X

X X

X X

X X X

X X

X X

X . X .

X . X . X . X . X .

X . X • X .X . X . X . X

MAXIMUM BENDING STRESS = I . H E 06 PSI

MAXIMUM SHEAR STRESS = I . O I E 03 PS I

MODE 4

CYCLES PER SECOND = 6.20

49

srcTios

23 27 26 75 24 23 7? 21 20 19 13 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

DEFLECTION

X X

X X X X X

X X X ,

X X X , . •

X X

X .X . X

X X X X

. X X

. X •x

• • •

SLGO^

X X X X

X X

X X X

X X X X

X X

X.

MAXIMUM DEFLECTION = ! • 3 1 6 - 0 2 FEET

MAXIMUM SLOPE = 1 . 8 8 r - 0 3 FOOT/FOOT

MODE

CYCLES PER SECOND = 6 , 2 1

50

SECT ION

28 27 26 75 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 3 7 6 5 4 3 2 1 0

X X X X X

X X

X

MOM«^NT

X. X .X . X

X X X X

. X

X .

X X. . X . X . X

X . X

X . X

• . • « •

SHEAR

X X X X

X X

X X

MAXIMUM MOMENT = 9 . 3 2 E 03 FT-LBS

MAXIMUM SHEAR = 5. 85E 02 L8 S

MODE

CYCLES PER SECOND = 6 . 2 3

51

S ECT I O N

78 27 76 25 24 23 22 21 2 0 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

B E N D I N G S T R E S S

X

X X

X X X X

X X

X X

. •

.X

. X X

X • X . X . X

• « «

X X

X X

X X

X X

X X

SHEAR STRESS

X X

X X X

X X

MAXIMUM RENDING STRESS = 1 .43E 03 P S I

MAXIMUM SHEAR STRESS = 5 . 6 7 E 00 PS I

52

Conclusions

The diagrams plotted by the computer program for the deflection,

slope, moment, and shear correspond to the patterns expected of these

quantities. Their maximum values also appear to be of the correct order

for the applied dynamic load. The large values in the third mode occur

because the frequency selected by the program for computation of these

bending diagrams corresponds to a natural frequency for the first three

decimal places as is shown in Table 1. For dynamic loads very close to

resonance, the analysis predicts structural failure as expected. The

bending stress diagrams for the first and second modes indicate that the

cross-sectional area of the pole decreases more rapidly than the moment

acting on it.

Comparison of Tables 1 and 2 indicates that coupling occurs

between bending and torsional oscillation. The lowest natural frequencies

of these tables correspond favorably with those determined by visual

observation of the structure.

LIST OF REFERENCES

1. Crandall, Stephen H., et a1. Dynamics of Mechanical and Electro­mechanical Systems. New York: McGraw-Hill Book Company, Inc., 1968.

2. Crandall, Stephen H., and Dahl, Norman C., eds. An Introduction to the Mechanics of Solids. New York: McGraw-Hill Book Company, inc., 1959.

3. Den Hartog, J. P. Mechanical Vibrations. 4th ed. New York: McGraw-Hill Book Company, Inc., 1956.

4. Pestel, Eduard C., and Leckie, Frederick A. Matrix Methods in Elastomechanics. New York: McGraw-Hill Book Company, Inc., 1963.

53

APPENDIX

A. Details of the Structure

B. Computer Programs

54

nf**i

APPENDIX A: DETAILS OF THE STRUCTURE

Dimensions

- /r

/S7

/o'^ ^"^^^^

^o' /^V>r^^ + /0\ f'vyALL

20, / i " ^ - ^ ^ ^

€9^1 /^^^^

1 / //

29' j^WAU

::_i-Jl"'""

/sr'

102.,

¥z.

