Analysis of twist-rate effects in a progressive-rifling barrel using finite element method

Abstract

In this study, a nonlinear finite element model of a 30 mm chain-gun barrel with progressive rifling was developed to investigate the influence of twist-law parameters on interior ballistic behavior and rifling-induced stresses. The progressive twist design was characterized using the twist exponent n and the muzzle exit angle $α_{E}$ . Their effects on projectile translation, spin evolution, and rotating-band engraving stresses were evaluated. The results reveal that muzzle velocity and axial acceleration are governed primarily by chamber-pressure loading and are largely insensitive to the rifling-geometry variations considered. In contrast, the twist exponent has a pronounced influence on spin-rate growth and on circumferential ( $θ$ -direction) stresses induced by rotating-band engraving. Circumferential stress exhibited strong sensitivity to twist intensity, with variations exceeding 30%, indicating that progressive twist design directly controls stress concentrations along the rifling lands. Among the examined cases, n = 1.6 produced a comparatively balanced stress response and stable spin-rate development. Based on a combined assessment using finite element stress analysis and the Miller gyroscopic stability criterion, a design range of n = 1.6 and $α_{E}$ = 7°–8° is recommended to achieve adequate gyroscopic stability while limiting excessive engraving stresses. The proposed modeling framework provides a quantitative basis for optimizing progressive-rifling barrels under high-pressure interior ballistic conditions.

Keywords

finite element method progressive rifling rotating band engraving stress interior ballistics

1. Introduction

The primary function of rifling is to impart rotational motion to a projectile to stabilize its trajectory. Rifling consists of helical grooves on the inner surface of a gun barrel; the raised portions are referred to as lands, and the recessed portions as grooves. The axial distance required for the rifling to complete one full revolution is defined as the twist pitch, whereas the number of calibers per revolution is termed the twist rate. The angle between the tangent to the rifling at any point and a line parallel to the barrel axis is defined as the twist angle.

During interior ballistic motion, the inner diameter across the lands is marginally smaller than the outer diameter of the rotating band; therefore, the rotating band undergoes engraving owing to radial interference. As the projectile is driven forward, the rifling induces rotational motion, enabling the projectile to maintain spin both inside the barrel and after muzzle exit, thereby enhancing flight stability and overall accuracy. Consequently, an appropriate rifling design is essential to ensure projectile stability and precision, as shown in Figure 1(a).

Figure 1.

Illustrations of rifling geometry.

The cross-sectional geometry of rifling lands and grooves can be categorized into several canonical profile types. The most prevalent configuration is the conventional rifling profile, which features sharp-shouldered rectangular lands and grooves. Beyond this traditional form, various alternative profiles are used in modern barrel design. Enfield/5R rifling and ratchet rifling correspond to symmetric and asymmetric trapezoidal cross-sections, respectively. Hybrid rifling incorporates rounded corners between rectangular and circular-arc contours. Polygonal rifling is characterized by an outer boundary formed by multiple circular-arc segments, producing a multi-arc (or chord-defined) polygonal profile. Representative examples of these rifling geometries are shown in Figure 1(b).

Rifling designs may also be categorized by twist-rate configuration, including uniform rifling and progressive rifling. Uniform rifling maintains a constant twist rate throughout the entire barrel length and is widely applied in conventional firearms to ensure stable projectile spin and a consistent rotational speed. In contrast, progressive rifling features a twist rate that gradually increases toward the muzzle; that is, the twist angle becomes larger as the projectile advances. This design enables the projectile’s spin rate to build progressively, thereby enhancing gyroscopic stability.

For gun barrels operating under high chamber-pressure conditions (e.g., a 30 mm chain gun), excessive torsional loading during the early stages of interior ballistics may cause severe engraving and rotating-band damage, or even structural failure of the projectile body, ultimately degrading flight stability and accuracy. A key advantage of progressive rifling is its gradually increasing twist rate from the breech toward the muzzle. This gradual increase mitigates the initial torsional impulse on the projectile, allowing the rotating band and projectile body to establish spin progressively during acceleration. As a result, abrupt stress peaks that could lead to material failure are reduced, and mechanical loading on the barrel near the breech is mitigated. Progressive rifling is commonly defined using a parabolic twist function (varied twist rate) with two parameters, including the twist-rate gradient and the muzzle twist angle, which together balance gyroscopic stability and structural integrity.

Recent developments in internal ballistic modeling have been reviewed by Ongaro et al.¹ covering lumped-parameter and computational fluid dynamics-based approaches. Regarding rifling studies, Kang et al.² proposed a systematic optimization framework for rifling design in 2012. By integrating design of experiments with simulation-based design, they established an optimal design procedure for uniform rifling and conducted high-fidelity simulations using LS-DYNA. This approach reduced the required number of experiments and identified improved design parameters. In 2014, Deng et al.³ performed a nonlinear finite element analysis to obtain the complete interior ballistic motion and stress history of a 9 mm pistol barrel. In 2017, Ding et al.⁴ developed a parameterized modeling and mesh-generation method for worn uniform-rifling barrels, demonstrating that bore wear decreases the frictional resistance on the rotating band, thereby reducing chamber pressure and degrading ballistic performance.

