# Shock Mechanism Analysis and Simulation of High-Power Hydraulic Shock Wave Simulator

Authors: Xiaoqiu Xu; Han Tao; Junwei Han
Journal: Shock and Vibration
 (2017)

---

## Abstract

The simulation of regular shock wave (e.g., half-sine) can be achieved by the traditional rubber shock simulator, but the practical high-power shock wave characterized by steep prepeak and gentle postpeak is hard to be realized by the same. To tackle this disadvantage, a novel high-power hydraulic shock wave simulator based on the live firing muzzle shock principle was proposed in the current work. The influence of the typical shock characteristic parameters on the shock force wave was investigated via both theoretical deduction and software simulation. According to the obtained data compared with the results, in fact, it can be concluded that the developed hydraulic shock wave simulator can be applied to simulate the real condition of the shocking system. Further, the similarity evaluation of shock wave simulation was achieved based on the curvature distance, and the results stated that the simulation method was reasonable and the structural optimization based on software simulation is also beneficial to the increase of efficiency. Finally, the combination of theoretical analysis and simulation for the development of artillery recoil tester is a comprehensive approach in the design and structure optimization of the recoil system.


---

## Body

## 1. Introduction

In modern shooting range construction process, the consumption was significantly increased due to the real firing practice. Four sorts of capability, that is, reliability, security, testability, and supportability, were the critical aims for the examinations [1]. Increasing development quality, shortening test cycle, and saving full-life research cost had been defined as important objectives of improving operational effectiveness of artillery equipment [2]. Since the 1950s, equivalent reliable artillery simulation test method had been initially explored at home and abroad. For example, an artillery fire shock simulation tester equipped with a hydraulic shock cylinder was developed by US researchers [3, 4]. According to the reports, this simulator was characterized by high accuracy, promoted safety performance, and pertinence, which successfully avoided the shortages as mentioned above.With the fast development of hardware and software resource, computational simulation is gradually introduced in the design of fire shock simulation tester. Yangwu innovatively reported the build of a numerical simulation model of artillery recoil system simulator with applying the classical internal ballistics and systematic dynamic related theories [5].A further optimization of the simulation model was proposed by Hang and Zhang which considered the load conditions and structure parameters of DS-II artillery recoil system [6]. Changchun et al. developed a virtual recoil tester in MSC in the ADAMS software for investigating a stereotype of ground artillery tester. The simulation of dynamic recoil process was achieved via applying homogeneous experiment design and influence study of shock parameters using Matlab [7, 8]. Yuliang et al. carried out finite element simulations and test researches of constitutive models of the rubber wave simulators and conducted optimization design of rubber wave simulators [9–13]. Jiao et al. analyzed shock wave characteristics of a hydraulic damper for shock test machine and achieved the influence of the annular gap, the diameter of the orifice pore, and the shock wave under different velocities [14]. Duym et al. carried out evaluation of shock absorber models and presented a nonexhaustive overview of shock absorber models [15]. Shams et al. analyzed hydraulic shock absorber valve behavior using CFD-FEA [16]. Jiuhong et al. presented a design method for fluid viscous dampers, derived mathematical model of viscous dampers, and carried out the shock test [17]. Hou et al. proposed a new model for nonlinear viscous fluid dampers with a simple annular orifice and analyzed its fluid dynamics behavior and shear-thinning effects [18–20]. In conclusion, the method where mass block with high velocity impacted artillery muzzle to simulate artillery firing process was proven as a feasible recoil simulation test method [21, 22], and the effects of structure parameters and shock parameters on dynamics and behavior had drawn attentions for hydraulic shock absorber as well as viscous fluid dampers. In order to verify the effectiveness of the simulation method, the similarity of time consequence was an important evaluation method; whatever the test data or the simulation data, both were seen as time consequence. The similarity measuring algorithm included time bend distance [23], slope distance [24], curvature distance [25], and radian distance [26]. Time bend distance was hard to calculate and slope distance and radian distance reflected the relationship within the adjacent time and were short of the adjacent time relevance. The curvature distance algorithm not only reflected the relationship but also expressed the relevance of the adjacent time. Wang et al. carried out credibility evaluation for guidance system dynamic simulation with curvature distance [27]. Yuliang et al. calculated accuracy evaluation of cannon dynamic recoil simulation based on the numerical and shape similarity and verified that rubber shock wave simulator could realize the cannon dynamic recoil motion [28].In shock simulation tests, the shock wave simulator played an important role in energy transferring and conversion. In addition, it could also achieve the recoil simulation of various cannons under various loading conditions with adjusting the structure parameters. For a regular wave test, a rubber shock wave simulator was usually used. However, practical shock wave was a complex shock curve; traditional rubber shock wave simulator was not fit for practical shock wave.Therefore, in the current work, a novel high-power hydraulic shock wave simulator was proposed to simulate the real shock wave. In the paper, the shock process and working principle were described and optimization design analysis of reasonable shock’s characteristic parameters was achieved. Finally, the similarity evaluation of shock wave simulation was achieved for evaluating the simulation method based on the curvature. The analysis results indicated that the novel high-power hydraulic shock wave simulator could solve the reappearance problem of practical high-power shock wave, which replaced a traditional rubber wave simulator, and also supplied theoretical references for designs of artillery fire shock simulation test.


