Pitching stability analysis of weapon based on the equivalent parameters of human joints
Shun Yan^{1} , Yunfeng He^{2} , Changming Wang^{3}
^{1, 2, 3}Nanjing University of Science and Technology, Nanjing, China
^{1}Corresponding author
Journal of Vibroengineering, Vol. 17, Issue 7, 2015, p. 39283935.
Received 10 July 2015; received in revised form 26 August 2015; accepted 6 October 2015; published 15 November 2015
In order to study the impact of human joint parameters on the pitching stability of a grenade launcher, a manweapon interaction model of two rigid bodies with five degrees of freedom was developed according to the characteristics of shooting. By simulating with different values of human parameters individually, the pitching movement curves of the weapon were obtained. The equivalent parameters of the shooter have different influence on the firing stability of the weapon. The equivalent damping coefficients of the waist and the shoulder are critical to the design of the firearm mount, and the equivalent damping coefficient of the waist is the most important parameter.
Keywords: equivalent parameter, manweapon interaction model, pitching stability.
1. Introduction
In addition to the factors of weapons, the shooter’s control factors can affect the stability of shooting. Early reports were concentrated on recoil analysis [1, 2] and manweapon interaction force relationships of shoulderfired weapons [3, 4]. The study of the simulation of the shooter and weapon was first appeared in the literature [3]. A three degrees of freedom mathematical model was developed that simulates the man and weapon as a coupled dynamical system. Through the sensitivity analysis of the parameters of the system and the experimental verification, the key parameters affecting the three degrees of freedom were obtained and the method of using the simplified model to study the system of mangun system has been proved to be feasible. In the research of recoil analysis [1, 2], the recoil analysis to assess several recoil mitigating technologies applied to shoulderfired weapons such as a grenade launcher or shotgun has been conducted, and three commercial recoil reducing devices were evaluated in the model to determine their specific effect on recoil motion, both on the weapon and on the soldier firing the weapon. In the research of the manweapon interaction [3, 4], the model analysis and experimental study of the 56 type of assault rifle were carried out, and the change rule of the interaction of the man and weapon based on three shot burst and eight shot burst was obtained.
The purpose of this paper was to study the influence of human equivalent parameters on the pitching stability which is very useful to the shooting training of the soldiers and the design of intelligent firearm mount. According to a new type of grenade launcher firing test, a mathematical model for describing the firing system of the man and the weapon was established. Calculated with different values of the equivalent parameters of the shoulder and the waist, the weapon pitching curves were obtained. A detailed description of the system and the analysis performed along with the results follows below.
2. Manweapon system modeling
As the human body is a nonhomogeneous composition of body segments with multiple degrees of freedom, the manweapon system is a complex multibody system [35]. Different system models had been developed for their respective research objectives in previous research [26]. In this paper, based on the movement characteristics of firing, the system was simplified to a mathematical model of two rigid bodies with five degrees of freedom as follows.
2.1. Simplifying assumption of system model
Compared with previous shooting experiments [35], the firing experiment of a new type of the grenade launcher appears the following movement characteristics of manweapon system:
1) The weapon has recoil motion compared with the upper torso.
2) The weapon has pitching motion compared with the upper torso.
3) The shooter’s upper torso has pitching motion compared with the lower torso.
4) The shooter’s upper torso has yawing motion in the transverse plane.
5) There almost has no relative motion between the lower torso and ground.
According to above movement features, the manweapon system model was simplified as:
1) In the dynamic equations, the breech pressure force of the grenade launcher was simplified to a pulse signal [3, 4].
2) The lower torso of the shooter was assumed to be fixed on the ground.
3) Relative to the lower torso, the pitching rotation and yawing rotation of upper torso were defined as two independent degrees of freedom. And the restrictive effect of the shooter's waist muscle on these degrees of freedom was simplified to the effect of two groups of equivalent springs and dampers.
