Section I-D: Forces Governing Trajectory
All of the equations listed below are in algebraic form, rather than integrals. This is done in order to present the physics in a easily graspable form.
For all of the calculations I made, I took the axis orientation such that the x-axis dealt with the forward/backward direction of the trajectory (i.e., parallel to the ground), the y-axis dealt with the up/down direction (i.e., perpendicular to the ground), and the z-axis dealt with any lateral movement of the BB (such as due to a crosswind). That is to say that I broke everything up into vector components, although the equations do not reflect this.
Section I-D-01: Drag Force
The force of drag FD can be calculated using the following formula:
FD = 1/2
FD = 1/2
where CD is the drag coefficient. For a non-rotating sphere, CD is constant. I originally estimated CD to be 0.47; however this is not a static number for the analysis as the projectiles are never in a true non-rotating state. In reality, CD tends to fall somewhere between 0.42 and 0.50, depending on the amount of spin imparted. For determining drag coefficient, see below (Section I-D-07: Drag Coefficient),
r (rho) is the air density in kg / m3 (as discussed in Section I-B: Air Density),
A is the cross-sectional area of the BB. In this case, itís simply calculated as a circleís area with a diameter of 6mm. For my calculations, I came up with A = 0.000028274 m2 = 0.000304314 ft2 = 0.043825132 in2
v is the instantaneous velocity of the projectile. Because velocity is derived from the previous drag calculation, an error is introduced. The best way to minimize this error is to use very, very small time intervals for calculating velocity, as well as using and ODE approximation such as the Runge-Kutta method. At time intervals of 0.001 seconds, I found that the error was completely nullified (i.e., by comparing results of velocities calculated using pre- and post-velocity results -- the results were less than 0.1 fps at 150').
Additionally, it is useful to have the equation for calculating Reynolds number for the projectile:
Re = D * r * v * u
where D is the diameter in meters,
r is the air density,
v is the instantaneous velocity, and
u (nu) is the air viscosity (which is 17.4 x 10-6 Pa*s at STP)
Again, I originally estimated a CD of around 0.47. At low speeds, CD is slightly lower, while at high speeds it increases to around 0.5. Technically speaking, a smooth sphere can surpass a critical velocity wherein the CD would be greatly diminished. This is most probable for larger objects, however, and is considered impossible in the realm of airsoft. (For this to happen with a 6mm BB, it would have to moving several times faster than 0.308 bullet.)
Ultimately, CD tends to stay between 0.43 and 0.47, with CD being around 0.47 for BB's with little to no spin, and about 0.43 for BB's with high spin. It is interesting to note that hop-up actually reduces drag when significant spin is applied. For aerodynamic shapes, lift incurs greater drag by inducing early separation of the boundary layer. However, for the rather non-aerodynamic sphere, spin actually increases laminar flow resulting in lowered pressure drag. This is explained in Section III: Effects of Hop-Up. While calculations are continuously made throughout the model run to determine CD accurately, I later found that such calculations were not paramount for determining trajectory nor energy dissipation. Model runs with a constant CD of 0.43 showed a similar trajectory to model runs using a CD of 0.47. Further, the velocity difference at 100 feet when using the two disparate drag coefficients was usually less than 5 fps (and the energy difference was essentially negligible).
To determine velocity, the following equation is used:
Here, the resultant velocity vf is calculated by taking the previous velocity measurement vi and adding it to the velocity change due to deceleration. Pretty straightforward in terms of physics equations. Do note that the acceleration is the average acceleration so, again, using small increments are necessary to minimize any errors.
One thing that should be noted is that velocity changes in a non-linear fashion. What this means is that if you measure your muzzle velocity as 300 fps and find that at 40 feet the BB is moving at 200 fps, you cannot assume that the BB is moving at 250 fps at 20 feet. This is because the velocity curve represents an exponential decay. Reality for the given example is that at 20 feet, the BB would be moving at around 235 fps.
