Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 1, pp. 81-91, Warsaw 2015
DOI: 10.15632/jtam-pl.53.1.81

MODELING AND NUMERICAL SIMULATION OF SEMI-AUTOMATIC

PISTOL DYNAMICS

Cyprian Suchocki, Janusz Ewertowski

Warsaw University of Technology, Department of Mechanics and Armament Technology, Warsaw, Poland

e-mail: c.suchocki@imik.wip.pw.edu.pl

In this study, amathematicalmodel of a semi-automatic hand-heldweapon is presented.The
dynamical equations ofmotion are derived by utilizing the formalismof Lagrange equations
of the second kind. Themodel is applied to describe motion of P-64 semi-automatic pistol.
The characteristics of rheological elements accounting for the mechanical reaction of the
human hand are identified and are further used to determine the force and torque which
act on the human hand during a shot. Some exemplary results of the conducted parametric
study are presented in order to illustrate sensitivity of the system.

Keywords: pistol, firearm, recoil, pitch, muzzle climb

Notations

a,b – distance between slide and receivermass centers perpendicular and parallel to
barrel axis, respectively

A4×4,B4×4,C4×4 – mass, damping and stiffness matrix
b4×1,F4×1 – force and external forces column matrix
bc1,bc2 – viscosity coefficient of elasto-plastic and plastic collision
bhx,bhy,bhϕ – hand damping coefficients
c – distance between slide mass center and barrel axis
D – Rayleigh dissipation function
e1,e2 – length of undeformed Kelvin-Voigt element no. 1 and of undeformed damper

no. 2
Fh,Mh – horizontal force and torque acting on human hand
g – gravitational acceleration
H – Heavisde step function
Jr,Js – receiver and slide moment of inertia
k,kc1 – recoil spring stiffness and stiffness coefficient of elasto-plastic collision
khx,khy,khϕ – hand stiffness coefficients
mr,ms – receiver and slide mass
Ps – breech force
q – generalized coordinates columnmatrix, q= [q1,q2,q3,q4]

T

δqj – j-th generalized coordinate variation (j=1,2,3,4)
Q,QNj – barrel pulling force and j-th non-potential generalized force (j=1,2,3,4)

rs,rr – slide and receiver mass center position vector
rr−s – position vector of slide mass center with respect to receiver mass center
T,V – total kinetic and total potential energy
Ti−j – transformation matrix from i-th to j-th coordinate system (i,j =0,1,2)
vs,vr – slide and receiver mass center velocity vector
vrs – slide velocity relative to receiver
xr,yr – horizontal and vertical coordinate of receiver mass center in inertial frame



82 C. Suchocki, J. Ewertowski

xs – horizontal coordinate of slide mass center in inertial frame
s,smax – slide translation andmaximal slide translation
ω – pistol angular velocity
ϕ – angle of pistol rotation
δA – virtual work variation

1. Introduction

The recoil is a term used for the phenomenon of weapon motion which is caused by a gunshot
(Fig. 1). Basically, two kinds of recoil can be distinguished (e.g. Hall, 2008):

• The horizontal recoil which refers to the weapon translation in the direction opposite to
projectile motion.

• The vertical recoil, also called the pitch or the muzzle climb, which refers to the weapons
rotation.

Onemay also adopt a different rule of distinction (e.g. Kochański 1979):

• The free recoil which takes place when shots are fired without physical contact of the
firearm and the shooter (for instance, when the firearm is hung up on strings or rests on
a surface which may roll on the floor with negligible friction).

• The braked recoil which takes place when the firearm is in contact with a part of the
human body (e.g. hand or arm) during a shot. It results in the appearance of the so-called
punch, i.e. a force is exerted by the weapon on the human organism during a gunshot and
shortly after.