In 2022, Deng et al.⁵ incorporated the discrete element method with finite element analysis to simulate the interior ballistics of a shotgun. In the same year, Wei et al.⁶ investigated stress-distribution characteristics during the projectile engraving stage for various uniform-rifling cross-sectional profiles. More recently, Deng et al.⁷ extended shotgun interior-ballistic simulations by introducing an equivalent aerodynamic model to capture continuous interior-to-exterior ballistic behavior of shot pellets, thereby enabling prediction of pellet-cloud velocity, dispersion angle, and hit probability. Zou et al.⁸ investigated wear in constant-twist rifling barrels using a temperature-corrected Archard model combined with finite element analysis and showed that sliding velocity and contact pressure play dominant roles in wear evolution. However, to date, no numerical analysis or simulation work has been published on nonuniform rifling, such as progressive rifling.

Regarding rotating-band studies, Silton and Weinacht⁹ investigated the aerodynamic effects of rifling engravings on projectiles fired from uniform rifling. Their results indicated that rifling grooves have only a minimal influence on aerodynamic behavior and flight stability. This suggests that rotating-band engraving primarily affects the interior ballistic phase, including projectile velocity, spin rate, and stress-related mechanical responses. In 2013, Eleiche et al.¹⁰ proposed replacing conventional copper rotating bands with polyamide 66. Their findings revealed that adding 2% glass fiber reduced friction and wear while maintaining sufficient strength and toughness, thereby improving barrel life and overall ballistic performance. In 2014, Wu et al.^11,12 conducted quasistatic and dynamic tests on rotating bands made of copper, aluminum bronze, and nylon. Their results indicated that strain rate and temperature strongly affect rotating-band deformation behavior. Copper and aluminum bronze produced substantially higher engraving forces than nylon, and copper exhibited the best performance under high-pressure and high-velocity loading, making it the most suitable material for operational firing.

In 2017, Chen et al.¹³ established a constraint model describing the interaction between rifling and the projectile body and derived geometric and mechanical relationships among twist-rate variation, projectile angular velocity, and angular acceleration. Their analysis indicated that although progressive rifling can reduce initial wear and improve sealing, it may increase angular acceleration and introduce potential spin instability in later stages. Therefore, careful design of the twist gradient and transition region is required to ensure overall projectile–barrel performance and service life. In 2019, Shen et al.¹⁴ demonstrated that barrel damage increases initial projectile disturbance and decreases spin rate, ultimately degrading ballistic stability and firing accuracy. This finding highlights the significant influence of barrel stress deterioration on weapon-system performance. Wang et al.¹⁵ developed a coupled two-phase-flow and finite element model to simulate the engraving process under constant-twist rifling conditions and reported good agreement between numerical predictions and experimental measurements. More recently, Silva-Rivera et al.¹⁶ used finite element simulations to compare polygonal rifling with conventional groove rifling. Their results showed that chamber pressure and muzzle velocity differed only marginally between the two rifling types; however, polygonal rifling produced a more uniform stress distribution, larger deformation regions, and shallower engraving impressions, suggesting improved durability and reduced localized stress concentrations. Nguyen et al.¹⁷ reported that a higher terminal twist increases projectile spin rate but requires greater driving force, potentially accelerating barrel wear and reducing service life.

Although these studies include both experimental and numerical investigations, they focus on rotating-band engraving under uniform (constant-twist) rifling. Because the twist angle continuously increases in progressive rifling, the rotating-band engraving process extends from the rifling origin to the muzzle. To date, no research has addressed rotating-band engraving in progressive-rifling barrels. In this study, a 30 mm chain-gun barrel with progressive rifling is used to analyze and simulate interior-ballistic performance under different twist rates and muzzle exit angles. The investigation evaluates muzzle velocity, acceleration, and spin rate as indicators of overall projectile flight stability and examines the effects of rifling on rotating-band engraving stresses.

2. Theoretical background

2.1. Progressive rifling

The defining feature of progressive rifling is that the twist angle increases gradually from the start of the rifling to the muzzle. Accordingly, the projectile initially travels through the bore under a relatively small twist angle; as it advances along the barrel, the twist angle increases progressively. This gradual increase raises the projectile’s spin rate, which reaches its maximum at the muzzle, as shown in Figure 2.

Figure 2.

Progressive rifling.

According to the U.S. Army design manual,¹⁸ the rifling twist law can be described using a parabolic-type formulation with exponent n and constant p:

{y = p x}^{n} .

(1)

Here, $p$ is the coefficient of the rifling twist equation. The variable $x$ denotes the axial distance along the rifling curve, and $y$ represents the circumferential distance traveled around the barrel. This $y$ can be interpreted as the arc-length parameter of the rifling when the cylindrical surface is unwrapped onto a plane. The geometric relationship is (see Figure 3)

y = R β,

(2)

where

R

is the barrel radius, and

β

is the angular displacement of the projectile. When

n

= 1, the rifling is uniform; when

n

> 1, the rifling is progressive. Differentiating Equation (1) gives the local slope of the rifling curve, which corresponds to the tangent of the twist angle:

\frac{d y}{d x} = p n x^{n - 1} = \tan α,

(3)

where

\tan α

is the slope of the unwrapped rifling curve, and

α

is the local twist angle (Figure 3). For progressive rifling (

n

> 1), the twist angle varies with axial position and is denoted as

α (x)

Figure 3.

Rifling geometry and twist law.

