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“Incoming fire has the right of way." - Unknown
Structured Barrel
Physics INSIGHTS
At TACOM HQ, we idealize physics, rooting our research and application in engineering discipline. It is for this reason we offer comprehensive physical and mathematical proofs and FEA (Finite Element Analysis) models and simulations using MSC/NASTRAN and LS-DYNA to describe Structured Barrels; independently peer-reviewed by a team of aeronautical engineers. This deep dive links complex subjects like motion, force, energy, dynamics of a rigid body, oscillations, waves, temperature, heat, and thermodynamics to barrels and internal and external ballistics. Various interactive fields are provided throughout to gain real-time insight to variable effects on a barrel. TACOM HQ offers the next level in barrel technology for sustained performance. Contact us today to learn more about our product and how we can assist to unlock next-level efficiency.
The Science Behind How Structured Barrels Work
1.1 Axial Stiffness
Structural stiffness is a measure of an object’s resistance to deflection when subjected to a force. Axial stiffness is one example that refers to a structure’s resistance to deformation along its longitudinal axis when a parallel force is applied. It is quantified by a coefficient that represents the relationship between an axial force (F) and the resulting axial displacement (x). A basic example, in finite terms, is an axial spring where deflection is defined as F = kx, also known as Hooke’s Law, where F is the applied force, k is the spring coefficient / stiffness constant, and x is the displacement. The following image demonstrates as load increases, deflection increases. Less visually obvious, though, is as spring stiffness increases, deflection decreases. In a rifle barrel, axial stiffness is a linear function of the cross-sectional area and its material stiffness (modulus of elasticity).
Axial stiffness in structural engineering analyses account for structural weight, external force(s), and in mechanical systems, like shafts and columns, determines the structure’s ability to maintain alignment and resist deflection when subjected to an axial load. Axial stiffness also includes vibration as this affects how an object elongates and contracts under load.
1.2 Bending Stiffness
Unlike axial stiffness, bending stiffness is the measure of an object’s resistance to deflection when subjected to a lateral load, or an end bending moment, and by increasing material distribution in the plane of the force will yield greater resistance to flexion. This concept is simplified by evaluating the below figure. Click here for a complete video animation.
Flexion caused by a bending load may be calculated using closed form equations from simple cross-sections, but for variable section beams, like Structured Barrels, a more complex method like finite elements is required. In general, though, bending stiffness is linearly related to the moment of inertia of the cross-sectional area. Therefore, comparing the relative bend resistance of two objects (conventional barrels versus a Structured Barrel) is done by contrasting their respective moments of inertia and as a function of section area.
1.3 Natural Frequency
An object’s resistance to bend is intrinsic to its mass distribution, structural stiffness, and material stiffness (modulus of elasticity). Tuners, like suppressors, can alter mass distribution, changes in length and diameter affect structural stiffness, while 416SS, 400 MOD, or chromoly exhibit different material stiffnesses. Each yields varying degrees of influence depending on the scale of the variable change and its relationship to the system’s overall mode.
These three functions drive an object’s natural frequency and its resistance to deflection. This can be clearly understood by considering the sound an object makes when struck. The pitch of that sound is the audible frequency at which the object is bending to some displacement value in oscillation. For example, when a tuning fork undergoes resonance, it vibrates at some frequency until energy dissipates through grounding and friction.
Similarly, barrels want to vibrate at their natural frequency(s) during a firing sequence, but in the case of rifles, it is not just barrel stiffness that determines its behavior. The attachment to the stock/chassis, the stiffness of the bipod/mount, the stiffness of the bipod interface with the ground, and the stiffness of the rifle interface with the shooter all serve as a unique interfaces comparable to a spring with its own stiffness constant, and since they are all attached to the same body, they all simultaneously affect the system’s dynamics. It is important to note, wave functions propagate ideally through uniform, homogeneous mediums, and those that diverge from the natural frequency of a body or individual component generate irregularities and non-repeating wave patterns that induce amplitudes perceived as vibration, which dissipates by grounding and friction.
1.4 P and S Waves
Vibration consists of S and P waves. S waves are sinusoidal functions while P waves are complex sheering functions generated from compressive and expansive forces. When a barrel undergoes flexion, S and P waves resonate at 8,000 and 13,000 mph, respectively, traversing the barrel’s entire length many times before the bullet even exits the muzzle. Examples of a S and P waves include barrel whip and the compression of the barrel acted by bullets engaging rifling and high bore pressure, respectively.