Fig. 1. Recoil of semi-automatic pistol

For semi-automatic weapons, one can also distinguish between the recoil and counterrecoil
(e.g. Benzkofer, 1990). The term counterrecoil refers to themotion of weapon components when
they return to their initial position, taken prior to the time instant when the shot took place.
The horizontal recoil is caused by the force which the burning propellant exerts on the gun.

This force is commonly called the breech force. On the other hand, the reason for the vertical
recoil is the existance of a certain distance between the barrel axis and the point at which the
gun is supported. The distance between the breech force and the reaction force of the support
results in the appearance of a torque which rotates the gun.
It is understandable that the human body has a limited mechanical strength and can bear

punches only up to a certain limiting magnitude. Thus, if the firearm is incorrectly designed,
the punchmay cause bruises or even injuries.
Themotion performed by aweapon during a shotmay be of interest itself even without any

investigation of the forces and torques acting on the hand holding the gun. During a gunshot,
the pistol components are exposed to large accelerations mainly caused by the breech force and



Modeling and numerical simulation... 83

impacts. An inadequatly designed pistol may be damaged under such dynamic loadings, thus
inflicting serious injuries on the person that is using it. A detailed motion and strength analysis
of the firearm becomes an issue of great importance as far as safety nad proper operation are
concerned.Anumericalmultibody simulation of semi-automatic pistolmotion provides valuable
input data for further local finite element analysis of a particular pistol part.

The literature concerning the recoil in semi-automatic pistols is relatively modest. A two-
-dimensional model of P-64 semi-automatic pistol was proposed by Czepukajtis (1974). The
Lagrange equationsmethodwas utilized to derive the dynamical equations ofmotion. The post-
-impact velocities of the colliding pistol parts were calculated using the conservation of linear
and angularmomentumequations.A solution to the equations ofmotionwas obtained only for a
simplified version of themodel. Somemajor discrepancies were observed between the simulation
results and the experimental data.

Benzkofer (1990) investigated both horizontal and vertical recoil for M9 semi-automatic pi-
stol. For that purpose, a two-dimensional multibodymodel was developed in DADS multibody
dynamics software. The weapon components were modeled as rigid bodies with the incorpo-
ration of several springs and dampers to account for elasto-plastic and ideally elastic impacts.
The pistol-human hand interaction was modeled by utilizing a single Kelvin-Voigt rheological
element. The pistol motion following a single shot was simulated. A number of discrepancies
between the experimental data and the simulation results were observed. In particular, the
predictions of slide and receiver displacements were unsatisfactory. The author concluded that
inadequate stiffness value assumed for the recoil spring is probably the major reason. What is
more, the values of stiffness and damping coefficients assumed for the pistol-hand contact were
considered to be too high, which resulted in very small receiver displacements.

A biomechanical model of a pistol grip handtool and human hand was developed by Lin et
al. (2001). For that purpose, the tool-hand contact was modeled by utilizing several rheological
elements. The model was mainly intended to simulate the motion of small hand tools such as
the drill driver, for instance.

In this paper, a discretemodel of a semi-automatic pistol is presented. Themodel takes into
account the pistol-hand interaction which takes place during a shot. The equations of motion
are derived utilizing the formalism of Lagrange equations of the second kind and represent
a mechanical system with unilateral constraints. The equations of motion are transformed to
the matrix form which is more suitable for performing numerical integration. The developed
model is further applied to simulate the recoil of P-64 semi-automatic pistol. The experimental
data gathered by Czepukajtis (1974) are used to calibrate the values of damping and stiffness
coefficients. Finally, some exemplary parametric study results are presented that have been
obtained for simulations of a series of subsequent shots.

2. Semi-automatic pistol definition and operation

Afirearm is called semi-automatic if it uses a part of the recoil energy to extract the shell case,
eject it andplace a newcartridge into the chamber (Kochański, 1989). Theprinciple of operation
of a semi-automatic pistol will be explained utilizing the example of P-64 semi-automatic pistol
(Fig. 2).