For $n$ = 1 (uniform rifling), Equation (3) reduces to $p = \tan α$ , indicating that the slope (and thus the twist angle) is constant along the barrel. For $n$ > 1 (progressive rifling), the slope varies with axial position. Let $x_{E}$ denote the muzzle exit position (i.e., the total rifling length), and let $α_{E}$ = $α (x_{E})$ be the muzzle twist angle. Substituting x = $x_{E}$ and $α$ = $α_{E}$ into Equation (3) gives

p {n x}_{E}^{n - 1} = \tan α_{E} .

(4)

Therefore,

p = \frac{\tan α_{E}}{{n x}_{E}^{n - 1}} .

(5)

Differentiating Equation (2) yields the relationship between angular velocity and twist angle:

\frac{d y}{d x} = R ω \frac{1}{v} = \tan α,

(6)

where v is the projectile axial velocity, and

ω

is the projectile angular velocity. Rearranging Equation (6) gives

ω = \frac{v \tan α}{R} .

(7)

2.2. Direct numerical integral (DNI) solution for interior ballistics of progressive rifling

During firing, the projectile axial velocity can be estimated by directly integrating the pressure-driven acceleration using a simple explicit time-marching scheme, while neglecting rifling–rotating-band interaction. The velocity at the (i + 1)^th time step is written as:

v_{i + 1} = \frac{P (t_{i + 1}) A_{b}}{m_{D}} d t + v_{i},

(8)

where

P (t)

is the chamber pressure (time-varying),

v_{i + 1}

and

v_{i}

are the projectile velocities at times

(t + d t)

and

t

, respectively,

A_{b}

is the projectile base area,

m_{D}

is the projectile mass, and

d t

is the time step.

Similarly, the angular velocity $ω$ is determined from the instantaneous geometric constraint between the projectile and the rifling (Equation (7)). Thus, the angular velocity at the (i + 1)^th time step is obtained by substituting the updated axial velocity $v_{i + 1}$ and the local twist angle $α_{i + 1}$ into Equation (7):

ω_{i + 1} = \frac{v_{i + 1} \tan α_{i + 1}}{R} .

(9)

Using this iterative formulation, reference curves for velocity–time (v–t), acceleration–time (a–t), and angular-velocity–time (ω–t) can be generated to support analysis and comparison with finite element simulation results for progressive-rifling interior ballistics.

2.3. Miller twist rule

The Miller twist rule¹⁹ is used to estimate the rifling twist rate required for a given projectile to achieve adequate flight stability under standard atmospheric and ballistic conditions (Mach 2.5, 2800 ft/s, 59 °F, 750 mm Hg, and 78% relative humidity). Equations (10)–(18) summarize Miller’s original formulation and its associated environmental correction factors; the notation is adjusted only for consistency with the present study.

τ = \frac{L_{t}}{d},

(10)

where

τ

is the twist expressed in calibers per turn,

L_{t}

is the rifling pitch (in/turn), and

d

is the projectile caliber. The projectile relative length parameter is defined as

l = \frac{L}{d},

(11)

where

l

is the ratio of projectile length to caliber, and L is the projectile length (in). Based on Miller’s empirical formulation, the twist-rate–stability relationship is

τ^{2} = \frac{30 m_{g}}{s d^{3} l (1 + l^{2})},

(12)

where

s

is the stability factor, and

m_{g}

is the projectile mass (grains). The corresponding expression for the stability factor is

s = \frac{30 m_{g}}{d^{3} l (1 + l^{2}) τ^{2}} .

(13)

Equation (13) shows that the stability factor

s

is inversely proportional to the square of the twist rate, indicating that a smaller

τ

(tighter twist) corresponds to a larger stability factor. To account for nonstandard environmental and ballistic conditions, correction factors for velocity, temperature, pressure, and altitude are applied. The velocity correction factor is

f_{v} = {(\frac{v_{f t}}{2800})}^{1 / 3},

(14)

where

f_{v}

is the velocity correction factor, and

v_{f t}

is the muzzle velocity (ft/s). The temperature–pressure correction factor is

f_{T P} = \frac{(460 + T_{F}) / (460 + 59)}{P / 750},

(15)

where

f_{T P}

is the temperature–pressure correction factor,

T_{F}

is the ambient temperature (°F), and

P

is the atmospheric pressure (mm Hg). The altitude correction factor is

f_{h} = \exp (- \frac{h}{15830}),

(16)

where

f_{h}

is the altitude correction factor, and

h

is the altitude (ft). When all correction factors are applied, the combined corrected twist rate becomes

τ_{f i n a l} = τ_{s t d} \sqrt{f_{v} f_{T P} f_{h}},

(17)

where

τ_{f i n a l}

is the combined corrected twist rate. The corresponding corrected stability factor is

s_{f i n a l} = s_{s t d} (f_{v} f_{T P} f_{h}) .

(18)

Based on the Miller twist rule, incorporating projectile mass together with correction factors for velocity, temperature, pressure, and altitude enables more accurate predictions of projectile stability under specific firing conditions, thereby providing practical guidance for selecting appropriate projectile–rifling combinations.

3. Finite element model development

Computer-aided engineering (CAE) involves the use of computational tools for engineering design, simulation, and analysis and is widely used in fields such as structural mechanics, mechanical systems, fluid dynamics, and heat transfer. The primary objective of CAE is to support performance prediction during the design phase, thereby enabling design optimization, failure diagnosis, and evaluation of alternative configurations. Advances in CAE provide a virtual testing platform capable of reproducing real-world physical processes, reducing experimental cost and time while improving design accuracy and efficiency.

