This project wants to model and simulate a Colt M1911A1, a semi-automatic pistol, with the software MSC Adams, during the cocking, loading and firing phases. It started from the study of the gun and his operations, and then it focused on its modelling into Adams, with the choices of the components, joints and approximations needed to represent faithfully, but at the same time shortly, the dynamics of the system. The results are then discussed and compared to data available from literature.
The aim of the project is to model and simulate the Colt M1911A1, newer version of the M1911, a single-action, semi-automatic, recoil-operated pistol, chambered for the .45 ACP cartridge, with the software MSC Adams.
The first step of this work is to model and simulate the cocking and loading phases of the gun, in which the first cartridge, initially into the tube of the magazine, is pushed into the barrel of the pistol.
The next step is to model and simulate the firing phase of the gun, in which, starting from the drive of the trigger, several components are moved in order to generate the explosion of the gunpowder, ejection of the shell and so consequently reloading of the gun.
In particular, because of the no presence of data about the impulse of the explosion of the gunpowder, it will be represented with some approximations: the results will be verified with respect to the real performance of the gun.
The modelling problem
The first part of the modelling is focused on the analysis of the system, in order to identify the essentials components necessary to represent it.
These ones are mentioned in the space below:
- Barrel – 1
- Bushing of the barrel – 2
- Link of the barrel – 3
- Disconnector – 5
- Sear – 64
- Slide – 66
- Ejector – 53
- Stop firing pin – 12
- Hammer – 56
- Strut of the hammer – 17
- Trigger – 51
- Tube of the magazine – 19
- Follower of the magazine – 57
- 2 Cartridges (2 bullets and 2 shells)
- Receiver – 61
- Recoil spring – 35
- Main spring – 59
- Magazine spring- 58
- Sear spring (modelled as equivalent torsional spring) – 42
It is mentioned a short explanation of the functions of the components, in order to clarify the next part.
If the gun is not cocked, it is necessary to manually move back the slide, so the hammer is rotated in the same direction, getting in touch with the sear, whose spring blocks the return of the hammer in its initial position.
Therefore, an equilibrium configuration between spring forces of the hammer (or main spring force) and the sear is reached.
When the slide is pushed back, also the barrel is moved in the same direction, rotating about the link, that can also rotate about the receiver: this motion finishes when the barrel gets in touch with the receiver, so the cartridge can be easier pushed into the barrel itself.
Moreover, the magazine spring, that is pre-compressed, pushes up the cartridges when the first cartridge loses contact with the slide: the gun is thus cocked.
The recoil spring is put in tension with the movement of the slide, so when the external force ceases, the spring pulls back the slide to the initial position.
This motion also pushes the cartridge into the barrel because of the contact between slide and cartridge, so every component returns to the initial position, except for the hammer, that remains in the equilibrium configuration. Now the gun is cocked and loaded.
Once loaded, if the trigger is driven it hits the disconnector, that is rotated together with the sear (disconnector and sear are fixed together).
The rotation of the sear releases the hammer that returns into its initial position forced by the mainspring, hitting the stop firing pin, on the back of the slide.
Therefore the firing pin (that is not modelled in this work) gets in touch with the detonator on the back of the shell, with resulting explosion of the gunpowder, shot of the bullet (that detaches from the shell) and movement of shell and slide in the opposite direction, as result of the reaction force.
Then the same actions of the cocking and loading phases are repeated, with the only difference that now the shell, pushed back from the explosion, hits the ejector and then is expelled thanks to the second cartridge that, at the same time, reaches the reloading position. So now the gun is again cocked and loaded.
The second part of this work is focused on the modelling of the joints, constraints, motions and forces acting on the system during the operations.
The Cad files of every component were obtained from the GrabCad website.
It’s important to point out that two different models have been developed for the cocking/loading and firing phases, because of different constraint configurations needed.
In order to make easier the description, components will be analysed after a division in three different sub-assemblies.
The sub-assembly consisting of barrel, bushing barrel, link barrel, slide and recoil spring is realised without any contacts, in order to get only 1 DOF.
A spatial RRPR (Revolute-Revolute-Prismatic-Revolute) mechanism was chosen, and an auxiliary mass (roughly in the middle of the bushing barrel, with negligible mass) is introduced in order to get it.
The constraint configuration is:
- Revolute joint, between link of the barrel and ground;
- Spherical joint, between link of the barrel and barrel;
- Spherical joint, between barrel and auxiliary mass;
- GCONS, between bushing of the barrel and auxiliary mass, in order to get only a relative rotation between them about the z axis without any redundant constraints;
- Fixed joint, between bushing of the barrel and slide
- Translational joint, between slide and ground, in order to get only the displacement along the x axis
This constraint configuration is the same in both the models of the gun.
The sub-assembly consisting of hammer, strut of the hammer, mainspring, sear, torsional sear spring, disconnector and trigger gets 4 DOF: sear and disconnector rotation, trigger displacement, hammer rotation and strut displacement.
