The derivation of the catenary can be split up in two parts. The first
part is the general derivation of the symmetric catenary cable
and the second part is more complicated derivation of the asymmetric catenary cable. In
order to understand the derivation of the
asymmetric catenary it is best to first try to understand to derivation
of the equation for the symmetric catenary. On this page the
general derivation of the symmetric catenary cable is explained.
With the equation of the symmetric catenary cable not only a totally
symmetric catenary cable can be described, but also a
catenary cable that is half of a symmetric catenary cable. For instance
a mooring line hanging down from a vessel onto the sea
floor and going into the anchor point on the sea floor horizontally.
Symmetric Catenary Cable
The following relations can be derived when using the trigonometric
relations and Pythagorean theorem;
T0 =
T cos φ (1)
μgs = T sin φ (2)
(3)
T2 = (μgs)2 + (To)2 (4)
(5)
The relation between the T, s and the constant a can be shown by
substituting the value of the constant a (5) in equation (4);
T2 = (μg)2 (s2
+ a)2 è (5+4) (6)
è (5+3) (7)
(8)
(9)
The purpose of this equation is to show the relations of the length of
the catenary and the vertical distance from the ancker to the suspension point
with the horizontal distance between the ancker and the suspension point,
meaning s and y as a function of x;
The relations between dimensions of the catenary particle can be derived
when using the trigonometric relations and Pythagorean theorem;
(10)
Equation (10) can be derived with the standard integral
So, the integral of (10) is;
(11)
Y’ can be isolated by aplying the following method;
(12)
By integrating y’ one can find y;
(13)
Therefore, by taking equation (8), one can find;
(14)
By taking the square of the equations (13) and (14) and subtracting then
from each other one can find the direct relation between y and s, with the
standard relation (cosh2 - sinh2 = 1).
y2 = s2
+ a2 (15)
The purpose of this equation is to show the relations of the length of
the catenary and the vertical distance from the ancker to the suspension point
with the angle between the ancker and the suspension point, meaning s and y as
a function of φ.
This can be achieved by substituting (8) into (15);
y2 = a2 tan2 φ + a2
(16)
To find the horizontal distance between theacker and suspention point as
a function of φ, one can
substitute (8) into (11).
(11)
(8)
x = a ln(tan φ + sec φ) (17)
After the
explanation of the symmetric catenary cable the explanation of the more
complicated asymmetric catenary cable can be
studied. With the
explanation of the asymmetric cable the equation of a cable hanging on two end
points at a different vertical
level can be
understood.
Besides the
general explanation of the catenary cable this site also contains “did you know
that?”s about catenaries. For the interested
readers who want
to know just a bit more than average, click this link: Did you know that???