The forced vibration of viscus damped, N-beam structures
by Daniel Franklin Prill
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE in Aerospace and Mechanical Engineering
Montana State University
© Copyright by Daniel Franklin Prill (1970)
Abstract:
The equation of motion for a linear beam with viscous damping and dynamic loading is presented. The
particular solution for a uniformly distributed load which varies sinusoidally in time is presented for a
beam with general boundary conditions. It is recognized that the boundary conditions are functions of
the first three spatial derivatives and thus recursion formulas in matrix form are developed.
For a system of beams which are interconnected, knowledge of the boundary conditions allows
development of algebraic equations which can be solved for the integration coefficients of the
particular solution for each beam of the system. Thus a method, including the appropriate computer
program, is presented which results in the exact particular solution for a structural system composed of
a large number of beams.
Also, the method is such that any number of the natural frequencies and mode shapes for the undamped
structural system can be approximated with a high degree of accuracy.
Example solutions for structural systems are presented for a cantilever beam and for a three beam
configuration. Statement of Permission to Copy
In presenting this thesis in partial fulfillment of the require­
ments for an advanced degree at Montana State University, I agree that:
the Library shall make it freely available for inspection.
I further
agree that permission for extensive copying of this thesis for scholarly
purposes may be granted by my major professor, or, in his absence, by
the Director of Libraries.
It is understood that any copying or publi­
cation of this thesis for financial gain shall not be allowed without
my written permission.
-
A
y/
Signature ,
Date
// — ~L
f
— 7O
■
J
<
THE FORCED VIBRATION OF VISCOUS,
DAMPED, N-BEAM STRUCTURES
•by
DANIEL FRANKLIN PRILL
A thesis; submitted to. the Graduate Faculty in partial
fulfillment o f .the requirements, for the degree
of
MASTER OF SCIENCE
in
Aerospace and Mechanical Engineering
Approved:
Head, Major/department
Chairman, Examining Committee
Graduate^Dean
MONTANA STATE UNIVERSITY
Bozeman, Montana '
December, 1970 '
ACKNOWLEDGEMENTS
The continuing guidance of Dr. D. 0. Blackketter is gratefully
acknowledged.
TABLE OF CONTENTS
Page
TITLE P A G E .................................
V I T A ......... ..
i
. . . _............................... ..
ii
A C K N O W L E D G E M E N T .................
iii
LIST OF T A B L E S ................'.......................................
v
LIST OF F I G U R E S ........................................................ vi
A B S T R A C T ................................................................ vii ■
CHAPTER I - I N T R O D U C T I O N .............................................
I
CHAPTER II - SOLUTION OF THE FORCED VIBRATION OF THE VISCOUS
DAMPED BEAM EQUATION
.................. . . . . . . .
2
Equation of Motion
........................................
2
Nondimensional Equation of Motion ........................
4
Solution
...................................................
5
CHAPTER III - APPLICATION TO AN N-BEAM S T R U C T U R E .................
11
M e t h o d ....................................................
.
11
P r o c e d u r e ...................................................... 11
CHAPTER IV - EXAMPLES
. ’............................................... 16
The Cantilever B e a m .............
16
A Three Beam S y s t e m ........................................
23
CHAPTER V - D I S C U S S I O N ................' ......... - .................. 36
APPENDICES - Appendix A, Computer Program
......................
.
39
LITERATURE CONSULTED . .................................................. 62
LIST OF TABLES
Number
4.1
Title
Three-Beam System Properties
LIST OF FIGURES
Number
Title
Page
2.1
Sign Convention . . . . .
.........
4.1
Cantilever B e a m .............
. . . . . .
.........
3
17
4.3
Q
Magnification Factor vs Frequency Ratio: -p— = 0
c
Phase Angle vs Forcing F r e q u e n c y ...........
4.4
Three-Beam S y s t e m ............................................. 24
4.5
End Elements
4.6
Nondimension Deflection at Z^=I vs Forcing Frequency: Cf O
4.7
Beam One
Mode S h a p e : First Natural F r e q u e n c y .............. 30
' 4.8
Beam Two
Mode Shape:
4.2
.........
21
22
.................................................. 27
First Natural Frequency
4.9
Beam Three Mode Shape:
4.10
Beam One
Mode Shape:
4.11
Beam Two
Mode Shape:
4.12
Beam Three Mode Shape:
. . . . . .
First Natural Frequency
29
31
.........
32
Second Natural Frequency
......
33
Second. Natural Frequency
. . . . .
34
Second Natural Frequency
35
ABSTRACT
The equation of motion for a linear beam with viscous damping and
dynamic loading is presented.
The particular solution for a uniformly
distributed load which varies sinusoidally in time is presented for a
beam with general boundary conditions.
It is recognized that the bound­
ary conditions are functions of the first three spatial derivatives and
thus recursion formulas in matrix form are developed.
For a system of beams which are interconnected, knowledge of the
boundary conditions allows development of algebraic equations which can
be solved for the integration coefficients of the particular solution
for each b e a m of the system.
Thus a method, including the appropriate ■
computer program, is presented which results in the exact particular
solution for a structural system composed of a large number of b e a m s .
Also, the method is such that any number of the natural frequencies
and mode shapes for the undamped structural system can be approximated
with a high degree of accuracy.
Example solutions for structural systems are presented for a
cantilever beam and for a three beam configuration.
CHAPTER I
INTRODUCTION
The method of solution to the free-vibration of linear beams is
well known
[1]^.
When damping and a forcing function are considered,
the complexity of the solution increases and, in some cases, an
approximation technique has been used to determine a solution [2].
If an undamped, multi-element structure is to be analyzed for freev i b rations, the eigenvalue problem may become complex [3].
This paper presents an exact solution to the problem of a viscous
damped beam with a distributed load,forcing function.
presented is similar to that of Stanek [4].
The solution
The solution, as presented,
is in matrix form and is applied to an N-beam structure.
Examples are worked for a cantilever beam and for a three-beam
structure.
Since the computer is used to obtain numerical results,
Fortran-4 level programs are presented.
A method of finding natural frequencies and the mode shapes of
these frequencies is also presented.
Numbers in brackets refer to literature consulted.
CHAPTER II
SOLUTION OF THE FORCED VIBRATION OF
THE VISCOUS DAMPED BEAM EQUATION
Equation of Motion
The derivation of the linear beam equation for dynamic motion is
well known [I].
When viscous damping and a forcing function are included
in the derivation, the resulting equation of motion has two terms which
are not in the linear equations.
These terms are the damping term,
and the forcing function, F q sinait.
Adopting the sign convention of Figure 2.1, the equation of motion
is
W
3 y
+ C i z + E-
L at
where:
F
o
( 2 . 1)
sinu)t
W = Total weight of beam (lb)
2
g = Acceleration due to gravity (in/sec )
L = Length of beam (in)
C = Total damping constant
(Ib-sec/in)
y = Lateral deflection of beam (in)
x = Axial coordinate of beam (in)
2
E = Modulus of Elasticity of material of beam (Ib/in )
4
I = Moment of Inertia of beam cross-section (in )
Figure 2.1.
Sign Convention
4
F q = Maximum load on beam (lb/in)
u) = Circular frequency of load (rad/sec)
t = time (sec)
Nondimensional Equation of Motion
The solution of Equation 2.1 may be presented in a more efficient
fashion in nondimensional form.
The equation of motion is put into
nondimensional form by use of the following quantities:
x
Z - L
Y ' T r r
o
C EI
o
.
<2 -2>
'
.
0 = U)t
where:
z = Nondimensional axial coordinate
Y = Nondimensional lateral deflection
0 = Nondimensional time
C
o
= Nondimensional constant
2
2
C
is a numerical nondimensional constant which is chosen for
o
convenience.
The constant may be any number, but is chosen to be eight
for the cantilever beam example of this paper and is chosen to be one
for the three-beam example.
See Chapter IV for reasons for these choices.
5
The nondimensional equation of motion is
C
o
sin0
(2.3)
W L 3OJ2
gEI
where:
(2.4)
and:
a =
Wm
Solution
For this study, only the steady-state solution is of interest.
The solution will, therefore, be a particular solution.
T h u s , assume a
solution of the form:
Y =
sinG +
where the Z-functions
al axial coordinate,
cos0
(2,5)
(Z® and Z^) are functions of only the nondimension-r ■
z.