The constraint configuration is:
- revolute joint, between hammer and ground
- spherical joint, between hammer and strut of the hammer
- inline joint, at the end of the strut, between strut and ground, along the main spring travel direction
- revolute joint, between sear and ground
- fixed joint, between disconnector and sear
- translational joint, between trigger and ground
This configuration is the same in both the models, except for the trigger, that is deactivated in the cocking/loading model because it is not necessary.
The sub-assembly consisting of the two cartridges (each consisting of a bullet and a shell) and the magazine follower must be treated separately in each model.
In the cocking/loading model at the beginning the sub-system gets 1 DOF: translational displacement of the magazine follower together with the cartridges, as a single component along the magazine spring travel.
The constraint configuration is:
- translational joint, between magazine follower and ground, along the magazine spring travel
- planar joint, for each shell
- GCONs for each shell, between shell and follower, in order to fix these components to the displacement of the follower itself
- Fixed joint, between each bullet and its shell
When the first cartridge is at the loading position, its GCONs are deactivated, leaving only the planar joint, so the slide can push the cartridge into the barrel.
So now the first cartridge gets 3 DOF until it is completely into the barrel and the gun is loaded.
In the firing model the difference is that the first cartridge is already into the barrel, so the sub-system gets 2 DOF: translational displacement of the magazine follower and the second cartridge, that are moving as a single component, translational displacement of the bullet of the first cartridge along the barrel axis.
The constraint configuration is:
- Same constraint configuration of the magazine follower as the cocking/loading model
- Planar joint to the shell of the second cartridge
- 3 GCONs, between the shell of the second cartridge and the follower, as in the previous model
- Fixed joint, between the bullet and the shell of the second cartridge
- Translational joint, between the shell of the first cartridge and the barrel
- Inplane joint, between the end of the shell of the first cartridge and its contact point with the slide, so the shell moves together with slide and has no DOF
- Translational joint between the bullet of the first cartridge and the barrel, with additional friction
The values of the friction coefficients were taken from the contact condition steel-steel (hypothesis of full metal jacket bullet (FMJ)).
– µs=0.60 with Vs=0.5 m/s
– µd=0.42 with Vd=1.5 m/s
When the shell of the first cartridge is going to hit the ejector, its translational joint with the barrel and inplane joint with the slide are deactivated, then a planar joint is activated, so the shell can be expelled.
The constraints relative to the second cartridge during the reloading phase are managed as the first cartridge in the cocking/loading model.
Receiver and magazine tube are fixed to the ground, in order to get the correct dimensions of the gun, while ejector and stop firing pin are fixed to the back of the slide.
During the simulation some contacts are needed to simulate the path of the cartridge from the magazine tube into the barrel and the interaction between hammer, sear and trigger.
In particular, the contacts are between:
- Hammer and stop firing pin (to stop the hammer motion)
- Hammer and Sear (to get the equilibrium between them)
- Disconnector and Trigger (to drive the shot mechanism)
- Shell1 and barrel (to introduce the shell into the barrel)
- Shell1 and slide (to push the shell into the barrel)
- Shell1 and ejector (to eject the shell after the shot)
- Shell1 and Shell2 (to support the shell1 during the reloading and to eject it during the firing)
- Bullet1 and receiver (to direct the shell into the barrel during the reloading)
- Bullet1 and barrel (same as the line before)
- Shell2 and Slide (to push the shell into the barrel)
- Shell2 and barrel (to introduce the shell into the barrel)
- Bullet2 and Receiver (to direct the shell into the barrel during the reloading)
- Bullet2 and barrel (same as the line before)
It’s important to point out that the continuous impact modelling is used, with default values of its parameters, reducing (if necessary) the values of maximum damping and stiffness.
Mainspring and recoil spring are modelled as spring-damper elements, whose values of stiffness and preload are taken from industrial technical drawings attached to the Cad files, while the values of the damping are chosen in order to remove possible vibrations during the simulation.
The flexible element linked to the sear is modelled as an equivalent torsion spring, whose values of stiffness and damping are chosen based on the simulation results.
Magazine spring is modelled as a spring-damper element too, but because of the absence of data about it, values of stiffness and damping are chosen in order to get a fast elastic response.
Preload is imposed to every spring: where not known, it is estimated based on the results.
It’s important to stress that, in order to avoid the displacement of the slide at the beginning of the simulation because of the recoil spring preload, its DOF is deactivated during that time step with an appropriate constraint.
Two motions are introduced in the models, one for each model.
A translational motion is used to control the displacement of the slide during the cocking phase, along the x direction: in particular, it follows a sinusoidal law in the time, with the maximum value (in module) at the end of the travel of the slide.
This motion is then deactivated so the recoil spring can pull back the slide.
A translational motion is used to control the displacement of the trigger during the firing phase, in order to hit the disconnector and remove the contact between sear and hammer. It is modelled with a step function.
Two forces are introduced in the firing model: these forces must simulate the explosion of the gunpowder into the shell of the first cartridge.
Because of the principle of action and reaction, these two forces are equal and opposite, and act respectively on the centres of mass of the bullet of the first cartridge and of the slide.
It’s necessary to point out that in Adams there is the possibility to create a single force that acts on a body and reacts on another one, but with two forces it’s easier to control the direction of them.
