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Rochester Institute of Technology Rochester Institute of Technology
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Theses
2-1-1992
Vibration analysis of a thin moving web and its finite element Vibration analysis of a thin moving web and its finite element
implementation implementation
Gavin Chunye Liu
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VIBRATION ANALYSIS OF A THIN MOVING WEB
AND ITS
FINITE ELEMENT IMPLEMENTATION
by
GAVIN CHUNYE LIU
THESIS ADVISOR: DR. H. GHONEIM
VIBRATION ANALYSIS OF A THIN MOVING WEBAND ITS
FINITE ELEMENT IMPLEMENTATION
GAVIN CHUNYE LIU
A Thesis Submitted
in partial Fulfillment
of the
Requirements for the Degree of
MASTER OF SCIENCE
In
Mechanical Engineering
ROCHESTER INSTITUTE OF TECHNOLOGY
Rochester New York
FEBRUARY 1992
Approved by:
Dr. H. Ghoneim (Advisor)
Dr. J. S. Torok
Dr. M. Kempski
Dr. Charles Haines
I, Gavin C. Liu, do hereby grant Wallace
permission to reproduce my thesis In whole
reproduction by Wallace Memorial Library will
for commercial use or profit.
ii
Memorial Library
or in part. Any
not to be used
ACKNOWLEDGMENTS
To my parents, and my wife, who have provided me the
supports and encouragement that I needed for all my work.
To Dr- Hany Ghoneim, my thesis advisor, I wish to express my
deepest gratitude for his boundless encouragement and
helpfulness, and his warmest friendship.
To Dr- Joseph Torok, who taught me finite element
formulation, and other graduate courses, my thanks for his
helps and admiration for his knowledge.
To Dr. Charles W. Haines, my academic advisor, my
appreciation for providing me with invaluable guidance during
my college years.
And finally to the members of my thesis defense committee, I
appreciate very much for the time they spent reading and
evaluating my work.
in
ABSTRACT
Equations of motion (lateral and axial) of an axially
moving web are developed based on the Newton's Second Law and
the Euler-Bernoul 1 i thin beam theory- The equations of motion
in the axial direction are solved by using the fourth-order
Runge-Kutta Method. The fourth-order, partial differential
equation for the lateral motion is solved using Galerkin's
finite element method and the Three-point Recurrence Scheme.
Effects of the flexibility of the end-supports, the weight of
the web, the axial web speeds, the eccentricities of the
rollers, and the applied torque to the web-roller system are
studied .
IV
TABLE OF CONTENTS
NOTATION vi
SYMBOLS vi i i
LIST OF FIGURES ix
1 INTRODUCTION 1
2 THEORY and DERIVATION 4
2.1 Governing Differential Equation 4
2.2 Supplementary Differential Equation 11
3 FINITE ELEMENT IMPLEMENTATION 14
3 . 1 Spatial Approximation 14
3.2 Time Approximation 18
4 RESULTS and DISCUSSION 20
4. 1 Free Vibration 25
4.2 Vibration of Constant Web Speeds 34
4.2.1 Effect of The Web Speed 34
4.2.2 Effect of Phase Shift Between
The Two Eccentricities of The Rollers ... 39
4.3 Vibration Due To The
Sudden Changing in The Applied Torque 44
5 CONCLUSION 50
REFERENCE 52
APPENDIX A 55
APPENDIX B 63
APPENDIX C 72
NOTATION
A cross-section area of the web
a lateral acceleration of element dx
B!,B2 viscous damping at rollers
b width of the web
b the global boundary condition vector
be
the element boundary condition vector
b*, boundary condition entries
c kinematic damping
[C] the global damping matrix
[C]e
the element damping matrix
Dl5D2 diameters of rollers
d thickness of the web
E modulus of elasticity of the web material
e eccentricity of unbalance rollers
Fy the resultant force of element dx in the
lateral direction
F force vector that includes the f and b
f the global external force vector on the web in
lateral direction
f the distributed external force on the web
fe
the element external force vector
fe, entries of external force matrix
g gravitational acceleration ( ss 9.81 m/s2)
h length of an element
I area moment of inertia of the web
<Ji > J2 polar moment of inertia of the web
K,Kl5K2 stiffness of the end-supports
k axial stiffness of the web
k, flexural stiffness of the web
[K] the global stiffness matrix
[K]e
the element stiffness matrix
1 length of the web
vi
[M] the global mass matrix
[M]*
the element mass matrix
m mass per unit length of the web
m0 unbalanced mass of the rollers
p tension of the web
Q , (X )shear force
R residual of the approximation
T applied torque
Tj frictional torque
V axial velocity of a material point on the web
Vx axial velocity of a material point at x/1 = 1
V2 axial velocity of a material point at x/1 = 0
u displacement in axial direction
w displacement in lateral direction
We
the element lateral displacement as a
function of time only
W the global lateral displacement as a function
of time only
X,Y fixed coordinates
x,y moving coordinates
/? Galerkin's coefficient ( /? = |)
7 Galerkin's coefficient ( 7 = |)
A a constant of a stiffness matrix
p density of the web material
strain
r equivalent axial force
a1,a2 angular accelerations of the rollers
Wj , w2 angular velocities of the rollers
0j , 02 angular displacements of the rollers
<?! , #2 angular displacements of the unbalanced mass
6 deflection angle of element dx
ip{ Hermite's interpolation functions (shape
functions)
(p{ the weight functions
vu
SYMBOLS
8 infinitesimal increment
A small increment
[ ] matrix
summation
| | determinant of a matrix
( ) derivative with respect to space
vm
LIST OF FIGURES
Figure Caption page
1 A simplified model of the web-roller system 2
2 An element of the moving web 4
3 Free body diagrams for end conditions 10
4 Free body diagrams for derivation of supplementary equations of axial motion 1 1
5 Model of" rigid-flexible"
setup 21
6 Model of"flexible-flexible"
setup21
7 Initial displacement for"rigid-flexible"
setup24
8 Initial displacement for"flexible-flexible"
set-up24
9 Response of free vibration for the"rigid-flexible"
setup at x/1 = 1.0 26
10 Response of free vibration for the"rigid-flexible"
setup at x/1 = 0.8 27
11 Response of free vibration for the"flexible-flexible"
setup at x/1 = 1.0 28
12 Response of free vibration for the"flexible-flexible"
setup at x/1 = 0.8 29
13 Response of free vibration for the"flexible-flexible"
setup at x/1 = 0.0 30
14 Response of system for the"flexible-flexible"
setup due to the weight
of the web at x/1 = 0.8 32
15 Displacement profile of the web with and without the weight of
the web at t = .501 sec 33
16 Response of the system at for the"rigid-flexible"
setup with
w = 20 rad/sec (V = 5 m/s) at x/1 = 0.8 36
17 Response of the system at for the"rigid-flexible"
setup with
w = 40 rad/sec (V = 10 m/s) at x/1 = 0.8 37
18 Response of the system at for the"rigid-flexible"
setup with
w = 60 rad/sec (V = 15 m/s) at x/1 = 0.8 38
19 Response of the system for"flexible-flexible"
setup with V = 10 m/s and
phase shift of
0"
at x/1 = 0.8 40
20 Response of the system for "flexible-flexible"
setup with V = 10 m/s and
phase shift of
90
at x/1 = 0.8 41
21 Response of the system for"flexible-flexible"
setup with V = 10 m/s and
phase shift of
180
at x/1 = 0.8 42
IX
22 Displacement profile of the web for three different phase angles at three
instants (t = .302, 1.201, 1.801 sec) 43
23 Plot of the changing torque for the transient vibration analysis 44
24 Response of the web tension due the changing torque 46
25 Response of the angular velocities of rollers 1 it. 2 due the changing torque 46
26 Response of the system for the"flexible-flexible"
setup due the changing
torque at x/1 = 1.0 47
27 Response of the system for the"flexible-flexible"
setup due the changing
torque at x/1 = 0.8 48
28 Response of the system for the "flexible-flexible"setup due the changing
torque at x/1 = 0.0 49
A-l Diagram for the derivation of equation (2.13) 56
A-2 Approximate function for the first and the last nodes 61
B- 1 Model for calculation of the natural frequencies of the roller-supports 64
B-2 Model for calculation of the natural frequencies of the web 66
B-3 Model for calculation of the natural frequencies of the web under the
assumption of rigid supports 67
1 INTRODUCTION
Vibration control on moving continua, such as moving
magnetic tapes, paper tapes, band-saw blades, pulley belts,
coating film, and other applications, is an active research
area for many manufacturing industries. Naguleswaran and
Williams [1] studied the vibration and stability of a moving
string excited paramet r ical ly by the periodic vibration of
tension in the string. Ulsoy and Mote [2] studied the
vibration of wide band-saw blades by modeling the blades as
moving plates. Mote and Naguleswaran [3] did research on the
linear free vibration of an axially moving thin beam using an
exact solution, Galerkin solution, and flexible band solution.
Chonan [4] investigated the steady state response of an axially
moving strip under a transverse point load. Effects of
t ranslat ional velocity on the natural frequencies and modes of
an axially moving beam between fixed ends were investigated by
Simpson [5] using the Euler beam theory. Wickert and Mote [6]
analyzed the response of an axially moving continua between two
fixed point using a second-order model and a fourth-order
model. Manor and Adams [7] analyzed the response of an elastic
beam moving with constant speed across a drop-out at a smooth
rigid foundation using the Euler-Bernoul 1 i beam theory.
Analysis of the vibration due to the tension variation of
moving continua together with effects of friction between the
moving continua and rollers were done by Whitworth and Harrison
[8] . Gottlieb [9] studied the non-linear vibration of a
constant-tension string. Daly did some studies on the effect
of air pressure on a moving web [10] . Brandenburg studied the
effect of the frictional force between the web and the roller
by assuming the web is stretchable and obeys the Hooke 's law
[11] . Most of these previous studies were done under the
assumption of rigid supports.
Figure 1 shows the model of the web-roller system to be
investigated in this study. The moving continuum is a thin
web, and it is modeled as a continuous elastic beam. The
driving and the driven mechanisms are simplified as rollers
which are supported by linear springs. Note that the
deflection of the roller-supports in the horizontal direction
are not taken into consideration in this investigation.
A, E, I
M J,
vv/v.v
Ml h
Figui* 1. A amplified Model of the web-roller syvtem.
Due to many factors such as the flexibility of supports,
the velocity of the web, the imperfection (eccentricities of
the rollers) of the machinery, the non-linear material
properties of the moving web, the frictional force between the
moving web and the stationary machine parts, and the
aerodynamics effects (for high speed only), vibration control
of the moving web is a very complicated process. For the sake
of simplicity, the effects of the non-linear material
properties, the frictional force, and the aerodynamics are not
taken into account in this work. Studies of the web are done
only on the effects of the flexibility of supports, the
velocity of the web, and the imperfection of the machinery (the
unbalance on the rollers) . Newton's second Law and Euler-
Bernoulli beam theory are applied in the formulation of the
flexural vibration of the continuous beam. The Fourth-order
Runge-Kutta method is employed to solve the equations of motion
in the axial direction. The Galerkin's finite element method
and the Three-point Recurrence Scheme [12] (the Newmark
Recurrence Scheme [13]) are used for the solution of the
flexural vibration.
2. THEORY and DERIVATION
The development of the basic differential equations
(governing and supplementary equations) in this study is based
on Newton's Second Law and Euler-Bernoul 1 i theory (the thin
beam theory) [14] . The governing differential equation for the
flexural vibration of the beam (in this case, the beam is a
flexible web) is developed first. Then supplementary
differential equations are derived for the axial motion of the
web .
2.1 GOVERNING DIFFERENTIAL EQUATION
An element of length dx of the web is shown in Figure 2
together with all forces and moments.
Figure 2. An element of the moving web.
Applying Newton's Second Law ( Fy = ma ) to the free
body diagram in Figure 2 gives
-psinfl + Q + fdx - (Q + |Qdx)<5X
7
+ (P + ^dx)sin(0 + Mdx) = ma (2.1)
Under the assumption of small 9, the following
relationships are valid.
sin0 as tan0 = ^ (2.2)
sin(0 + gdx) a 0 + Mdx
tan0 + f^dx>5x
= fe + ^dx (2-3)
Substituting equations (2.2)-(2.3) into equation (2.1),
taking into consideration that m = pAdx, and a =
2-Wancj
Dt2
neglecting the second order terms yield
f ' S + " - >* C*.4>
where p is assumed to be time but not space dependent for
s impl ic ity .
Summing the moment about point 0 (Figure 2) , and assuming
the moment of inertia of the element to be negligible gives
Mo = _M + fdx^ + (M + |Mdx) _ (Q + |9dx)dx = 0 (2.5)
Once again, neglecting the second order terms, equation
(2.5) reduces to
or
Mdx - Qdx = 0
= (2.6)
Substituting equation (2.6) into equation (2.4) results in
* - (!5) + p S = m6xv5x
PiwDt2 (2.7)
Applying the Euler - Bernoulli beam theory (the thin be<
theory) [14], M(*,t) = EI()W(*'t}
, to equation (2.7) gi
<5x
ves
f - A_[EI(,)d-Sli] + p X() = pA g^ (2.8)5x <5x (5x
v.5x' Dt"
Assuming the cross-sectional area of the web is constant
along the web, the moment of inertia I(i) is equal I, and
equation (2.8) can be rewritten as
EI^ + PA) + f = pA a^fix dx Dt(2.9)
Where
and
E
I
P
f
P
A
. .the lateral displacement of the centroid of
element dx,
..the modulus of elasticity of the web material,
..the cross sectional area moment of inertia,
. .the axial force (tension) in the web,
..the lateral distributed force per unit length,
..the density of the web material,
. .the cross sectional area of web.
6
r\2
The term ^-^ in equation (2.9) is the material derivative
of w with respect to time [15] , and it can be expressed as:
Dfw_
Dw n
{2 10^1Dt2 Dt1^ Dt
> ^,iUj
where
Dw_ 6w , 6_w x
Dt St"*"
<5x St
or
where V is the axial velocity of a material point on the web,
and it is a function of both time and position, x. It is
defined, at any given instant, as
V = V2 + (Vx - V2) 2f (2.12)
where Vx and V2 are the web velocity at the right end (x/1 =
1.0) and the left end (x/1 = 0.0), respectively, and they are
defined as
Vx = ^ Ul (2.13)
and
V2 = u,2 (2.14)
where Dj and D2 are the diameters of the rollers #1 and #2,
respectively, uil and w2 are the angular velocities of the
rollers #1 and #2, respectively, and they can be found from the
supplementary differential equations.
