Rochester Institute of Technology
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1994
Dynamic stability of a self-excited elastic beam
Francisco L. Ziegelmuller
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Recommended Citation
DYNAMIC STABILITY
OF A
SELF-EXCITED ELASTIC BEAM
Francisco L. Ziegelmuller
A Thesis Submitted in Partial Fullfillment
of the Requirements for the Degree of
Master of Science
in
Mechanical Engineering
Approved by:
Professor
J.S..Torok (Thesis Advisor)
Professor
R. G. Budynas
Professor
R. H. Hetnarski
Professor
C. W. Haines (Department Head)
Department of Mechanical Engineering
College of Engineering
Rochester Institute of Technology
Rochester, New York
PERMISSION GRANTED:
I, Francisco L. ZiegelmuJler, hereby grant permission to the Wallace
Memorial Library of the Rochester Institute of Technology to
reproduce my thesis entitled "Dynamic Stability of A Self-Excited
Elastic Beam" in whole or in part. Any reproduction will not be for
commercial use or profit.
July 15, 1994
ABSTRACT:
An
elastic cantileverbeam
under pressure contact with amoving
web undergoes self-excited vibration
that
may lead
to
unstable and even selfdestructive
behavior.
The
equations of motionfor
the
system arederived
from
the
principle of virtual work and
Hamilton's
principleusing
the
techniques
ofthe
calculus of variations.The
beam,
being
a continuum withinfinite
degrees
offreedom,
is
approximatedby
a model with afinite
number ofdegrees
offreedom
using
Galerkin's
method.The
characteristic equationfor
the
modelis
examinedto
determine
its
dynamic
criterionfor
stability.
A
parametricstudy is
performedto
determine
the
effects ofthe
beam
properties such asbeam
length
(L),
extension(W),
thickness
(h),
elastic modulus(E),
stiffness(El),
beam
inclination
angle(0)
withrespect
to
moving
web,
the
static andkinematic
coefficient offriction
fas.
^k)-The
beam
responsedue
to
the
motion ofthe contacting
webis
TABLE OF CONTENTS
[1]
Introduction
[2]
Background
2.1
Introduction
2.2
Principle
ofVirtual
Displacements
2.3
Energy
Method using Hamilton's Principle
2.4
Structural
(Static)
Stability
2.5
Dynamic
Stability
[3]
The
Governing
Dynamical
System
3.1
Derivation
ofGoverning
Equations
3.2
Modal Analysis
ofA
Cantilevered
Beam
3.3
Numerical
Values
ofBeam Parameters
andConstants
3.4
Governing
Equations
[4]
Dynamical
Stability
Analysis
4.1
Bifurcation
andEigenvalue Approaches
4.2
Solution
ofCharacteristic
Equation
4.3
Dynamic
Stability
Criterion
4.4
Parametric
Evaluation
ofStability
LIST
OF
SYMBOLS
x coordinate
along
the beam length
axist
time
coordinateqi
generalized coordinates(i
=1,2,
... ,N)
N
number of modes ordegrees
offreedom
DCF
degrees
offreedom
NDOF
number ofdegrees
offreedom
3qj/9t
partialderivative
ofqj
with respectto
t
3q/3x
partialderivative
ofqj
with respectto
xQj,
p(t)
generalizedforce
Uj(t)
axial generalizeddisplacement
Vj(t)
transverse
generalizeddisplacement
u(x,t)
beam
axialdeformation
as afunction
of x andt
v(x,t)
beam
transverse
deflection
as afunction
of x andt
u\u",... partial
derivatives
ofu(x,t)
with respectto
x(3u/3x,
32u/3x
2,
... orux,
uxx,...)
v'.v",... partial
derivatives
ofv(x,t)
with respectto
x(3v/3x,
32v/3x
2,
... orvx, vxx,
...)
ut>
uttt... partialderivatives
ofu(x,t)
with respectto
t
(3u/dt,
32u/3t2,
...)
vt,
vlt)... partialderivatives
ofv(x,t)
with respectto
t
(3v/9t,
32v/at2,
...)
8
variational operator or virtual operatorT
total
kinetic energy
ofthe
systemV
total
potentialenergy
ofthe
systemUb
elastic or strainenergy
ofthe beam
in
bending
wcons
workdone
by
external conservativeforces
(=-V)
Wpc
workdone
by
external nonconservativeforces
W
combined workby
the
real andinertial
forces
$,
<>(x)
transverse
displacement
ofthe beam
0i(x)
characteristic modalfunction
of a uniform cantileveredbeam (mode
shape)
<J>'j(x)
partialderivative
of<>j(x)
with respectto
x(3<>/3x)
j
naturalfrequency
corresponding
to
tyt
X\
spatialfrequency
constantaj
constantin
the
mode shapes of a cantileveredbeam
Y,y(x)
axialdisplacement
ofthe
beam
Yj(x)
characteristic modalfunction
of a uniform cantileveredbeam
(mode
shape)
V'j(x)
partialderivative
ofvj(x)
with respectto
x(3\|/j/3x)
E
Young's
modulus or elastic modulusI
moment ofinertia
ofthe beam
E*I
flexural
rigidity
orbending
stiffnessL
beam
length
h
beam
thickness
b
beam
widthA
cross-sectional area ofthe beam
p
mass per unitlength
ofthe
beam
co
forcing frequency
\i coefficient of
friction
between the
beam
and web\is
static coefficient offriction
6
inclination
anglebetween
the
beam
andthe moving
webor
inclination
angle ofthe
undeflectedbeam
withthe
tangential
line
atthe
point of contact withthe
webN
normalforce
appliedby
the
web onthe
beam
T8
webtension
f
frictional
force,
equalsto
|is*Nup to
the
onset of motionand equals
to |xk
*N afterwards.Its
direction
is
alwaysopposite
to the
motion.M
moment atthe
free
end ofthe
beam
fx, fy,
M
x- and y- components ofN
andf,
andM
magnitudesC
angledefined between N
andthe
blade
surfacefacing
the
weba attack angle of
the beam
onthe
rollerP,
y
anglesdefined
by
the
deflected
webti
angleincluded between
tension
forces (=180
-p-y)
s,
s1
web spanbetween
rollers andbeam
s a root
to the
characteristic equation which equals ojco.a
is
the
real component and cois
the
imaginary
component. e webdeflection
r,
e roller radius and rollerexcentricity
vw
webvelocity
vb
velocity
ofthe
free
end ofbeam
in
contact withweb
(-
{
[ut(L,t)]2 + [vt(L,t)]2 }.s)
sv
slip
velocity
(=
vw
-vb)
sgn(x)
sign of xc
damping
coefficient[1]
Introduction
The
application of a cantileverbeam
in
pressure contact with amoving
web resultsin
nonlinear vibrationsdue
to
self-excited motion or acondition
commonly
referredto
as"slip-stick
motion".Such
vibrationscan reach
levels
that
lead
to
unstablebehavior
resulting
in
a conditionreferred
to
as"tuck-under",
wherethe "free
" end ofthe beam
experiences aninversion
in
its deformed
shape(figure
1).