The subscripts and superscripts of Equation 2.5
are explained as follows:
1) The superscript on the Z-function in question will be
either an "s" or a " c " .
An "s" will designate that the Z-
function is a coefficient of the term sin0; and a "c" will
designate that the Z-function is a coefficient of the term
COS0.
2) The subscripts of the Z-functions will designate the order
6
of the derivative of the Z-functions with respect to the
axial coordinate, z .
For example, the nondimensional slope
9Y
of the beam, — , would be written as:
Y 1 = Z® sin6 + Z^ cos0
(2.6)
• »
'
Differentiating Equation 2.5, substituting the results into Equation
2.3 and equating coefficients of sin6 and cos6 yields
- B 4 Z8 - ct B4 Z c + Z® = C
(2.7)
- B 4 Z c + ct B4 Zs + Z^ = O
(2.8)
O
0
O
4
0
0
4
Equation 2.5 becomes a solution of Equation 2.3 if the Z-functions
satisfy Equations 2.7 and 2.8 simultaneously.
Combining Equations 2.7
g
and 2.8 to eliminate Z^ yields the following eighth-order nonhomogeneous
differential equation.
Zg - 2S4 Z/ + B8 Z C (I +
O
4
0
a2) =
-C
O
ct B4
(2-. 9)
The following symbols are introduced for writing the homogeneous
solution of Equation 2.9.
H -e(l + a 2)1/8
♦ -
tan"1 a
a = y cost))
(2.10)
7
b = ysin<j)
(2.10 con't)
The complete solution of Equation 2.9 is
Z c = Icosh az
o
+
sinh azj M c
-j O
[cosh bz
sinh bzj
fcosbz
smbz
c
[cos az I + 2 C
sin az
(2 .11)
where Z , the particular solution of Equation 2.9, is
4
C a B
(2 .12)
zP = -
^
x
-
The eight integration constants are the elements of the
and
matrices where
A
D
C
B
n o
and
(2.13)
-J o
The subscripts and superscripts of Equation 2.13 have the same meanings
as those of the Z-functions.
The form of the Z
S
o
C
function is the same as that for the Z function.
,.
o
There are eight additional integration constants for the Z® function.
Since the Z-functions contain sixteen integration constants, the eight
integration constants of one Z-function must be dependent upon the
8
integration constants of the other Z-function.
To find this relation­
ship the first four derivatives of the Z-functions must be developed.
These derivatives are
Z^ =
[cosh az
+
cos b z I
sin b z I
sinh azj M'j
[cosh bz
sinh bz)
cos az
i Isin az
(2.14)
i - 1,2,3,.. .n
j = s or c
The difference between successive derivatives will be the elements of
*i
the M~[ and
i
matrices.
When derivatives of the Z-functions are taken,
recursion relationships for the
and
matrices become evident.
These relations are
= a J
, + b
i
i-l
N*] = b J
, K
i-l
, + a N*] , K
i-l
i-l
i = 1,2,3,...n
j = s or c
(2.15)
where the J and K matrices are defined as
0
I
-I
1
0
0
Expansion of Equations 2.15 enables one to write the
(2.16)
and
matrices which occur in the first four derivatives of the Z-function as
= y (cos<#> J
+ sin<j>
K)
(2.17)
9
Mg = M 2 (cos20
+ sin2<}) J
K)
Mg = M 3 (cos 34, J M^ + sin3(j) M^ K)
M^ =
(cos4<}> M^ + sin4<j) J M^ K)
(2.17 con't)
j =s
or c
and:
Ng = U (sin* J
+ cos*
K)
Ng = M2 (-cos2* N^ + sin2* J
K)
Ng = - W 3 (sin3* J N^ + cos3* N^ K)
i
A
4
ji
N^ = M
(cos4* N “ - sin4* N^ K)
(2.18)
Substitution of Equations 2.17 and 2.18 into Equation 2.17 gives the
Zo function, in terms of the integration constants of the
Z® = -
[coshaz
sinhazj J M C K
-j
O
+
[coshbz
sinhbzj J N^K
0
function, as
Jcosbz
Asinbz
,/cosaz|
(2.19)
The relationships between the integration constants of the Zfunctions a r e :
Ms = - J M c K
0
0
N8 = J N C K
0
0
(2.20)
10
or in expanded form as
(2 .21)
The particular solution of Equation 2.3 is now complete.
CHAPTER III
APPLICATION TO AN N-BEAM STRUCTURE
Method
The solution to the equation of motion developed in Chapter II is
adaptable to an N-beam structure.
A structure is first divided into
substructures, each being a beam.
Each beam has a separate equation of
motion and a separate solution.
The determination of the integration
constants of each solution is done through the use of equations which
identify the boundary conditions of deflection, slope, moment and
shear.
The system will be defined such that the boundary conditions
will be evaluated at the origin of each coordinate system, z = 0, or
at the point where z = I in the coordinate system.
Note that these
two points locate.the ends of the beam in the coordinate system.
The
number of boundary conditions required for a solution is 4 N .
Procedure
General boundary conditions for deflection, slope, moment, and
shear will be developed such that a general N-beam structure may be
analyzed.
Twisting of beams will not be allowed.
■ In order to insure that the boundary conditions are satisfied, each
Z-function will be required to satisfy an appropriate set of boundary
12
conditions.
The following quantities are introduced for brevity:
S2 =
sinh az
Il
Il
Tl =
cos bz
S3
cn
cosh az
H
IO
sI =
T3 =
sin bz
T4
cosh bz
sinh bz
cos az
sin az
(3.1)
and:
S 1 ^ = S 1S 0
1,3
I 3
C/2
IT.
S1 ,
1,4
T
„ = TT
I 3
1,3
T1 , = TT,
I 4
1,4
S0 „ = S 0S 0
2,3
2 3
T0 „ = S S
2,3
2 3
S 0 , = S 0S,
2 4
2,4
T0 , = T T ,
2,4
2 4
Il
I 4
(3.2)
The boundary conditions may now be written in general for the
Z-functions.
They a r e , starting with nondimensional deflection,
,j
;1,3 a O + S2,4 Eo + =2,3 c O + =1,4
+ T. . Fj + T. . Gj + T 1 . Hj + Z2,4
o
2,3 o
1,4
o
j
where:
C a B
+ 1 I l3 E o
(3.3)
j=s or c
4
(3.4)
and:
cO 6
for nondimensional slope as
(3.5)
13
(S2
Zj
I =
3M cos* - S 1 ^ M sin*)
+
(S1^ M
+
(S13 \
i cos* _
S2 4
sin*)
+
(S0 .y
^ ,4
S y
X jJ
sin*)
+
(T2
+ (T
cos* ■*-
cos* +
^2 3 U sin*)
o
3 M sin* - T 1 ^ y cos*)
. y sin* +
-1,4
T
+
(T1
3 y sin* - T 2
+
(T0
,y
sin* +
y
/, J
T
cos*)
o
y cos*)
y
cos*)
(3.6)
for nondimensional moment as:
Z2 = (Sl 3
cos 2* - S 2 ^ y 2 sin 2*)
+
(S2
^ y 2 cos 2* + S 1 g y 2 sin 2*)
+
(S2
3y 2 cos 2* - S 1 ^ y 2 sin 2*)
+
(S1
^y 2
+
(-T 0 . y 2 sin2* - T
y 2 cos2*)
^ 4
XjJ
+
(T1
+
(-T 1 ^ y 2 sin 2* -
+
(T2
cos 2* + S 2 g y 2 sin 2*)
O
3 y 2 sin 2* - T 2 ^ y 2 cos 2*)
T2 3 y 2 cos 2*)
3 y 2 sin 2* - T 1 ^ y 2 cos 2*)
j = s or c
(3./)
14
and for nondimensional shear a s :
3
(^2 ^ ^ cos3<J) -
+ (slj4 M3 cos3<J) +
^ y-3 sin3<fi)
S2 3
o
M3 sin3<J>)
3 ^ cos3<}) "^ 2 4 ^^ sin3<J>)
+
+
(S2
^ M 3 cos34> +
S13
M 3 sin3<j))
+
(T1
4 M 3 cos3* “
T2 3
sin3^)
+
(-T 0 , M 3 cos3# - T 1 ,
2,3
1,4
sin3#)
+ (t2j4 ^3 cos3* “ T1 3 h3 sin3<}>)
+ (-Tj 3 P
cos3<j) - T2 ^
sin3(j>)
s or c
(3.8)
Matching of dimensional boundary conditions is necessary, thus
the nondimensional boundary conditions developed in Equations 3.3
through 3.8 must be multiplied by an appropriate constant to be changed
to dimensional form.
