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Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

-A novel and practical approach for computing the Spin Drift perturbation-

James A. Boatright & Gustavo F. Ruiz

▪ Introduction

▪ Framework of the Analytical Solution

▪ The Horizontal Tangent Angle

▪ PRODAS Simulated Flight Data

▪ Yaw of Repose

▪ Estimating the Yaw of Repose

▪ Analysis of the Spin Drift

▪ Analytic Calculation of the Spin Drift at the Target

▪ Estimating the Ratio of Second Moments of Inertia for Rifle Bullets

▪ Estimating the Spin Drift Scale Factor ScF

▪ Calculating the Spin Drift at the Target

▪ Example Calculations of ly/lx

▪ Example Calculations of Spin Drift

▪ Sensitivity analysis & Model comparisons

▪ Closing summary

Introduction

The Yaw of Repose angle R is a very small, but gradually increasing, horizontally rightward,

aircraft-type yaw--

coning axis

of a right-hand spinning

bullet. The Yaw of Repose reverses sign and angles leftward for a left-hand spinning bullet.

We discuss only right-hand spinning bullets here for clarity. It can be shown that for right-

hand twist, the yaw of repose lies to the right of the trajectory. Thus the bullet cones around

with an average attitude offset to the right, leading to increasing side drift to the right caused

by a small rightward net aerodynamic lift-force. As we shall show, the yaw of repose is

caused by the downward curving of the trajectory due to gravity. The yaw of repose is

constrained to lie in a plane perpendicular to the gravity gradient.

For spin-stabilized bullets, this small rightward yaw attitude bias creates the well known

rightward Spin Drift displacement. The small horizontally rightward yaw of repose angle

causes a small rightward aerodynamic lift force which, in turn, causes a slowly increasing

horizontal velocity of the bullet.

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This effect occurs independently of the presence of surface wind of any force or from any

direction. Bear in mind that the yaw of repose represents the horizontal yaw attitude of the

average yaw of the coning bullet itself.

The acceleration of gravity acting upon the flat-fired bullet in free flight is the original cause

of this small yaw-attitude bias angle. The downward curving of the trajectory due to gravity

causes the airstream passing over the bullet to approach from below the nose of that bullet.

FIGURE: Extreme TDC and BDC Positions of Coning Bullet

This wind shift during each coning cycle causes an increased aerodynamic angle of attack

which peaks as a

maximum

when the center of gravity (CG) of the bullet is at the Bottom

Dead Center (BDC) position in each coning cycle where its nose is oriented maximally

upward.

Since the coning angle always exceeds this small change in the approaching airstream

direction during each coning cycle, the aerodynamic angle of attack for a bullet at Top Dead

ownward is at a

minimum

for

that coning cycle, but that airstream continues to approach the bullet from below its axis of

symmetry, even when the bullet is flying with its minimum possible coning angle.

The airflow approaching the coning bullet from beneath its coning axis does produce

overturning torque vectors also lying in the horizontal plane when the coning bullet is

located in the horizontal plane of the coning axis. These moments continually enlarge the

coning motion to allow the coning axis to reorient itself into the new apparent wind

direction. It is only the vertical-direction modulations of overturning moments which must

be considered here, and they change sign and go through zero in the plane of the coning axis.

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Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

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These modulations of aerodynamic angle of attack during each coning cycle produce small

differential rightward-acting increments in the aerodynamic overturning moment

experienced by the bullet which are centered upon the BDC or TDC positions of the bullet

during each (lower or upper) half coning cycle.

In physics these recurring differential torques

pushes the forward-pointing angular momentum vector of the right-hand spinning bullet

horizontally rightward without affecting its spin-rate or magnitude. Each torque impulse is

evaluated by integration over the time of its half-coning cycle.

Both, the rotating overturning moment vector M due to the coning angle of attack and its

differential torque impulse vector inherently point

positive

rightward for the bullet at its

BDC location. While the moment vector M itself points leftward at TDC, its

negative

differential torque vector is

positive rightward

as well at TDC. The differential torque is

negative leftward

at TDC because the aerodynamic angle of attack when the bullet is located

there (or anywhere in the top half of the coning cycle) is

less

than the coning angle itself. The

aerodynamic overturning moment is an

odd function

in the signed aerodynamic angle-of-

attack, but we do not need to use that concept here.

Each rightward torque impulse tugs the forward-pointing angular momentum vector L

slightly rightward along with the nose of the spinning bullet. The angular momentum vector

L points

forward

along the spin-axis for a right-hand spinning bullet, as discussed here, and

rearward for a left-hand spinning bullet. Each torque impulse is constrained to lie in the

horizontal plane perpendicular to the gravity gradient because of the vector cross-product

physical definition of torque.

The 175.16-grain M118LR 30-caliber bullet used as an example here is experiencing its 88th

coning half-cycle when it reaches its target distance of 1,000 yards. The reinforcing

cumulative effect of these rightward torque impulses occurring twice per coning cycle is the

mechanism by which the downward arcing of the trajectory due to gravity causes the slowly

increasing rightward yaw of repose attitude bias of the flying bullet.

The epicyclic motion of the spin-axis direction of a typical right-hand spinning rifle bullet is

shown below for the first hundred yards, or so, of its flight. The gyroscopic stability Sg of this

bullet at launch is about 1.33

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Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

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Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

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General form of the yaw of repose, as described by BRL.

If the 3-dimensional mean trajectory of a rifle bullet in nearly horizontal flat firing is

projected down onto a horizontal plane, the rightward deviation T of its tangent direction

from the firing azimuth essentially defines this yaw of repose angle R throughout the flight,

except for an even smaller horizontal dynamic tracking error angle H as the trajectory

curves to the right

following

(but lagging behind) the slightly larger yaw of repose angle R:

R T H (1)

We will formulate a good approximation for T as an aid in formulating R accurately. The

horizontal tracking error angle H is

inherently non-negative

(H) for right-hand spinning

bullets. The angle T also defines the horizontal-plane orientation of a mean CG-centered

coordinate system moving with the +V direction of the flying bullet, with respect to the

firing-point-centered earth-fixed coordinate system in which the trajectory is measured.

The yaw of repose has two effects on the trajectory of the projectile: 1) it produces a lateral

lift-force that results in a the projectile drifting rightward (for a right-hand spinning

projectile); and 2) it increases the total drag due by a small additional yaw-drag component.

The additional lift is of a very small magnitude, but cumulatively causes the rightward

horizontal spin--range trajectory. The accompanying

additional yaw-drag component is an even smaller second-order term; thus, it is omitted.

Any aerodynamic lift is always accompanied by some increase in aerodynamic drag.

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Framework of the Analytical Solution

The horizontal spin-drift SD which we observe in long-range shooting is due to a horizontally

acting aerodynamic

lift force

attributable to the increasing yaw of repose attitude angle R of

the coning axis of the rightward-spinning rifle bullet.

We will use the principles of linear aeroballistics in formulating the yaw of repose R and its

resulting spin-drift SD.

Detailed analyses of PRODAS 6-DoF simulation runs show that in flat firing the magnitude of

the spin-drift SD in any given simulated firing is, beyond the first 150 yards or so,

very nearly

equal

to some invariant scale factor ScF of about 1.0 to 2.4 percent, more or less, times the

drop from the projected bore axis

:

SD(t) = -ScF*DROP(t) (2)

In other words, the horizontal spin-drift trajectory looks just like a small fraction ScF of the

vertical trajectory rotated 90 degrees about the extended axis of the bore with each

curvature ultimately caused by the same gravitational effect. The ratio of drift to drop rapidly

approaches some particular ScF value for any rifle bullet asymptotically beyond the first 150

yards of that

Our task is to formulate the scale factor ScF so that it can be accurately evaluated for any

given bullet type and firing conditions.

Then using Eq. 2

axis at the target distance to calculate an accurate spin-drift at any long-range target.

Existing 3-DoF trajectory programs specialize in the accurate calculation of this

bullet drop at the target distance in any firing conditions.

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Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

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The Horizontal Tangent Angle

The instantaneous tangent to the horizontal-plane projection of the

mean trajectory

forms

the angle T(t) to an X-axis which defines the launch azimuth of the fired bullet in that

horizontal plane. The mean trajectory of the bullet is the 3-dimensional path of the mean CG

location, and which path would have been followed by the CG of the bullet if it were not

coning about that mean trajectory. The mean trajectory is the path of the mean CG of the

coning bullet.

This horizontal tangent angle T(t) is always defined by the horizontal projection of the

V(t), but these mean velocity components are not calculated in

cross-track velocity components

are modulated by the helical coning motion of the CG of the bullet in flight.

Another important use for this horizontal tangent angle function T(t) in ballistics is in

plotting the horizontal yaw-attitude of the spin-axis of the bullet -

versus-yaw plots long used by ballisticians.

-axis direction is corrected by

subtracting out the total change since launch Total(t) = (t) - (0) in the vertical-plane

mean flight path angle before plotting the pitch data for a 6-DoF simulated flight, the total

change since launch in the mean -angle T(t) should also be

subtracted out before plotting -- in the interest of logical

consistency. However, this correction is not being done in contemporary aeroballistics. The

resulting logical inconsistency stems from not fully understanding the coning motion of the

flying bullet. The error persists because the angles involved are usually quite small.

With this change being made, the origin of the wind axes plots could truly be defined

(horizontally as well as vertically) as the instantaneous +V

mean

trajectory

. Only the horizontal dynamic tracking error angle H(t) would remain in the

plotted yaw-attitude values instead of the entire yaw-of-repose angle R(t). A similar small

positive upward

vertical-direction dynamic tracking error angle V(t)

is

currently shown in

these wind-axes plots of 6-DoF simulation results.

Because the scale factor ScF in Eq. 2 is essentially invariant over time t and distance x(t) at

long ranges, we can evaluate the s horizontal-plane tangent angle T(t) directly

-plane DROP data in suitable distance units at any time t during its

flight, by utilizing the small angle approximation that in radians:

T(t) = dSD/dx = -ScF*[d(DROP)/dx ]= - = -(t) (3)

The x-derivative of DROP(t) can easily be seen to equal the Mean Flight Path Angle in

horizontal firing when , but it is clearly also when , whenever

(0) is small as in the flat-firing cases being considered here.