Substitution from equation (2.11) into equation (2.10)
with further expansion yields
7
Pi = Pi + 2vf- + &g +SW |w|Vv (215)Dt2
St StSx. oxot8x.2 6x.6x
v J
for small deformation
W=k^ (2-16)
where e, which is the strain [16] in the web due the web
tension p, is defined as
e = & (2.17)
Derivation of equation (2.16) is shown in Appendix A.
Substituting equations (2 . 15) - (2 . 17) into equation (2.9),
and rearranging terms yields
m<5!w
+ 2c^ + k^- + rpi + Af^ = f (2.18)S-t2 6t,8x 6x4 <5x2 *x
where m = pA (2.19)
c = pVA (2.20)
k = EI (2.21)
r =pAV2
-
p (2.22)
and A =pArSV
+ _V_*E) (2.23)
The term fx can be defined as
ot
SV _D2 6u2 fD1 Sux D_2 Su^.
x r9 94n
Tt~
It+ ^2 6t 2 6t ; 1 <^.^;
where and -^can be found from the supplementary
r5t Ot
differential equations.
The boundary conditions of equation (2.18) are derived
from free body diagrams in Figure 3. Figure 3 shows that the
continuous beam is postulated to be simply supported at each
end. For simplicity, two rollers are assumed to be identical
with unbalanced m0e, and are supported by two identical elastic
springs with stiffness K as shown in Figure 3. The unbalance
m0e in each roller is assumed to be identical. It should be
pointed out that the phase angle between the two eccentricities
is zero in Figure 3. However, the phase angle could be any
value .
By summing all forces in the lateral direction at each end
of the free body diagram one at a time, the boundary conditions
[14] are found to be
1) At the left end:
n - FTo"3w(:M)|y(o,)
~
ti"7x3~
l(o,)
= Mw(o,t) + Kw(o,t) + m0ew22sin# - m0ea2cos# (2.25)
where K = Kx = K2 in this study.
2) At the right end:
n - FT^'1'0!yC'*>
~
<5x3l(M)
= - Mw(i,) - Kw(i,t) - niQeu^sin^ + mgeo^cosd} (2.26)
Where m0 is the unbalanced mass.
m0ew3
m0ee*j
fBoewj
nioei
F i gu re 3Ffee body diagram for derivation of end conditions.
For simply-supported beams, the ends
take any moment at any time, i.e.
of the beams do not
8 W(r,t)i_ Q
M(o,t)~ Ei
6x2 l(o,t)~ u (2.27)
M(i,t)
= EI62w(>,0|5x2 'l('.t)
= 0 (2.28)
Equation (2.18) is the equation of motion of the element
dx in the lateral direction with the boundary conditions as
defined by equations (2 . 25)- (2 . 28) . The equations of motion in
,1direction for providing
the values of V, p, ^,and
the axial
dpdt
are to be derived in Section 2.2,
10
2.2 SUPPLEMENTARY DIFFERENTIAL EQUATIONS
In order to solve the fourth order partial differential
equation (equation (2.18)), variables V, p, |^, and-jS in the
axial motion have to be found first. Figure 4 shows the free
body diagrams of the two rollers of the web-roller system.
.5m
roller #2 roller #1
Figure 4 Free body diagram of rollers for derivation of
supplementary differential equations
Summing the moment about the center of roller #1, Oj ,
gives
or
J^ = T - B^ -
p
8ui1 _ \_
Di
Di
w- ^( T - BlWl
-
P^ ) (2.29)
where Jj ... the polar moment of inertia of the roller #1,
11
Bj ... the viscous damping at the roller #1,
Wj . . . the angular velocity of the roller #1,
Dx ... the diameter of the roller #1,
T ... the applied torque at the driving roller.
Summing the moment about the center of roller #2, 02 ,
gives
or
J2"2 =
P^-
T, B->u>2w2
St J2^ p2( P-y-
T,- B2w2 ) (2.30)
where J2
B2
w2
D,
the polar moment of inertia of the roller #2,
the viscous damping at the roller #2,
the angular velocity of the roller #2,
the diameter of the roller #2,
the fictional torque at the driven roller.
Assuming that
p = k Al (2.31)
where k is the axial stiffness of the web in this study and Al
is the change in the length of the web due to the different
angular displacement of the two rollers. Under the assumption
of no slipping between the rollers and the web, Al is defined
as
2Al = 0, -
2Uz (2.32)
where Qx , 02 are the angular displacements of rollers.
Substitution from equation (2.31) into equation (2.32)
yields
P = k( Sfe, - ^e2) (2.33)
Taking the derivative of equation (2.33) with respect to
12
time gives
g= j|( DlWl - D2u,2 ) (2.34)
Equations (2.29), (2.30), and (2.34) represent the
supplementary differential equations that are required to
evaluate the values of V, p, ^, and -jx in the axial directionot dt
of the web. Initial conditions for equations (2.29), (2.30)
and (2.34) are to be defined in the chapter of Results and
Discussion in this paper.
Solution of equation (2.18) is to be approximated by
applying the finite element method with the boundary conditions
that are established in equations (2 . 25) - (2 . 28) . The
supplementary differential equations are to be solved by the
fourth-order Runge-Kutta method. The finite element
formulation of equation (2.18) is to be presented in the next
chapter .
13
3. FINITE ELEMENT IMPLEMENTATION
Solution of the partial differential equation (equation
(2.18)) is accomplished in two stages: (1) Spatial
approximation which is achieved by using the Galerkin's method
(the Weighted Residual Method). (2) Time approximation which is
approached by using the Three-point Recurrence Schemes [12]
(the Newmark recurrence scheme [13] ) . The first stage
approximates the partial differential equation (2.18) by a set
of simultaneous, ordinary differential equations, and the
second stage approximates the set of ordinary differential
equations by a set of linear algebraic equations, which can be
solved at each time step. Such a procedure, which finds the
spatial approximation first, then the time approximation, is
known as a semi -d iscrete approximation [17] .
3.1 SPATIAL APPROXIMATION
The Galerkin's method is used for the approximation of the
partial differential equation.
Assuming the solution of equation (2.18) for each
element, wc(x, ) ,
to be
we(*,0 = 1>l*)j WV);, N = 4 (3.1);'=i
where xp(x) is the shape function within each element, We(*)j is
the nodal lateral displacement of the element, and N is the
order of the interpolation function. For simplicity, ip(x)j and
14
We(t);- are to be written as tp- and WO, respectively, in the later
descr ipt ion .
Substituting equation (3.1) and its derivative with
respect to time and space into equation (2.18) gives
N N N
mEV-jWe;. +
2ct/>;.' We;- +kl>;""
W%j=i j=i i=i
+rV;" W%- + At/y
W%- - f = R ^ 0 (3.2).
j=i j=i
Note that expression (3.2) is no longer exactly equal to zero
since the approximate representation for we(x,t) is used. Hence R
is called the residual of its original equation.
Setting the integral of a weighted residual of the
approximation over the whole domain of each element to zero
results in
J"
<t>{ R dX = 0 (3.3)o
where </>, are the weight functions (which, in general, are
not the same as the shape function ipj) .
Substituting equation (3.2) into equation (3.3), and after
some manipulation, gives
(m iM,-dX) '\f'j + (2c V/^dX) W%j=i o j=i o
+ (kE ^ W'dx) w%. + (rE / w^'dx)vi'
jj=l 0 j=l o
+ (AE / V-/^dX) We,. - / <^,fedX = 0 (3.4)j=i o o
15
Integrating the third term of equation (3.4) by parts, it
becomes
(k ^/'"dX)we; = (k V^"dX)Wi" M " *i (3-5)
i=i o j=i o o
Nwhere M = k xp
We;- is the bending moment of each element
i=i
N
and Q = k V/" We;- is the shear force of each element.
;'=i
Substituting equation (3.5) into equation (3.4), gives
Me{jW%. + Ce(j
W%- + K'{jW%- = f% + b% (3.6)
where
and
Ke0- = Keoij + KeUj + K%,;. (3.7)
M8,- = | rPjm <, dX (3.8)
o
Ce{j = /V c <p{ dX (3.9)
Keotj =jV' k ^" dX (3.10)o
K^j=jV'
T +i dX (3-11)0
Ke2ij= jV A *,- dX (3.12)
o
f. = J*f 0,- dX (3.13)
o
andb\- = [M 0,'
- Q <,]
"
(3.14)o
In this study, the Hermite cubic interpolation functions
(shown in Appendix A) are adopted for the shape function.
16
Applying the Galerkin's method, the weight functions <p{ are the
same as the shape functions xp);. The corresponding element
matrices, the external force vector, and the boundary condition
vector are evaluated based on the Hermite cubic interpolation
functions in Appendix A.
The global equations (a set of ordinary differential
equations) are obtained via the standard assembly method [18],
i.e.
[M]e We
+[C]e We
+[K]e We
=fe
+be
or [M] + [C] W + [K] W = f + b (3.15)
where n is the number of elements, W is the global lateral
displacement vector, and [M] , [C] , [K] ,f , and b are the global
mass matrix, damping matrix, stiffness matrix, external force
vector, and boundary condition vector, respectively.
It should be pointed out that the global boundary
condition vector, b, has zero components except for those at
the end nodes (the first and the last nodes), i. e.
b = ( Q(0), - M(0), 0, 0,..., 0, 0, -Q(i), M(/) ) (3.16)
where the shear forces of the first and the last nodes are to
be determined from equations (2.25) and (2.26), and the moments
of the first and the last nodes are zero according to equations
(2.27) and (2.28).
Equation (3.15) is the global ordinary differential
equation which needs further approximation in the time domain.
17
3.2 TIME APPROXIMATION
Equation (3.15) is exactly the same as equation (21.1) in
"The Finite Element Method"
by 0. C. Zienkiewicz [12] . The
Three-point Recurrence Scheme (the Newmark Recurrence Scheme)
was used by Zienkiewicz for the time approximation of equation
(21.1). Following Zienkiewicz 's derivation procedure, the time
approximation of equation (3.15) takes the form of
[[M] + 74t[C] + /?4t2[K]] Wt+1
+ [-2 01] + (l-27) *t[C] + (i-2/?+T)^t2[K]] W,
+ [[M] - (l-7)*t[C] + (i+/?-7)at2[K]] W(_x +FAt2
= 0 (3.17)
where [M] , [C] ,and [K] are defined as in equation (3.15) ,
F is
the force vector that includes both the global force vector and
the boundary condition vector, 0 and 7 are the Galerkin's
coefficients for time approximation, and the subscripts t+i, i,
and (-1 indicate three consecutive instants. The values of 0
and 7 are | and |, respectively, which are evaluated by
Zienkiewicz .
According to Zienkiewicz's derivation, the F is
interpolated as
F = F(+1/3 + F((i-2/?+7) + F(_1(|+/?-7) (3.18)
Equations (3.17) and (3.18) are the final forms for the
solution of equation (2.18), and they are used in the finite
18
element program. The values of W(-1 , Wt , Ft-1 , F, , and Ft+1, which
are used to start the program, are described in the program in
Appendix C. This chapter described the whole process of
solving equation (2.18) starting from implementing the element
matrices, to assembling the global matrices, and then to
finding the final form of approximation. The numerical results
of equation (3.17) for this study are shown in the next
chapter .
19
4. RESULTS and DISCUSSION
All results are generated by three computer programs:
AXIAL, WEB1 , and CONVER. The AXIAL program, which was written
by Dr. H. Ghoneim, utilizes the fourth-order Runge-Kutta method
to solve the supplementary differential equations of the axial
motion, and it has been updated to include the investigation of
different web speeds and different phase angles between the two
unbalanced rollers. The AXIAL program has to be run first to
provide the WEB1 program with the necessary values of V, -^ , p,
and - at each time step. The WEB1 program is a finite element
program, which was originally written by Dr. H. Ghoneim in
FORTRAN code. The WEB1 program utilizes the final
approximation equations (equations (3.17) and (3.18)) with the
data of V, j^ , p, and-r^
from the AXIAL program to solve the
partial differential equation (equation (2.18)) for the
flexural vibration of the web-roller system. The WEB1 program
has been updated to incorporate the gravitational force of the
web. The CONVER program is used to convert the time domain
response from the WEB1 program into the frequency domain. The
CONVER program calls a sub-program, FFTRF, from the IMSL
library of the RITVAX system [19]. The FFTRF employs the Fast
Fourier Transform algorithm. All programs are listed in the
Appendix C.
The web-roller system is investigated for two different
setups (Figure 5 and 6) :
1) A"rigid-flexible"
setup ( Figure 5, ), where one
roller-support is treated as a rigid support and the other is
20
out
WW.
F igu re 5 a simplified model of the"rigid-flexible"
web-roller system.
Figure 6A simplified model of the
"flexible-flexible"
web-roller system.
21
Studies are conducted for the following cases:
1. Free vibration of the web when it is not axially
moving ( V = 0 ) for both"rigid-flexible"
and
"flexible-flexible"setups. This is done to verify
the validity of the computer programs and to study
the effect of the flexibility of the roller-supports.
Effect of the weight is also considered for the
"flexible-flexible"
setup when the web is not moving.
2. Vibration of the web when it is moving with constant
speeds. Two factors are investigated:
a. The effects of different web speeds for the
"flexible-rigid"setup.
b. The effects of the phase shift between the
eccentricities of the rollers for the "flexible-
flexible"
setup.
3. Vibration of the web under the effect of a sudden
change of the applied torque at the driving roller-
The following values of the system parameters are adopted
for the investigation.
p =103
kg/m3, Dx = D2 = 0.5 m,
E = 4 x109
pa, Jx = J2 = 8.283kgm2
,
d = 1.25 x10-4
m, Bi = B2 = 0.5 Nms ,
b = 1.35 m, I<! = K2 = 7.36 x109
N/m,
1 = 1.52 m, Mt = M, = 8.27 x103
kg,
22
b = 1.35 m, K! = K2 = 7.36 x109
N/m ,
1 = 1.52 m, M! = M2 = 8.27 x103
kg,
k = 4.44 x105
N/m, m0e = 10 kgm .
where d, b, 1, and k are the thickness, width, length, and
axial stiffness of the web, respectively.
