Tsing Ma Bridge

 

Have you ever been to Tsing Ma Bridge in Hong Kong ? Beautiful, isn’t it?

 

Have you been fascinated by the shape of the curve of the hanging suspension steel cord? (shown in blue arrow in the photo in the right)

 

Is the curve a parabola?

 

This “hanging chain problem” was a famous challenging mathematical problem raised by Jacob Bernoulli (1654 ~ 1705).

 

 

 

tsing ma bridge at sunset

 

Short History

 

 Huygens was the first to use the term catenary to describe the shape of a perfectly flexible chain suspended by its ends and acted on by gravity. Galileo (1564~1643) mistakenly thought that is a parabola. . Its equation was obtained by Leibniz, Huygens and Johann Bernoulli in 1691. Let us investigate how to find this curve. Surprisingly it is connected with the famous constant e.

 

 

The left figure shows a static hanging chain.

Let the y-axis passes through the lowest point Po of the curve and s be the arc length from Po to a variable point P(x,y).

 

Let the tension at points Po and P be To and T respectively. If wo is the density of the chain, then  wos  is the weight of the arc s.


 

The equation

 

Now, since s is static, the resultants of all forces in x and y directions are 0, therefore:

                                                     (1)

                                                                              (2)

      Since    and let    be a constant, we get:

                                                                                      (3)

 

     In order to eliminate s, differentiate (3) and use the arc length formula, we have:

                                                          (4)

 

     This is a second order differential equation with initial conditions :

                                                                    (5)

 

 

How to solve

 

Let  ,  (4) becomes:

                                                                                   (6)

 

     Separate the variables, we get:

                                                               (7)

 

     Using trigonometric substitution  , we get:

          (8)

 

 By (7) and (8), we get:          

                                                                     (9)

 

 By the initial conditions in (5), y’(0) = 0, when x = 0, p = 0, therefore c1 = 0. (9) becomes:

              

    \     

    Put p to the left hand side, squaring, solve a quadratic equation in p, we get:

           

        \  

 

    For simplicity, we adjust the y axis such that   ,  then c2 = 0.

    We finally get the equation of the catenary:

                                                                       (10)

 

 

 

Parabola Vs Catenary

 

Equation (10) is a kind of transcendental functions. Nowadays, we let  and the catenary is:

 

   In the left, you can see a catenary in red and a parabola in blue.

 

       Next time, if you visit Tsing Ma Bridge, tell your friend that the suspended cord is a catenary, not a parabola!

 

 

Finale

 

          The Gateway Arch of St. Louis, Missouri is an inverted catenary. It is a beautiful landmark. The Arch is 630 ft wide at the base and 630 ft tall. The formula is shown on display inside the arch:

 

          y = 68.8 cosh [0.01x – 1]

 

 

 

GatewayArch.jpg (11726 bytes)St. Louis Gateway, Missouri

 

Finale

 

     The graphs of the catenary of equation:

 

         

 

with different a is shown on the left.

 

Hope you enjoy the drawings.