These constants
(in parentheses) are:
/f L 4 \
o
C EI
o
/
F L'
o
C EI
o
1F L 2 \
Ely" = Y"
(3.9)
%
15
/f L
Ely'" =Y'" -S-
(3.9 con't)
\ °
Similarly for the fourth derivative:
(3.10)
The equations necessary for determining the 16N integration constants
for an N-beam structure may now be developed.
Eight-N of the relation­
ships are given by Equations 2.20; recall that these equations relate
the integration constants of the Z-functions. The remaining equations
come directly from boundary conditions.
Since both Z-functions are
required to satisfy boundary conditions, each boundary condition will
yield two equations.
The resulting equations are linear, hence the
integration constants may be solved for using matrix methods.
CHAPTER IV
EXAMPLES
The Cantilever Beam
The analysis of a cantilever beam is presented to illustrate the
method of formulating the matrix of boundary condition equations.
A
method of illustrating the results of the analysis is also presented.
As stated in Chapter II, C q is chosen to be e i g h t .
The reason for
this is that the nondimensional deflection can then be referred to as
the magnification factor.
Anderson [ 5 ] uses this term in his analysis
of the forced vibration of a single degree of freedom system.
this example, the magnification factor is the deflection, y
T h u s , in
(x,t), divided
by the static deflection of the end of the beam under the load F
o
[ 6].
The boundary conditions for the cantilever beam of Figure 4.1 a r e :
y(0) = 0
y*
(0) = 0
Ely"
Ely'"
(L) = 0
(L) = 0
(4.1)
These boundary conditions are insured when:
(4.2)
/////////////
F
o
sin u)t
L
cross-section
Figure 4.1.
Cantilever Beam
18
3
Z3
1
(O) = 0
j = s or c
(4.2 con't)
The equations relating integration constants are written in the
matrix form:
[A] (B) = ( C ) .
These are Equations 4.3.
The first eight rows of the A matrix are derived by choosing the
appropriate conditions from Equations 3.3 through 3.8.
rows of this matrix are Equations 2.20.
columns are associated with the
The B matrix is a column matrix,
sixteen integration constants.
Note that the first eight
functions
eight columns are associated with the
The second eight
(1=0,1,2,3), and the last
functions
(i=0,l,2,3).
the elements of which are the
The C matrix is a column matrix
whose elements are either zero or the constants zjj (j=s or c) .
The use of certain properties of the free-vibration solution to the
viscous damped beam of Stanek [ 4 ] and the free vibration solution
presented by Anderson
[ 5 ] results in the derivation of a frequency ratio
and a damping ratio for the beam of Figure 4.1.
6
The frequency ratio is
2
2
X
(4.4)
(F L 4 /8EI)(ZC (0) - Z=)
O
O
P
(FoL ttZSEI) (Z® (0) -
Zp
(FoL 3ZSEI) Z= (0)
(FoL 3ZSEI)
Z5
1
(0)
(F L Z /8) Z= (I)
O
L
Z5
2
(Fq L z ZS)
(I)
(Fq LZS) Z= (I)
(Fq LZS)
Z3
3
(I)
I
I
I
-I
I
I
-I
I
-I
I
I
I
-I
I
* Blanks Indicate Zeros.
I
I
20
■where:
= First natural frequency
A = First eigenvalue
and the damping ratio i s :
£
C
(4 .5)
c
where:
= Critical damping constant.
Specification of the frequency ratio and the damping ratio allows
B and
a to
be determined.
The results may then be presented in the
same manner as Anderson [5 ] presents the results of a single degree
of freedom system.
Figure 4.2 is a plot of magnification factor versus
frequency ratio for the end of the beam.
When the magnification factor increases to infinity. Figure 4.2,
the beam is being forced at a natural frequency.
Also, when the
forcing frequency is very near a natural frequency, the mode shape
associated with the frequency becomes dominant.
Thus the analysis
results in an approximation of the natural frequencies and the associated
mode shapes.
As the system becomes more complex, the value of plots
similar to Figure 4.2 increases in importance.
The phase angle between the forcing function and motion is defined
as
(4 .6)
ulHj
Figure 4.2.
Magnification Factor vs Frequency Ratio:
C
=1l-e-0.5
Figure 4.3.
Phase Angle vs Forcing Frequency
23
The normalized phase angle
(-^) versus the forcing frequency is
plotted in Figure 4.3.
The curves of Figures 4.2 and 4.3 show a similarity to the curves
presented,by Anderson [5] in his analysis of the single degree of
freedom lumped-mass system.
A Three Beam System
Consider the three beam system of Figure 4.4.
three beam system are given in Table 4.1.
the system are expressed as follows.
The properties of the
The boundary conditions of
The deflection at each ,origin is
2
zero; this requires
:
/ F O1
0, L?\
Ei 3X
o»i
(O) = 0
i = 1,2,3
j = s or c
(4.7)
The slope of each beam at its origin is zero; this requires:
3
F °i L i
Ei
.The
1U
zL
( 0)
=
0
deflections of the ends of the beams.
i = 1,2,3
j = s or c
(4,8).
z^ = I, are equal; this
requires:
3
The introduction of the new subscript indicates beam number.
ho
4N
Figure 4.4.
Three-Beam System.
25
Table 4.1.
Three Beam System Properties
Beam Number
Property
I
2
3
Length
(in)
100
50
200
Height
(in)
6
2
3
3
2
4
Width (in)
Moment of Inertia
(in4 )
Material
54
1.333
9
Steel
Aluminum
Magnesium
5.112
0.40
0.792
Weight per Unit
Length
(Ib/in)
Modulus of
.
Elasticity (Ib/in ]
30xl06
10.3xl06
6.5xl06
0
0
0
20
30
Damping Coefficient
(lb-sec/in)
Maximum Load
(Ib/in)
Forcing Frequency
(Rad/sec)
40
0-140
0-140
0-140
26
zO 1I
(1) -
E„ I0
2 2/
1
and:
F°i 2 4
E2 1Z
H-}
E 1 I1
< r CN
CN
O
Pu
fF=i 4 1I
F°»3 4 '
zO
2 <»
0,2
"I|
v E 3 1S j
The moment at the end of beam o n e ,
2) Z^ i
(F01
5I L1) Zo
z ,1
=0,2 (I) " °
Zj . (I) = 0
o,3
(4.9)
j = s or c
= I, is zero; this requires:
(I) = O
j = s or c
(4.10)
Figure 4.5 shows a free-body diagram of the end elements of the three
b e a m s , z. = I.
The moment relationship at the ends of beams two and
three (z^ = Z^ = I) are given as:
(Fo2 L 2) Z^ 2 (I) +
(Fq3 L 3) Z ^
3 (I)
O
(4.11)
j = s or c
The slope relationship at the ends of these beams
(z2
Z3 = I) is
given a s :
F=2 L2
E2 1Z
1 ,2
(I) +
(4.12)
s or c
The shears at the ends of the beams
<F„1
Ll>2L
(1) +
(z^ = I) are related as follows:
<Fo 2 L 2> Z 3,2 (1) +(F0 3 L 3)2J3,3 (1) = 0
j = s or c
(4.13)
Figure 4.5.
End Elements.
28
The boundary condition equations may now be put into matrix form.
The A matrix will be a 48 x 48 matrix.
matrix, as will be the C matrix.
The B matrix will be a 48 x I
The method of writing the equations
is identical to the method presented for the cantilever beam.
The
digital computer programs presented in Appendix A can be used to
determine numerical results for an N beam system.
Figure 4.6 is a plot of the nondimensional deflection at the end
of beam one,
= I, versus the forcing frequency.
The first two
natural frequencies may be closely approximated by inspection of this
curve.
Approximations for the mode shapes of the system associated
with the first two natural frequencies are shown in Figures 4.7 through
4.12.
I
Figure 4.6.
Nondimension Deflection at
z Jl
-
I
vs
Forcing Frequency: C = O
Figure 4.7.
Beam one mode shape:
First natural frequency.
Figure 4.8.
Beam Two Mode Shape:
First Natural Frequency.
Figure 4.9.
