國立交通大學
機械工程學系
碩士論文
制退復進機之動態模擬分析與最佳化
Dynamic Simulation Analysis and Optimization of
Recoil Mechanisms
研究生：楊筑妃
指導教授：徐瑞坤 曾錦煥
共同指導教授：林聰穎
制退復進機之動態模擬分析與最佳化
研究生:楊筑妃 指導教授:徐瑞坤 林聰穎 國立交通大學機械工程學系摘要
制退復進機的主要功能是用於吸收火炮射擊時所產生的後座力，並且利用壓縮氣 體或彈簧將砲管回復至射擊前的位置。換言之，制退運動是火砲在射擊時，彈頭因氣 體作用而向前發射，另外部分氣體作用於燃燒室而使機構向後移動，且制退完成後， 砲管與其連接部份會恢復至擊發狀態。近幾年來，制退復進機的發展趨勢專注在車載 系統之整合上。因此，短制退行程、高制退效率與低成本制退復進機的需求，漸漸受 到重視。 本論文針對減少制退復進機的制退距離，提出了一個動態模擬分析的模型。在動 態模型的建立上，主要是建立制退復進機、砲管與支撐結構。而針對所建立出的動態 模型的特性，設計一目標函數並介紹所使用的設計參數。此外，藉由最佳化原理分析 其目標函數及所考慮的設計參數，使射擊時制退復進機的制退距離有更符合設計需求 的表現。最後，分析制退距離與控制孔面積的關係，提出藉由控制孔的概念，使制退 復進機在限定的制退距離內停止。Dynamic Simulation Analysis and Optimization of
Recoil Mechanism
Student: Zhu-Fei Yang Advisor: Ray-Quan Hsu, Tsung-Yin Lin
Institute of Mechanical Engineering National Chiao Tung University
ABSTRACT
The recoil mechanism is mainly used to absorb the recoil force during firing, and furthermore it can use compressed gas or springs to return the gun tube to its original position for artillery weapons. In other words, the recoil motion is the rearward movement of the gun during and after firing. The recoil motion is caused by the reaction of the projectile and the propellant gases. After recoil, the gun and connecting parts return to the original firing position. In recent years, the development trends of artillery weapons with recoil mechanisms focus on the vehicular integration. Therefore, a small volume, high recoil efficiency, and low cost of recoil mechanism are necessary.
This thesis presents dynamic analysis and simulation on recoil mechanism, which is used to reduce the recoil length of the recoil mechanism. The first objective is to create a dynamic model of the recoil system, which includes the recoil mechanism, the barrel, and supporting parts. The second objective uses the created dynamic model to design an objective function for the recoil mechanism, some design parameters are introduced here. The third objective is the optimization; some parameters of the recoil mechanism are optimized to give a better result on the recoil length of the device. Finally, the relation
between the recoil length and the orifice area is analyzed. The recoil mechanism can be stopped at a limit recoil length by controlling the orifice area.
TABLE OF CONTENTS
摘要... i
ABSTRACT ...ii
TABLE OF CONTENTS ... iv
LIST OF TABLES ...vii
LIST OF FIGURES...viii NOTATIONS... xi CHAPTER 1 INTRODUCTION... 1 1.1 Background ... 1 1.2 Recoil Mechanism... 2 1.3 Motivations... 5 1.4 Literature Reviews ... 6 1.5 Thesis Outlines... 7 CHAPTER 2 PRELIMINARY... 9
2.1 Component of the Recoil Mechanism ... 9
2.2 The Main Types... 10
2.2.1 Hydrospring ... 11
2.2.2 Hydropneumatic ... 13
2.3 Forces and Procedures during Firing ... 15
CHAPTER 3 FORCE ANALYSIS ... 17
3.1 Introduction ... 17
3.2 Equation of Motion... 19
3.3 Analysis of Interior Ballistics... 24
3.3.1 Basic Equations... 25
3.3.3 Projectile Velocity and Breech Force ... 28
3.3.4 Relationship between Projectile Travel and Time ... 28
3.4 Determination of Total Resistance to Recoil ... 30
3.5 Forces Contributing to the Total Resistance to Recoil ... 35
3.5.1 The Recuperator Force ... 36
3.5.2 Frictional Force of Sliding Surfaces... 38
3.5.3 Frictional Resistance of Packings... 40
3.5.4 Hydraulic Baking Force... 41
3.5.5 Effective Area of the Equivalent Orifice... 42
3.6 Remarks... 44
CHAPTER 4 SYSTEM DYNAMIC MODELING... 46
4.1 Introduction ... 46
4.2 Dynamic Analysis of the Recoil Mechanism ... 46
4.2.1 Structure of Dynamic System ... 47
4.2.2 Recoiling Parts ... 48
4.2.3 Recoil Brake ... 50
4.2.4 Counterrecoil Mechanism... 50
4.3 Dynamic Model Creation... 51
4.3.1 Matlab Simulink ... 51
4.3.2 Module Creation ... 52
4.3.3 Parameter Settings ... 59
4.3.4 Module Combination... 61
CHAPTER 5 DYNAMIC SIMULATION AND OPTIMIZATION... 63
5.1 Introduction ... 63
5.2.2 Simulation of Recoil Mechanism... 64
5.2.3 Leakage Area... 68
5.2.4 Model Adequacy Checking ... 72
5.3 Optimization ... 75 5.3.1 Sensitivity Analysis ... 75 5.3.2 Cost Function ... 78 5.3.3 Design Variables... 79 5.3.4 Constraints ... 80 5.3.5 Optimization Results ... 80 5.4 Remarks... 83
CHAPTER 6 DYNAMIC CONTROL OF THE ORIFICE AREA... 84
6.1 Introduction ... 84