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If the scale factor ScF is known, t

tangent angle in flat firing from either the ratio of its vertical drop rate to its forward velocity

at any point during its flight or from the total change (t) in the vertical-plane mean flight

path angle since launch.

DROP(t) data rapidly fades to an

DROP distance from the axis of the bore as the flight

progresses.

The differential DROP-rate and remaining forward velocity V(t) are readily found from any

simulated flight data. Calculation of the invariant scale factor ScF for any particular flight

trajectory is discussed later in this paper.

Alternatively, one could evaluate T(t) directly in the horizontal plane. As the bullet drifts

horizontally due solely to spin-drift SD(t), the intersection point with the X-axis of the

in the horizontal plane moves forward in the +X

direction, but at a faster velocity than the forward velocity V(t) of the bullet itself. This

intersection point starts about 150 yards later, but never quite catches up with the X-

coordinate of the bullet in flat firing.

If we assume a continually increasing curvature of the horizontal trajectory so that this

velocity ratio varies exponentially with range X(t), we can estimate T, the dominant portion

of R, as:

T T) = SD(t)/{X(t)*0.825*exp[-0.925*X(t)/X(max)]} (4)

This hand-fitted estimator function agrees quite well with T(t) angular values extracted

from available trajectories generated by PRODAS 6-DoF simulations for the 1000-yard flight

of our particular long-range rifle bullet by ratioing an extracted rightward VR(t) to V(t) for

each millisecond of the PRODAS trajectory reports.

The horizontally rightward velocity component data VR(t) is extracted by applying a

ta converted

into linear units.

Comparing the two T functions for each millisecond over the 1.6923-second simulated flight

time yields a mean difference of 1.12 micro-radians with a population standard deviation of

0.0514 milliradians.

Extraction of the small rightward horizontal velocity VR from the trajectory drift data is

complicated by the superimposed epicyclic swerving of the CG of the bullet which accounts

for most of the variance between these two functions.

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The two approaches in Eq. 3 and Eq. 4 for evaluating T(t) agree reasonably closely for

PRODAS data as the epicyclic swerve modulations in SD(t) and DROP(t) dynamically damp

out and fade into insignificance with ongoing flight time t.

We formulate these approximations for T(t) so that they can be used as reasonableness

checks on calculations of the yaw of repose R(t) which is not itself reported by PRODAS.

We will eventually need an accurate formulation for R(t) in order to calculate the scale

factor ScF and thence the spin-drift SD(t) for other rifle bullet trajectories without relying

upon 6-DoF simulation data.

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PRODAS Simulated Flight Data

In this paper we will use as our example bullet the 30-caliber 175.16-grain

bullet as was loaded in their M118LR Special Ball (7.62x51 mm NATO) long-range

sniper and match ammunition in 2011.

We do this because we have several PRODAS 6-DoF simulation runs on hand from 2011 for

this 7.62 mm NATO ammunition, reporting the linear ballistic results (including spin-drift)

for each millisecond of its 1.6923-second total simulated flight time to 1000 yards.

The bullet weight actually used in these PRODAS runs is 175.16 grains. The simulated firing

conditions are 1) flat firing, 2) standard sea-level ICAO atmosphere, 3) no wind, 4) no Coliolis

effect calculated, 5) muzzle velocity of 2600.07 feet per second, and 6) barrel twist is right-

handed at 11.5 inches per turn.

---drift SD is the only

secular horizonted by PRODAS.

However, the PRODAS reported drift and drop data necessarily include the oscillating

horizontal and vertical components

trajectory throughout its simulated flight. We also have PRODAS runs available for this same

bullet fired through constant left and right 10 MPH crosswinds as well as left-hand twist runs

in each of the three constant wind conditions.

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Yaw of Repose

We will show that in flat firing the continual downward arcing of the flight path angle due

to gravity causes repeated rightward differential aerodynamic torque impulses centered

about the extreme top-dead-center (TDC) and bottom-dead-center (BDC) positions of the

CG of the bullet in its coning motion.

These double-rate yaw attitude-changing horizontal torque impulses cause the forward-

pointing angular momentum vector L of the right-hand spinning bullet to shift horizontally

evermore rightward during its flight. In light of Coning Theory, we should more precisely say

coning axis

drifts horizontally rightward in its yaw attitude throughout the flight.

Ballisticians term this accumulating yaw-yaw of reposeR of the

flying bullet and classically formulate its horizontal component from calculus as [Eq.10.83 in

R = P*G/M (5)

This expression is the horizontal part of the

particular solution

for the differential Equations

of Motion which determine each trajectory in terms of the classic aeroballistic auxiliary

parameters:

P = 1 2)*d/V (6)

G = g*d*Cos/V2 2 (7)

M = (m*d21 22*d2/V2 (8)

after converting each classic auxiliary parameter from dimensionless (canonical) arc-length-

rates into the time-rate units used in our physical analyses of flat-firing a spin-stabilized rifle

bullet.

The change-of-variables in Eq. 6 uses one of the gyroscopic relationships from Tri-Cyclic

Theory [Harold Vaughn of Sandia Labs and Dr. John D. Nicolaides, 1953] that:

(I1 2 = 2*(R + 1) (9)

where R = 1/2 is called the stability ratio which is perfectly 1:1 corelated with the

gyroscopic stability Sg = (R + 1)2/(4*R).

McCoy defines the spin-rate p of the bullet as used here to be a circular frequency given in

-rate p is sometimes given elsewhere in aeroballistics in

units of revolutions per second (or hertz), and is sometimes given in radians per foot of bullet

travel, or even in radians per caliber d of bullet travel.

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To avoid this confusion we use the more conventional symbol here for the circular

frequency of the spin-rate of the bullet given in radians per second. We also use the symbol f

for rotation rates as frequencies in revolutions per second or hertz.

The change of variables in Eq. 8 uses the fundamental magnitude relationship from Coning

Theory that:

22 2 (10)

as well as the Tri-Cyclic relation in Eq. 9 again. Note that this Coning Theory relationship

implies that the slow-mode coning rate 2 in radians per second is always given by

(M/P)*V/d in terms of the canonical aeroballistic auxiliary parameters M and P.

With these changes of variables, the classic formulation for the yaw of repose angle R

reduces to:

R 22(t)*V(t)] (11)

While this formulation for R is classic, it

does not inherently yield

zero

at

t = 0, and it is

about a factor of too small at long ranges when compared to T as formulated above [Eq. 1

and Eq. 3].

Let us say the mean flight path angle s downward by a

small decrement due solely to the pull of gravity (as with a vacuum trajectory) during

one-half of the period T2 of a particular coning cycle. As a continuous variable in flight time

t, this angular decrement at t = 0 by definition.

In flat firing, the small decrement in the nearly horizontal flight path angle during

the time interval T2/2 of a particular half-coning cycle can be expressed as:

-(g*T2)/[2*V(t)] = -g/[2*f2(t)*V(t)] = -2(t)*V(t)] (12)

where f2(t) is the instantaneous coning rate, or gyroscopic precession rate, of the bullet given

in revolutions per second, or hertz. [Here we are ignoring the significant cross-bore-axis

(upward) component of the real s ballistic drag force FD in interest of formulating a

simple SD estimator. This oversimplification will be explained and dealt with later.]

Comparing our version of the classic formulation for the steady-state yaw of repose R(t) in

Eq. 11 with the change in flight path angle due solely to gravity

per half-coning cycle

above, we note that:

R(t) = (- (13)

Thus, our formulation in Eq. 12 above for , the change in flight path angle per half-

coning cycle T2/2 which

does

inherently equal zero at t = 0, actually looks like a more

suitable formulation for R(t) than does the classic form.

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We will now investigate the aerodynamic and gyroscopic causes of R(t) so that we can

formulate its value at any time t during the flight of any rifle bullet.

At each extreme vertical location, TDC and BDC, the coning bullet experiences a peak rate of

differential change in its aerodynamic overturning moment vector M due to this differential

change in its vertical-direction (upward airflow) aerodynamic angle-of-attack. Each of

these two differential torque impulse vectors points

horizontally

rightward

as seen from

behind the right-hand spinning bullet.

Here these two differential torque

impulse

vectors are to be evaluated by integrating the

differential torque over each half (upper or lower) of the coning period T2, giving them units

of torque multiplied by time which correspond with the units of angular momentum.

Owing to the increased aerodynamic angle-of-attack of the apparent wind experienced by

the bullet at its BDC position, the differential torque impulse at BDC is

inherently positive

rightward

, temporarily increasing the overturning moment M acting upon the bullet at this

BDC location in its coning motion.

While the overturning moment vector M itself points

leftward

at the TDC position of the

bullet, the differential torque impulse vector is

inherently negative

due to the

reduced

aerodynamic angle-of-attack experienced by the coning bullet at that upper location, and so

the differential torque impulse vector itself points

positive rightward

, once again, at TDC.

Thus, the alternating sequences of TDC and BDC differential torque impulses are

mutually

reinforcing

t

Recall that in Coning Theory the spin-axis of the bullet is pointing maximally

upward

when

the CG of the bullet is at its BDC position in any coning cycle; i.e., its aerodynamic pitch

attitude is a relative maximum during that coning cycle.

As formulated in linear aeroballistics, the instantaneous magnitude {M} of the overturning

moment M at time t is:

{M} =

Where

2 = Dynamic Pressure in lbf/square foot.

Density of the atmosphere = 0.0764742 lbm/cubic foot for the standard sea-level ICAO

atmosphere used here. This value of the density must be divided by the acceleration of

gravity g = 32.174 feet per second per second to convert its units into proper density units,

mass (slugs) per cubic foot.

V = Airspeed of the bullet in feet/second.

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S = Reference (frontal) area of the bullet at the base of its ogive in square feet = 2.

d = Diameter of the bullet in feet.

Coning angle (and aerodynamic angle-of-attack) of the bullet in radians at any time t

during its flight.

