MATHEMATICAL is the curve obtained by the
MATHEMATICALMODEL OF SUSPENSION BRIDGEIntroduction: A Mathematical Modelling and Presentationcompetition was held on 1 November, 2017 at Amity International School, Saket. TheDPS RKP presentation comprised both physical and mathematical (simulation based)models of a suspension bridge. A Special Mention Trophy and certificate wasawarded by the Organisers for this.
The mathematical model was based onsimulations carried out in Wolfram Mathematica, which is a mathematical symbolic computation programming languageused in many scientific, engineering, mathematical and computing fields. Mathematical model: The mathematical modelthat was simulated comprised two parts – a catenary and a parabola, both being relevantto suspension bridges. a.
Parabola A conic section is the curve obtainedby the intersection of a plane with the surface of a cone. A parabola is a conic section that can beformed by the intersection of a right circular conical surface with a planeintersecting the conical surface as depicted in the figure below: The above figure has beentaken from the mathematical simulation of the model carried out in Mathematica;in this a parabola wasobtained by the intersection of a plane y=k with the conic surface. b. Catenary: A catenary symbolizes a chain with itstwo ends supported using vertical structures i.
e. columns; mathematically, thecurve of a catenary resembles a hyperbolic cosine. The hyperbolic cosinefunction is defined as Thefollowing images depict the physical model of a catenary presented at thecompetition, and the suitable hyperbolic cosine function, fitted usingMathematica, superimposed on the same: However, a catenary resembles a parabola when it is usedto support weights which are much heavier than the mass of the chain or cableused! The images below show a suitable quadratic function (an algebraicrepresentation of a parabola), fitted using Mathematica, superimposed on thephysical model, wherein a heavy object is suspended from the catenary. As seenin the image, the part of the catenary where the weight is suspended has abetter superimposition than the remaining part. A better fit would have beenobtained if the weight was suspended uniformly along the length of thecatenary. Suspension BridgeModel: Finally, the following hasbeen taken from the simulation of the mathematical model of a suspension bridgethat was also created in Mathematica: In the simulation, the following parameterscan be manipulated: i.
Catenarysupport: Varies the curve of the catenaries modeled using parabolic functions(since the bridge deck is supported by the catenaries).ii. Diameter of thesupports: This varies the diameters of both the catenaries supporting thebridge deck.iii. Support Density:This varies the number of vertical supporting members per unit length hangingfrom the catenaries.iv.
Rotationalong y-axis: This rotates the virtual model for better viewing. Conclusion: The physical andmathematical (simulation-based) models depict the application of mathematics incivil infrastructure. A suspension bridge actually utilizes steel cables forsuspending the bridge deck. Suspension bridges usually have large spans. Someexamples of suspension bridges in India are the Lakshman Jhula and HowrahBridge. The Golden Gate Bridge, San Francisco (USA), is also an example of asuspension bridge.
Acknowledgment: I would like to expressmy gratitude to Ms. Naga Laxmi and Ms. Vandana Seth, teachers of the MathematicsDepartment, for their constant support and motivation and for giving me thisopportunity Thank you forreading! Anmol SinghClass 9 D