55

-r; '1 . !• Ill

56

Photographs

Figure 1 Figure 2

Figure 3 Figure 4

i^JI^

57

APPENDIX B: COMPUTER PROGRAMS

Bending Analysis

INTEGER SECT, INCo , RESt '-'ODE, L I N i r , x , DOT, BLANK DOUBLE PRECISION CPS ,W , E, L , OD (28 I , ID (2 8 ) , IX ( ?8 ) ,

2 MASPOL( 2 8 ) , YBAR( 2 8 ) , I Z Z , F 1 ( 4 , 2 ) , P ( 4 , 4 ) , 3 E( 4 , 4 ) , f ' F ( ^ , 2 ) , Z 0 { 2 ) , A X ( 2 3 ) , D E L , D E I C L D , 4 PROD, A, F I X T , V 0 , Y 0 , R 3 , M l l , M 17 , V 2 l , V 2 ? , 5 T i l ,T12 , T 7 1 , T 7 7 , P I , M A S L H T

D I -^ENSIGN Z( 4 , 2 3 ) ,T ( 23) ,V ( 28) , L l NF (40 ) .C (2 0) DAT A I NCR, MODE, R E S / 3 * 0 / , R 8, CP S, DELOL 0 / 3 * 0 . D O / ,

2 YMAX, S M ^ X , T M A X , V M A X , T S M A X , V S M A X / 6 * 1 0 . E - 1 5 / , 3 E , P I / 4 3 2 0 . D 6 , 3 .1415926535897930 0 / , 4 0 0 T , X , B L A N K / 1 H . , 1 H X , I H /

C r, BEGIN GENERATING MATRICES C

1 SECT = 1 L = 5 . 0 W = 7 . * P I * CPS OD(SECT) = 1 2 . / 1 2 . I D ( S E C T ) = Qn(SECT) - ( 0 . 5 / 1 2 . 0 ) I X ( S E C T ) = ( P I / 6 4 . ) * ( 0 D ( S E C T ) * * 4 - I D ( S E C T ) * * 4 ) AX (SECT ) = ( ( 0 0 ( S E C T ) / 2 . ) * * 2 - ( ID( StCT) / 2 . ) * * 2 ) * P I MASPOK SECT) = AX (SECT) * ( L*0 . 2 8 4 * ( 12 . * * 3 ) / 32 . 1 7 4 )

C C GENERATE MATRIX F SUB 1 C

F l ( 1 , 1) = L Fl ( 2 , 1 ) = 1 .0 F 1 ( 3 , 1) = 0 . 0 F l ( 4 , 1 ) = 0 . 0 F i d , 2 ) = ( L * * 3 ) / ( 6 . 0 * E * I X ( S E C T ) ) F l ( 2 , 2 ) = ( L * * 2 ) / ( 2 . 0 * E * I X ( S F C T ) ) F l ( 3 , 2 ) = L F l ( 4 , 2 ) = 1 . 0

10 I D ( S E C T ) = OD(SECT) - ( 0 . 5 / 1 2 . 0 ) IX ( SECT) = ( P I / 6 4 . ) * ( 0 D ( S E C T ) * * 4 - I 0 ( S E C T ) * * 4 ) AX( SECT) = ( ( 0 0 ( S E C T ) / 2 . ) * * 2 - ( I 0 ( S E C T ) / 2 . ) * * 2 ) * P I MASPGL(SECT) = A X( SEC T ) • ( L * 0 . 2 8 4 * ( 1 2 . * * 3 ) / 3 2 . 1 74)

C C GENERATE MATRIX P SUB N C

l l

0 0 DO IF IF

11 l l ( I ( I

1 = 1 , 4 N = I , 4

. E O . N ) P( I ,N ) = 1 . 0

.NE. N) P(I ,N) = 0 . 0 t > ( 4 , l ) ^ MASPOLC S E C T ) * ( W**2)

58

C MULT P ( 4 X 4 ) X F 1 ( 4 X 2 ) = ^F(^»X2)

no 12 J = 1 ,2 00 12 I = 1 .4 PE( I , J ) = 0 . 0 on 12 K = 1 ,4

l ^ P F ( I , J ) = P F ( I , J ) 4. P( I , K ) * F l ( K , J ) C

C GENERATE MATRIX F SUB N C

15 DO 16 1 = 1 , 4 DO 16 N = 1 , 4 I F ( I . E Q . N) F( I , N ) = 1.0

16 I F ( I . N F . N) F ( I ,N) = 0 . 0 F( 1 , 2 ) = L F ( l , 3 ) = ( L * * 7 ) / ( 7 . 0 * E * I X ( S F C T ) ) F ( l , 4 ) = ( L * * 3 ) / ( 6 . 0 * E * IX (SECT ) ) F( 2, 3) = L / ( E ^ I X( SECT)) F ( ? , 4 ) = ( L * * 2 ) / (7 . 0 * E * I X (SECT ) ) F( 3 , 4 ) = L