Dimensionless overturning moment coefficient in linear aeroballistics theory.

Here we are ignoring the fast-mode gyroscopic nutation 1(t) -axis for

several reasons: 1) It does not normally move the CG of the bullet by any measurable amount,

2) Its aeroballistic effects tend to average out to zero rather rapidly, and 3) It rapidly damps

to insignificance for most rifle bullets after any flight disturbance.

As the flat-firing trajectory of the coning bullet, flying essentially horizontally near the X-axis

(with and with its coning axis aligned into the approaching windstream), arcs

downward due to gravity, the aerodynamic angle-of-attack increases by the magnitude

of at its BDC location in this coning cycle.

The cosines of small coning angles , the flight path angle , and the small

change in flight path angle all remain essentially equal 1.00. From trigonometry, the peak

magnitude PEAK of this differential overturning torque with the bullet at its BDC

location can be expressed as:

Sin( + ) = Sin()*Cos() + Cos()*Sin(in() + Sin()

M + {M}PEAK = q*S*d*Sin( + )*CM in()*CM

PEAK = q*S*d*Sin (14)

This expression can also be well approximated as:

PEAK (15)

The instantaneous vertical-direction aerodynamic angle-of-attack is actually the vector sum

of three small angles in complex wind-axes coordinates (ignoring the fast-mode 1 motion):

1. Vertical component of the slow-mode coning angle, os20)

2. Downward change in flight path angle , and

3. Very small vertical-direction tracking error angle V (upward in wind-axes plots). This

vertical-direction tracking error angle V -of-

The primary overturning moment M is due to (1.) the coning angle-of-attack . This

rotating torque vector M produces the slow-mode circular coning motion of the CG of the

bullet at the coning rate 2(t) of the bullet as a gyroscopic preces-

axis.

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Examination of several different PRODAS runs shows that even for a

dynamically stable

bullet with any early coning motion fully damped down, (t) always exceeds by some

margin all the way to maximum supersonic range and beyond.

From Coning Theory, the vertical component of the complex coning angle

(t) given by the

real part

of the complex , again neglecting the fast-mode motion:

Re

os20) (16)

Whenever {}, only the portion of equal in magnitude to produces the

differential overturning moment impulse which drives the spin-axis of the bullet

Yaw of Repose angle R, and the overturning moment

impulses at BDC and TDC can be modeled as having the same form.

The excess of (t) over goes toward enlarging the coning angle , counteracting any

frictional aerodynamic damping of that slow-mode coning motion of the bullet.

The instantaneous differential overturning moment { is then due to the vertical-

direction differential aerodynamic angle-of-attack (t)*Cos20).

This modulation at the coning-rate 2 looks like a full-wave-rectified sine wave over each

coning cycle. The time-average over each quarter wave is just times the peak value.

The average value of itself over each half-coning cycle is just because the flight path

angle varies almost linearly over the small interval T2/2. Averaged over the top or bottom

one-half of a coning cycle, the average effective angle-of-attack is then (.

The vector sum of (2) and (3) V varies only gradually with ongoing time-of-flight t. The

magnitudes

of these two small angles

sum

to an average vertical-direction aerodynamic

angle-of-attack which drives the coning-axis direction

continually downward

according to

Coning Theory, dynamically tracking (but lagging behind) the downward-curving trajectory.

The time-integrated

torque impulse

centered at TDC or BDC must equal the differential

torque due to the time-average of the modulated aerodynamic angle-of-attack

multiplied by the total time interval T2/2 for each half-coning cycle. The interval T2 increases

gradually as the coning rate 2(t) slows throughout the flight.

The effective differential torque impulse integrated over a particular

half-coning cycle

thus becomes:

2 (17)

Substituting the unsigned

magnitude

of the first expression for from Eq. 12 yields:

2/2)2 (18)

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This differential torque impulse has units of lbf-feet-seconds which can be converted into

slug-feet squared per second, a proper set of units for angular momentum.

The right-hand spinning bullet alters its pointing direction

rightward

in gyroscopic reaction

to each of these two differential torque impulses during each coning cycle. However, it

does so in an unusual way.

When a constant-magnitude, rotating overturning moment vector M is applied to a spinning

gyroscope, its spin-axis direction soon begins moving in precession and nutation in reaction

to that steadily rotating torque vector.

However, the first motion of its spin-axis is always in the direction of the eccentric force

producing the overturning moment M while those epicyclic motions are getting started.

For the spinning bullet, the eccentric force is the total aerodynamic force F, a line vector

acting through the aerodynamic center-of-pressure CP of the bullet at any instant during its

flight. For spin-stabilized, rotationally symmetric rifle bullets, the CP is nearly always located

forward of the CG along the spin-axis of the bullet.

In response to each small torque

impulse

, the spin-axis of our bullet moves

initially

rightward

, but each impulse ceases well before any vertically upward or downward

movement of the spin-axis can become established.

When the torque impulse vector is expressed in the same units as the angular momentum

vector L of the spinning bullet, having physical dimensions of mass times length squared over

time, their

direct vector sum

defines the resulting angular momentum L of the spinning

bullet after the torque impulse has been applied.

For a right-hand spinning bullet the angular momentum vector L points forward along its

spin-axis. Here, since is always acting perpendicularly to L, the

direction

of the angular

momentum vector L is shifted rightward by an incremental angular amount (in radians),

which we term R, but its

magnitude

remains unchanged.

Of course, the nose of the right-hand spinning rifle bullet in stable supersonic flight always

points in the direction of its angular momentum vector L.

The incremental increase R in the yaw of repose angle R during each

half coning cycle

is

thus:

R anR2/2)2*q (19)

Recalling Eq. 22 from the Coning Theory paper, we note that the right-hand side of Eq. 19

above contains the fundamental expression from Coning Theory for determining the

magnitude

of the circular coning rate 2(t) for a spin-stabilized bullet coning at non-zero

angles of attack :

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2 , L

Due to the acceleration of gravity, the coning angle cannot be zero in flat firing except

perhaps very briefly at t = 0, where this magnitude relationship still holds true.

After this change of variables,

R 222(t)/[4*V(t)] (20)

This change of variables is critically important in formulating an analytical calculation of R

because it simultaneously eliminates from the formulation both the overturning moment

coefficient and the angular momentum L of the bullet, each of which is difficult to

calculate for a new bullet. The coning rate 2(t) is more readily obtainable from Tri-Cyclic

Theory.

Also recall that by definition T22 = 1/(f2)2 2/22. After this substitution we have:

R 2(t)*V(t)] = g/[2*f2(t)*V(t)] = g*T2/[2*V(t)] = - (21)

While this expression is dimensionless, the increment in the yaw of repose angle R for each

half-coning-cycle T2/2 is calculated here in radians. The proper algebraic sign depends upon

-rate.

In linear aeroballistics theory, the instantaneous aerodynamic lift-force driving the spin-drift

SD of the bullet horizontally rightward from the X-axis is

linearly

proportional to the

aerodynamic angle-of-attack for the very small yaw of repose angle R.

Thus, the linear dependence of R upon shown in Eq. 21 explains the remarkable

similarity in shape of the horizontal-plane and vertical-

-space.

We could have arrived at the result shown in Eq. 21 more directly by assuming unrealistically

that the flying bullet was not coning, but simply spinning about its coning axis direction with

a zero coning angle , or by assuming that a non-zero coning angle does not matter. But

then we would have to validate either of those assumptions as we have done above.

For minimum coning angle flight, when as becomes the case eventually in most

-DoF simulations, the average torque impulses are no longer precisely

symmetrically equal at BDC and TDC. In fact, , and their combined

average effect would be slightly smaller (by about 5 percent) than these estimates here

yielding a maximal yaw of repose angle.

The yaw of repose angle R(t) can be found by

summing

the increments R divided by T2/2

for each

half coning cycle

which has occurred from t = 0 to time t, starting with R(0) equal

zero.

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for this 30-caliber bullet, yields

R(1.430 sec) = 0.67208 milliradians, which exceeds our fitted value of T(1.430 sec) =

0.61019 mrad by 10.14 percent.

We term this difference, R(t) T(t), the

horizontal tracking error angle

H(t). We are

comparing these angles here at t = 1.430 seconds after launch when this M118LR bullet has

slowed to Mach 1.20 or 1340 feet per second at 888.5 yards downrange in these simulated

firing conditions.

Using the PRODAS-calculated velocity and coning-rate data, our adjusted version of the

classic formulation of the yaw of repose yields R

milliradians, which exceeds our fitted value of T(1.430 sec) = 0.61019 mrad by 15.76 percent

for the horizontal dynamic tracking error angle H. We believe this adjusted classic

formulation for R better matches the case for a significantly coning bullet than for this

particular minimally coning PRODAS trajectory.

Projection of the Mean Trajectory onto the Horizontal Plane.

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Estimating the Yaw of Repose

In the absence of having 6-DoF simulation data available, we could approximate the yaw of

repose angle R(t) by assigning readily integrable (in closed form) continuous functions of

time t to represent the variables 2(t) and V(t) in Eq. 21 so that we could then approximate

this

summing

operation by performing the definite integration of R(t) over time from t =

0 to time t and dividing the integrated result by the total time interval t:

R2(t)*V(t)]-1 dt (22)

Here the extra factor of 2 in this expression for R(t) in Eq. 22 versus the expression for R(t)

in Eq. 21 is due to integrating continuously rather than using its

average value

over each half-coning cycle.

Note that the size of the yaw of repose angle R(t) whenever depends only on the

velocity V(t) and coning rate 2(t) of the bullet as functions of time. In particular, R(t) in

this formulation is independent of the coning angle itself in this analysis.

Since the spin-drift displacement SD(t) is caused directly by this yaw of repose angle R(t)

as an aerodynamic lift effect, evaluation of the spin-drift SD(t) does not require detailed

. This independence of R(t) is significant because

the coning angle is a

free variable

in Coning Theory and is thus difficult to evaluate

analytically except in special cases.

