Study on kinetic characteristics of a reciprocating mechanism with clearances

Clearances in mechanism are inevitable due to the errors from assembly, manufacture and wear. To study the kinetic characteristics of a typical automatic mechanism and the influence of clearances, an experimental device based on a reciprocating mechanism with clearances was set up, and a high speed camera system (HSCS) was used to record the mechanism motions. While the kinetic characteristics were analyzed based on the experimental data from HSCS, a coefficient of initial pressure of gas chamber used to characterize the randomness of the gunpowder gas force was proposed and its distribution was obtained. Meanwhile, the stiffness coefficient and damping coefficient of counterrecoil spring were identified. The virtual prototype model of this automatic mechanism was established. It’s found that the kinetic characteristics from the simulation and the experimental results of stochastic parameters and identification parameters are coincident approximately. Finally, the influence of guide rail’s clearances on kinetic characteristics based on the virtual prototype model was studied.


Introduction
Clearances in mechanism are inevitable due to assembly, manufacturing deflection and wear [1].Moreover, mechanical systems with clearance joints can have a predictable nonlinear dynamic response [2], and the presence of clearance in the joints degrades the performances of the mechanism, leading to high frequency vibration, joint wear [3] and noise [4].
Over the last few decades, dynamic behaviors of mechanisms with clearances have been studied by many researchers.Tian et al. [5] presented a new computational methodology to model and analyze the planar flexible multi-body system with clearance and lubricated revolute joints.The methodology was validated by two numerical examples including a planar flexible double pendulum and a slider-crank mechanism.A slider-crank mechanism with two revolute clearance joints was set up to study the behavior of planar rigid body mechanical systems within the existence of the dynamic interaction of multiple revolute clearance joints [6].Similarly, the dynamic behaviors of a planar flexible slider-crank mechanism with clearance were studied by Khemili and Romdhane et al. [4] through simulation and experimental tests.Erkaya and Uzmay [7] set up an experimental test rig of a slider-crank mechanism to investigate the effects of joint clearances on vibration and noise characteristics of mechanism.The obtained results show that vibration and noise levels and amplitudes increase with the clearance.Moreover, the dynamic response of a four-bar mechanism having two joints with clearance was investigated by the numerical model with a continuous contact mode and an optimization strategy was proposed to reduce the vibration effect due to the joint clearance [9].Flores et al. [8] presented a comprehensive combined numerical and investigation on the dynamic response of a slider-crank mechanism with revolute clearance joints.The paper also presented and discussed the correlation between the numerical and experimental results.Chen et al. [10], Lu et al. [11] and Chen [12] presented the dynamic analysis of spur gear pair system with backlash and other parameters based on their nonlinear dynamic model, respectively.Chen et al. [10] studied the effect of backlash on instability of vibration amplitude and dynamic meshing force of the system; Lu et al. [11] discussed the relationship between the dynamic stability of the system and the variance of assembling backlash; Chen and Tang [12] analyzed the effects of different kinds of backlash on nonlinear dynamic response, including constant backlash, time-varying backlash and stochastic backlash.Cui et al. [13] used a numerical integration method to study the dynamic responses of geared rotor systems, including the effect of clearance on jump phenomena of amplitude.Moreover, all the presented results show that the mechanism responses are greatly influenced by the clearance.Thus, it is necessary to study the dynamic behaviors of mechanisms considering clearances in joints.
The automatic mechanism in automatic weapon is a reciprocating motion system with clearances that contains multi-components, and the motion of automatic mechanism has a direct impact on the operation, firing accuracy and safety.Thus, kinetic characteristics analysis is an important procedure in the design process of automatic weapon.The automatic mechanism in AK-47 or a similar structure was applied to other guns in the world due to its reliability, simplicity and durability, e.g.Russian AK-74, Israeli Galil assault rifle, Chinese 81-type and 03-type automatic rifles.To study the kinetic characteristics and the effect of clearances on the motion of this typical automatic mechanism, the experiment and virtual prototype simulation are investigated in this paper.
The paper is organized as follows: Section 2 shows the experimental investigation with a high-speed camera system (HSCS) and the results.Section 3 describes the establishment of the virtual prototype model and comparison with the experimental results.Section 4 is devoted to study the effect of clearances on the motion based on the simulation model.A conclusion ends the paper.

