A Multifactor Combination Optimization Design Based on Orthogonality for a Two-Degree-of-Freedom Floating Machine Gun Vibration System

Tis paper introduces a novel type of foating machine gun that can be simplifed as a self-balancing two-degree-of-freedom mechanical system with distinct vibration characteristics. Te model accounts for intricate motion patterns and encompasses numerous potential infuencing factors. Multifactor combination optimization of the system represents a pressing engineering challenge. After establishing a simulation model for the machine gun and validating it experimentally, seven factors were chosen as optimization variables. Te maximum recoil displacement of the inner receiver (MRD) and the fring rate were chosen to be indicators. Orthogonal combinations and variance analyses were used, and the efects of multiple factors were analyzed using SPSS software; these processes led to a determination of the optimal combination. Te results indicated that the piston cylinder pressure, the bi-directional bufer spring energy storage, and the inner receiver mass signifcantly afected the MRD. Furthermore, the automaton mass and the reset spring energy storage were found to substantially afect the fring rate. Careful analysis of the variance results facilitated the determination of the optimal combination of parameter values. Remarkably, the optimal combination chosen resulted in an MRD reduction of approximately 20.2% and a fring rate increase of approximately 26.6%.


Introduction
Te foating principle is an innovative design approach used for frearms.It is predominantly applied to open-bolt machine guns and small-caliber cannons.A traditional frearm can be simplifed as a single-degree-of-freedom system, in which an automaton moves back and forth inside the receiver and generates impacts.Tese impacts are ultimately transmitted to the frearm tripod or a human body, thereby causing the frearm to vibrate along with the tripod or body.In contrast, a foating frearm possesses an additional inner receiver between the automaton and the receiver, and it is therefore a two-degree-of-freedom system.Te impacts from the automaton are transmitted to the inner receiver.Ten the vibrations that would normally be transmitted to a tripod or a human body are largely transformed into vibrations of the inner receiver.As a result, the tripod or human body vibrations are signifcantly reduced, which is benefcial for increasing the shooting accuracy and stability of the frearm.Notably, several types of frearms, such as the General Dynamics Lightweight Medium Machine Gun, the SIG Arms MG338 machine gun, and the new XM250 machine gun developed for the Next Generation Squad Weapons program by the US Army, have adopted the foating principle [1][2][3].
During the fring process of a foating frearm, the inner receiver vibrates between extreme recoil and return points without colliding with the receiver [4]; this is how the frearm earns its "foating" designation.During the mechanical design of a foating frearm, the foating (or vibration) amplitude should be kept as small as possible.In practical applications, since the recoil energy of a frearm is signifcantly greater than its return energy, the extreme return displacement is often more than sufcient in application.
Consequently, the primary assessment criterion used for achieving ideal foating is the maximum recoil displacement (MRD) of the inner receiver.Ideally, the MRD should be minimized [5].In addition, for frearms utilizing the foating principle, the vibration frequency of the entire two-degreeof-freedom system, or the fring rate, tends to be low; thus, it is necessary to increase the fring rate.Terefore, the MRD and the fring rate are crucial design considerations for twodegree-of-freedom foating frearm systems.
Currently, research regarding foating frearms predominantly utilizes a methodology that involves modifying specifc input parameters to investigate the efects of parameter variations on the results.For example, Lu et al. studied the efects of muzzle brakes and piston parameters on foating frearms [6][7][8][9].However, though muzzle brakes and piston parameters are factors that commonly infuence traditional machine guns, foating frearms may be infuenced by many unique parameters that have not yet been addressed.Wang investigated the efects of the foating lock position [10], which is a specifc factor of foating technology.However, this study only focused on examining the impacts of this single factor.Additionally, the foating lock is a mechanism typically used in earlier foating technology, while the new foating mechanism described in this paper does not possess a foating lock.In addition, Yongjian et al. explored the use of foating technology in modifed rifes for unmanned aerial vehicles [11] while studying recoil reduction in foating frearms.None of the abovementioned studies addressed multifactor optimization of foating machine guns.
Floating machine guns are two-degree-of-freedom systems and thus introduce novel and distinct infuencing parameters that interact in intricate ways, thereby yielding complex relationships.As a result, the research methods mentioned earlier are not equipped for analyzing two-degree-of-freedom systems.Using the MRD and fring rate as examples, a multifactor analysis method was employed during the current study to examine the efects of various parameters on a frearm foating mechanism.Te study identifed the optimal combination of parameters that would enhance the foating performance of the frearm.
Tis paper contains three innovative aspects.First, it introduces a new type of foating machine gun model.In the foating machine gun industry, this model represents an entirely new technology.Second, new technologies inevitably bring new potential infuencing factors.Te impacts of these factors on the foating mechanism were explored during this study, and this exploration encompassed an area not addressed in previous research.Finally, the paper presents joint optimization results involving multiple factors.Tese kinds of results also remained unexplored during prior research.

