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# Complex Numbers

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Application of Complex Number In Mechanical Engineering

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### Complex Numbers

1. 1. COMPLEX NUMBERS APPLICATION OF COMPLEX NUMBER IN MECHANICAL ENGINEERING 1
2. 2. INTRODUCTION A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. Complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics as well as in physics, including hydrodynamics and thermodynamics and also in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. 2
3. 3. Application in Mechanical Engineering 1. Air Foils 2. Control Theory 3. Quantum Mechanics 4. Relativity 5. Fluid Mechanics 6. Heat Flow The concept of complex geometry and Argand plane is very much useful in constructing cars and 2-D designing of cars. It is also very useful in cutting of tools. Another possibility to use complex numbers in simple mechanics might be to use them to represent rotations. 3
4. 4. CONTROL THEORY Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs, and how their behavior is modified by feedback. 4
5. 5. CONTROL THEORY An example of a control system is a car's cruise control, which is a device designed to maintain vehicle speed at a constant desired or reference speed provided by the driver. The controller is the cruise control, the plant is the car, and the system is the car and the cruise control. The system output is the car's speed, and the control itself is the engine's throttle position which determines how much power the engine generates. 5
6. 6. CONTROL THEORY Robust control deals explicitly with uncertainty in its approach to controller design. Controllers designed using robust control methods tend to be able to cope with small differences between the true system and the nominal model used for design. Automation or automatic control, is the use of various control systems for operating equipment such as machinery, processes in factories, boilers and heat treating ovens, switching in telephone networks, steering and stabilization of ships, aircraft and other applications with minimal or reduced human intervention 6
7. 7. Air Foil  The Joukowsky transform, named after Nikolai Zhukovsky is a conformal map historically used to understand some principles of air foil design.  A large amount of airfoil theory has been developed by distorting flow around a cylinder to flow around an airfoil.  The essential feature of the distortion is that the potential flow being distorted ends up also as potential flow.  The most common Conformal transformation is the Joukowsky transformation which is given by  ζ = X + iɳ is a complex variable in the original space. 7
8. 8. Example of a Joukowsky transform. The circle above is transformed into the Joukowsky airfoil below 8
9. 9. RELATIVITY  General Relativity, one of the two pillars of modern physics General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915.  General relativity generalizes special relativity and Newton's law of universal gravitation.  It’s providing a unified description of gravity as a geometric property of space and time, or space-time. 9
10. 10. RELATIVITY  In special and general relativity, some formulas for the metric on space time become simpler if one takes the time variable to be imaginary. (This is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) 10
11. 11. QUANTUM MECHANICS  Quantum mechanics provides a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter.  The complex number field is relevant in the mathematical formulation of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. 11
12. 12. QUANTUM MECHANICS  The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers.  Expressing wave function as complex number Schrödinger's idea was to express the phase of a plane wave as a complex phase factor: Ψ(x,t) = Aei(k.x-wt) 12
13. 13. Thank You…!!! Ashwini Gupta Mahendra Rathod Siddhant Mangla C-2 Tarun Yadav Vikrant Singla 13