1 Dept. of Agricultural & Biological Engineering University of Illinois ABE223: ABE Principles – Machine Systems ABE223: ABE Principles – Machine Systems Air cannon as a pneumatic safety testing device Tony Grift

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 (Re>2000)

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); -K*x_state(2)*sqrt(x_state(2)^2+x_state(4)^2);... x_state(4); -K*x_state(4)*sqrt(x_state(2)^2+x_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

Fall test to determine aerodynamic properties of fertilizer particles (1994) 17

Fall test to determine aerodynamic properties of fertilizer particles (1994) 18

Required video (prof. Mattuck) 19 Euler's Numerical Method for y'=f(x,y) and its Generalizations

20 Dept. of Agricultural & Biological Engineering University of Illinois ABE223: ABE Principles – Machine Systems ABE223: ABE Principles – Machine Systems Air cannon as a pneumatic safety testing device The End