## 2. Structure and Working Principle

### 2.1. Principle of the Live Firing Muzzle Shock Test

Breech resultant forceFpt that acted on breech axis direction in live firing was an active force that drives artillery recoil motion, which reaches 107N, and the whole process just requires 5–11 ms; this process belongs to instant strong shock. (1)mh=d2xdt2=Fpt-FR,where mh was the mass of recoil parts; x was recoil displacement; t was recoil time; Fpt was breech resultant force; FR was recoil resistance. Breech resultant force Fpt was expressed as formula (2), when the projectile moves in the bore [29].(2)Fpt=1φ1+12mϖmqsp,where φ was secondary work calculation coefficient, mϖ was explosive payload, mq was projectile mass, s was cross-sectional area of line chamber, and p was powder gas pressure, which changes with time and projectile trip. Shock of the projectile on the bore disappeared at the moment when projectile rushed out of the artillery muzzle, which led to a sudden increase of breech resultant force, a phenomenon where transient muzzle-exit Fptg jumped to Fg was formed at the beginning of after-effect period.(3)Fptg=1φ1+12mϖmqspR,Fg=1φφ1+12mϖmqspg,where pg was explosive gas’s average pressure in artillery bore at the moment of projectile out from the artillery muzzle and φ1 was secondary coefficient considering projectile rotation and friction. At explosive gas’s after-effect period, breech resultant force involves a complex phenomenon of explosive gas flowing out from an artillery muzzle. For convenient calculation, Fpt was expressed as formula (4) with exponential empirical formula.(4)Fpt=Fge-1/h,where h was time constant of breech resultant force damping speed at after-effect period and t was after-effect time at the beginning of after-effect period. In simulation test, an active force that drives artillery recoil motion was FC, generated by mass block with high velocity impacting an artillery muzzle.(5)mh=d2Xdt2=FC-FR.An alternative model of breech resultant force, where shock force FC replaced breech resultant force Fpt, played a similar explosive and promoting role for propellant burning. To be certain that shock simulation test possessed higher precision, we need to design right shock parameters and wave simulator structure forms to ensure the equivalence between shock force and breech resultant force. Therefore, for making sure that dynamic behavior of artillery fire shock simulation test was similar to live firing, similar simulation method was chosen, which could generate instant strong shock effect.


### 2.2. Shock Simulation Tester Description

Firing shock simulation test system (shown in Figure1) was a distributing measurement and control system based on industrial Ethernet and fieldbus technology, and the system was composed of six subsystems which were artillery locating subsystem, shock simulation subsystem, test control subsystem, shock locating subsystem, parameter measurement subsystem, and hydraulic power subsystem. In the shock simulation subsystem, shock simulation tester was the key component.Figure 1
Firing shock simulation test system.The shock simulation tester used in the firing shock simulation test system was shown in Figure2 for artillery firing process simulation by computer control. The tester took liquid as storage medium to simulate the shock wave based on controlled hydraulic technology. In the past shock test, the rubber shock simulator was used for simulating the shock wave, and the rubber shock simulator is composed by several rubber blankets with certain stiffness that was calculated for meeting shock wave requirements, and these rubber blankets were installed on subpanel.Figure 2
Shock simulation tester with rubber shock simulator.Controlling firing rate, firing angle and shock load simulation were realized, which were used to test reliability and durability of the big bore ground and self-propelled and tank artillery. Hydraulic subsystem drove big mass block with shock wave simulator; then the high-velocity big mass block impacted artillery muzzle and simulated the deflagration effect. The simulations made artillery shock test system generate similar recoil motion with artillery live firing.Shock and restoration processes were shown in Figure3. Shock wave simulator was accelerated by velocity generator and then impacted muzzle to simulate the required shock wave; after a shock was finished, the shock wave simulator was restated. After the above steps, a shock test was finished, repeating the above process to carry out many experiments. In shock simulation tests, the shock wave simulator was a key component which played an important role in transfer and conversion of shock energy.Figure 3
Shock and restoration process.The rubber shock simulator was usually used for the regular shock wave reappearance such as a half-sine shock wave. Additionally, practical shock force wave simulating the breech force had the characteristic of steep prepeak and gentle postpeak relatively (dotted line in Figure10); the traditional rubber wave simulator was not fit for the simulation. Therefore, for simulating the practical shock force wave and solving the distortion problem, a new idea about high-power hydraulic shock wave simulator was proposed based on hydraulic buffering technology.


### 2.3. Structure and Principle of the High-Power Hydraulic Shock Wave Simulator

Shock simulation system adopted the principle of the momentum transfer; that is, a shock wave simulator with a certain velocity transferred momentum into cannon barrel during the shock, and the large shock force and the shock acceleration were formed on cannon barrel, and the shock wave could be adjusted by the damping and rigidity of the shock wave simulator. Structure sketch of the high-power hydraulic shock wave simulator was shown in Figure4.Figure 4
Structure stretch of the high-power hydraulic shock wave simulator.The inner diameter of the fixed block was 440 mm, and its initial length was 100 mm. The inner diameter of the moving block was 310 mm, and its initial length was 80 mm; there were two throttle holes in the throttle spindle. The piston chambers were fully filled with silicone oil.The shock wave simulator was accelerated to certain velocity before shock, and then shocking the cannon barrel through the high-power hydraulic shock wave simulator produced instantaneous shock force. During the shock, the moving block moved to the right relative to the fixed block, which led to the oil pressure rise; silicone oil was squeezed to the left side of moving cylinder block through orifice plunger; then isolating piston was pushed to compress Nitrogen for the energy storage. The damping force varied with oil cavity pressure changes.When an external shock force was applied to the left of the shock wave simulator, the moving block moved toward the right. The oil in the right fixed block chamber was compressed and flew through the throttle gap into the left chamber (as the red arrow shown in Figure4). In this process, damping force was raised to balance the external shock force.