Other parameters, that are needed for obtaining the
results of this paper from the WEB1 and AXIAL program, are
listed as the following:
start time = 0.0 second,
end time = 1.93 second,
time increment = 0.001 second,
and number of elements = 21 elements.
However, the values of the above parameters can be changed to
different values.
Results are displayed in Figures 9-22, and 24-28. Note
that for Figures 9-14, 16-21, and 26-28, each figure is
composed of three graphs: a, b, and c. Part a of each figure
represents the temporal response of the system in the time
domain, while parts b and c show the corresponding response of
the system in the frequency domain for the linear and
logarithmic amplitudes respectively. Logarithmic plot for all
frequency responses of the system are displayed to give better
visibility of the peaks of the frequency spectrum.
23
Hl> WW
\-L.
Tcsua.iMu nsirrcn
c- >
Figure 7 A aimplified model of the"rigid-flexible"
web-roller system.
cajuamw narwm
<H
F i gu re 8 A simplified model of the"flexible-flexible"
web-roller system.
24
4.1 FREE VIBRATION
The following parameters are considered for the free
vibration studies of the web-roller system when the web is not
in motion (wx =w2
= 0 rad/sec), and the pre-tension, p = 200 N.
The system is excited from rest via a linear, initial
displacement in the lateral direction as shown in Figure 7 and
8. The values of the displacement at the left and the right
ends are w(o,o) = 0 m, and w(/,o) = 4E-6 m, respectively.
Samples of the free vibration results of the web-roller
system are shown in Figures 9 and 10 for the"rigid-flexible"
setup, and in Figures 11, 12, and 13 for the "flexible-
flexible"setup. Figure 9 shows the response of the system at
the position of x/1 = 1.0. Figures 9b, and 9c show the natural
frequency of the roller-support is about 140 Hz, which crudely
agrees with the natural frequency of the roller support.
Analytical results (shown in Appendix B) of the natural
frequencies of the roller-supports and the web are found to
verify the numerical solution obtained from the program.
Figures 10b and 10c show the numerical values of the natural
frequencies of the web from the programs. Those numerical
values are in excellent agreement with the analytical results
in the Appendix B. Figures 10b and c also indicate very
clearly that the vibration of the web-roller system is
dominated by the natural frequency of the roller-supports.
Comparison of Figure 11, 12 with Figure 9, 10 respectively
indicates that the flexibility of the left end has very little
effect on the responses of the web-roller system. Figure 13
shows the responses of the system at x/1 = 0.0, and it
indicates that the natural frequency of the left roller-support
to be 140 Hz. That is because the two roller-supports are
25
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exactly the same for the "flexible-flexible"setup. An
interesting phenomenon is observed in Figure 13a. The left
roller support, which is initially at rest, is gradually
picking up vibration that is transferring from the right roller
through the web. This is happening because the web-roller
system is acting as a predominantly two degrees of freedom
system with the two rollers linked together by a very soft
spring (the web) as shown in Figure 8. Since the stiffness of
the web, in this analysis, is far smaller than those of the
roller supports (calculations of stiffness for support and web
are shown in Appendix B) , the energy transfer from the right
roller to the left roller is happening in an extremely slow
manner. (Note that after 2 sec, the left roller picked up a
vibrating amplitude of10-8
m which is about 450 times smaller
than the vibrating amplitude at the right roller.) The
significant difference between the stiffness of the roller-
supports and the flexural stiffness of the web is the reason
why most previous studies in this field were done under the
assumption of rigid supports.
When the web is not in motion (V = 0 m/s) ,the damping of
the system is equal to zero as represented by c in equation
(2.17). The beating phenomena (Figure 10a, and 12a) occur when
the system with very little damping (in this case the system
has no damping) is subjected to the excitation frequency (the
natural frequency of the support) that is very close to the
thirteenth mode natural frequency of the web [20] (analytical
solution for the natural frequencies of the roller-supports and
the web are shown in Appendix B) .
Responses at the position of x/1 = 0.8 due to the weight
of the web are shown in Figure 14. Due to the downward
direction of gravitational force, the temporal response (Figure
14a) of the web shifts toward the negative side. Comparing the
plots in Figures 14b and 14c with the plots in Figure 12b and
12c, reveals that the vibrating amplitude is bigger for the web
31
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32
with the weight than for the web without the weight. The
weight of the web tends to promote the lower frequencies.
Figure 15 shows the displacement profile of the web, with
and without the weight of the web, at the instance of t = 0.501
second after the right end has been released from the initial
displacement of 1E-4 meter. The lower curve is the
displacement profile of the web with the weight of the web, and
the other curve is the displacement profile of the web
excluding the weight.
RcturaL shape of the ueb at t-
.501 sec
'a o
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figure 15 Displacement profile of the web with and without the weight of
the web at t = .501 sec
Upper curve: without the effect of the weight.
Lower curve: with the effect of the weight.
33
4.2 VIBRATION OF CONSTANT WEB SPEEDS
The web speeds, that are going to be used in this section,
are actually the axial velocity of a material point on the web
(equations (2 . 12) - (2 . 14) ) since the angular velocities of the
rollers are the same.
4.2.1 Effects of the Web Speed
The system is studied, when the initial displacements at
the left and the right ends are w(o,o) = 0 m, and w(*,o) = 4E-6 m,
respectively, for the following cases:
1) V = 5 m/s (w1 =u>2
= 20 rad/s) ,and p = 160 N (constant) ,
2) V = 10 m/s (u*! =w2
= 40 rad/s) ,and p = 200 N (constant) ,
3) V = 15 m/s (u1 =u>2
= 60 rad/s) ,and p = 240 N (constant) .
Figure 16, 17, and 18 show the responses of the system for
the"flexible-flexible"
setup with the web speeds of 5 m/sec ,
10 m/sec, and 15 m/sec at x/1 = 0.8, respectively. A closer
look at the Figures (b, and c of Figures 16, 17, and 18)
reveals that the natural frequencies of the web decrease as the
web speed increases. That is because the stiffness of the web
decreases as the web moves faster in the axial direction.
Three factors contribute to the stiffness of the web-roller
system as described by equations (3.7). These factors are: 1)
the flexural stiffness of the web, k(EI) in equation (3.10),
2) the axial velocity and tension of the web, as represented by
t in equation (3.11), and 3) the time rate of change of the
34
equation (3.12). For constant web axial speed and tension, the
third factor drops out (A = 0). As illustrated in Appendix B,
the value of r decreases with increasing web axial speed, and
consequently, the total stiffness of the web decreases as well.
It should be pointed out that the axial speed introduces a
"kinematic"
damping to the system through the parameter c in
equation (2.18). As the speed increases, the damping factor, c,
(equation (2.20)) increases. It appears, however, from the
figures that the change in c with the speed is too small to
introduce any significant effects.
Again, the beating phenomenon occurs in all the three
cases as one of the higher natural frequencies ( the 13 ,the
14'A, and the16"1
mode for V = 5 m/sec, 10 m/sec, and 15 m/sec
respectively) modulates with the natural frequency of the rotor
support .
35
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38
4.2.2 Effects of Phase Shift Between The Two
Eccentricities of The Rollers
Investigation on the effects of 0, 90, and180
phase
shift between the two eccentricities of the rollers is done
when the web speed is 10 m/sec (wj = w2 = 40 rad/s) ,and the
tension of the web is 200 N.
For the "flexible-flexible"
system, the phase angle of the
eccentricities of the rollers plays an important roll in the
vibrating modes of the moving continua [1] . Figures 19, 20,
and 21 show the responses of the system with phase angles of
0, 90, and 180, respectively. Comparison of the results of
the three cases shows that the0
phase angle-difference case
produces the smoothest vibration as shown in Figure 22a. For
this case, since both end-rollers move together, only the lower
modes are excited, and the higher modes are relatively
suppressed as shown in Figure 19b. That also explains why the
amplitude of vibration of the first peak is slightly higher for
the 0 case than the other two cases(90
and 180) . For the
90
and180
phase difference, the opposite movement of the two
rollers excites the higher modes and triggers the natural
frequencies of the roller supports as demonstrated in Figures
20b, 20c, 21b, and 21c. It appears that there is no
significant difference between the responses for the90
and
the180
phase shift case. These suggest that beyond a certain
phase shift the degree of shift does not have a significant
influence on the dominant vibrating frequencies.
Figure 22 shows the displacement profiles of the web for
the three different phase angles (0, 90, and 180) at three
different instants(t = .302 sec, t = 1.201 sec, and t = 1.801
sec). Clearly, for the0
phase angle case, thel"
mode
dominates the rest (Figure 22a) as a result of the synchronized
motion of the end rollers, while in the other two cases (Figure
22b, 22c) higher frequencies are well excited.
39
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43
4.3 VIBRATION DUE TO THE SUDDEN CHANGE IN THE APPLIED TORQUE
Studies of the effect of a sudden change in the applied
torque from 70 NM to 110 NM (Figure 23) are done when the web
is initially moving at 10 m/sec (wt = w2= 40 rad/s) and has an
initial tension of 200 N. The phase angle in this case is 0.
?8-
,8-
cr
O 5
Toroue vs TLms
i | I I I I I I i i i | i i i i . . I r r r i i r i | I I I I | i I I I | I I I r | . . i I | i .
0.2 0.1 0.6 O.S l 1. 2 1.1 1. 6 I.S 2 2.2
Time (sec)
Figure 23
Figures 24-25 show the responses of tension, and the
angular velocities of rollers due to the sudden change of the
applied torque. As the applied torque is suddenly increased,
the tension of the web follows almost instantaneously and
increases to an average steady state value of 280 N (Figure
24). Because of the assumption of an elastic web, the steady
44
state tension in the web is oscillatory. As for the angular
velocities of the end rollers, they follow "sluggishly"and
increase very slowly (Figure 25) . This is attributed to the
high inertia of the rollers which resists any sudden change in
speed and delays the response.
Figures 26, 27 and 28 show the responses of the system,
for the"flexible-flexible"
setup due to the sudden change of
the applied torque at three points on the web. Figures 26a and
28a demonstrate that the amplitudes of vibration of the roller-
supports are gradually increasing. This is because the angular
velocities of both rollers are gradually increasing and
consequently the unbalanced forces are also gradually
increasing. The same, but less obvious, can be observed about
the web vibration in Figure 27a. Plots b and c of Figures 26,
27, and 28 clearly show that the response of the system is
dominated by the excitation frequency of the rotating
unbalanced rotors.
45
Figure 24 Response of tans, on due to the sudden charge
of torqu* wlUi t_h i_rii.li.al voLu of 200 N
Figure 25a
f?MD008 of W2 due to the sudden chanat'
of Corqu w<.th ch Lni.ti.ol. voLu of 40 rod/sac
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TLm (sc)
Response of HI due to the sudden change
of Corqu* wi-Ch Lh Lr-n.(.i.oL voLut of 40 rod/sac
46
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49
5. CONCLUSION
From the foregoing analysis, it can be concluded that the
web-roller system is affected by the flexibility of the roller
supports, the weight of the web, the axial speed of the web,
the phase shift between the unbalance masses, and the sudden
change of the applied torque on the driving roller.
1) Flexibility of the driven roller-supports has very little
effect on the free vibration of the system for a short period
of time. However, for a long run, the free vibration indicates
that the energy transfer phenomenon (beating phenomenon)
between the two roller-supports can occur.
2) Increasing the speed of the moving web increases the
"kinematic damping", which reduces the vibration. However, it
turns out that the increased damping is too small to be
significant .
3) Increasing the speed of the moving web reduces the overall
lateral stiffness of the system and consequently decreases the
values of the natural frequencies of the web. This indicates
that there is a higher possibility for one of the natural
frequencies of the web to match the angular velocities of. the
rollers (excitation frequencies).
4) Increasing the speed of the web (which means increasing the
angular velocity of the rollers) amplifies the unbalanced
forces at the rollers and consequently introduces more
vibration to the system.
50
5) The phase shift between the unbalanced forces of the rollers
excites the higher modes of vibration and aggravates the
vibration of the system.
6) A sudden increase in the applied torque introduces
fluctuation in the web tension.
This work can be extended to study the effects of other
factors such as the viscoelast ic ity of the web, the change in
the dimensions of the web and the material properties, the
aerodynamic forces on the web, and the frictional force between
the web and the rollers.
51
REFERENCES
[1] S. Naguleswaran and J. H. Williams, "Lateral vibration
of band-saw blades, pulley belts andlike,"
International Journal of Mechanical Sciences, 1968, Vol.
10, pp 239-250.
[2] A. G. Ulsoy and C. D. Mote, Jr., "Vibration of wide band
sawblades,"
Transactions of the American Society of
Mechanical Engineering, Series B, Journal of Engineering
for Industry, 1982, Vol. 104, pp 71-77.
[3] C. D. Mote, Jr. and S. Naguleswaran, "Theoretical and
experimental band sawvibrations,"
Transactions of the
American Society of Mechanical Engineering, Series B,
Journal of Engineering for Industry, 1966, Vol. 88,
pp 151-156.
[4] S. Chonan, "Steady state response of an axially moving
strip subjected to a stationary lateralload,"
Journal
of Sound and Vibration, 1986, Vol. 107, pp 155-165.
[5] A. Simpson, "Transverse modes and frequencies of beams
translating between fixed endsupports,"
Journal of
Mechanical Engineering Science, 1973, Vol.15, ppl59-164.
[6] J. A. Wickert and C. D. Mote, Jr., "Classical vibration
analysis of axially movingcontinua,"
Transactions of
the American Society of Mechanical Engineering, Journal
of Applied Mechanics, 1990, Vol. 57, pp 738-744.
52
[7] H. Manor and G. G. Adams, "An elastic beam moving with
constant speed across adrop-out,"
International Journal
of Mechanical Science, 1983, Vol. 25, No. 2, pp 137-147.