4) Because of having the same movement in the firing process, the shooter’s arms and the weapon were simplified to an equivalent mass. Relative to the upper torso, this equivalent mass involves three independent degrees of freedom including the recoil motion, the pitching rotation and yawing rotation. The restriction effect of the shooter's shoulder muscle and arms on these degrees of freedom was simplified to the effect of three groups of equivalent springs and dampers. The shooting model was shown in Fig. 1.
Fig. 1. Simplified model of standing shooting
The $OXYZ$ coordinate system fixed with the earth was defined at the rotation center of the upper torso. The coefficients corresponding equivalent parameters were defined as follows:
${K}_{x}$, ${C}_{x}$ – equivalent elastic coefficient and damping coefficient of the shoulder joint in the recoil direction;
${K}_{\theta}$, ${C}_{\theta}$ – equivalent elastic coefficient and damping coefficient of the shoulder joint in the pitching direction;
${K}_{\gamma}$, ${C}_{\gamma}$ – equivalent elastic coefficient and damping coefficient of the shoulder joint in the yawing direction;
${K}_{\alpha}$, ${C}_{\alpha}$ – equivalent elastic coefficient and damping coefficient of the waist joint in the pitching direction;
${K}_{\beta}$, ${C}_{\beta}$ – equivalent elastic coefficient and damping coefficient of the waist joint in the pitching direction.
2.2. Dynamic equations of Manweapon system
Corresponding to the five degrees of freedom about the simplified model, the generalized displacements of the system were:
$x$ – the generalized displacement of weapon recoil;
$\theta $ – the generalized displacement of the pitching of the weapon compared with the upper torso;
$\gamma $ – the generalized displacement of the yawing of the weapon compared with the upper torso;
$\alpha $ – the generalized displacement of the pitching of the upper torso compared with the lower torso;
$\beta $ – the generalized displacement of the yawing of the upper torso compared with the lower torso.
The centroid motion of the weapon and that of upper torso were converted into the inertial coordinate system $OXYZ$, the system kinetic energy function, the system potential energy function and the dynamic differential equation were deduced by the Lagrange method as follows:
where: $L$: The Lagrange function of system; $P$: The dissipation function of system; ${Q}_{x}$, ${Q}_{\theta}$, ${Q}_{\gamma}$, ${Q}_{\alpha}$, ${Q}_{\beta}$: The generalized forces corresponding to the generalized displacements; $L=TV$ and:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+mg{r}_{cm}\left(\mathrm{c}\mathrm{o}\mathrm{s}\alpha \mathrm{c}\mathrm{o}\mathrm{s}{\alpha}_{0}\right)+{m}_{r}g\left[{L}^{\mathrm{\text{'}}}\left(\mathrm{c}\mathrm{o}\mathrm{s}\alpha \mathrm{c}\mathrm{o}\mathrm{s}{\alpha}_{0}\right)+x\mathrm{s}\mathrm{i}\mathrm{n}\left(\alpha +\theta \right){x}_{0}\mathrm{s}\mathrm{i}\mathrm{n}\left({\alpha}_{0}+{\theta}_{0}\right)\right],$
${Q}_{\beta}=\left(h\mathrm{c}\mathrm{o}\mathrm{s}\beta +\eta +{\eta}^{\mathrm{\text{'}}}\right)F\left(t\right),\mathrm{}\mathrm{}\mathrm{}{Q}_{\gamma}=\left(\eta +{\eta}^{\mathrm{\text{'}}}\right)F\left(t\right),$
where: ${I}_{cm}$ – the moment of inertia of the upper torso equivalent mass relative to $OZ$ axis; ${I}_{r}$ – the moment of inertia of the weapon equivalent mass relative to $OZ$ axis; ${I}_{cm1}$ – the moment of inertia of the upper torso equivalent mass relative to $OY$ axis; ${I}_{r1}$ – the moment of inertia of the weapon equivalent mass relative to $OY$ axis; $m$ – the equivalent mass of the upper torso; ${m}_{r}$ – the equivalent mass of the shooter’s arms and the weapon; $L\text{'}$ – the projection of distance between shoulder points and point $O$ in $XOY$ plane; $h$ – the projection of distance between shoulder points and point $O$ on $OZ$ axis; $\delta $ – the projection of distance between the centroid of weapon equivalent mass and shoulder points on $OY$ axis; $\delta \text{'}$ – the projection of distance between the acting point of gunpowder gas and the centroid of weapon equivalent mass on $OY$ axis; $\eta $ – the projection of distance between the centroid of weapon equivalent mass and shoulder points on $OZ$ axis; $\eta \text{'}$ – the projection of distance between the acting point of gunpowder gas and the centroid of weapon equivalent mass on $OZ$ axis; $F\left(t\right)$ – the function of gunpowder gas force.
In the System dynamics equation, the values of human inertia parameters were calculated by statistical equations according to the shooter’s height and weight [4, 7]; the range of the equivalent elastic coefficient and damping coefficient were obtained by the identification experiment of model parameters [4, 5]; the function of gunpowder gas force was obtained by the pressure test experiment of the grenade launcher. The estimation of model parameters was shown in Table 1.
Table 1. Estimation of model parameters
Parameters

