The ballistics engine of Genesis is a proprietary aeroballistics model based on a Kalman modified filter, which includes numerous proprietary extensions. The only resemblance to the method developed by the late Professor Arthur J. Pejsa ( who finished his research circa 1953 ) is just partially related to the way it handles the drag slope. This proprietary method has no velocity or range restrictions, covering the full spectrum from supersonic, transonic and subsonic trajectories at extended ranges with muzzle velocities starting at any sonic boundary, giving a continuous, non-linear solution. This engine is the result of over 10 years of original research in the field of aeroballistics.
It has been proved time after time its superior predictive capability over the traditional “Point Mass” method that is used on almost any other program, that requires the use of the G7 drag function to yield reasonable solutions in and beyond the transonic range. Even using G7 the results may be not satisfactory, usually calling for the so called “custom drag curves” and the use of “truing” procedures. None of these shortcomings are needed in ColdBore© to produce outstanding predictions.
Many enhancements to the algorithms were developed, using both iterative and recursion approaches to solve the problem of computing downrange figures for velocity, time of flight and drop, the main variables of the solution from which others parameters are derived.
The model is an analytic, closed-form solution that doesn’t use tables generated for standard-shaped projectiles, a fact of paramount importance, since while it does not rely on the G1/G7 drag model used in most of the available programs, (some following the Mayeveski-Ingalls tables), the present method can use a readily available, G1/G7 ballistic coefficient as published, incorporating a custom drag function in order to model the specific projectile.
Since it effectively uses an analytic function ( of Mach number ) in order to match the drag curve of the specific bullet, it does not need to rely on other drag tables to approximate a better fit.
A ballistics software foundation rests in its ability to predict accurate trajectories, matching field results as the ultimate test.
Nearly all ballistic software take for granted that a specific drag function correctly describes the drag characteristics of a bullet as related to its ballistics coefficient, consequently, whether the bullet style is a flat-based, spitzer, a boat-tail, ULD, etc they take care of them to the same drag function as indicated by the published BC, usually the G1/G7 function.
The state of the art mathematical approach taken here, is quite different and a major improvement, of special application to the long range shooter.
Being practical, the use of the published BCs data as available, by almost every major bullet manufacturer is almost mandatory. However this program does not use it to calculate the changes to the drag function over the supersonic velocity range. In other words, this program ballistics engine uses a drag coefficient (DC) that affects the calculated rate of bullet deceleration.
Applying this DC, the resultant trajectory calculations are very accurate for that particular bullet because the drag function has effectively been tailored for that particular bullet.
As the drag coefficient can be modeled as a simple function of the Mach number, the particular functional form of the drag coefficient allows variations in the shape of the drag curve to be considered parametrically.
It’s easily demonstrated that the projectile trajectory and velocity history can be characterized in terms of three parameters: the projectile’s muzzle velocity, a parameter related to the muzzle retardation, and a single parameter defining the shape of the drag curve.
Other approaches ( on the market ) base their predictions on trying to make a match of a specific bullet ( yours ) to like drag data which is available to examine the variation of the drag coefficient with Mach number in the standard axisymmetric ( having symmetry around an axis ) projectile shapes, that were tested in the early to mid-1900’s.
These resulting drag curves are referred to as the Ingalls, G1, G2, G5, G6, G7, and G8 drag curves.
Most bullet manufacturers assume and publish the G1 drag function shape. Examination of the form of a G1 standard-projectile, will to the obvious conclusion that it doesn’t apply to the modern boattail bullet shapes in common use today. The G7 or G8 shape is much more appropriate but general availability of these ballistic coefficients is almost nonexistent.
Particularly, the G1, G5, and G7 curves model a specific bullet form, since they were deduced from, and it’s common for some programs to adapt ( modify ) the G1 drag curve in an effort to fit the G5 to G7 bullet shapes, and in other words, these approaches are, at best an improvement to model modern bullet shapes in order to make predictions more accurate.
That’s why the BC must be defined as a function of velocity. The present solution uses the actual drag, and not rely on an average fit to a non-representative standard or table-based drag function.