With continued improvements in computational power and numerical algorithms, CAE can address increasingly complex problems, including nonlinear finite element analysis, dynamic response, and thermomechanical coupling. As a result, CAE plays an important role in advanced manufacturing and in the development of aerospace, automotive, energy, and defense systems. For applications requiring high precision, numerical simulation provides strong predictive capability and supports iterative design refinement and performance optimization.

3.1. Nonlinear finite element analysis

Nonlinear transient finite element simulations were performed in LS-DYNA^® to model projectile–barrel interaction during the interior ballistic process. An explicit central-difference time-integration scheme was used because of its robustness in problems involving large deformation, high strain rates, and complex contact conditions. The interior ballistic event includes multiple sources of nonlinearity, including (i) material nonlinearity associated with elastoplastic deformation of the rotating band, (ii) geometric nonlinearity arising from large deformation during engraving, and (iii) contact nonlinearity between the rotating band and the rifling lands, as well as between projectile components.

The numerical framework follows the approach proposed by Deng et al.³ The penalty method is used to model contact interactions between the projectile and the rifling and between the projectile core and jacket. An isotropic elastic–plastic material model with failure is implemented to represent material behavior, enabling realistic simulation of rotating-band engraving and associated plastic deformation.

3.2. Computer-aided design (CAD) models

Simulations were conducted for a 30 mm chain-gun system. The overall barrel length is 2700 mm, including a 2550 mm rifled section with sixteen right-hand progressive rifling grooves. The barrel geometry is divided into three regions: (i) a smooth-bore chamber section, (ii) a forcing-cone transition section where rifling initiates, and (iii) a fully rifled section.

In the first stage of model generation, three rifling-twist profiles corresponding to twist-law exponents n = 1.2, 1.6, and 2.0 were constructed, as shown in Figure 4(a). In the second stage, four muzzle twist-angle configurations with $α_{E}$ = 6°, 7°, 8°, and 9° were generated, as shown in Figure 4(b). The full barrel geometry was created by circumferentially replicating a single rifling segment according to the total number of grooves (Figure 4(c)). The projectile assembly was modeled as separate components, including the detonator, projectile body, and rotating band (Figure 4(d)).

Figure 4.

CAD model.

3.3. Finite element models

The CAD geometries were converted into finite element models for simulation. The mesh configuration was explicitly defined instead of relying on default software settings. Eight-node hexahedral solid elements were used throughout. In the barrel model, the element layout was aligned with the rifling helix to capture the geometric twist and accurately represent projectile–rifling contact. Hexahedral elements were selected to reduce element count, improve computational efficiency, and enhance numerical accuracy.

Mesh partitioning was performed according to contact interfaces, particularly in the rifling–rotating-band interaction region, to ensure proper element alignment and accurate contact computation. Because the full barrel model was simulated, the total element count was relatively large. Therefore, a graded mesh strategy was applied in which the element size gradually increases from the inner bore surface to the outer diameter. This approach maintains a fine mesh in the rifling region and rotating band to resolve contact interactions while controlling the total element count, as shown in Figure 5(a) and (b). The rotating band, as the primary contact component, was further refined to accurately capture engraving and deformation.

Figure 5.

Finite element model.

The complete barrel model consists of 439200 elements and 500064 nodes (Figure 5(c)). The projectile assembly comprises the detonator (5996 elements; 6617 nodes), projectile body (6248 elements; 8377 nodes), and rotating band (129060 elements; 143100 nodes). In total, the projectile model contains 141304 elements and 158094 nodes (Figure 5(d)).

3.4. Material properties

Because the 30 mm gun barrel is designed to withstand high chamber pressures, rolled homogeneous armor (RHA) steel was adopted as the barrel material.²⁰ The detonator was modeled using 7075-T6 aluminum alloy,²¹ the projectile body using AISI 4140 medium-carbon steel,²² and the rotating band using copper.²³

During interior ballistic motion, the projectile experiences extremely high acceleration and deformation rates, which fall within the intended application range of the Johnson–Cook (J–C) constitutive model. The J–C model describes material behavior under large strain, high strain rate, and elevated temperature. Its constitutive form is given by²⁴

σ_{0} = (A + B ε^{N}) (1 + C \ln (\dot{\frac{ε}{\dot{ε_{0}}}})) {(1 - T^{*})}^{Q},

(19)

where

σ_{0}

is the flow stress. Parameter

A

is the quasistatic yield stress,

B

is the strain-hardening modulus,

ε

is the plastic strain, and

N

is the strain-hardening exponent. The coefficient

C

governs strain-rate sensitivity,

\dot{ε}

is the strain rate, and

\dot{ε_{0}}

is the reference strain rate (typically quasistatic). The homologous temperature is

T^{*}

= (T−T₀)/(T_m−T₀), where T is the current temperature, T₀ is room temperature, and T_m is the melting temperature. The exponent

Q

represents the thermal softening coefficient.

Because the firing duration in this study is approximately 4 ms and each case represents a single-shot interior ballistic event, thermal softening was neglected. Accordingly, the analysis focuses on strain hardening and strain-rate effects, and the simplified J–C model was adopted²⁴:

σ_{s} = (A + B ε^{N}) (1 + C \ln (\dot{\frac{ε}{\dot{ε_{0}}}})) .