"Tuck-under"may
resultin
self-destructive
behavior
ofthe
beam
and/or web.Normal
cantileverbeam
Inverted
cantileverbeam
In
actualapplications,
the
beam
is
in
reality
aplate,
commonly
referred
to
as a"doctor
blade".
It
is
usually
made of polymeric ormetallic material and
it
is
usedin
copiersto
clean amoving
web offine
powder.
In the
absence of suchfine
powder,
the
friction force increases
causing
the
blade
to
changeits
deformed
shape.If
a criticalload
is
reached,
the
blade may
tuck
underthe
web.This
behavior
generally
occurslocally
andthen
rapidly expanding along
the
blade
width.The
abovebehavior
canbe
repeatedfor
shorter plate width andcantilever
beam
as a model.The
analysisis
performedby
first presenting
two
versions ofthe
problem andthen
by describing
the
problem as onein
which
the
beam is inclined from
the
webby
a given angle.The
normal andfrictional
forces
willbe
defined
with respectto
this
angle.Then
the
equations of motion
for
the
system arederived
from
the
principle ofvirtual work and
Hamilton's
principleusing the
techniques
ofthe
calculusof variations.
The
beam,
being
a continuum withinfinite
degrees
offreedom,
is
approximatedby
a model with afinite
number ofdegrees
offreedom
using
Galerkin's
method.The
characteristic equationfor
the
modelis
examinedto
determine
its
dynamic
criterionfor
stability.The
goal of
this investigation
is
a parametricstudy
ofthe
effects ofthe
beam
properties such as
beam
length
(L),
extension(W),
thickness
(h),
elasticmodulus
(E),
stiffness(El),
beam
inclination
angle(6)
with respectto
moving
web,
static andkinematic
coefficient offriction
(\is,
nk).The
beam
response
due
to the
motion ofthe
contacting
webis
analyzedto
evaluate critical propertiesto
be
used as guidein
the
design
of stablebeam for
such applications.
The
first
problem presented examinesthe
case whenthe
cantileverbeam
engagesthe
web over arelatively hard
roller(figure
2)
sothat
the
normal
force,
N
,is
in
the
radialdirection
andthe
friction
force,
f
,perpendicular
to
the
normalforce
and relatedto
it
by
the
expression:Jf]=]jj.*N|,
where \iis
the
coefficient offriction
between
the
contacting
surfaces.The
beam
is
inclined
from
the
tangential
line
atthe
point of contactby
an angle9.
The
beam
contactsthe
web at an attack angle aalong
the
roller.These
forces
canbe decomposed
along
x- andy-axis
components as:
fx
=N*[sin8
+ n*cos6]
andfy
=N*[cos8
(A)
CANTILEVER
BEAM WITH
FREE
END IN
PRESSURE
CONTACT
WITH A
MOVING
WEB OVER A HARD ROLLER
Tangential
line
atNormal
line
at/
point of contact
point of contact
Figure
2
f
=H
*N
Due to
the
fact that
fx
is
an eccentricload,
we can replaceit
by
aload
fx
acting
along
the
neutral axis and a counterclockwise momentM
acting along
the
neutral axis atthe
free
end ofthe
beam (figure
3).
The
magnitude of
this
momentis:
M
=fx*h/2
=[N
*h/2]
*[sin 6
+ n*cos6]
(1.2)
The
beam
has
rectangular cross-section withthe
following
dimensions: length
L,
widthb,
thickness
h,
and moment ofinertia
I
which equalsb*h3/12.
The
materialsto
be
investigated
willbe
stainless steel and polyurethane withthe
following
valuesfor
their
respectiveYoung's
modulus,
E:
29*1
06 and1000
psi.The
coefficient offriction
valuesto
be
Cantilever
Beam:
L=length,
h=thickness
b=width
l=Moment
ofInertia
E=Young's
modulus(a)
MODEL
STAGES:
r
2
(b)
v\
Figure
3
(C)
y,V4
#
x,u
x,u
fx
fy
M=
fx*h/2
"^^N
x.uThe
web moves with anapproximately
constantlinear
velocity
vw.The drive
roller radiusis
r andit may have
a smalleccentricity
e whichleads
to
anoscillating deflection
ofthe
cantileverfree
end(figure 4).
The
second version ofthe
problem placesthe
cantileverbeam
in
pressure
contact
withthe
moving
webin
a sectionbetween
two
hard
rollers
(figure
5).
One
ofthe
rollersis spring loaded
to
provide enoughtension
for
the
webto
be
friction
driven
by
fixed
roller.The
beam
loads
the
webcausing it
to
deflect
by
e.The
spanbetween the
rollersis
s.The
beam
contactsthe
web at adistance st
from
the
spring
loaded
roller.In
this
casethe
load
N
willbe in the direction
ofthe
resultant ofthe
webtension
components,
Ts,
as shownin figure
6.
The
friction force
willbe
perpendicular
to
the
load
N.
The
beam
is
inclined from
the
undeflected webby
an angle0.
The
angleincluded between
the tension
forces is:
Tl-180-p-y,
(1.3)
where:
y
=tan-1[e/(s-s1)]
andp
= tan-1(e/s^.
(1.4)
The load
N
makes an angle equalto half rj
withthe
tension
force. The
beam
makes an angle equalto:
C=-n/2-0+y
(1.5)
with
the beam
surfacefacing
the
web.But,
C
= 90 -p/2
-y/2 -
0
+y
= 90 -p/2
+ y/2
LU
cr
This
geometricconfiguration
is
shownin
figure
7.
Notice
that if
p
is
equal
to
y,
that
is,
the
beam
contactsthe
web at midspan and/orthese
angles are
very
smallin
magnitude,
then
we would endup
withthe
samegeometric configuration of
the
first
version ofthe
problem.In
summary,
we willbe
investigating
a cantileverbeam
in
pressurecontact with
the
web under a normalload
N,
making
an angleC
vviththe
blade
surface
facing
the
web and africtional
force
f
perpendicularto
N.
