CHAPTER 4 SYSTEM DYNAMIC MODELING
4.3 Dynamic Model Creation
4.3.4 Module Combination
Combine both input and output parameters of modules created above, a system model of recoil mechanism can be obtained. The system includes modules of the breech force, the hydraulic braking force, the recuperator force, the weight force, the sliding track friction, and the packing friction. The combined system is shown in Figure 4.3-11. The main logic of module combination is simple and its procedure can be shown in Figure 4.3-12.
Figure 4.3-11 Combined system model of the recoil mechanism
Figure 4.3-12 Flowchart of modules combination
CHAPTER 5
DYNAMIC SIMULATION AND OPTIMIZATION
5.1 Introduction
In this chapter, the dynamic models of individual components and overall system created in CHPATER 4 will be simulated and analyzed. Because there is no experiment data available at this moment, the simulation results are verified with previous studies and qualitative analyses. The assumptions proposed in previous sections are confirmed again to make sure the simulation results will be reasonable. The effects of individual parameters on the recoil mechanism are discussed in section 5.2, and optimization of the entire model is introduced in section 5.3.
Design objective of the recoil mechanism is to decrease the maximum recoil length.
After the main forces, the breech force, the hydraulic braking force, and the recuperator force, are combined, the recoil length is expected to decrease during the recoil time within
0.14sec.
5.2 Dynamic Simulation and Results
5.2.1 Simulation Assumptions
In this study, there are some assumptions supposed. Before simulation, these assumptions have to be defined. The detail descriptions on these assumptions are mentioned in section 3.1 and 3.5.5. Most important of all, the fluids including oil and air are assumed to be incompressible.
5.2.2 Simulation of Recoil Mechanism
The relations of breech force and projectile velocity versus projectile travel are shown in Figure 5.2-1 and Figure 5.2-2. In bore period, the time only retains . The input value of breech force versus time is shown in
0.0123sec
Figure 5.2-3. When the projectile travel is up to , the projectile exits the muzzle. At this moment, the breech force still exists. After exiting the muzzle, the force decreases to zero gradually. Because there is no muzzle brake in this model, the breech force is not changed from positive to negative value as the projectile exits the muzzle.
200in
Simulation result of the recoil mechanism model is shown in Figure 5.2-4. The relation is between recoil acceleration and time. Because no muzzle brake exists, the moment of negative recoil acceleration occurs later than the system with a muzzle brake. Around , the recoil acceleration is changed to negative. Thus the recoil velocity will decrease at this moment.
0.0220sec
The recoil velocity is the integration of the recoil acceleration relative to the time, as shown in Figure 5.2-5. The maximum value happens about 0.0220 . This time is the same that the recoil acceleration is changed from positive to negative. It means that the breech force is smaller than original value. After that, the recoil acceleration is changed to negative and velocity decrease gradually. Finally, the recoil velocity becomes to zero at . The recoil travel is finished, and the counterrecoil travel will begin.
sec
0.1160sec
Similarly, the recoil length can be obtained by the integration of the recoil length, as shown in Figure 5.2-6. This result is for long travel simulation. The maximum displacement of recoil length happens at 0.1160sec. At that time, the recoil velocity is zero, and the
the counterrecoil travel starts.
Figure 5.2-1 Breech force versus projectile travel
0 50 100 150 200 250 300
Figure 5.2-2 Projectile velocity versus projectile travel
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Figure 5.2-3 Breech force versus time
0 0.02 0.04 0.06 0.08 0.1 0.12
0 0.02 0.04 0.06 0.08 0.1 0.12
Figure 5.2-5 Recoil velocity versus time
0 0.02 0.04 0.06 0.08 0.1 0.12
Figure 5.2-6 Recoil length versus time
5.2.3 Leakage Area
Under the original design, the relation between orifice area and recoil length is shown in Figure 5.2-7. The simulation in section 5.2.3 is made according to the original values.
However, the orifice area is required to add the leakage area, as shown in Figure 5.2-7, during the action of the recoil mechanism. Therefore, the actual orifice area will be used here.
The existence of the leakage area is due to the manufacturing tolerance. When the gun stops, the recoil piston is hermetically sealed with orifice. But because of the tolerance shown in Figure 5.2-8, the leakage affect disappears after the backward movement of the recoil piston. In general design experience, the leakage area is about , and will be constant in all the recoil travel.
0.5in2
Figure 5.2-8 Diagram of leakage area
In the previous simulation, the orifice area is zero at start position and maximum displacement. But it is required to add the leakage area, and the minimum area is . In other words, if the area is smaller than , it is set to be , as shown in
0.5in2
0.5in2 0.5in2 Figure
4.3-8.
