Example Engineering Application of ODEs
A numerical integration project:
Sounding Rocket Performance – Phase 1
Numerical Integration of ODEs
Using the computer to calculate a solution time-step by time-step
The engineering professional standard –how engineers deal in
practice with most ODEs
Homework Assignment 6
Read:
(Guide to numerical integration program of student’s or instructor’s
choice)
Work :
We will use a numerical integration program to explore further the
sounding rocket trajectory analyzed in Classes 3 and 4. This
Homework Assignment 6 (2)
We will employ the computer program to analyze the trajectory of a
small model rocket for which the simple model should work well.
We will compare the results the computer numerically integrates
with the closed form solutions we derived. This comparison will
test that we have programmed the basic equations of motion and
the basic equations for thrust and drag correctly.
Homework Assignment 6 (3)
Output:
Run the numerical integration program for rocket number 1 with the
equations given on the pages following. Print out and hand in a copy of
the computed rocket trajectory and your program, including user-
defined functions. Save your program for use in the later assignment.
Hand in copies of the “data” pages at the appropriate times to record
Homework Assignment 6:
Definition of Terms
=
=
=
=
m
m
V
h
0
Rocket altitude
Rocket vertical velocity
Rocket mass
Rocket mass at launch
Homework Assignment 6:
Definition of Terms (2)
=
=
=
0
W
D
T
Thrust (lbs)
Drag (lbs)
Rocket initial weight (lbs)
Homework Assignment 6:
Sounding Rocket Equations
/)(
gmDTV
Vh
−−=
=
Basic equations of motion:
Homework Assignment 6:
Sounding Rocket Equations (2)
Basic Thrust and Drag Equations
Thrust:
0
2
=
=
T
AcTrg
b
mm
b
mm
Homework Assignment 6:
Sounding Rocket Equations (3)
For this assignment we will program the following to describe the
drag coefficient and air density:
0
0
aa
DD CC
=
=
Homework Assignment 6:
Sounding Rocket Equations (4)
Constant parameters:
Parameters changing with rocket design:
2.32
=
g
ft/sec/sec
Homework Assignment 6:
Sounding Rocket Equations (5)
Rocket number
Parameters
1 2 3
8120 1500
1.15 1.5 5
0
W
MR
Initial weight (lbs)
Mass ratio
)/( 0b
mm
* unique for each student
c
Homework Assignment 6: Helpful Hints
Expect numerically computed solutions to be within 2% or better of
closed-form-calculated values.
*Write thrust and drag equations as “User–defined functions”.
Pass all required parameters to the user-defined functions.
Homework Assignment 6: Helpful Hints (2)
*To print out an ODE Architect Tool page:
Press “Ctrl” and “PrntScn” keyboard keys simultaneously
*These are for ODE Architect, given as examples to the instructor to replace with
appropriate guidance for his or her computing environment
Homework Assignment 6: Student
Parameter Assignments
Rocket 1
Student gas exit speed
#1 4500
#2 4600
Rocket 1
Student gas exit speed
#6 5100
#7 5200