From
figure
8,
we canderive
the
x-and
y-components of
the
forces
andthe
end moment as:fy
=N*sinC
-f*sin(90-C)
=N*(sinC
-u*cosC),
(1.7)
fx
=N*cosC
+f*cos(90-C)
=N*(cos
+u*sinC),
and(1.8)
M
=fx*h/2
=(N*h/2)*(cosC
+ H*sinC).(1.9)
The
latter
version ofthe
problemis
moregeneralized,
but
usually
the
web anglesp
andy
are small.We
will assumethey
are negligiblefor
Figure
7
90-5
[2]
Background
2.1
Introduction
The
cantileverbeam
problem presentedin the
previous sectionhas
very
low
transverse
stiffness
andhigh
axial stiffness.Runout
oreccentricity
ofthe
roller,
variationin
frictional
forces
due
to
dry
webcondition and
differential
levels
oflubrication
will resultin
harmonic
forcing
functions
to
be
applied atthe
free
end ofthe
beam.
These
variousloading
conditions
may
resultin
unstablebehavior
ofthe
beam.
Elastic
beam
stability
has
been
extensibly
studied as modelsfor
such problems as a
flexible
missile under an endthrust
[1],
buckling
of amagnetoelastic system
[2],
elasticstability
underfollower
forces
[3]
such as a structural part of an aircraft under aerodynamic
loads,
acantilever
beam
conveying
fluid
whichloses
stability
by
flutter,
andothers.
The study
ofthe
dynamic stability
of an elasticbeam
under anharmonically
varying
load
with a pinned end and an end constrainedto
move
axially
has
been
presentedby
Mettler,
Timoshenko,
Bolotin
andothers
[4,5,6].
Approximate
techniques
using
numerical methods are often requiredto
solvethese types
of problems.Galerkin's
technique
wasfirst
appliedby
Leipholz
(1962a)
to
calculatebifurcation
loads
for
anonconservative
system
[1,7].
Timoshenko
[5],
Mettler
[4]
andBolotin
[6]
have
usedGalerkin's
methodto
convertthe
equations of motionto
time
differential
equation which are studied
for
their
stability
using
Floquet
theory
[8,9,10,11,12].
Levinson
[7]
showedthat
the
conventionalHamilton's
principle with nonconservative
forces
may
be
treated
as a variationalproblem with some
constraints,
andthen
the Ritz
method canbe
employed.
problem without
any
constraint
conditions.The
finite
element methodhas
been
usedbased
onthe
variational approach suggestedby
Levinson,
andbased
onLeipholz's
variational principle.The
ordinary
finite
difference
method was used with
the
concept of atransfer
matrixto
solve anonconservative problem
by
Leipholz.
Guran
andOssia
investigated
the
dynamic
stability
of a uniformfree-free
rod under an endthrust
using
finite
difference
techniques
withthe
concept of atransfer
matrix.In
nonconservative
stability
problems,
the
dynamic
stability
criteria mustbe
applied.The
bifurcation
or criticalload
is
the
smallestload
which,
when
slightly
disturbed,
causes a changein
the
equilibrium configuration ofthe
system.A
forced
vibration analysis mustbe
performedto
determine
this
load.
The
derivation
ofthe
governing
partialdifferential
equations ofmotion and
the
evaluation of criticalloads
by
using
Hamilton's
principlefollows
the
work presentedby
Guran,
Ossia
[1]
andLevinson
[7].
Based
on a set of generalizedcoordinates,
qj,
the
equations ofmotion are
determined
from
variational principles.Virtual
displacements
and virtual work concepts are utilized
in
the
evaluation ofenergy
transformations.
The
beam
transverse
deflection,
v(x,t)
is
approximatedby
the
sum ofthe
N lowest
mode shapes of a uniformbeam
underfree
vibration
[13].
The
beam deflection
may
then
be
written as:N
v(x,t)
=5>i00*vi(t)
(2.1)
i=l
where
N
= number ofdegrees
offreedom,
nodes or mode shapes<{>j(x)
= mode shape associated with nodei
The
aboveexpression
is
then
usedin
association withHamilton's
principle
to
derive
the
equations
of motion ofthe
forced
system andits
natural and
geometric
boundary
conditions.The
equations of motion arethen
manipulated
to
arrive atthe
characteristic equationfor the
systemby
using
Galerkin's
method.Roots
ofthe
characteristic equation areanalyzed
using
the
dynamic
criterionfor
stability
to
determine
the
critical or
bifurcation
load.
This
load
willthen
be
usedin
a parametricevaluation of
stability
for the
cantileverbeam.
2.2
Principle
ofVirtual
Displacements
andVirtual
Work
The
application of virtualdisplacements
is
usedto
define
energy
transformations
in
a system.This
technique
is
usedto
approximatecontinuous
bodies
by
modelling
a system with aninfinite
number ofdegrees
offreedom
(DOF) by
onehaving
only
afinite
number ofdegrees
offreedom [13].
A
changein
the
configuration of a systemis
specifiedby
adisplacement
coordinate.The
systemmay
be
under a certain number ofconstraints which are
kinematical
restrictions on possible configurationsthat the
systemmay
undertake.A
virtualdisplacement is
animaginary
infinitesimal
change of configuration of a systemconforming
to
its
constraints.
An
arbitrary
configuration of a systemmay
be
described
using
a set of generalized coordinates which arelinearly
independent
displacements,
consistent withthe
constraintsimposed
onthe
system.The
symbolqt (where
i=1,2,...,N)
are usedfor
the
generalizedcoordinates
of an
N-DOF
system.A
virtualdisplacement
is
then
denoted
by
Sq;.
The
virtualwork,
8W,
is
the
work ofthe
forces acting
on a systemN
8W
=XQi*&_i
(2.2)
i=l
where
Qj=
the
generalized
force
8qj
= virtualdisplacement
by
Qj
Qi
is
the
virtual workdone
when8qj=1
and5qj=0
for j*i. This
conceptof generalized
force is
usedextensively
in
structuraldynamics
andin the
study
of structural stability.The
principle of virtualdisplacements
may be
stated as:"For
any
arbitrary
virtualdisplacement
of a systemin
equilibrium,
the
combined virtual work of real
forces
andinertia forces
must vanish".This
may be
represented as:5W
=8Wrea,
forces +
SWinertia
forces =0
(2.3)
where
8W
=the
sum ofthe
virtual workdone
by
realforces
andinertia
forces.