C C MULT F ( 4 X 4 ) X P F ( 4 X 2 ) = F 1 ( 4 X 2 ) C

00 17 J = 1 ,2 DO 17 I = I , 4 F l ( I , J ) = 0 . 0 DO 17 K = 1 ,4

17 F 1 ( I , J ) = r i ( l , j ) ^ F ( I , K ) * P F ( K , J ) C ' C DETERMINE BENDING FOR EACH SECTION IF NEAR RESONANCF C

IF (RES . E Q . 0) GO TO 19 C C MULT F l ( 4 , 2 ) X Z 0 ( 2 , l ) = Z ( 4 , 2 8 ) C

DO 13 I = 1 , 4 Z( I ,SECT) = 0 . 0 DO 18 J = 1 ,2

18 Z ( I t S E C T ) = Z d t S F C T ) + F l ( I , j ) * Z 0 ( J ) T(SECT) = Z ( 3 , S F C T ) * O 0 ( S E C T ) / ( 2 . * I X ( S E r T ) * 1 4 4 . ) YRAP(SECT) = 2 . 0 * ( 0 D ( SECT) + I D( SEC T) ) / ( 2 . 0 * PI ) V( SFCT) = Z (4 ,SECT )*AX(SECT )*YBAR(SECT ) / ( I X I S E C T ) * 2 8 8 . ) I F ( ABS(Z( l . S E C T ) ) . G T . YMAX) Y MAX =ABS ( Z ( 1 • SECT ) ) IF { A B S ( Z { 2 , S E C T ) ) .GT . SMAX) SM AX= ABS( Z ( 2 , SEC T ) ) IF ( A B S ( Z ( 3 ,SECT) ) . GT . TMAX) T M AX = ABS ( Z ( 3 , S ECT ) ) IF ( ABS(Z( 4 , SECT) ) . G T . VMAX) VMA X=AB S( Z( 4 , SEC T ) ) I F ( A 8 S ( T (SECT ) ) .GT . TSMAX) TSM AX= AfiS( T( SECT ) ) IF (ABS( V(SFCT) ) . G T . VSMAX) VSMAX =ABS (V (S ECT ) )

19 NSET = SECT SETT = SECT + 1

59

c c c

DETERMINE HUW TO ALTER TMF NEW MAT . ICES

IF (S ECT .LT . a ) GO TO 70 IF ( SET r . E Q . 8) GO TC 30 IF ( SECT , L T . 29) GO TO 40 IF (SECT , F Q . 29 ) GO TO 50

20 nD(SECT) = OD(NSET) i- ( 7 . 5 / 1 2 . ) GO TO 10

3 0 P F ( l , n = 0 . 3 P F ( 2 , l ) = ( F l ( l , 2 ) V F l ( 2 , l ) - E l ( l , l ) * F l ( 2 , 2 ) ) / F l ( l , 2 ) P F ( 3 , 1 ) = ( F l ( L,2)<cF U 1 , u - F U 1, 1)=<'E 1( ^ , 2 ) ) / F l ( 1 ,2) P F ( 4 , 1 ) = ( F l ( l , 2 ) ' : ' F l ( 4 , l ) - F l ( I , l ) * F l ( 4 , 2 ) ) / F l ( l , 2 ) PF( 1 , 2 ) = 0 . 0 P F { ? , 2 ) = 0 .0 PF( 3 , 2 ) = 0 . 0 P F ( 4 , 2 ) = 1 .0 G = - F l ( 1 , 1 ) / F l ( 1 , 2 ) Z0( 2) = R8 On(SECT ) = 3 2 . / 1 2 . I D ( S E C T ) = OD(SECT) - ( 0 . 5 / 1 2 . 3 ) AX(SECT) = ( ( Q D ( S E C T ) / 2 . ) * * 2 - ( I 0 ( S E C T ) / 2 . ) * * 2 ) * P I IX (SECT) = ( P I / 6 4 . ) * ( 0 D ( S f c C T ) * * 4 - ID( S E C T ) * * 4 ) GO TO 15