(20)

The material parameters used in this study are listed in Table 1.

Table 1.

Material parameters of the barrel and projectile components.

Item	Barrel	Projectile
Item	Barrel	Detonator	Projectile body	Rotating band
Material	RHA steel²⁰	7075-T6 aluminum alloy²¹	AISI 4140 steel²²	Copper²³
Density (g/mm³)	0.00784	0.0027	0.00785	0.0088022
Young’s modulus (MPa)	205000	70000	205000	117000
A (MPa)	1832	520	1100	80
B (MPa)	1685	477	768	500
N (-)	0.754	0.52	0.2092	0.605
C (-)	0.00435	0.001	0.0137	0.29

3.5. Chamber pressure calculation

In ammunition performance evaluation and weapon-system design, chamber pressure refers to the pressure generated by high-temperature, high-pressure gases produced during propellant combustion. This pressure directly affects muzzle velocity, flight stability, and structural safety. Following the empirical formulation of Vallier–Heydenreich,²⁵ the average chamber pressure is expressed as

P_{m e a n} = \frac{(m_{D} + 0.5 m_{y})}{2 g A x_{E}} v_{0}^{2},

(21)

where

P_{m e a n}

is the average chamber pressure,

m_{y}

is the propellant mass,

m_{D}

is the projectile mass,

g

is the gravitational acceleration (9.8 m/s²),

A

is the bore cross-sectional area,

x_{E}

is the total rifling length of the barrel, and

v_{o}

is the muzzle velocity. The base pressure is obtained by converting the chamber pressure using the method proposed by Deng et al.^5,26:

P_{D} = \frac{P_{m e a n}}{1 + \frac{m_{y}}{3 φ_{1} m_{D}}},

(22)

where

P_{D}

is the base pressure, and

φ_{1}

is the projectile mass coefficient.

In this study, a 30 mm projectile with a mass of 378.0 g, a propellant mass of 151.5 g, and a measured muzzle velocity of 1063 m/s was used as input. The average chamber pressure and corresponding base pressure were calculated using Equations (21) and (22), and the resulting pressure histories are shown in Figure 6.

Figure 6.

Base pressure and average chamber pressure.

3.6. Boundary conditions

Boundary conditions were defined to represent mechanical constraints and loading during interior ballistic motion. The outer surface of the barrel was fully constrained to isolate the structural response, enabling focused evaluation of stresses induced by chamber pressure and by interaction between the rotating band and the progressive-rifling configurations (Figure 7(a)). Chamber pressure was applied to the projectile base to provide the driving force for forward motion and to reproduce the loading acting on the projectile base (Figure 7(b)). In addition, the internal bore and rifling regions were assigned the corresponding pressure distribution and contact definitions to evaluate the effects of chamber pressure on barrel stresses and to simulate engraving/contact behavior between the rotating band and rifling (Figure 7(c)). Gravitational acceleration was set to 9.8 m/s².

Figure 7.

Boundary conditions.

4. Simulation results and analysis

A full-barrel finite element simulation was performed in this study using the same chamber-pressure curve as the driving load for projectile motion. The interior ballistic process was evaluated primarily through the von Mises stress distribution. At t = 0.0 ms, the system is in its initial state, with the projectile at rest and negligible stress in the barrel. At t = 0.2 ms, the chamber pressure begins to rise in the chamber region, initiating forward acceleration of the projectile and generating low-level stress in the barrel. At t = 0.9 ms, the rotating band passes the forcing cone and fully engages the rifling, which increases contact stress and initiates noticeable plastic deformation. At t = 1.25 ms, the maximum von Mises stress reaches 1443.1 MPa, concentrated primarily in the chamber and the forward portion of the barrel. Significant plastic deformation occurs in the rotating band owing to engraving by the progressive rifling. By t = 3.8 ms, both chamber pressure and barrel stress begin to decrease. At t = 4.05 ms, the projectile exits the muzzle, resulting in a rapid reduction in barrel stress. The simulation sequence is shown in Figure 8.

Figure 8.

Barrel stresses after firing.

4.1. Analysis of twist-exponent variation

To investigate the influence of the progressive-rifling twist-law exponent (n) on projectile motion, full-barrel simulations were conducted with a fixed muzzle twist angle of $α_{E}$ = 7.5°. Three rifling configurations were examined (n = 1.2, 1.6, and 2.0). The results show that, under the same muzzle twist angle, varying n produces nearly identical projectile velocity profiles within the bore and at the muzzle. All cases yield a muzzle velocity of 1071.0 m/s, which differs from the DNI solution (1063.0 m/s) by 0.75%, as shown in Figure 9(a).

Figure 9.

Interior ballistic performance for different twist exponents n at a fixed muzzle twist angle ( $α_{E}$ = 7.5°).

The acceleration histories show a gradual increase during the early stage of the interior ballistic event. Between 0.5 and 0.9 ms, the rotating band begins to engage the forcing cone, causing the simulated acceleration to fall marginally below the DNI prediction. After 0.9 ms, the rotating band has fully passed the forcing cone and entered the rifled section. Peak acceleration occurs at approximately 1.2 ms, followed by a decrease until muzzle exit. This behavior indicates that the projectile reaches its maximum velocity at muzzle exit; once the projectile leaves the barrel, acceleration drops rapidly because the base pressure is no longer applied. The acceleration profiles for the three twist exponents exhibit similar trends. Based on the exit velocity, the simulated muzzle-exit time is approximately 4.01 ms, compared with 4.07 ms from the DNI solution (error: 1.47%), as shown in Figure 9(b).

