What is Catenary: Definition and 40 Discussions

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.

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  1. P

    I Why is my derivation of the catenary wrong?

    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
  2. tjhollis1

    I Unsolvable Catenary Problem

    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})##...
  3. M

    I How can I solve catenary problems?

    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?
  4. C

    Sag of elastic / non-elastic catenary

    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
  5. BubblesAreUs

    Solve Catenary Problem: Minimize Potential Energy

    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)...
  6. zwierz

    I History of brachistochrone and catenary

    Could someone please give me references on historical literature devoted to brachistochrone and catenary problems Thanks.
  7. DuckAmuck

    I Minimal Surface shape with gravity

    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...
  8. Vajhe

    What shape does a deformed body take after being stretched?

    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...
  9. DuckAmuck

    A Shape of a pinned canvas w/ Lagrange Multipliers

    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...
  10. A

    Catenary lighting canopy -- Working out reactions at columns

    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...
  11. Coffee_

    A Lagrange multiplier approach to the catenary problem

    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...
  12. A

    Current flow in infinite sheet, and shape of isocurrents

    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...
  13. M

    Hanging a point load from a catenary

    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...
  14. O

    Find centroid of catenary (y=cosh(x))

    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...
  15. X

    Catenary shape of a towed underwater cable

    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...
  16. J

    Tension in a catenary curve - Cable Camera

    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...
  17. J

    Finding m_c: Tangent to Catenary Curve

    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...
  18. B

    Center of mass and moment of inertia of catenary

    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...
  19. E

    Tension force in catenary

    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...
  20. J

    Catenary supporting a beam problem

    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...
  21. D

    Proving T=H+q*h in a Catenary Spanned Cable

    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...
  22. D

    Derivation of equation for catenary

    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...
  23. M

    Wire sag calculation Catenary?

    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...
  24. V

    Tension in a catenary

    How do I calculate the tension at the middle of a rope (weight=W) suspended at two points as a catenary? It's ends make angle A with the horizontal.
  25. S

    Catenary Realism: Calculating Weight of Electric Wire

    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...
  26. P

    Hi i'm an applied maths student i'd like to study about catenary

    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?
  27. S

    Catenary and coiled rope - for dummies.

    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...
  28. B

    Catenary - find rope length Maple must be used

    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...
  29. M

    Real-life catenary examples

    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.
  30. B

    Need help solving the differential equation for the shape of a catenary?

    (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!
  31. Jonnyb42

    Catenary - rope hanging between poles

    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...
  32. K

    Get the equation catenary using variational method

    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...
  33. N

    Mathematica Solving Implicit Function of Catenary in Mathematica

    I need to solve this implicit function of the catenary k=cosh(k/sqrt(k^2-1)) how do i Do this in mathematica?
  34. A

    Point loading a catenary.

    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...
  35. T

    Catenary Correction formula

    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...
  36. J

    Hanging cable (catenary)

    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...
  37. F

    Angle between catenary and line

    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...
  38. A

    Exploring the Derivation of the Catenary Equation

    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.
  39. I

    Catenary Problem: Find Formula for Any Point

    anybody here have any idea about catenary problem?i tried to find the formula of any point on the catenary.it involves some derivatives too
  40. C

    Find Value of C for Catenary Cable | Trial and Error Method

    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