Themost important components of the pistol are depicted inFig. 3. One can distinguish the
following phases of the pistol operation (Czepukajtis, 1974):

1. The trigger is pulled by the shooter which causes the firing pin to fire the primer. The
propellant gases begin to expandwhichmakes the projectile startmoving down the barrel.
Simultaneously, the gases exert the so-called breech force on the bottom surface of the



84 C. Suchocki, J. Ewertowski

Fig. 2. P-64 semi-automatic pistol

Fig. 3. P-64 pistol componentry: 1 – slide, 2 – recoil spring, 3 – receiver, 4 – magazine, 5 – projectile

cartidge case causing the backward movement of the slide. During its recoil, the slide
compresses the recoil spring.

2. As the projectile opens the barrel, the breech force practically drops to zero. At the same
time the slide continues to move backward. The spent cartridge is extracted and ejected.

3. The slides collides the receiver.

4. The impact, together with the energy accumulated in the compressed recoil spring, cause
the slide to start moving to its forward position.

5. The slide continues its counterrecoil and picks up a new cartridge from the magazine.

6. The slide along with the new cartridge collide the receiver. The cartridge is pushed into
the chamber. The slide stops in the forward position.

The system of operations listed above is typical for the group of blowback operated pistols.
Another commondesign are recoil operated pistols. The only difference between a blowback and
a recoil operated pistol is that in the case of the latter the slide is initially locked and unlocks
after the bullet has left the barrel. The locking of the slide ensures that it will not open the
breech while the projectile is still in the barrel. A premature opening of the breech would result
in the propellant gases expanding in the shooter’s hand and possibly disassembly the pistol.

3. Physical model of a semi-automatic pistol and kinematics

For the purpose of developing a physicalmodel of a semi-automatic pistol, a number of assump-
tions is adopted, i.e.:

• The pistol displacement and rotation in the horizontal plane are assumed to be negligible.
The motion of the mechanical system is two-dimensional and takes place in the vertical
plane which contains the barrel axis and all mass centers.

• Themechanical system includes the receiver, the slide and the recoil springwith negligible
mass. The slide is in motion with respect to the receiver (Fig. 4).

• The receiver and the magazine are modeled as a single rigid body which is referred to
as receiver further in the text. The mass and the moment of inertia correspond to the
magazine loaded with three cartridges.



Modeling and numerical simulation... 85

Fig. 4. Physical model and assumed coordinate systems

Fig. 5. Forces acting on the system and the model of contact

Fig. 6. Modeling of impact

• Thehumanhand - pistol interaction ismodeled byutilizing horizontal and verticalKelvin-
-Voigt elements alongwith an angular spring and an angular damper (Fig. 5). The strains
exhibited by soft tissues of the human hand are assumed to be small, thus all the utilized
springs and dampers are linear.

• The external forces acting on the pistol are the breech force Ps(t) and the force Q(t)
which is exerted on the barrel by the projectile (Fig. 5). The force value estimations given
in Czepukajtis (1974) are used.

• The slide-receiver impacts are simulated using very stiff rheological elements (Fig. 6), cf
Kukla et al. (1992).



86 C. Suchocki, J. Ewertowski

• After the slide collides the receiver during its counterrecoil, they continue motion as a
single rigid body.

To determine the motion of the pistol, the following orthogonal coordinate systems are
introduced (Fig. 4):

Ox0y0 – the inertial coordinate system,

Crx1y1 – the system parallel to the inertial coordinate systemwith the origin fixed at the
mass center of the receiver,

Crx2y2 – the system fixed to the receiver with the origin at the receiver mass center.