Research object
A diagrammatic graph of an automatic mechanism motion model is shown in Fig. 1.The model includes barrel extension, bolt head, bolt carrier, hammer, counterrecoil spring, receiver and hammer spring.The driving force of the mechanism's motion is from the gunpowder gas acting on the bolt carrier.The bolt carrier reciprocates along the receiver guide rail by the force from the gunpowder gas and counterrecoil spring.Two typical motions exist: • Recoil motion: bolt carrier moving from barrel extension to receiver rear end by the gunpowder gas.
• Counterrecoil motion: bolt carrier moving from receiver rear end back to barrel extension during energy release of the counterrecoil spring.A complete cycle process can be depicted as follows: The recoil motion of bolt carrier begins with a high velocity under the action of gun powder gas, bolt head is pushing to unlock.
Bolt carrier and bolt head move together until colliding with receiver rear end.Meanwhile, the hammer is pressed down by bolt carrier in the recoil motion.Bolt carrier and bolt head carry out counterrecoil motion together under the action of counterrecoil spring force.After bolt head completes the locking action, the counterrecoil motion of bolt carrier would go on until colliding with receiver.
The topological graph of this automatic mechanism is shown in Fig. 2. The bolt carrier is an important research object of this system.Rotational and translational motion both happen between bolt carrier and bolt head when bolt head is unlocking or locking, translational motion also happen between bolt carrier and receiver, and bolt carrier contacts and collides with other components in a whole cycle.

Experimental method
The experimental devices include a high-speed camera system (HSCS), an experiment table and an automatic mechanism.This automatic mechanism was fixed on experiment table, and the motions were recorded by a high-speed camera system (HSCS).In order to obtain the motion of parts in the mechanism, some identification points were marked such as points and on the receiver, points and on the bolt and points and on the counter-recoil spring (shown in Fig. 3).Thus, the motion parameters (displacement, velocity) of every part versus time could be obtained through image recognition on the pictures from HSCS.The experimental device is shown in Fig. 3.

Fig. 3. Experimental device
The experimental device may rock slightly as suffering strong impact in the experiment, so data processing is required in order to guarantee the accuracy of experimental data.The Cartesian coordinate system is shown in Fig. 3.The -axis is horizontal and the -axis is vertical.For rigid plane motions, it has been validated that if the movements of any two points of a rigid are known, the movement of others can be calculated through coordinate transformation.Therefore, displacements of identification points on receiver ( and ) and identification points on bolt carrier ( and ) were obtained through image recognition.Then, the movement of the point A attached to bolt carrier was calculated through coordinate transformation.

Stochastic kinetic characteristics analysis of bolt carrier
Fig. 4 shows the experiment displacement curves of the point A attached to bolt carrier during continual firing.It obviously shows that six cyclical peaks appear on the -displacement curve and each cycle is about 90 ms as well as on the -displacement curve.It means six firing cycles happen.The maximum -displacement is about 135 mm, which appears about 30 ms later after beginning.It can be deduced that time cycle of the recoil motion is about 30 ms.The appearance time of a maximum -displacement is nearly 3 ms later after the maximum -displacement in each cycle, besides the maximum values are different.It can be concluded that some vibrations appear along the direction during bolt carrier's motion due to the existence of clearances through the analysis of Fig. 4. Comparing with the -displacement, it is found that the maximum -displacement appears on the beginning of counterrecoil motion, and it is close to the maximum recoil position.By analyzing the structure, a slot designed at the end of receiver guide rail is used to assembly bolt carrier; bolt carrier is unconstrained in direction as moving to the slot.In addition, bolt carrier gets an upward velocity after colliding with receiver rear end.The maximum -displacement appears due to the existence of a slot and impact.The kinematic parameters' values of each bolt carrier in Table 1 are dynamic and various.The difference value between maximum and minimum of , and are 1.13 m/s, 1.01 m/s and 0.51 m/s, and the ratios of the range and the mean are 13.2 %, 20.7 % and 15.8%, respectively.What's more, the variance range of reaches 56.7 %.It is clear that range's changes have a strong influence on the stability of automatic mechanism motion, and all kinematic parameters have significant randomness.It is indicated that , , and are approximately normally distributed due to their Shapiro-Wilk Significances are greater than 0.05.
Figs. 5 to 8 present comparisons between and other kinematic parameters of bolt carrier, respectively.Fig. 5 shows that generally increases with increasing, but this trend is not obvious in a small range.The correlation between and is actually randomness when observing in a small range.There isn't any obvious trend between , , and in Figs. 6 to 8, it is indicated that there is no obvious correlation between and these kinematic parameters.

Determination on force from gunpowder gas
Bolt carrier recoil is powered by gunpowder gas, which is randomness in actually.Gunpowder gas force is closely related to charge mass, combustion efficiency of gunpowder, environmental factors, etc., and it is difficult to determine the value due to the impact of multiple factors.As the maximum recoil velocity ( ) represents the recoil energy well, gunpowder gas force is obtained through statistical analysis on the maximum recoil velocities.Table 1 shows that can be considered to have a normal distribution due to Shapiro-Wilk Significance (0.532) which is greater than 0.05.Similarly, it can be also assumed that gunpowder gas force is approximately normally distributed.In order to characterize the randomness of the gunpowder gas force, the coefficient of initial pressure of gas chamber ( ) is proposed in this paper, which is normally distributed with (1,0.292) based on the analysis in above (Section 2.3).According to the Brauen empirical formula [14], gunpowder gas force is calculated using the following expression: where is gunpowder gas force; is the area of piston lateral section; is in-bore average pressure when bullet arrives at gas port; is starting time when bullet arrives at gas port; is a time coefficient related to in-bore pressure impulse; is a structure coefficient related to structure parameters of gas operated device.