## 2.1. Principle of the Live Firing Muzzle Shock Test

Breech resultant forceFpt that acted on breech axis direction in live firing was an active force that drives artillery recoil motion, which reaches 107N, and the whole process just requires 5–11 ms; this process belongs to instant strong shock. (1)mh=d2xdt2=Fpt-FR,where mh was the mass of recoil parts; x was recoil displacement; t was recoil time; Fpt was breech resultant force; FR was recoil resistance. Breech resultant force Fpt was expressed as formula (2), when the projectile moves in the bore [29].(2)Fpt=1φ1+12mϖmqsp,where φ was secondary work calculation coefficient, mϖ was explosive payload, mq was projectile mass, s was cross-sectional area of line chamber, and p was powder gas pressure, which changes with time and projectile trip. Shock of the projectile on the bore disappeared at the moment when projectile rushed out of the artillery muzzle, which led to a sudden increase of breech resultant force, a phenomenon where transient muzzle-exit Fptg jumped to Fg was formed at the beginning of after-effect period.(3)Fptg=1φ1+12mϖmqspR,Fg=1φφ1+12mϖmqspg,where pg was explosive gas’s average pressure in artillery bore at the moment of projectile out from the artillery muzzle and φ1 was secondary coefficient considering projectile rotation and friction. At explosive gas’s after-effect period, breech resultant force involves a complex phenomenon of explosive gas flowing out from an artillery muzzle. For convenient calculation, Fpt was expressed as formula (4) with exponential empirical formula.(4)Fpt=Fge-1/h,where h was time constant of breech resultant force damping speed at after-effect period and t was after-effect time at the beginning of after-effect period. In simulation test, an active force that drives artillery recoil motion was FC, generated by mass block with high velocity impacting an artillery muzzle.(5)mh=d2Xdt2=FC-FR.An alternative model of breech resultant force, where shock force FC replaced breech resultant force Fpt, played a similar explosive and promoting role for propellant burning. To be certain that shock simulation test possessed higher precision, we need to design right shock parameters and wave simulator structure forms to ensure the equivalence between shock force and breech resultant force. Therefore, for making sure that dynamic behavior of artillery fire shock simulation test was similar to live firing, similar simulation method was chosen, which could generate instant strong shock effect.


## 2.2. Shock Simulation Tester Description

Firing shock simulation test system (shown in Figure1) was a distributing measurement and control system based on industrial Ethernet and fieldbus technology, and the system was composed of six subsystems which were artillery locating subsystem, shock simulation subsystem, test control subsystem, shock locating subsystem, parameter measurement subsystem, and hydraulic power subsystem. In the shock simulation subsystem, shock simulation tester was the key component.Figure 1
Firing shock simulation test system.The shock simulation tester used in the firing shock simulation test system was shown in Figure2 for artillery firing process simulation by computer control. The tester took liquid as storage medium to simulate the shock wave based on controlled hydraulic technology. In the past shock test, the rubber shock simulator was used for simulating the shock wave, and the rubber shock simulator is composed by several rubber blankets with certain stiffness that was calculated for meeting shock wave requirements, and these rubber blankets were installed on subpanel.Figure 2
Shock simulation tester with rubber shock simulator.Controlling firing rate, firing angle and shock load simulation were realized, which were used to test reliability and durability of the big bore ground and self-propelled and tank artillery. Hydraulic subsystem drove big mass block with shock wave simulator; then the high-velocity big mass block impacted artillery muzzle and simulated the deflagration effect. The simulations made artillery shock test system generate similar recoil motion with artillery live firing.Shock and restoration processes were shown in Figure3. Shock wave simulator was accelerated by velocity generator and then impacted muzzle to simulate the required shock wave; after a shock was finished, the shock wave simulator was restated. After the above steps, a shock test was finished, repeating the above process to carry out many experiments. In shock simulation tests, the shock wave simulator was a key component which played an important role in transfer and conversion of shock energy.Figure 3
Shock and restoration process.The rubber shock simulator was usually used for the regular shock wave reappearance such as a half-sine shock wave. Additionally, practical shock force wave simulating the breech force had the characteristic of steep prepeak and gentle postpeak relatively (dotted line in Figure10); the traditional rubber wave simulator was not fit for the simulation. Therefore, for simulating the practical shock force wave and solving the distortion problem, a new idea about high-power hydraulic shock wave simulator was proposed based on hydraulic buffering technology.


## 2.3. Structure and Principle of the High-Power Hydraulic Shock Wave Simulator

Shock simulation system adopted the principle of the momentum transfer; that is, a shock wave simulator with a certain velocity transferred momentum into cannon barrel during the shock, and the large shock force and the shock acceleration were formed on cannon barrel, and the shock wave could be adjusted by the damping and rigidity of the shock wave simulator. Structure sketch of the high-power hydraulic shock wave simulator was shown in Figure4.Figure 4
Structure stretch of the high-power hydraulic shock wave simulator.The inner diameter of the fixed block was 440 mm, and its initial length was 100 mm. The inner diameter of the moving block was 310 mm, and its initial length was 80 mm; there were two throttle holes in the throttle spindle. The piston chambers were fully filled with silicone oil.The shock wave simulator was accelerated to certain velocity before shock, and then shocking the cannon barrel through the high-power hydraulic shock wave simulator produced instantaneous shock force. During the shock, the moving block moved to the right relative to the fixed block, which led to the oil pressure rise; silicone oil was squeezed to the left side of moving cylinder block through orifice plunger; then isolating piston was pushed to compress Nitrogen for the energy storage. The damping force varied with oil cavity pressure changes.When an external shock force was applied to the left of the shock wave simulator, the moving block moved toward the right. The oil in the right fixed block chamber was compressed and flew through the throttle gap into the left chamber (as the red arrow shown in Figure4). In this process, damping force was raised to balance the external shock force.


## 3. Coupling Model Building of a High-Power Shock Wave Simulation Test System

In the high-power shock simulation test, the whole simulation process could be divided into two stages. The sketch of the whole model is shown in Figure5. In the figure, the model can be considered as two coupling systems: the left one is the recoil system and the right one is the damping system including a damper.Figure 5
Coupling model for the high-power shock simulation test.The shock was the first stage; in this process, the rubber pad located on the pounding head of a high-power hydraulic shock wave simulator collides with the canon muzzle and supplies velocity to the cannon barrel and accelerates the canon barrel including recoil buffer device and the cannon suffered shock force from the shock system and the resistance from the recoil system. In the second process, the moving block part will be braked by the damper part of the high-power hydraulic shock wave simulator.