The basis vectors of the systemsCrx1y1 andOx0y0 are related as in

{

i1
j1

}

=T0−1

{

i0
j0

}

T0−1 =

[

1 0
0 1

]

(3.1)

whereas for the basis vectors of the systemsCrx2y2 andCrx1y1

{

i2
j2

}

=T1−2

{

i1
j1

} {

i1
j1

}

=T2−1

{

i2
j2

}

(3.2)

with

T1−2 =

[

cosϕ −sinϕ
sinϕ cosϕ

]

T2−1 =T
T
1−2 (3.3)

and

T0−2 =T1−2T0−1 T2−0 =T
T
0−2 =T

T
0−1T

T
1−2 (3.4)

The position vector of the receiver mass center Cr (Fig. 4) is

rr =xri1+yrj1 (3.5)

whereas the position vector of the slide mass center Cs is

rs = rr +rr−s rr−s =(b+s)i2+aj2 (3.6)

The velocity ofCr is given by the relation

vr = ẋri1+ ẏrj1 (3.7)

while the velocity ofCs

vs =vr+ω×rr−s+vrs ω=−ϕ̇k2 vrs = ṡi2 (3.8)

After calculating the cross product in (3.8)1, the following equation is found

vs =
(

ẋr cosϕ− ẏr sinϕ+ ϕ̇a+ ṡ
)

i2+
(

ẋr sinϕ+ ẏr cosϕ− ϕ̇(b+s)
)

j2 (3.9)

Themechanical system has four degrees of freedom, i.e. xr, yr,ϕ and s.



Modeling and numerical simulation... 87

4. Dynamical model of the semi-automatic pistol

The Lagrange equations of the second kind in the presence of potential and dissipative forces
take the form (e.g. Gutowski, 1971)

d

dt

(∂T

∂q̇j

)

−

∂T

∂qj
=
∂V

∂qj
−

∂D

∂q̇j
+QNj j=1,2, . . . ,4 (4.1)

where

q1 =ϕ q2 =xr q3 = yr q4 = s

According to König’s theorem, the total kinetic energy of the system is

T =
1

2
mrvr ·vr +

1

2
msvs ·vs+

1

2
(Jr+Js)ϕ̇

2 (4.2)

After substituting (3.7) and (3.9) into (4.2), the energy takes form

T =
1

2
mr(ẋ

2
r + ẏ

2
r)+
1

2
(Jr +Js)ϕ̇

2

+
1

2
ms

[

(

ẋr cosϕ− ẏr sinϕ+ ϕ̇a+ ṡ
)2
+(ẋr sinϕ+ ẏr cosϕ− ϕ̇(b+s)

)2
]

(4.3)

By substitution of (4.3) into (4.1) and performing differentiations, one obtains
[

Jr +Js+ms
(

(b+s)2+a2
)]

ϕ̈+ms[acosϕ− (b+s)sinϕ]ẍr

−ms[asinϕ+(b+s)cosϕ]ÿr+msas̈+2ms(s+ b)ϕ̇ṡ=Qϕ

ms[acosϕ− (b+s)sinϕ]ϕ̈+(mr+ms)ẍr+mscosϕs̈

−ms[asinϕ+(b+s)cosϕ]ϕ̇
2
−2msϕ̇ṡsinϕ=Qxr

−ms[asinϕ+(b+s)cosϕ]ϕ̈+(mr+ms)ÿr−ms sinϕs̈

+ms[(b+s)sinϕ−acosϕ]ϕ̇
2
−2msϕ̇ṡcosϕ=Qyr

msaϕ̈+mscosϕẍr −ms sinϕÿr +mss̈−ms(s+b)ϕ̇
2 =Qs

(4.4)

The potential energy of the system is

V =mrgyr+msg[yr − (b+s)sinϕ+acosϕ]+
1

2
ks2 (4.5)

and the Rayleigh dissipation function is

D=
1

2
bhϕϕ̇

2+
1

2
bhxẋ

2
r +
1

2
bhyẏ

2
r (4.6)

The non-potential generalized forces, not included in the dissipation function, are determined
using the principle of virtual work, i.e.