Symbols and units

Value

Parameters

Symbols and units

Value

Mass

${m}_{r}$ / Kg

6

Damping coefficient

${C}_{x}$ / N·S·M^{1}

300

Mass

$m$/ Kg

30

Damping coefficient

${C}_{\alpha}$ / N·M·S·Deg^{1}

0.54.5

Moment of inertia

${I}_{cm}$ / Kg·M^{2}

1.0411

Damping coefficient

${C}_{\theta}$ / N·M·S·Deg^{1}

0.54.5

Moment of inertia

${I}_{cm1}$ / Kg·M^{2}

0.8185

Damping coefficient

${C}_{\beta}$ / N·M·S·Deg^{1}

0.54.5

Moment of inertia

${I}_{r}$ / Kg·M^{2}

0.3056

Damping coefficient

${C}_{\gamma}$ / N·M·S·Deg^{1}

0.54.5

Moment of inertia

${I}_{r1}$ / Kg·M^{2}

0.2738

Geometric length

$L\text{'}$ / M

0.4

Elastic coefficient

${K}_{x}$ / N·M^{1}

2000

Geometric length

$h$ / M

0.03

Elastic coefficient

${K}_{\alpha}$ / N·M·Deg^{1}

424

Geometric length

$\delta $ / M

0.02

Elastic coefficient

${K}_{\theta}$ / N·M·Deg^{1}

15

Geometric length

$\eta $ / M

0.02

Elastic coefficient

${K}_{\beta}$ / N·M·Deg^{1}

424


Elastic coefficient

${K}_{\gamma}$ / N·M·Deg^{1}

424

3. Influence of human equivalent parameters on pitching stability
As a principal influence factor for the firing stability of weapons [3, 8, 9], the pitching movement was discussed by calculating with different values of equivalent parameters (${K}_{\theta}$, ${C}_{\theta}$, ${K}_{\gamma}$, ${C}_{\gamma}$, ${K}_{\alpha}$, ${C}_{\alpha}$, ${K}_{\beta}$, ${C}_{\beta}$) individually. The results showed that the parameters, ${K}_{\gamma}$, ${C}_{\gamma}$, ${K}_{\beta}$, ${C}_{\beta}$, have no effect on pitch motion, while the parameters, ${K}_{\theta}$, ${C}_{\theta}$, ${K}_{\alpha}$, ${C}_{\alpha}$, have different influence on the pitching movement. The motion curves of the weapon were presented in Figs. 29 as follows.
Fig. 2. Pitching angular displacement curves vs. ${K}_{\theta}$
Fig. 3. Pitching angular velocity curves vs. ${K}_{\theta}$
(1) Relative to different values of ${K}_{\theta}$, the angular displacement curves and angular velocity curves of pitching movement were shown in Fig. 2 and Fig. 3. Along with the increase of ${K}_{\theta}$, the peak value of the pitching angular displacement curves and the time corresponding to the maximum value decrease slightly. The pitching angular velocity curves, which are almost identical in the ascending phase, appear some difference only in the descent phase. Therefore, in the vicinity of the peak value of the pitching angular displacement curves, the slope of these curves appears corresponding difference.
