CHAPTER 5 DYNAMIC SIMULATION AND OPTIMIZATION
5.3 Optimization
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.
With the two figures, the changes of orifice area versus time or recoil travel at different elevation can be known. And if the orifice area can be controlled according to the curves,
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
With low elevation θ=45o With high elevation θ=75o
tr=0.0636 t
r=0.0864
Figure 6.3-1 Orifice area versus time at different elevation
0 5 10 15 20 25 30 35 40
With low elevation θ=45o With high elevation θ=75o
Xmax=24.5665 X
max=36.2671 (recoil length) (recoil length)
Figure 6.3-2 Orifice area versus recoil travel at different elevation
Figure 6.3-3 shows the 3D curves of orifice area versus recoil travel at different elevation from thirty degrees to seventy-five degrees. Figure 6.3-4 and Figure 6.3-5 show the surface plots of the orifice area versus the recoil travel and the orifice area versus time at different elevation. And from these figures, the tendency states that the orifice area will shrink faster when the elevation is higher and higher.
30 40 50 60 70 80 90
Figure 6.3-3 3D curves at different elevation
Figure 6.3-4 Surface plot of the orifice area versus recoil travel at different elevation
Figure 6.3-5 Surface plot of the orifice area versus time at different elevation
6.4 Discussions
From the simulation results, it is obvious that the desired orifice areas vary according to the recoil travel and time. Therefore, the design of the orifice area is the most important parameter to control the total braking force. Through the study of this chapter, the relation of the orifice area versus recoil travel and time at different elevation can be known.
Traditionally, designers usually designed the structure to cater to the change of the orifice area in the recoil mechanism. But in the present, there are so many electrical and mechanical control methods and apparatus which can get the goal. By integrating sensors and actuators, environmental situation can be obtained easily and input to a control center.
The desired orifice area can be found, and the actuator can work according to the calculated output. This control method can increase the efficiency of the recoil mechanism, and keep the recoil length (maximum recoil travel) in a desired distance. As mentioned before, it is very important when firing at a high elevation, and also important when designing the vehicular application.
CHAPTER 7
CONCLUSIONS AND FURTHER WORKS
7.1 Conclusions
Owing to the advancement of military views and scientific techniques, all countries in the world do their best to develop the war industry. One of the most important things is the development of guns, which are the backbone of the ground protection. A recoil mechanism can reduce the mass recoil force during firing, and push the gun body back to the original position after firing. Because the recoil mechanism was invented, the gun performance got unprecedented improvement. Therefore, the purpose of this research is to find a mathematical model for the recoil force, and it can be implemented by computer program for simulation. The results can provide a clear understanding for designing the mechanism or improving the performance of recoil ability. And some conclusions can be made as follows:
1. On the recoil mechanism, all forces and important parameters are listed clearly. A complete and reliable mathematical model of the recoil mechanism is obtained. Up to the present, there is less information on the complete description of the recoil mechanism. Before analyzing or designing, it can check the theory and the mathematical model. The proposed checking process ensures that the product can be designed correctly according to the original information.
2. The dynamic model can be built by Simulink® instinctively, and has different combinations under different hypothesis. It is very flexible. Besides, 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. It means
that the simulation result predicts the changes of recoil length. It can save the cost to do many real experiments. Furthermore, the maximum recoil length is improved by adjusting the simple design variables. And it makes the space of the artillery use effectively.
3. In fact, the orifice area is also important to the maximum recoil length. So the concept to control the change of the orifice area is a subject. After understanding the relation between orifice area versus time and orifice area versus recoil travel, the control method is generated. By controlling the orifice area, the maximum recoil length can also reduce to meet the objective.
7.2 Future Works
In the future works of this study, some points can be focused on as listed below.
1. There are two equipments which can absorb the breech force. One is the recoil mechanism, and the other is the muzzle brake. In this study, the muzzle brake is ignored. But if it can be considered, the mathematical model of recoil mechanism can become more complete, and the model can be close to the real condition.
2. All the results in the study are not absolute results. Most of the data in this study are theoretical and some hypotheses are supposed. Modifications with experiments are necessary for providing a set of practical results and confirming the hypotheses.
3. The artillery will combine not only mechanism but also electrical parts. So how to use control methods or tools to change the change of orifice area can be studied.
4. The landing gear is similar to the recoil mechanism. The working principle is to
also can make the launching and landing of an airplane smoothly.
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