1.5 Temperature and Waves
The speed with which these waves travel through the barrel though varies by temperature. Greater the temperature yields greater atomic vibration, and this increases the distance between atoms and thus the time for one atom to collide with another atom to relay a wave at some frequency. Since all atoms vibrate, a consequence of having a temperature above absolute zero, temperature drives the five material states: solid, liquid, gas, plasma, and Bose-Einstein condensate. Wave functions travel more efficiently through solids than liquids and in liquids than gases (with minimal energy loss in the vibration of the atom itself) due to atomic vibration and atomic arrangement density. In other words, atoms are more closely packed in a solid and so on. Waves are described by the equation λf = v. Click on the following video animations on Sound Waves and Sound Propagation in Different Mediums.
1.6 Barrel Flexion and Waves
As heat flows from a high to low energy state, from low to high entropy, incremental increases in temperature induce infinitesimally small changes in the efficiency of the medium to conduct waves as the medium itself undergoes infinitesimally small phase shifts from solid to liquid. Sinusoidal S and compressive P waves are generated by gas pressure, recoil, and the interaction between the bullet and the barrel in way of rifling, bore concentricity and straightness, and gravitational torque imparted on the barrel itself. These waves are the derivative of immense forces working to simultaneously push, pull, stretch, compress, and twist a barrel. The speed and magnitude of each flexion is relayed as a frequency of displacement, and they respond to temperature gradients. Since these waves travel many times faster than the bullet, they create amplitudes that can further exaggerate the bullet’s interaction and effect(s) on the barrel, causing even greater deflection and accelerations about its center axis that is perceptible as barrel whip (a consequence of highly dynamic constructive and destructive interference patterns).
1.7 Ladder Tests
Conventional barrels exude sufficient stiffness, but since no object is absolutely rigid, flexion affects axial or lateral bend amplitudes, causing different shot patterns. Applying a Gaussian distribution, 67% and 95% of groups will fall within a 1 and 2 standard deviation, respectively. Bell curves like this are common to predict the likelihood of specific outcomes in various scientific disciplines. In other words, a rifle that yields a 0.5-moa group will do so most of the time. In theory, ladder tests should produce purely vertical groups with points of impact increasing in relation to increases in velocity if all other variables are kept constant. However, ladder tests almost always yield horizontal dispersion.
Given rifling only marginally affects BC (1-2% on a “shout-out” barrel) and hand loads further reduce error found in principal axis tilt or pressure curves, contrasts in group size and or position suggest other forces are at work. Shooters can compensate using their scope, zeroing to an observed position of precision. For example, if A-bullet shot an average (x,y) and B-bullet shot (x+1,y+1), technically, one could dial out the “+1” movement and make that zero, but this zero is now calibrated to those conditions, and changes in barrel temperature due to ambient air, sun soak, or long, shot strings introduce a variable. Thus, barrels that undergo horizontal dispersion from small variable change are likely reacting to one or more forces affecting flexion, which is exacerbated by heat as it changes material properties.
Focusing solely on the vertical plane, extremely small flexion values effect down-range results as depicted in the illustrations below. The image on the left is a 1.35″ DIA barrel. The image in the middle is the same barrel on the left, but magnified 8x. The light, thin gray line above the barrel represents 0.001″ of flexion. Therefore, measuring the DIA of the first image in real-life, and multiplying this value by 8x, totals the DIA of the barrel in the middle image to see the light, thin gray line above the barrel representing only 0.001″ of flexion. The image on the right depicts a barrel with varying degrees of flexion ranging from 0.0005″ to 0.002″ and its corresponding impacts at 100-yds on a scaled target. These calculations realize a 24″ barrel with a fulcrum 6″ from the chamber. This illustration further appreciates a nominal difference in trajectory arc for displaced ϴ ranging 0.0016° to 0.0064°. This is equivalent to 0.0954 MOA or 0.0278 MIL or 0.3822 MOA or 0.1112 MIL, respectively, which are discernible within a scope.
We see the difference between 0.0005″ and 0.001″ of flexion is only 0.100″, which is not only challenging to isolate among other variables, like CGI offset and or heat driving driving thermal expansion in consecutive 5-shot groups, but it would also be difficult to discern in a ladder test, giving rise to Gaussian distribution curves. So while the first order of dispersion may be visible movement, these small, near invisible movements about the barrel can have profound significance down-range. Again, this focuses solely on the vertical plane though for illustrative purposes, despite bullet-barrel interaction imposing flexion in multiple planes and affecting the magnitude of the offset, given their xyz vector components.
