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ABE223: ABE Principles

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Title: ABE223: ABE Principles


1
ABE223 ABE Principles Machine Systems Air
cannon as a pneumatic safety testing device
Tony Grift
Dept. of Agricultural Biological
EngineeringUniversity of Illinois
2
Agenda
  • Derivation of dynamics in equation form
  • What happens to a flying projectile in air?
  • Solving non-linear equations in MatLab
  • Linking MatLab and Excel

3
This work is (at least partially) used to
accelerate our projectile
  • Assuming that all energy from the gas is
    converted into kinetic energy of the projectile
    (this is a major assumption) we get
  • The exit velocity of the projectile would now be

4
In the computer lab we will develop a spread
sheet for the complete cannon
5
Ballistic model of a projectile in flight
6
The drag coefficient (Cd) value for a sphere is
about 0.4 for turbulent flow regimes (Regt2000)
7
Force balance in x-direction
8
Force balance in y-direction
9
Terminal velocity (only in y-direction) can be
calculated by setting acceleration to zero
10
The complete ballistic model is a set of two
coupled differential equations
  • This is a system of ordinary (not partial)
    differential equations
  • The differential equations are coupled (x in y
    and y in x)
  • The differential equations are non-linear due to
    the square roots
  • These types of equations cannot be solved
    analytically We need to resort to numerical
    methods such as the Runge-Kutta solver in MatLab
  • These equations are only valid for spherical
    particles. For non-spherical particles,
    adjustments need to be made See the link.

11
Solving this system requires translating the
system into a set of first order (non-linear)
state equations as follows
  • Define
  • Result

12
MatLab needs a file (here aerodyn.m) that defines
the system of equations
  • function xp aerodyn(t_sim,x_state)
  • global g K
  • xp x_state(2)
  • -Kx_state(2)sqrt(x_state(2)2x_state(4)2
    )...
  • x_state(4)
  • -Kx_state(4)sqrt(x_state(2)2x_state(4)2
    )-g

13
This is what your program should produce
14
Simulation can yield a graph that relates the
exit velocity to the air time
15
The total height achieved can also be graphed
(assuming vertical trajectory)
16
Lab
  • Outdoor part
  • Fire the cannon with various pressure settings
    and various projectile
  • Indoor part
  • Build complete Excel spreadsheet from a template
    to model the cannon assuming adiabatic expansion
  • Use MatLab to simulate a ballistic model
  • Connect MatLab and Excel using a Dynamic Data
    Exchange (DDE) link and make them work together

17
Fall test to determine aerodynamic properties of
fertilizer particles (1994)
18
Fall test to determine aerodynamic properties of
fertilizer particles (1994)
19
Required video (prof. Mattuck)
Euler's Numerical Method for y'f(x,y) and its
Generalizations
http//www.youtube.com/watch?vLbKKzMag5Rc
20
ABE223 ABE Principles Machine Systems Air
cannon as a pneumatic safety testing device
The End
Dept. of Agricultural Biological
EngineeringUniversity of Illinois
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