6.2 Model Creation ... 85
6.3 Dynamic Simulation and Results ... 86
6.4 Discussions... 90
CHAPTER 7 CONCLUSIONS AND FURTHER WORKS ... 91
7.1 Conclusions ... 91
7.2 Future Works ... 92
LIST OF TABLES
Table 2.2-1 List the features of hydrospring system...12
Table 2.2-2 List the features of hydropneumatic system...15
Table 3.6-1 List all force and parameters of the recoil mechanism ...45
Table 4.3-1 Parameter table of the breech force ...55
Table 4.3-2 Parameter table of the hydraulic braking force...57
Table 4.3-3 Parameter table of the recuperator force ...58
Table 4.3-4 List of parameters assigning in the recoiling parts...59
Table 4.3-5 List of parameters assigning in the recoil brake and recuperator ...60
Table 4.3-6 Parameters from measurement data...61
Table 5.3-1 The range of parameters ...76
Table 5.3-2 Design variables ...80
LIST OF FIGURES
Figure 1.1-1 Artillery structure ...1
Figure 1.2-1 Recoil mechanism [1]...3
Figure 1.2-2 Recoil and counterrecoil system [2]...4
Figure 2.2-1 Classification of recoil mechanism [3]... 11
Figure 2.2-2 Hydrospring recoil mechanism [3] ...12
Figure 2.2-3 Hydropneumatic recoil mechanism (independent type) [3]...13
Figure 2.2-4 Hydropneumatic recoil mechanism (dependent type) [3] ...14
Figure 3.2-1 1-D mode of gun ...19
Figure 3.2-2 Forces on the parts moving in elevation...20
Figure 3.2-3 Curves of recuperator force ...21
Figure 3.2-4 A Functional relation between B t( ) and time ...23
Figure 3.3-1 Three periods of recoil motion ...24
Figure 3.3-2 Moving projectile and barrel ...25
Figure 3.4-1 Practical shape of total resistance to recoil...30
Figure 3.4-2 The curves of B t( ), K t( ), and Wrsinθ [13]...34
Figure 3.5-1 Curve for the braking force [3] [6] ...35
Figure 3.5-2 Sliding track friction on mechanism ...38
Figure 4.2-1 Simple structure of recoil mechanism ...48
Figure 4.3-1 Breech force in Simulink® model...53
Figure 4.3-2 Breech force module ...53
Figure 4.3-3 Relation between time and projectile travel in Simulink® model...54
Figure 4.3-4 Time and projectile travel transfer module...54
Figure 4.3-5 Time and breech force transfer module...55
Figure 4.3-7 Hydraulic braking force in Simulink® model...56
Figure 4.3-8 Hydraulic braking force module ...57
Figure 4.3-9 Recuperator force in Simulink® model ...58
Figure 4.3-10 Recuperator force module...58
Figure 4.3-11 Combined system model of the recoil mechanism...62
Figure 4.3-12 Flowchart of modules combination ...62
Figure 5.2-1 Breech force versus projectile travel ...65
Figure 5.2-2 Projectile velocity versus projectile travel ...65
Figure 5.2-3 Breech force versus time...66
Figure 5.2-4 Recoil acceleration versus time...66
Figure 5.2-5 Recoil velocity versus time ...67
Figure 5.2-6 Recoil length versus time...67
Figure 5.2-7 Orifice area versus recoil length ...68
Figure 5.2-8 Diagram of leakage area...69
Figure 5.2-9 Orifice area versus recoil length ...69
Figure 5.2-10 Recoil acceleration versus time at different leakage area ...70
Figure 5.2-11 Recoil velocity versus time at different leakage area ...71
Figure 5.2-12 Recoil length versus time at different leakage area ...72
Figure 5.2-13 Breech force versus time at different charge weight...73
Figure 5.2-14 Recoil length versus time at different charge weight...73
Figure 5.2-15 Breech force versus time at different muzzle velocity ...74
Figure 5.2-16 Recoil length versus time at different muzzle velocity...75
Figure 5.3-1 Sensitivity analysis of all variables ...77
Figure 5.3-2 Sensitivity analysis of partial variables...78
Figure 6.2-1 Flowchart of modules combination ...85
Figure 6.2-2 Combined system model of the recoil mechanism ...86
Figure 6.3-1 Orifice area versus time at different elevation ...87
Figure 6.3-2 Orifice area versus recoil travel at different elevation ...87
Figure 6.3-3 3D curves at different elevation ...88
Figure 6.3-4 Surface plot of the orifice area versus recoil travel at different elevation...89
NOTATIONS
A effective area of the recoil piston
1
A contact area of packing on cylinder wall
b
A bore area
c
A cross-section area of the control rod
cr
A effective area of the recuperating piston
R