8x
1.8 Second Moment of Area
The S and P waves produced from gas pressure, recoil, and the bullet’s interaction with the barrel transfer enormous energy into the compression, tension, and torque of the barrel. As these linear and radial functions traverse the barrel, amplitudes manifest as acceleration and thereby a velocity about its center axis. This velocity is described by its moment of inertia while its flexion is described as its second moment of area.
To better visualize a barrel’s physical response to waves, the following models and simulations are accredited to Al Harral, in charge of performing engineering and structural analyses on complex systems for 30 years at Lawrence Livermore National Laboratory, a government R&D facility. Analyses utilized LS-DYNA Finite Element Code, a powerful program for analyzing the dynamic and static loading of structures. It is important to note, the following FEA simulations assume a perfectly straight bore and exhibit displacements that are greatly amplified so changes in shape can be easily perceived. The simulations exhibit “forced deformations from the high pressure gas and recoil cause the muzzle to change where it is pointing at the target when the bullet exits the muzzle (….) The recoil and bullet motions ‘pulls’ the rifle barrel to a new shape.” – Al Harral’s 3-D Finite Element Analysis of a Barrel’s First Few Vibration Mode Shapes and Frequencies. Hence, a barrel with superior stiffness will exhibit less flexion just from gas pressure and recoil alone. Further, the depicted simulations shows “The vertical amplitude of vibration is more heavily excited than the horizontal vibration, because the center of gravity of the rifle is located below the barrel’s center line and the bullet’s travel down the barrel causes a vertical turning moment about the rifles center of gravity. [Making] the vertical vibration most important” – Al Harral’s Barrel Tuner Vibration Analysis.
The simulated 6mm rifle assumes a 22″ 416 stainless barrel with no rifling and straight taper from 1.24″ at the breach to a 0.935″ DIA at the muzzle. The modeled barrel, action, stock, scope, scope mounts, brass weight, etc. came within approximately 7oz of the real thing. Mesh detail is provided, consisting of 17,196 nodes and 12,080 elements. Despite only half of the mesh being used in calculations, each shot calculation still took about an hour to set up and simulate, outputting about 2.5Gb of data.
Mode 1 – Cantilever Bending
82Hz
Mode 2 – 1 Node Bending
406Hz
Mode 3 – 2 Node Bending
1050Hz
Mode 4 – Axial Extension
1756Hz
Mode 5 – 3 Node Bending
1984Hz
The above simulations demonstrate how different frequencies manifest as physical movement along the barrel’s length. It is important to note, while modes 1, 2, and 3 are shown in a single plane, they may exist in other planes, and exemplary modes 1-5 are – all – excited, simultaneously when firing a round. While the higher frequency modes have extremely small amplitudes, it is difficult to visualize just five modes acting at the same time, let alone many more.
The amount of resistance a shape can exert to bending about its geometry on a particular axis is its second moment of area, and this is dependent on the area and material distribution to the axial moment. Cross-sections of an object that locate the majority of the material far from the bending axis have larger moments of inertia, making said cross section considerably more difficult to bend when force is applied. This is one reason I-beams, and not rectangular bars, are commonly used in structural applications, because they position sufficient mass in both x-y planes to resist force and deflection. Since Structural Barrels are analogous to I-beams reinforcing the bore in multiple planes, this creates a geometry that resists bending in multiple planes.
1.9 Mathematical Proofs for Increased Stiffness
Second Moment of Area (J) for a cross section of a barrel
J = ∫ r² dA; where r² = x² + y² and dA = 2π • r dr
J = ∫ r² • 2π • r dr = πR⁴/2
Perpendicular Axis Theorem Jz = Ix + Iy; barrels are a symmetric object
Ix = Iy = πR⁴/4
Second Moment of Area per Unit of Area | Rod (conventional barrel) versus Tube (Structured Barrel)
SRod = IRod / A; where A is the area πR² = R²/4
STube= (IRod / A) – (ITube / A) = R²(1+x²)/4
We see as the x limit approaches a 1:1 ratio to the radius R in the STube, a tube exhibits twice the stiffness, but this also requires an infinitely small radius. Nevertheless, these equations indisputably demonstrate a body of equal mass positioned farther from the bending axis has a larger moment of inertia, making it significantly more difficult to bend. This is further corroborated by an independent aeronautical CEL analysis that found +50% greater stiffness in a simulated barrel of the same weight.