Beam Three Mode Shape:
First Natural Frequency.
Figure A.10.
Beam One Mode Shape:
Second Natural Frequency.
LO
4^
I
Figure 4.11.
Beam Two Mode Shape:
Second Natural Frequency.
Figure 4.12.
Beam Three Mode Sh a p e :
Second Natural Frequency.
CHAPTER V
DISCUSSION
The formulation presented is applicable to complex systems;
limitations in complexity are primarily a result of computer capacity.
It is estimated that the XDS Sigma 7 at Montana State University has
the capacity to perform the numerical analysis for a 14 beam system
with the present core storage available; use of additional equipment
can greatly increase this capacity.
Since the deflection of a system becomes large when it is forced
to vibrate at a natural frequency, the natural frequencies may be
approximated using a plot similar to Figure 4.2.
Approximations of
the natural mode shapes are also found by forcing with a frequency
very near a natural frequency. ■
' Persons involved with approximate methods of analysis may also
find use of this solution.
The Galerkin Method, for example, requires
that deflection shapes be assumed.
This method of solution can be used
to determine an infinite number of deflection shapes which satisfy a
specified set of boundary conditions.
The parameters which affect the
deflection shape include the forcing frequency, damping coefficients,
and the time for which a solution is selected.
It is recognized that the solution as presented is applicable to
structures with elements in various planes; however, the fact that
37
neither axial loading nor twisting effects are included will be a
limitation.
There is need for a development which will allow these
two effects to be included.
APPENDIX
Appendix A
Computer Program
6
C
C
C
C
C
C
7
C
1
2
4
5
8
12
C
C
21
22
23
24
25
26
27
28
29
30
31
32
33
*
WRITTEN
*
ASMr
EY
DANIEL
DEPT.
F,
STATE
T HE
FORCED
structure
SUMMER
P r ILL
MONTANA
TO
1970
university
*
*
*
*
*
*
r
C
14
15
16
17
13
15
23
PROGRAM f o r f i n d i n g t h e SOLUTION
V I B R A T I O N OF AN N-F-EAM ■
*******************+**********4************************^
16
11
13
*****************************4**************************
#
*
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
N IS THE. N U M B E R SF b e a m s IN t h e S T R U C T U R E
A R R A Y IS T ^ e m a t r i x RF S I Z E N X 1 6 , n X 1 6 + 1
P U T IN C B M X S N S T A T E M E N T S OF THE F q RM!
C e v ''eN/7RRAV/F ( 16XN, 16 XN + 1 )
this
statement
also
goes
in s u b r o u t i n e s
Be,Re s u l t s ,
M.ATV'RITE, AN D M A T IN I T
C O ^ ' e N / P A R T IC Z Z P C ( N ) , Z P S ( N )
C O M 'QNVCPNSTZH(IAXN)
C O M M g N X P A R A N / A ( N ) , B ( N ) , M E W ( N ) , PHI (N)
these
also
go
in
subroutine
results
N S 1,1 D I M E N S I O N THr F O L L O W I N G W i J H A D I M E n S I ON S T A T E M E N T :
A L P W A ( N ) ........ D a m p i n g f a c t o r
BETA(N) , . . . . , , . B E A M PARAMETER
C D ( N ) ........ .C O N S T A N T S BY W H I C H N O N D In EN S I O N a L
D E F L E C T I O N M U S T BE M U L T I P L I E D BY TO
Pr C H A N G r D TO D I M E N S I O N A L D E F L E C T I O N
C S L ( N ) ....... ... C O N S T A N T S W I T H W H I C H N O N O l M E N S l O N A L
S L O P E M U S T BE M U L T I P L I E D BY TO
BE C H A N G E D TO D I M E N S I O N A L S L O P E
C M ( N ) ............ c o n s t a n t s w i t h w h i c h n o n d i m l n s i q n a l
moment
M U S T BE M U L T I P L I E D BY TO
Br C H A N G E D TC D I M E N S I O N A L M O M E N T
C S ( N ) . . . ........ C O N S T A N T S W I T H W H I C H N O N D I M E N S IO N A L
S H E A R M U S T BE M U L T I P L I E D BY TO
->
O
C
C
C
C
C
............. '"'AXI1IuH LOAD QN LACH 5£AM (LB/IN)
L(N)...... ,..LENGTH GF EACH BEAM (IN)
E(N)...... .
OF ELASTICITY SF EACH BrAM (PgI)
............. VISCOUS DAMPING COEFFICIENT (LB-SEC/IN)
CE(V)..,..,.,...C1GHT Gf EACH BEAM PER INCH (L3/IN)
N 1(........... .
MOMENT 2F INERTIA (
EXTERNAL CHjgH
C5'<i'eN/ARRAY/F(48,49)
i p h i
( T ) / r A R T I C / z r c < 3 ) / Z P s ( 3 ) / C 9 N S T / 'H < 4 8 ) / P A R A f 1 / A < 3 > / B ( 3 ) i M E w (3,i
REAL
MELjMI,L
1E ! ^ c [ ^ S ^ ; ? r A < 3 , ' C D ( 3 ) ' c S l 3 , ' C M < 3 , ' c s L t 3 l ' F 0 ( 3 >'L ( 3 >'
C H ( X ) = ( E X P ( X ) > E X P ( - X ) )/?,
S H ( X ) = ( E X P ( X ) - E X P t - X ) )/2.
C-I
GIVE N A VALUE NOW
C-I
C
N =S
N 9 X R E A P T HE Q U A N T I T I E S
T HE R E A D I N G F O R M A T IS?
H
F O , L , E / M I, Cr F O R
( 5 F1 5. 0)
EACH
BEAM
R E a D ( 1C5/10)(F0( I),L( I )/E(I ) / M I (I )/DE(I)/I=IjN)
F9RMAT(5F15.0)
c : , 10 NS-'-' R E A D C f o r E A C H b e a m .
t h e FORMAT
IS:
(F l o * O )
R E A r'(I Co, 2 0 ) ( C d ) , I = l,N)
20 F S R v A T ( F 1 5 . o )
C-I
N O N R E A D FF, T H E F O R C I N G F R E Q U E N C Y (RAD/SEC)
C
T HE R E A D I N G F O R M A T IS:
(F15.Q)
R E A D ( 1 0 5 , 2 0 ) FF
C
NS--' C A L C U L A T E a n d WRITE ALL CONSTANTS
DS
SC
1=1,N
CD(I)=FO(I)*L(I)**4/E(I)/MI(I)
CSL(I)=CD(I)ZLd)
C M (I ) = C S L d ) # E d )*Mi (I )Z L( I)
C S ( I ) = C M ( I ) ZL(I).