0 5 10 15 20 25 30 35 40
0 0.5 1 1.5 2 2.5 3 3.5
Recoil Length(in) Orifice Area(in2 )
Figure 5.2-9 Orifice area versus recoil length
Because the recoil velocity is positive proportion to the orifice area, the initial recoil velocity will raise after adding leakage area if the breech force keeps constant. At the same time, the orifice area is when recoil travel finished. It causes that the recoil length exceeds the original value. Besides, the hydraulic braking force is direct proportional to the square of recoil velocity. Hence, the addition of recoil velocity makes the hydraulic braking force increase.
0.5in2
After adding the leakage area, the initial values of the recoil acceleration, the recoil velocity, and the recoil length are increased. Figure 5.2-10 shows that the recoil acceleration increases after considering leakage area. And when the recoil travel is ending, there exists a large recoil acceleration making the velocity to zero.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Figure 5.2-10 Recoil acceleration versus time at different leakage area
As a result of the change of the recoil acceleration, the recoil velocity also changes.
The recoil velocity is direct proportional to the orifice area. And the hydraulic braking force is also direct proportional to the square of the recoil velocity. Thus, the total braking force increases. Because the orifice area cannot drop into zero when the recoil travel finishes, the moment of the recoil velocity dropping into zero is extended to 0.1220sec Figure 5.2-11 is . the comparison of recoil velocity when leakage area exists or not.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Figure 5.2-11 Recoil velocity versus time at different leakage area
Similarly, the recoil length increases because of the existence of the leakage area.
However, the hydraulic braking force is an inverse proportion to the orifice area. Therefore, if the orifice area is less than , it will set to be in order that the hydraulic braking force will decrease rapidly. The recoil motion can’t stop at original maximum position. After joining leakage area, the time of the maximum displacement increases to
0.5in2 0.5in2
0.1220sec , and displacement increases to 37.7573in.
Figure 5.2-12 Recoil length versus time at different leakage area
5.2.4 Model Adequacy Checking
In this section, the simulation model is required to be confirmed that it is reasonable.
Because there is no experimental data to support the simulation results, the model will be confirmed by physical analysis.
First, if the charge weight increases (more powder), the generated energy also increases. It causes bigger forward force of the projectile, and the breech force also becomes bigger relatively. Therefore, the recoil length will increase because of the bigger breech force. The simulation model shows the result in Figure 5.2-13 and Figure 5.2-14.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Figure 5.2-13 Breech force versus time at different charge weight
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Figure 5.2-14 Recoil length versus time at different charge weight
Second, the energy generated from the charge weight will push the projectile moving forward. When projectile exits the muzzle, there is a projectile velocity generating. It calls muzzle velocity of projectile. If muzzle velocity of projectile is faster, it means the forward force is bigger. And it also increases breech force. Therefore, the recoil length will increase because of the bigger breech force. Figure 5.2-15 and Figure 5.2-16 show this phenomenon.
The above mentioned is a simple principle to confirm the model adequacy, and the property of each parameter can be explained by physical phenomenon. The tendency is reasonable and correct. Then, the optimization method can be performed in the following section.
Breech Force(pound) Increasing muzzle velocity of the projectile Original muzzle velocity of the projectile
Figure 5.2-15 Breech force versus time at different muzzle velocity
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Increasing muzzle velocity of the projectile Original muzzle velocity of the projectile
Figure 5.2-16 Recoil length versus time at different muzzle velocity
5.3 Optimization
In this section, the recoil mechanism is optimized to get a shorter recoil length. In the following subsections, cost function, design variables, and constraints are defined according to the foregoing objective. Finally, optimization is executed.
5.3.1 Sensitivity Analysis
The recoil length will change when related variables are altered. The benefit of sensitivity analysis can investigate how the variables affect the recoil length, and Figure 5.3-1 and Figure 5.3-2 show the results. In Figure 5.3-1 and Figure 5.3-2, the percentage of variable change is from 0% to 100% in the X-axis and the change of maximum recoil
length is in Y-axis. Eight geometrical parameters are investigated in this sensitivity analysis.
Originally, there are many parameters in this mechanism. But maximum bore pressure, muzzle velocity of the projectile, and charge weight force are determined by specific kind of ammunition. Therefore, these parameters seldom change. And the recuperator usually uses the nitrogen gas, and its polytropic exponent is fixed. Because of these reasons, there are only eight parameters suitable to do sensitivity analysis to decide the more influenced parameters on the maximum recoil length.