[8] D. P. D. Whitworth and M. C. Harrison, "Tension
Variation in pliable material in productionmachinery,"
Applied Mathematical Modeling, 1983, Vol. 7, June, pp
189-196.
[9] H. P. W. Gottlieb, "Non-linear vibration of a constant-
tensionstring,"
Journal of Sound and Vibration, 1990,
Vol. 143, No. 3, pp 455-460.
[10] D. A. Daly, "Factors Controlling Traction Between Webs
And Their CarryingRolls,"
Tappi, 1965, Vol. 48, No. 9,
pp 88A-90A.
[11] G. Brandenburg, "The Dynamics of Elastic Webs Threading
a System ofRollers,"
Newspaper Techniques, 1972,
September, pp 12-25.
[12] 0. C. Zienkiewicz, "The Finite ElementMethod,"
3rd
edition, McGraw-Hill Book Company (UK) Limited, 1977, pp
581-589.
[13] D. S. Burnett, "Finite Element Analysis From Concepts to
Applications,"
Add i son-Wesley Publishing Co., 1987,
pp 526-531.
[14] S. S. Rao, "MechanicalVibration,"
Addison-Wesley
Company, Inc. 1984.
53
[15] T. J. Chung, "Continuum Mechanics", Prentice Hall, New
Jersey, 1988.
[16] F. P. Beer and E. R. Johnston, Jr., "Mechanics of
Materials,"
McGraw-Hill Book Company, Inc. 1981, p42 .
[17] J. N. Reddy, "An Introduction to The Finite Element
Method,"
McGraw-Hill Publishing Company, Inc. 1984,
p 50.
[18] D. L. Logan, "A First Course In The Finite Element
Method", PWS-KENT Publishing Company, Inc, 1986.
[19] "User's Manual: IMSLMath/Library,"
IMSL Company, 1989,
Version 1.1, January, p 707.
[20] M. L. James, G- M. Smith, J. C. Wolford and P. W-
Whaley, "Vibration of Mechanical and Structural system:
With MicrocomputerApplication,"
Harper k Row,
Publishers, New York, 1989.
54
APPENDIX A
Al DERIVATION OF EQUATION (2.13) IN CHAPTER 2
A2 SOME DETAILS OF THE FINITE ELEMENT IMPLEMENTATION
55
Al DERIVATION OF EQUATION (2.13) IN CHAPTER 2
Figure A-l shows a section of undeformed web, O'o ,and a
section of deformed web oox with a moving coordinate system xy
in the space of a fixed coordinate system XY. Note that
section O'o has been straightened as section Oo for better
understanding of the diagram.
-**
k
-J\
oEjSsrrEai
Figure A-l
At time equal to zero, point A is measured from system XY
with Z, so Z is time independent. After a period of time, At,
the moving system xy moves from 0, the origin of fixed system
XY, to o, and point A moves to A'. For the sake of simplicity,
the motion of the web is considered in the x-direction only.
At any time, the following relationship holds:
Z + u = r + x
56
(Al.l)
where Z is the material coordinate of point A, r is the
distance traveled by the origin of the moving system, u is the
displacement of point A, and x is the coordinate of point A'.
Taking the time derivative of equation (Al.l) gives
dZ , i5u_ i5r , i5x /a-i oa
dt+
It~
St+
St{AL.^)
where the first term ^ is zero. Hence, equation (A1.2) can be
rewritten as
& =
K~ & <A1-3>-
Equation (A1.3) gives the velocity of a material point.
Considering the left hand side of equation (2.13) ,
= <&> <A1-4>
Under the assumption of small deformation, substitution of
equation (Al . 3) in to equation (A1.4) gives
8W_ __8_
(8u_
Sr\
<5x
~
<5x ^-St 6tJ
jL fiu\ _ A. (f>'r\~
<5x^<5t;
-5xKSt}
- A. ff>n\-
-5x^St}
- A. (>u\~
Stkc5x;
The term ( ) is zero because r is time dependent only.
<5xy8ty
57
A2 SOME DETAILS OF THE FINITE ELEMENT IMPLEMENTATION
A2-1 The Hermite interpolation function and the element
matrices
Hermite cubic interpolation functions are used to
interpolate the spatial approximation, and they take the form
as :
^ = 1 -
3(g)2
+2(g)3
V>2 = x ~ T" +
,2 3
(A2.1)
(A2.2)
V>3 =
V-4 =
3(g)2 2(g)3
h+
h2
(A2.3)
(A2.4)
here h is the length of each element
Substitution of equation (A2 . 1) - (A2 . 4) and their
derivatives into equation (3 . 8) - (3 . 13) ,all element matrices
are defined as the following:
r\M~\ e mh
[M] -
420
156 22h 54 -13h
22h4h2
13h
54 13h 156 -22h
13h-3h2
-22h
4h'
(A2.5)
58
[C]e= 2c
-.5 .lh .5 -.lh
-.lh 0 .lh
-.5 -.lh .5 .lh
hi"60
lh ^i-.lh 0
60
(A2.6)
[KoJ'= h
12 6h -12 6h
6h4h2
-6h
2h2
12 -6h 12 -6h
6h2h2
-6h
4h2
(A2.7)
[KJ6=
30h
36 3h -36 3h
3h4h2
-3h
-36 -3h 36 -3h
3h -3h
4h*-
(A2.8)
[K2]e=
30h
-36 -33h 36 -3h
-3h 3hh'
36 3h -36 33h
-3h
h2
3h
(A2.9)
59
m hfj2
h
6
(A2.10)
h
'6
wherefe
in equation (A2.10) is the distributive, external
force along an element of the web. In this study, the only
external force, that is taken into account for one case, is the
gravitational force of the web. As a result, the external force
acting on an element is
fe= Ml (A2.ll)
where p is the density of web, A is the cross-sectional area of
the web, and g is the acceleration due to gravity.
With equation (A2.ll), the external element force matrix
is expressed as
MMgh
h
6
(A2.12)
h
'6
60
A2-2 The end conditions and its approximated function
The only boundary conditions that needed to be evaluated
are those at the first and the last nodes, because those end
conditions between elements will cancel each other out when the
global matrices are being assembled.
Figure A-2 shows the approximate functions for the
solutions at both ends, and those approximate function and
their derivatives take the values of
<*1 = 1, V = o at x = 0 (A2.13)
4>2 = 0,<p2'
= i at x = 0 (A2.14)
K-i = i-K-i'
= o at x = 1 (A2.15)
4>n = 0,4>n'
= 1 at x = 1 (A2.16)
F.rst Nod*
F igure A-2 Approximate function of first and last nodes.
Applying the approximate function equat ions (A2 . 13) - (A2 . 16)
61
to equation (3.14)
b,- = [M<6,.'
- Q <6,-] (3.14)
for the first and the last elements at the first and the last
nodes gives
bi -
(0)
M(0)
br-l =
b =
1(0
M(')
The shear forces Q and the moments M can be found by
usingequations (2 . 21) - (2 . 24) .
62
APPENDIX B
Bl . Calculations of natural frequencies of the roller-supports
and the lateral stiffness of the web.
B2 Analytical solution for the natural frequencies of web
B3 Calculation of r with w = 20, 40, 60 rad/sec.
63
Analytical calculations of the natural frequencies of the
supports, the lateral stiffness of the web, the natural
frequencies of the web, and the value of r with different
angular velocities are carried out in this appendix.
Bl . Calculations of natural frequencies of the roller-supports
and the flexural stiffness of the web.
The system is modeled as shown in Figure Bl .
VV^
7777T777 777777777
Figure Bl Model for calculation of n*tuial frequmeki of
roller-support*.
The energymethod [15] is used for the formulation of the
equat ions .
M 0
0 M
yi
y'2
+
K + kf -kf
-kfK + k; Y2
= 0 (B.l)
or
64
1 0
0 1
yi
y2
+
r + s -s
-s r + s
yi
y2(B.2)
[K] e<j
r + s -s
-s r + s
(B.3)
where = I7 and s = -d-
M M (B.4)
Finding the value of kf (the flexural stiffness of the web)
E (the modulus of elasticity of the web) is 4E9 Pa,
I (the area moment of inertial of the web)
l =(1.35)
q.25E-4)
= 2.1973E_13m4
k, =3EI
=3(4E9)(2.1973E-13)
= 7.50g2E_4 N/r5 l3 1.523
The [A] matrix is
[A] = [M]-1^] = [K]e,
r + s -s
-s r + s
(B.5)
the eigenvalue A of [A] will give the natural frequencies
of the system. The natural frequency of the system is
Wn = ^A (B.6)
+ s- A - s
+ s - A
= 0
or (r + s -
A)2-
s2
= 0
(B.7)
(B.8)
Solve equation (B.7) for A
65
Ax = r
A, = r + 2s
so
Wi = -jr
or nfx =
^ = 943.38rad
sec
150 Hz
where nf means natural frequency.
w9 \v + 2s = ^889963.724 + 2(9.0788E-8)
* 943.38 rad/sec
nf2 = 150 Hz
B2 Analytical solution for the natural frequencies of the web
Consider the problem described in Figure B2 with tension p
in the beam, the governing differential equation of the beam is
EI<S4w
5x4 K<5x2"
H~8t2PH + Pk-52w
_ 0
HTTTmrm
A, B, I, P
(B.9)
muwmn
F igu re B2 Model for calculation of natural frequencies of
the web.
66
Equation (B.9) is the same as equation (2.15) when V is
zero .
Because of the significant difference between the
stiffness of the end-supports and the flexural stiffness of the
web (K = 7.36E9 N/m, k; = 7.5082E-4 N/m), the system of Figure
B2 can be model as the system in Figure B3 .
1gu re 3 Model for calcuIation of natura, frequencies Qf
the web under the assumption of rigid supports .
The simple support boundary conditions for the system in Figure
B3 are
1) at the left hand side:
w(o,t) = 0, and Elf^(o.t) = 0.^
ox
67
2) at the right hand side:
w(*,o = 0, and EI^(/,t) = 0,8^
Assuming the solution of equation (B.9) to be
w(i,t) = W(r)(A1sinwt + Bjcoswt) (B.10)
where W(x) is a function of space only, Al and Bx are constants
which can be found from the initial conditions, and the w is
the natural frequencies of the web in rad/sec. Substituting
equation (B.10) and its time derivative into equation (B.9)
gives
PTd4W nd2W
dx dxpAw2W = 0 (B.ll)
By assuming the solution of W to be
W(s)= ^ (B.12)
the auxiliary equation of equation (B.ll) can be obtained:
r -
-z-t
p_e2pAw2
_ 0EI
(B.13)
Solving equation (B.13) for gives
*ai,*aa +2EI
- \ 4E2I2+
pAw
~ET (B.14)
Equation (B.12) can be rewritten as (with absolute value of f2)
W, x = C,cos2x + C2sinf2x + C3cosh^x + C4sinh^x (B.15)\X)
68
where C^ , C2 , C3 and C4 are to be determined by the boundary
condit ions .
Applying the left hand side boundary conditions to
equation (B.15), constant C1 = C3 = 0. Hence equation (B.15)
becomes
wO)= C2sin2x + C4sinh^x (B.16)
Applying the right hand side boundary conditions to
equation (B.16) gives
and
C2sin2l + C4sinh^l = 0
- C222sin2l + C^i'sinh^l = 0
(B.17)
(B.18)
Equations (B.17) and (B.18) are a set of simultaneous
equations for unknowns C2 and C4 ,and they can be put into the
matrix form as
sin2l
-22sin2l
sinh^l
^sinh^l
C, 0
0
x and 2 can be found by setting the determinant of the
coefficient matrix equal to zero.
(sinhf1l)(sint2l) = 0 (B.19)
Since sinh^l > 0 for all values of c^l ^ 0, the only roots for
equation (B.19) are
<t2l = mr n = 0, 1, 2, (B.20)
69
Thus the natural frequencies of the system in Figure B3 (using
equations (B.14) and (B.20) after algebraic manipulation) are
Uln
12VAav" +
_2r,TI
2 i 2n pi
ir2EI
(B.21)
The natural frequencies for the first few modes are found
when p = 1000kg/m3
, 1 = 1 . 52 m, E = 4E9 Pa, I = 2.1273E-13
m4
(calculation is shown previously in this Appendix), p = 200 N,
and A = 1.6875E-4m2
, and listed on Table Bl .
Table Bl
mode
n
Frequenc ies
(Hz)
0 0
1 11.32
2 22.65
3 34.0
4 45.3
5 56.64
6 68.0
7 79.31
8 90.65
9 102.0
10 113.35
11 124.71
12 136.08
13 147.45
70
B3 Calculation of r with w = 20, 40, 60 rad/sec .
The relationship of ulf w2 and p is described in equation
(2.25), (2.26), and (2.30) in Chapter 2. If the values of ws
are changed, the value of p would also be changed accordingly.
Assuming ul and w2 are the same, equation (2.30) is zero.
Inputting 20, 40, and 60 rad/sec into equation (2.25) for
w2 ,the values of p are 160, 200, and 240 N, respectively.
Converting the values of angular velocities of 20, 40, and
60 rad/sec into linear velocities, the linear velocities are
5, 10, and 15 m/sec, respectively.
Using equation (2.24)
r =pAV2
-
p (2.24)
for V = 5 m/sec (w = 20 rad/sec) ,and p = 160
r =103
(1.6875E-4)(52) - 160 = -155 N;
for V = 5 m/sec (u = 20 rad/sec) ,and p = 160
t =103
(1.6875E-4)(102) - 160 = -183 N;
for V = 5 m/sec (w = 20 rad/sec) ,and p = 160
r =103
(1.6875E-4)(152)- 240 = -202 N.
71
APPENDIX C
All programs, which are used for this work, are listed in
this appendix.