The
principle of virtualdisplacements
can produce a generalizedparameter model of a continuous system
in
a mannerthat
approximatesits
flexible
behavior.
This
procedureis
commonly
referredto
asthe
assumed-modes method.
A
continuous systemis
one whosedeformation
is
described
by
one or morefunctions
ofone,
two
orthree
spatial variablesand
time.
The
deformation
of a cantileverbeam in
figure 6 is
specifiedin
terms
ofthe
deflection
curvev(x,t)
ofthe
neutral axis.Kinematical
constraints placed on
the
displacement
and/or slope on portions ofthe
boundary
ofthe
continuousbody
are called a geometricboundary
imaginary
infinitesimal
change
in
the displacement functions
with allGBC
enforced.For
the
cantileverbeam,
the
GBC
are:v(0,t)
=v'(0,t)
=0
where v'=
3v/3x
(2.4)
The dotted
curvein
figure
9
shows a possible virtualdisplacement,
5v(x,t),
ofthe
beam. The only
conditions on8v(x,t)
is
that
it
satisfiesthe
homogeneous form
ofthe
sameGBC
as v(x,t).That
is:
Sv(0,t)
=8v'(0,t)
=0.
(2.5)
5v(x,t)
is
not afunction
oftime,
but it is
to
be
understood as anarbitrary
small change of
configuration,
relativeto the
configuration ofthe
beam
attime
t.
8 v(x,t)
*x,
u(x,t)
Figure 9:
Cantilever
Beam,
GBC,
virtualdisplacements
system.
An
assumed-mode
or shapefunction
is
an admissiblefunction
that
is
usedto
approximate
the
deformation
of a continuous system.To
approximate a continuous system with a generalized-parameter
N-DOF
model,
N
assumed-modes
are used.The
deflection
ofthe
beam
may
be
approximated
by:
N
v(x_t)
=<|>i(x)*vi(t)
(2.1)
i=l
Any
admissiblefunction
may
be
used as<J>j(x),
but
a shapethat
closely
resemblesthe
deformed
shape ofthe
beam
shouldbe
selected.vs(t)
Is
called a generalizeddisplacement
for
the
N-DOF
system.It
is
determined
asthe
solutionto
anordinary differential
equation.The
principle of virtualdisplacements is
employedhere
to
create aN-DOF
generalized-parameter modelbased
on(2.1).
For
a continuoussystem,
it is
convenientto
introduce
potentialenergy,
that
is,
the
workdone
by
conservativeforces.
8Wrea,
forces =8Wcons
+8Wnc
(2.6)
where
SWcons
is
the
virtual work of conservativeforces
and8Wnc
is that
of nonconservativeforces.
From the definition
of a conservativeforce:
8Wcons
= -8V(2.7)
where
8V
is
the
change
in
the
potentialenergy
asthe
conservativeforces
move
through
a virtualchange
in configuration, for
example,
8v(x,t)
for
abeam.
Consequently,
we can writethat
8W
=8Wnc
-8V
+8Winertia
forces =0
(2.8)
Strain energy in
abeam undergoing
pure axialdeformation
u(x,t)
andin
abeam
undergoing
puretransverse deflection v(x,t)
may
be
expressedrespectively
as:*-"[
A*E*(u')2*dx
(2.9)
and-i
Ub
=E*I*(v")2*dx
(2.10)
where
Ua,
Ub
arethe
axial andthe transverse
strainenergy respectively
A
is
the
cross-sectional area ofthe beam
E
is
the
elastic modulus ofthe beam
materialI
is
the
moment ofinertia
ofthe
beam
aboutthe
bending
axis u'is
the
first
partialderivative
ofu(x,t)
with respectto
x or3u/3x
v"
is
the
second partialderivative
ofv(x,t)
with respectto
x or32v/3x2.
Consequently,
the
variational operator8
appliedto the
aboveexpressions
result
in:
8Ua
=I
(A*E*u')*8u'*dx
(2.9-a)
and
5Ub
=I
(E*I*v")*8v"*dx
(2.10-a)
f
If the
beam is deformed in both
the
axial andtransverse
direction
asshown
in figure
10,
the
strainalong
the
neutral axis canbe
written as(Appendix):
e =
u'
+ .5*v'2.
(2.11)
The
strainenergy along
the
neutral axismay
then
be
written as:a__l|
o*e*dV=l|
o*e*A*dx =i-|
E*A*e2*dx
=
J
E*A*(u'+.5*v'2)2*dx
(2.12)
where a and e are
the
axial stress and axial strainrespectively
andV
represents
the
volume ofthe beam.
Applying
the
variational operatorto the
aboveexpression,
we get:5Ua=
I
E*A*(u,+.5*v'2)*(u'+.5*v,2y*dx
(2.12-a)
<-[
We
assumein
these
derivations that
the
beam
materialis
linearly
elastic,
homogeneous,
of constant cross section andthat
it
follows
Hooke's
law relating
stressto
strain as a =E*e.
Planes
ofdeformation
along
the
longitudinal
axis will remain perpendicularto
the
neutral axis(no
twisting).
X
J
<__
!-UJ
UJ
_r
vD
2
9
<3
Z
h
0)
3
y
v
X
2
UJ
The
damping
of abeam may be
the
result ofinternal
damping
in
the
material and
Coulomb,
ordry
friction,
damping
between the beam
free
endand
the
moving
web.These
types
ofdamping
generally
resultsin
nonlinearbehavior
ofthe
beam.
The internal
damping
of abeam
may be
modelled as a viscoustype
ofdamper.
If
welet
c representthe
viscousdamping
coefficient,
then
the
work
done
by
this
nonconservative
damping
force,
c*vt,
is:
Wc=
I
c*vt*v(x,t)*dx(2.13)
where
vt
=dv(x,t)/3t.
Similarly,
we get:-[
5Wnc=l
c*[
vt*Sv(x,t) + v(x,t)*8vj*dx(2.14)
An
axialdamping
force
could alsobe
addedto
the
beam
model andits
strainenergy
wouldbe
arrived atby
substituting
ufor
vin
the
aboveexpressions.
In
this
analysis,
we will neglectthe
structuraldamping
ofthe
beam.
Mettler
[4]
provided someinsight
into
the
damping
onthe
structural
stability
ofthe
beams.
The
frictional
force
always opposes motion.When
there
is
slip
between the beam
andthe
moving
web,
the frictional
force is
oppositeto
that
ofthe
slip
velocity
sv
[14,15]. The slip velocity
may be defined
as:sv
=vw-vb,
(2.15)
and
vb
=velocity
ofthe
free
end ofthe beam.
