-
Jounal of matenals processing technology 63 (1997) 881-886
Experiment and study into the axial drifting of the cylinder
of a welding rollerbed
Fenggang shen ,xide pan ,jin xue
Welding
research
instiute
,xi`an
jiaotong
university.
Xi`an
.shaanxi
province
Abstract
The
basic
theory
of
the
axial
drifting
of
the
cylinder
of
a
welding
roller
bed
is
introduced in the paper,and at the same time experiment and study on the mechanism
of the axial drifting of the cylinder have been done on an experimental model of the
welding roller bed . It is shown that the main cause of the axial drifting of the cylinder
lies in the existence of a spiral angle between the cylinder and the cylinder and the
roller . the relative axial motions between the roller and the cylinder are compose of
spiral
motion,elastic sliding and frictional sliding.
The theory
of
compatible motion
and non-compatible motion is put forward for the axial motions of the cylinder .the
relative axial motions of the cylinder . The relative axial motion between the rollers
and the cylinder is coordinated by elastic sliding and frictional sliding between them
Keywords: welding roller bed; cylinder roller axial motion spiral angle
1
Introduction
In welding production, the assembly and circular seam welding of rotary workpieces,
such as a
boiler, a
petrochemical pressure vessel and
so on, are conducted on a
welding
roller
bed.
When
rotating
On
a
welding
roller
bed.
The
cylinde
will
inevitably
produce
axial
drifting
due
to
manufacturing,
assembling
tolerance
of
the
welding
roller
bed
and
the
cylinder’s
surface
irr
egularity
(divergiug
froman
ideal
rotary workpiece),
thus the welding procedure may not be carried out successfully.
It is necessary, therefore, to study the mechanism of the axial
drifting of the cylinder
to solve the problem of the axial drifting of the cylinder in circumferential welding.
The
results
of
the
research
will
benefit
the
studying
and
designing
of
antidrifting
welding roller
bed. especially the analysis of the applied forces on the bed, and lead
to
determining
the
manufacturing
and
assembling
tolerance
of
the
bed,
and
providing
the
basis
of
theory
for
the
mechanical
adjusting
mode
to
avoid
axial
drifting,
the
adjusting
mode
of
closed
circuit
in
the
control
circuit,
and
the
selection of
the
adjusting
value.
2.
Theoretical analysis
2.1. Welding roller bed and cylinder
A
welding
roller
bed
is
generally
composed
of
four
rollers.
Driven
by
the
driving
roller,
the
cylinder
makes
a
rotary
uniform
motion
around
its
axis(shown
in
Fig.
I),
during
which
the
circumferential
welding
procedure
is
carried
out
In
Fig.1, a is the central angle, S is the supporting distance, L is the span of the
roller.
and V
, is the circular linear velocity of the cylinder, also named the welding velocity2.
The axis of the cylinder will be not parallel to that of a roller if the roller is deflected
by a certain
angle from the deal position, or if
the centers of the four rollers lie in
the vertices of a simple
quadrilateral, or if the centers of the four rollers are not on
the same plane, or if the circu- larity
of the cylinder is irregular because of deviation
in manufacturing and assembling. Thus. the
cylinder will nevitably move along its
axis
when
rotating
on
a
bed
the
contact
of
the
cylinder
and
a
roller
can
be
cansidered as point contact if cytinder’s axis and roller’s axis do not lie in the
same
plane. Suppose P is the point of contact. the cylinder’s normal plane A is def
ined by
the
plane
on
which
are
the
cylinder’s
axis
and
generatrix
n
across
the
point
of
tangency on the
cylinder (shown in
fig2) makea cylinder’s tangent plane B across
point P.