Simulation Model of the Two-Degree-of-Freedom Floating Machine Gun
2.1.Simulation Model of the Floating Machine Gun.Te core mechanism of a traditional machine gun can be simplifed as a single-degree-of-freedom model, as shown in Figure 1(a).
Te automaton moves back and forth within the receiver, and when it reaches the left end or the right end of the receiver, it violently impacts the receiver.Figure 1(b) shows that, in the foating machine gun described in this paper, there is an additional inner receiver between the automaton and the receiver.Te inner receiver is connected to the automaton by a reset spring, while it is connected to the receiver by a bi-directional bufer spring.In a traditional machine gun, when the automaton reaches its rearmost or foremost position, it impacts the receiver and transmits the impact to the human shooter or the gun mount.In the foating machine gun described in this paper, however, the impacts are transmitted to the inner receiver and are absorbed by the bi-directional bufer spring.Te marked points on the receiver in Figure 1(b) are denoted as O 1 , O 2 , and O 3 , while those on the inner receiver are denoted as O 4 , O 5 , and O 6 .O 2 and O 5 represent the points of zero displacement of the automaton and the inner receiver, respectively, while O 1 and O 3 represent the extreme recoil and extreme return points, respectively, of the inner receiver.Figure 2 depicts the correlation between the extreme recoil and extreme return points, as well as the corresponding displacements, namely, the extreme recoil and extreme return displacements.Figure 2 also illustrates their relationships with the MRD.
Te foating machine gun model presented in this paper is similar to the car tire-ground model, which is one of the most common two-degree-of-freedom models [12].Car models are subject to random vibrations from the ground, while the external forces on the foating machine gun are triggered when the automaton moves to a certain location.Car models must pursue smaller human body or vehicle body vibration amplitudes to ensure comfort, while machine gun models must pursue smaller inner receiver vibration amplitude.
In this paper, motion in the recoil direction, such as that which occurs when the automaton travels from O 6 to O 4 , is defned as positive, while motion in the opposite direction, which is defned as the return direction, is defned as negative.Figure 3 illustrates the internal ballistic force and the piston cylinder force on the inner receiver, the piston cylinder force on the automaton, and the feeding resistance using purple, blue, green, and yellow arrows, respectively.It is important to note that the piston cylinder forces acting on the inner receiver and the automaton have equal magnitudes but opposite directions.
Figure 3(a) shows that when the automaton return motion is initiated by the reset spring, resistance is imposed by the feeding mechanism.In Figure 3(b), the automaton has just reached position O 6 and has thus collided with the inner receiver.Figure 3(c) illustrates that this collision impels the inner receiver, which is positioned at O 2 , to commence the return motion by fring a bullet.
Because the internal ballistic force has a greater magnitude than the piston cylinder force, the inner receiver recoils under the combined efect of both forces.In contrast, the automaton recoil is solely attributable to the piston cylinder force.

Recoil
Return

Shock and Vibration
due to the action of the bi-directional bufer spring.Ultimately, the automaton and inner receiver return to their initial states depicted in Figure 3(a).After establishing the three-dimensional structural assembly model of the foating machine gun on the UG CAD platform, the model is imported into the Adams mechanical simulation software.Motion joints and contact relationships between gun parts are defned within the software.Te motion joint between the automaton and the inner receiver is a translational joint; they are connected by the reset spring, and contact-impact efects occur when the recoil and return are in place.Te motion joint between the receiver and the inner receiver is also a translational joint, connected by the bi-direction bufer spring.Contact-impact efects occur between the inner receiver and the receiver when the recoil is in place.By fxing the receiver and applying the same motion relationships between other components as in reality and handling contact-impact constraints while applying the main loads, a simulation model of the foating machine gun with the same motion principles as shown in Figure 4 can be obtained.Te relevant model parameters are listed in Table 1.