The velocity
ofthe
free
end ofthe
beam
may
be
expressed as:vb
-{
[ut(L,t)]2 + [v,(L,t)]2 }.s(2.16)
where
ut
andvt
arethe
partialderivatives
of u and vwith
respect
to
t.
The
coefficient offriction
between
sliding
surfacesmay
be
expressed as:
Ji
-H8
-kr(vw
-vb)
-ns
-krsv
(2-17)
where
|xs
= static coefficient offriction
k1
= aconstant,
usually
a small numberThe frictional
force
may
then
be
expressed as:f
= ji*N *sgn(sv)
(2.18)
where
sgn(sv)
meansthe
sign ofthe
slip
velocity.We
will assumein the
analysisthat the friction
force
will notvary
with
the
slip
velocity
to
simplify
the
model.The frictional
force
willbe
assumed
to
be
constant with a small pertubation addedto
it.
This
pertubation
may be
assumedto
vary
harmonically
withtime
and with afrequency
to whichis
much smallerthan
the
fundamental
frequency
ofthe
2.3
Energy
Method
using
Hamilton's
Principle
Newton's
laws
andthe
principle of virtual work are employedto
derive
the
equations
of motion of mechanical systems.Hamilton's
principle,
whichis
directly
relatedto
the
principle of virtualdisplacements,
employskinetic
and potentialenergy,
which are scalarquantities,
ratherthan
the
virtual work ofinertia
forces
and elasticforces
to
derive
the
equations of motion.For
continuoussystems, the
useof
Hamilton's
principle provides an additionalbonus
which arethe
naturalboundary
conditions.In
the
formulation
ofHamilton's
principle,
virtualdisplacements,
orvirtual changes of
configuration,
are used.The
virtual change ofconfiguration must
satisfy
all geometricboundary
conditions.Hamilton
also assumed
that
the
configurationis
specified attimes
t1
andt2.
For
the
cantileverbeam
in figure
9,
this
wouldimply
that:
Svfx.tO
=8v(x,t2)
=0.
(2.19)
Hamilton's
principlemay
be
stated as:Ps(T-V)*dt
+p8Wnc*dt
=0Jtl
J,
(2.20)
where
T=
total
kinetic
energy
ofthe
systemV
= potentialenergy
ofthe system,
including
the
strainenergy
andthe
potentialenergy
of conservative externalforces
8Wnc=
virtual workdone
by
nonconservativeforces,
including
damping
forces
and externalforces
not accountedfor
in V.
t1ft2
=times
at whichthe
configuration ofthe
systemis known.
2.4
Structural
(Static)
Stability
Structural
stability
relatesto
the
study
ofstability
of structuresunder static equilibrium.
Hence,
there
is
no changein kinetic energy in
the
system or
the
inertial forces
are neglected(8T
=0).
This
methodis
validonly
for
structural systemsthat
are elastic andconservative and members are assumed
to
be geometrically
perfect.For
ageometrically
perfectmember,
the
lateral
deflections
of acentrally
loaded
beam
subjectedto
in-plane forces
will not occur untilthe
appliedload
reaches a critical value.At
this
criticalload
value,
a smalldisturbance
onthe
beam
will generate alarge lateral deflection [16,17].
To
obtainthe
criticalload,
wefirst have to
arrive at adifferential
equation
for
the
beam
in
aslightly deformed
state.The
solutionto
the
characteristic equation
derived
from
this
governing
differential
equationyields
the
criticalload
ofthe
beam[16].
There
aretwo
waysto
derive
the
governing
equationsfor
the
structure:
the
vector approach(Newton's
Laws)
andthe
energy
method.Both
methods willlead
to
eigenvalue,
i.e.
bifurcation,
analysis.We
willutilize
the
energy
method approach.The
total
potentialenergy
of asystem,
n,
is:
n
=U
+V
(2.21)
where
U
=internal
strainenergy
V
= potentialenergy
of appliedloads
= -WW
= workdone
by
externalloads.
n
=U
-W
(2.22)
If
we restrictthe
analysis
to
elastic systems subjectedby
conservative
forces,
we canapply
the
principle ofstationary
potentialenergy
which canbe developed from
the
principle of virtual work[16].
The
principle of
stationary
potentialenergy
statesthat:
"A
structureis
in
.static, eouilibriumif
and onlyif
the
total
potentialenergy
is
stationary with respectto
admissibledisplacements".
That
is,
an/aqj
=o,
i-1,2
n
(2.23)
Furthermore,
this
equilibrium stateis
stable,if
n
is
a relative minimumthere,
and unstable,if
n
is
a relative maximumthere.
For
n
to
be
a minimum.32n/3qj2>0
(2.24)
and
the
equilibrium stateis
stable.For
n to
be
a maximum.a2n/3qj2<0
(2.25)
and
the
equilibriumis
unstable.For
a condition of neutral equilibrium.32n/3qj2
=0.
(2.26)
For
continuous
systems(elastic
bodies),
the
principle ofstationary
potential
energy is
writtenin
variationalform
as:811
=0
or8(U
+V)
=0.
(2.27)
Likewise,
for
a condition of stableequilibrium,
82n
>0.
For
acondition of
instability,
82n
<0
andfor
neutralequilibrium,
82n
=0.
The
use ofthe
above principleto
establishthe
equilibrium conditions ofthe
system requires usto
resortto the
calculus of variations.Calculus
of variationsdeals
withthe
evaluation ofstationary
valuesor extremals
(maximum
orminimum)
offunctionals.
Functionals
aredefinite
integrals
having
functions
as arguments.The
functions
in
these
integrals
are unknowns andthe
calculus of variationsis
usedto
determine
the
conditions under whichthese
functionals
will assume astationary
value.
In
applying
the
calculus of variationsto
extremize afunctional,
one obtains conditions
that the
functions
mustsatisfy
to
ensurethat the
functional
will assume astationary
value.In
constrast,
in
applying
the
ordinary
calculusto
extremize afunction,
one obtainsthe
value ofthe
independent
variablefor
whichthe
function
will assume an extremumvalue.
U
andV
shouldbe
written asfunctionals
ofthe
deflection
(or
deformation)
andthen,
weapply
the
calculus of variationto
makeIT
stationary
with respectto
admissable variations[16].
2.5
Dynamic
Stability
ln
dynamic
stability, there
are changesin
the
kinetic energy
ofthe
system.