Thus, plane A is
vertical to plane B. lc is a cylinder’s tangent across P
and
lies in plane
B. Ir is the roller’s
tangent across the same point P, and lies in plane B
also. In
general, θ is
defined
as
the
axial
deviation
angle
between
the
rol!er’saxis
and
the
cylinder’s
axis; β is defined as the spiral
angle between generat
rix n and m’ . a projective line
obtained
by
projecting
the
roller’s
generatrix
m
across
point
p
on
plane
B
and
γ
is
defined
as
the
projective
angle
between
n
and
m
,
a
projective
line
obtained
byprojecting m on plane A. Fig. 3 indicates that the rela- tionship amongst the three
angles is tanβ = tan2θ
-
tan 2γ
In
Fig.
3,
SB,
Sθ
and
Sγ,
are
called
the
spiral
displace
-ment
vector,
the
axial
deviation
displacement
vector
and
the
projective
displacement
vector
respectively.
their
relationship being:
Fig. 2 Geometric relationship between the cylinder and an
individual roller
Fig. 3 Relationship between the angle vector and the
displacement vector
2.2.2 relative axial motions relationship
(1)spiral motion
2.
Fig. 4 Component of axial velocity
Because the roller’s axis is not parallel to the cylin
-
der’s central line, there is a
spiral angle
β
between Vr,. and Vc, on the point of contact (shown
in
Fig.
2).
When the roller and cylinder rotate synchronistically around their own axes,
driven
by tangential
frictional
force.
a spiral
effect
will occur because
of
the
different
linear velocity direction between the roller and the cylinder at point P of contact The
cylinder has a component of axial velocity,
where
Vc
is
the
circular
linear
velocity
of
the
cylinder.
is
the
cylinder’s
axial
component velociry exerted by single roller, and j can be 1. 2, 3, 4, representing the
four rollers, respectively.
(2) Elastic sliding
Because of the existence of a spiral angle, an axial force Faj acts on cylinder.
When
the
force
is
less
than
the
maximum
axial
frictional
force
fNj
(where
f
is
the
frictionfactor, and Nj is the normal pressure between a single roller and the
cylinder),
the
cylinder
will
slide
elastically
over
the
roller
along
the
axial
direction[2~3]
The component of the sliding velocity is.
where e is the elastic sliding factor for metallic roller. e=
~~
0.005.
(3) Frictional sliding
When Faj is greater than the maximum frictional force fNj, the cylinder will make a
frictional sliding over the roller. The sliding resistance is
fNj[3]. The component of
the
frictional
sliding
velocity
on
Cylinder
is
Vaj
the
magnitude
and
direction
of
which
can
be
determined
by
the
universal
relationship
between
the
cylinder
and
the four rollers Frictional sliding will lead to the wear and tear of the surface of
the
cylinder
and
the
rollers.
which
is
unexpected
in
welding
production
When
the
cylinder drifts, above three kinds of motion
do not occur
simultaneously ‘I’hereforc. the axial drifting velocity o
f the cylinder is
not
the
algebraic
sum
of
the
three
components
of
velocity
In
the
case
of
elastic
sliding, the axial velocity is.
2.3 axial motion of the cylinder on a welding roller bed
2.3.1 Axial
compatible motion
Under
ideal
conditions,
when
spiral
angles
β
j
between
the
cylinder
and
the
four
rollers are all the same, that is:
β
1=
β
2=
β
3=
β
4=
β
the
cylinder
will
move
its
compatible
spiral
motion.
Two
categories
can
be
cl
assified
to
analyze
the
axial
motion
of
the
cylinder:
(I)
When
there
does
not
exist
an
axial
component
due
to
gravity.
the
cylinde
r’s
axial
drifting
velocity
is:
Va=Vc * tan
β
(2)
When there exists an axial component
of gravity Ga there exists an axial force
on the
cylinder.
Now, the axial forces
exerted on the four
rollers
have
the
same directional
And magnitude, the value being equal to Ga besues the component of spiral vetocity,
there exist component of elastic on the cylinder the cylinder`s axial drifting velocity is
non-compatible motion
In general. spiral angles
β
j between the cylinder and the four rollers are not equal
to
each
other
in
size
and
direction.
i
.e.
the
geometric
relationships
between
the
cylinder and the four rollers
are all inconsistent Therefore, the components of
the
cylinder’s axial velocity
against four rollers (i.e Vc *ta
β
j) are not identical to each
another. The cylinder will move with axial nompatible motion The axial
velocities
of the cylinder
againsa
the
four
rollers
should
be
the
same
because
the
cylinder
is
considered
as
a
rigid
body
as
a
whole and it has only one axial velocity.
