In physics and geometry, a catenary (US: , UK: ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends.
The catenary curve has a U-like shape, superficially similar in appearance to a parabolic arch, but it is not a parabola.
The curve appears in the design of certain types of arches and as a cross section of the catenoid—the shape assumed by a soap film bounded by two parallel circular rings.
The catenary is also called the alysoid, chainette, or, particularly in the materials sciences, funicular. Rope statics describes catenaries in a classic statics problem involving a hanging rope.Mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the catenary curve, the catenoid, is a minimal surface, specifically a minimal surface of revolution. A hanging chain will assume a shape of least potential energy which is a catenary. Galileo Galilei in 1638 discussed the catenary in the book Two New Sciences recognizing that it was different from a parabola. The mathematical properties of the catenary curve were studied by Robert Hooke in the 1670s, and its equation was derived by Leibniz, Huygens and Johann Bernoulli in 1691.
Catenaries and related curves are used in architecture and engineering (e.g., in the design of bridges and arches so that forces do not result in bending moments). In the offshore oil and gas industry, "catenary" refers to a steel catenary riser, a pipeline suspended between a production platform and the seabed that adopts an approximate catenary shape. In the rail industry it refers to the overhead wiring that transfers power to trains. (This often supports a lighter contact wire, in which case it does not follow a true catenary curve.)
In optics and electromagnetics, the hyperbolic cosine and sine functions are basic solutions to Maxwell's equations. The symmetric modes consisting of two evanescent waves would form a catenary shape.
Important note: I only derived the differential equation, I did not solve it.
What I think caused the mistake:
- the tangent approximation (tan(theta+dtheta) ~ tan theta + d theta
Hi all,
I am no structural engineer and I have toiled extensively over this for far too long and at this point I am wondering if this problem has a solution with the given variables...
It is my understanding that the standard equation for a Catenary is: ##y = a \cdot cosh(\frac {x}{a})##...
If you are on site and you only can get the weight per unit length of cable, the distance between 2 points and the difference in elevation of the said 2 points. How can I solve for the location along the cable for the lowest sag and the tensions exerted at the two points?
Hi,
I would like to ask about the difference of sag elastic and non-elastic catenary.
When these catenaries (their mathematic model) have a same horizontal tension, it is bigger sag elastic or non-elastic catenary?
I have my own calculations and I want to verify them.
Thanks
Joseph
Homework Statement
Suppose we have a rope of length L and total mass M. Suppose we x its ends at points
(xA; yA) and (xB; yB). We want to determine the shape the rope makes, hanging under the
influence of gravity. The rope is motionless, with a shape parametrised by y(x) or equivalently,
x(y)...
Minimal surfaces are sort of the "shortest path" but in terms of surface shapes.
So I figured I could characterize the shape of a hammock by adding the influence of gravity, much like you can get the shape of a catenary cable (y=cosh(x)).
The equation of motion I get from the Lagrangian is...
Hi, I have a problem that I'm trying to solve and I'm not sure at all how can I proceed.
If we have a metal rectangle (or maybe a rectangle-shaped hole in a piece metal), and we stretch this body a small distance, what is the shape of the top and bottom lines of the body? Can we say it is a...
I'm basically trying to understand the 2-D case of the catenary cable problem. The 1-D case is pretty straightforward, you have a functional of the shape of a cable with a constraint for length and gravity, and you get the explicit function of the shape of a cable.
But if you imagine a square...
Hi guys, first time poster so play nice :)
Outline of problem:
The situation I have is that there are 4 supporting columns. These columns support support a wire and this wire is then used to hang wires which support festive lights. (see diagram below)
What I'm looking for as an end point is...
In general, when dealing with mechanics problems using a function ##f(q1,q2,...)=0## that represent constraints one is minimizing the action ##S## while adding a term to the Lagrangian of the not-independent coordinates ##L + \lambda f ##. One can show that this addition doesn't change the...
I am trying to determine the size of a conductive 2-D sheet that has a specified degree of increased resistance (or reduced conductivity) compared to an infinite sheet.
Imagine that electrons enter the infinite sheet and exit the sheet at 2 points which are 1 unit of distance apart and aligned...
Homework Statement
I have attached a scanned file of the problem outline, including all known data. For the initial curve, H, (horizontal force) is constant, while for the final curve, H', is a combination of the initial H and the stress added by the point load, P.
I run into issues trying to...
Homework Statement
A uniform chain hangs in the shape of the catenary y=cosh(x) between x=−1 and x=1
Find \bar{y}
Homework Equations
∫\bar{y}\rhods=∫y\rhodsThe Attempt at a Solution
I can define ds (some small segment of the arc) by \sqrt{1+sinh(x)^2}dx.
Also, y is given as y=cosh(x)
If I...
Hello,
I'm designing a towed underwater vehicle (towfish) and I am having trouble with determining cable tension at the length of the cable at given depths and speeds.
I have estimates of vehicle drag at speeds up to 4 m/s and the vehicle weight in water is known. Its really just the...
Hi, my problem is this; I am designing a cable camera system to film downhill pursuits. I need to calculate how much tension is required to hang a cable over a 100m span with no more that 0.5m sag in the middle, when is it fixed at two point, both at the same hight. From the research I have done...