δA=
4
∑

j=1

QNj δqj (4.7)

which yields

QNϕ =(a− c)[Ps(t)−Q(t)] Q
N
xr
= [Ps(t)−Q(t)]cosϕ

QNyr =−[Ps(t)−Q(t)]sinϕ

QNs =
[

−kc1
(

s− (smax−e1)
)

− bc1ṡ
]

H
(

s− (smax−e1)
)

− bc2ṡH(e2−s)+Ps(t)

The impact forces arising from unilateral constrains are incorporated by utilization of certain
Heaviside functions (Kukla et al., 1992). After the collision during the counterrecoil, the system
continues its motion as a single rigid body with three degrees of freedom, i.e. xr, yr and ϕ.



88 C. Suchocki, J. Ewertowski

The computer algebra software wxMaxima has been used to write a script further utilized
to generate the equations of motion of the system (Ney de Souza et al., 2004).

5. Numerical integration

For the purpose of numerical integration, the equations ofmotion arewritten in thematrix form
(e.g. Grabysz, 2002), i.e.

A4×4q̈4×1+B4×4q̇4×1+C4×4q4×1 =F4×1 (5.1)

Introducing a column matrix

b4×1 =F4×1−B4×4q̇4×1−C4×4q4×1 (5.2)

Thus

A4×4q̈4×1 =b4×1 (5.3)

The matrix components are found in wxMaxima symbolic algebra system by taking advantage
of coeff command (Ney de Souza et al., 2004), i.e.

Aij = coeff
( d

dt

(∂T

∂q̇i

)

, q̈j

)

bi =Q
N
i −
∂V

∂qi
−

∂D

∂q̇i
+
∂T

∂qi
−

[

d

dt

(∂T

∂q̇i

)

−

4
∑

j=1

coeff

( d

dt

(∂T

∂q̇i

)

, q̈j

)

q̈j

]

Equation (5.3) takes an explicit form











A11 A12 A13 A14
A21 A22 0 A24
A31 0 A33 A34
A41 A42 A43 A44





























ϕ̈

ẍr
ÿr
s̈



















=



















b1
b2
b3
b4



















with the following matrix components

A11 = Jr +Js+ms[(s+ b)
2+a2] A12 =A21 =ms[−(b+s)sinϕ+acosϕ]

A13 =A31 =−ms[(b+s)cosϕ+asinϕ] A14 =A41 =msa

A22 =mr+ms A23 =A32 =0 A24 =A42 =mscosϕ

A33 =mr+ms A34 =A43 =−ms sinϕ A44 =ms

and

b1 =−2ms(b+s)ϕ̇ṡ+msg[(b+s)cosϕ+asinϕ]+(a− c)[Ps(t)−Q(t)]−khϕϕ− bhϕϕ̇

b2 =2msϕ̇ṡsinϕ+ms[(b+s)cosϕ+asinϕ]ϕ̇
2+[Ps(t)−Q(t)]cosϕ−khxxr− bhxẋr

b3 =2msϕ̇ṡsinϕ+ms[−(b+s)sinϕ+acosϕ]ϕ̇
2
− (mr+ms)g

−[Ps(t)−Q(t)]sinϕ−khyyr− bhyẏr

b4 =
[

−kc1
(

s− (smax−e1)
)

− bc1ṡ
]

H
(

s− (smax−e1)
)

− bc2ṡH(e2−s)

+ms(b+s)ϕ̇
2+msg sinϕ−ks+Ps(t)

The given system of nonlinear ordinary differential equations has been solved in Scilab. For that
purpose, a program has been written utilizing the Scilab programming language (e.g. Brozi,
2007).



Modeling and numerical simulation... 89

6. Model calibration

The developed mathematical model has been applied to simulate the motion of P-64 semi-
-automatic pistol. The experimental data gathered in Czepukajtis (1974) are utilized to deter-
mine values of the stiffness and damping coefficients responsible for slide-receiver collisions and
pistol-hand contact. The results of the curve-fitting can be seen in Fig. 7. The time instant at
which the bullet leaves the barrel has beenmarked with a vertical solid line.