However. for some roller, Vc . tan
β
j and
the
cylinder’s real axial velocity are
not likely to be the same, so an
axial frictional
force
almost
certainly
appears
between
this
roller
and
the
cylinder
The
following
two
categories
can
be
classified
to
discuss
the
non-compatible
axial
motion of the cylinder according to Ihe frictional force’s magnitude:
(I)
When
the
axial
frictional
forces
erected
by
each
roller
and
the
cylinder
are
all
less
than
the
maximum
axial
the
action
of
the
cylinder
against
the
frictional force
the
action of the cylinder against the
rollers produces elastic
sliding The axial
motion betweenan individual roller and the
cylinder
is
coordinated by their elastic sliding
when
the
axial
velocity
of
the
cylinder
is
constant,
the
algebraic
sum
of
cylinder’s
axial
forces
erected
by
four
rollers
should
be
zero
if
rhe
axial
component
of
gravity is ignored. i.e.
and there is little difference amongst Nj, against the four rollers, so that they can
be
approximately regarded as the same. Thus:
according to the above two equations, the axial drifting velocity of the cylinder is.
Where
0.25∑Tanβt
represents
the
intrinsic
attributes
of
the
welding.
Other
bed
under the condition that only the cylinder against all rolls produces elastic sliding this
may be called the spiral rate of the cylinder`s spiral motion
(2) When the axial frictional force erected by some roller and the cylinder is
greater
than the maximum axial frictional force, frictional sliding occurs between the cylinder
and this roller Then. the maximum axial force is acting on the
bearing of the roller,
its value being
Ffmax=fFNfmax
Because
of
the
esistence
of
this
frictional
sliding.
the
Axial
motion
between
an
individual roller and the cylinder is
not
coordinated by
their
elastic
sliding
Now
the axial
non-compatible motion of the cylinder is determined by
the
relative
relationships
between
the
cylinder
and
the
four
rollers.
It
is
difficult
to
write a general compatible equation of the cylinder’s axial drifting velocity
because
this
kind
of
condition
is
very
complex.
The
following
is
further
analysis
and
discussion
of
the
problem
At
first,
for
ease
in
analyzing
problem,
the
spiral
angle average is defined as
and the relative spiral angle as
Arrange
,β1
in
the
order
from
big
to
smll
and
then
from
posirive
to
negative,
expressed as β(j). then
β1≥β2≥β3≥β4
Similarly,
the
normal
force
between
the
cylinder
and
a
roller
can
be
expressed
as
N(j). and the axial force as
Fj≤fNj
In general, the axial motion of the cylinder determined by the spiral angle average β is
definel as the compatible component of the axial motion,
is velocity
being
The
axial
motion
of
the
cylinder
determined
by
the
relative
spiral
angle
βj
is
defined
as
the
non-compatible
component
of
axial
motion,
its
velocity
being
expressed
asVan
Analysis
shows
that
Va``
is
determined
by
the
equilibrium
condition
the
four
roller
axial
forces
when
the
cylinder
moves
along
axial
direction at
a constant
velocity. where not
taking
into account
of
the function of
gravity’s
axial
component.
Supposing
that
the
cylinder
makes
a
non
-compatible
component
of
ax
ial
motion
with
the
maximum
relative
spiral
angle
β(I).
its
velocity is
Then the four axial forces can not be in equlibrium .i.e
F1-
(F2+F3+F4) ≤0
Because
there
is
little
difference
amongst
four
normal
forces,
the
four
axial
farces
are
also
determined
by
normal
force
and
the
friction
factor
any
axial
force
undoubtedly
being
less
than
the
sum
of
the
other
three
forces.