Hi all, this question stems from a homework question but is not the homework question itself, more a discussion on something I found, hence why I have put it here.
The question involved using variational calculus to minimise the surface area of a soap bubble to find the shape it would take. The...
Homework Statement
A homogeneous catenary ##z=acosh(x/a)##, ##y=0## and ##x\in \left [ -a,a \right ]## is given. Calculate the center of mass and moment of inertia
Homework Equations
The Attempt at a Solution
I started with ##x=at##, for##t\in \left [ -1,1 \right ]##, therefore...
Hello all,
I have yet another mathematical quandary that is robbing me sleep, so I return to gather ideas.
This problem (fuelled from self-study) involves the equation of a catenary.
The profile of a catenary can be expressed as y(x) = T/w*(cosh(w/T*x) – 1). My inputs into this equation...
A cable weighing 10kg/m supports a 250kg beam(AC) hinged at the wall(BC). The beam is 4.5m long.
Find the length of cable AB.
BC is the wall. AC is the beam. Distance AC is 4.5m. Cable runs from A to B and sags forming a catenary. The angle of the cable at A is 30 degrees from the horizontal...
Homework Statement
A cable is spanned between two points at the left touching the ground and at the right smoothly spanned over a large roller. Assume that there's no friction present. The tension left at the bottom is H and the tension T at the top is such that T>H. The cable has a weight per...
I am a bit confused on one part of the derivation of the catenary equation. At one point my book says ds2 = dx2 + dy2 and thus \frac{ds}{dx}=\sqrt{1 + {y'}^2} .
however that doesn't seem very rigorous to me and i am a little wary of accepting that explanation. i know that s = \sqrt{{x'}^2...
Hi all!
I have an odd question that i thought somebody with a degree could help with. I am looking to find the formula to describe wire sag over a given length and given horizontal tension and given wire diameter. I believe the formula is a catenary but am unsure how to apply it to my...
Homework Statement
when a cable with non-zero mass is connected to a pole at both ends, the shape it assumes is called a catenary.
it can be shown that for an electrical wire whose linear mass density is .9 kg/m strung between poles 30m apart(and making a 22 degree angle at each end) the...
hi I'm an applied maths student i'd like to study about catenary.. where can i get details of catenary and its applications in detail?which book i can refer or is there any website i can go for?
situation: boat moored at the dock
fact (so to say) 1 - why the longer the rope and his catenary shape the stronger it will become? assuming it is true.
fact 2 - why a man can keep a heavy boat stopped with just a coiled rope in the mooring?
ps..if there's a better place to put the thread...
Homework Statement
Utilities companies usually use a separation between poles about 340 feet. These poles are 34 feet tall and due to restrictions due to maximum height of trucks using interstates and state routes, the minimum clearance is 20 feet (and also considering the increase of sag...
What are good sources to find a model for catenary used to build suspension bridges?
I know that the equation for catenary is a*sin(x/a). what does a stand for?
Thanks.
(Moderator's note: thread moved from "Differential Equations")
The DE is y''=a*sqrt(1+(y')^2)
I have no idea how to go about integrating it, I just started taking diff eq's and haven't taken calc in over a year. Any help would be appreciated, thanks!
Homework Statement
Problem is regarding approximating the value of a in y= a cosh(x/a) using Newton's method, and then use a to find the length of the rope.
That equation represents the curve formed by a rope hanging with it's ends attached to poles at a distance 2b.
(cosh() = hyperbolic...
I tried to solve the equation of catenary by variational method the other day. The integral we want to minimize is the potential energy:
U = \int_{{x_2}}^{{x_1}} {\rho gy\sqrt {1 + y{'^2}} } dx
Then I got stuck at the constraint problem, and in this...
I have been asking this question of everyone I meet and can't find a straight answer.
Imagine a wire rope anchored at both ends, spanning 50ft [15m].
That line is 'pre-tensioned' to... say 500 lbs. [2kN].
The idea is that it is pulled fairly 'flat'.
What happens to the line tension when I...
Sag formula
When measuring a distance with a 100 foot steel tape supported at both ends the sag formula needs to be applied to correct the distance. Does anybody know why an overall distance does not work in this sag formula? For example If you measure a distance of 735 feet you would have...
hi~
i need to determine the tension a in the equation of the hanging cable y=acosh(x-h)/a+k with a known length and known location of end points. i figured out how to determine the horizontal translation h but i need a in order to do it and to have the complete equation.
this is for a...
Homework Statement
Find the angle between the line x=7 and the catenary y=20\cosh{(\frac{x}{20})}-15
The Attempt at a Solution
I found the tangent has gradient \sinh{(\frac{7}{20})}
Then I used \tan{\theta}=|\frac{1}{m}|
where m=\sinh{(\frac{7}{20})}
And evaluated using inverse...
I am curious as to how the equation of a catenary was derived:
x = t
y = a*cosh(t)
Does anyone have any insight on this or know a good webpage that can explain it?
Thank you for the assistance.
Can somebody teach me a better way to solve the equation of catenary cable to find the value of c? If there is no other better alternative, can you tell me how to estimate a value for the trial and error method.
Thanks