Determination of the Main Load.
As shown by the purple, blue, green, and yellow arrows in Figure 3, the model presented in this paper contains four forces.Te internal ballistic pressure can be expressed by the following equation [13][14]: In equation (1), t i represents the interior ballistic time, l is the bullet displacement in the barrel, p is the gas pressure, v represents the bullet velocity, ψ is the mass percentage of the burned propellant, Z is the relative burned thickness of the propellant, S denotes the equivalent area of the barrel cross-section, and W 0 is the equivalent volume of the piston chamber.In addition, m p represents the propellant mass, δ is the propellant density, m i is the bullet mass, l 0 denotes the initial equivalent volume of the barrel, △ is the propellant charge density, I k is the impulse of the propellant gas pressure, α represents the residual volume of the propellant gas, f is the propellant force, φ is a coefcient, and p 0 denotes the extrusion pressure.Furthermore, χ, λ, and μ represent the features of the powder shape.
Te after-efect period of the interior ballistic pressure can be mathematically expressed by the following equation: where p a represents the after-efect period pressure, v 0 is the muzzle velocity, P k is the mean muzzle pressure at the instant when the bullet traverses the muzzle, P e is 1.8 times the atmospheric pressure, t h denotes the duration time calculated from the moment the bullet exits the muzzle, and β is the after-efect coefcient.
Te piston cylinder pressure can be calculated using Bravin's formula, as shown in the following equation: In equation (3), P s represents the piston cylinder pressure, P d is the mean pressure within the barrel at the instant when the bullet traverses the gas port, t ′ is the duration time calculated from the moment the bullet exits the gas port, a is a structural coefcient, c is a time constant, and t dk denotes the duration of the bullet travel from the gas port to the muzzle.Figure 5 illustrates the internal ballistic pressure and the piston cylinder pressure.Te internal ballistic force and the piston cylinder force can be calculated by multiplying the corresponding pressures and efective areas, as shown in the following equation: In equation ( 4), F i represents the force exerted by the internal ballistic process, F ch denotes the force exerted by the piston cylinder, and S s is the efective area of the piston cylinder.To ensure accurate modeling of the dynamic behavior of the belt, a rigid-fexible coupling model was constructed using the MPC-BRE2 method [15].Experimental measurements of the feeding resistance are presented in Figure 6.

4
Shock and Vibration

Model Verification
Te experiment takes place in an indoor fring range.Te experimental system consists of a foating machine gun, a FASTCAM AX200 high-speed camera, and PCC 2.8 image processing software.Te machine gun is mounted on a fxed stand, and markers are placed on the inner receiver and automaton.High-speed photography is used to capture the movement of frearm components.Te experiment involves a 5-round burst with a frame rate of 5,000 frames per second.Te PCC 2.8 image processing software is then used to extract the positions and times of the markers in the captured videos, allowing the derivation of displacement-time curves for the inner receiver and automaton.Further data processing yields velocity-time curves.
Figures 7 and 8 depict comparisons between the experimentally and simulation automaton velocity-time curves and inner receiver displacement-time curves, respectively.Table 2 compiles statistics on factors such as fring rate, automaton maximum recoil speed, maximum return speed, MRD, and inner receiver maximum return displacement.In comparison with experimental results, the maximum error of the simulation model is 8.0%, confrming the accuracy of the simulation model.Te close agreement between the simulation and experimental results demonstrates the accuracy of the simulation model.
Figure 7 shows that the fring rate of the frearm was 367.3 rounds/min.Figure 8 depicts the displacement-time curves for the inner receiver, which can also be understood as the vibration of the inner receiver.An MRD value of 19.1 mm was obtained.
It is noteworthy that a certain degree of error was present in both the experimental and simulation curves.In the simulation model, an equivalent feeding resistance of the belt chain was determined using fexible bodies, and the piston cylinder and internal ballistic pressures were calculated using empirical formulas, which resulted in constant force values.
In reality, variations existed in the individual belt-chain conditions, and the feeding resistance was not the same for each round of ammunition.Te combustion conditions of the gunpowder were also not identical for each round of ammunition, which caused variations in the gas pressure and internal ballistic forces.Furthermore, the locking, extraction, and ejection process of the frearm all involved subtle material deformations, which were simplifed for the simulation model.Hence, there was a certain degree of error between the simulation and experimental results.