Projectile spin rate is critical for aerodynamic stability, range, and accuracy. The simulations predict muzzle spin rates of 9299 rad/s for n = 1.2, 9285 rad/s for n = 1.6, and 9281 rad/s for n = 2.0; these values deviate by 0.33%, 0.48%, and 0.53%, respectively, from the DNI solution of 9330 rad/s (Figure 10(a)–(c)). The growth histories also differ marginally: n = 1.2 shows a faster initial increase, n = 2.0 increases more gradually, and n = 1.6 exhibits an intermediate trend. In the early stage, the simulated spin rate increases more slowly than the DNI curve because of the additional resistance associated with rotating-band engagement in the forcing cone; thereafter, the finite element predictions exhibit a trend similar to that of the DNI result (Figure 10(d)).

Figure 10.

Projectile spin rate for different twist exponents n at a fixed muzzle twist angle ( $α_{E}$ = 7.5°).

4.2. Analysis of muzzle twist-angle variation

To examine the influence of the rifling-twist constant p on projectile motion, simulations were performed with a fixed twist exponent of n = 1.6. Because p is primarily determined by the muzzle twist angle $α_{E}$ , four configurations were analyzed: $α_{E}$ = 6°, 7°, 8°, and 9°. The results show that varying $α_{E}$ has a negligible effect on projectile velocity under the same chamber-pressure loading. All cases yield essentially identical velocity profiles within the barrel and a muzzle velocity of approximately 1071.0 m/s, corresponding to a deviation of approximately 0.75% from the DNI solution (1063 m/s), as shown in Figure 11(a).

Figure 11.

Interior ballistic performance for different muzzle twist angles at a fixed twist exponent (n = 1.6).

The acceleration histories show the same overall behavior as discussed in Section 4.1. Acceleration increases gradually up to approximately 0.5–0.9 ms while the rotating band begins engaging the forcing cone, which causes the finite element results to fall marginally below the DNI curve. After 0.9 ms, the rotating band fully enters the rifled section, and peak acceleration occurs near 1.2 ms, followed by a decrease until muzzle exit. The simulated muzzle-exit time is approximately 4.01 ms, compared with 4.07 ms in the DNI solution (error: 1.47%), as shown in Figure 11(b). Overall, variations in muzzle twist angle do not significantly affect projectile velocity or acceleration under the same chamber-pressure loading; the curves show nearly identical trends and the same muzzle velocity.

The projectile spin rate increases with muzzle twist angle: a larger $α_{E}$ produces a higher muzzle spin. For $α_{E}$ = 6°, the simulated muzzle spin rate is 7423 rad/s (70884 rpm), compared with the DNI value of 7448.42 rad/s (71123 rpm), corresponding to an error of approximately 0.34%. For $α_{E}$ = 7°, the simulated maximum spin rate is 8670 rad/s (82792 rpm), compared with 8701 rad/s (83088 rpm) from the DNI solution (error: 0.36%). For $α_{E}$ = 8°, the simulated maximum spin rate is 9900 rad/s (94538 rpm), while the DNI solution gives 9960 rad/s (95111 rpm), yielding an error of approximately 0.60%. For $α_{E}$ = 9°, the simulated maximum spin rate is 11200 rad/s (106952 rpm), compared with the DNI value of 11224 rad/s (107181 rpm), corresponding to an error of approximately 0.21%. These results are shown in Figure 12(a)–(d). The spin-rate growth trend is consistent across all muzzle angles, and the muzzle spin rate increases with $α_{E}$ , as shown in Figure 12(e).

Figure 12.

Projectile spin rate for different muzzle twist angles at a fixed twist exponent (n = 1.6).

4.3. Engraving stress analysis in progressive rifling

Previous interior ballistic studies have primarily emphasized numerical prediction of projectile motion (e.g., displacement, velocity, and spin rate), which can be compared with analytical solutions or simplified numerical models to evaluate deviations and trends.¹³ However, the rotating band experiences complex contact- and deformation-driven stress interactions with the rifling during interior ballistics, and no closed-form analytical model is available to directly compute these stresses. This limitation is more pronounced for progressive rifling, in which stress evolution persists throughout the interior ballistic cycle because the twist angle continuously increases along the barrel. Systematic investigations of stress distributions and their evolution in progressive-rifling barrels, therefore, remain limited.^9–12,27 Accordingly, this section focuses on numerical analysis of rifling stresses in a progressive-rifling barrel.

The rotating band engages the rifling through contact while being propelled by base pressure. As the rifling engraves the band, the projectile is driven forward and simultaneously induced to spin. Throughout this process, the rotating band ensures controlled projectile motion within the barrel.

As shown in Figure 13, the transient deformation and engraving evolution of the rotating band under progressive rifling are illustrated. The engraving geometry transitions from an initial localized axial indentation to a fully developed helical pattern consistent with the twist-law evolution. At t = 0.0 ms, the rotating band remains intact. At t = 0.3 ms, initial contact occurs between the band and the forcing cone at the rifling origin, producing localized engraving and corresponding stress concentrations. At t = 0.5 ms, the band begins to plastically deform under rifling engagement. Because the band is made of copper, plastic deformation can transmit load to the projectile body, producing stress responses even in regions not directly contacting the barrel. By t = 0.9 ms, the band has passed the forcing cone and fully entered the rifled section, generating engraving marks that are initially nearly parallel to the projectile travel direction. Owing to the progressive twist, projectile rotation remains relatively small at this stage. At t = 1.0 ms, the band is fully engaged, and the engraving marks are clearly developed. After t = 2.0 ms, the engraving pattern becomes increasingly slanted, consistent with the increasing twist angle of the progressive rifling as the projectile advances.

