Applied Mathematical Modelling of Engineering Problems
The subject of the book is the "know-how" of applied mathematical modelling: how to construct specific models and adjust them to a new engineering environment or more precise realistic assumptions; how to analyze models for the purpose of investigating re
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Applied Optimization Volume 81 Se ries Editors:
Panos M. Pardalos University 0/ Florida, US.A. Donald W. Hearn University 0/ Florida, US.A.
Applied Mathematical Modelling of Engineering Problems By N atali Hritonenko Prairie View A&M University Department ofMathematics Prairie View, TX 77446
US.A. Yuri Yatsenko Houston Baptist University College ofBusiness and Economics Houston, TX 77074 US.A.
Library of Congress Cataloging-in-PubUcaüon CIP info or: Title: Applied Mathematical Modelling ofEngineering Problems Author: Hritonenko, Yatsenko ISBN 978-1-4613-4815-3 ISBN 978-1-4419-9160-7 (eBook) DOI 10.1007/978-1-4419-9160-7
Copyright 340 rn/sec).
Chapterl
4
Three examples in the following subsections describe model (1.2) in more specific cases.
1.1.1
Vertical Projectile Problem.
Consider a body of mass m that is radially projected upward from the Earth's surface with an initial speed Vo. Let R denote the radius ofthe Earth and y(t) denote the sought-for radial distance from the Earth's surface at time t. If one neglects the air resistance, the model of the body's dynamics (1.2) is described by the following non linear differential equation of the second order:
(1.3) with the initial conditions :
y(O)
=
0,
dy/dt (0)
=
Vo,
(104)
The constant g = yM/R2 is called the gravitation acceleration on the Earth's surface (aty=O) where Mis the Earth's mass (see (1.1)). If the displacement y is small compared to R (it is true when the initial speed Vo is not too large), then it leads to the simplified linear problem:
cty/dP = -g,
y(O) = 0,
dy/dt(O) = Vo,
with the well-known elementary solution: y(t) = -gP/2 + Vot.
1.1.2
Free Fall with Air Resistance
In this case, the air resistance is considered but changing gravitational attraction is not taken into account. Let us consider a free fall with the initial speed of zero. In this case, there is no horizontal component in the velocity v and in the displacement x, and the only motion is the vertical one. Let us choose the y-axis as the vertical axis upwards. Then equations (1.2) lead to
Same Basic Models Of Physical Systems
cfyldt2 = y(O)
=
5
g - j(v),
Yo,
(1.5)
dyldt(O)
=
0,
or to the nonlinear differential equation ofthe first order
dvldt=-g-j(v),
v(O)
=
0,
(1.6)
with respect to the speed v. Whether the air resistance force j(v) is suggested to be proportional to the speed v or to the to the square of v, the solution v(t) of (1 .6) tends to a constant speed V"' called the terminal speed, when t~oo. In the casej(v) = kv 2 , the tenninal speed is V", = (g/k)l/2 and the (1.6) solution is v(t) = -V", tanh (gtIV",).
1.1.3
Plane Projectile Problem
Let us consider the motion of a projectile that is launched to an acute angle () to the horizontal, so that the motion can be described in a plane. Choose the Cartesian coordinates with the origin at the point of launching the projectile, the x-axis in the horizontal direction of the initial speed, and the y-axis in the vertically upward. We restrict ourselves with the case of a moderate initial s