Think of it this way: for a gun firing at 300 fps with a 0.20g BB, at 5í the BB has slowed by 18 fps. While traversing the distance from 5í to 10í, it only slows 16 fps, and during the 10í to 15í interval, it only slows 15 fps. Once the BB approaches a speed under 100 fps, itís only losing about 4 fps for every 5í traveled. Granted, itís moving so slow at that point that itís actually moving downward faster than it is moving horizontally.
Velocity at range is different for BB's shot with and without hop-up. BB's with high spin will experience less drag and will ultimately have a higher velocity downrange. This velocity difference is small, however it is noticeable. At 100 feet, a spinning BB may be moving as much as 15 fps faster than a BB with little or no spin. All velocity listings in Section VIII are for BB's with no spin.
To determine distance, the following equation is used:
* t + 1/2 * a * t 2
Here, distance is incrementally calculated where resultant distance xf is calculated by taking the previous distance measurement xi and adding it to the average velocity and acceleration change. Again, pretty straightforward in terms of physics equations. Do note that here both velocity and acceleration are the averages so, again, using small increments are necessary to minimize any errors, as well as ordinary differential equation techniques.
To calculate Magnus force, I used the following equation:
FM = CL * r * v 2 * A
where CL is the lift coefficient (explained in Section I-D-08: Lift Coefficient),
where r is the air density in kg/m3,
where v is the average velocity in meters per second,
and where A is the cross-sectional area of the projectile.
Keep in mind that the force is going to be orthogonal to the velocity.
Terminal velocity is the maximum velocity an object can reach in freefall through atmosphere. It is calculated by determining what velocity (in the y-, or up/down, axis) is necessary to create enough drag such that drag force (again, in the y-direction) is equal to the force of gravity.
vt = ( ( Fg ) / ( 1/2 * CD * r * A) ) ^1/2
For example, a 0.20g 6mm BB would have a terminal velocity of about 35 mph (or 52 fps) at sea level at room temperature.
Spin decay is the rate at which a solid, spherical object, slows down from a given rotational velocity. For instance, a CD-ROM might have a CD spinning at 15,000 rpm (revolutions per minute). If the CD-ROM were turned off (and the brake disabled), it might take a minute for it to stop spinning with only air friction working to slow it down (there would be mechanical friction, but this is a hypothetical situation). Spin decay governs how quickly factors such as air friction act to overcome rotational inertia and cause the spin imparted upon a BB by the hop-up mechanism to degrade.
To determine spin decay, it's necessary to determine how much toque is induced by air friction. Unfortunately, this is one of the difficult things to determine and some estimations had to be made by looking at trajectories of BB's from a side view. While calculating torque is simple, calculating the friction coefficient is not. BB's, like bullets, exit the barrel with a high amount of spin (though BB's are not nearly as high as bullets). Unlike bullets, which have significant rotational inertia in comparison to the torque induced by air friction, the spin rate of the BB begins to degrade rather rapidly. This is one section that I'll have to come back to once I've determined the actual constants to use for spin decay.
In the interim, I've used modified torque equations to estimate the effects of hop-up. While the estimations are close to reality, they're not as accurate as I'd like (simply because they've been derived from empirical observations as opposed to straight physics). I'm going to hold off on posting the complete set of equations until I've had a chance to fill in all of the coefficients. Once I've determined the proper equations and coefficients, I'll post the estimated constants as well as the correct ones, and will replot the charts only if necessary. I will add that if you're looking to perform graduate work in mechanical or aeronautical engineering and haven't found a topic, there is a dearth of information concerning spin decay for spherical objects. Hint, hint.
For the time being, the current calculation for spin decay depicts trajectories close to reality. Further, the results are very close to other calculations for relatively smooth spheres experiencing high Reynolds numbers (though there is a fourth source that seems to disagree with the other three).
In terms of simplified equations for determining spin decay, the following equations were used:
The angular acceleration a is calculated as
a = t / I
where t is the torque and I is the moment of inertia for a solid sphere.