40 OD(SECT) = OD(NSET) - ( 1 . 1 8 / 1 2 . ) L = 5 . 2 GO TO 10

50 IF (RES . E Q . 0 ) GO TO 80 CALL P lOT(Z fT ,V ,YMAX,SMAX,TMAX,VMAX,TS ' - lAX ,VSMAX,

2 DOT,BLANK,X ,CY,MODE) RES = 0 CPS = CPS ^ 0 . 1 DELOLD = DEL YMAX = l O . F - 1 5 SMAX = l O . E - 1 5 TMAX = 1 0 . F - 15 VMAX - l O . E - 1 5 TSMAX = l O . E - 1 5 VSMAX = l O . E - 1 5 GO TO 1

RO F I X T = 6 2 . 7 3 2 . 1 7 4 I 7Z = 1 6 . * F I X T * ( 1 . 2 5 * * 2 • 3 . 7 5 * * 2 + 6 . 2 5 * * 2 ) A = 7 . 5 MASLHT = 4 8 . • F I X T T i l = M A S L H T * A * ( W * * 2 ) T12 = ( I Z Z ^ MASLHT* CA**2 ) ) * ( W * * 2 ) T 2 1 = M A S L H T * ( W * * 2 ) T22 = M A S L H T * A * ( W * * 2 )

60

M i l M12 V2 l V2? DEL

F i( 3,n F l ( i , 2 ) Fl ( 4 , 1 ) «" 1( 4 , 2 )

4- T l l * F l ( l , l ) • T l l * F l ( 1 , 2 ) • T 2 l * F l ( 1 , 1 ) ^ T 2 l * F U l , 2 )

- r i 2 » - l (2 ,1 ) - T12*F 1( 2 , 2) - T 2 2 * F 1 ( 2 . n - T 2 2 * F 1 (7 , 2 )

= Ml 1*V?2 - M17*V21

DETERMINE WHAT NEW VALUE OF CPS TO USE

IF (CPS . G E . 8 . ) GH TO 199 IF ( INCR . E g . 0 ) GO TO 160 I F ( INCR . L T . 1 1 ) GO TO 171 I F ( INCR . E Q . 11 ) INCR = 0

160 IF (CPS . F O . 0 ) DELOLD = DEL PROD = DEL * DELOLD IF (PROD . L T . 0) GO TO 170 CPS = CPS ^ 1.0 GO TO 1

170 CPS = CPS - 1 .0 171 INCR = INCR *• 1

CPS = CPS <- 0 . 1 IF (INCR . E O . 1) GO PROD = DEL * DELOLD IF ( PROD . L T . 0) GO TO I

1 73 CY = CPS - 3 . 1 R8 = 5 0 0 . / ( V 2 2 - M 1 2 * V 2 1 / M 1 1 ) YO = ( - M 1 2 / M l l ) * R 8 VO = G*Y0 Z 0 ( 1 ) = YO Z 0 ( 2 ) = VO RES = 1 MODE = MODE + I CPS = CPS - 0 . 1 GO TO 1

199 CALL EXIT END

TO I

GO TO 173

c c r

61

SURROUTINf^ OL0T(7 , T , V , Y M \ X , SMAX, TMAX ,V MAX ,TSMAX,VS'^Ay, ^ DOT, BLANK, X,CY, '^ODE)

IMTFGER '^LANK, DOT, X , L I N E , SECT DIMENSION L I N F l 40) , Z ( 4 , 2 3 ) , T ( 2 8 ) , V ( 2 8 )

PLOT DEFLECTlUN AND SLOPE

WRITE ( 6 , 6 0 ) MODE, CY 6 3 FORMA T( I H I , / / / / / 1 H 3 ,39X ,4 HMODE , I 4 / / 1 H 0 , 3 1 X ,

2 70HCYCLFS PER SECOND = ,F 4 . 2 / / I H O ,1 6X , 7H SEC T I ON , 3 6 X , 1 0 H D F F I FCT I 0 N , 1 2 X , 5 H S L 0 P r / l H )

DO 63 NSFCT = 1 , 2 3 SFCT = 29 - NSFCT DO 62 J = I , 4 0

62 L I N E ( J ) = BLANK L I N E d l ) = OOT L I N E (31 ) = DOT J = 8 . * ( Z( 1 , SECT) /YMAX) 4 - U . 5 K = 3 . * ( Z ( 2 , S F C T ) /SMAX) + 3 1 . 5 L I NF ( J) = X L INE(K ) = X