If we find the values of 2(t) and V(t) at t = 0 and at a much later flight time t = T, and we

assume for approximation purposes that each function decays exponentially with time t,

then the definite integral for R(t) can be expressed as:

R(t) 2-V)*t/T] dt (23)

with

22(0)]

22 (24)

The decay in bullet spin-rate d/dt is due to skin friction, a torque S about the x-axis of the

bullet, which opposes and slows its spin-rate :

S = dL/dt = d/dt[Ix*] .

This skin friction torque S itself is proportional to the circumference of the bullet .

The second moment of inertia of the spinning bullet about its spin-axis is given by

Ix = m*d2 *kx2

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So, the time rate of change in spin-rate is inversely proportional to the caliber d-1 of

the bullet.

The decay in the spin-rate (t) for any modern rifle bullet in flight is closely approximated

by the exponential expression:

(0)*exp[-(0.0321/d)*t] (with d = caliber in inches)

The indicated spin decay-rate coefficient -0.0321/d closely matches the spin-rates shown

throughout the PRODAS runs for the M118LR bullet having 0.308-inch diameter d.

From the Tri-Cyclic Theory, we can evaluate the coning rate 2(t) as

2(t) = (Ix/Iy)*/[R(t) + 1]

22(0) = [T)/0)]*{[R(0) +1]/[R(T) + 1]}

ln[22(0)] = [-(0.0321/d)*T] + ln{[R(0) +1]/[R(T) + 1]}

-(0.0321/d)*T] +[-0.585*T] = -(0.585 + 0.0321/d)*T

This approximation provides a better than 5-percent fit to the coning rates 2(t) calculated

in PRODAS for this M118LR example bullet for each millisecond of its flight to 1000 yards,

so we shall use this approximation as well for other rifle bullets pending further analysis.

We also approximate the decay-rate in bullet velocity V(t) as exponential in time t:

kV = ln[V(T)/V(0)]

V(t) V(0)*exp[kV*t/T] (25)

Here we are using t/T as a dimensionless canonical variable in the exponential decay

expressions and as a dummy variable in the (summing) integration.

After the definite integration from t = 0 to t =T, the expression for R(t) is:

R(t) = {-2V)]}*{exp[-V)*t/T] - 1} (26)

Note that R(0) = 0.00 as we require here.

anywhereor this M118LR bullet show that V(t) slows from an

initial velocity of 2600.07 feet per second to 1340 FPS (Mach 1.20) at 888.5 yards downrange

with a time-of-flight (T) of 1.430 seconds, and that the coning rate 2(t) of the bullet slows

from ians per second to over this same interval T.

The yaw of repose angle R(T) at T = 1.430 seconds, and at 888.5 yards downrange, would

then be calculated as:

-(0.585 + 0.0321/d)*T = -0.98559

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kV = ln[V(T)/V(0)] = -0.66328

R(1.430 sec) = [1.6469x10-4]*[4.2011] = 0.69186 mrad = 0.039641 degrees (27)

While PRODAS does not report R, this small 0.040-degree angle is not unreasonable for this

bullet at 888.5 yards downrange.

A smoothed value of 0.5040 milliradians (or 0.02888 degrees) can be directly calculated for

the

tangent angle

T

extracted horizontally rightward velocity VR(t) to the forward velocity V(t) of the bullet at

t=1.430 seconds.

However, this velocity ratio is very sensitive to the ongoing epicyclic swerving motion

included in the PRODAS Drift reports, and its smoothed value probably should be somewhat

larger here at t = 1.430 seconds.

Our fitted algorithm, mentioned above, for estimating the tangent angle T at 888.5 yards

yields 0.61019 milliradians or 0.034961 degrees. This would indicate a reasonable horizontal

tracking error angle H of 0.08167 milliradians, or 13.38 percent of T at that point in the

flight.

This closed-form integration yields a value of R about midway between our numerically-

integrated value and the adjusted classic value of R as calculated from the same PRODAS

data at t = T.

We shall use this closed-form algorithm (Eq. 26) for estimating the yaw of repose angle R(t)

without relying upon any 6-DoF simulation data in formulating the spin-drift SD(t) of any

rifle bullet at long ranges.

Think of these incremental yaw-attitude changes as occurring twice per coning cycle at the

TDC and BDC positions of the coning bullet throughout the flight.

The double-coning-rate sequence of small torque impulses produces a reinforcing chain

-axis direction of the spinning bullet

evermore rightward.

The initial yaw of repose angle at bullet launch R(0) must be zero by definition.

These calculations serve to validate our analysis of the gyroscopic and aerodynamic causes

of the yaw of repose.

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Analysis of the Spin Drift

The horizontally rightward spin-drift SD(t) of the trajectory is caused by a net horizontal

aerodynamic lift-force attributable to this small, but ever increasing, rightward yaw of

repose angular bias R(t) in the yaw-attitude of the coning-axis of the spinning bullet.

in the approaching apparent wind direction within one half of a coning cycle, just as with any

other type of wind change.

As the horizontal projection of the

mean trajectory

traced by the

mean CG

of the bullet

gradually accelerates rightward with this spin-drift SD(t), its tangent +V direction defining

the origin of wind-axes plots drifts slowly rightward also,

following

(but dynamically

lagging) the increasing yaw of repose attitude angle R(t) of the bullet.

We formulated this tangent angle T(t) earlier. Logically, only the horizontal tracking error

angle HR(t) T(t) should appear in these wind-axes plots in place of R(t), itself.

In formulating the effective net (time-averaged) aerodynamic lift-force accelerating the CG

of the coning bullet rightward, we must consider the coning modulation of the aerodynamic

effect as the CG of the bullet moves throughout its circular coning cycle.

Here the modulation is horizontally left-to-right, and the effect being modulated is an

aerodynamic lift force.

However, as we saw above for the modulation of the overturning moment, for the uniformly

coning rifle bullet, analysis of the modulation of this lift force can be greatly simplified by

making use of Coning Theory. We can express the average effective aerodynamic lift-force

on the coning bullet arising from the yaw of repose angle R(t) as if the bullet were

not

coning, but simply flying with the spin-axis always aligned with the attitude of its coning axis

. After all, it is the attitude of that coning axis which properly defines this yaw of

repose angle R(t).

The actual average aerodynamic angle of attack in a coordinate system moving with, and

oriented with the mean trajectory of the coning bullet, is just the tracking error angle H(t).

The lift-force attributable to this H(t) angle of attack keeps increasing the rightward

curvature of the mean trajectory. In earth-fixed coordinates, not oriented to T(t) with the

yawing bullet, the average horizontal angle of attack driving the mean trajectory away from

the original firing azimuth, the X-axis, is H(t)+T(t) = R(t).

The magnitude of the small net rightward aerodynamic lift force {FL}R attributable to the

rightward yaw attitude bias R(t) of the coning axis is given in linear aeroballistics as:

{FL}R = q*S*{CL*Sin[T(t) + H(t)] CD*SinT(t)]}

Or

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{FL}R (t)*R(t) q(t)*S*CD(t)T(t) (29)

The small rightward aerodynamic lift force acting horizontally on the bullet is actually

counteracted partially by an even smaller cross-

aerodynamic drag force FD given by q*S*CD*T(t).

Here the coefficient of lift CL(t) and coefficient of drag CD(t) are evaluated for the very small

aerodynamic angle-of-attack R(t). However, they still vary with the Mach-speed of the

slowing bullet. The dynamic pressure q(t) also reduces with the square of its airspeed V(t)

as the bullet slows.

This small rightward horizontal force {FL}R acting on a bullet of mass m for one half the

period T2 of each coning cycle produces a rightward horizontal bullet velocity increment R

given here in feet per second per half-coning cycle as:

R = {FL}R*T2/(2*m) = {FL}R/[2*m*f2(t)] L}R2(t) (30)

Where m is the mass of the bullet expressed in slugs. Here, m = 175.16/(7000*g) =

0.00077774 slugs. We are using g = 32.174 feet per second per second for the standard

effective acceleration of gravity on or near the surface of our rotating earth.

These rightward velocity increments R accumulate (sum) from zero at t = 0 for each

half

coning cycle

which occurs from launch to time t to form the horizontally rightward velocity

VR(t) of the CG of the bullet which is caused aerodynamically by the yaw of repose angle R(t).

The incremental rightward horizontal spin-drift of the bullet, in feet, during one

particular half-coning cycle T2/2 is then:

R(t)*T2/2 = VR(t)/[2*f2R2(t) (31)

The horizontal spin-drift SD(t) at time t is then found by

summing

these incremental

displacements for each

half coning cycle

starting with zero at t = 0. Our subject

M118LR bullet experiences 87 complete half-coning cycles during its flight to 1000 yards.

Numerical integration of using PRODAS data for each millisecond of the simulated

SD(1.6923 sec) = 9.7019 inches. PRODAS itself calculates a total drift

of 9.5407 inches at 1000 yards. The PRODAS drift includes the horizontal component of the

minimal coning motion of the spinning bullet.

This level of agreement verifies our aeroballistic analysis of the causes of spin-drift.

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Analytic Calculation of the Spin Drift at the Target

If we formulate a reliable estimation of the scale factor ScF for any given bullet in any given

firing conditions, this scale factor ScF can then be used together with a reliably calculated

DROP from the bore axis at the target

to calculate analytically the spin-

drift SD(t) at the target for any given rifle bullet in any firing conditions according to Eq. 2:

SD(t) = -ScF*DROP(t) (2)

-DoF simulations

that the scale factor ScF needs to be 0.0219685 -(minimum coning

angle) runs, and 0.0222219 (or 1.154 percent larger) for the somewhat more realistic

M118LR bullet fired in these simulated

conditions to 1000 yards.

The unrealistic no-PRODAS case represents the

minimum possible

SD(t)

values for this bullet fired at this muzzle velocity and spin-rate in this atmosphere and flying

with the minimum possible coning motion throughout its ballistic flight.

Most

dynamically stable

rifle bullets fired outdoors at long ranges will likely suffer only the

minimal 1.154 percent -drift SD(t) at

long ranges.

However, a

dynamically unstable

rifle bullet, such as the infamous mid-range 30-caliber 168-

grain Sierra International, might experience about 5 percent greater spin-drift (SD) when

fired to long ranges (up to 800 meters) through

any

non-zero crosswinds.