CI PROGRAM WEB1
C2 PROGRAM AXIAL
C3 PROGRAM CONVER
72
PROGRAM WEB1
73
00001
00002
00003
00004
00005
00006
00007
00008
00009
00010
00011
00012
00013
00014
00015
00016
00017
00018
00019
00020
00021
00022
00023
00024
00025
00026
00027
00028
00029
00030
00031
00032
00033
00034
00035
00036
00037
00038
00039
00040
00041
00042
00043
00044
00045
00046
00047
00048
00049
00050
00051
00052
00053
00054
00055
00056
00057
* *
*
*
*
*
*
*
* *
C
c*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
**********************************
*
FINITE ELEMENT PROGRAM *
**********************************
*
MOVING WEB *
WITH VIBRATING BOUNDARIES *
*
**********************************
THIS PRRGRAM IS USED TO CALCULATE THE ELEMENT STIFFNESS MATRIX,DAMPING MATRIX, AND MASS MATRIX OF THE WEB, TO CALCULATE THE
GLOBAL STIFFNESS MATRIX, DAMPING MATRIX, AND MASS MATRIX OF
THE WEB BASED ON THE ELEMEMT MATRICES, AND IT IS ALSO USED TO
TRANSFORM A SYSTEM OF DIFFERENTIAL EQUESTIONS IN A FORM OF:
[GM]*YDD + [GC]*YD + [GK] *Y BD
INTO A SYSTEM OF LINEAR EQUATIONS IN A FORM OF
[A]*Y + [B]*Y1 + [D]*Y0 - R
SUB-PROGRAM DESCRIPTION:
ELEMENT . . . PROVIDES ELEMENT MATRICES
GLOBFX. . . .ASSEMBLES FIXED GLOBAL MATRICES
GLOBVR. . . .ASSEMBLES VARIABLE GOLBAL MATRICES
NUMARK. . . .CLACULATES THENEWMARK'
S COEFFICIENT [A], [B] , [D]MULT PREPARES ARRAYS FOR ITERATION.
SEIDEL OR GUAS FINDS THE APPROXIMATION.
OUTPUT. . . .WRITES DATA FOR PLOTS.
VARIABLES DESCRIPTION:
HT THE THICKNESS OF THE WEB
WD THE WIDTH OF THE WEB
LNGTH . . . THE LENGTH OF THE WEB
RO THE DENSITY OF THE WEB
EE MODULUS OF ELATICITY OF THE WEB
DY INITIAL DISPLACEMENT
G GRAVITY
DT TIME INCREMENT
BETA. . . .GALERKIN'SCOEFFICIENT
GAMMA. . .GALERKIN'SCOEFFICIENT
IX MASS MOMENT OF INERTIA
H LENGTH OF ELEMENT
SLOPE ... SLOPE OF THE LINEAR INITIAL DISPLACEMENT
VI AXIAL VELOCITY OF WEB
Y(I) . . ..Y-DIRECTION
DISPLACEMENT ARRAY
M ELEMENT MASS MATRIX
C ELEMENT DAMPING MATRIX
K1,K2,K3. . .ELEMENTSTIFFNESS MATRICES
EF ELEMENT FORCE MATRIX
74
00058
00059
00060
00061
00062
00063
00064
00065
00066
00067
00068
00069
00070
00071
00072
00073
00074
00075
00076
00077
00078
00079
00080
00081
00082
00083
00084
00085
00086
00087
00088
00089
00090
00091
00092
00093
00094
00095
00096
00097
00098
00099
00100
00101
00102
00103
00104
00105
00106
00107
00108
00109
00110
00111
00112
00113
00114
*
*
*
*
*
*
*
*
*
*
*
GM GLOBAL MASS MATRIX
GC GLOBAL DAMPING MATRIX
GK1 , GK .. GLOBAL STIFFNESS MATRICES
GF GLOBAL FORCE MATRIX
VI,V2. . .LINEAR VELOCITIES OF ROLLER 1 & 2
V1D,V2D. .LINEAR ACCELERATION OF ROLLER 1 & 2
Vin CONSTANT INPUT VELOCITY
P TENSION OF WEB (CONSTANT OR VARIABLE)PD TIME RATE OF TENSION IN WEB
P0 PRE-TENSION (FOR FREE VIBRATION ANALYSIS ONLY)
PARAMETER (NB-6, N-4, NN-168)
INTEGER CONTROL
REALM A(NN,NB) ,B(NN,NB) ,D(NN,NB)REALM GM(NN,NB) ,GC(NN,NB) ,GK(NN,NB) ,GK1(NN,NB) , GF (NN)
REALM M(N,N) ,C(N,N) ,K1(N,N) ,K2(N,N) ,K3(N,N) , EF (N)
REALM R (NN) , R0 (NN) , Rl (NN)REALM Y(NN) , Y0 (NN) , YI (NN) ,V0(NN) ,SIG(NN/2)
REALM ZER(NN) , LNGTH, IX,P0,Vin
DIMENSION IBD(4) , FBD (NN) ,FF(3,4) ,EN(3,NN)*
COMMON/MATDIM/RO,EE,AREA, IX,HT,P, G,P0,Vin
COMMON/NEWMRK/BETA, GAMMA
COMMON/DELTAS /DT, H
COMMON/SEIDEL/ERRCRT,MAXITR
COMMON/CONTR/NFRQ, IWRT, CONTROL* TITLE
WRITE (6, 600)600 FORMAT (///10X, 'VIBRATION OF A MOVING WEB',
+ /10X,-
--'
)* INITIALIZATION
DO 10 I-1,NN
R(I)-0.0
R0(I)-0.0
Rl ( I ) -0 . 0
ZER(I)-0.0
DO 10 J-1,NB
A(I, J) -0.0
B(I, J) -0.0
D(I, J) -0.0
GM(I, J) -0.0
GC(I,J)-0.0
GK(I, J) -0.0
10 END DO
* BASIC DATA
READ ( 5 ,* ) CONTROL
READ(5,*) P0, Vin
READ (5,*) HT,WD, LNGTH,RO,EE,G
READ (5,*) DT, BETA, GAMMA
READ (5,*) ERRCRT,MAXITR
READ (5,*) TERM, KODBND,KODLIN, KODINT
READ (5,*) NFRQ,IWRT
*
AREA-HT*WD
IX-WD*HT**3/12.0
NE-NN/2-1
75
00115 H =LNGTH/NE
00116 N4=NN-4
00117 T = DT
00118 NBOUND=4
00119 KOUNT=2
00120 IFRQ-1
00121 STOP=-1.0
00122 ZERO=0.0
00123 *NEWMARK'S METHOD FORMULATION ON P584 OF THE FEM
00124 *BY O. C. ZIENKIEWICZ.
00125 DT2=DT*DT
00126 A2=0.5-2.0*BETA+GAMMA00 127 A1=0 . 5+BETA-GAMMA
00128 *. . . .WRITE DATA
00129 WRITE(6,610)00130 610 FORMAT (/5X, 'INPUT DATA',/ 5X,
'
)00131 WRITE'6,620) RO,EE
00132 620 FORMAT //5X, 'MATERIAL: ',00133 1 /5X, 'DENSITY =',1PE12. 3,00134 2 /5X, 'MODULUS =',1PE12. 3)00135 WRITE(6,630) HT, WD, LNGTH, IX, AREA00136 630 FORMAT (/5X, 'GEOMETRY: ',00137 1 /5X, 'HEIGHT =',1PE12.3,
138 2 /5X, 'WIDTH =',1PE12.3,
00139 3 /5X, 'LENGTH =', 1PE12 . 3,00140 4 /5X, 'IXX =',1PE12.3,
00141 5 /5X, 'AREA =',1PE12.3)
00142 WRITE (6, 640) GAMMA, BETA, ERRCRT,MAXITR
00143 640 FORMAT (/5X, 'GUASS-SEIDEL: ',00144 1 /5X, 'GAMMA =',1PE12.3,
00145 1 /5X, 'BETA =',1PE12.3,
00146 1 / 5X, 'ERRCRT =',1PE12.3,
00147 2 /5X, 'MAXITR =', 15)
00148 WRITE(6,650) DT,TERM,H,NE
00149 650 FORMAT(/5X, 'OTHER INFORMATIONS: ',
00150 1 /5X, 'TIME INCREMENT =', 1PE12.3,
00151 1 /5X, 'TERMINATION T =', 1PE12.3,
00152 2 /5X, 'ELEMENT LENGTH =', 1PE12.3,
00153 3 //5X, 'NUMMER OF ELEMENTS =',I5)
00154 IF(IWRT.EQ.O) WRITE(6,654) NFRQ
00155 IF(IWRT.EQ.l) WRITE(6,656)
00156 654 FORMAT (//5X, 'FREQUENCY OF PRINT OUT IS ',14)
00157 656 FORMAT (//5X, 'JUST WRITE EVERY THING !')
00158 IF(KODBND.EQ.O) WRITE(6,660)
00159 IF(KODBND.EQ.l) WRITE(6,670)
00160 660 FORMAT (//5X, 'PRESCRIBED BOUNDARIES')
00161 670 FORMAT (//5X, 'VIBRATORY BOUNDARIES')
00162 IF(KODLIN.EQ.O) WRITE(6,674)
00163 IF(KODLIN.EQ.l) WRITE(6,676)
00164 674 FORMAT (//5X, "LINEAR PROBLEM')
00165 676 FORMAT (//5X, 'NONLINEAR PROBLEM')
00166 * IBD=BOUNDARY CODE (0 FIXED, 1 FREE)
00167 READ(5,*) (IBD(I) , I-l,NBOUND)
00168 WRITE(6,710) (IBD (I) , I=l,NBOUND)
00169 710 FORMAT (//2X, 'BOUNDARY CONDITIONS: ',/5X, "IBD (I) : '.413)
00170 READ(5,*) EQM1, EQC1, EQK1,EQMN,EQCN, EQKN
00171 WRITE(6,730) EQM1, EQC1, EQK1, EQMN, EQCN, EQKN
76
00172 730 FORMAT(//2X, 'EQUIVALENT MASS, DAMPING, STIFNESS: ',
00173 1 /5X, 'AT FIRST (LHS) END :'
, 1P3E12 . 3,00174 2 /5X, 'AT LAST (RHS) END:
'
,1P3E12 . 3)
00175 IF(KODINT.EQ.O) WRITE(6,678)
00176 IF(KODINT.EQ.l) WRITE(6,679)
00177 678 FORMAT (//5X, 'INITIAL DISPLACEMET AND VELOCITIES')
00178 679 FORMAT (//5X, 'INITIAL DISPLACEMENT AT THE FIRST TWO STEPS')
00179 * INITIAL CONDITIONS
00180 WRITE(6,680)
00181 680 FORMAT (///5X, 'INITIAL CONDITIONS',
00182 + /5X,'
')
00183 WRITE(6,690)
00184 690 FORMAT </5X, 'DISPLACEMENTS: ')
00185 READ(5,*) (Y0 (I) , 1=1, NN)
00186 WRITE(6,*) (Y0 (I) , 1=1, NN)
00187 IF(KODINT.EQ.O) THEN
00188 WRITE(6,700)
00189 700 FORMAT </5X, 'VELOCITIES: ')
00190 READ(5,*) (V0 (I) , 1=1, NN)
00191 WRITE<6,*) (V0 (I) ,I=1,NN)
00192 DO 20 1=1, NN
00193 20 Y1(I)=DT*V0 (I)+Y0 (I)
00194 END IF
00195 DO 25 1=1, NN
00196 Y(I)=Y1(I)
00197 25 ENDDO
00198 * WRITE THE FIRST TWO STEPS TO OUTPUT FILE
00199 WRITE (12,*) ZER0,Y1(1)
00200 WRITE (12,*) DT,Y1(1)
00201 WRITE(33,*) ZER0,Y1(33)
00202 WRITE(33,*) DT,Y1(33)
00203 WRITE (41,*) ZERO, YI (41)
00204 WRITE (41,*) DT,Y1(41)
00205 * FIXED MATRICES, [GM] AND [GK1]
00206 CALL ELEMENT(N,H,M,C,K1,K2,K3,EF)
00207 CALL GLOBFX (N, NE, NN, NB,M, Kl, EF, GM, GK1, GF)
00208 *. ...INSERT B.C. (KOD=l)
0020 9 * INITIAL FORCED B. C.