63 WRITE ( 6 , 6 4 ) SECT, L INE 64 FORMAT! I H , 1 9X , I 2 ,2 X ,40A 1)

WR ITE ( 6 , 6 8 ) YMAX, SMAX 68 FORMATdH , 1 9 X , 7 H 0 , 1 0 X , 5 H , I 5 X , 5 H / / / / I H O ,

2 26X , 2 1HMAXIMUM DEFLECTION = , I P E 8 . 2 , 5 H F E E T / / 3 I H 0 , 2 6 X , 1 6 H M A X I M U M SLOPE = » 1 P E 8 . 2 , lOH FOOT/FOOT)

C C PLOT MOMENT AND SHEAR C

WRI TE ( 6 , 7 0 ) MODE, CY 70 FORMAT(I H I , / / / / / I H O , 3 9 X , 4 H M 0 D E , I 4 / / I H O , 3 1 X ,

7 20HCYCLES PER SECOND = ,F 4 . 2 / / IHO, I 6X, 7H SEC TI ON , 3 8 X , 6 H M 0 M E N T , 1 4 X , 5 H S H E A R / I H )

DO 73 NSECT ^ 1 , 2 8 SECT = 29 - NSFCT DO 7 2 J = 1 , 4 0

72 L I N E ( J ) = BLANK L I N E d l ) = DOT L I N E ! 3 1 ) = DOT J = 3 . * ( Z ( 3 , SECT) /TMAX) + 1 1 . 5 K = 8 . * ( Z ( 4 , S E C T ) / V M A X ) > 3 1 . 5 L I N E ( K ) = X L I N E ( J ) = X

73 WRITE ( 6 , 7 4 ) SECT, L INE 74 F0RMAT(1H , 1 9 X , I 2 , 2 X , 4 0 A I )

WRITE (6 , 7 8 ) TMAX, VMAX 78 F O R M A T d H , 1 9 X , 2 H 0 , 1 0 X , 5 H ,15X ,5H / / / / I H O ,

2 26X, 17HMAXIMUM MOMENT = , 1 P E 8 . 2 , 7 H F T - L B S / / 3 I H 0 , 2 6 X , 1 6 H M A X I M U M SHEAR = , I P E 3 . 2 , 4 H L B S )

62

c r

PLOT BENOING AND SHEAR STRESS

WRITE ( 6 , 3 0 ) MODE, CY 30 FORMA T( H I , / / / / / I H O , 39X ,4H MODE , I 4 / / I HO , 31 X ,

7 20HCYCLES PER SECOND = , F^ . 2 / / I K J , I 6X, 7H SfC TI ON , 3 4X ,14HBFNDING ST RESS ,7 X, I 2 HSHE A^ S T ^ E S S / I H )

DO 33 NSEC T = 1 , 28 SECT = 29 - NSFCT o n 82 J = 1 , 4 0

3 2 L I N E ( J ) = BLANK I I N C d l ) = DOT L I N r ( 31) = DOT J = 8 . * ( T ( S F C T ) / T S M A X ) + 1 1 . 5 K = 8 . * ( V ( S E C T ) / V S M A X ) • 3 1 . 5 L I NE(K) = X L INE( J ) = X

33 WOITE ( 6 , 3 4 ) SECT, L I N E 34 FORMATdH , 1 9 X , I 2 ,2 X , 4 0 Al )

WRITE ( 6 , 8 8 ) TSMAX, VSMAX 33 F O R M A T d H , 1 9 X , 2 H 3 , 1 3 X , 5 H , 1 5 X , 5 H / / / / I H O ,

26X,?5HMA XIMUM BENDING STRESS = , I P E 3 . 2 , 4 H P S I / / 7 3 I H 0 , 2 6 X , 2 3 H M A X I M U M

RETURN END

SHFAP STPESS = , l ' ' E 3 . ? , 4 H P S I )

Viwm

63

Torsional Analysis

DOUBLE PRECISION Z (2 ) , E (2 ,2 ) , P ( ^ , e ) i f'Z ( 2 ) , 2 J T , Q D , I D , I X , W , L , M A S S , D F L ,CPS,CY

INTEGER COUNT , INCR, MODE DI MENSION C ( 2 0 ) DATA INCR, MODE / C , 0 / , W , C P S , G / 2 * 0 . 0 0 , 1 . 6 5 60 9 /

1 e cu NT = 0 W = 2 . * 3 . 1 4 1 5 S 2 6 5 * CPS Z( I ) = 0 . 0 l{2) = 1 .0 L = 5. 0 OU = 1 2 . 0 / 1 2 . 0