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Estimating the Ratio of Second Moments of Inertia for Rifle Bullets

We need an accurate estimation of the ratio Iy/Ix for our rifle bullet so that we can find the

coning rate 2(t) of that bullet at any time t during its flight from Tri-Cyclic Theory.

We will use the

reference diameter

for our subject bullet d(in inches) = 1.00 calibers as the

distance metric throughout these calculations.

Input parameters are needed which describe the bullet.

We need the weight Wt of the bullet in grains, and we need the average density P of the

bullet based on its type of construction: 2235.6 grains/cubic inch for monolithic copper

bullets; 2681 gr/in3 for a thick-jacketed, lead-alloy-cored bullet having no appreciable

hollow cavities; 2750 gr/in3 for a thin-jacketed, pure-lead-cored match bullet; and 2120

gr/in3 for monolithic bullets constructed of C360 brass.

We need the actual length L of the bullet in calibers. We need the actual length LN of the nose

of the bullet in calibers. We need the diameter of the meplat DM at the front of the bullet in

calibers. We also need the RT/R circular-arc head-shape design ratio for the ogive (nose) of

the bullet, termed RTR here.

We then calculate the generating radius RT for a tangent ogive for this bullet, the full length

LFT of a pointed

tangent ogive

, the full length LFC of a

conical ogive

, and the full nose length

LFN actual ogive shape if it went all the way to a pointed tip.

RT = [LN2 + ((1 DM)/2)2]/(1 DM) (32)

LFT = SQRT(RT 0.25) (33)

LFC = LN/(1 DM) (34)

LFN = LFT*RTR + LFC*(1 RTR) (35)

We then calculate The (full-ogive) total length LL of the bullet in calibers and shape factor h

describing a cone-on-cylinder model of this rifle bullet:

LL = L LN + LFN

h = LFN/LL (36)

We now calculate the weight Wtcalc in grains for the cone-on-cylinder model of this bullet:

P*d3 *LL*(1 2*h/3) (37)

We evaluate the polynomial f1(LL,h) as:

f1(LL,h) = 15 12*h +LL2 *(60 160*h +180*h2 - 96*h3 + 19*h4)/(3 2*h). (38)

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We are now ready to calculate the ratio of the second moments of this bullet about its crossed

principal axes:

Iy/Ix = (Wt/Wtcalc)0.894 *f1(LL,h)/[30*(1 4*h/5)] (39)

This estimator matches within 1 percent the Iy/Ix ratios calculated by numerical integration

for many different solid monolithic rifle bullet designs. Applying this estimator to data for

the old 30-caliber 168-grain Sierra International bullet [from McCoy, page 217 MEB] yields

an Iy/Ix ratio of 7.7748 or 4.48 percent greater than the value 7.4413 reported by McCoy.

This over-estimation is to be expected due to the significant hollow cavity within the nose of

that bullet.

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Estimating the Spin Drift Scale Factor ScF

With all this in mind, we formulate an estimator for an ScF value for

of coning motion of our example bullet which can be duplicated for any

other

dynamically stable

rifle bullet in any likely firing conditions.

In flat firing, we can formulate the scale factor ScF in terms of the

ratio

of 1) the horizontal

aerodynamic lift-force acting on the flying bullet due to its yaw of repose R as that bullet

nears its long-range target to 2) the vertically downward-acting weight of that bullet. In this

manner, we can formulate ScF for any given bullet and likely wind conditions as:

ScF = 1.01154*0.383703*[q(t)*S]*SinR(t)]*CL(t)/Wt

R(t)*CL(t)/Wt (40)

with Wt = 175.16/7000 representing the weight of this example M118LR bullet given in

pounds-force lbf.

Here again, as in the earlier simplified formulation of , we are ignoring the upward force

on the free-flying bullet caused by the cross-bore component of its aeroballistic drag force.

The force offset effects of these two simplifications cancel out here in forming this ratio for

evaluating ScF.

We define the scale factor ScF as the ratio of the magnitudes of the net rightward horizontal

and downward vertical forces acting upon the flying bullet as a

free body

:

ScF = {F}H/{F}V (41)

From Eq. 29 above, the net rightward horizontal force is {F}H :

{FL}R (t)*R(t) q(t)*S*CD(t)T(t) (29)

The corresponding net downward vertical force {F}V is given by:

{F}V = Wt - D = Wt - (42)

Substituting these expressions into Eq. 41 and simplifying by utilizing Eq. 3 above,

ScF*{F}V = {F}H = q(t)*S*CL(t)*R(t) q(t)*S*CD(t)T(t) = (43)

ScF*{F}V = ScF*Wt ()*q(t)*S*CD(t) = ScF*Wt q(t)*S*CD(t)T(t) (44)

After adding the small quantity q(t)*S*CD(t)T(t) to these equal expressions, Eq. 43 and Eq.

44, we have:

ScF*{F}V + q(t)*S*CD(t)T(t) = q(t)*S*CL(t)*R(t) = ScF*Wt

ScF= q(t)*S*CL(t)*R(t)/Wt (45)

which is the expression which we will actually evaluate for ScF at time T far downrange.

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The initial constants (0.383703 and 0.388132) have been empirically determined from

several PRODAS runs and should be the same for any

dynamically stable

rifle bullet in any

firing conditions likely to be encountered in long-range shooting. The PRODAS runs show

the M118LR bullet to be dynamically stable.

These PRODAS simulations, together with the classic formulation for the yaw of repose angle

R(t), indicate that the scale factor ScF might need to be increased by about 5 percent for

dynamically unstable

bullets which will fly with significant coning angles throughout their

flight when fired through

any crosswinds at all

.

Each of these functions of time t should be evaluated at the time T when the bullet has slowed

to an airspeed of 1340 feet per second (or approximately Mach 1.20, depending upon

ambient conditions).

This flight time T and the flight distance Rg at which it occurs are completely independent of

the actual range to the target. The time T and range Rg to 1340 fps

airspeed

can be

determined by using any current 3-DoF point-mass trajectory calculator.

The airflow over any good long-range rifle bullet should remain safely above the turbulent

transonic region at this 1340 fps airspeed in almost any reasonable atmospheric conditions.

our lowest-drag long-range rifle bullet designs will not

encounter transonic buffeting until they slow to about Mach 1.10 airspeed. The needed

coefficient of lift CL is particularly difficult to estimate for any bullet in the transonic

airspeed regime.

Most experienced long-range riflemen select their shooting equipment so that whenever

possible their bullets will impact the target at airspeeds above Mach 1.2.

For similar best-accuracy reasons, we base our calculation of ScF upon bullet data at 1340

fps airspeed regardless of the actual range to the intended target.

We calculate the potential drag-force q(T)*S using the calculated density of the ambient

atmosphere in slugs per cubic foot, and the airspeed V(T) = 1340 feet per second:

q(T)*S = 2/4)2 (46)

This potential drag-force value should be about 1.1 lbf for a 30-caliber bullet at this airspeed

depending on air density. The potential drag-force at 1340 fps varies most strongly with the

square of the caliber d of the bullet (in feet).

The analytic estimate of R(T) is calculated per Eq. 26 above with t = T and V(T) = 1340 fps.

With these simplifications Eq. 26 becomes:

R(T) = -g*{exp[-V)] - 1}/[f2V)] (47)

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If we know the

initial

gyroscopic stability Sg of the bullet, we can calculate the initial Stability

Ratio R of its epicyclic rates f1/f2 from:

R = 2*{Sg + SQRT[Sg*(Sg 1)]} 1 (48)

The

initial

coning rate f2(0) in hertz can then be found from Tri-Cyclic Theory as:

f2(0) = V(0)/[Tw*(Iy/Ix)*(R + 1)] (49)

where

Tw = Absolute value of the Twist Rate of barrel in feet per turn.

Iy/Ix = Ratio of transverse to axial second moments of inertia for this bullet as estimated

above.

Substituting back into Eq. 47, we have:

R(T) = -g*Tw*(Iy/Ix)*(R + 1)*{exp[-V)] - 1}/[V2V)] (50)

where

V(0) = Launch velocity of this particular bullet in feet per second. [V(0) is assumed to exceed

Mach 2.0]

-(0.585 + 0.0321/d)*T, as approximated earlier and used here for any long-range

bullet, and

kV = ln[1340/V(0)].

This value R(T) in radians is an estimate of the yaw of repose angle for this bullet where it

slows to an airspeed of 1340 fps.

The CL(T) value is estimated based on an estimate of the initial CL(0) for the Mach-speed

of the bullet at launch (here Mach 2.3289) evaluated from Rob

program for the nose-length effect, but using our own boat-tail effect lift reduction for these

long-range bullets.

-length estimated CL by the square root of 0.2720/BC7 for each

bullet, reasoning that about half of any differing drag for bullets having higher or lower

ballistic coefficients BC7 (relative to the G7 Reference Projectile) than our example M118LR

bullet is due to having a more or less effective boat-tail design.

If a more reliable BC1 value (relative to the G1 Reference Projectile) is available for your rifle

bullet, use the square root of 0.5310/BC1 for this estimated CL adjustment for variations in

bullet drag.

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The full nose-length (LFN) for the 30-caliber M118LR bullet is 2.5955 calibers. The initial

coefficient of lift CL(0) at this Mach 2.3289 airspeed calculates to 2.720 using our adjusted

INTLIFT estimate.

-studied 30-caliber 168-grain

Sierra International bullet to this lift coefficient 2.720 at this Mach 2.3289 airspeed yields a

coefficient of lift CL(T) of 1.877 at Mach 1.20 in this standard sea-level ICAO atmosphere.

CD0 determines its time-rate of decay in Mach-speed. The

supersonic lift-to-drag ratio FL/FD for any given angle-of-attack tends to be an invariant

aerodynamic characteristic of each basic bullet shape.

Since the coefficients of lift and drag are highly correlated at any given Mach-speed over the

population of long-range rifle bullets, the same exponential time-decay coefficient:

kL = ln[CL(T)/ CL(0)] = -0.3711 (51)

can be used in propagating the coefficients of lift CL(T) for any long-range rifle bullets of

interest here.