Otherwise,
if
the
cylinder
makes
a
non-compatible
component
of
axial
motion
with
the
minimum
relative spiral
angle
β(4).
its
velocity
is.
Va” = Vc * tanβ(4)
Similarly. four axial forces can not be in equilibrium also, i.e. :
[F(l)
+ F(2)
+ F(3) J
- F(4)
> 0
Therefore,
the
cylinder
can
only
be
approximately
considered
as
making
a
non-compatible component of axial motion with the second or third
relative
spiral
angle,
i.e.:
In
whatever
case
as
expressed
above.
when
the
cylinder
make
a
non- compatible
component of axial
motion, thetwo rollers having a greater velocity
are
driving
rollers,and
the
other
two
rollers
having
a
lesser
velocity
are
resistant
rollers,
the
equilibrium
condition of axial forces being operative, i.e.:
F(1) + F(2) = F(3) + F(4)
According
to
the
analysis
above,
and
because
of
the
unstability
of
friction
factor
f
that is affected by the factors of load, material, condition of the contact surface, and
circumstance, the non-compatible component Va of the axial velocity of the
cylinder
is undefined. When the cylinder
makes a non-compatible axial motion,
its
axial
velocity is composed of a compatible
component Va0 and a non-compatible component Van
i.e
Va=Va0+Van
Va=Va0+Van
The
most
optimal
adjustment
of
the
axial
motion
is
to
make
the
non- compatible
component as small as possible according to the stability of adjustment and decrease
in axial force. No matter whether the cylinder
makes compatible or non- compatible
motion, supposing that the cylinder
is ideal, its axial velocity is
always
existent
and definable for a particular
bed, its magnitude and direction refle
cting the bed’s
inherent property.
3.
Experiment
3.1.
Descriphm of experment
The
experimental
model
is
shown
in
Fig
5.
Experiments
were
done
to
study
two
factors: the spiral angle and the cylinder’s
circular linear velocity, which affect the
axial drifting of the cylinder.
In the experimenting process. the axial
displacement
Sa
and
the
axial
drifting
velocity
Va
of
the
cylinder
were
measured
by
the
variation of the two
factors
described
above.
The
measuring
method
is
shown
in
Fig.
5,
and
is
carried
out by means of bringing an axial displacement sensor into contact with one end of
the
cylinder.
with
the
sensor
being
connected
to
an
X-Y
recorder
to
record
the
cylinder’s
axial
displacement
every
5s.
Linearly
regressing
the
plot
Sa--t
(t
expresses
time),
the
average
drifting
velocity
Va,
at
every
deflecting
angle
can
be calculated.
Before
experimenting.
the
experimental
model
is
initialised
as
follows:
first.
the
height
of
the
four
rollers
is
adusted
by
means
of
a
level
to
put
the
centers
of
the four
rollers in the same horizontal plane, and at the four
vertexes of the rectangle.
then
the
rollers
are
deflected
so
that
the
rotating
cylinder
is
at
the
relative
equilibrium
position. Then the cylinder does not drift over a long time. or periodically drift
over
a very small axial range
3.2
experiment results and discussion
3.2.1 Effect of spiral angle (I) Fig. 6 shows that change of Va with the variation ofThe
testing condition is: positive rotalion, Vc=35m/h
L=422mm,
α
=60”
The
Va-tan
β
4
curve
shows
that
Va
is
directly
proportional
to
tan
β
4
when
β
4
is
relatively
small
(1~~~6c
).
The
slope
of
the
line
being
3.
06
mm/s,
Va
is
no
longer
direclly
proportional
to
tan
β
4
when
β
4,
is
greater
than
6C
The
curve
is
an
arched
curve.
i.
e .
with
the
increment
of
β
4,.Va,
increases.
but
with
the
increment of Va gradually becoming smallet Because only one driven
roller (roller
No.
4)
is
deflected,
i.e
β
4
can
be
changed
whilst
the
others
remain
zero,
the
cylinder
makes
a
non-compatible
motion.
When
β
4
is
relatively
small,
Va
is
small
also.