15 ID = GD - ( . 5 / 1 2 . ) MASS = ( ( { C D / 2 . ) * * 2 - ( I C / 2 . ) * * 2 ) * 3 . 1 4 159)

2 * ( L * 0 . 2 8 4 * 1 2 . * * 3 ) / 3 2 . 174 JT = 3 . 1 4 1 5 9 * ( G C * * 4 - I C * « 4 ) / 3 2 . C I X = 0 . 5*MASS*( ( C 0 / 2 . 0 ) * 2 + ( l C / 2 . 0 ) * * 2 )

90

100

110

F( 1 , 1) F ( 2 , l ) F( 1 ,2 ) F ( 2 , 2) P ( l , l ) P( 2 , I ) P ( l , 2 ) P (2 , 2 ) MUL T . CC 90 FZ( I ) DO 90 fZil) MULT. CO 110

2 0 0

3 0 0

4 0 0

= l . C = 0 .0 = L / ( J T * G ) = 1.0 = I . 0 - - V«.**2 * I X = 0 .0 = 1 . 0

FXZ=FZ I = 1 ,2 = 0 . 0 J = I , 2 = F Z d ) + F d , J ) * Z ( J ) PXFZ=Z

I = 1 ,2

I ' * - • •

Z ( I ) = 0 . 0 DO l i e J = l f 2 l ( l \ = Z ( l ) + P ( I , J ) * E Z ( J ) COUNT = CCUNT + I IF IF IF IF 00 GO

OD GO FZ(

(COUNT (COUNT (CCUNT (COUNT = CO • TO 15

= CD -TO 15 I ) - Z(

. L T . . L T . . E C . . G T .

8) 2 8 ) 2 8 ) 28)

GC GO GC GO

TC TO TC TO

200 300 400 500

( 2 . 5 / 1 2 . 0 )

( 1.18/12.0)

1) F Z ( 2 ) = Zi2) MTOP = 4 8 . * 6 2 . / 32 IX= MTUP * ( 2 . 0 * * 2 •

. 1 7 4 1 5 . * * 2 ) / 1 2 . C

64

500

6 0 0

609 6 1 0

P ( l , 1 ) = 1 . 0 P( 2 , 1 ) = -Vi**2 * I X P ( l , 2 ) = 0 . 0 P(2 , 2 ) = i . O GO TO 100 DEL = l{2) IF (CPS . G E . IF ( INCR .EQ If^ ( INCR . L T I F ( INCR .ECJ IF (CPS . L T . PROD = DEL * IF (PROD . L T . 0 ) CPS = CPS + 1 . 0 GO TC I CPS = CPS - i . O INCR = INCR • I CPS = CPS + 0 . 01 IF ( INCR . E Q . 1 ) GO PROD = DEL * CELCLD

30) GC TC 700 0 ) GO TO fcOO

100 ) GO TO 6 10 IOC) INCR = 0

0 . 0 0 0 1 ) DELOLD = DELCLD

GO TG tc;

DEL

TO 1

IF (PRUO . L T . 0) GO TO 65C GO TO 1

6 5 0 MODE = MCDfc 4- 1 C(MOD£) = CPS - C . O l DELOLC = DEL GO TO 1

700 WR ITE( 6 , 710) 710 FORMAT ( I H l , / / / / / / / I E , 3 0 X , 9 H T A 8 L E 2 / / 1 H ,

2 2 0 X , 2 9 H TORSI UNAL NATURAL ERE CUENCI E S / / I H , 2 4 X , 3 21HGE THE PULE STR UC TUR E / / I H , 1 6 X , 4 3 8H / I 7 >, 5 I H - , 3 6 X , 1 H - / 1 H , 1 6 X , 4 H - , 4HMCDL ,10X , 6 20FCYCLES PER SECOND - / I H , 16X , I N - , 36 X, I H - )

DO 718 J = 1,MCDE W R I T E ( 6 , 7 15)

715 F O R M A T d H , 16X , 1 H - , 36X , I N - ) 718 WRI TE ( 6 , 720) J , C ( J ) 720 FORMAT( lH , 16X, 4 H - , I 2 , 1 8X ,F 5 . 2 , 8X ,1 H-)

WRITE ( 6 , 72 5 ) 725 FORMAT( IH , 1 6 X ,

2 38H ) CALL EXIT END