ABE223: ABE Principles

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ABE223 ABE Principles Machine Systems Air
cannon as a pneumatic safety testing device
Tony Grift
Dept. of Agricultural Biological
EngineeringUniversity of Illinois
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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

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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

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In the computer lab we will develop a spread
sheet for the complete cannon
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Ballistic model of a projectile in flight
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The drag coefficient (Cd) value for a sphere is
about 0.4 for turbulent flow regimes (Regt2000)
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Force balance in x-direction
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Force balance in y-direction
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Terminal velocity (only in y-direction) can be
calculated by setting acceleration to zero
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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.

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

• Define
• Result

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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

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This is what your program should produce
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Simulation can yield a graph that relates the
exit velocity to the air time
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The total height achieved can also be graphed
(assuming vertical trajectory)
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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

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