The maximum pitching angle is 0.126°when ${K}_{\theta}$ is 1 (N·M·Deg^{1}), while the maximum pitching angle is 0.107° when ${K}_{\theta}$ is 5 (N·M·Deg^{1}), and the pitching angle of the weapon at the moment of projectile muzzleleaving decrease from 0.0633° to 0.0618° (shown as Table 2). By the shooting deviation conversion, corresponding to the decrease of the pitching angle at the moment of projectile muzzleleaving (0.0015°), the shooting deviation of the 100 m target is 0.26 centimeters.
Table 2. Pitching angular displacement of weapon vs. ${K}_{\theta}$
${K}_{\theta}$ (N·M·Deg^{1})

Max pitching angle of weapon (Deg)

Pitching angle of the weapon at the moment of projectile muzzleleaving (Deg)

1

0.126

0.0633

2

0.121

0.0629

3

0.115

0.0626

4

0.111

0.0622

5

0.107

0.0618

Fig. 4. Pitching angular displacement curves vs. ${C}_{\theta}$
Fig. 5. Pitching angular velocity curves vs. ${C}_{\theta}$
(2) Relative to different values of ${C}_{\theta}$, the angular displacement curves and the angular velocity curves of pitching movement were shown in Fig. 4 and Fig. 5. Along with the increase of ${C}_{\theta}$, the peak value of the pitching angular velocity curves decrease obviously, and the time corresponding to the maximum value decrease slightly. Accordingly, there are noticeable differences in the trend of the angular displacement curves.
The maximum pitching angle is 0.238°when ${C}_{\theta}$ is 0.5 (N·M·S·Deg^{1}), while the maximum pitching angle is 0.032° when ${C}_{\theta}$ is 4.5 (N·M·S·Deg^{1}), and the pitching angle of the weapon at the moment of projectile muzzleleaving decrease from 0.0890° to 0.0224° (shown as Table 3). By the shooting deviation conversion, corresponding to the decrease of the pitching angle at the moment of projectile muzzleleaving (0.0666°), the shooting deviation of the 100 m target is 11.62 centimeters.
Table 3. Pitching angular displacement of weapon vs. ${C}_{\theta}$
${C}_{\theta}$ (N·M·S·Deg^{1})

Max pitching angle of weapon (Deg)

Pitching angle of the weapon at the moment of projectile muzzleleaving (Deg)

