MATHEMATICAL

MODEL OF SUSPENSION BRIDGE

Introduction: A Mathematical Modelling and Presentation

competition was held on 1 November, 2017 at Amity International School, Saket. The

DPS RKP presentation comprised both physical and mathematical (simulation based)

models of a suspension bridge. A Special Mention Trophy and certificate was

awarded by the Organisers for this.

The mathematical model was based on

simulations carried out in Wolfram Mathematica, which is a mathematical symbolic computation programming language

used in many scientific, engineering, mathematical and computing fields.

Mathematical model: The mathematical model

that was simulated comprised two parts – a catenary and a parabola, both being relevant

to suspension bridges.

a. Parabola A conic section is the curve obtained

by the intersection of a plane with the surface of a cone. A parabola is a conic section that can be

formed by the intersection of a right circular conical surface with a plane

intersecting the conical surface as depicted in the figure below:

The above figure has been

taken from the mathematical simulation of the model carried out in Mathematica;

in this a parabola was

obtained by the intersection of a plane y=k with the conic surface.

b. Catenary: A catenary symbolizes a chain with its

two ends supported using vertical structures i.e. columns; mathematically, the

curve of a catenary resembles a hyperbolic cosine. The hyperbolic cosine

function is defined as

The

following images depict the physical model of a catenary presented at the

competition, and the suitable hyperbolic cosine function, fitted using

Mathematica, superimposed on the same:

However, a catenary resembles a parabola when it is used

to support weights which are much heavier than the mass of the chain or cable

used! The images below show a suitable quadratic function (an algebraic

representation of a parabola), fitted using Mathematica, superimposed on the

physical model, wherein a heavy object is suspended from the catenary. As seen

in the image, the part of the catenary where the weight is suspended has a

better superimposition than the remaining part. A better fit would have been

obtained if the weight was suspended uniformly along the length of the

catenary.

Suspension Bridge

Model: Finally, the following has

been taken from the simulation of the mathematical model of a suspension bridge

that was also created in Mathematica:

In the simulation, the following parameters

can be manipulated:

i. Catenary

support: Varies the curve of the catenaries modeled using parabolic functions

(since the bridge deck is supported by the catenaries).

ii. Diameter of the

supports: This varies the diameters of both the catenaries supporting the

bridge deck.

iii. Support Density:

This varies the number of vertical supporting members per unit length hanging

from the catenaries.

iv. Rotation

along y-axis: This rotates the virtual model for better viewing.

Conclusion: The physical and

mathematical (simulation-based) models depict the application of mathematics in

civil infrastructure. A suspension bridge actually utilizes steel cables for

suspending the bridge deck. Suspension bridges usually have large spans. Some

examples of suspension bridges in India are the Lakshman Jhula and Howrah

Bridge. The Golden Gate Bridge, San Francisco (USA), is also an example of a

suspension bridge.

Acknowledgment: I would like to express

my gratitude to Ms. Naga Laxmi and Ms. Vandana Seth, teachers of the Mathematics

Department, for their constant support and motivation and for giving me this

opportunity

Thank you for

reading!

Anmol Singh

Class 9 D