00210 L=NN-1
00211 K=NB-1
00212 IF (CONTROL . EQ . 3 .OR. CONTROL . EQ . 4) THEN
00213 READ(8,*) ( (FF(I, J), J=l, 4) , 1=1, 2)
00214 ELSE
00215 DO 1=1,2,1
00216 DO J=l, 4, 1
00217 FF(I,J)= 0.0
00218 ENDDO
00219 ENDDO
00220 ENDIF
00221 DO 101 1=1,2
00222 EN(I,1)=FF(I,1)+GF(1)
00223 EN(I,2)=FF(I,2)+GF(2)
00224 EN(I,L)=FF(I,3)+GF(L)
00225 EN(I,NN)=FF(I,4)+GF(NN)
00226 DO 102 J=3,NN-2
00227 EN(I,J)=GF(J)
00228 102 ENDDO
77
(RO(I) , 1=1, NN)
(Rl(I) ,I=1,NN)
(R(D , 1=1, NN)
00229 101 ENDDO
00230 * BOUNDARY CONDITIONS
00231 IF(KODBND.EQ.l) THEN
00232 GM(1,1)=GM(1,1)+EQM1
00233 GM(L,K)=GM(L,K)+EQMN
00234 END IF
00235 * TIME MARCHING
00236 DO 100 WHILE (T.LE.TERM)
00237 T=T+DT
00238 KOUNT=KOUNT+l
00239 IFRQ=IFRQ+1
00240 CALL GLOBVR (N, NE, NN, NB, LNGTH, C, K2, K3, GK1, GC, GK)
00241 * INSERT B.C. (KOD=l)
00242 IF (KODBND.EQ. 1) THEN
00243 GC(1,1)=GC(1,1)+EQC1
00244 GK(1, 1)=GK(1, D+EQK1
00245 GC(L,K)=GC(L,K)+EQCN
00246 GK(L,K)=GK(L,K) +EQKN
00247 END IF
00248 CALL NUMARK (N, NB,NN, GM, GC, GK, A, B, D)
00249 CALL MULT (NN, NB,NE,D, Y0, R0)
00250 CALL MULT(NN,NB,NE,B,Y1,R1)
00251 DO 30 1=1, NN
00252 30 R(I)=-(R1(I)+R0(I) )
00253 IF(IWRT.EQ.l) THEN
00254 WRITE(6,930)
00255 WRITE'6,940)
00256 WRITE(6,940)
00257 WRITE(6,940)
00258 END IF
00259 * INSERT B.C. (KOD=0)
00260 * FBD=FORCED BOUDARIES (Q0, -M0, -QN, MN)
00261 IF (CONTROL . EQ . 3 .OR. CONTROL .EQ. 4) THEN
00262 READ(8,*) (FF (3, I) , 1=1, NBOUND)
00263 ELSE
00264 DO 1=1, NBOUND, 1
00265 FF(3,I)= 0.0
00266 ENDDO
00267 ENDIF
00268 EN(3,1)=FF(3,1)+GF(1)
00269 EN(3,2)=FF(3,2)+GF(2)
00270 EN(3,L)=FF(3,3)+GF(L)
00271 EN(3,NN)=FF(3, 4)+GF(NN)
00272 DO 103 J=3,NN-2
00273 EN(3, J)=GF(J)
00274 103 ENDDO
00275 DO 35 1=1, 4
00276 35FBD(I)=-DT2*(BETA*EN(3,I)+A2*EN(2,I)+A1*EN(1,I) )
00277 DO 104 1=1, NN
00278 R(I)=R(D+FBD(I)
00279 104 ENDDO
00280 *
00281 R(L)=R(L)+FBD(3)
00282 R(NN)=R(NN)+FBD(4)
00283 N4=NN-N
00284 DO 60 1=1, NBOUND
00285 IF(IBDd) .EQ.0) THEN
78
00286 IF(I.LE.2) THEN
00287 DO 40 J=1,N
00288 A(I,J)=0.0
00289 A(J,I)=0.0
00290 40 END DO
00291 A(I,I)=1.0
00292 R(I)=0.0
00293 ELSE
00294 I2=N4+I
00295 IP2=I+2
00296 DO 50 J=1,N
00297 J2=N4+J
00298 JP2=J+2
00299 A(I2, JP2)=0.0
00300 A(J2,IP2)=0.0
00301 50 END DO
00302 A(I2, IP2)=1.0
00303 R(I2)=0.0
00304 END IF
00305 END IF
00306 60 END DO
00307 IF (IWRT.EQ. 1) THEN
00308 WRITE(6, 910)
00309 WRITE (6, 940) ( (A (I, J) , J=1,NB) ,I=1,NN)
00310 WRITE(6, 920)
00311 WRITE(6,940) (R(I) , 1=1, NN)
00312 910 FORMAT (/5X,'
[A]'
)
00313 920 FORMAT (/5X,'
R')
00314 930 FORMAT(/5X, 'R0,R1,R')
00315 940 FORMAT (5X, 1P6E12.3)
00316 END IF
00317 IF(KODLIN.EQ.O) CALL GUAS (NN, NB, NE, A, R, Y)
00318 IF (KODLIN.EQ. 1) CALL SEIDEL (NN, NB, NE, A, R, Y, Y0)
00319 IF (IFRQ.EQ.NFRQ.OR.NFRQ.EQ.l) THEN
00320 IFRQ=0
00321 CALL OUTPUT (NN, NE, T, Y, SIG)
00322 END IF
00323 DO 70 1=1, NN
00324 YO (I)=Y1(I)
00325 Y1(I)=Y(I)
00326 70 END DO
00327 DO 8 0 J=l, NBOUND
00328 EN(1, J)=EN(2, J)
00329 EN (2, J)=EN(3, J)
00330 80 END DO
00331 100 END DO
00332 WRITE(9,*) STOP, (ZER(I) ,1=1, NN)
00333 WRITE (10,*) STOP, (ZER(I) ,1=1, NN)
00334 STOP
00335 END
79
00001 *
00002 SUBROUTINE ELEMENT (N, H,M, C,K1, K2, K3, EF)
00003 *
00004 ******************************** ***
00005 *
00006 * THIS SUBROUTINE IS USED TO CALCULATE THE ELEMENT STIFFNESS
00007 *MATRIX, THE ELEMENT DAMPING MATRIX, AND THE ELEMENT MASS MATRIX.
00008
00009 ******************************* ****
00010 *
00011 REAL*4 M(N,N) ,C(N,N) ,K1(N,N) ,K2 (N,N) ,K3(N,N) ,EF(N)
00012 COMMON/MATDIM/RO,EE,AREA, IX, HT, P, G, P0,Vin
00013 COMMON/CONTR/NFRQ, IWRT, CONTROL
00014 H2=H*H
00015 H3=H*H2
00016 *MASS MATRIX [M]
00017 RA=RO*AREA
00018 M(l,l)=156.0
00019 M(1,2)=22.0*H
00020 M(l,3)=54.0
00021 M(1,4)=-13.0*H
00022 M(2,2)=4.0*H2
00023 M(2,3)=-M(l,4)
00024 M(2, 4)=-3.0*H2
00025 M(3,3)=M(1,1)
00026 M(3, 4)=-M(l,2)
00027 M(4,4)=M(2,2)
00028 DUM=H/420.0
00029 DO 10 1=1, N
00030 DO 10 J=1,I
00031 M(J,I)=DUM*M(J,I)
00032 M(I,J)=M(J,I)
00033 10 END DO
00034 * DAMPING MATRIX [C]
00035 C(l,l)=-0.5
00036 C(1,2)=0.1*H
00037 C(l,3)=0.5
00038 C(l,4)=-C(l,2)
00039 C(2,2)=0.0
00040 C(2,3)=C(1,2)
00041 C(2,4)=-H2/60.0
00042 C(3,3)=0.5
00043 C(3,4)=C(1,2)
00044 C(4,4)=0.0
00045 DO 20 1=2, N
00046 DO 20 J=1,I-1
00047 20 C(I,J)=-C(J,D
00048 * STIFNESS MATRICES [Ki] i=l,2,3.
00049 Kl(l,l)=12.0
00050 K1(1,2)=6.0*H
00051 Kl(l,3)=-12.0
00052 K1(1,4)=K1(1,2)
00053 K1(2,2)=4.0*H2
00054 K1(2,3)=-K1(1,2)
00055 K1(2,4)=2.0*H2
00056 K1(3,3)=K1(1,1)
00057 K1(3,4)=K1(2,3)
80
00058 K1(4,4)-K1(2,2)00059 *
00060 K3(l,l)6.0/(5.0*H)
00061 K3(l,2)=-l.l00062 K3(l,3)=-K3(l,l)00063 K3(l,4)0.100064 K3(2,2)=-2.0*H/15.000065 K3(2,3)=0.100066 K3(2,4)=H/30.000067 K3(3,3)=K3(1,1)00068 K3(3,4)=l.l00069 K3(4,4)=K3(2,2)00070 *
00071 DO 30 I-1,N
00072 DO 30 J-1,100073 K1(J,I)=K1(J, I)/H300074 K1(I, J)-K1(J, I)00075 K2(J, I)-C(J,I)00076 K2(I,J)-C(I, J)00077 K3(I, J)-K3(J,I)00078 30 END DO
00079 K3(2,l)=-0.1
00080 K3(4,3)=0.1
00081 * ELEMENT FORCE MATRIX
00082 Fa=RA*G*H/2
00083 EF(l)=Fa
00084 EF(2)=Fa*H/6
00085 EF(3)=EF(1)00086 EF(4)EF(2)00087 * WRITE [M],[C],[Ki]00088 IF(IWRT.EQ.l) THEN
00089 WRITE (6, 610)00090 WRITE(6,640) ( (M (I, J) , J-l, 4) , 1=1, 4)00091 WRITE (6, 620)00092 WRITE(6,640) ( (C (I, J) , J-l, 4) , 1=1, 4)00093 WRITE (6, 630)00094 WRITE(6,640) ( (Kl (I, J) , J-l, 4) , 1-1, 4)00095 WRITE (6, 640) ( (K2 (I, J) , J=l, 4) , 1-1, 4)00096 WRITE(6,640) ( (K3 (I, J) , J-l, 4) , 1-1, 4)00097 610 FORMAT <//5X, 'ELEMENT MASS MATRIX [M] :
'
)00098 620 FORMAT (//5X, 'ELEMENT DAMP MATRIX [C] :
'
)00099 630 FORMAT (//5X, 'ELEMENT STIF MATRIX [K] :
'
)00100 640 FORMAT (2X,1P4E13. 4)
00101 END IF
00102 RETURN
00103 END
81
00001 *
00002 SUBROUTINE GLOBFX (N, NE, NN, NB,M, Kl, EF, GM, GK1, GF)
00003 *
00004 ***********************************
00005 *
00006 *THIS SUBROUTINE IS USED TO ASSEMBLE THE GLOBAL MATRICES BASED
00007 *ON THE ELEMENT MATRICES FROM THE PREVIOUS SUBROUTINE.
00008 *
00009***********************************
00010 *
00011 COMMON/CONTR/NFRQ, IWRT
00012 REAL*4 M(N,N) ,K1(N,N) , IX,EF(N)
00013 REAL*4 GM (NN, NB) ,GK1 (NN, NB) , GF (NN)
00014 DIMENSION Bl (2, 6) ,B2 (2,6)
00015 COMMON/MATDIM/RO,EE,AREA, IX, HT, P,G, P0,Vin
00016 * INITIALIZATIO
00017 DO 5 1=1,2
00018 DO 5 J=l,6
00019 B1(I,J)=0.0
00020 B2(I,J)=0.0
00021 5 END DO
00022 * PRELIMINARIES
00023 RA=RO*AREA
00024 FX=EE*IX
00025 DO 10 1=1, N
00026 DO 10 J=1,N
00027 M(I, J)=RA*M(I, J)
00028 Kl (I, J)=FX*K1 (I, J)
00029 10 END DO
00030 * FIRST AND LAST PAIRS
00031 DO 20 1=1,2
00032 K=NN-2+I
00033 IP2=I+2
00034 DO 20 J=1,N
00035 JP2=J+2
00036 GM(I,J)=M(I,J)
00037 GK1(I,J)=K1(I, J)
00038 GM(K,JP2)=M(IP2,J)
00039 GK1(K, JP2)=K1(IP2, J)
00040 20 END DO
00041 * BASIC UNIT
00042 DO 30 1=1,2
00043 IP2=I+2
00044 DO 30 J=1,NB
00045 JM2=J-2
00046 IF(J.LE.4) THEN
00047 B1(I,J)=M(IP2,J)
00048 B2(I,J)=K1(IP2,J)
00049 END IF
00050 IF(J.GT.2) THEN
00051B1(I,J)=B1(I,J)+M(I,JM2)
00052B2(I,J)=B2(I,J)+K1(I,JM2)
00053 END IF
00054 30 END D
00055 * REPEAT BASIC UNIT
00056 NEM1=NE-1
00057 DO 40 I=1,NEM1
82
00058 DO 40 K=l,2
00059 L=2*I+K
00060 DO 40 J=1,NB
00061 GM(L, J)=B1(K, J)00062 GK1(L, J)=B2(K, J)
00063 40 END DO
00064 * GLOBAL FOR MATRIX
00065 GF(1)=EF(1)
00066 GF(2)=EF(2)00067 GF(NN-1)=EF(3)00068 GF(NN) = EF(4)
00069 DO 50 I=3,NN-3,2
00070 GF(I) = EF(3)+EF(1)
00071 GF(I+1) = EF(4)+EF(2)
00072 50 ENDDO
00073 *
00074 *PRINT OUT
00075 IF(IWRT.EQ.l) THEN
00076 WRITE(6, 610)
00077 WRITE (6, 630) ( (GM(I, J) , J=1,NB) , 1=1, NN)
00078 WRITE(6, 620)
00079 WRITE (6, 630) ( (GK1 (I, J) , J=1,NB) , 1=1, NN)
00080 610 FORMAT (//5X, 'GLOBAL MASS MATRIX [GM]'
)
00081 620 FORMAT (//5X, 'GLOBAL STIF MATRIX [GK1]')
00082 630 FORMAT (2X, 1P6E12 .3)
00083 END IF
00084 RETURN
00085 END
83
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SUBROUTINE GLOBVR(N,NE,NN,NB, X, C,K2,K3, GK1, GC, GK)* *
* *
**********************************
THIS PROGRAM IS USED TO ASSEMBLE THE VARIABLE GLOBAL
MATRICES BASED ON THE ELEMENT MATRICES FROM SUBROUTINE
'ELEMENT'.