This
analysis
is
performed
when we areinterested
in
the
stability
ofthe
system orbeam
undertime-varying
force
or whennonconservative
forces
are present.The energy
transformations
aretime
dependent
andHamilton's
principle mustbe
fully
implemented
to
derive
the
equations of motion.The
solutionto
the
equations of motionis
accomplished
by
assuming
an admissible response which canbe
aneigenfunction
from
anNDOF
free
vibration analysis or another admissiblefunction that
properly
reflectsthe
geometric constraints ofthe beam.
The
response
is
then
substitutedinto
the
equations of motion and symplifiedinto
a system ofN
linear
homogeneous
equations whose nontrivialsolutions are
determined
by
setting
the
determinant
ofthe
coefficientmatrix
to
zero.The
resulting
equationis
calledthe
characteristicequation
for
the
system.The dynamic stability
criterionis
then
appliedto
this
equationto
determine
the
bifurcation
or criticalload.
The
dynamic
criterion requires
that
the
rootsto
the
characteristic equationhave
nopositive real
terms
andconsequently,
no unbounded responses.In
otherwords,
the
critical value ofthe
loading
parameterfx
is
its
smallestpositive value which
leads
to
multiple roots ofthe
characteristicequation with no positive real
terms.
Figure
11
illustrates
a plot ofthe
response of second ordersystems
depending
onthe
location
ofthe
roots ofthe
characteristicequations and
the
damping
factor[18].
Several
methodshave
been
appliedto
determine
the
existance of rootsin the
righthalf
ofthe
complex plane.A
root ofthe
characteristic equationmay
be defined
by
s = ajco,
where smagnitude of
the
imaginary
part.Figure
12
showsthe
rootlocus
of a second orderlinear
system as afunction
ofthe
damping
ratioC[18].
Notice
that
the
roots willbe
onthe
right-hand side ofthe
s-planefor
negative
damping
values.These
roots resultin
unbounded andthus
c 3 + c 3 . CM + + o
5
_V
*- t t \ I ?v "E.<2
E 2 T3 0) c co > OT ?-OT 0) OT h . 3 (5 TT C s s_ \ V ii-0> p?
ca <x> _= o o a) OT co o w2
o OTs
3?
ca to c8
_s
c 0) J ? .3 * K
[3]
The
Governing
Dynamical
System
3.1
Derivation
ofGoverning
Equations
The
equations of motiongoverning
the
loaded
cantileverbeam
described in
section1
are obtained viaHamilton's
principle:P
8(T
- V)*dt
+P
8Wnc*dt
=0
(3-1)
where
T=
total
kinetic energy
ofthe
systemV=
potentialenergy
ofthe
system,
including
the
strainenergy
and
the
potentialenergy
of conservative externalforces
8Wnc=
virtual workdone
by
nonconservativeforces,
including
damping
forces
and externalforces
not accountedfor
in V.
t1tt2
=times
at whichthe
configuration ofthe
systemis
known.
At
a point xalong
the
neutral axis ofthe
beam,
adisplacement
is
defined
by
orthogonalcomponents,
u(x,t),
in
the
axialdirection
andv(x,t),
in the
transverse
direction
(figure
9).
We
assumethat
the
beam
displacements,
u(x,t)
andv(x,t),
canbe
written as
the
sum ofdeflections
in
the
N
lowest
modes of a uniformcantilever elastic
beam.
That is:
N
u(x,t) =
SViW*Ui(t)
(3.2-a)
i=l
N
v(x,t) =
^(x)*vi(t)
(3.2-a)
This
procedureis known
asGalerkin's
method,
whichis
widely
usedto
approximate
continuous
systems
by
an equivalent systemhaving
afinite
number ofdegrees
offreedom[19].
In the Galerkin's
method,
u(x,t)
and
v(x,t)
are selectedin
such away
asto
satisfy
both
the
naturalboundary
conditions
andthe
geometricboundary
conditions.Another
method called
Raleigh-Ritz
selectsu(x,t)
andv(x,t)
that
needto
satisfy
only
the
geometricboundary
conditions[7,16].
The
total
kinetic energy
ofthe
systemis
givenby:
T
=0.5*p
"
[(Ut)2
+
(v,)2]*dx
(3.3)
where
p is
the
beam
density
per unitlength.
The
total
elastic or strainenergy
due
to
bending
and compressionis
given
by:
UT
=0.5*f
[E*I*(v"?]*dx
+0.5*f
[E*A*(u'+0.5*v2)2]*dx
(3.4)
Jo
JoThe
moment,
axial andtransverse
component ofthe load
are assumedto
be
concentrated atthe
free
end ofthe
beam.