Each parameter has its reasonable range, and the comparative standard regards as the same percentage of the variable range as shown in Table 5.3-1. Although many parameters influence the recoil length, the sensitivity analysis only considers one parameter change at a time.
Table 5.3-1 The range of parameters
Description Unit Notation Range
Effective area of the recoil piston in2 A 29.6784 ~ 36.2736in2
Bore area in2 A b 29.4525 ~ 30.0475in2
Effective area of the recuperating cylinder in2 A R 8.7516 ~ 10.6964in2 Recuperator gas pressure in battery psi Pi 585 ~ 715psi
Barrel length in U 0 140 ~ 260in
Recuperator gas volume in battery in3 Vi 913.5 ~ 1116.5in3
Projectile weight force lb W P 90 ~ 140lb
Density of fluid lb in/ 3 ω 0.030845 ~ 0.031815lb in/ 3
In Figure 5.3-1, it can be seen that there are three sensitive factors influencing the
the projectile weight force W . Effective area of the recoil piston, and barrel length are P negatively correlated with the maximum recoil length. And projectile weight force is positively correlated with the maximum recoil length. Other five parameters are not obvious in this figure.
Figure 5.3-1 Sensitivity analysis of all variables
Figure 5.3-2 shows the changes of the five parameters in a larger scale. Effective area of the recuperating cylinder is the most sensitive factor which is negatively correlated with the maximum recoil length. The other four parameters are slight sensitivity in the results.
According to the above discussion, there are four most sensitive parameters treated as design variables in optimization, the effective area of the recoil piston A, the barrel length
U , the projectile weight force 0 W , and the effective area of the recuperating cylinder P AR.
0 10 20 30 40 50 60 70 80 90 100
35.825 35.83 35.835 35.84 35.845 35.85 35.855
Percentage of Variable Range(%)
Maximum Recoil Length(in)
AR Ab Pi Vi ω
Figure 5.3-2 Sensitivity analysis of partial variables
5.3.2 Cost Function
After the model construction of the recoil mechanism, this section uses this model to perform optimization. Large recoil distance is the most important problem that hinders from designing the space requirement of other mechanisms. Therefore, optimization will focus on how to reduce the maximum recoil length. Therefore, the maximum recoil length of the recoil mechanism which is needed to be minimized is defined as the cost function of the optimization problem.
5.3.3 Design Variables
All parameters in the model are listed in Table 4.3-4 and Table 4.3-5, and the influences of these parameters on cost function are discussed in Section 5.3.1. There are four design variables: the barrel length, the projectile weight force, the effective area of the recoil piston, and the effective area of the recuperating cylinder.
The first variable, barrel length, is the geometrical parameter from the recoiling parts.
The barrel length affects the value of the breech force. Since the artillery has different length of barrel, adjustment of barrel length most modifies the required breech force.
The second variable, the projectile weight force, is the weight of the projectile. It also affects the value of breech force. Since the artillery has different weight of projectile, adjustment of the projectile weight can control the required breech force. The design value can be adjusted as long as the gunshot is still satisfied.
The third variable, the effective area of the recoil piston, is the geometrical parameter from the recoil brake. Effective area of the recoil piston affects the value of hydraulic braking force. It affects the total braking force indirectly. And it provides negative relation of recoil length.
The fourth variable, the effective area of the recuperating cylinder, is the geometrical parameter from the recuperator. Effective area of the recoil piston affects the value of recuperator force. Although it doesn’t affect the recoil length too much, it is still improved to find a better design on the recoil length.
The design variables are listed in Table 5.3-2.
Table 5.3-2 Design variables
Design Variables Unit Notation Component
Barrel length in U 0 Recoil parts
Projectile weight force lb W P Recoil parts
Effective area of the recoil piston in2 A Recoil brake Effective area of the recuperating cylinder in2 A R Recuperator
5.3.4
5.3.5
Constraints
In order to avoid unimplemented optimization result, some constraints are defined in this subsection. First, since the cost function of the optimization is to minimize the maximum recoil length, improper parameter setting may cause the optimization results impracticable. Thus, it needs some suitable range to constraint the geometrical parameters in the mechanism. And the range of design variables is listed in Table 5.3-1.
Optimization Results
According to the definitions in previous subsections, optimization of the mechanical structure of the recoil mechanism is executed by using Matlab®.
In this study, optimization Toolbox Matlab® is used to solve the problem. Since the optimization problem is defined as a nonlinear constrained multivariable problem,
“fmincon”, which is used to find a minimum of a constrained multivariable function, is chosen to solve the problem.