Counterrecoil spring parameters identification
Stiffness coefficient and damping coefficient of counterrecoil spring have great influence on the performance of automatic mechanism.In order to obtain more accurate parameters, displacements of identification points ( and , shown in Fig. 3) on the counterrecoil spring were obtained, then the values of stiffness coefficient and damping coefficient were calculated using the least square identification method [15].Fig. 9 shows the diagrammatic graph of motion model with bolt carrier free recoil, when displacement of bolt carrier is from 90 mm to 135 mm.In this state, the forces acting on bolt carrier in the x direction are counterrecoil spring force ( ) force and friction ( ).The equation of motion is given by: where is the total mass of moving parts; is the damping coefficient of counterrecoil spring; / is the velocity/acceleration of bolt carrier at time .The counterrecoil spring force is written as: where is the preload of counterrecoil spring; is the stiffness coefficient of counterrecoil spring; is the displacement of bolt carrier at time .
According to the experimental results, bolt carrier is shaking at this stage, and the contacting time between bolt carrier and receiver guide rail are transient.Thus, friction can be ignored, Eq. ( 2) transforms into: where = − − .According to the least square identification method, Eq. ( 5) can be derived: Based on experimental data and Eq. ( 5), it is concluded that is 0.00163 N•s/m and is 229.1 N/m.

Virtual prototype modeling and validation
The virtual prototype model is established under ADAMS.The model includes 10 rigid bodies, 6 fixed joints, 2 revolute joints, 13 contacts, 2 springs, 1 torsion spring and 12 forces (gunpowder gas force and base pressure in each cycle).The model is shown in Fig. 10.
In this model, gunpowder gas force is calculated by Eq. ( 1) in Section 2.1, and identification results in Section 2.2 are used to set the parameters of counterrecoil spring.The parametric model of slot related to the receiver guide rail on bolt carrier is used to set different clearances in different cases.
Fig. 11 shows the topological graph of bolt carrier.In order to simulate the real conditions, the joints connecting bolt carrier with bolt head or receiver are replaced by contacts, and the contact force is computed by contact method based on the impact function [4].The force between ground and bolt carrier is gunpowder gas force; the spring-damper between bolt carrier and counterrecoil spring base fixed on receiver is counterrecoil spring.
Fig. 12 shows the simulation displacement curves of the point A attached to bolt carrier.The model was simulated continuously 6 times.In each cycle, there is a maximum x-displacement and a maximum -displacement and the maximum -displacement appears after the maximum -displacement.The maximum -displacement is about 135 mm, while the values of maximum -displacement are different.By comparing Fig. 4 and Fig. 12, it can be concluded that the trends of experiment displacement and simulation displacement are identical.The simulation curves are much smoother than experiment curves due to the experimental device is vibrating while fixed well in simulation model.
Table 2 provides a comparison of statistical simulation results and statistical experiment results on kinematic parameters of bolt carrier.It is clear that the errors of and are small and less than 5 %; the errors of , and are larger but less than 14 %, however, both errors are acceptable.
In general, the virtual prototype model considered stochastic gunpowder gas force and identification counterrecoil spring parameters can simulated the motion and the stochastic kinetic characteristics of automatic mechanism well.Thus, it is a good choice to use this simulation model for further analysis of kinetic characteristics.

Influence of vertical guide rail clearance
The influence of clearance in vertical guide rail ( , the clearance in direction) is studied based on the virtual prototype model established in Section 3. The horizontal guide rail clearance ( , the clearance in direction) is 0.42 mm; is 0.3 mm, 0.5 mm, 0.7 mm, 0.9 mm, 1.1 mm and 1.3 mm, respectively.Simulations with different are calculated many times.The virtual prototype model works well in each case while studying effect on the motion.It is concluded that the range of in this paper has no significant effects on the function of automatic mechanism.is small (such as 0.5 mm), there is a maximum -displacement in each cycle.Meanwhile, the value of maximum -displacement is also increasing as increasing.When becomes large (such as 1.3 mm), the characteristic of maximum -displacement almost disappears.
Figs. 14 to 16 show statistics of bolt carrier related to .As shown in Fig. 14, generally decreases with increasing.On the contrary, the standard deviation of increases.It is concluded that the state (i.e.serious collisions, large energy consumption, great randomness) caused by large clearance will happen when bolt carrier does recoil motion with high speed.As shown in Fig. 15, when increases, generally increases but the standard deviation of decreases due to bolt carrier does counterrecoil motion driven by counterrecoil spring force and the speed increases slowly.By comparing with Fig. 14, the influence of on recoil motion is contrary to the influence on counterrecoil motion.
As shown in Fig. 16, there is no obvious correlation between the change of and the change of .However, is more stable as well as standard deviation of when is small.