Fig. 7. Model predictions and experimental data

Fig. 8. Force and torque exerted on hand

7. Exemplary results of parametric study

The sensitivity has been investigated by performing simulations of a 4-shot series. The subsequ-
ent shots have been assumed to occur after∆t=0.25s. InFig. 9. the simulation results obtained
for various magnitudes of the breech force Ps are presented. In Fig. 10, the simulation results
obtained for different values of the recoil spring stiffness k are presented.



90 C. Suchocki, J. Ewertowski

Fig. 9. Simulation results for four subsequent shots with varying magnitude of the breech force

Fig. 10. Simulation results for four subsequent shots with varying stiffness of the recoil spring



Modeling and numerical simulation... 91

8. Conclusions

The results presented above allow one to make the following observations:

• The projectile leaves the barrel after approximately 0.375ms. At this time instant the
pistol displacements caused by both horizontal and vertical recoil are negligible (Fig. 7).
Thus, as far as a single shot is concerned, the recoil has no serious influence on shooting
accuracy.

• A break of about 0.25s is sufficient to almost completely damp out the pistols angular
motion (Fig. 9a and Fig. 10a).

• The longitudinal displacements of the pistol in the case of several subsequent shots do
accumulate (Fig. 9b andFig. 10b). There is not enough time for the humanhand to damp
the longitudinal motion out.

• Themagnitude of the breech force has a very strong influence on pistol displacements and
the pistol-hand interaction (Fig. 9).

• The simulation results are in a good agreement with the experimental observation that
the receiver displacements aremuch larger than the total displacements of the slide (Cze-
pukajtis, 1974). In fact it is the receiver that hits an almost motionless slide during the
last phase of the operation cycle, see Figs. 7b and 7f.

The aforementioned conclusions represent some exemplary information that themathemati-
calmodel of a semi-automatic pistol proposed in this studymayprovide aweapondesignerwith.
The developed model is general and may be useful for simulating the recoil of both blowback
operated and recoil operated handguns.

References

1. Benzkofer P.D., 1990, Dynamic analysis of a hand-held weapon, Proceedings of the Sixth U.S.
Army Symposium on Gun Dynamics, 2, 62-79

2. Brozi A., 2007, Scilab in Examples (in Polish), NakomPublishing, Poznań

3. Czepukajtis W., 1974, Influence of operator on motion of light mechanical devices (in Polish),
PhD thesis, Institute ofMechanical EquipmentDesign,WarsawUniversity of Technology,Warsaw

4. Grabysz W., 2002,Selected Topics in Symbolic-Numerical Simulation of Mechanisms (in Polish),
Silesian University of Technology Publishing, Gliwice

5. Gutowski R., 1971,Analytical Mechanics (in Polish), Polish Scientific Publishers,Warsaw

6. Hall M.J., 2008, Measuring felt recoil for sporting arms, International Journal of Impact Engi-
neering, 35, 540-548

7. Kochański S., 1979, Braked recoil of firearms (in Polish), Scientific Surveys of the Institute of
Mechanical Equipment Design, 2, WarsawUniversity of Technology,Warsaw

8. Kochański S., 1989,Low Caliber Automatic Weapons (in Polish),WarsawUniversity of Techno-
logy Publishing House,Warsaw

9. Kukla S., Posiadała B., Skalmierski B., Tylikowski M., 1992, Dynamics problem of me-
chanical systems with unilateral constraints, Journal of Theoretical and Applied Mechanics, 30,
401-418

10. Lin J.-H., Radwin R.G., Richard T.G., 2001, Dynamic biomechanical model of the hand and
arm in pistol grip power handtool usage,Ergonomics, 44, 295-312

11. Ney de Souza P., Fateman R.J.,Moses J., YappC., 2004,TheMaxima Book, on laws of the
manuscript

Manuscript received April 29, 2014; accepted for print July 21, 2014