### 3.1. Model of the Recoil System

m
1 represents the canon muzzle and it suffers the shock force FC and recoil resistance R in the axis direction; recoil resistance R includes recuperator force FFJ, recoil brake force FZT, and constant friction Fφ.(6)R=FFJ+FZT+Fφ.Recuperator force could be expressed as follows in a recoil process:(7)FFJ=50000×2.152.15-x11.3,where x1 is the displacement of m1. Recuperator force is a function related to canon recoil displacement. Recuperator force could be expressed as follows in recoil process.(8)FZT=1800×150+axax2+130dx1dt2,where ax is equivalent area of recoil brake ventage and dx1/dt is canon recoil velocity.(9)ax=23-π4dx2,where dx is throttling bar diameter.Although the corresponding throttling bar diameters under different recoil displacement are different, throttling bar diameter remains constant.(10)FZT=1580dx1dt.Recoil brake force is a function related to canon recoil velocity. Constant friction Fφ could be expressed as follows:(11)Fφ=f1mgcos⁡φ-mgsin⁡φ+F0.where f1 is the friction coefficient between canon recoil and cradle, φ is canon firing angle, and F0 is the friction generated by plug component of counter-recoil mechanism, which is composed by two parts.(12)F0=F01+F02,where F01 is the friction generated by plug component in recuperator and F02 is the friction generated by plug component in the recoil brake. When canon firing angle remains constant, constant friction could be seen as constant value. Fφ=10500N when canon firing angle φ equals 0°.In conclusion, the force balance equation of the shocked cannon is expressed as follows when firing angle is 0°:(13)m1d2x1dt2=FC-FFJ-FZT-Fφ=FC-50000×2.152.15-x11.3-1580dx1dt-10500,where FC is the shock force acting on the canon muzzle and d2x1/dt2 is recoil acceleration.


### 3.2. Model of the Damper System

For the shock wave simulator, considering oil compressibility, formula (14) is expressed according to flow continuity function in the damper system.(14)Qjl+Qys+Qxl=Qz,where Qjl is the throttled flow, Qys is the compressed flow, Qxl is the leakage flow, and Qz is the total flow.(15)Qys=A1L-A1x2-x3βdP1dt,where L is the original length, β is the bulk elastic modulus of oil, x2 is the displacement of the moving block, x3 is the displacement of the fixed block, A1 is the area of the right end surface of the moving block, and P1 is the pressure imposed on the left end surface.(16)Qz=dx2dt-dx3dtA1,where dx2/dt is the velocity of the moving block and dx3/dt is the velocity of the fixed block.Throttling groove adopts slender hole; flow state is laminar flow because the flow is not free when viscous fluid flows through the throttling groove. The flow of slender hole is related to the viscosity of the fluid. The viscosity of oil changes with its temperature.When the areaA of throttling hole is πd2/4, the throttling flow could be expressed as follows:(17)Qjl=d232μlAΔP=πd4ΔP128μl=πd4P1-P2128μl,where μ is the dynamic viscous, l is the through-flow length, and d is the hole diameter. Formula (18) is achieved by plugging formulas (15), (16), and (17) into formula (14).(18)πd4P1-P2128μl+A1Ymax-A1x2-x3βdP1dt=dx2dt-dx3dtA1.The ideal gas adiabatic compression equation is as follows:(19)P2P0=V0V0-x2A0k,where P0 is the initial pressure in nitrogen gas cavity, V0 is the initial volume in nitrogen gas cavity, A0 is the inertial effective working area in the nitrogen gas cavity, and k is the adiabatic coefficient.


## 3.1. Model of the Recoil System

m
1 represents the canon muzzle and it suffers the shock force FC and recoil resistance R in the axis direction; recoil resistance R includes recuperator force FFJ, recoil brake force FZT, and constant friction Fφ.(6)R=FFJ+FZT+Fφ.Recuperator force could be expressed as follows in a recoil process:(7)FFJ=50000×2.152.15-x11.3,where x1 is the displacement of m1. Recuperator force is a function related to canon recoil displacement. Recuperator force could be expressed as follows in recoil process.(8)FZT=1800×150+axax2+130dx1dt2,where ax is equivalent area of recoil brake ventage and dx1/dt is canon recoil velocity.(9)ax=23-π4dx2,where dx is throttling bar diameter.Although the corresponding throttling bar diameters under different recoil displacement are different, throttling bar diameter remains constant.(10)FZT=1580dx1dt.Recoil brake force is a function related to canon recoil velocity. Constant friction Fφ could be expressed as follows:(11)Fφ=f1mgcos⁡φ-mgsin⁡φ+F0.where f1 is the friction coefficient between canon recoil and cradle, φ is canon firing angle, and F0 is the friction generated by plug component of counter-recoil mechanism, which is composed by two parts.(12)F0=F01+F02,where F01 is the friction generated by plug component in recuperator and F02 is the friction generated by plug component in the recoil brake. When canon firing angle remains constant, constant friction could be seen as constant value. Fφ=10500N when canon firing angle φ equals 0°.In conclusion, the force balance equation of the shocked cannon is expressed as follows when firing angle is 0°:(13)m1d2x1dt2=FC-FFJ-FZT-Fφ=FC-50000×2.152.15-x11.3-1580dx1dt-10500,where FC is the shock force acting on the canon muzzle and d2x1/dt2 is recoil acceleration.