The function “fmincon” deals with the constrained problem using Sequential Quadratic
SQP method attempts to compute the Lagrange multiplier directly. Constrained quasi-Newton method guarantees super linear convergence by accumulating second order information regarding the KT equations using quasi-Newton updating procedure [14].
There are three main stages to implement SQP method. The first is updating of the Hessian matrix. At each major iteration, a positive definite quasi-Newton approximation of the Hessian of the Lagrangian function is calculated that is using BFGS (Broyden-Fletcher-Goldfarb-Shanno) method. The second is to compute Quadratic Programming QP solution. At each major iteration, a QP problem, which is a sub problem generated from Hessian of the Lagrangian function calculated before, is solved, and the solution is used to form a search direction for a line search procedure. The third is line search and merit function calculation. Using the search direction produced in QP problem, a step length which is sufficient to decrease a merit function is determined, where the merit function is in the form defined by Han [16] and Powell [14] [17].
Using “fmincon” as the implement program, and combining the dynamic model built in Simulink® as cost function, the optimization results can be obtained as follows.
From Table 5.3-3 and Figure 5.3-3, it is obviously that the maximum recoil distance is reduced from to . It decreases 1.8456 (about ) from the original length. And from the results of optimization, all design variables go on the boundary of the variable range. It is because there are no extra constraints to limit the magnitude of each force expect the variable boundary.
37.7573in 35.9117in in 4.89%
Table 5.3-3 Optimization results
Design Variables Initial Design Optimum Results
Barrel length U0 200in 260in
Projectile weight force W P 96lb 90lb
Effective area of the recoil piston A 32.976in2 36.2736in2 Effective area of the recuperating cylinder AR 9.724in2 10.6964in2
Cost Function Xmax(recoil length) 37.7573in 35.9117in
Corresponding Time 0.1220sec 0.1260sec
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Figure 5.3-3 Optimization result
5.4 Remarks
1. The simulation of the dynamic model shows that the maximum recoil length can be computed specifically. Under different conditions, there are changes on the recoil length.
2. The tendency of the model is reasonable from physical meanings.
3. Before optimization, design variables can be determined by the sensitivity analysis. It can not only figure out the properties of all parameters, but also reduce the computation on optimization. Because too many design variables are not necessary for optimization or mechanism design.
4. After deciding the cost function, design variables, and constraints, optimization is applied. And optimum result can be obtained.
5. By adjusting the simple design variables, the maximum recoil length can be reduced. This phenomenon can make the recoil mechanism design simpler.
CHAPTER 6
DYNAMIC CONTROL OF THE ORIFICE AREA
6.1 Introduction
In this chapter, the importance of the orifice area is studied. In previous chapters, the orifice area is a pre-determined parameter. But it is better to design an ideal one according to the total braking force. But, the total braking force is difficult to keep constant in fact; the curve of the force will be treated as an ideal trapezoid, like Figure 3.4-1.
Figure 6.1-1 shows the main ideal of this chapter. When total braking force K t( ) is determined, the original recoil mechanism has been redesigned to meet the objective. That is, the change of orifice area versus recoil travel and time must be found by importing the different elevation and the recoil length. Therefore, control methods can make the change of the orifice area fill the bill.
In order to find the relation between orifice area and elevation, the new model is created. Besides, the model will be simulated and analyzed later. Finally, the concepts of controlling the orifice area will be discussed.
Figure 6.1-1 Block diagram for control orifice area
6.2 Model Creation
Figure 6.2-1 shows the logic flow of controlling the orifice area. It is different form Figure 4.3-12. Here, the curve of the total braking force is designed as an ideal trapezoid, and it can be determined by experiments in real applications. Other force components like the packing friction, the recuperator force, and the hydraulic braking force are calculated in successive. Then the orifice area versus recoil travel and time can be found by the known hydraulic braking force.
The dynamic equations of the total braking force can be form at section 3.4. And the other forces are as same as the foregoing model. The combined system is shown in Figure 6.2-2.
Figure 6.2-1 Flowchart of modules combination
Figure 6.2-2 Combined system model of the recoil mechanism
6.3 Dynamic Simulation and Results
The relation between the orifice area and time is shown in Figure 6.3-1. The real line with low elevation is stopped at . The other virtual line with high elevation is stopped at . The results show that high elevation causes the change of the orifice area faster. Besides, the relation of orifice area versus recoil travel is shown in
0.0864sec 0.0636sec
Figure 6.3-2.
In the same way, because there is less space, the high elevation causes shorter recoil length.
In the same way, because there is less space, the high elevation causes shorter recoil length.