Application of Complex Number In Mechanical Engineering

1. COMPLEX NUMBERS
APPLICATION OF COMPLEX NUMBER IN MECHANICAL
ENGINEERING
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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.
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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.
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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.
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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.
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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
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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.
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8. Example of a Joukowsky transform. The circle above is
transformed into the Joukowsky airfoil below
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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.
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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.)
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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.
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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)
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13. Thank You…!!!
Ashwini Gupta
Mahendra Rathod
Siddhant Mangla C-2
Tarun Yadav
Vikrant Singla
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