## 3.2. Model of the Damper System

For the shock wave simulator, considering oil compressibility, formula (14) is expressed according to flow continuity function in the damper system.(14)Qjl+Qys+Qxl=Qz,where Qjl is the throttled flow, Qys is the compressed flow, Qxl is the leakage flow, and Qz is the total flow.(15)Qys=A1L-A1x2-x3βdP1dt,where L is the original length, β is the bulk elastic modulus of oil, x2 is the displacement of the moving block, x3 is the displacement of the fixed block, A1 is the area of the right end surface of the moving block, and P1 is the pressure imposed on the left end surface.(16)Qz=dx2dt-dx3dtA1,where dx2/dt is the velocity of the moving block and dx3/dt is the velocity of the fixed block.Throttling groove adopts slender hole; flow state is laminar flow because the flow is not free when viscous fluid flows through the throttling groove. The flow of slender hole is related to the viscosity of the fluid. The viscosity of oil changes with its temperature.When the areaA of throttling hole is πd2/4, the throttling flow could be expressed as follows:(17)Qjl=d232μlAΔP=πd4ΔP128μl=πd4P1-P2128μl,where μ is the dynamic viscous, l is the through-flow length, and d is the hole diameter. Formula (18) is achieved by plugging formulas (15), (16), and (17) into formula (14).(18)πd4P1-P2128μl+A1Ymax-A1x2-x3βdP1dt=dx2dt-dx3dtA1.The ideal gas adiabatic compression equation is as follows:(19)P2P0=V0V0-x2A0k,where P0 is the initial pressure in nitrogen gas cavity, V0 is the initial volume in nitrogen gas cavity, A0 is the inertial effective working area in the nitrogen gas cavity, and k is the adiabatic coefficient.


## 4. Numerical Simulation and Result Analysis

Using Matlab/Simulink software, firstly, import into the characteristic parametersρ,μ,β,k,fmc the test parameters M,m,P0,V0,L and the controlled parameters l,d as well as the initial velocity v0; then solve the above motion functions according to the fourth-order Runge-Kutta method adopting variable step arithmetic. Shock force curve, acceleration curve, velocity curve, and displacement were received in the high pressure cavity. Finally, model accuracy was verified by the comparison of simulation test and live firing test, and parameters were optimized and corrected. The optimized model could use debugged shock wave simulator for simulating various shock force waves. The specific method was that practical shock force curve was imported as a known condition, throttling area curve was deduced, and required shock force curve was simulated by adjusting the throttling area.From the building process of mechanical model, the factors affecting shock wave characteristics were mainly divided into three classes: the first class was test parameters includingρ,μ,β,k, and fmc; the second class was characteristic parameters including M,m,P0,V0, and L, and the third class was controlled parameters including d and v0.The influences of typical parameters on the shock force wave were analyzed, and these typical parameters included initial velocityv0, dynamic viscosity μ, the bulk elastic modulus β of oil, and the diameter d of the throttling hole.


### 4.1. The Influence of Initial Velocityv0 on the Shock Behavior

The influence of initial velocityv0 on the shock behavior in controlled parameters could be inferred in that the increase of initial velocity would increase the total flow Qz and then affect the shock force according to (17) and (6); the shock force waves were shown in Figure 6 under different initial velocities. In comparison, it was obvious that the shape trend of shock force wave had similar values under different initial velocity v0, and the peak value increased with the initial velocity increase, which accorded with the theory analysis as well as the mechanical performances of the shock wave simulator.Figure 6
Shock force curve of the high-power hydraulic shock wave simulator under various initial velocityv0.


### 4.2. The Influence of Dynamic Viscosityμ on the Shock Behavior

In characteristic parameters, the dynamic viscosityμ affected the throttled flow Qjl and the throttled flow Qjl increased with μ declining according to function (18). Hence, shock force wave was simulated under various dynamic viscosities μ for describing the influence of the dynamic viscosity μ on the shock behavior. The curves were shown in Figure 8, and it was obvious that the shape of shock force wave curves changed obviously in the postpeak period under different dynamic viscosity μ in Figure 7; the shock force decreased seriously with dynamic viscosity μ declining and the curve’s shape fluctuated seriously with dynamic viscosity μ declining.Figure 7
Shock force curve of the high-power hydraulic shock wave simulator under various dynamic viscosityμ.Figure 8
Shock force curve of the high-power hydraulic shock wave simulator under various bulk elastic modulusβ of oil.


### 4.3. The Influence of the Bulk Elastic Modulusβ of Oil on the Shock Behavior

According to the building process of mechanical model, the bulk elastic modulusβ of oil in characteristic parameters affected the compressed flow Qys, and the compressed flow Qys increased with β declining according to function (16). Hence, shock force wave was simulated under bulk elastic modulus β of oil for describing the influence of the bulk elastic modulus β of oil on shock behavior. The comparison curves were shown in Figure 8, and it was obvious that the shape of shock force wave curves changed gently under bulk elastic modulus β of oil, the shock force decreased, and the impulse increased with the decline of bulk elastic modulus β of oil in Figure 8.


### 4.4. The Influence of the Diameterd of the Throttling Hole on the Shock Behavior

In controlled parameters, the diameterd of the throttling hole directly affected the throttled flow Qjl and the throttled flow Qjl increased with the increase of the diameter d of the throttling hole according to function (18). For describing the influence of the diameter d of the throttling hole on the shock behavior, the shock force wave was simulated under varying diameter d of the throttling hole. The curves were shown in Figure 9, and it was obvious that the shape of shock force wave curves changed obviously in the postpeak period, the shock force decreased seriously with the increase of the diameter d of the throttling hole, and the wave’s shape fluctuated seriously.Figure 9
Shock force curve of the high-power hydraulic shock wave simulator under varying throttle diameterd.Figure 10
Shock force curve comparison generated by the high-power hydraulic shock wave simulator and live firing test, respectively.By the optimization design analysis of typical shock parameters, a set of reasonable shock parameters were received as shown in Table1.Table 1
Main shock parameter list.
Main shock parameter
M (kg)
m (kg)
v
0 (m/s)
μ (m2/s)
β (MPa)
l (m)
L (m)
Value
3000
2540
15
0.5
800
0.08
0.27The throttle diameterd curve of the optimized model was debugged shock wave simulator for simulating various shock force waves. The specific method was that practical shock force curve was imported as a known condition, throttling diameter d curve was deduced, and the curve required subsection discretization in practical engineering design. Required shock force curve was simulated by the optimization of throttling diameter d curve.The shock force wave was simulated adopting the optimized shock parameters in Figure10.The simulation curve was similar to the test curve under the live firing. It was indicated that the above mathematical model had good precision by optimization and correction of numerical simulation tests under various working conditions, and the high-power hydraulic shock wave simulator adopting the optimized shock parameters realized the shock wave reappearance of the shock force on artillery muzzle in the live firing test.


