Catapult Simulator

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Problem Description

Pull-down type after firing 

Pull-up "Oops" type after firing 

Here's a "cup" type catapult simulator (no sling).   It simulates tensional or torsional  driving forces with user specified design parameters. 

Background & Techniques

Jim Corley,  a retired  engineer friend of mine, sponsors an annual competition for the high school Physics classes  in his home  town of Lucedale Mississippi.   Past year's competitions have included bridge building and trebuchets and,  last year, cup type catapults.  He had asked me then about writing a simulator program the to help students in their design phase, but I just didn't get to it.  I did promise him one for this year, so here it is.    

Early catapults looked like most catapults seen in movies with a cup or bucket on top of the beam containing the projectile until it was fired..  The development of the sling proved to be more efficient and apparently came into common use.  For current purposes, we'll stick with the "sling-free" type. 

Real catapults evolved with two types of driving force, tensional were spring type and commonly took on the look of giant crossbows.  Torsional  versions used multiple strands of rope wrapped around the axle of the beam in such a way that pulling the beam into a horizontal position twisted the ropes to provide the driving force.   Most models today will probably use springs or rubber bands and be of the tensional type.  The simulator models either.   Since the end of the beam must move upward to fling the projectile, spring forces may be applied on the same side of the beam as the projectile and pull upward, or on the opposite side of the pivot and pull downwards.    Our simulator accommodates both "pull-up" and "pull-down" spring arrangements.   

By convention, our catapults throw from left to right with horizontal (X) coordinates increasing to the right and vertical (Y) coordinates increasing upward.   The beam pivot point is the origin of the coordinate system.  

With that background, here are the parameters required to define a catapult:

Catapult Parameters

Pivot height above ground (so we can tell when the projectile hits the ground).
Beam Length
Distance from Pivot to projectile end of beam (negative values =  left of pivot)
Distance from Pivot to Force point  - may be negative, (left of pivot), or positive (right of pivot)
Mass of projectile
Force type (torsion or tension)
For Tension (spring) force
Spring Fixed end point coordinates
Spring Constant : Force per unit length of the driving spring.  Compression springs may be modeled by specifying a negative spring constant.i   
For Torsion force 
Force value.
Mass of beam (On Advanced tab sheet)
Air frictional drag coefficient (On Advanced tab sheet)

Units Choices

Large Metric:  Length in Meters (m); Mass in kilograms (kg); Force in newtons (N).  One newton is the force required to accelerate one kilogram at one meter per second per second.
Small Metric: Length in  centimeters (cm), Mass in grams (g), Force in grams force (gf)      Note that 1 gram force is approximately equal to 1 centinewton.   (1 gram force  = 0.980665 centinewton =  0.00980665 newtons = approximate force of earth's gravity on a 1 gram mass ).
Large English: Length in feet (ft), Mass in pounds (lb); Force in pounds-force (lbf).  One pound force = 4.44822 newtons =  approximate force of earth's gravity on a 1 pound mass ). 
Small English: Length in inches (in); Mass in ounces (oz), Force in ounces force  (ozf). . One ounce force = 28.35 gf = 0.278 N = approximate force of earth's gravity on a 1 ounce mass ).

Here's a link to the force conversion chart  I used.  When changing unit systems, the user can convert existing values or leave them numerically unchanged. 

Simulation parameters

Each of the following parameters can be set for either phase of the simulation: Firing while the projectile is being accelerated, and Free flight after the projectile leaves the beam.  

Max simulated run seconds:  Simulation stops after this many simulated seconds.  Prevent infinite loops in case of program bugs or design errors (for example setting gravity <=0).
Returned samples per second:  This controls the resolution of the simulated results.  Higher values will more accurately determine the stopping point.
Calculated points per returned point:  Acceleration values due to outside forces acting on the projectile (the beam, air drag, and gravity) are assumed to be constant throughout the calculation interval.   Calculating more points for each returned point will increase accuracy  slightly. The default values of 20 returned samples per second and 10 calculated points for each returned point provide a reasonable compromise between  run speed and accuracy. 

The program can save and reload catapult definition files (with ".cat" extension).   The downloadable files include several sample catapult designs.  

I need to go work on my real model catapult now.  Have some fun!  

Non-programmers are welcome to read on, but may want to skip to the bottom of this page to download executable version of the program.

The Runge-Kutta unit previously introduced here on DFF is used  to calculate the acceleration forces.   During the firing phase we need to calculate an angular acceleration value at each calculation time.  Without the details, acceleration is a function of the driving force, the angle that the force is applied and it's distance from the pivot, the mass of the projectile and it's distance from the pivot point, the moment of inertia of the projectile and the  inertia of  the beam.   See the source code for details..  

For each returned point, we get to check the status of the simulation and decide when the projectile is fired.  We'll stop when the beam hits it's stop or when the velocity of the projectile decreases (i.e. acceleration becomes negative).

The free flight phase evaluates differential equations for  horizontal and vertical  movement of a projectile fired and some initial angle and velocity.  This initial trajectory plus gravity and the air drag coefficient influence the path followed.   Simulation stops when the projectile reaches ground level. 

Each of the returned point exits save some data concerning the state of the projectile in an array of position records.  When simulations end, the saved results are used to animate the firing sequence and to display the values at each time interval.

Clicking a Units conversion radio button calls an exit that recalculates unit values, conversions factors and unit names.  All values are converted from user input fields to large metric units internally and results are converted back to the selected units system before display to the user.  

Addendum August 21, 2005:  I completed my model catapult and got to fire the first test shots today!  It's still clamped rather than screwed together and I just approximated the firing angle, spring constant and distance, so there may still be some computational errors in the program, but my approximations  were different enough from the test cases to uncover a couple of minor bugs: Projectile location on the animation was offset vertically between the firing phase to the flight phase.  Also the units label was incorrect for the spring constant -  it showed mass per unit length instead of force per unit length.   There may be additional updates in the coming weeks as I test it against my real catapult. 

Addendum: August 24, 2005:  The version from a few days ago introduced some problems that had been fixed in the original posting -  converting from one unit's system to another changed the catapult's operation.  Fixed today as Version 4.11.   

 I'm still having problems verifying the calculations against my physical model.  Either I'm  measuring the spring constant incorrectly or there's an error in calculating the Moment of Inertia for the beam and mass.  Observed flight distances are about 1/10th the calculated values.     

 Addendum: August 24, 2005:  

Completed the first round of catapult testing this morning.  The results are in the ball park with program results, although there are some anomalies.  Version 4.12 of the program was posted with a small change or two. 

I defined efficiency as average observe throw distance divided by program distance. For the spring constant as I originally measured it, some efficiencies were over 100%, obviously not possible.  I suspect that the inertia of the throwing beam may be underestimated. When I increased the input Spring constant value by 25%, the efficiencies ranged from 77% to 97% with the extremes occurring when using the small (5 gram) weight.   So there may be some issue with the pin that holds the weight on the beam.   I'm not done experimenting, but it looks like results are ballpark, at least for this version.   I can provide some construction and usage tips for anyone planning to build a cat similar to this one.    Just drop me a line.  

Addendum November 7, 2006:  I had some interesting exchanges with Joe L. and his daughter as they worked on building a catapult  for her high school Physics project this fall.  Their construction was quite different from my model, so I asked Joe to write it up for me.  The resulting page is here. 

Running/Exploring the Program 

bulletDownload source
bulletDownload  executable

Suggestions for Further Explorations

 Add pivot bearing friction.
Add second driving force.
Add additional beam shapes for Moment Inertia calculation.
Add sling option. 

 

Original Date: August 01, 2005 

Modified: May 18, 2009

 

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