0.5

0.238

0.0890

1.5

0.092

0.0532

2.5

0.057

0.0369

3.5

0.041

0.0279

4.5

0.032

0.0224

(3) Relative to different values of ${K}_{\alpha}$, the angular displacement curves and angular velocity curves of pitching movement were shown in Fig. 6 and Fig. 7. The variation characteristics of these curves are similar to that of curves presented in Fig. 2 and Fig. 3.
The maximum pitching angle is 1.92° when ${K}_{\alpha}$ is 4 (N·M·Deg^{1}), while the maximum pitching angle is 1.41° when ${K}_{\alpha}$ is 24 (N·M·Deg^{1}), and the pitching angle of the weapon at the moment of projectile muzzleleaving decrease from 0.814° to 0.775° (shown as Table 4). By the shooting deviation conversion, corresponding to the decrease of the pitching angle at the moment of projectile muzzleleaving (0.039°), the shooting deviation of the 100 m target is 6.81 centimeters.
(4) Relative to different values of ${C}_{\alpha}$, the angular displacement curves and the angular velocity curves of pitching movement were shown in Fig. 8 and Fig. 9. The variation characteristics of these curves are similar to that of curves presented in Fig. 4 and Fig. 5.
The maximum pitching angle is 6.96° when ${C}_{\alpha}$ is 0.5 (N·M·S·Deg^{1}), while the maximum pitching angle is 1.33° when ${C}_{\alpha}$ is 4.5 (N·M·S·Deg^{1}), and the pitching angle of the weapon at the moment of projectile muzzleleaving decrease from 1.53° to 0.63° (shown as Table 5). By the shooting deviation conversion, corresponding to the decrease of the pitching angle at the moment of projectile muzzleleaving (0.9°), the shooting deviation of the 100 m target is 157.1 centimeters.
Fig. 6. Pitching angular displacement curves vs. ${K}_{\alpha}$
Fig. 7. Pitching angular velocity curves vs. ${K}_{\alpha}$
Table 4. Pitching angular displacement of weapon vs. ${K}_{\alpha}$
K_{α} (N·M·Deg^{1})

Max pitching angle of weapon (Deg)

Pitching angle of the weapon at the moment of projectile muzzleleaving (Deg)

4

1.92

0.814

9

1.73

0.804

14

1.60

0.794

19

1.50

0.785

24

1.41

0.775

Fig. 8. Pitching angular displacement curves vs. ${C}_{\alpha}$
Fig. 9. Pitching angular velocity curves vs. ${C}_{\alpha}$
Table 5. Pitching angular displacement of weapon vs. ${C}_{\alpha}$
${C}_{\alpha}$ (N·M·S·Deg^{1})

Max pitching angle of weapon (Deg)

Pitching angle of the weapon at the moment of projectile muzzleleaving (Deg)

0.5

6.96

1.53

1.5

3.32

1.13

2.5

2.19

0.89

3.5

1.65

0.74

4.5

1.33

0.63

As stated above, the effects of the equivalent elastic coefficients and the equivalent damping coefficient on the pitching movement of weapons were different significantly:
1) With the increase of the equivalent elastic coefficients, the pitching angle of the weapon at the moment of projectile muzzleleaving decreases a little, and the maximum value of the pitching angular velocity curves in the positive direction has a very small decrease, however, the maximum value of these curves in the opposite direction increases slightly.
2) With the increase of the equivalent damping coefficients, the pitching angle of the weapon at the moment of projectile muzzleleaving decreases obviously, and the maximum value of the pitching angular velocity curves both in the positive direction and the opposite direction decreases evidently.
3) In the same interval, the effects of ${C}_{\alpha}$ on the pitching angular displacement curves and pitching angular velocity curves are more obvious than that of ${C}_{\theta}$.
The authors declare that there is no conflict of interests regarding the publication of this article.
4. Conclusions
Based on the dynamic model of manweapon system consisted of two rigid bodies with five degrees of freedom, the effect of human equivalent parameters on the pitching stability of the grenade launcher was discussed. The simulation results showed that the fundamental parameters for the stability of pitching movement were ${K}_{\theta}$, ${C}_{\theta}$, ${K}_{\alpha}$ and ${C}_{\alpha}$, and the conclusions were as follows:
1) To increase the value of ${K}_{\alpha}$ or ${K}_{\theta}$ individually is beneficial to the stability of the pitching angular displacement, but it is not conducive to the stability of the pitching angular velocity. And the influence of these parameters on pitching movement is small.
2) To increase the value of ${C}_{\alpha}$ or ${C}_{\theta}$ individually is very useful to the stability of pitching movement. Moreover, ${C}_{\alpha}$ is the most critical parameter and the range of it should be defined strictly.
Acknowledgements
The authors wish to express their deep appreciation to Mr. Liu bing, Mr. Gong Penghan and to the test crew at weapon shooting laboratory of Nanjing University of Science and Technology, for their contribution in the test experiment and data reduction efforts, in each of the tests conducted in support of this study.
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