68:
B E T M I)= (Dr<I )* U J)**4*22**2/385./ M I (!)/[(I))**(0,25)
fcs:
A L 3 - A d )=C( I ) * 3 8 6 . / F R / (DE d )*L( I ) )
M E u ( D = - E T A d ) ^( l + A L pHA{ D * » 2 ) * * ( 0 ' 1 2 5 )
PHI ( I )= A T A N E ( A L p HAt I ) > 1 . 0 ) / 4 » 0
At I ) =v Fld I ) * C 9 S ( P H I ( D )
B(I)=ME', ( I ) * 5 I \ ( P H ! ( I ))
Z p Ct I ) = A L P H A ( I ) * B [ T A ( I ) * * 4 / M E W ( I )**8
Z p St I >= B E T A t I > » * 4 / M E W ( I ) **8
W R D E t 103/ 40 ) D n ETA { I D ALpHAt I ),%[%( I )/p HI ( I )/F0{ I )/L( I )/
70:
7i:
72:
73:
74;
7?:
76:
77:
30
78:
40
79:
so:
si:
82:
83:
84:
85:
86:
87:
58;
89:
SO:
SI i
sz:
52:
94:
55;
96:
97:
SB:
59:
ico:
loi:
I F (I ) , H I ( 1 ) , C D ( I ) , C S L ( I ) , C M ( I ) / C S ( I ) / F O ( I )
F S 3 A T ( / / / 2 0 X / 'THE p R S p E p T I E S SF B E A M ' / I X , 1 1 , I X , ' A R E : * /
l 2 6 X / lpE T A = ' , C I S , 6 ///
? 2 5 X , ' A L p H A = D E l S . 6,/,
3 2 7 X , ' M E ' . = D E I S , 6,/,
4 r 7 X / 'p H I = ' , C I S . 6,/,
5 2 S X , ' F O = D C l S . 6 / 1 X / 'LB P E R I N C H ' , / ,
6 2 9 X » 'L= D E I S . 6« IX, ' I N C H E S ' , / ,
72 9 X , ' E = ' , E 1 5 , 6 , I X , ' P S I D / ,
R?SX, f I = D E l S , 6, IX, ' I N C H E S * * 4 D / ,
9 2 8 X , ' C D = D f 15.6,/,
9 2 7 X, 'CSL= D E I S . 6,/,
9 28X, 'C M= D 2 1 5 , 6 , / ,
9 2 8 X , 'CS= D E I S . 6,/,
9 2 7 X , ' C L r'= ' , F.15.6,/' I ' )
WR IT F. ( I 0 3 , 9 0 0 FF
9 0 0 F O R ' A T t ' I ' , / / / / / / , 2 C X , ' T H E F O R C I N G F R p O' IS D C 1 5 . 6 , I X , iC p S D / ' I
N S W C A L L S U B R O U T I N E M A T I Ni T
(MATRIX I N I T I A L I Z E )
ITS O N L Y A p G U E M E N T IS THE N U M B E R QF B E A M S , N
C A L L M A t I N I T (N )
-I
N S W P U T B O U N D A R Y C O N D I T I O N S IN TH r M A T p IX L A B E L E D A R R A Y
the
S U B R O U T I N E M a t IN I T U S E S r o w s 9 - 1 6 , 2 5 - 3 2 , 4 1 - 4 8 / , , . , N L g T H R U N
OS D O T U SE T H E S E R O W S F O R B O U N D A R Y C O N D I T I O N E Q U A T I O N S
S EE S U B R O U T I N E BC F O R ITS N O T A T I O N
C A L L B C ( 0 , 0 , I , I , I , M E W ( I ) , p H I ( I ) , C D ( I >)
102
103
104
105
106
107
108
105
H O
111
112
113
114
115
116
117
1 18
115
C- I
120
121
122
123
124
125
126
127
122
129
130
131
132
133
134
135
C-I
C
C-I
C
C
C A L L S C ( I , 0 , 1 , 3 , I * v E W ( I ) , P H I ( I ) , CsLi I) )
C A L L B C ( 0 , I , 1 , 5 , I, v E W ( I y , RH I (I ), C D ( I ) )
C ALL B C ( 2 , 1 , 1 , 7 , 1 , v E W ( I ) , P H I (I),CM(I))
C A L L B C ( 3 , I , I , 3 9 , I , m e W (I ),RH I (I ),C S ( I ))
CALL B C (0,1 , - 1 , 5 , 2 , M E W ( 2 ) , P H I ( 2 ) , C D ( 2 ) )
C A L L B C ( 0 , 0 , I , 1 7 , 2 , M E W ( 2 ) , PH I ( 2 ) , C D (2))
CALL B C ( 1 , 0 , 1 , 1 9 , 2 , M E W ( 2 5 , P h I ( 2 ) , C S L ( 2 ) )
C A L L B C (1 , 1 , 1 , 2 1 , 2 , M E W ( 2 ) , P H I ( 2 ) , C S L (2))
C A L L Be (2, I, I, 23, 2, M E W (2), P m I (2),C'<(2))
CALL B C ( 0 , I,I,37,2,M E W ( 2 ) , P H I ( 2 ) , C D ( 2 ) )
C A L L SC (3, I , I , 3 9,2» ME W (2), PH 1 ( 2 ) , CS( 2.) )
C A L L B C (I , i, I , 2 1 , 3 , v E W ( 3 ) , RH I ( 3 ) , C S L (3))
C A L L B C ( 2 , I , - I , 2 3 , 3 , M E W ( 3 ) , P H I ( 3 ) , C M (3))
CALL B C ( 0 , 0 , I , 3 3 , 3 , v E W (3),P H I ( 3 ) , C D ( 3 ) )
CALL B C ( 1 , C / 1 , 3 5 , 3 , M E W (3),P H I ( 3 ) , C S L ( 3 ) )
C A L L B C (o , I , - I , 3 7 , 3 , M E W (3), RH I ( 3 ) , C D ( 3 ) )
C A L L S C ( 3 , I , I , 3 9 , 3 , M E W ( 3 ) , R H I ( 3 ) , C S (3))
N S W B U I L D t h e 16 XN + 1 C B L U M N QF A R R A Y
F(1/49)=CD(1)*ZPC(1)
F ( 2 , 4 9 ) = C D ( 1 5 " ZP g ( I )
F(5,49)=CD(1)»ZPC(1)-CD(2)*ZPC(2)
F (6,49) =CD ( I H Z c"'3(1 ) - CD (2 ) * Z P S < 2 )
F(17,45)=CD(2)«ZPC(2)
F( 1 8 , 4 9 ) = C D ( P ) " Z P S ( 2 )
F(33,49)=CD(3)*ZPC(3)
F ( 3 4 , 4 9 ) = C D ( 3 ) " ZPg(3)
F ( 3 7 , 4 9 ) = C D ( E ) * Z P C ( 2 ) - C D ( 3 ) * ZP C ( 3 )
F ( 3 ” , 4 9 ) = C D ( 2 ) " Z P S ( 2 ) - C d (3)*ZPS(3)
S U B R O U T I N E M a T W R I T E M A Y BE C A L L E D AT T H l S P S I N T
A L L IT W I L L CS IS W R I T E T H E M A T R I X , A R R A Y
N S W C A L L T H E S U B R O U T I N E R E S U L T S (N)
ITS O N L Y A R G U E M E N T IS T HE N U M B E R QF B E A M g (N)
R E S U L T S c a l c u l a t e s A N D W R I T E S E V E R Y T H I N G F S R T h E EN D
CALL RESULTS(N)
SF B E A M
I
is
2;
3!
4:
5:
6:
7:
8:
2:
IC!
11 J
12!
13:
I ti:
15:
16:
17:
18:
19!
PC:
21:
22:
23:
24!
25:
26:
27:
28:
29:
SU3?eUTl\'E B C ( \ ' l , L , J l , l R , N B , M E W , F H l , C )
C
C
C
C
C
c
c
C
C
C
C
C
c
C
C
C
C
C
C
C
C
C
C
C
W R I T T E N OY D A N I E L E, P R l L L
S U M M E R 1 9 70
C H A N G E ThT C R M vrLN C A R D TO TH E A P P R O P R I A T E F O R M .
SEE M A I N L I N E
C9M-3N/ARRAY/F(48,43)
T H I S S U B R O U T I N E C A L C U L A T E S B O U N D A R Y C O N D I T I O N S BY
C A L C U L A T I N G i s C o l u m n s a n d I ROW FOR e a c h C a l l .
A R R A N G E TNP M A T R I X A R R A Y S3 T H AT B E A M I E G 1
J A T l g N S ARE IN
COLUMN'S l-s F O R T HF ZC F U N C T I O N S C o l U sivS 9 - 1 6 F g R
T H E ZS F U N C T I O N .
then
B F A M 2 E Q U A T I O N S Gg INTO TH E
N E X T 16 C O L U M N S ( 17 - 3 2 «
T H E N TH E ZC F U N C T IS NS F O R
B E A M 2 CO I N'T 5 C O L U M N S 1 7 - 2 4 5 TH E ZS F U N C T I O N IN
C O L U M N S 2 5 - 3 2 ........ E T C T tjRU B E A M N
THE A R G L E M E N T S
OF T H E S U B R O U T I N E A^E:
N I . . . . O R D E R OF
D E R I V A T I V E , IS E I T h E R 0 ' 1 , 2 , 3
.......P O I N T ON B e a m AT W H I C H E V A L U A T I O N O C C U R S ; O OR I
J l . . . . S I G N OF B O U N D A R Y C O N D I T I O N ; I OR -I
IR . . . . R o W IN W H I C H B O U N D A R Y C O N D I T I O N IS TQ BE P L A C E D
N B , ...BEAM NUMBER; 1 , 2 , 3 , " * , N
M E - v . . . J U S T PiJT
IN M E W ( N b ) ; W H E R NB IS B E A M N U M s E R
P H I . . . JUST PUT
IN P H I ( N B )) W H E R E NB IS B E A M N U M B E R
....... C O N S T A N T W H I C H M A K E S TH [ B O U N D A R Y C O N D I T I O N
DIMENSIONAL
IF N l = C i U S E C D( N B )
IR N l = I i U S E C S L ( N B )
Ir Nl=-Ei U S E C M( N Q )
IF N i = 3 ; U S E C S( N B )
REAL m e n
IF(L.EG. 0)7=c,C
IF(L.EG.U Z = I O
30 I
A=MRW* C S ( PHI )
31:
32:
33:
B =M E W * S I N ( P H I )
S13=CH(A*Z)*C0S(B*Z)*J1
S14«CH(A*Z)*SIN(B*Z)*J1
34;
35:
36:
37:
3s:
35:
40:
4i:
42:
43:
44:
45:
46:
47;
48:
49:
so:
5i:
52:
53:
54:
5556:
57:
Sg;
59!