**********************************
* *
* *
REALM C(N,N) ,K2(N,N) ,K3(N,N)REALM GK1 (NN,NB) ,GC(NN,NB) ,GK(NN,NB)
REALM LMD(2000) ,V(2000) ,VD(2000) ,Q(2000) ,TAU(2000)
COMMON/MATDIM/RO, EE,AREA, IX, HT, P , G, P0 ,Vin
COMMON/DELTAS /DT , H
COMMON/CONTR/NFRQ, IWRT, CONTROL* INITIALIZATION
DO 5 1=1, NN
DO 5 J=1,NB
GC(I, J) =0.0
GK(I, J) =0.0
5 END DO
* PREREQUISITES
IF (CONTROL .EQ. 4) THEN
READ(7,*) V1,V2,V1D,V2D,P,PD
SLOPV=(V2-Vl) /X
SLOPD- (V2D-V1D) /X
DV-H*SLOPV
DD=H*SLOPD
V(l)-Vl+DV/2.0
VD(l)-VlD+DD/2.0
DO 10 1-2, NE
V(I)-V(I-1)+DV
VD(I)=VD(I-1)+DD
10 END DO
ELSE
DO I-1,NE
V(I)=Vin
ENDDO
P - P0
EPSD =0.0
ENDIF
* COMPUTE TAU,Q,LMD PER ELEMENT
RA=RO*AREA
FX=EE*IX
EPSD-PD/ (EE*AREA)
DO 20 1=1, NE
Q(I)-RA*V(I)
QDI-RA*VD(I)
TAU(I)=Q(D*V(D-P
LMD (I) -QDI+Q (I) *EPSD
20 END DO
*. ..FIRST
PAIRS OF ROWS
DO 30 1-1,2
84
00058 DO 30 J=1,N
00059 GC(I,J)=2.0*Q(1)*C(I,J)00060 GK(I,J)=LMD(1)*K2(I,J)+TAU(1)*K3(I,J)00061 30 END DO
00062 * LAST PAIRS OF RAWS
00063 DO 40 1=1,2
00064 K=NN-2+I
00065 IP2=I+2
00066 DO 40 J=1,N
000 67 JP2=J+2
000 68 GC(K,JP2)=2.0*Q(NE)*C(IP2, J)000 6 9 GK(K, JP2)=LMD(NE)*K2(IP2, J)+TAU(NE) *K3(IP2, J)00070 40 END DO
00071 *IN BETWEEN
00072 NEM1=NE-1
00073 DO 50 I=1,NEM1
00074 IP1=I+1
00075 DO 50 K=l,2
00076 L=2*I+K
00077 KP2=K+2
00078 DO 50 J=1,NB
00079 JM2=J-2
00080 IF(J.LE.4) THEN
00081 GC(L, J)=2.0*Q(I) *C(KP2, J)
000 82 GK(L, J)=LMD(I) *K2 (KP2 , J) +TAU (I) *K3 (KP2, J)
00083 END IF
00084 IF(J.GT.2) THEN
000 85 GC(L, J)=GC(L, J) +2.0*Q(IP1) *C(K, JM2 )
0008 6 GK(L, J)=GK(L, J) +LMD(IP1) *K2 (K, JM2) -t-TAU(IPl) *K3(K, JM2)
00087 END IF
00088 50 END DO
00089 DO 60 1=1, NN
00090 DO 60 J=1,NB
00091 60 GK(I, J)=GK(I, J)+GK1(I, J)
00092 * WRITE
00093 IF(IWRT.EQ.l) THEN
00094 WRITE(6,610)
00095 WRITE(6,630) ( (GC ( I, J) , J=l, NB) , 1=1, NN)
00096 WRITE(6,620)
00097 WRITE(6,630) ( (GK ( I, J) , J=l, NB) , 1=1, NN)
00098 610 FORMAT (//5X, 'GLOBAL DAMPING MATRIX [GC]')
00099 620 FORMAT (//5X, 'GLOBAL STIFNESS MATRIX [GK]'
)
00100 630 F0RMAT(2X,1P6E12.3)
00101 END IF
00102 RETURN
00103 END
85
00001 *
00002 SUBROUTINE NUMARK (N,NB,NN,GM, GC, GK, A, B, D)
00003 *
00004 **********************************
00005 *
00006 * THIS SUBROUTINE IS USED TO TRANSFORM A SYSTEM OF DIFFERENTIAL
00007 * EQUATIONS INTO A SYSTEM OF LINEAR ALGEBRA EQUATIONS.
00008 *
00009 **********************************
00010 *
00011 REAL*4 A(NN,NB) ,B(NN,NB) ,D(NN,NB)
00012 REAL*4 GM (NN, NB) , GC (NN, NB) , GK (NN, NB)
00013 COMMON/NEWMRK/BETA, GAMMA
00014 COMMON/DELTAS /DT,H
00015 COMMON/CONTR/NFRQ, IWRT, CONTROL
00016 *PREREQUISITES
00017 D2=DT*DT
00018 Dll=1.0
00019 D12=DT*GAMMA
00020 D13=D2*BETA
00021 D21=-2.0
00022 D22=DT*(1.0-2.0*GAMMA)
00023 D23=D2* (0 . 5-2 . 0*BETA+GAMMA)
00024 D31=1.0
00025 D32=DT* (-1.0+GAMMA)
00026 D33=D2* (0 . 5+BETA-GAMMA)
00027 *....COMPUTE [A] , [B] , [C]
00028 DO 10 1=1, NN
00029 DO 10 J=1,NB
00030 A (I, J)=D11*GM(I, J) +D12*GC(I, J)+D13*GK(I, J)
00031 B(I, J)=D21*GM(I, J) +D22*GC (I, J)+D23*GK(I, J)
00032 D(I, J)=D31*GM(I, J)+D32*GC(I, J) +D33*GK (I, J)
00033 10 END DO
00034 * WRITE [A],[B],[D]
00035 IF(IWRT.EQ.l) THEN
00036 WRITE(6,610)
00037 WRITE(6,640) ( (A(I, J) , J=1,NB) , 1=1, NN)
00038 WRITE(6,620)
00039 WRITE(6,640) ( (B (I, J) , J=l, NB) , 1=1, NN)
00040 WRITE(6,630)
00041 WRITE(6,640) ( (D (I, J) , J=l, NB) , 1=1, NN)
00042 610 FORMAT(/5X,'[A]')
00043 620 FORMAK/5X,'
[B]'
)
00044 630 FORMAK/5X,'
[D]'
)
00045 640 FORMAT(5X, 1P6E12.3)
00046 END IF
00047 RETURN
00048 END
86
00001 *
00002 SUBROUTINE MULT (N,M,NE, A, X, Y)
00003 REALM A(N,M) ,X(N) ,Y(N)
00004* FIRST PAIR
00005 DO 20 K=l,2
00006 SUM=0 . 0
00007 DO 10 J=1,M
00008 10 SUM=SUM+A(K, J) *X(J)
00009 Y(K)=SUM
00010 20 END DO
00011 * LAST PAIR
00012 NM2=N-2
00013 NMM=N-M
00014 DO 40 K=l,2
00015 SUM=0 . 0
00016 I=NM2+K
00017 DO 30 J=1,M
00018 30 SUM=SUM+A ( I , J) *X(NMM+J)
00019 Y(I)=SUM
00020 40 END DO
00021* IN-BETWEEN PAIRS
00022 DO 100 11=2, NE
00023 I0=2*(II-1)
00024L0=2* (II-2)
00025 DO 60 K=l,2
00026 SUM=0 . 0
00027 I=I0+K
00028 DO 50 J=1,M
00029 L=L0+J
00030 SUM=SUM+A(I, J) *X(L)
00031 50 END DO
00032 Y(I)=SUM
00033 60 END DO
00034 100 END DO
00035 RETURN
00036 END
87
00001 *
00002 SUBROUTINE SEIDEL (N,M,NE,A,R, X, XO)
00003 *
00004 ************************** *******
00005 *
00006 * THIS SUBROUTINE IS USED TO SOLVE A SYSTEMOF LINEAR ALGEBRAIC
00007 * EQUATIONS USING GAUSS-SEIDEL METHOD.
00008 *
00009 ******************* **************
00010 *
00011 DIMENSION A(N,M) ,R(N) ,X(N) , X0 (N)
00012 COMMON/SEIDEL/ERRCRT,MAXITR
00013 COMMON /CONTR/NFRQ, IWRT, CONTROL
00014 KOUNT=l
00015 *ITERATION
00016 100 CONTINUE
00017 DO 10 1=1, N
00018 10 X0(I)=X(I)
00019 *... .FIRST PAIR
00020 DUM=A(1,1)
00021 SUM=0.0
00022 DO 20 J=2,M
00023 20 SUM=SUM+A(1, J) *X(J)
00024 X(1)=(R(1) -SUM) /DUM
00025 DUM=A(2,2)
00026 SUM=A(2,1)*X(1)
00027 DO 30 J=3,M
00028 30 SUM=SUM+A(2, J) *X(J)
00029 X(2)=(R(2)-SUM) /DUM
00030 * IN-BETWEEN
00031 DO 60 11=2, NE
00032 I0=2*(II-1)
00033 L0=2*(II-2)
00034 DO 50 K=l,2
00035 SUM=0.0
00036 I=I0+K
00037 IF(K.EQ.l) IPIV=3
00038 IF(K.EQ.2) IPIV=4
00039 DUM=A(I,IPIV)
00040 SUM=0.0
00041 DO 40 J=1,M
00042 IF(J.EQ.IPIV) GO TO 40
00043 L=L0+J
00044 SUM=SUM+A(I, J) *X(L)
00045 40 END DO
00046 X(I)= (R(U-SUM)/DUM
00047 50 END DO
00048 60 END DO
000 4 9 * LAST PAIRS
00050 NM1=N-1
00051 NM2=N-2
00052 NM7=N-M
00053 MM1=M-1
00054 MM2=M-2
00055 DUM=A(NM1,MM1)
00056 SUM=0.0
00057 DO 70 J=1,M
88
00058 IF(J.EQ.MMl) GO TO 70
00059 L=NM7+J
00060 SUM=SUM+A(NM1, J) *X(L)00061 70 END DO
00062 X(NM1)=(R(NM1)-SUM)/DUM
00063 DUM=A(N,M)
00064 SUM=0.0
00065 DO 80 J=1,MM1
00066 L=NM7+J
00067 SUM=SUM+A(N, J) *X(L)00068 80 END DO
00069 X(N)=(R(N)-SUM) /DUM
00070 * CHECK CONVERGENCE
00071 ERRMAX=0.0
00072 DO 90 1=1, N
00073 ERRI=X(I) -X0 (I)
00074 ABSERR=ABS (ERRI)
00075 ERRMAX=AMAX1 (ABSERR, ERRMAX)
00076 90 END DO
00077 IF ( ERRMAX. LE. ERRCRT) KOUNT=0
00078 IF ( ERRMAX. GT. ERRCRT) THEN
00079 KOUNT=KOUNT+l
00080 IF (KOUNT.GT.MAXITR) THEN
00081 KOUNT=0
00082 WRITE(6, 610)
00083 610 FORMAT (//5X,' **** NO CONVERGENCE ****';
0008 4 END IF
00085 END IF
00086 IF(IWRT.EQ.l) THEN
00087 WRITE(6,620) KOUNT
00088 620 FORMAT (/2X, 'ITERATION NO. :', 14)
00089 WRITE(6,630) (X(I),I=1,N)
00090 630 FORMAT (2X,1P10E11. 3)
00091 WRITE(6,640) ERRMAX
00092 640 FORMAT ( 2X,' ERROR= '
,1PE12 . 3)
00093 END IF
00094 IF(KOUNT.NE.O) GO TO 100
000 95 RETURN
00096 END
89
00001 *
00002 SUBROUTINE GUAS (N,M,NE, A, R, Y)
00003 *
00004 a**********************************,
00005 *
00006 * THIS SUBROUTINE IS USED TO SOLVE A SYSTEM OF LINEAR ALGEBAIC
00007 * EQUATIONS USING GAUSS-ELEMINATION METHOD WITH BACK SUBSTITUTION.
00008 *
00009************************************
00010 *
00011 DIMENSION A(N,M) ,R(N) ,Y(N)
00012 * PRELIMINARIES
00013 NM1=N-1
00014 MM1=M-1
00015 MM2=M-2
00016 * ELEMINATION
00017 * FIRST PAIR
00018 DO 10 K=l,2
00019 KP1=K+1
00020 DO 10 I=KP1,MM2
00021 C=A(I,K)/A(K,K)
00022 DO 5 J=KP1,M
00023 5 A(I, J)=A(I, J)-C*A(K, J)
00024 R(I)=R(I)-C*R(K)
00025 10 END DO
00026 * IN-BETWEEN
00027 DO 50 11=2, NE-1
00028 10=2* (II-l)
00029 DO 50 K=l,2
00030 KP1=K+1
00031 IPV=I0+K
00032 JPV=2+K
00033 IF(K.EQ.l) THEN
00034 IPVP1=IPV+1
00035 JPVP1=JPV+1
00036 C=A(IPVP1,JPV)/A(IPV, JPV)
00 037 DO 20 J=JPVP1,M
00038 20 A(IPVP1, J)=A(IPVP1, J)-C*A(IPV,J)
00039 R(IPVP1)=R(IPVP1)-C*R(IPV)
000 40 END IF
00041 DO 40 IP=1,2
00042 I=(l0+2)+IP
00043 C=A(I,K)/A(IPV,JPV)
00044 DO 30 J=KP1,MM2
00045 JJ=JPV+J-K
00046 30 A(I,J)=A(I,J)-C*A(IPV,JJ)
00047 R(I)=R(I)-C*R(IPV)
00048 40 END DO
00049 50 END DO
00050 * LAST PAIR
0005110=2* (NE-1)
00052 DO 70 K=l,3
00053 IPV=I0+K
00054 JPV=2+K
00055 IPVP1=IPV+1
00056 JPVP1=JPV+1
00057 DO 70 I=IPVP1,N
90
00058 C=A(I, JPV)/A(IPV,JPV)
00059 DO 60 J=JPVP1,M
00060 60 A(I, J)=A(I, J)-C*A(IPV, J)
00061 R(I)=R(I)-C*R(IPV)
000 62 7 0 END DO
00063 * BACK SUBSTITUTION
00064 * FIRST PAIR
00065 R(N)=R(N)/A(N,M)
00066 R(NM1)=(R(NM1) -A(NM1,M) *R(N) ) /A(NM1,MM1)
00067 * IN-BETWEEN
00068 DO 90 11=1, NE-1
00069 IB=2*(NE-II)
00070 DO 90 K=l,2
00071 KB=3-K
00072 I=IB+KB
00073 IF(K.EQ.l) IPV=4
00074 IF(K.EQ.2) IPV=3
00075 SUM=0.0
00076 IPVP1=IPV+1
00077 DO 80 J=IPVP1,M
00078 L=I+J-IPV
00079 80 SUM=SUM+A(I, J) *R(L)
00080 R(I)=(R(I)-SUM) /A(I,IPV)
00081 90 END DO
00082 * FIRST PAIR
00083 DO 100 K=l,2
00084 KB=3-K
00085 SUM=0.0
00086 DO 95 J=KB+1,4
00087 95 SUM=SUM+A(KB, J) *R(J)
00088 R (KB) =(R (KB) -SUM) /A (KB, KB)
00089 100 END DO
00090 * OK NOW TO Y
00091 DO 110 1=1, N
00092 110 Y(I)=R(D
00093 RETURN
00094 END
91
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00003
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00007
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00010
00011
00012
00013
00014
00015
00016
00017
00018
00019
00020
00021
00022
00023
00024
00025
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00027
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00029
00030
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00043
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00045
00046
SUBROUTINE OUTPUT (N,NE, T, Y, SIG)
** **************************** ******
THIS SUBROUTINE IS USED TO WRITE THE OUTPUT RESULTS FOR PLOTTING.