The
potentialenergy
attained
by
these
loads
canbe
described
as:VT
= -W =fx*u(L,t)
+M*v'(L,t)
-fy*v(L,t)
(3.5)
where
W is
the
total
workdone
by
the
externalforces
and moments.The
total
potentialenergy
ofthe
beam,
V
orIT,
is
the
sum ofthe
strain
energy,
UT,
andthe
potentialenergy
ofthe
externalloads, VT,
that
is:
II
=V
=UT
+VT
=0.5*J
[E*I*(v")2]*dx
+0.5*I
[E*A*(u' +0.5*v'2)2]*dx
+
fx*u(L,t)
+M*v'(L,t)
-fy*v(L,t)
(3.6)
Applying
the
variational operatorto T
andV,
we get:ST=
p*[
[(ut)*(8Ut)
+(vt)*(8vt)]*dx
(3.7)
SV=
f
[E*I*(v")*6v")]x
+f
[E*A*(u'+
0.5*v;2)*8(u'+0.5*va)]*dx
Jo
Jo
+
f
*8u(L,t)
+M*8v'(M)
-fv*8v(L,t)
(3.8)
Integrating
8V
by
partstwice,
wehave:
8V
=E*I*(v")*(8v')fe
-E*I*(v'")*8v
fe
+I
[E*I*(v"")*8v]*dx
+E*A*(u'+
0.5*v'2)*8u
Iq
-I
[
E*A*(u" + 0.5*v'2)'*8u
]*dx
+E*A*(u'+
0.5*v,2)*v**8v
lo
-I
{
E*A*[(u' + 0.5*v,2)*v']'*8v
}*dx
+
fx*8u(L,t)
+M*Sv'(L,t)
Substituting
eq.(3.7)
and eq.(3.9)
into
eq.(3.1),
we get:r*
j
[(u0*(8ut
8(T-V)
={
p*|
[ (Ut)*(8ut)
+(vt)*(8v,)
]*dx
}*dt
(
{E*I*(v")*(8v')lfr
-E*I*(vm)*8v
fe
+[E*I*(v"")*8v]*dx
*+f
+E*A*(u' +
0.5*v,2)*8u
fe
-I
[
E*A*(u'+ 0.5*v'2)'*8u
]*dx
+E*A*(u' +
0.5*v'2)*v'*8v
Iq
-I
{
E*A*[(iT
+ 0.5*v,2)*v']'*8v
}*dx
+
fx*8u(L,t)
+M*8v'(L,t)
-f *8v(L,t)}*dt
=0
(3.10)
Notice
that the
first
term
in the
above expression canbe integrated
by
partsto
get:/"'2
j
8T*dt
=I
{p*l
[ut*8ut
+vt*8vj*dx}*dt=J
[
p*ut*8ult=t2
- p*ut*8ult=tl]*dx -I
I
r
I
[
p*vt*8vlt=t2
- p*vt*8vlt=tl]*dx-I I
p*utt*8u*dx*dt
p*vtt*8v*dx*dt
(3.11)
Substituting
eq.(3.11)
into
eq.(3.10),
wehave:
[p*ut*5ulW4-p*ut*8ult_tl]*d__
-I
p*utt*8u*dx*dt
r
+
J
[
p*vt*8vlt=t2
-p*vt*8vlt=t.]*dx-
I I
p*vtt*8v*dx*dt/
j*i{E*I*(v")*8v,l^
-E*I*(vm)*8v
\\s
+[E*I*(v"")*8v]*dx
^
Af
+E*A*(u' +
0.5*v,2)*8u
ifr
-I
[
E*A*(u* + 0.5*v*2)'*8u
]*dx
Jo
+E*A*(u' +
0.5*v'2)*v'*8v
lj,
-|
{
E*A*[(u'
+ 0.5*v'2)*v']'
*8v}*dx
+
fx*8u(L,t)
+M*8v'(L,t)
-fy*8v(L,t)}*dt
=0
(3.12)
Grouping
like
terms,
we get:j
[p*ut*8ult=t2-p*ut*8ult=tl]*dx+
I
[p*vt*8vlt=t2
- p*vt*8vlt=tl]*dx[f
{
p*utt
-E*A*(u'
+0.5*v*2)'
}*8u*dx*dt
ff
{
p*v
+ E*I*v""-E*A*[(u'
+
0.5*v*2)*v']'
-I
{
E*I*v"
+
M}*8v1x=L*dt
-J
E*I*v"*8v'lx=o*dt
Jn
Jti
J
|
{- E*I*vm+ E*A*(u' + 0.5*v,2)*v'
-fy
} *8vlx=L
*dt
-
P
{-E*I*v"'+ E*A*(u'+
O^v'Vv'}*8vlx=0
*dt
-
f
{E*A*(u'
+ 0.5*vVfx}*8ulx=L*dt-
P
{E*A*(u'+0.5*v'2)
}*8ulx=0*dt
=0(3.13)
A
Jn
For
the
above equationto
be
satisfied, the
following
conditions mustbe held:
p*ut*8u|t=t2
=0
(3.1
3-i)
p*ut*8u|t=t1
=0
(3.13-ii)
p*
v,*8v|t=t2
=0
(3.13-iii)
p*
vt*8v|t=t1
=0
(3.13-iv)
{
p*u
-E*A*(u'
+
0.5*v'2)'
}*8u
=0
(3.13-v)
{p*vtt
+ E*l*v""-E*A*[(u'
+
0.5*v'2)*v']'
}*8v
=0
(3.13-vi)
{E*l*v"
+
M}*8v'|x=L
=0
(3.13-vii)
E*l*v"*8v'|x=0
*0
(3.13-viii)
{- E*l*v"'
+ E*A*(u' + 0.5*v'2)*v'
-fy
}
*8v|x=L
=0
(3.1
3-ix)
{-
E*l*v"'+
E*A*(u'+ 0.5*v'2)*v'
}
*8v|x=0
=0
(3.13-x)
{
E*A*(u' +0.5*v'2)
+fx
}*8u
|X=L
=0
(3.13-xi)
E*A*(u'
+
0.5*v'2)*8u
|x=0=
0
(3.13-xii)
In using
Hamilton's
principle,
it
is
assumedthat the
configurationis
specified at
times
^
andt2.
For the
cantileverbeam,
this
meansthat
Sufx,^)
=8u(x,t2)
=8v(x,t1)
=8v(x,t2)
=0
(3.14)
which
takes
care ofthe
first
four
conditions above[13].Since
8u,
8v
are notarbitrarily
equalto
zero,
it follows
from
eq.(3.1
3-v)
and(3.13-vi)
that:
p*utt
-E*A*(u'
+
0.5*v'2)'
=
0
(3.15)
and
p*v
+ E*l*v""-E*A*[(u'
+
0.5*v'2)*vT
=0
(3.16)
which are
the
equations of motionfor
axial andtransverse
The
otherconditions
are called natural and geometricboundary
conditions
for
the
cantileverbeam.
At
the
fixed
end(x=0),
8u
and8v
arezero,
therefore
from
equations(3.13-xii,-x):
u(0,t)
=0
(3.17)
and
v(0,t)
=0.
(3.18)
At
the
free
end(x=L),
8u
and8v
are notnecessarily
zero,
hence
from
equations
(3.13-xi,-ix):
E*A*(u'
+
0.5*v'2)
+fx
=0
or
fx
=-E*A*(u'
+
0.5*v'2)
(3.19)
and
-E*_*v'"
+ E*A*(u' +
0.5*v'2)*V
-f
y =
0
(3.20)
Finally,
atthe
fixed
endthe
slope 8v*is
zero and atthe
free
end,
the
slope
is
notnecessarily
equalto
zero.Therefore from
equations(3.13-vii,
%
-Vlll)
v'(0,t)
=0
(3.21)
and
E*l*v"
+
M
=0
or v"= -M/(E*I) at x=L
(3.22)
We
may
linearize
the
equations of motionby
striking
outthe
nonlinear
term
0.5*v'2to
arrive at alinear
differential
equation withlongitudinal
motionis
generally
calledthe
primary
motion andit
is
calculated on a
linearized
basis
accordingly
to
the
usual procedurein
stability
theory.
A
solutionfor
the
primary
motionis
appliedto
the
transverse
equation of motion which willlead
to
an equationfor
v(x,t)
alone.