Because there aren’t data about the values of the forces, a qualitative trend of the pressure barrel behind the bullet was found.
It is proposed an example, related to another kind of gun.
The ideal value of the pressure peak for the colt is 21000 psi (about 1500 bar), that corresponds to roughly 18000 N, once you note the area of the back surface of the bullet.
However, considering the losses associated to the heat transfer, the rotational kinetic energy, considering that the trend of the force in Adams is obtained using step functions, a peak value of about 13000 N is estimated.
The duration of the impulse and the time taken by the force to reach the peak are obtained from semi empirical formula found in the literature, based on the mass and geometry of the cartridge and the length of the barrel.
In particular, the travel time of the bullet into the barrel is about 0.6 ms, with 0.12 ms taken to reach the peak value.
The impulse must be measured, together with the velocity of the bullet, in order to evaluate the validity of the estimates done.
Simulations and analysis of results
Cocking-loading and firing are simulated with two different scripts with Adams solver.
A third script is used to measure the impulse of the force of the explosion.
It’s important to stress that the second model is obtained from the final configuration of the first model at the end of its simulation.
Since the constraint configurations related to the various components vary during the simulation, several sensors are used: they permit to stop the simulation upon the occurrence of a given event, in order to change the constraints configuration with the commands ‘ACTIVATE’ and ‘DEACTIVATE’.
The simulation settings are:
- GSTIFF solver
- I3 formulation, with EMAX up to 1.0E-7
- SI2 formulation, with EMAX: 1.0E-3
Simulations have been tested both with I3 (with several decreasing EMAX) and SI2 formulations, in order to obtain the same results and so reach the convergence.
Contacts are set with the Parasolid method.
Short descriptions of the simulations are reported below.
The cocking-loading simulation starts with a first phase in which slide is forced by a translational displacement with sinusoidal law towards the back of the gun, in order to cock the gun.
A first sensor is activated when the barrel touches the receiver, so the barrel is fix and the slide continues to go back.
A second sensor is activated when the slide exceeds a given longitudinal coordinate, so the magazine follower, that was fix at the beginning in order to lighten the simulation, can move up: the first cartridge can thus reach the loading position.
A third sensor is activated when the slide is next to the end of the travel, so it’s possible to remove its motion.
A fourth sensor is activated just before the contact between slide and cartridge, in order to release the cartridge itself from the magazine follower, getting 3 DOF and leaving only the planar joint between cartridge and ground.
A fifth sensor is activated when the slide hits the barrel, so it’s possible to release the barrel and to let it move.
The last sensor is used to stop the simulation when the barrel returns at the initial position.
The firing simulation starts with a first phase in which the trigger is forced by a translational motion to hit the disconnector, so the sear loses the contact with the hammer that quickly hits the stop firing pin.
This impact activates the action and reaction forces that simulate the explosion of the gunpowder.
As in the previous model, a couple of sensors are used to manage the kinematic of the barrel, while a third sensor is used to release the magazine follower from the fix condition.
However, in this case both the translational joint (between the shell of the first cartridge and the barrel) and the inplane joint (between shell and slide) are deactivated, while a planar joint is activated: in this way the shell can hit the ejector and be pushed away from the second shell.
It’s important to stress the approximation of the planarity condition applied to the shell during the ejection, that is not true in general.
The last sensors are used as in the previous model, because it’s necessary to reload again the gun.
Results show that the models simulate quite accurately the operation of the gun.
In particular, the firing model gives a muzzle velocity (that is, the velocity of the bullet at the end of the barrel) of 254 m/s, while the literature states 253 m/s.
The integration of the force due to the explosion gives an impulse of about 3.84 Ns, while from the variation of quantity of motion of the bullet (m=15 g, ΔV=253 m/s) an impulse of 3.79 Ns is obtained.
The value of the simulation is obviously higher because of the presence of friction.
Moreover, the impulse of the friction is quite low because it’s related to a very short time step.
The travels of the force and its impulse are shown below.
It’s important to stress that, during the design, the influence of the rotation of the bullet imposed by the rifling of the barrel has been tested: however, it resulted negligible.
In fact, the rifling has a pitch of 1:16’’, equal to a complete rotation of the bullet every about 400 mm (instead the free length of the barrel, that is the length of the barrel minus the length of the shell, is about 100 mm): the angular velocity is thus about 3500 rad/s, from which it’s possible to evaluate a rotational kinetic energy of about 0.2 J.
The translational kinetic energy at the muzzle is about 480 J, so the rotational effect is negligible.
The aim of this work is to model and simulate the phases of cocking, loading and firing of a gun, starting from no data about the explosion of the gunpowder, so many approximations have been taken.
Results shows that the models simulate quite well the operations of the pistol, with a muzzle velocity of 254 m/s and an impulse of the explosion of 3.84 Ns.
These values are roughly the same as expected, therefore the approximations taken can be considered correct.
Rio Benson, ‘Drawings of the Government Model M1911-A1 Semi-Automatic Pistol’, Benson Consulting, LLP, ©2010 (file from GrabCad)