************************
COMMON /MATDIM/RO, EE, AREA, IX, HT, P, G, P0 ,Vin
COMMON/DELTAS/DT,H
REAL*4 Y(N) , SIG(NE)
REAL LEN
WRITE(6,610) T
610 FORMAT (//5X,'TIME='
, 1PE12 . 3, /5X,'
WRITE(6,620) (Y(I) ,I=1,N-1,2)
620 FORMAT (2X,'DISP'
, 1P10E11. 3)
WRITE(6,640) (Y(I) , 1=2, N, 2)
640 FORMAT (2X, 'SLOP', 1P10E11. 3)* FOR POST PROCESSOR
IF (T .GT. 0.301 .AND. T . LT . 0.302) THEN
WRITE(6,*) T
DO 77 1=1, N, 2
LEN = H*( (1-1) /2)
WRITE(97,*) LEN, Y(I)
77 CONTINUE
ENDIF
IF (T .GT. 1.201 .AND. T . LT . 1.202) THEN
WRITE (6,*) T
DO 78 1=1, N, 2
LEN = H*( (1-1) /2)
WRITE(98,*) LEN, Y(I)
7 8 CONTINUE
ENDIF
IF (T .GT. 1.801 .AND. T . LT . 1.802) THEN
WRITE(6,*) T
DO 79 1=1, N, 2
LEN = H*((I-l)/2)
WRITE(99,*) LEN, Y(I)
***** ******
',/2X>
7 9 CONTINUE
ENDIF
WRITEU2,*) T,Y(1)
WRITE(33,*) T,Y(33)
WRITE(41,*) T,Y(41)
RETURN
END
92
PROGRAM AXIAL
93
00001 *
00002 * This program is used to provide the program WEB1 with
00003 * those variables in the axial motion and the boundary00004 *
conditions at the two end3 for the vibration analyses
00005 *of phase shift and transient effects.
00006 *
00007 * Variable desciption:
00008 * J1,J2 polar moment of inertia of rollers
00009 * B1,B2 viscous damping of rollers 1 & 2
00010 *HD1,HD2 radii of rollers 1 & 2
00011 *TF frictional torque of driven roller
00012 * K axial stiffness of web
00013 *UNBL1,UNBL2 unbalanced forces of roller 1 & 2
00014 *TRQ0 initial applied torque
00015 * TRQ1 final torque applied (for transient analysis only)
00016 *OMG01,OMG02 initial angular velocities of rollers 1 & 2
00017 *P0 initial tension in the web
00018 * TEND end time
00019 * DT time increment
00020 * T0,T1 start & end time of changing torque
00021 * TSLP time rate of changing torque
00022 *VI, V2 web velocities at the contact point with roller 1 &
00023 * V1D,V2D time rate of web velocities at the contact point
00024 *with rollers 1 & 2
00025 * P web tension
0002 6 * PRAT time rate of web tension
00027 * Q1,Q2 shear forces at the left and right ends
00028 * M1,M2 moments at the left and right ends
00029 *
00030 C
00031 C V1,V2,V1D,V2D,P,PRAT FILE 7
00032 C Q1,M1,Q2,M2 FILE 8
00033 C
00034 INTEGER TRANS
00035 PARAMETER NMX=2000
00036 PARAMETER NEQ=3
00037 DIMENSION XX (NEQ) ,F (NEQ)
00038 DIMENSION Rl (NEQ) , R2 (NEQ) , R3 (NEQ) , R4 (NEQ)
00039 REAL*4 J1,J2,K
000 40 COMMON/PARAM/J1, J2, Bl, B2, HD1, HD2
00041 COMMON/OTHER/TF,K,UNBLl,UNBL2
00042 COMMON/INPUT/TRQ0,TRQl,TSLP,T0,Tl
00043 C READ/WRITE DATA
00044 READ (5,*) TRANS ! This is a control number for
00045 C transient analysis.
00046 READ (5,*) Jl, J2, Bl, B2, HD1, HD2
00047 READ (5,*) TF, K, UNBL1,UNBL2
00048 READ (5,*) OMG01,OMG02,P0
00049 READ (5,*) TEND,DT
00050 READ (5,*) TRQ0 , TRQ1, TO, Tl
00051 WRITE(6,600)
00052 600 FORMAT (//2X,'INPUT DATA:
'
)
00053 WRITE(6,610) Jl, J2, Bl, B2, HD1, HD2
00054 610 FORMAT(/2X, 'SYSTEM PARAMETERS:',
00055 1 /2X, 'Jl=', 1PE12.3,'
(KG-M**2)',
00056 2 /2X, 'J2=', 1PE12.3,'(KG-M**2)',
00057 3 /2X, 'Bl=', 1PE12.3,'
(N-S-M)'
,
94
00058 4 /2X, 'B2=\ 1PE12.3,'
(N-S-M)',
00059 5 /2X, 'Rl=', 1PE12.3,'
(M)'
,
00060 6 /2X, 'R2=', 1PE12.3,'
(M)'
)
00061 WRITE(6,620) TF, K,UNBL1,UNBL2
00062 620 FORMAT (/2X, 'OTHER INFORMATIONS: ',
00063 1 /2X, 'FRICTIONAL TORQUE= '
, 1PE12 . 3,'
(N-M)'
,
00064 2 /2X, 'WEB AXIAL STIF K =',1PE12.3,*(N/M)',
00065 3 / 2X, 'UNBALANCE INROLl='
, 1PE12 . 3,'
(KG-M)',
00066 4 / 2X, 'UNBALANCE INROL2='
, 1PE12 . 3,'
(KG-M)')
00067 WRITE(6,630) OMG01, OMG02, P0
00068 630 FORMAT (/2X, 'INITIAL CONDITIONS: ',
00069 1 /2X, 'OMEGAl=',lPE12.3,'
(RAD/S)',
00070 2 /2X, 0MEGA2=',1PE12.3,'
(RAD/S)1,
00071 3 /2X, 'P(TEN)=',1PE12.3,'
(N)'
)
00072 WRITE(6,640) TEND,DT
00073 640 FORMAT (/2X, 'TEND, DT. ...', 1P2E12. 3)
00074 TSLP=(TRQ1-TRQ0) / (T1-T0)
00075 WRITE(6,650) TRQ0, TRQ1, TO, Tl, TSLP
00076 650 FORMAT (/ 2X, 'TRQ0=', 1PE12. 3,'
(NM)'
,
00077 1 /2X,'TRQ1='
00078 2 /2X,'TIM0='
00079 3 /2X,'TIM1='
00080 4 /2X,'TSLP='
00081 C INITIALIZATION
00082 WRITE(6,660)
00083 660 FORMAT (//10X, 'TIME', 5X, 'OMEGAl',6X, 'OMEGA2'
, 10X, 'P',
00084 *10X, 'Q1',10X, 'Q2')
00085 XX(1)=OMG01
00086 XX(2)=OMG02
00087 XX(3)=P0
00088 T=0.0
00089 CALL FUN (T, NEQ, XX, F)
00090 CALL OUTPUT (T, NEQ, XX, F)
00091 C NUMERICAL INTEGRATION
00092 DO 20 WHILE (T.LE.TEND)
00093 CALL RK4(T,DT,NEQ,XX,R1,R2,R3,R4,F)
00094 CALL OUTPUT (T, NEQ, XX, F)
00095 20 END DO
000 96 STOP
00097 END
1PE12,.3,'
(NM) ',
1PE12..3,
'
(SEC) ',
1PE12 3,'
(SEC) ',
1PE123,'
(NM/S) ')
95
00001 *
00002 * The fourth-order Runge-Kutta method
00003 *
00004 SUBROUTINE RK4 (T, DT, N, XX, Rl, R2, R3, R4, F)
00005 DIMENSION XX (N) , F (N) , Rl (N) , R2 (N) , R3 (N) , R4 (N)
00006 DIMENSION DUM(10)
00007 DO 10 1=1, N
00008 DUM(I)=XX(I)
00009 10 END DO
00010 CALL FUN(T,N,XX,F)
00011 DO 20 1=1, N
00012 Rl (I)=DT*F(I)
00013 XX(I)=DUM(I)+R1 (I) /2.0
00014 20 END DO
00015 T=T+DT/2.0
00016 CALL FUN(T,N,XX,F)
00017 DO 30 1=1, N
00018 R2 (I)=DT*F(I)
00019 XX(I)=DUM(I)+R2 (I) /2.0
00020 30 END DO
00021 CALL FUN(T,N,XX,F)
00022 DO 40 1=1, N
00023 R3(I)=DT*F(I)
00024 XX(I)=DUM(I)+R3(I)
00025 40 END DO
00026 T=T+DT/2.0
00027 CALL FUN (T, N, XX, F)
00028 DO 50 1=1, N
00029 R4 (I)=DT*F(I)
00030xX(I)=DUM(I)+(Rl(I)+2.0*R2(I)+2.0*R3(I)+R4(I))/6.0
00031 50 END DO
00032 RETURN
00033 END
96
00001 c
00002 SUBROUTINE FUN(T,N,X,F)
00003 DIMENSION X(N) ,F(N)
00004 REALM J1,J2,K
0000 5 COMMON/PARAM/J1, J2, Bl, B2, HD1, HD2
00006 COMMON/OTHER/TF,K,UNBLl,UNBL2
00 007 COMMON/INPUT/TRQ0,TRQl,TSLP,T0,Tl
00008 CALL INPUT (T,TRQ)
00009 F(l) =(TRQ-B1*X(1)-HD1*X(3) ) /Jl
00010 F(2) =(HD2*X(3)-B2*X(2)-TF) /J2
00011 F(3)=K* (HD1*X(1) -HD2*X(2) )
00012 RETURN
00013 END
00001 C
00002 SUBROUTINE INPUT (T,Y)
00003 COMMON/INPUT/TRQ0,TRQl,TSLP,T0,Tl
00004 IF(T.LE.TO) THEN
00005 Y=TRQ0
00006 ELSE IF(T.GT.TO.AND.T.LT.Tl) THEN
00007Y=TRQ0+TSLP* (T-T0)
00008 ELSE IF(T.GE.Tl) THEN
00009 Y=TRQ1
00010 END IF
00011 RETURN
00012 END
97
00001 C
00002 SUBROUTINE OUTPUT (T, N, X, F)
00003 DIMENSION X(N) , F (N)
00004 REALM J1,J2,K
00005 COMMON/PARAM/Jl , J2 , Bl , B2 , HD1 , HD2
00006 COMMON/OTHER/TF, K, UNBL1 , UNBL2
00007 C. . . . HISTORIES OF OMGl,OMG2,P TO FILES
00008 OMGl=X(l)
00009 OMG2=X(2)
00010 P =X(3)
00011 ALF1=F(1)
00012 ALF2=F (2)
00013 PRAT=F(3)
00014 IF (TRANS .EQ. 1) THEN
00015 WRITE (56,*) T,OMGl
00016 WRITE (55,*) T,OMG2
00017 WRITE(24,*) T,P
00018 ENDIF
00019 c. . , , OTHERS FOR WEB1
00020 c V1,V2,V1D,V2D,P,PD TO FILE 7
00021 VI =HDl*OMGl
00022 V2 =HD2*OMG2
00023 V1D=HD1*ALF1
00024 V2D=HD2*ALF2
00025 WRITE(7,610) V1,V2,V1D,V2D,P,PRAT
00026 c Q1,M1,Q2,M2 TO FILE 8
00027 OTl=OMGl*T
00028 OT2=OMG2*T
00029 SI =SIN(OTl)
00030 S2 =SIN(OT2)
00031 CI =COS(OTl)
00032 C2 =COS (OT2)
00033 Ql=+UNBLl* (0MG1*0MG1*S1-ALF1*C1)
00034 Q2=+UNBL2* (0MG2*0MG2*S2-ALF2*C2)
00035 ZER=0.0E0
00036 WRITE (8,*) Q1,ZER,Q2,ZER
00037 610 FORMAT (4X, 1P6E12 .3)
00038 C. ..LAST
INFORMATION
00039 IF (TRANS .EQ. 1) THEN
00040 CALL INPUT (T,TRQ)
00041 WRITE (27,*) T,TRQ
00042 ENDIF
00043 RETURN
00044 END
56,55,24
98
PROGRAM CONVER
99
00001 *
00002 * This program is used to convert the dita of the responses
00003 *of the web-roller system from the time domain into the
00004 *frequency domain.
00005 *
00006 * Variables Description:
00007 *
00008 * N number of data point
00009 * IN file control number
00010 * DT time increment
00011 *T(I) tmie array
00012 *SEQ(I) displacement array
00013 *FREQ(I) frequency array
00014 * COEF(I) amplitude in the frequency domain
00015 *
00016 *
00017 INTEGER N, I, IN
00018 REAL PI,DFREQ,DT, TN
00019 PARAMETER (N=1930, PI=3. 14159)
00020 REAL COEF(N) , SEQ(N) ,FREQ(N) ,T(N) ,seql(N)
00021 EXTERNAL FFTRF
00022 C
00023 READ (5,*) IN !Get control number
00024 IF (IN .EQ. 0) THEN
00025 C
0002 6 C reading input file by unit number
00027 C file unit 12 is the first node data (X/1 = 0.0)
00028 C file unit 33 is the node 33 data (X/1 =0.8)
00029 C file unit 41 is the node 41 data (X/1 =1.0)
00030 C
00031 DO 10 1=1, N
00032 READU2,*) T(I),SEQ(I)
00033 10 CONTINUE
00034 ELSE IF (IN . EQ . 1) THEN
00035 DO 11 1=1, N
00036 READ(33,*) T(I),SEQ(I)
00037 11 CONTINUE
00038 ELSE IF (IN .EQ. 2) THEN
00039 DO 12 1=1, N
00040 READ (41,*) T(I),SEQ(I)
00041 12 CONTINUE
00042 ENDIF
00043 DFREQ= 1/(2*T(N))
00044 C Compute the Fourier transform of SEQ
000 45 CALL FFTRF (N, SEQ, COEF)
00046 DO 30 1=1, N
00047 COEF (I) = ABS(COEFd) )
00 0 48 30 CONTINUE
00049 WRITE (98,*) FREQ(l), COEF(l)
00050 DO 40 1=2, N
00051 FREQ(I) = FREQd-D + DFREQ
00052 IF (FREQ(I) . LT . 220) THEN
00053 WRITE (98,*) FREQ(I), COEF (I)
00054 ENDIF
00055 40 CONTINUE
00056 STOP
00057 END
100