60:
61:
62:
63:
64:
65:
6 6:
67:
S23=SH(A*Z)«CBS(9*Z)*J1
S 2 4 =s H ( A * Z - ) * S I N ( B * Z ) * J 1
T13=CH(P*Z)4CQS(A#Z)*U1
T14=CH(5*Z)*SI\'(A*Z)*J1
T23=SH(0*Z)*C8S(A*Z)*J1
T24=SH(?*Z)*SI\'(A*Z)*J1
10=^5*16-15
IRc=IR+!
I C ? = iC+8
I F ( ^ 1 , E C . 0 ) G 3 T9 40
G3 T Q ( I r , 2 0 , 3 3 ) , NI
10 A l = S 2 3 * A - S 1 4 * S
A2=S14*A+S23*S
A 3 = 5l3« a - 5 2 4 * B
A4=524*A+S13*3
A5=T23*5-T14*a
A 6 = T 14 * B + T 2 3 * A
A 7 = T 13 * 3 - T 2 4 * A
AgcT24*n+Ti3+A
GS TO 5 0
20 A = <'E'A= * * 2 * C R S ( 2 » P H I )
G='1EW**2*SI\(2*PHI)
Al = ' 1 3 * A - S 2 4 * B
A 2 s R24*A+S13*B
A3 = -323*A-Sl4*B
A 4 s 5 i 4 * a + S23*B
A5=-T24*B-T13*A
A6=Ti3*H-T24*A
A7=-T14*B-T23*A
A 8 = T23*b -T14*A
G9 TO 50
30 A = K/E W * * 3 * C 9 S ( 3 * P H I >
B«MEW**3*SIN(3*PHI)
Al=S23*A-314*B
Ln
A2=?l4*/.+S23*B
A 3 s S 1 3 * a -S24*F3
A4»-£4* a + 2 1 3 * b
A5=T14*a -T23*B
A6=-T23>A-T14*R
A 7 = T 2 4 * a - T 13*B
AS=-T13*A-T24*B
GS Te SC
i
40 A l =513
A2=5?4
A 3 =523
A4 = 5 14
AS=Ti3
A6=T24
A7=T23
AS = T i A
50
F (1%,I C i = F (IRS,ICS)=A1*C
F ( I , I C + 1 ) = P { IRS, ICS + 1 ) =A 2 * C
F ( I I C + 2 )= F ( I R S , I C 5 + 2 ) = A 3 * C
F(IF,IC+3)=F(IRS,ICS+3)=A4*C
F (I
10 + 4 ) = F ( I R S , I C S + 4 ) = A 5 * C
F ( I", IC + 5 ) = F ( I R S , I C S + 5 ) = A 6 * C
F ( 1°,IC+6)=F(IRS,ICS+6)=A7*C
F d - : , IC+7) =F( IRS, I C s + 7 ) = A 8 * C
RETURN
END
ON
SU3^?UTike
NR IT T £ s; B Y
CALCULATES
REAL M/N
Z F U N (Mi K'/A l / Z )
D A N I E L F. P R I L L
the
!.FUNCTIONS
SUMMER
1970
D Iv E N S IDN F I (I / ? ) / M ; 2 / 2 ) / N (E / 2 ) / G l ( 2 / I >/F 2 (I i 2 ) / G 2 (2/ I )J a
1,0(2,2),Z1(1/2),Z2(1,1)
Fl(Izl)=Al(I)
F l ( 1 / 2 ) = A l (2)
G l (I / I )= A l (3)
G l ( 2 / I ) = A l (4)
F 2 M , I ) =Al (5)
F 2 (I / 2 ) aA l (6)
G 2 ( 1 / 1 ) = A l (7)
G2(2,D=Al(R)
C A L L K a v u l (F I / M / Z I / 1/2/2)
C A L L M A V U L ( Z 1 / G 1 / Z 2 , 1/2,1)
Z * Z 2 (I / I )
C A L L M A V U L ( F 2 / N / Z 1 / 1/2/2)
CALL M A M U L ( Z 1 / G 2 / Z 2 , 1 / 2 , 1 )
Z =Z +Z 2 (I / I )
RETURN
END
i
{s )
SUBROUTINE RESULTS(MBS)
W R I T T E N BY D A N I E L Fe P R l L L
S U M M E R 1970
C A L C U L A T E S A L L R E S U L T S F O R T HE E N D S SF T H E B E A M S
N E E D TS C H A N G E T^ E C o M M 8 N C A R D .
SE E M A I N L I N E
CBH"9N/ARRAY/F(4g,4q)
i C e x ^ o N / P A R T ! C Z z PC (3'}, Z p S ( 3 ) / c 8 N S T/H( 4 3 ) / P A R A M / A O
JjMEW(S)i
D IMENS IE N iM (2,2),N(2,2),M l (2,2),N I (2,2)/X(EOl),Al(S),Dl(2/2),
1D2(2,2),S(4),S1(4)
REAL
DS
M E W , M , N , M l , NI
J= I, 201
X ( J ) = ( J - I ) /200.
5 C E N T INUE
I =Mr-S* 16
11=1+1
CALL
SSLTN(F,H,I,II)
H =I
12 = 2
13 =3
15 =B
14 = 4
I6 =6
17 =7
I8 = ?
19 = 9
IlO =IO
Ill=Il
112=12
113=13
114=14
115=15
116=16
DB IOO I - I , MBS
IF * I .E3.1)Ge
T9
10
11=11*16
36!
37!
35!
35!
40!
4i:
42!
43!
44;
45!
46!
47!
12=12+16
13=13+16
14=14+16
15=15+16
16=16+16
17=17+16
18=15+16
19=19+16
.
T1 0 = 1 1 0 + 1 6
111=111+16
112=112+16
113=113+16
114=114+16
1 15 = 115 + 16
116=116+16
10
M d ' U= W ( H )
4R ;
49!
50!
51 !
52!
53!
54!
55!
56!
57!
58!
59!
60!
61!
62:
63:
64:
65:
66 !
67;
M ( 1 / 2 ) = H ( 14)
M ( 2 ' I )= H (13)
M ( 2 ' ? ) = H ( 12)
N d ' I ) = H( I S)
N( UP) =H (IP )
N(2'1)=H(I7)
N (2 ' 2 ) = H ( I 6 )
M l ( U l ) = H f 19)
M l ( U 2 ) c H ( 112)
M l ( 2 , I >= H ( I U )
M l ( 2 , 2 ) =H( 110)
N l ( U l ) . H{ 113)
N I (1,2 ) = H ( 1 16 )
N i (2,I ) =H(IlS)
NI ( 2 , 2 ) = H ( 114)
WRITE(103,60)I
ND
68:
65:
70:
71;
72:
73:
74;
75:
76:
77:
78:
755
so;
s i :
82:
83:
84:
85;
86:
87:
62:
60
F e r yA T { / / / / / / 4 C X , ' B E A M ' , I X / 1 3 , I X , ! C O N S T A N T S
Jl=J-IS
W R I T E ( 1 0 8 , 7 0 ) (H(J2),J2=J1,J)
70 F S = ^ A T ( 2 E 1 5 , 6)
WRIt E (103,^0)I
80 F O R M A T ; ' I ' , 5 X , ' T H E D E F L E C T I B N A N D D E R I V A T I V E N j M S E R S I T H R j 4
1 'r D 7 B E A M ' I X , 1 3 , I X , ' A R E P R I N T E D B E L B W , U / g X , ' E V E R Y T H I N G IS IN
2 ' N S N D I M E N S I B N A L F 9 R M , ' / , 5 X , ' N U M B E R S S T A R T AT N G N C l M E N S l B N A L '
3 ' L E v G T H BF Z E R B , ' / / )
IF ( I . GT . I )GB TB 30
Cl = A ( I )
C2=B(I)
A i d ) =CH(Cl)
A l (2)= S H ( 0 1 )
Al (3) = C O S ( C S )
Al ( 4 ) = S I N ( C Z )
A l ( 5 ) =CH(CZ)
A l ( 6 ) =SH(CZ)
A l ( 7 ) =COS(Cl)
Al(S)=SIN(Cl)
85:
YMAX=Q.Q
so;
C A L L Z F J N ( M , N , A1,'ZC)
ZC=ZC-ZFC(I)
CALL ZFJN(M1,N1,A1,ZS)
ZS=Zs-ZPS(I)
CS ?0 v a l , 181
91:
52:
93:
94:
95!