The
axial
frictional
forces
between
the
cylinder
and
rollers
are
less
than
the
maximum
axial
frictional
force,
and
the
cylinder
produces
an
elastic
sliding
against
rollers.
Axial
motion
between
each
roller
and
the
cylinder
is
coordinated by elastic sliding. thus Va is:
in the theoretical curve, the
slope K’ can be calculated by
the following equation:
K=3.06mm/s in the experimental curve. Thus, in taking account of the
experimental
tolerance, the two slopes can be considered to be approximately equal.
When
β
4
is relatively large, the axial frictional forces between the cylinder and the rollers are
larger
than
the
maximum
axial
frictional
Force,
and
cylinder
produces
frictional
sliding against the
rollers Because of
Ihe
existence of sliding
frictional
resistance.
Va
is
no
longer
lincarty
increased
with
the
increment
of
tan
β
4
With
the
increment of tan
β
4 the increment of V a; with gradually become smaller
(2)
The
following
three
experiments
were
arranged
to
study
t
he
cylinder’s
non- compatible
axial
motion
further,
deflecting
positively
one
roller.
two
rollers
and three rollers by the same spiral angle to measure three curves
between Sa and
v The experimental results are shown in Fig 7. With
the increment
in the
number
of deflected rollers, Va becomes greater. i e Va 3 > Va 2 > Va1
When
the
number
of
driven
rollers
deflected
is
varied,
the
degree
of
the
cylinder’s
non
-compatible
axial
motion
will
be
changed.
With
the
increment
of
the
number
of
lollers
deflected
by
the
same
spiral
angle.
the
compatible
component
becomes
greater,
but
the
non-compatible
component
becomes
smaller.
In
other
words,
the
cylinder’s
axial
motion
will
be
transformed
from
noncompatible
motio
n
to
compatible
motion.
Thus,
Va
becomes
greater
also,
ultimately,
being
equal
to
the
compatible
axial
velocity
determined
by
the
spiral
angle
β
Now.
the
four
rollers have the same spiral anyle
β
. So that Va is:
3.2.2 effect of circular linear velocity
Deflecting driven roller No 4 to a spiral angle of +2”from the equilibrium position,
the
cylinder
will
suffer
axial
drifting,
Fig.
8
shows
the
Va-Vc
curve,
which
latter
indicates
that
Va
is
directly
proportional
to
Vc,
the
slope
of
the
curve
being
approximately
0.00708
because
β
4=+2
is
too
small,
the
cylinder
does
not
make
frictional
sliding
against
each
roller.
Thus,
the
relative
axial
motion
between
the
roller and the cylinder is completely coordinated by their elastic sliding, so that
Va
is
I.
e .Va is directly proportional to Ve For the theoretical Curve the slope
K *
c
an
be
calculated
by
the
following
equation
K”=
0.25tan
β
4=
0.25tan2'=0.00873
where
K=0.00708mm/s
in
the
experimental
curve.
Thus, in taking account of the experimental tolerance, the two slopes can
be
considered to be approximately equal.
4
Conclusions
1.
Because
of
the
deviat
ions
due
to
manufacturing
and
assembling.
the
cylinder’s
central
line
and
the
roller’s
a
xis
are
not
parallel.
i.
e
,
they
are
not
in
the
same
plane,
and
there
is
a
spiral
angle
β
at
thc
point
of
contact
between
the
cylinder
and
the
roller
in
the
circular
linear
velocity
direction.
The
existence
of
β
is
the
basic
reason
for
the
occurrence
of
axial
drifting.
The
effect
of
gravity
in
cylinder’s axial direction is also one of reasons
for drifting.
2.
The
relative
axial
motions
between
an
individual
roller
and
the
cylinder
are
composed
of
spiral
motion.
elastic
sliding
and
frictional
sliding
When
axial
frictional sliding does not occur between the cylinder and a single
roller,
the
relative
axial
motion
between
the
rollers
and
the
cylinder
is
completely
coordinated by their elastic sliding, Va is directly proportional to
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