In short, what these models can, is to compute accurate trajectory values if range is moderate, up to approximately 600 yards, beyond that, you need a different way to model air drag and to account specifically for a drag curve that matches your bullet dynamics. This is what this method does precisely.
The method used here takes on a different approach, since if the shooter takes the necessary effort to actually measure the Drag Coefficient, the model will make an almost perfect fit, under different environmental conditions, thus yielding very precise downrage values for Velocity, Time of Flight and Drop.
The analysis and prediction of the trajectory of projectiles has been a subject investigated for many centuries and is still a topic of interest today.
The interest and advancement of the problem has come from two fields: mathematics and physics. As a mathematical problem, the focus has been on the methods for solution of the governing equations.
Some of the most noted mathematicians and physicists of the past several centuries, such as Galileo, Bernoulli, and Euler, have investigated the mathematical solution of this problem and have obtained technical advances important for trajectory prediction.
To accurately determine the trajectory of a projectile, one must also properly account for the physical effects such as gravity and the air resistance or drag of the projectile. In additional to his laws of motion, Newton has also been credited for the development of the quadratic law of resistance characterizing the aerodynamic drag of a body.
Further advances have shown that more sophisticated characterization of the drag is required to predict the trajectory across the complete flight regime of many projectiles.
Over a century ago, Mayevski found that it was possible to express the drag of a projectile as proportional to a power of the velocity within restricted velocity regimes. This has been described as the Mayevski Law of Resistance. Piecing together the drag in adjacent velocity regimes using this approach allowed the drag to be characterized over the complete flight regime. Mayevski’s advance led to further developments in trajectory prediction methods.
One of the more famous methods is the Siacci method. The Siacci method was widely used to predict the flat-fire trajectories for decades after its initial development. The method still has some adherents in the sporting and ammunition community.
With the advent of the computer, the Siacci method has been replaced by more modern numerical methods.
These methods allow rapid and accurate computation of the projectile’s trajectory provided the physical characteristics of the projectile (such as the drag) and the atmosphere are appropriately modeled. Modern aerodynamic analysis and trajectory programs provide an excellent means of accurately determining the flight behavior of specific designs.
The increased sophistication of trajectory prediction methods has some unfortunate consequences. The user must often provide an array of details that may or may not be relevant to the answer that is being sought. Additionally, these methods often obscure critical insights into the relationship between important parameters that produce the physical behavior of interest.
For example, there are occasions where a ballistician would like to be able to predict aspects of the flight trajectory without completely defining the geometry or aerodynamic characteristics of a projectile, such as in preliminary design or experimental testing.
In these cases, simplified analyses can provide accurate results with the minimum amount of relevant input from the user and provide the designer with a clearer understanding of the primary design variables.
Drift of a gun projectile is defined as the lateral displacement of the projectile from the original plane of fire due only to the effect of the rotation of the projectile. The principal cause of the drift lies in the gyroscopic properties of the spinning projectile. According to the laws of the gyroscope, the projectile seeks, first of all, to maintain its axis in the direction of its line of departure from the gun.
The center of gravity of the projectile, however, follows the curved path of the trajectory, and the instantaneous direction of its motion, at any point, is that of the tangent to the trajectory at that point. Therefore, the projectile’s tendency to maintain the original direction of its axis soon results in leaving this axis pointed slightly above the tangent to the trajectory.
The force of the air resistance opposed to the flight of the projectile is then applied against its underside.
This force tends to push the nose of the projectile up, upending it. However, because of the gyroscopic stabilization, this force results in the nose of the projectile going to the right when viewed from above.
The movement of the nose of the projectile to the right then produces air resistance forces tending to rotate the projectile clockwise when viewed from above. However, again because of the gyroscopic stabilization, this force results in the nose of the projectile rotating down, which then tends to decrease the first upending force.
Thus, the projectile curves or drifts to the right as it goes toward the target. This effect can easily be shown with a gyroscope.
A more detailed analysis of gyroscopic stability would show that a projectile is statically unstable, but because of the gyroscopic effects, becomes dynamically stable. These effects actually cause the axis of the projectile to make oscillatory nutations about its flight path. Also the Magnus Effect, where a rotating body creates lift (figure 20-9) must be considered. The Magnus Effect is what causes a pitched baseball to curve.