This resulting
transverse
motionis
calledthe secondary
motion andit is
representedby
alinear
differential
equation with either constantcoefficients or
time-dependent
coefficients[4].The
elimination of 0.5*v'2 willlead
to
the
following
set ofequations:
p*utt
-E*A*u"
=
0
(3.15a)
p*vtt
+ E*l*vM"-E*A*[u'*vT
=0
(3.16a)
or
p*vtt
+E*l*v""
-E*A*[u'*v"
+
u"*v']
=0
(3.16b)
u(0,t)
=0
(3.17)
v(0,t)
=0
(3.18)
fx
=-E*A*u'(L,t)
(3.19a)
-E*l*v'"
+
E*A*u'*V
-fy
=0
at x=L(3.20a)
or
E*l*v'"(L,t)+
fx*v'(L,t)
+fy
=0
(3.20b)
v'(0,t)
=0
(3.21)
E*l*v"
+
M
=0
orvM
= -M/(E*I) at x=L
(3.22)
The
above set of equations canbe divided
into
two
sets: one whichis
dependent
onu(x,t)
only
andthe
other set whichis
dependent
onboth u(x,t)
and v(x,t).
The first
setis
based
onthe
following
equations:p*utt
-E*A*u"
=
0
(3.15a)
u(0,t)
=0
(317)
fx
=Once
a solutionis found
for this set,
onemay
substitutethe
resultinto
the
second
setto
makeit
dependent
ononly
v(x,t).The
second setis based
onthe
following
equations:p*vtt
+ E*l*v,,M - E*A*[u*v"+
u"*v']
=0
(3.16b)
v(0,t)
=0
(3.18)
E*l*vM,(L,t)+
fx*v'(L,t)
+fy
=0
(3.20b)
v*(0,t)
=0
(3.21)
v"(L,t)
=-M/(E*I)
(3.22)
To simplify
the
problem,
wemay
assumethe
load
fx
to
be
eitherconstant or a
slowly varying
function
oftime
(figure
13)
suchthat:
*x
=fxo
+fxi*cos(co*t)
with constantsfx0,
fxi
(3.23)
and with
fx0fxi
(3.24)
and
that
co
fundamental
frequency
oflongitudinal
vibration[4].(3.25)
For
such acase,
onemay
considerfx
asapproximately
a constant and usethis
approximationto
solvethe
first
set of equations.The
resulting
solution
is
then:
u(x,t)
=-fx
* x/(E*A
)
(3.26)
since
it
satisfies allthree
equationsin
the
first
set.Notice
alsothat the
following
conditions arethen
true:
u'(x,t)
=-fx
/
(E*A)
(3.27)
I/)
a;
E
r-
4-0
v!)
and
u(x,t)
=0
(3.29)
We
can nowsubstitute
the
above resultsinto
the
second set of equationsto
uncoupleit from u(x,t)
variable.The resulting
equations willthen
be:
p*vtt
+ E*l*v"" + fx*v" =0
(3.30)
v(0,t)
=0
(3.18)
E*l*v'"(L,t)
+fx*v'(L,t)
+fy
=0
(3.20b)
V(0,t)
=0
(3.21)
v"(L,t)
=-M/(E*I)
(3.22)
This
set of equations are now uncoupledbut
equation(3.30)
is
notintegrable
in
its
final
form.
We
have
to
recourseto
Galerkin
variationalmethod[4,6,7]
to
solvefor
equation(3.30).
Assuming
that
the
beam
transverse deflection
canbe defined
as:v(x,t)
=?(x)*v(t),
(3.31)
where
<)>(x)
is
an admissiblefunction
whose shapeclosely
approximate
the
deformed
shape of a cantileveredbeam
and
v(t)
is
afunction
oftime.
The
admissiblefunction,
<|>(x),
couldbe defined
asany
one ofthe
eigenfunctions
for the
free
vibration of a cantileveredbeam
or alinear
combination of
them.
The
function
<)>(x)
can also representthe
mode shapesassociated with
the
beam
mode shapefrequency
co[6,7,16,20].
Levinson
[7]
showed a solutionto
a similar problemusing
Galerkin's
method and chose as a
trial
function:
<>(x)
=a^M
+a2M>2(x)
(3.32)
where
a^ a2
are constantsand
(^(x)
and<|>2(x)
arethe
first
two
eigenfunctions ofthe
free
vibration problem which are
derived
in
the
Appendix.
Another
set of admissiblefunctions
often used comesin
the
form
ofa polynomial whose shape
closely
resemblesthat
ofthe
deformed
shape ofthe beam.
As
an example of admissiblefunctions
for
a cantileverbeam,
wehave[1 6]:
4>(x)
=k*[
1
-cos(
0.5*tc*x/L)
],
k=constant
(3.33)
Substituting
equation(3.31)
into
the
set ofgoverning
equationsfor
the
transverse
motion,
we get:p*vtt*<j> +
E^v^"" +
fx*v*<}>"
=
0
(3.30a)
<}>(0)
=0
(3.18a)
E*l*v*<J>'"(L)+
fx*v*(|>'(L)
+fy
=0
(3.20c)
$'(0)
=0
(3.21a)
v*(J)"(L)
= -M/(E*I)(3.22a)
In
the
last
equation,
if
we wereto
neglectthe
effect ofthe
moment atthe
end of
the
beam,
then
we would arrive atthe
final form:
Since
the integral
overthe length
ofthe beam
of equation(3.30a)
is
not equal
to
zerodue
to
the
approximation usedfor
v(x,t),
we needto
apply Galerkin's
method.In
this
method,
we wouldmultiply
the
integrand
by
the
admissiblefunction <|>(x)
and requirethe
integrand
and<|>(x)
to be
orthogonal.
This resulting
integration
wouldthen
be
equalto
zero andits
computation would result
in
adifferential
equationin
time
only (or
a setof
differential
equationsin time
only
that
canbe
treated
as an eigenvalueproblem
to
be
solved).This
procedureis
illustrated below
by
the
Galerkin
equation:
f
[
p*vtt*<|)
+E*I*v*<t>""+
fx*v*(j)"]*(|)*dx
=0
(3.34)
If
we wereto
useLevinson's
admissiblefunction,
we would arrive attwo Galerkin
equations:f
[
p*vtt*<|>
+
E*I*v*<j>""+
fx*v*<J)"]*<|)i*dx
=0
(3.35)
and
f
I
p*vtt*(J) +E*I*v*(J)""+
fx*v*<T]*<l>2*dx
=(3-36)
which are solved
to
find
a set oftime differential
equationsthat
needto