Differential Equations
for Engineers:
the Essentials
Supplement to Class 14A
Homework Assignment 14:
Example Engineering Application of ODEs
Sounding Rocket Performance
Phase 2
Homework Assignment 14
Read:
Refresh your memory by skimming (the guide to the instructor’s or
student’s choice of numerical integration computer programs.)
Work :
We will use the numerical integration computer program once again
Homework Assignment 14 (2)
Programming:
Either recover your own stored program or type in the program given
out by the instructor in reviewing Homework Assignment 6. Program
the equations for air density and drag coefficient described on the
next charts.
Output:
Run the numerical integration computer program for each of the
Due: (date)
Homework Assignment 14:
Definition of Terms
=
=
=
=
=
mSCk
c
m
h
a
bDa
b
)2/(
Rocket altitude
Rocket mass at burnout
Speed of rocket exit gases
Air density
Homework Assignment 14:
Definition of Terms (2)
=
==
=
=
0
0
/
)(
b
A
mmMR
W
hTemp
Air temperature at altitude h
Rocket initial weight (lbs)
Rocket mass ratio
Area of rocket nozzle
Homework Assignment 14:
Sounding Rocket Equations
Vh
=
Basic equations of motion (as before):
Homework Assignment 14:
Sounding Rocket Equations (2)
Basic Thrust and Drag Equations (as before):
Thrust:
Drag:
2
2
1
VSCD
Da
=
0V
0V
Homework Assignment 14:
Sounding Rocket Equations (3)
Constant parameters:
Parameters changing with rocket number:
2.32
0
=
g
gWm
/
00
=
ft/sec/sec
Homework Assignment 14:
Sounding Rocket Equations (4)
Improved gravity model:
Modify list of equations as follows:
Replace
Homework Assignment 14:
Sounding Rocket Equations (5)
D
C
M
Drag coefficient as a function of Mach Number
Homework Assignment 14:
Sounding Rocket Equations (6)
Homework Assignment 14:
Sounding Rocket Equations (7)
Speed of sound:
Air density:
Homework Assignment 14:
Sounding Rocket Equations (8)
Graph 1: Air temperature vs altitude
Coordinates:
(altitude (k ft), temp (deg R))
0, 519
Homework Assignment 14:
Sounding Rocket Equations (9)
Graph 2: Air density reference
altitude vs altitude
)(/
0
hHh
aa e−
=
)(hH
Homework Assignment 14:
Sounding Rocket Equations (10)
Rocket number
1 2 3
8120 1500
W
Initial weight (lbs)
* unique for each student
c
Homework Assignment 14: Student
Parameter Assignments
Rocket 1
Student gas exit speed
#1 4500
Rocket 1
Student gas exit speed
#6 5100
Homework Assignment 14:
Helpful Hints
In look-up tables, remember that data is given in kft, not feet.
*Example logic for User-Defined Function:
If X < 10 then
return 6