Combination Optimization Design
Te physical prototype manufacturing process of frearms is exceedingly lengthy, often requiring one to two years.In addition, some experiments pose dangers, and conducting parameter variation tests is challenging and costly.As a novel principle machine gun, the presence of inner receiver makes the foating frearm a typical dual-degree-of-freedom system.Moreover, within the system, there are numerous impact and momentum transfer processes, leading to a plethora of potential infuencing factors.Te coupling efects of multiple parameters are highly complex and peculiar, given that most of these factors have not been previously studied, and their impact levels remain unknown.Tis presents numerous challenges in achieving a rational parameter matching design for current foating machine guns.Te challenge is addressed by employing the Combination Optimization Design method in this paper.
In this study, a comprehensive optimization design approach was developed to enhance the performance of the foating machine gun.Tis approach accounted for the effects of multiple factors.A schematic representation of the design process is depicted in Figure 9. Te initial step involved selecting the infuential factors for the orthogonal design; the MRD and the fring rate were chosen as the evaluation criteria.Subsequently, the simulation model, which was discussed in Section 2, was used to assess the performance of the foating machine gun under various combinations of factors.
Te SPSS software is used to conduct analyses of variance and generate F values.Te F value serves as an indicator for assessing the degree of parameter impact, where a larger F value indicates a more signifcant infuence.Te statistical signifcance level is a conversion index for the F value, used to evaluate the meaningfulness of the results.It is generally considered that when the statistical signifcance level is less than 0.1, the impact is considered meaningful.Finally, the optimal levels of the highly signifcant factors were identifed for integration, resulting in an optimized combination scheme [16,17].

Factor Selection and Orthogonal Design.
Factors that commonly infuence traditional frearm operation, such as the automaton mass, the reset spring energy storage, and the piston cylinder pressure, as well as factors specifc to the foating machine gun, such as the inner receiver mass, the bidirectional bufer spring energy storage, the internal ballistic pressure, and the reset spring position, were used as factors in the orthogonal design [18][19][20].Te reset spring position was assigned two levels, A1 and A2, which indicate inner receiver-automaton and receiver-automaton positions, respectively.Tree levels were chosen to represent the states of the remaining factors: minimum, medium, and maximum.Te specifc parameter values are summarized in Table 3, while the orthogonal scheme is presented in Table 4. Due to practical engineering considerations, the variation ranges were not uniform for factors B through G.
Te selection of factors and levels in this paper is primarily based on design rules and engineering experience.For example, the choice of reset spring position is based on diferent designers' perspectives.Similarly, internal ballistic pressure is a common variable in frearm design, but it directly determines the velocity and power of the bullet, and its range of variation is very limited.A common engineering variation range is from 0.9 times to 1.1 times.

Analysis of Simulation Results
. Te results of the variance analysis are presented in Figures 10 and 11.Equation ( 5) presents the variance analysis calculation process [21,22]:  Shock and Vibration where n represents the total number of tests, x ij is the result value for the i th level of the j th factor, x i denotes the mean value of level i, n i and k represent the total number of factors and maximum number of levels, respectively, and � � x is the mean value of all 18 test results.
With regards to the MRD indicators, the F values corresponding to parameters A, B, C, D, E, F, and G were determined to be 1.9, 2.467, 8.069, 12.139, 1.921, 6.394, and 2.289, respectively.After conversion, the statistical signifcance level of the piston cylinder pressure (C) was determined to be 0.039.In addition, the signifcance level of the bi-directional bufer spring energy storage (D) was found to be 0.02, while the signifcance level of the inner receiver mass (F) was determined to be 0.057.For all these cases, the signifcance levels were determined to be less than 0.1.
Te mechanism by which piston cylinder pressure (C), energy storage of the bi-directional bufer spring (D), and inner receiver mass (F) infuence MRD is as follows: for a single-degree-of-freedom system composed of the inner receiver and bi-directional bufer spring, piston cylinder pressure (C) serves as the excitation to the system.Te energy storage of the bi-directional bufer spring (D) and inner receiver mass (F) act as the equivalent spring stifness and mass of the system, directly afecting the inherent amplitude and frequency of the single-degree-of-freedom system.Although the actual amplitude is highly infuenced by impacts between the automaton and inner receiver, the inherent amplitude and frequency also have a signifcant impact on MRD.
Tese signifcance levels indicate that three parameters most signifcantly afect the MRD.Notably, the MRD is minimized when the piston cylinder pressure is at its minimum level (C1), the bi-directional bufer spring energy storage is at its maximum level (D3), and the inner receiver mass is at its medium level (F2).
With regards to the fring rate indicator, the F values corresponding to parameters A, B, C, D, E, F, and G were determined to be 2.131, 3.959, 1.7, 0.852, 35.888, 0.753, and 79.756, respectively.After conversion, the signifcance level of the automaton mass (G) was determined to be 0.003, while the signifcance level of the reset spring energy storage (E) was found to be 0.001.For both cases, the signifcance levels were determined to be less than 0.1.
Te factors afecting fring rate, reset spring energy storage (E), and automaton mass (G) are explained as follows: the reset spring is responsible for propelling the automaton to complete the return motion.Terefore, a larger reset spring energy storage (E) and a smaller automaton  Shock and Vibration   It was therefore concluded that the fring rate is primarily infuenced by the automaton mass and the energy storage of the reset spring.Notably, the fring rate is maximized when the reset spring energy storage is at its maximum level (E3) and the automaton mass is at its minimum level (G1).