We propagate this

initial

coefficient of lift CL(0) estimate forward to its value at time T as:

CL(T) = CL(0)*exp{-0.3711*[V(0)/2600 fps]2 *(1.430 sec/T)} (52)

The coefficient of lift CL(T) for any very-low-drag (VLD) or ultra-low-drag (ULD) long-range

rifle bullet should be smaller than 1.90 at this airspeed of 1340 fps. Bullets designed for lower

aerodynamic drag will also produce less aerodynamic lift. Conversely, one cannot produce

more lift without also increasing drag in aerodynamics.

The exponential propagation function [Eq. 52] estimates a larger fraction of the initial

coefficient of lift CL(0) remaining at 1340 fps airspeed when the time-of-flight T

to that

airspeed

is increased due to firing a higher-drag bullet, but the initial velocity correction

factor [V(0)/2600 fps]2 prevents this increase when time-of-flight T to 1340 fps increases

simply due to firing that same bullet with a higher muzzle velocity V(0).

That is, if the

same bullet

is fired at different muzzle velocities, its estimated coefficient of lift

CL(T) when it has slowed to an airspeed of 1340 fps should remain the same.

The muzzle velocity V(0) is assumed to exceed Mach 2. This coefficient of lift propagation

yields the expected CL(T) = 1.8769 at 1340 fps for the M118LR bullet, and varies by less

than 1 percent over any reasonable launch speeds V(0) for this one bullet type.

The scale factor ScF is now calculated from Eq. 40 using the values of the time-functions at

time T as calculated in Eq. 46, Eq. 50, and Eq. 52 above:

R(T)*CL(T)/Wt (53)

31 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

where

0.388132 = An empirically determined constant (from PRODAS data) for all firings of

dynamically stable

rifle bullets through any non-steady

(non-diabolical) crosswinds.

This constant is numerically necessary for several likely reasons, among them that the

driving horizontal lift-force FLR(t)] is actually attributable only to the dynamic horizontal

tracking error attitude angle H(t) instead of the entire yaw-of-repose angle R(t).

If we might be slightly misestimating the yaw of repose angle R(T) or coefficient of lift

CL(T) used here in any systematic ways for these minimal-coning-

6-DoF flight simulations, the empirically-determined initial constant factor 0.388132, from

that same PRODAS data, tends to absorb any net systematic difference.

32 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

Calculating the Spin Drift at the Target

The spin-drift SD(tof) at the

target distance

is calculated from Eq. 2 above using the invariant

Scale Factor ScF, as calculated in Eq. 53 above for the bullet slowed to 1340 fps, and the total

DROP from the axis of the bore for the actual time-of-flight (tof) to the target:

SD(tof) = -ScF*DROP(tof) (54)

The Spin-Drift SD at the target is calculated here in Eq. 54 in the same

distance

or

angular

units

DROP from the bore axis is given. Again, the proper algebraic sign

rotation.

The bullet DROP and time-of-flight (tof) to the target are accurately calculated in many

existing 3-DoF point-mass trajectory propagators. After all, the accurate calculation of bullet

DROP at the target distance is the basic figure of merit for these software aids.

The time-of-flight (tof) to the target is used in the Litz SD estimator and is nice to know even

if we do not actually use it explicitly in these calculations.

ting your scope height equal zero,

setting the angle-of-

but perhaps 5 or 10 yards if

made necessary by input limit constraints), and by specifying that the trajectory calculations

go oor measured range.

In other words, we want to calculate the DROP from the bore axis at the target distance as if

--range target.

The smoothed spin- -

simulations with this 175.16-grain M118LR bullet fired in these conditions is 9.5407 inches.

The spin-drift SD at 1000 yards estimated via this algorithm using PRODAS data values (and

without the factor of 1.01154 increase in ScF) is 9.5635 inches.

Comparing the two results millisecond-by-millisecond, throughout the flight of 1692.3

milliseconds, yields a mean difference of 0.0043 inches, with a population standard deviation

of 0.0207 inches.

This level of agreement between our analytical estimator for spin-drift for each millisecond

and the PRODAS numerical (non-analytical) simulation results is rather astonishing. The

rounding error for drop and drift data given in angular units in the PRODAS report format is

0.180 inches at 1000 yards, and we are not even estimating the horizontal component of the

33 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

The agreement of this spin-drift SD

runs is also excellent when the Scale Factor ScF includes the 1.154 percent increase as

formulated above.

This 1.154-percent-augmented version of the ScF estimator in Eq. 48 should be calculated for

outdoor firing of any other

dynamically stable

rifle bullets.

Summary

I. We fit an exponential tangent angle function T(t) to extracted velocity-ratio data

from a PRODAS simulation which minimizes the epicyclic swerve complications

in measuring the yaw of repose angle R(t). We discovered that the spin-drift SD

at long range is affected slightly (about 5 percent) by the magnitude of coning

motion experienced by the bullet en route to the target, with consistently larger

coning angles (t) producing slightly more spin-drift SD(t).

II. We define the horizontal and vertical direction dynamic tracking error angles, H

and V -axes plots

resulting from 6-DoF flight simulations. Just as with the flight path angle , the

yaw of repose angle R(t) logically should not appear in those wind-axes plots

which reference as their origin the +V direction of the

vector V, which is always tangent to its 3-dimensional mean trajectory. We

provide an analytic formulation in Eq. 3 for T(t), the horizontal tangent angle,

which logically should -axis yaw attitude data

before plotting.

III. We explain the aerodynamic causes of yaw of repose and spin-drift and

numerically verify those explanations using data from PRODAS 6-DoF simulations

together with the principles of linear aeroballistic theory.

IV. We reformulate the classic aeroballistic yaw of repose angle as R , which

holds for a significantly coning bullet with throughout its flight.

Furthermore, R(0) = 0.00 at launch by definition. For a minimally coning bullet

with , R T H .

V. We formulated an accurate analytic estimator for the ratio Iy/Ix of the second

moments of inertia for any long-range rifle bullet so that the sum of its two

epicyclic rates (1 2) can be calculated via Tri-Cyclic Theory from the circular

spin-rate of the bullet (in radians per second) remaining at any time t during its

flight. We noted that the spin-rate (t) decreases very nearly exponentially with

time t for modern rifle bullets:

34 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

(0)*exp{[(-0.0321/d(inches)]*t}.

VI. We note that in flat firing the spin-drift displacement SD of the bullet at any long

range is essentially an invariant scale factor ScF distance

from the projected bore axis at that range. The scale factor ScF runs about 1.0 to

2.3 percent for the various long-range rifle bullets in typical flat firing

drop from the axis of the bore is accurately computed in any 3-DoF trajectory

propagation program. This same scale factor ScF defines the ratio of the horizontal

and vertical angular deviations of the tangent to the

mean trajectory

from the axis

of the bore at firing time (when t = 0). The angular deviation in the horizontal

plane T(t) is always equal to the Scale Factor ScF times the vertical-direction

deviation Total(t) .

VII. We present an analytic calculation of that invariant scale factor ScF so that an

accurate and reliable calculation of spin-drift SD(t) can be computed for any long-

range rifle bullet flat-fired in any likely conditions without relying upon 6-DoF

simulations. This dimensionless Scale Factor ScF can also be used as part of a

collection K of invariant values from Eq. 40 such that the yaw of repose angle R(t)

can be calculated for any flight time t as:

R(t) = K/[V2(t)*CL(t)] (55)

with

(56)

This formulation of R(t) is very similar to the classic formulation for R(t), and

this formulation also does

not

evaluate to zero at t = 0. This calculated non-zero

initial yaw of repose attitude angle R(0) is just the initial yaw

attitude which would be required to produce a hypothetical horizontal lift force

of ScF*Wt at muzzle velocity V(0). Of course, no such side-force exists at bullet

launch.

35 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

Example Calculations of ly/lx

These parameters and calculations are needed to determine the crucially important ratio of

Iy/Ix.

Four different 30-caliber rifle bullets are selected in addition to our example M118LR bullet

in these parallel (spreadsheet) calculations for variety and based on availability of bullet

measurements for estimating Iy/Ix ratios. Two of the additional 30-caliber rifle bullets are

included because they were tested in s

The obsolete 168-grain Sierra International bullet (for which McCoy supplies the needed

data) is similar to their current, improved 30-caliber 168-grain MatchKing (SMK). We

tabulated the calculations of Iy/Ix because this ratio was measured and reported by McCoy.

We have several PRODAS runs for the bullet used in 2011 in the US Army M118LR 7.62 mm

NATO Special Ball ammunition. The 175.16-grain M118LR bullet used by PRODAS has an

Iy/Ix ratio which can be determined very accurately from the PRODAS reports. We are using

reasonably estimated bullet shape parameters scaled from images which produce

approximately that PRODAS calculated Iy/Ix value in lieu of the actual bullet shape data on

the M118LR bullet until such data can be obtained.

The 173-grain solid (monolithic) brass Ultra-Low-Drag (ULD) bullet design has not yet been

tested, but its numerical design description allows accurate modeling of its flight

aeroballistic estimators. We calculated its mass

characteristics, including its Iy/Ix ratio, using accurate numerical integration.

The dimensional data on the Berger 175-grain Open-Tip Match (OTM) Tactical bullet and

their 185-grain Long Range Boat Tail (LRBT) bullets, as well as the test conditions during

their 1000-yard drift firings, were taken from Bryan Litz

An Iy/Ix ratio of 7.4413 is published by McCoy for the old 168-grain Sierra International

bullet. Our estimate of 7.7748 is 4.48 percent This over-

The target Iy/Ix ratio of 13.4733 for the new 173-grain monolithic brass ULD bullet was

calculated by numerical integration of its elements of mass. Our estimated value here of

13.4975 is just 0.180 percent larger than this value.

The data used here for the two Berger 30-caliber bullets selected by Bryan Litz in his drift

firing experiments at 1000 yards are taken from his publications. Long-range drift firings are

a traditional method for measuring horizontal spin-drift. No information is available

concerning their Iy/Ix ratios.

