mathematical description
TRANSCRIPT
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Mathematical description[edit] quation[edit]
tenaries for different values of a
ree different catenaries through the same two points, depending horizontal force being and λ mass per unit length.
e equation of a catenary in Cartesian coordinates has the form[27]
where cosh is the hyperbolic cosine function. All catenary curves are similar to each other, having eccentricity = √2.
Changing the parameter a is equivalent to a uniform scaling of the curve.[30]
The Whewell equation for the catenary is[27]
Differentiating gives
and eliminating gives the Cesàro equation[31]
The radius of curvature is then
which is the length of the line normal to the curve between it and the x -axis.[32]
Relation to other curves[edit]
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When a parabola is rolled along a straight line, the roulette curve traced by its focus is a
catenary.[33]
The envelope of the directrix of the parabola is also a catenary.[34]
The involute from the
vertex, that is the roulette formed traced by a point starting at the vertex when a line is rolled on a
catenary, is the tractrix.[33]
Another roulette, formed by rolling a line on a catenary, is another line. This implies that square
wheels can roll perfectly smoothly if the road has evenly spaced bumps in the shape of a series of
inverted catenary curves. The wheels can be any regular polygon except a triangle, but the catenary
must have parameters corresponding to the shape and dimensions of the wheels.[35]
Geometrical properties[edit]
Over any horizontal interval, the ratio of the area under the catenary to its length equals a, independent
of the interval selected. The catenary is the only plane curve other than a horizontal line with this
property. Also, the geometric centroid of the area under a stretch of catenary is the midpoint of the
perpendicular segment connecting the centroid of the curve itself and the x-axis.[36]
Science[edit]
A charge in a uniform electric field moves along a catenary (which tends to a parabola if the charge
velocity is much less than the speed of light c ).[37]
The surface of revolution with fixed radii at either end that has minimum surface area is a catenary
revolved about the x-axis.[33]
Analysis[edit] Model of chains and arches[edit]
In the mathematical model the chain (or cord, cable, rope, string, etc.) is idealized by assuming that it is
so thin that it can be regarded as a curve and that it is so flexible any force of tension exerted by the
chain is parallel to the chain.[38] The analysis of the curve for an optimal arch is similar except that the
forces of tension become forces of compressionand everything is inverted.[39]
An underlying principle is
that the chain may be considered a rigid body once it has attained equilibrium.[40]
Equations which define
the shape of the curve and the tension of the chain at each point may be derived by a careful inspection
of the various forces acting on a segment using the fact that these forces must be in balance if the chain
is in static equilibrium.
Let the path followed by the chain be given parametrically by r = ( x , y ) = ( x (s), y (s))
where s represents arc length and r is the position vector . This is the natural parameterizationand has
the property that
where u is a unit tangent vector .
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Diagram of forces acting on a segment of a catenary from c to r . The forces are the tension T0 at c, the
tension T at r , and the weight of the chain (0, −λgs). Since the chain is at rest the sum of these forces must be
zero.
A differential equation for the curve may be derived as follows.[41]
Let c be the lowest point on the
chain, called the vertex of the catenary,[42]
and measure the parameter s from c. Assume r is to the
right of c since the other case is implied by symmetry. The forces acting on the section of the chain
from c to r are the tension of the chain at c, the tension of the chain at r , and the weight of the chain.
The tension at c is tangent to the curve at c and is therefore horizontal, and it pulls the section to theleft so it may be written (−T 0, 0) where T 0 is the magnitude of the force. The tension at r is parallel to
the curve at r and pulls the section to the right, so it may be written T u=(T cos φ, T sin φ), where T is
the magnitude of the force and φ is the angle between the curve at r and the x -axis (see tangential
angle). Finally, the weight of the chain is represented by (0, −λgs) where λ is the mass per unit
length, g is the acceleration of gravity and s is the length of chain between c and r .
The chain is in equilibrium so the sum of three forces is 0, therefore
and
and dividing these gives
It is convenient to write
which is the length of chain whose weight is equal in magnitude to the tension
at c.[43]
Then
is an equation defining the curve.
The horizontal component of the tension, T cos φ = T 0 is constant and the
vertical component of the tension, T sin φ = λgs is proportional to the length of
chain between the r and the vertex.[44]
Derivation of equations for the curve[edit]
The differential equation given above can be solved to produce equations for
the curve.[45]
From
the formula for arc length gives
Then
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and
The second of these equations can be integrated to give
and by shifting the position of the x -axis, β can be taken to
be 0. Then
The x -axis thus chosen is called the directrix of the
catenary.
It follows that the magnitude of the tension at a
point T = λgy which is proportional to the distancebetween the point and the directrix.
[44]
The integral of expression for dx /ds can be found
using standard techniques giving[46]
and, again, by shifting the position of the y -axis, α
can be taken to be 0. Then
The y -axis thus chosen passes though the
vertex and is called the axis of the catenary.
These results can be used to eliminate s giving
Alternative derivation[edit]
The differential equation can be solved
using a different approach.[47]
From
it follows that
and
Integrating gives,
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and
As before,
the x and y -axes can
be shifted so α and β
can be taken to be 0.
Then
and taking the
reciprocal of both
sides
Adding and
subtracting
the last two
equations
then gives the
solution
and
Determiningpar ameter s[edit]
In
gener al the
para
meter
a an
d the
positi
on of
the
axisand
direct
rix
are
not
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given
but
must
be
deter
mine
d
from
other
infor
matio
n.
Typic
ally,
the
infor
matio
n
given
is
that
the
caten
ary is
susp
ende
d at
given
point
s P 1
and
P 2 an
d
with
given
lengt
h s.
The
equat
ion
can
be
deter
mined in
this
case
as
follow
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s:[48]
Rela
bel if
nece
ssary
so
that
P 1 is
to the
left
of P 2
and
let h
be
the
horiz
ontal
and v
be
the
vertic
al
dista
nce
from
P 1 to
P 2. Tr
ansla
te the
axes
so
that
the
verte
x of
the
caten
ary
lies
on
the y-
axis
and
itsheigh
t a is
adjus
ted
so
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the
caten
ary
satisfi
es
the
stand
ard
equat
ion of
the
curve
a
n
d
le
t
t
h
e
c
o
o
r
d
i
n
a
t
e
s
o
f
P
1
a
n
d
P
2
b
e
(
x
1,
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y
1)
a
n
d
(
x
2,
y
2)
r
e
s
p
e
c
ti
v
e
ly
.
T
h
e
c
u
r
v
e
p
a
s
s
e
s
t
h
r
o
u
g
ht
h
e
s
e
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p
o
i
n
t
s
,
s
o
t
h
e
d
if
f
e
r
e
n
c
e
o
f
h
e
i
g
h
t
is
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