## 4.1. The Influence of Initial Velocityv0 on the Shock Behavior

The influence of initial velocityv0 on the shock behavior in controlled parameters could be inferred in that the increase of initial velocity would increase the total flow Qz and then affect the shock force according to (17) and (6); the shock force waves were shown in Figure 6 under different initial velocities. In comparison, it was obvious that the shape trend of shock force wave had similar values under different initial velocity v0, and the peak value increased with the initial velocity increase, which accorded with the theory analysis as well as the mechanical performances of the shock wave simulator.Figure 6
Shock force curve of the high-power hydraulic shock wave simulator under various initial velocityv0.


## 4.2. The Influence of Dynamic Viscosityμ on the Shock Behavior

In characteristic parameters, the dynamic viscosityμ affected the throttled flow Qjl and the throttled flow Qjl increased with μ declining according to function (18). Hence, shock force wave was simulated under various dynamic viscosities μ for describing the influence of the dynamic viscosity μ on the shock behavior. The curves were shown in Figure 8, and it was obvious that the shape of shock force wave curves changed obviously in the postpeak period under different dynamic viscosity μ in Figure 7; the shock force decreased seriously with dynamic viscosity μ declining and the curve’s shape fluctuated seriously with dynamic viscosity μ declining.Figure 7
Shock force curve of the high-power hydraulic shock wave simulator under various dynamic viscosityμ.Figure 8
Shock force curve of the high-power hydraulic shock wave simulator under various bulk elastic modulusβ of oil.


## 4.3. The Influence of the Bulk Elastic Modulusβ of Oil on the Shock Behavior

According to the building process of mechanical model, the bulk elastic modulusβ of oil in characteristic parameters affected the compressed flow Qys, and the compressed flow Qys increased with β declining according to function (16). Hence, shock force wave was simulated under bulk elastic modulus β of oil for describing the influence of the bulk elastic modulus β of oil on shock behavior. The comparison curves were shown in Figure 8, and it was obvious that the shape of shock force wave curves changed gently under bulk elastic modulus β of oil, the shock force decreased, and the impulse increased with the decline of bulk elastic modulus β of oil in Figure 8.


## 4.4. The Influence of the Diameterd of the Throttling Hole on the Shock Behavior

In controlled parameters, the diameterd of the throttling hole directly affected the throttled flow Qjl and the throttled flow Qjl increased with the increase of the diameter d of the throttling hole according to function (18). For describing the influence of the diameter d of the throttling hole on the shock behavior, the shock force wave was simulated under varying diameter d of the throttling hole. The curves were shown in Figure 9, and it was obvious that the shape of shock force wave curves changed obviously in the postpeak period, the shock force decreased seriously with the increase of the diameter d of the throttling hole, and the wave’s shape fluctuated seriously.Figure 9
Shock force curve of the high-power hydraulic shock wave simulator under varying throttle diameterd.Figure 10
Shock force curve comparison generated by the high-power hydraulic shock wave simulator and live firing test, respectively.By the optimization design analysis of typical shock parameters, a set of reasonable shock parameters were received as shown in Table1.Table 1
Main shock parameter list.
Main shock parameter
M (kg)
m (kg)
v
0 (m/s)
μ (m2/s)
β (MPa)
l (m)
L (m)
Value
3000
2540
15
0.5
800
0.08
0.27The throttle diameterd curve of the optimized model was debugged shock wave simulator for simulating various shock force waves. The specific method was that practical shock force curve was imported as a known condition, throttling diameter d curve was deduced, and the curve required subsection discretization in practical engineering design. Required shock force curve was simulated by the optimization of throttling diameter d curve.The shock force wave was simulated adopting the optimized shock parameters in Figure10.The simulation curve was similar to the test curve under the live firing. It was indicated that the above mathematical model had good precision by optimization and correction of numerical simulation tests under various working conditions, and the high-power hydraulic shock wave simulator adopting the optimized shock parameters realized the shock wave reappearance of the shock force on artillery muzzle in the live firing test.


## 5. Similarity Evaluation

Using the above data, similarity evaluation was carried out for the simulation method based on the numerical similarity and shape similarity.


### 5.1. Numerical Similarity

Assume that the test data consequence wasDT=(dt(1),dt(2),…,dt(n)) and the simulation consequence was DS=(ds(1),ds(2),…,ds(n)). Weighted mean of the relative error in the two time sequences was used for expressing numerical similarity. The calculation method was as follows.Whendt(i) and ds(i)(i=1,2,…,n) were both not zero, numerical similarity of each point s1(i) was expressed as follows:(20)s1i=1-dti-dsimax⁡dti,dsidtidsi>00dtidsi<0.When at least one parameter was zero in dt(i) and ds(i), numerical similarity of each point s1(i) was as follows:(21)s1i=0dti=0ordsi=01dti=0,dsi=0.Take the ratio of each point’s amplitude and all sample points’ sum as the weight γi of the point.(22)γi=dti∑i=1ndti.The total numerical similarity S1 of the two time sequences was shown as follows:(23)S1=∑i=1ns1iγi,i=1,2,…,n.