56:
97:
96:
99:
loo:
loi:
AR E! ’/// /)
J=16*I
T = 3 . 1 4 1 5 9 * ( J - I )/iSO,
Y0=ZC*CBS(T)+ZS*SIN(T)
I F ( A E S ( Y O ) . G T , Y M A X ) Y M A x = A B S ( Y O ) ;T leT
20 C S N T I N U E
T = Tl
30 CS 1 00 J= 1 , 201
Cl=A(I)^X(J)
Ln
O
1C2
1C3
I C4
I CS
106
107
108
109
H O
Hl
112
113
114
HS
116
117
H S
H s
12C
121
122:
123?
124:
125;
126:
127:
128;
129:
130:
131J
132;
I 33 J
134:
135:
C2 = °..( I ) *X( J)
Al(I)=Cu(Cl)
A l (2)=SH(Cl)
A l (3) = C 0 S ( C 2 )
Al (4 ) = S T V ( C 2 )
A l ( S ) = C H (C2)
Al (6) = S u ( C S )
A l ( 7 ) =CCS(Cl)
Al (3 ) = S I N C C l )
C A L L Zr U N ' t ^ A ' / A H Z C )
ZC = Z c - Z U C d )
C A L L Z f u n (m H N H A H Z S )
ZS=Zs-ZPS(I)
DS 40 V l = 1,4
C A L L RECUR ( Jl A E W ( I ) , P H f ( I ) A A , D H D 2 )
C A L L Z F L N ( D H D 2 , Al, Z j O
s ( J i )= Z jc
C A L L R E C V R ( J l A E W d ) A H i ( I ) A l A l , D l , 02)
.
C A L L ZFiv V (D H D2, Al, Z j O
40 S l ( J l ) = Z J C
Y0=Zr*C33(T)+ZS*SIN(T)
Y l = S ( I ) * C e S ( T )+ S i {I X S l N ( T )
Y 2 = S ( 2 ) * C 9 S ( T ) + S 1 (E)^SlN(T)
Y3=S(3)*C9S(T)+S1(3>*SIN(T)
Y4=3(4)*C3S(T)+Sl(4)*SlN(T)
W R I T E d C S , 4 0 0 ) Y 0 , Y H Y2/Y3,Y4
4 00 F9R'"AT(2X, 5 2 1 5 , 6 )
ICO C S N T INUE
WR I T f (1 0 8 , 9 0 ) Tl
SO F 5 R " A T ( ' 1 ' / / / / , 3 0 X , ' T H E M A X I M U M E N D D E F L E C T I O N BF B E A M
I ' B C C U R S AT THE N Q N D I M E N S IB N A L T I M E B F i ' / , 3 0 X , E l 3,6)
R E W I N D 100
RETURN
end
S U - d S U T I N E M A T X R IT E (N B S )
WRITTEN b y CANIEL F . PRILL
WRITES t h e MATRIX, ARRAY
N E E D Tp C H A N G E T h e C P M M g N C A R D
C e M - ' Q N / A R R A Y / F (48,49)
SUMMER
1970
K = N B s * 16
Kl=1
Ol
W R J T E ( I C S , 1 0 ) K , Kl
30
40
20
50
60
DS PO I * 1 , K
WR I T e ( 1 0 8 , 3 0 ) I
F P R v A T ( / / , 4 0 X , •R O W ’, I X , 13,/)
W1R I T e ( I CS, 40 ) (F ( I, J ) , J a I , K )
F Q R m A T (8 E 1 5 16 )
C S N T IN U r
WR I t E ( I C S , 5 0 ) Kl
F S R M a T ( / / / / , 4 Q X , *T H E ' , l X , 1 3 , I X , 'TH C S L U M N
WR I T E ( 1 0 8 , 4 0 ) ( F ( J , K i ) , U = 1 , K )
W R I T e ( 1 0 3 ,6 0)
F S R v A T ( 'I')
return
END
Ln
NO
IS:'/)
I:
2:
3:
4;
5:
6?
7:
85
9:
1C:
11:
12:
13:
14:
15:
16 J
17:
is:
19:
20:
21:
22:
23:
24;
25:
26:
27:
28:
29:
30:
31:
32:
33:
S U B p S U T I N E R E C U R ( N I , M E W , P H I / M , N / D l * D2)
W R I T T E N BY D A N I E L E * P R I L L
S U M M E R 1970
C A L C U L A T E S i n t e g r a t i o n c o n s t a n t s FSR e a c h d e r i v a t i v e
REAL J,<,M,N,MEW
DVENSIGN
J(2,2);K(2,2),M(2,2),N(2,2),D(2,2),E(2,2),D1(2,2),
102(2,2),03(2,2),04(2,2)
'
J ( V l ) =c*o
J ( 1,2)=1,0
U (2< I ) = 1 ,0
J(2,2)=C*0
K(Vl)=CO
K(VE)=-VO
K (2' I ) = V Q
K (2,2)=0.0
S3 T Q d c , 2 0 , 3 0 , 40), NI
10 A s M E W * C 2 S (RHI)
B=n e w *SIN(FHI)
C A L L MAf'UL (J, M, D, 2 , 2 , 2 )
CALL M A M U L (M,K,£,2,2,2)
CALL M A T C (A,0 , 03,2,2)
CALL M A T C ( E , E , 04,2,2)
CALL MA T A t 03,04,01,2,2)
CALL M A M U L t J , N , 0,2,2,2)
CALL MAMUL(M,K,E,2,2,2)
CALL M A T C (9,0,03,2,2)
C A L L M A T C (A , £ , 0 4 , 2 , H )
CALL M A T A (03,04,02,2,2)
GS TQ SC
20 A = M E w * * 2 » C G S ( 2 * R H I )
B = ME W * * 2 * S IN (2 * P H I )
C A L L M A M U l (J , M , D , 2 , 2 , 2 )
CALL M A M J L (D,K,2,2,2,2)
C A L L M A T C (A , M , 0 3 , 2 , 2 )
Ln
LO
CALL M A T C(B,E,34,2,2)
CALL M A T A (03,04,01,2,2)
A = -A
CALL MAMUL(J , N , D , 2 , 2 , 2 )
CAl L MAMUL(D,K,E,2,2,2)
C A L L M A T C ( A,,v / 0 3 , 2 , 2 )
CALL MATC(B,E,04,2,2)
CALL M A T A (03,04,02,2,2)
G9 Mg 5c
30
A = M2;\i**3#cgS(3*PHI >
D = Mr A * # 3 »S I \!( 3 *
I)
CALL MAMUL(J , M , i , 2 , 2 , 2 )
CALL MAMUL(M , < , E , 2 , 2 , 2 )
CALL M A T C (A,0,03,2,2)
CALL M A T C ( 3 , 2,04,2,2)
CALL M A T A ( 0 3 ,04,01,2,2)
A = -A
r =-B
CALL M A M U L (J,N,0,2,2,2)
C A L L M A M U L (M ,K , £ , 2 , 2 , 2 )
CALL M A T C t B , 0 ,03,2,2)
CALL M A T C (A,[ , 04,2,2)
CALL M A T A ( 0 3 ,04,02,2,2)
^
os Tg 5:
40 A = m :' W * * 4 * C ? S ( 4 * p MI )
P 3 M EW**4*SIN'( 4'*PHI )
CALL M A M U L (J,M,0,2,2,2)
C A L L M A w U L ( 0 , K , E , 2 ,2 , 2 )
C A L L M A T C {A , M , 0 3 , 2 , 2 )
CALL M A T C (B,5,04 , 2 , 2 )
CALL M A T A (03,04,01,2,2)
B = -S
CAuL
11A M U L (J , N , D , 2 , 2 , 2 )
CALL
M A w U L t O , K , E , 2,2,2)
Ln
-C~
68 ;
6 ?!
