chapter 1 slide
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numeircal methodsTRANSCRIPT
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Chapter 1MEEN 221-01
Sam Ibekwe, Ph.D., P.E.355 Pinchback Engineering Building
Email: samuel_ibekwe.subr.edu
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Mathematical Modeling and Engineering Problem solving
Chapter 1
• Requires understanding of engineering systems– By observation and experiment– Theoretical analysis and generalization
• Computers are great tools, however, without fundamental understanding of engineering problems, they will be useless.
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Fig. 1.1The engineering
problem-solving process.
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A mathematical model is represented as a functional relationship of the form
Dependent independent forcingVariable = f variables, parameters, functions
Dependent variable: Characteristic that usually reflects the state of the system
Independent variables: Dimensions such as time ans space along which the systems behavior is being determined
Parameters: reflect the system’s properties or compositionForcing functions: external influences acting upon the system
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Newton’s 2nd law of MotionStates that “the time rate change of momentum of a body
is equal to the resulting force acting on it.”The model is formulated as
F = m a (1.2)
F = net force acting on the body (N)
m = mass of the object (kg)
a = its acceleration (m/s2)
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Formulation of Newton’s 2nd law has several characteristics that are typical of mathematical models of the physical world:It describes a natural process or system in
mathematical termsIt represents an idealization and simplification of
realityFinally, it yields reproducible results,
consequently, can be used for predictive purposes.
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Some mathematical models of physical phenomena may be much more complex.
Complex models may not be solved exactly or require more sophisticated mathematical techniques than simple algebra for their solution
Example, modeling of a falling parachutist:
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Figure 1.02.
Schematic diagram of the forces acting on a falling parachutist. – Upward and Downward Forces.
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m
cvmg
dt
dv
cvF
mgF
FFF
m
F
dt
dv
U
D
UD
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Expression the acceleration as the time rate of change of the velocity yield Eq. 1.4
The net force includes both downward and upward forces.
FD = force due to gravityFU = force due to air resistanceC = drag coefficient
Net Force is the difference between the downward and upward force.
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This is a differential equation and is written in terms of the differential rate of change dv/dt of the variable that we are interested in predicting.
If the parachutist is initially at rest (v=0 at t=0), using calculus
tmcec
gmtv )/(1)(
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vm
cg
dt
dv
Independent variable
Dependent variableParametersForcing function
Simplifying the equation-
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Example 1.1Figure 1.3
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Example 1.1Figure 1.4
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Example 1.2 – Numerical SolutionUsing equation Eq. 1.11 to yield to following:
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)()()()( 11 iiiii tttvm
cgtvtv
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Example 1.1Figure 1.5
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Section 1.2Conservation Laws and EngineeringConservation laws are the most important and
fundamental laws that are used in engineering.
Change = increases – decreases (1.13)Change implies changes with time (transient). If the
change is nonexistent (steady-state), Eq. 1.13 becomes
Increases =Decreases
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Figure 1.6
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For steady-state incompressible fluid flow in pipes:Flow in = Flow out
or100 + 80 = 120 + Flow4
Flow4 = 60
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Fig 1.6
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Problems 1.4, 1.5, 1.7
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Problem 1.4For a free falling parachutist with linear
drag, assume a first jumper is 70 kg and has a drag coefficient of 12kg/s and a mass of 75 kg, if the second jumper drag coefficient of 15kg/s. how long will it take him to reach the same velocity the first jumper reached in 10 seconds?
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tmcec
gmtv )/(1)(
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Problem 1.5
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Compute the velocity of a free falling parachutist using Euler’s method for the case where m= 80 kg and c = 10kgs. Perform the calculation from t = 0 to 20s with a step size of 1 second. Using an initial condition that the parachutist has an upward velocity of 20 m/s at t = 0. At t=10 seconds, assume that the chute is instantaneously deployed sot that the drag coefficient jumps to 50 kg/s
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Problem 1.7 Storage tank contains a liquid at depth y where y = 0 when the tank is full.
Figure P1_07.jpg
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Problem 1.7 results
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t
Y