### 5.2. Shape Similarity

The shape similarity was calculated according to the curvature distance between the two time sequences [27]; the calculating process was as follows.Firstly, primitive sequenceX(0)=[x(0)(1),x(0)(2),…,x(0)(n)] was translated into nonnegative sequence X(1), and make sure that the sequence shape was constant.(24)x1i=x0imin⁡X0≥0x0i-min⁡X0min⁡X0<0.For improving the precision and decreasing the calculation amount, the time sequence needs sector-by-sector solution.X
(
1
)
=
[
X
1
(
1
)
,
X
2
(
2
)
,
…
,
X
m
(
1
)
] and Xj(1)=[xj(1)(1),xj(1)(2),…,xj(1)(lj)], where Xj(1) was the jth nonnegative subsequence (j=1,2,…,m), m was the number of the subsequences, and lj was the length of jth subsequence.Nonnegative subsequence was generated according to AGO (Accumulated Generating Operation), and the consequence presents approximate exponential increase law, which decreased the randomness, possessed statistical significance, and kept the relevance of primitive consequence. Nonnegative subsequenceX(1) carried out accumulation calculation, generating the consequence X(2)=[X1(2),X2(2),…,Xm(2)]. Take subsequence Xj(2) as example.(25)xj2p=∑a=1pxj1a,j=1,2,…,m,where xj(2)(p) was the pth data in jth generated subsequence.We can getxj(2)(1)=xj(1)(1). Time consequence X(2) carried out subtraction calculation; original nonnegative subsequence X(1) could be achieved as follows:(26)xj1p=xj2p-xj2p-1p=1,2,…,lj.Consequences X(1) and X(2) could be converted into one another, and X(2) reflected change law in consequence X(1), so we can build the model of X(2) to analyze X(1).For increasing the fitting model accuracy of time consequence, the subsequenceXj(2) model was built.(27)xj2λ=c1eγλ+c2λ3+c3λ2+c4λ+c5,where ci (i=1,2,…,5) and γ were model parameters; λ=1,2,…,lj. The fitting model could represent the exponent, the parabola, and the third power curve according to the value of model parameters ci (i=1,2,…,5) and γ, which is the precise expression of the consequence.Assume that the curvature of each section in time consequence wasK=[K1,K2,…,Km]; the curvature was Kj=[kj(1),kj(2),…,kj(lj)] of each subsequence.(28)kjλ=xj2′′λ1+xj2′λ23/2λ=1,2,…,lj.Assume that the curvatures of the two time consequence wereK=[K1,K2,…,Km] and K-=[K-1,K-2,…,K-m]; taking subsequences Kj and K-j as example, the shape similarity was calculated: Kj=[kj(1),kj(2),…,kj(lj)] and K-j=[k-j(1),k-j(2),…,k-j(lj)].The curvature distance of subsequencesKj and K-j was as follows:(29)dj=1lj∑p=1ljkjp-k-jp.Assume that curvature distance threshold value of the two subsequences is thDC. When the curvature distance was less than threshold value, the two subsequences were similar, while when the curvature distance was greater than threshold value thDC, the two subsequences were not similar, where threshold value thDC was related to curvature of each point in each subsequence. Use the ratio of all the similar subsequences’ length and the consequence total length as measurement of shape similarity of consequence.(30)S2=∑j=1nLjLX,where S2 was the shape similarity of time consequences X and Y, L(X) was the total length of time consequence X, L(j) was the jth similar subsequence of X and Y, and n was the number of similar subsequences of X and Y.


### 5.3. Integrated Similarity of Time Consequence

Numerical similarity and shape similarity were, respectively, calculated based on the two time consequences. Based on the multiattribute integrated evaluation theory, we use multiplication relationship to obtain comprehensive similarity.So the comprehensive similarity ofS was as follows between live firing and simulation firing. (31)S=S1S2,where S1 and S2 were, respectively, numerical similarity and shape similarity.


### 5.4. Shock Wave Simulation Evaluation

In cannon recoil motion process, recoil time consequence was divided into recoil section and counter-recoil section. Recoil section was from original point to displacement peak point; counter-recoil section was from displacement peak point to the end point. In recoil section, cannon motion velocity was high and had serious vibration and the maximum velocity existed in the section; cannon motion velocity was low and had small vibration. The weights of recoil section and counter-recoil section were, respectively, 0.67 and 0.33.The two long subsequences were divided into multiple short subsequences. The length of each short subsequence was 8; numerical similarity and shape similarity of each short subsequence were calculated by the above method; then the numerical similarity and shape similarity of each long subsequence and the total consequence were calculated. Using the data in simulation, the integrated similarity of the shock wave was calculated as in Table2.Table 2
Similarity of shock wave simulation.
Subsequence 1
Subsequence 2
Total consequence
Numerical similarity
0.95
0.78
0.894
Shape similarity
0.92
0.73
0.857The evaluation result indicated that the high-power hydraulic shock wave could simulate the dynamic process of live firing accurately. Therefore, the optimization of shock characteristic parameters could improve the simulation accuracy.


## 5.1. Numerical Similarity

Assume that the test data consequence wasDT=(dt(1),dt(2),…,dt(n)) and the simulation consequence was DS=(ds(1),ds(2),…,ds(n)). Weighted mean of the relative error in the two time sequences was used for expressing numerical similarity. The calculation method was as follows.Whendt(i) and ds(i)(i=1,2,…,n) were both not zero, numerical similarity of each point s1(i) was expressed as follows:(20)s1i=1-dti-dsimax⁡dti,dsidtidsi>00dtidsi<0.When at least one parameter was zero in dt(i) and ds(i), numerical similarity of each point s1(i) was as follows:(21)s1i=0dti=0ordsi=01dti=0,dsi=0.Take the ratio of each point’s amplitude and all sample points’ sum as the weight γi of the point.(22)γi=dti∑i=1ndti.The total numerical similarity S1 of the two time sequences was shown as follows:(23)S1=∑i=1ns1iγi,i=1,2,…,n.