Figure 13.

Engraving process of the rotating band under progressive rifling (n = 1.6, $α_{E}$ = 7.5°).

The relationship between engraving direction and muzzle twist angle is governed by the progressive twist law. At t = 0.9 ms, the rotating band has passed the forcing cone and entered the fully rifled region. Because the projectile is still in the early portion of the progressive rifling, the engraving marks are nearly parallel to the projectile axis. The engraving width is 2.69 mm, which matches the designed rifling-land width (2.69 mm). At t = 4.1 ms, when the projectile exits the barrel, the engraving produced by the progressive rifling forms a funnel-shaped pattern; the front and rear ends of the engraved region widen, while the central portion remains close to 2.69 mm. The final engraving direction forms an angle of approximately 7.5° relative to the projectile axis, consistent with the prescribed muzzle twist angle (Figure 14).

Figure 14.

Rotating-band engraving geometry at early and final states (n = 1.6, $α_{E}$ = 7.5°).

The stresses acting on the rifling arise from rotating-band engraving contact, chamber-pressure loading, and torque generated by projectile rotation. To examine the stress response in a physically meaningful manner, the original global Cartesian coordinate system (x, y, z) is transformed into a local cylindrical coordinate system (z, θ, r) through an intermediate local Cartesian system (X, Y, Z), as shown in Figure 3(a). The maximum stresses in the z-, θ-, and r-directions are extracted from the elements on all sixteen rifling lands and then averaged to evaluate the overall stress response of the rifling.

To evaluate the effect of rotating-band engraving stress on the rifling, simulations were performed at a fixed muzzle twist angle of $α_{E}$ = 7.5° for three twist exponents (n = 1.2, 1.6, and 2.0). Because the projectile is not laterally constrained, the contact load is not distributed uniformly among the rifling lands. For comparison across cases, the average absolute rifling stress $σ_{X}$ is defined as

σ_{X} (x) = \frac{1}{m} \sum_{j = 1}^{m} | {\bar{σ}}_{X} (x) |, x_{j} \leq x \leq x_{E},

(23)

where

| {\bar{σ}}_{X} (x) |

is the maximum absolute stress at axial position x,

x_{j}

and

x_{E}

are the starting and ending axial positions of the rifling, respectively, m is the number of rifling lands, and X denotes the stress component. Here,

σ_{X}

∈ {

σ_{r}, σ_{z}, σ_{θ}, σ_{v m}

}, corresponding to radial, axial, circumferential (normal), and von Mises stresses, respectively.

The simulations show that once the projectile begins to move at the rifling origin, the rotating band immediately enters the forcing cone. During this stage, the band is subjected to substantial engraving stress as the lands penetrate the rotating-band material. When the band fully traverses the forcing cone, the engraving-induced stresses reach their peak. The peak stresses are 2515.2 MPa for n = 1.2, 2557.1 MPa for n = 1.6, and 2562.0 MPa for n = 2.0, corresponding to increases of 1.67% and 1.86% relative to the n = 1.2 case. After the peak, stress levels decrease; however, because the twist angle continues to increase along the barrel, engraving remains active and sustains comparatively high stress over the remaining length (Figure 15(a)).

Figure 15.

Average absolute rifling stress.

In the local cylindrical system, the dominant component in the direction of motion is the axial (z) stress, which is driven by band–rifling interaction along the projectile travel path. Because the twist gradient is steeper for n = 1.2, the early-stage axial stress is higher than that for n = 2.0. After the rotating band clears the forcing cone, the peak axial stresses are 1373.4 MPa (n = 1.2), 1364.0 MPa (n = 1.6), and 1352.9 MPa (n = 2.0), corresponding to decreases of 0.68% and 1.49% relative to n = 1.2 (Figure 15(b)).

The circumferential ( $θ$ ) stress component is strongly influenced by the local twist angle and is therefore sensitive to the twist exponent. Because the twist angle is larger near the rifling origin for n = 1.2, the $θ$ -direction stress is also higher in the early stage. The maximum $θ$ -stress occurs near the midpoint of the forcing cone and is 984.5 MPa for n = 1.2, 766.4 MPa for n = 1.6, and 658.5 MPa for n = 2.0, representing reductions of 22.15% and 33.11% relative to n = 1.2. This result indicates that the $θ$ -direction stress is highly sensitive to the twist exponent n (Figure 15(c)).

The overall stress response was also evaluated using the von Mises stress. The peak von Mises stresses are 1909.2 MPa for n = 1.2, 1860.7 MPa for n = 1.6, and 1859.1 MPa for n = 2.0, corresponding to reductions of 2.54% and 2.62% relative to n = 1.2. Overall, n = 1.2 produces the highest peak stress, whereas n = 2.0 yields the lowest peak stress but a marginally higher post-peak stress level than n= 1.6. The n = 1.6 case remains intermediate in both peak magnitude and overall stress level (Figure 15(d)).