Using R²/4, or the stiffness per unit area, we can evaluate this to a barrel that is X% larger, and the difference between a big and small barrel may be simplified as R²(1-X²)/4. Since X is the difference between two barrel diameters as a percent, it is important to note this value is driven by a square. So, procuring a 1.65″ blank versus a 1.35″ blank yields 22.50% more (free) stiffness. Compound this with added structural rigidity, and this value becomes rather significant. It is no secret bull barrels are typically more precise than a standard profile. Mathematically, this is one reason why.
2.1 Flexion and Energy
Gas pressure, recoil, and the interaction between the bullet and the barrel are all exemplary forces that act on a barrel The equation to describe force is the multiple of mass and acceleration, the rate velocity changes in respect to time. Since barrels have mass, the work done to accelerate a barrel onto a sinusoidal wave about its center axis via tensive and compressive forces requires energy. These relationships are described by the below equations:
v = x/t
a = dx/dt (rate of change)
F = m•a
W = F•Δx
ΔK = W
K = Energy (kinetic)
2.2 Energy and Temperature
Since energy is proportional to the work that acts on a system from an initial state to a final state (the First Law of Thermodynamics) and the Law for Conservation Energy dictates energy remains constant and is neither created nor destroyed, the work done on a barrel to accelerate it about its center axis due to flexion requires massive energy. Friction caused between atoms as the barrel accelerates into flexion generates heat, thereby raising barrel temperature and affecting material stiffness (modulus of elasticity). Even though each atom is not vibrating at the same rate, described by Maxwellian distribution, it is important to note the increase in thermal energy raises all individual atom vibrational speeds as an average, even though the medium as a whole will consist of multiple heat gradients along the barrel’s entire length.
2.3 Energy and Temperature
Bull barrels maintain a lower temperature than a standard profile, and a standard profile than a pencil barrel, a reason typically accredited to thermal mass. While this is correct, it fails to define the function and thereby other primary variables. Physics tells us the rate of heat transfer is directly proportional to the temperature difference and transfer area, but inversely proportional to the area’s thickness.
Q = kA • ΔT/ΔX; where k is the thermal conductivity coefficient
A = 2πrL (still L x W)
ΔX = Δr (for the radius)
In this evaluation, since temperature and radius are changing with respect to bore center, we use an integral ( ∫ ) to solve an infinite number of evaluations from r1 to r2. Rewriting and solving the above equation yields:
Q = 2πrkL • (T1 – T2) / Ln(r2/r1); (W)
This equation mathematically defines the amount of heat energy required to be conducted through the wall of a tube from bore r1 to bring the exterior r2 to temperature T2. If we increase wall thickness L by increasing r2 or temperature disparity ΔT, we see Q, the energy to reach temperature T2, also increases.
2.4 Thermal Expansion
The increase in vibration speeds affect the kinetic energy of individual atoms and thereby the properties of the material. In solids, atoms are closely packed together, held firmly by strong attractive forces. Since these atoms primarily vibrate about their fixed positions without moving past each other, solids have a definite shape and volume. These vibrations are described by Lennard-Jones Potential, a physics equation that evaluates the interaction and resulting potential energy between two atoms based on their distance of separation. Visualized as a spring linking two neighboring atoms, the potential energy is asymmetrical, increasing more steeply as two atoms get closer, a consequence of it taking more energy to compress two atoms together than to repel them apart, and this causes atoms to spend more time apart than together when oscillating. Thus, as the kinetic energy of atoms increases, the average distance between each atom also increases, thereby causing the material to expand. Since solids are isotropic, there is no preferred direction, so while proportions stay the same, the overall size changes by some fraction per degree (thermal expansion coefficient). This is called thermal expansion, or the tendency of matter to change in shape, volume, and area in response to changes in temperature.
Thermal expansion is defined by the physics equation:
ΔL = αL • ΔT
ΔL is the change in length
α is the thermal expansion coefficient
L is the starting length
ΔT is the change in temperature
Using 6.5 x 10-6 inches per °F as the thermal coefficient for 416 stainless steel, changes in ΔT will affect barrel length, diameter, and even bore diameter. Premiere bullet manufacturers segregate below a 0.0001″ margin of error, and simple air temperature, sun soak, or extended shot strings can easily accumulate +0.0001″ of bore movement. Using this equation assumes a uniform temperature about its entire longitudinal and radial length, so it is important to note temperature gradients realize different thermal expansion rates and material properties as layers undergo phase shifts.
