Differential calculus & its Practical Applications In Field of Mechanical engineering

BY:OMAR EBADUR RAHMANMOHD. SHARIK KHANGAMPALA VAMSHIDHAR RAO

INTRODUCTION Differential calculus is a subfield of calculus concerned with the study

of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus.

The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value.

TERMS

FUNCTION DIIFRENTIATION LIMITS TANGENTS NORMALS MAXIMA & MINIMA & etc

HISTORY

GREEKS:Euclid (c. 300 BCE),Archimedes (c. 287–212 BCE)

INDIANS:  Aryabhata (476–550), Bhāskara II (1114-1185); PERSIAN: Sharaf al-Dīn al-Tūsī (1135-1213). MODERN DEVLOPERS:  Isaac Newton (1643 – 1727)

and Gottfried Leibniz (1646 – 1716)

The Motion of Artificial satellites The motion of artificial satellites is physically governed

by the following vector differential equation: x is the position vector of the satellite in the

inertial frame. GM is the product of the Earth's mass M and the gravitational constant G, r=||x||, and T is the disturbing potential of the Earth’s gravity

field

𝑇=(𝐺𝑀𝑟 ) ∑𝑙=2

𝑁𝑚𝑎𝑥

∑𝑚=0

𝑙

¿ ¿¿

R is the mean radius of the Earth Clm and Slm are the normalized, dimensionless harmonic

coefficients, λ and θ are the longitude and colatitude of the satellite,

respectively Plm(t) is the normalized Legendre function.

TURBOMACHINARY Many types of secondary flows occur in turbomachinery, including inlet prerotation

(intakes vorticity), tip clearance flow (tip leakage), flows at off-design performance (e.g. flow separation), and secondary vorticity flows.

 Although secondary flows occur in all turbomachinery, it is particularly considered in axial flow compressors because of the thick boundary layers on the annulus walls.

For such axial-flow compressors, consider a set of guide vanes with an approach velocity c1.

The velocity profile will be non-uniform due to friction between the annulus wall and the fluid.

The vorticity of this boundary layer is normal to the approach velocity c1 and of magnitude

Where z is the distance to the wall. As the vorticity of each blade onto each other will be of opposite directions, a secondary vorticity will be generated.

If the deflection angle, e, between the guide vanes is small, the magnitude of the secondary vorticity is represented as

This secondary flow will be the integrated effect of the distribution of secondary vorticity along the blade length

 

ENGINE FUEL EFFICIENCY Since the total force opposing the vehicle's motion (at constant

speed) multiplied by the distance through which the vehicle travels represents the work that the vehicle's engine must perform, the study of mileage

The amount of energy consumed per unit of distance travelled

For a vehicle whose source of power is a heat engine (an engine that uses heat to perform useful work), the amount of fuel energy that a vehicle consumes per unit of distance (level road) depends upon:

The thermodynamic efficiency of the heat engine; The forces of friction within the mechanical system that delivers

engine output to the wheels; The forces of friction in the wheels and between the road and

the wheels (rolling friction); Other internal forces that the engine works against (electrical

generator, air conditioner etc., water pump, engine fan etc.); External forces that resist motion (e.g., wind, rain); Non-regenerative braking force (brakes that turn motion energy

into heat rather than storing it in a useful form; e.g., electrical energy in hybrid vehicles).

By the conservation of energy, the sum of the two energies is zero.

The quantity is known as the "log mean temperature difference" and is a measure of the effectiveness of the heat exchanger in transferring heat energy.

 U is the term use for energy