The peak von Mises stress in Figure 15 occurs as a localized transient concentration near the forcing cone during early-stage engraving. Because the barrel material is modeled using a J–C formulation (Table 1), strain hardening and strain-rate sensitivity increase the effective flow stress under interior ballistic loading, making such localized peaks mechanically plausible.

Based on these results, the rifling experiences its highest von Mises stress during initial engraving. Therefore, selecting an appropriate twist exponent n is important for reducing transient peak stresses while maintaining a favorable overall stress response throughout the interior ballistic process. Under the conditions examined here, n = 1.6 provides a comparatively balanced stress response relative to n = 1.2 and n = 2.0.

4.4. Projectile flight-stability analysis

The muzzle twist angle $α_{E}$ directly influences projectile spin rate, as reflected by the Miller twist rule (Section 2.3). For high-velocity projectiles, insufficient or unstable spin can degrade aerodynamic stability, effective range, and accuracy. To quantify stability under different twist configurations, the stability factor proposed by Miller¹⁹ was adopted. This criterion incorporates projectile caliber, mass, muzzle velocity, and twist (spin) parameters, providing a quantitative measure that accounts not only for standard-condition stability but also for stability degradation under adverse environments (e.g., cold or high-density air).

The computed results indicate that muzzle twist angles of 7°, 8°, and 9° satisfy the military stability requirement of s = 1.5–2.0. Considering Miller’s recommendation that the stability factor approach s = 2.0 when environmental influences are included, muzzle twist angles in the range of 7°–8° meet the recommended stability criterion in the present evaluation (Table 2).

Table 2.

Muzzle twist angle and corresponding spin-rate stability factors.

Muzzle twist angle (°)	Projectile weight (grains)	Caliber (in)	Projectile length (in)	Twist (calibers/turn)	Simulated spin rate (rpm)	Stability factor (s)	Velocity-corrected stability (s $f_{v}$ )
6	5833.447	1.1811	5.394	35.691	70884	1.165	1.257
7	5833.447	1.1811	5.394	30.558	82792	1.590	1.716
8	5833.447	1.1811	5.394	26.761	94539	2.073	2.237
9	5833.447	1.1811	5.394	23.655	106952	2.653	2.863

Note. Temperature, atmospheric-pressure, and altitude corrections are environmental factors and are not included in the present analysis.

5. Conclusions and future perspectives

In this study, a nonlinear transient finite element framework was developed to investigate the influence of progressive-rifling parameters on interior ballistic behavior and engraving-induced stresses in a 30 mm chain-gun barrel. The effects of the twist exponent (n) and muzzle twist angle ( $α_{E}$ ) were systematically evaluated and compared with a DNI reference solution and the Miller stability criterion. The main conclusions are as follows:

1. Axial (translational) interior-ballistic response: Projectile translational motion, represented by muzzle velocity and acceleration, was governed primarily by chamber-pressure loading and was largely insensitive to variations in progressive-rifling geometry within the investigated ranges. Across all configurations, the predicted muzzle velocity was approximately 1071 m/s, differing from the DNI reference value by 0.75%, which supports the internal consistency of the numerical framework.

2. Rifling stress response and the role of n: In contrast to the axial response, the twist exponent had a clear influence on rifling stress evolution. The circumferential ( $θ$ -direction) stress varied by more than 30% across the examined n values, indicating that progressive twist intensity affects torsional load transfer and local stress concentration during engraving. Among the investigated cases, n = 1.6 provided a comparatively balanced stress response while maintaining stable spin-rate development.

3. Spin rate, stability, and the role of $α_{E}$ : The muzzle twist angle primarily controlled the projectile spin rate and thus gyroscopic stability. Based on the Miller stability criterion, muzzle twist angles in the range of 7°–8°produced stability factors close to the recommended value (s = 2.0), indicating sufficient rotational stability under the standard conditions considered in this study.

Overall, the results show that progressive-rifling parameters mainly affect rotational mechanics and engraving-induced stresses rather than axial ballistic performance. Combining finite element stress analysis with stability evaluation provides a consistent quantitative basis for assessing progressive-rifling designs.

Future work will focus on optimization-oriented progressive-rifling design to achieve a balanced combination of spin rate, gyroscopic stability, structural durability, and stress reduction. By treating the twist exponent (n) and muzzle twist angle ( $α_{E}$ ) as continuous design variables, the trade-off between rotational performance and engraving-induced stress can be quantified. This approach can help identify rifling configurations that maintain adequate stability while reducing circumferential stress concentrations and improving long-term barrel durability under high-pressure firing conditions.

Footnotes

Acknowledgments

The authors acknowledge the technical support of the 205^th Arsenal, Materiel Production and Manufacturing Center, Armaments Bureau, MND, for this research. The authors also thank Editage () for English language editing.

ORCID iDs

Chien-Chih Lai

Shigan Deng

Chun-Cheng Lin

Author contributions

Chien-Chih Lai: Model preparation, Writing–Original Draft Preparation, Validation.

Shigan Deng: Conceptualization, Methodology, Review & Editing.

Chun-Cheng Lin: Supervision, Validation.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was financially supported by the 205^th Arsenal, Materiel Production and Manufacturing Center, Armaments Bureau, MND, under the National Defense Advanced Science Technology Research Program (Project No. 183).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Data will be made available upon request.*

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