Torque is calculated as:
t = 1/2 * CT * r * r 3 * w 2
where CT is the torque coefficient,
r is the radius of the sphere,
and w is the angular velocity.
The torque coefficient CT is generally calculated as
CT = 6.45 / ((Rew)^1/2) + 32.1 / Rew
where Rew is the Reynolds number for centerline rotation.
The Reynolds number for centerline rotation Rew is generally calculated as
Rew = r * r 2 * w / hf
where hf is the viscous friction coefficient.
As stated earlier, the drag coefficient is not a static number. As the rotational velocity V changes with respect to linear velocity U, so does the drag coefficient. Both the drag coefficient and lift coefficient have been studied and determined, most notably through the research of Achenbach* and Mehta**. Fortunately for us, Dr. Gary Dyrkacz has has taken the older plots and, using SigmaPlot, determined the polynomials necessary to calculated both CD and CL using data from Davies' study of golf balls***.
(If you get a chance, visit Dr. Dyrkacz's page on The Physics of Paintball as his page describes in detail what happens to a projectile such as a spinning paintball as it moves through the air, as well as providing the calculus-based equations which are more useful to us even though they're much harder to type!)
CD is initially calculated without spin as CD0, and is determined by the equation:
CD0 = ( 0.4274794 + 0.000001146254 * Re - 7.559635 x 10-12 * Re2 - 3.817309 x 10-18 * Re3 + 2.389417 x 10-23 * Re4) / (1 - 0.000002120623 * Re + 2.952772 x 10-11* Re2 - 1.914687 x 10-16 * Re3 + 3.125996 x 10-22 * Re4)
Where Re is the Reynolds Number.
With spin, CD is calculated using the equation:
CD = ( CD0 + 2.2132291 * V/U - 10.345178 * ( V/U )2 + 16.157030 * ( V/U )3 - 5.27306480 * ( V/U )4) / (1 + 3.1077276 * ( V/U ) - 13.6598678 * ( V/U )2 + 24.00539887 * ( V/U )3 - 8.340493152 * ( V/U )4 + 0.07910093 * ( V/U )5);
where V is the rotational velocity and U is the linear velocity.
Notice that a sphere with a high amount of spin would have a CD of around 0.43, marginally less than the 0.47 I used for non-spinning spheres.
Again, I had to use Dr. Dyrkacz's polynomial to calculate CL.
CL = (-0.0020907 - 0.208056226 * ( V/U ) + 0.768791456 * ( V/U )2 - 0.84865215 * ( V/U )3 + 0.75365982 * ( V/U )4) / (1 - 4.82629033 * ( V/U ) + 9.95459464 * ( V/U )2 - 7.85649742 * ( V/U )3 + 3.273765328 * ( V/U )4);
Gravity can normally be assumed as a constant mass times acceleration where the gravitational acceleration is about 9.8 m /s 2. However, gravity can be more accurately calculated (and was done so in the model) by using the following equation:
Ag = 9.7803185 * [ 1 + ( 0.005278895 * ( sine (Lat) ) 2 ) - 0.0000589 * ( sine (2 * Lat) ) 2 ) ]
where Lat is the latitude in degrees.
The acceleration due to gravity varies from about 9.78 m /s 2 at the equator to about 9.83 m /s 2 at the poles. In truth, while the model did take into account the variation of the gravitational acceleration in accordance with latitude, it was not necessary to do so. The effects of gravity at various latitudes has a miniscule effect on trajectory; using a constant of 9.8 m /s 2 would have been sufficient.
* "Experiments of the flow past spheres at very high Reynolds numbers," Achenbach. E., in American Journal of Physics, 54, 565-575 (1972).
** "Aerodynamics of sports balls," Rabindra D. Mehta, in Annual Review of Fluid Mechanics, 17, pages 151-189 (1985).
*** Davies, J.M., The Aerodynamics of Golf Balls, Journal of Applied Physics, 20, pages 821-828.
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