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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