70:
71:
72:
50
CALL M A T C i 4 , S 3 , 2 , 2 )
CALL M A T C O , E , D 4 , 2 , 2 )
w A L L fiATA(D3,0 4 , 0 2 , 2 , 2 )
RETURN
END
Ln
Ln
I:
2:
2:
4:
5:
6!
7;
3!
55
1C:
115
c
c
c
c
c
C
c
C
C
C
C
12: c
13: c
14:
c
15:
16:
17:
is:
19:
20:
21:
22;
c
c
c
c
c
C
c
c
23: c
24; c
SSL''TISv 9P S I sI N L T A N E e U S E Q U A T I O N S BY G A U S S I A N E L I M I N A T I O N
N
k U^ber
or s Im U l t a n e s u s e q u a t i s n s
N U v n ER AF C O L U M N S i n THr A U G M E N T E D M A T R I X
L = N -I
A(IzJ)
E L E m E N T S SF T H E A U G M E N T E D M A T R I C E S
I
m A TRIX RSN N U M B E R
J
m ATRIX COLUMN NUMBER
J J T A X E S SN V A L U E S SF T H E R S W N U M B E R S W H l c H AR r P O S S I B L E P l V S T
R S NS, E V E N T U A L L Y T A K I N G ON T HE V A L U E I D E N T I F Y I N G THE RQ w H A V I N G
THE L A R G E S T P IVOT e l e m e n t
BIG
T A K E S SN V A L U E S OF THE E L E M E N T S IN T HE C S L U M N C Q N T A I N I N G
P S S F l S L E R I V 5 T E L E M E N T S , E V E N T U A L L Y T A K I N G QN T H E V A L U E B F THE
ELEMENT USrD
TEmP
temporary
name
used
fsr
the elements
sf the
rsw selected
ts
B c C S M E THE P I V O T RSW, B r F Q R E TH E I N T E R C H A N G E IS M A D E
K
I n d e x o f a o s l o o p t a k i n g q n v a l u e s f r q m i t s n - i ; i t IDr N i i F i r s S
THF C O L U M N C O N T A I N I N G P O S S I B L E P lV^T E L E M E N T S
KPl
K-r I
AB
A B S O L U T E V A L U E SF A (I , K )
GUOT
QUOTIENT A(I,K)/A(K/K)
x(i)
unknowns
of the
set of e q u a t i o n s b e i n g
solved
SUM
S U M M A T IO O N QF A ( I ; J ) * X ( J ) F R q M J=l + 1 TO N
N N i n d e x OF a d o l o o p t a k i n g o n v a l u e s p r o m i t s n - i
i9 i
i+i
,
25:
SUBROUTINE
26:
DIm Pns i o n
27:
25:
29:
30:
31:
32:
23:
L = N-I
DS
12
5 0 L T N (A z X z N z M )
a
( n , m ) , x (n )
K sI, L
JJ =1
K
B I o = A B S ( A ( K , K )>
K 3 I =K + !
DS 7 I = K P l , N
AB=ASS(AfIzK))
34
35
36
37
38
39
40
41
42
43
44
45
46
47
45
4o
50
51
52
53
54
55
56
57
53
IF
( S I u -A B)
6 SIG=AB
6,7,7
JJ= I
7 C O V T IN V E
IF (JJ-K) 8 , 1 0 , 8
8 CO 9 J = K , M
TE'-'P = A ( J J , J )
A ( J J, J ) B A ('<, J )
9 A(KiJ)= TEMp
10 CS U
I=KPliN
C J 9 T “ A ( I#K)/A(K,K)
CO 11 J = K P I , M
11 A ( I ' J 5 * A ( I , J ) - QUfiJ * A (K, J )
CS 12 I = K P 1 , n
12 A ( I, K )=C»
Xv v ) = A ( \ , M ) / a (N,N)
CS 14 N N = I z U
S U M =0»
I=N-NN
IPl=I+!
CS 13 J = I P 1 , N
13 S U m = S U M + a ( I , J ) * X ( J )
14 X ( I ) = ( A ( I i M ) - S U M ) Z A ( I i I )
RETURN
E ND
Ln
sur-™guTi\'E m a t i n i k n b s )
W R I T T E N BY D A N I E L F • P R l L L S U M M E R 1 970
I N I T I A L I Z E S T HE m a t r i x l a b e l e d a r r a y
NEED T 5 C H A N G E C 9M M 5 N CARD,
SEE M A I N L I N E
C S M 11C W A R R A Y Z r ( A S / 49)
M = N E 5 * 1 6
Mle1-Ul
DO 10 1=1,M
DS IC J=I,Ml
10
F(IfJ)=C,3
DS ICO I = I j NB S
M= I * 16
F (M - 7 / M - 1 4 )= F ( M - 5 , M - 1 2 ) = 1.0
F ( 1-2, M - H ) = F ( M , M - 9 ) = IeQ
F (M - 6 , M - 1 5 ) = F ( M- a , M - 1 3 ) = - 1 » C
F (M - 3 > m . I Q ) = f ( M - I j M - S ) = - 1 , 0
CS ICO J = l , 8
Jl=J-S
JP =iUJl
100
F ( J P j J S ) = I tC
RETURN
END
C
r
c
C
O
to c\
IC
11
12
13
S U B R O U T I N E V A M U L (A , B , C , M , N , M M )
W R I t TEK BY D A N I E L F * PR ILL
S U M M E R 1970
T H I B S U B R O U T I N E IS FOR T H E C O M P U T A T I O N GF M A T R I X C
F R fU-1 C = M B W H E R E C IS AN M * M M M A T R I X , A IS AN M*N,
A N D B IS a n M lv Y,
D I M E N S I O N A ( M j N ) , B ( N v M M ) , C ( M z MM)
CS 3 I = i , M
DG 3 J= I,MM
C(IfJ)=C.
DG 3 K = I , M
3 C( Iv J ) = C ( I,J)+A( I, K ) *B (K, J )
RETURN
END
Ln
VO
I:
2?
3:
5:
6:
7:
SU3ReUTTNE
c
C
MATCCA,C,D,I,J)
Daniel f , prill
SUP-SUTINr MULTIPLIES CdNSTANT
D I m E N S ISN C ( 2 / 2 ) , D ( E / 2 )
DB 10 1 1=1,1
io u i = i , j
written
by
summer
970
A TIMES
matrix
c
10 D(IlyUl)=A^C(IliJl)
8:
RETURN
9!
end
o>
o
I:
2
:c
3: C
L:
5:
6:
7:
8:
s;
SU?>'eUTl\!E M A T A( A , B , C , I# J )
v r i t t e v by dani cl r , p r i l l
summer 197 o
THIS SL-RSUTI NE A D O S MATRIX A TS MATRIX B, C =A+ 3
DI yLNS I SN A ( 2 , 2 ) , S ( 2 , 2 ) , C ( 2 , 2 )
DS IO y =lil
. DS IO N = I i J
10 C ( M/N) =A(MiV)+B(MifN)
RETURN
END
ON
H
LITERATURE CONSULTED
1.
C. R. Wylie, J r . : Advanced Engineering Mathematics; McGraw-Hill
C o . , Third Edition.
2.
Schwesinger, G:
"On One Term Approximations of Forced NonHarmonic
Vibrations"; ASME Journal of Applied Mechanics, 1950.
3.
I . U. Ojaluo and M. Newman:
"Vibration Modes of Large Structures
by an Automatic Matrix-Reduction M e t h o d " ; AIAA Journal, July 1970,
P. 1234.
4.
F. J . Sta n e k : "Free and Forced Vibrations of Cantilever Beams
With Viscous D a m p i n g " ; NASATN D-2831.
5.
R. A. Anderson:
Fundamentals of Vibrations; The Macmillan Co.,
First Edition, 1967.
6.
M. F. S p o t t s : Design of Machine Elements; Prentice-Hall, Inc.,
Third Edition.
MONTANA STATE UNIVERSITY LIBRARIES
$
<
P93&
cop.2
P r i l l , Daniel
The forced vibration
of viscous damped, Nb e a m structures
N A M * A ND AODR***