36 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

30-Caliber Example

Bullets:

168-gr

International

175.16-gr

M118LR

173-gr

ULD(SB)

175-gr

Berger

Tactical

185-gr Berger

LR-BT

Reference Diameter

(inches)

0.3080

0.3080

0.3002

0.3080

0.3080

Bullet Length L (cal)

3.9800

4.4000

5.4368

4.1169

4.3929

Nose Length LN (cal)

2.2600

2.4500

2.8368

2.3701

2.5747

Diameter of Meplat

DM (cal)

0.2500

0.2175

0.1000

0.1948

0.2013

Length of Boat-Tail

LBT (cal)

0.5100

0.6000

0.7012

0.6331

0.5844

Diameter of Base DB

(cal)

0.7645

0.8000

0.8420

0.8409

0.8182

Ratio of Ogive

Generating Radii

RT/R

0.9000

1.0000

0.5000

0.9000

0.9500

Rho-P, Bullet Density

in grains/cu. in.

2750

2600

2128

2750

2750

Rho-P, Ave. Specific

Gravity (gm/cc):

10.8742

10.2811

8.4147

10.8742

10.8742

Wt, Bullet Weight in

grains

168

175.16

173

175

185

Calc. Tangent Ogive

Radius RT (cal)

6.9976

7.8666

9.1666

7.1779

8.4993

Calc. Full Tangent

Ogive Length LFT

(cal)

2.5976

2.7598

2.9861

2.6321

2.8722

Calc. Full Conical

Nose Length LFC

(cal)

3.0133

3.1310

3.1520

2.9435

3.2236

Calc. Full Nose

Length LFN (cal)

2.6392

2.7598

3.0690

2.6632

2.8897

Length LL with LFN

(cal)

4.3592

4.7098

5.6690

4.4100

4.7079

h, Ratio LFN/LL

0.6054

0.5860

0.5414

0.6039

0.6138

Wtcalc, Calc Wt in

grains

164.0601

171.2331

163.8186

166.2542

175.5264

f1(LL,h)

117.7480

141.1386

218.6318

120.6627

133.9879

37 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

Cone-on-Cylinder

Estimated Iy/Ix=

7.7748

9.0376

13.4975

8.1466

9.1976

Target Iy/Ix

7.4413

9.0673

13.4733

Iy/Ix Error:

0.3335

-0.0297

0.0242

Percent Error:

4.481

-0.328

0.180

38 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

Example Calculations of Spin Drift

The remaining parameters needed to calculate yaw of repose R and spin-drift SD are

calculated for these same five example bullets in another spreadsheet shown below. A 3-DoF

trajectory program was used to compute the time-of-flight (tof) and flight distance to an

airspeed of 1340 FPS and tof to a 1000-yard target for both the 168-grain SMK bullet and the

new 173-grain ULD bullet. PRODAS trajectory data was used for the 175.16-grain M118LR

bullet.

The initial gyroscopic stability factor Sg was taken from McCoy for the old 168-grain Sierra

International bullet, and Sg is calculated GYRO program for the new 173-

grain ULD bullet.

PRODAS reports the Sg-value for each millisecond of the flight of the M118LR bullet, but we

just used their initial value. Bryan Litz gives the initial Sg values for the two Berger bullets

used in his drift firings.

The ULD bullet is a dual-diameter design with the base of the ogive measuring 0.3002 inches

in diameter (1.0-calibers for this bullet design). It has a rear driving-band measuring 0.3082

inches in diameter (or 1.02665 calibers). The midpoint (CG) of the rear driving-band is

located 1.6 calibers behind the base of its 3-caliber secant ogive, and the width of this driving

band is 0.6 calibers.

Our calculated yaw of repose angles R for the first three example bullets when each has

slowed to an airspeed of 1340 FPS shows an interesting progression.

The estimated yaw of repose angles R of the three trajectories at the 1340 FPS airspeed

points are 0.471231 milliradians for the obsolete 168-grain International bullet at 816 yards

downrange, and 0.693417 milliradians at 888.5 yards downrange for the M118LR bullet, but

just 0.434382 milliradians for the new 173-grain monolithic brass ULD bullet fired at 3200

fps, and this occurs way beyond the 1000-yard target at 1457 yards downrange.

The assumed 3200 fps muzzle velocity of this new ULD bullet is based on firing it from a 300

Remington UltraMag cartridge. Each of the other example 30-caliber bullets is assumed to

be fired from a much less powerful 7.62 mm NATO or 308 Winchester cartridge.

For comparison purposes the spin-drift SD at 1000 yards is calculated in inches for each of

our five example bullets using the SD estimator published by Bryan Litz:

SD(inches) = 1.25*(Sg + 1.2)*(tof)1.83 (57)

Our estimates of SD at 1000 yards are smaller than Bryaestimates for each of these five

example bullets. Our estimate of spin-drift SD at 1000 yards for the M118LR bullet of 9.5111

inches matches the SD computed by PRODAS (9.5407 inches) quite closely (error: -0.0296

39 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

inches, or -0.003 percent). The Litz-estimated SD of 10.2791 inches for this M118LR bullet at

1000 yards exceeds the PRODAS value by 0.7156 inches, or +7.483 percent.

Our estimate of 6.8061 inches of spin-drift SD at 1000 yards for the old 168-grain Sierra

International bullet from a 12-inch twist barrel is approximately 2.344 inches less than the

9.15 inches shown graphically by McCoy in Figure 9.8 of his MEB, and is 3.214 inches less

than the 10.020 inches calculated by the Litz estimator for this bullet.

We expected our estimate to be 5 percent (or 0.340 inches) too small for this

dynamically

unstable

bullet. We cannot readily explain the remainder of this difference. Perhaps we

should concede that this formulation inherently assumes that the rifle bullet is

dynamically

stable

.

Our estimate of 4.2983 inches of spin-drift SD at 1000 yards for the radical new 173-grain

monolithic brass ULD bullet design, versus the value of 5.1696 inches calculated by the Litz

estimator for this bullet, indicates the need for our more elaborate SD calculation in

predicting the long-range flights of current and future ultra-low-lift rifle bullets, even when

fired from faster twist-rate barrels.

The Litz spin-drift estimator is closer than our estimator to reported spin-drift values for the

old 168-grain Sierra International bullet and for the Berger 175-grain OTM Tactical bullet.

Our estimator seems closer for the remaining three bullets, especially for the two very-low-

drag (and correspondingly very-low-lift) bulletsthe copper 173-grain ULD bullet and the

Berger 185-grain Long Range Boat-Tail (LR-BT) bullet.

Of course, our predictive agreement with the PRODAS calculations for the M118LR bullet is

best of all. We expect that if 6-DoF simulations could be run for the other four bullets, our

estimator would match those results more closely.

-DoF simulation

results particularly well if they could be computed.

The aerodynamic responses of real rifle bullets are non-linear enough to affect the

calculation of these small second-order effects. Real bullets are also subject to other types of

aerodynamic jump phenomena in real firingssome of which might be at least partially

systematic.

40 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

Spin-Drift Example

Calculations:

30-Caliber

Example Bullets:

168-gr

Internation

al

175.16-gr

M118LR

173-gr

ULD

175-gr

Berger

Tactical

185-gr

Berger

LR-BT

Bullet Length L (cal)

3.9800

4.4000

5.4368

4.1169

4.3929

Nose Length LN (cal)

2.2600

2.4500

2.8368

2.3701

2.5747

Diameter of Meplat

DM (cal)

0.2500

0.2175

0.0808

0.1948

0.2013

Length of Boat-Tail LBT

(cal)

0.5100

0.6000

0.7012

0.6331

0.5844

Diameter of Base DB

(cal)

0.7645

0.8000

0.8420

0.8409

0.8182

Ratio of Ogive

Generating Radii RT/R

0.9000

1.0000

0.5000

0.9000

0.9500

Calc. Tangent Ogive

Radius RT (cal)

6.9976

7.8666

8.9846

7.1779

8.4993

Calc. Full Tangent

Ogive Length LFT (cal)

2.5976

2.7598

2.9554

2.6321

2.8722

Calc. Full Conical Nose

Length LFC (cal)

3.0133

3.1310

3.0862

2.9435

3.2236

Calc. Full Nose Length

LFN (cal)

2.6392

2.7598

3.0208

2.6632

2.8897

V0=Launch velocity

(FPS):

2800.00

2600.07

3200.00

2660.00

2630.00

Initial Mach-Speed

2.5079

2.3289

2.8662

2.4332

2.4057

Initial B-value

2.3000

2.1032

2.6861

2.2182

2.1880

Ballistic Coef (G1 Ref)

0.4260

0.5460

0.6290

0.5060

0.5530

Ballistic Coef (G7 Ref)

0.2180

0.2720

0.3220

0.2580

0.2830

INTLIFT CL(0)

3.1015

2.6759

2.5670

2.8145

2.6189

Time T to 1340 FPS

(sec)

1.2723

1.4300

2.1070

1.3972

1.5030

Range at 1340 FPS

Airspeed (yards)

816.00

888.50

1457.00

881.54

839.10

Est CL(T) at 1340 FPS:

1.9120

1.8463

1.7528

1.8913

1.8248

41 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

Twist Rate

(inches/turn, RH)

12.0000

11.5000

8.2500

10.0000

10.0000

Initial Sg

1.7400

1.9400

1.5940

2.2400

1.9100

Initial Stability Ratio

(R)

4.7494

5.5808

4.1341

6.8132

5.4567

Calculated (Iy/Ix) Ratio

7.7748

9.0376

13.4975

8.1466

9.1976

Initial Coning Rate f2

(hz)

62.6387

45.6182

67.1674

50.1486

53.1437

kv=LN(1340/V(0)

-0.73695

-0.66287

-0.87048

-0.68566

-0.67431

komega+kv

-1.61385

-1.64845

-2.32837

-1.64866

-1.71021

Beta-R at time T

(mrad)

0.4572

0.6909

0.5954

0.6144

0.6098

Ref. Diam. (1.0 cal. in

inches):