## 5.2. Shape Similarity

The shape similarity was calculated according to the curvature distance between the two time sequences [27]; the calculating process was as follows.Firstly, primitive sequenceX(0)=[x(0)(1),x(0)(2),…,x(0)(n)] was translated into nonnegative sequence X(1), and make sure that the sequence shape was constant.(24)x1i=x0imin⁡X0≥0x0i-min⁡X0min⁡X0<0.For improving the precision and decreasing the calculation amount, the time sequence needs sector-by-sector solution.X
(
1
)
=
[
X
1
(
1
)
,
X
2
(
2
)
,
…
,
X
m
(
1
)
] and Xj(1)=[xj(1)(1),xj(1)(2),…,xj(1)(lj)], where Xj(1) was the jth nonnegative subsequence (j=1,2,…,m), m was the number of the subsequences, and lj was the length of jth subsequence.Nonnegative subsequence was generated according to AGO (Accumulated Generating Operation), and the consequence presents approximate exponential increase law, which decreased the randomness, possessed statistical significance, and kept the relevance of primitive consequence. Nonnegative subsequenceX(1) carried out accumulation calculation, generating the consequence X(2)=[X1(2),X2(2),…,Xm(2)]. Take subsequence Xj(2) as example.(25)xj2p=∑a=1pxj1a,j=1,2,…,m,where xj(2)(p) was the pth data in jth generated subsequence.We can getxj(2)(1)=xj(1)(1). Time consequence X(2) carried out subtraction calculation; original nonnegative subsequence X(1) could be achieved as follows:(26)xj1p=xj2p-xj2p-1p=1,2,…,lj.Consequences X(1) and X(2) could be converted into one another, and X(2) reflected change law in consequence X(1), so we can build the model of X(2) to analyze X(1).For increasing the fitting model accuracy of time consequence, the subsequenceXj(2) model was built.(27)xj2λ=c1eγλ+c2λ3+c3λ2+c4λ+c5,where ci (i=1,2,…,5) and γ were model parameters; λ=1,2,…,lj. The fitting model could represent the exponent, the parabola, and the third power curve according to the value of model parameters ci (i=1,2,…,5) and γ, which is the precise expression of the consequence.Assume that the curvature of each section in time consequence wasK=[K1,K2,…,Km]; the curvature was Kj=[kj(1),kj(2),…,kj(lj)] of each subsequence.(28)kjλ=xj2′′λ1+xj2′λ23/2λ=1,2,…,lj.Assume that the curvatures of the two time consequence wereK=[K1,K2,…,Km] and K-=[K-1,K-2,…,K-m]; taking subsequences Kj and K-j as example, the shape similarity was calculated: Kj=[kj(1),kj(2),…,kj(lj)] and K-j=[k-j(1),k-j(2),…,k-j(lj)].The curvature distance of subsequencesKj and K-j was as follows:(29)dj=1lj∑p=1ljkjp-k-jp.Assume that curvature distance threshold value of the two subsequences is thDC. When the curvature distance was less than threshold value, the two subsequences were similar, while when the curvature distance was greater than threshold value thDC, the two subsequences were not similar, where threshold value thDC was related to curvature of each point in each subsequence. Use the ratio of all the similar subsequences’ length and the consequence total length as measurement of shape similarity of consequence.(30)S2=∑j=1nLjLX,where S2 was the shape similarity of time consequences X and Y, L(X) was the total length of time consequence X, L(j) was the jth similar subsequence of X and Y, and n was the number of similar subsequences of X and Y.


## 5.3. Integrated Similarity of Time Consequence

Numerical similarity and shape similarity were, respectively, calculated based on the two time consequences. Based on the multiattribute integrated evaluation theory, we use multiplication relationship to obtain comprehensive similarity.So the comprehensive similarity ofS was as follows between live firing and simulation firing. (31)S=S1S2,where S1 and S2 were, respectively, numerical similarity and shape similarity.


## 5.4. Shock Wave Simulation Evaluation

In cannon recoil motion process, recoil time consequence was divided into recoil section and counter-recoil section. Recoil section was from original point to displacement peak point; counter-recoil section was from displacement peak point to the end point. In recoil section, cannon motion velocity was high and had serious vibration and the maximum velocity existed in the section; cannon motion velocity was low and had small vibration. The weights of recoil section and counter-recoil section were, respectively, 0.67 and 0.33.The two long subsequences were divided into multiple short subsequences. The length of each short subsequence was 8; numerical similarity and shape similarity of each short subsequence were calculated by the above method; then the numerical similarity and shape similarity of each long subsequence and the total consequence were calculated. Using the data in simulation, the integrated similarity of the shock wave was calculated as in Table2.Table 2
Similarity of shock wave simulation.
Subsequence 1
Subsequence 2
Total consequence
Numerical similarity
0.95
0.78
0.894
Shape similarity
0.92
0.73
0.857The evaluation result indicated that the high-power hydraulic shock wave could simulate the dynamic process of live firing accurately. Therefore, the optimization of shock characteristic parameters could improve the simulation accuracy.


## 6. Conclusions

Theoretical analysis and numerical simulation result analysis demonstrated that the high-power hydraulic shock wave simulator could be adopted for simulating the shock force generated on the artillery muzzle; the similarity evaluation of shock wave simulation indicated that high similarity existed between the shock force in simulation test and the practical shock force in live firing test. In other words, the simulation method was reasonable. Finally, the analysis supplied references for the utility of artillery fire simulation technology. The novel high-power hydraulic shock wave simulator replaced a traditional rubber wave simulator to realize practical high-power shock wave reappearance, which also supplied references for the artillery dynamic recoil simulation test.



---

*Source: 4361256-2017-08-02.xml*