Influence of horizontal guide rail clearance
The influence of clearance in horizontal guide rail is studied in this section.The vertical clearance in guide rail is 0.42 mm, is 0.42 mm, 0.62 mm, 0.82 mm, 1.02 mm and 1.22 mm, respectively.Simulations with different are calculated many times.The virtual prototype model works well in each case.It is concluded that the ranges of in this paper also have no significant effects on the function of automatic mechanism.Fig. 17 presents the statistics of maximum -displacement of the point A attached to bolt carrier.It shows that if is 0.42 mm or 0.62 mm, the maximum -displacement is about 1mm; if is larger than 0.62 mm. the maximum -displacement decreases by 10.3 %, 15.7 % and 8.9 %, respectively.It is concluded that vibration amplitude in vertical direction will effectively reduce as increasing.Figs.18 to 20 show statistics of bolt carrier related to , respectively.By observing Fig. 18 and Fig. 19, the influence of on recoil motion and counterrecoil motion is similar to the influence of .Small clearance is of benefit to recoil motion, but large clearance is of benefit to counterrecoil motion.As is shown in Fig. 20, if is less than 0.82 mm, is almost unchanged as increasing but the standard deviation of increases.if is larger than 0.82 mm, increases as increasing but the standard deviation of decreases.In a word, is small and stable when is small.

Conclusions
To study the kinetic characteristics of a typical automatic mechanism with clearances, an experimental device of reciprocating mechanism with clearances was set up; the motions of this experimental device were recorded by a high-speed camera system (HSCS).Based on experimental data, it is found that bolt carrier vibrates in the y direction and there is a maximum -displacement in each cycle due to the existence of clearances.All kinematic parameters ( , , , and ) have randomness through statistical analysis.It is indicated that , , and are approximately normally distributed due to their Shapiro-Wilk Significances are greater than 0.05.There is no obvious correlation between and other kinematic parameters except which generally increases as increasing.Based on experimental data, a coefficient of initial pressure of gas chamber was proposed and its regularities of distribution was obtained, the stiffness coefficient and damping coefficient of counterrecoil spring were identified.The virtual prototype model of this automatic mechanism was established.The kinetic characteristics of the automatic mechanism were matched well by comparing the simulation results of the virtual prototype model with stochastic parameters and identification parameters from experiments.The virtual prototype model can be used for further analysis of kinetic characteristics.
By analysis of the influence of guide rail clearances on kinetic characteristics, it is concluded that the ranges of and have no significant effects on the function of automatic mechanism.The influence of guide rail clearances on recoil motion is contrary to the influence on counterrecoil motion, small clearance is of benefit torecoil motion while large clearance is of benefit to counterrecoil motion.There is no obvious correlation between and , the increasing of causes the increasing of .However, is stable when clearance is small.YAN ZOU, CHENG XU, SHAO MIN LUO

Fig. 4 .
Fig. 4. Experiment displacement curves of the point A attached to bolt carrier ON KINETIC CHARACTERISTICS OF A RECIPROCATING MECHANISM WITH CLEARANCES.YAN ZOU, CHENG XU, SHAO MIN LUO

Fig. 9 .
Fig. 9. Diagrammatic graph of bolt carrier free recoil motion model

Fig. 13 .
Fig. 13.Simulation -displacement curves of the point A attached to bolt carrier Fig. 13 presents -displacement curves of point A attached to bolt carrier under different .It shows that whenis small (such as 0.5 mm), there is a maximum -displacement in each cycle.Meanwhile, the value of maximum -displacement is also increasing as increasing.When becomes large (such as 1.3 mm), the characteristic of maximum -displacement almost disappears.Figs.14 to 16 show statistics of bolt carrier related to .As shown in Fig.14, generally decreases with increasing.On the contrary, the standard deviation of increases.It is concluded that the state (i.e.serious collisions, large energy consumption, great randomness) caused by large clearance will happen when bolt carrier does recoil motion with high speed.

Fig. 17 .Fig. 18 .Fig. 19 .
Fig. 17.Statistics of maximum -displacement of bolt carrier identification point A ON KINETIC CHARACTERISTICS OF A RECIPROCATING MECHANISM WITH CLEARANCES.

Table 1 .
Statistical results of kinematic parameters of bolt carrier

Table 2 .
Comparison of bolt carrier kinematic parameters