2.5 Energy and Waves
As thermal energy increases, the barrel’s atomic arrangement density also decreases due to thermal expansion. This expansion is uniform per calorie, but not uniform about the barrel length due to many temperature gradients. Since the mass of the barrel remains unchanged but the overall distance between each atom increases, thermal expansion continuously changes the tendency for the barrel to conduct waves as the distance and associated time for one atom to collide with another atom to relay a frequency changes gradient to gradient. Section ‘Temperature and Waves’ aides to provide more detail on the concept. Thermal expansion also fundamentally alters the barrel’s response to resulting waves generated by gas pressure, recoil, bullet-barrel interaction, and gravitational torque as barrel dimensions are continuously changing with heat.
2.6 Variable Heat (Q)
While heat variable Q has minimal effects on the system’s rigidity at minimal round counts, extended firing sequences adds non-linear heat values Q₁ + Q₂ + Q₃ + Qₙ into the system, causing the barrel to become increasingly susceptible to flexion due to changes in material stiffness. An example of this is identifying groups opening, losing both precision and accuracy, on extended round counts. The prime temperature gain Q₁ describes detonation, pressure, and friction, but each subsequent Qₙ segways into greater and greater contributions from atomic friction and the work done to accelerate the barrel to greater and greater amplitudes. This increasing Q variable affects barrel stiffness and thereby reduces precision and accuracy.
2.7 Heat and Material Failure
The work converted into thermal energy not only increases atomic vibration, the time an atom oscillates from its neighboring atom per Lennard-Jones Potential, but it changes material stiffness (modulus of elasticity) and other properties that affect its response to force load. Once a force impulse wrought by flexion exceeds the material’s stiffness provided by its temperature moment, material failure ensues.
3.1 Non-linear Acceleration
The work done to accelerate a barrel about its center axis can also cost valuable kinetic energy (K = 1/2•m•v²). Knowing no system is absolutely rigid with zero energy loss, projectiles must accelerate through a dynamic system, causing irregular accelerations given the bullet must momentarily straighten an imperfectly straight bore under non-uniform load. “The barrel [itself] is initially slightly deflected downward due to gravity [torque]. When the round is fired, the pressure also tends to straighten the barrel. As the barrel straightens, it over shoots in the upward direction and this adds to the excitation of the Mode 1 vibration. As a side note, the axial extension vibration mode is also probably heavily excited.” – Al Harral’s Barrel Tuner Analysis. This concept is further expanded on in Ep. 057 of the Hornady Podcast discussing ‘One Hole Groups? Dispersion’.
3.2 Ballistic Coefficient (BC)
Accelerating the barrel about its axis not only costs energy, but it can also affect BC, the ability for a projectile to resist drag. The following is an exert from Sierra Bullets 2.4 Lessons Learned from Ballistic Coefficient Testing:
“Theoretically, the BC of a bullet depends only on its weight, caliber and shape. But in a practical sense, the measured BC of a bullet also depends on many other effects. The gun can affect the measured BC value in two important ways: spin stabilization and tip-off moments. A bullet is gyroscopically stabilized by its spin, which is imparted by the rifling in the barrel. If a bullet is perfectly stabilized by its spin, then its longitudinal axis (which is also its spin axis) is almost perfectly aligned with its velocity vector. If a bullet is not perfectly stabilized (which is usually the case), the bullet may not be tumbling, but its point undergoes a precessional rotation as it flies. In previous editions of Sierra’s Reloading Manuals we have described this precessional rotation and have called it “coning” motion to aid in mental visualization of the motion. As the bullet flies, the point rotates in a circular arc around the direction of the velocity vector. Coning motion results in increased drag on the bullet, and any firing test method then yields an effective BC value for the bullet that is lower than the theoretical value. We have never been successful in accurately predicting BC values, or determining these values by any method other than firing tests.”
1. Longitudinal axis
2. Velocity vector
3. Nutation
4. Precession
ψ Yaw
Image credit
“Not only is the muzzle exit angle changing in time, but the muzzle is also moving in the vertical direction while the bullet is traveling down the barrel. When the bullet exits the muzzle, the bullet will have the same velocity as the muzzle.” – Al Harral’s Barrel Tuner Analysis. One of the most basic evaluations, in finite terms, is given bullets have length, they cannot be evaluated as a finite point. Thus, bullets will almost always have nutation, precession, or yaw of some degree due to a barrel’s acceleration about its center axis from gas pressure, recoil, and the interface between the bullet and the barrel. This is the precise reason load development embodies finding an amplitude where barrel velocity and the resulting tip-off moment is minimal.
