### MATHEMATICAL is the curve obtained by the 0 Comment

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.

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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

Anmol Singh

Class 9 D

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