Comparison of Results.
To enhance the performance of the foating machine gun designed during this study, reducing the MRD was a crucial objective.Te C1D3F2 combination was selected as the preliminary optimal confguration.However, during the initial test, an unacceptably low fring rate of 367.3 rounds per minute was obtained.To improve the fring rate, the E3G1 combination was identifed as the optimal confguration.Te remaining factors were maintained at their default settings of A1B2.Consequently, the fnal optimal combination was determined to be A1B2C1D3E3F2G1.Table 5 compares the results for the fnal optimal combination with the initial test data.
Using the optimized parameter combination for the frearm resulted in an MRD reduction from 19.1 mm to 15.24 mm and an increase in the fring rate from 367.3 rounds/min to 465 rounds/min.Tese values represent a 20.2% reduction in the MRD and a 26.6% increase in the fring rate from the values produced by the initial combination.
For a foating machine gun, designers often ensure suffcient space redundancy to achieve foating successfully.Unsuccessful foating can be detrimental to the frearm's shooting accuracy and imposes signifcant spatial and structural constraints, afecting reliability and adding weight.Terefore, reducing the MRD is advantageous for improving reliability, reducing weight, and efectively enhancing the frearm's foating performance.In addition, foating machine guns often have lower fring rates, resulting in lower frepower density, which is unacceptable for machine guns.In practical engineering applications, the common issue in foating machine gun design is the synchronous occurrence of larger MRD and lower fring rates.Terefore, the ability to simultaneously reduce MRD and increase fring rates signifcantly improves the frearm's performance.

Conclusions
In this study, a two-degree-of-freedom simulation model was developed for a foating machine gun.Te simulation results agreed well with experimental observations, accurately depicting the motion characteristics of the inner receiver and the automaton.By combining experimental and orthogonal simulation methods, a thorough examination of the efects of seven parameters on the foating performance was conducted.
A combination optimization revealed that the piston cylinder pressure, the bi-directional bufer spring energy storage, and the inner receiver mass all signifcantly afect the MRD.In contrast, the energy storage of the reset spring and the automaton mass notably afect the fring rate.Te other parameters investigated do not exhibit obvious correlations with the foating machine gun performance.
Te optimal combination of parameter values was determined by applying an optimization approach.It included a reset spring placement between the inner receiver and the automaton, a medium interior ballistic pressure, a small piston cylinder pressure, large bi-directional bufer spring and reset spring energy storage values, a medium inner receiver mass, and a small automaton mass.Tis optimized combination resulted in a remarkable MRD reduction of approximately 20.2% and a substantial fring rate increase of approximately 26.6%; these improvements efectively enhanced the overall performance of the foating machine gun.Similar methods can be applied to address other performance aspects of foating machine guns or similar engineering problems.
Figure 3(d)  indicates that the reset spring causes the automaton to decelerate and that the inner 2Shock and Vibration receiver also slows because of the action of the bi-directional bufer spring.In Figure3(e), the automaton has just recoiled to O 4 , and has thus collided with the inner receiver, thereby propelling it to recoil further.

Figure 3 (Figure 1 :Figure 2 :
Figure 3(f ) illustrates the maximum recoil positions attained by both the inner receiver and the automaton.Te automaton then initiates its return motion, which is propelled by the reset spring, while the inner receiver decelerates

Figure 3 :
Figure 3: Schematic of the fring cycle of the foating machine gun.

Figure 4 :
Figure 4: Dynamic simulation model of the foating machine gun.

Table 1 :Figure 5 :Figure 6 :
Figure 5: Pressure-time curves of the interior ballistic and piston cylinder pressures.

Figure 10 :Figure 11 :
Figure 10: F values of the seven parameters that infuence the MRD.

Table 2 :
Variation ranges of various factors.

Table 3 :
Variation ranges of various factors.

Table 5 :
Comparison between the optimal combination results and the experimental results.