0.3080

0.3080

0.3002

0.3080

0.3080

Frontal Area at Base of

Ogive S (square feet)

0.0005174

0.0005174

0.0004915

0.0005174

0.0005174

Potential Drag Force at

1340 fps (lbf)

1.1041

1.1041

1.0489

1.1137

1.1137

Bullet Weight (grains)

168.00

175.16

173.00

175.00

185.00

Bullet Weight (lbf)

0.02400

0.02502

0.02471

0.02500

0.02643

Calculated Scale Factor

ScF

0.01561

0.02185

0.01719

0.02009

0.01820

DROP from Bore Axis

at 1000 yds (inches)

436.0450

435.3450

250.0250

428.4970

414.8350

Time of Flight (tof) to

1000 yds (seconds)

1.7300

1.6923

1.2390

1.6870

1.6400

Remaining Velocity at

1000 yds (FPS)

1145.00

1213.99

1836.00

1197.00

1278.00

Calculated 1000-yd

Spin-Drift (inches

rightward)

6.8061

9.5111

4.2983

8.6095

7.5496

SD from McCoy Figure

9.8

9.1500

SD from PRODAS runs

9.5407

42 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

SD (inches) from Drift

Firings

11.4000

6.7000

Litz Est. Spin-Drift

(inches rightward)

10.0203

10.2791

5.1696

11.1967

9.6125

Litz Est. Spin-Drift

Minus Our Calc. SD

(inches)

3.2142

0.7679

0.8713

2.5871

2.0628

43 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

Sensitivity Analysis & Model comparisons

Sensitivity analysis was significant in studying and assessing the uncertainty in the output of

our model, which can be attributed to different sources of error of the input parameters.

Sensitivity analysis is an integral part of model development and involves analytical

examination of input parameters to aid in model validation and provide guidance for future

research.

We used it to determine how different values of one or more independent variables, impact

a particular dependent variable under a given set of conditions.

In other words, it helped us to investigate the robustness of the model predictions and to

explore the impact of varying input assumptions.

We chose to set on what is known as local (sampled) sensitivity analysis, which is derivative

based (numerical or analytical). The use of this technique is the assessment of the local

impact of input factors' variation on model response by concentrating on the sensitivity in

vicinity of a set of factor values.

Such sensitivity is often evaluated through 2-dimensional gradients or partial derivatives of

the output functions at these factor values, (the values of other input factors are kept

constant) when studying the local sensitivity of a given input factor.

One of the critical objectives was to stress-test the model as well as to study its fidelity to

known experimental and model-based 6-DoF runs.

Unfortunately there are many results and accompanying charts to add, but in order to make

pair of them, which are

significant in terms of reliability of the underlying numerical algorithm.

The following charts compare the outputs of three models to estimate SD, namely Hornady

4-DoF, Litz and the B&R method presented in this paper.

In the case of Hornady-DoF, the reader must take into consideration that three major

variables, Sg, DROP and ToF, are different than the ones used to calculate the B&R and Litz

outputs because it produces different DROP and ToF values as well as a varying Sg.

On the other hand, all DROP and ToF figures are the same for both Litz and B&R, and were

calculated with a common 3-DoF point mass software with a fixed-muzzle-only Sg based on

Indeed neither model is intended to work with a progressively increasing static

stability.

44 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

As can be easily seen, the response to a varying Sg with a fixed ToF is clearly linear for both

models. Same behavior for a varying ToF with a fixed Sg. Bear in mind that the variation

ranges are quite narrow, which is the normal and expected uncertainty of these inputs.

1.5

3.5

5.5

7.5

9.5

11.5

13.5

15.5

17.5

19.5

1.00 1.50 2.00 2.50 3.00

Berger 215gr Hybrid - SD (inches) with a varying Sg and fixed TOF

Litz

B&R

4.0

5.0

6.0

7.0

8.0

9.0

1.500 1.550 1.600 1.650 1.700

Berger 215gr Hybrid - SD (inches) with a varying TOF and fixed Sg

Litz

B&R

45 / 51

Calculating Yaw of Repose and Spin Drift for Firing Point Conditions Boatright & Ruiz rev.

September/2018

The slight decrease in SD with increasing ToF shown above for the B&R model is explained

by not adjusting the velocity V(t) of the bullet as ToF is varied. The B&R model uses V(t)

explicitly in many places.

In the 4-DoF case, the model response to a varying initial Sg with a fixed ToF, is quasi-linear

and also exhibits a quite similar magnitude of the delta variation of Sg as the Litz and B&R

models.

The 4-DoF (Modified Point-Mass, Lieske & Reiter, 1966) provides an estimate of the yaw of

repose. This model considers the bullet rolling motion around its longitudinal axis of

symmetry, called spinning motion. Therefore, this model presents four degrees of freedom:

three translational coordinates for describing position and one for angular speed.

Some may argue that the underlying phenomena calls for a more elaborated multi-

parameter analysis and while the concept is right, we chose to perform a single parameter

analysis in order to compare to the Litz model which is a simple 2D model and as such does

not relate the influence of one parameter over the other as the bullet goes down range,

namely the aerodynamic coefficients.

4.8

5.8

6.8

7.8

8.8

9.8

1.00 1.50 2.00 2.50 3.00 3.50

Hornady 178 BTHP - SD (Inches) with a varying Sg and fixed TOF

4DOF

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September/2018

The Spin Drift is expressed inches, while each bullet is compared with the three different

estimators, and grouped at 1000, 1500 and 2000 yards, which are typical ranges for

extended Long Range shooting.

The Litz estimator does fair work, given its simple inputs, but its reliability is dictated by the

underlying aerodynamics characteristics of the bullet, which are not accounted for in this

simple linear approach.

Consequently, as soon as the bullet does not exhibit certain properties that cannot be

encompassed by Sg alone, its predictive accuracy is decidedly affected. In general terms, the

Litz model tends to over predict SD in a significant way.

0.0

20.0

40.0

60.0

80.0

100.0

120.0

1000 1500 2000 1000 1500 2000 1000 1500 2000

Sierra 168 Intl. / 175 M118LR / Sierra 220 SMK

B&R Hornady 4DOF Litz

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September/2018

As can be appreciated, as the range increases, the difference among the estimators becomes

larger. The practical side of this is that the correct method is of paramount importance when

dealing with Extreme Long Range (ELR) shooting.

0.0

10.0

20.0

30.0

40.0

50.0

60.0

1000 1500 2000 1000 1500 2000

Berger 215 Hybrid / Hornady .338 285 BTHP

B&R Hornady 4DOF Litz

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0.0

20.0

40.0

60.0

80.0

100.0

120.0

100 600 1100 1600

Sierra 168gr Int. / SD (inches) / range in yards

B&R

Hornady 4DOF

Litz

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

100 600 1100 1600

175gr M118LR / SD (inches) / range in yards

B&R

Hornady 4DOF

Litz

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0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

100 600 1100 1600

Boatright 173gr ULD / SD (inches) / range in yards

Litz

B&R

0.0

10.0

20.0

30.0

40.0

50.0

100 600 1100 1600

Sierra 220gr SMK / SD (inches) /range in yards

B&R

Hornady 4DOF

Litz

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Closing Summary

Taken together, the implications of Eq. 8 and Eq. 19 determine the bullet and rifle

characteristics which affect the size of the horizontal spin-drift SD(t) which will be seen in

flat firing at a long-range target.

First

, we see from Eq. 8 that SD(t)

DROP(t) in distance units from the projected axis of the bore at firing.

This implies that modern lighter-

velocities V(0) and retaining more velocity farther downrange (higher ballistic coefficient,

lower drag bullets) will produce much less spin-drift SD(t) at any target distance compared

to slower, higher-drag bullets. That is, SD(t) is roughly proportional to time-of-flight t to the

target distance.

Second

, according to Eq. 19, the size of the scale factor ScF, and thence the size of the spin-

drift SD(t)2(t)*S/2 in

pounds. The ambient atmospheric density varies with shooting conditions.

V(t) depends upon its muzzle velocity V(0), its mass m,

and the integrated drag function CD -sectional area S =

2/4 d.

Third

, the spin-drift SD(t) of the bullet is proportional to its yaw of repose angle R(t)

throughout its flight:

R2(t)*V(t)]-1 dt

Both the coning rate 2(t) and the forward velocity V(t) of the bullet always gradually

decrease, continually increasing R(t) e 2(t) is

Iy/Ix and by the remaining spin-rate and

slowly increasing gyroscopic stability Sg of the flying bullet.

The forward velocity V(t) of the flying bullet depends on its launch velocity V(0) and its

coefficient of drag profile.

The yaw of repose attitude angle R(t) is

increased

for bullets having larger numerical Iy/Ix

ratios and higher initial stability Sg, but R(t) is

decreased

by using faster twist-rate barrels

and higher muzzle velocities V(0) to achieve that higher gyroscopic stability Sg.

Fourth

, the spin-drift SD(t) is directly proportional to the small-yaw coefficient of lift CL(t)

of the bullet. Very-Low-Drag (VLD) and Ultra-Low-Drag (ULD) bullet designs usually have

correspondingly reduced coefficient-of-lift functions at all supersonic airspeeds.

Fifth

, and lastly, the spin-drift SD(t) of the bullet is inversely proportional to the weight Wt

(or mass m) of that bullet.

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All else being equal, bullets made with lower average material densities, such as turned brass

or copper bullets, will weigh less and will suffer greater spin-drift.

These five SD effects combine multiplicatively in this analysis.

Some bullet and rifle design parameters recur in several of these different SD effects, and not

always working in the same direction.

As modern long-range rifles and their bullets seem to be evolving toward lighter-weight,

smaller-caliber, lower-drag bullets fired at higher muzzle velocities, these related

incremental variations in design parameters combine algebraically to

reduce the spin-drift

SD occurring on long-range targets

.

Disclaimers & Notices

The findings in this report are not to be construed as an official position by any individual or

organization, unless so designated by other authorized documents.

approval of the use thereof.

Free and public distribution of this document is unrestricted and encouraged.