A effective area of the recuperating cylinder
B breech force or gas force at the breech
B average breech force
( )
B t breech force or gas force at the breech
o
C orifice coefficient
P
C constant pressure specific heat
V
C constant volume specific heat
( )
D t recoil impulse of time
F
∑
total force on the recoil mechanisma
F inertia force of recoiling parts
o
R
F recuperator force
H hydraulic braking force
( )
H t total resistant impulse of time
*
I total impulse
*
B
I impulse of the recoil force
K total braking force
p
K pressure factor
R
K rod pull force
( )
K t recoil braking force
L recoil distance r M recoil mass 1 N guide force 2 N guide force P recoil pressure a
P axial pressure in packing
b
P pressure in recoil cylinder
g
i
P recuperator gas pressure in battery
m
P maximum fluid pressure
max
P maximum bore pressure
o
P fluid pressure on packing at any recoil position
R
P radial pressure in packing
rx
P gas pressure at any position of recoil
( )
P x
Δ pressure drop across recoil orifice
r
Q rate of flow
x
P recuperator gas pressure at x
R frictional force
c
R packing friction of the recoil brake
P
R packing friction
r
R packing friction of the recuperator
SL
R sliding track friction
0
U barrel length
V projectile velocity
g
0
V muzzle velocity of the projectile
x
V recuperator gas volume at x
x
V
Δ increase in V x
C
W charge weight force
P
W projectile weight force
r
W weight of recoiling parts
X recoil length
X recoil velocity
X recoil acceleration
R
X displacement of the control rod
e
a effective area of orifice area
o a orifice area R a braking retardation g acceleration of gravity ( )
h x velocity head at x position
R
m recoil mass
n polytropic exponent
b
t projectile transit time
r
t time of recoil stroke
u projectile travel
v absolute velocity of the gun body
0
v recoil velocity of the gun body when the projectile exits the muzzle
( )
v x recoil velocity ( )
o
v x velocity of flow through orifice
x absolute displacement of projectile
θ elevation
( )t
α centroid of the area
( )t
β centroid of the area
μ frictional coefficient
ν leakage factor
CHAPTER 1
INTRODUCTION
1.1 Background
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. However, gun protection is still insufficient for air attack because the airplane performance goes better and the developments of the air-to-ground weapon grow up day after day. Hence, artillery weapons, as shown in Figure 1.1-1, become the important ground firepower on the modern battlefield.
Figure 1.1-1 Artillery structure
The artillery weapons have been developed from the thirteenth century to the present. Generally, it includes a gun body and a gun mount. The gun body consists of a barrel, a breech, a breechblock, and a muzzle brake. In addition, the gun mount is composed of recoil mechanisms, elevating mechanisms, traversing mechanisms, and supporting parts. Among
these parts, the muzzle brake and recoil mechanism can reduce the mass recoil force during firing, and push the gun body back to the original position after firing.
Before the mid-nineteenth century, general guns didn’t assemble any device having the cushioning effect. For this reason, people had to design bulky guns to avoid the powerful recoil force. But the design led to a terrible mobility. It took much time to return the gun tube to original position and aim again. In order to solve above problems, the designers tried to install a buffer in the base which could generate a retarding force to stop the recoil motion. Finally, the recoil mechanism was created until the ninth decade of the nineteenth century.
Because the recoil mechanism was invented, the gun performance got unprecedented improvement. Therefore, the nineteenth century is a milepost in the developed history of artillery weapons. In recent years, the development trends of artillery weapons with recoil mechanisms focus on the vehicular integration. It can increase the mobility of artillery weapons. Therefore, a small volume, high recoil efficiency, and low cost of the recoil mechanism which can be easily installed on vehicles is very important.
1.2 Recoil Mechanism
The recoil mechanism, shown in Figure 1.2-1, is mainly used to absorb the recoil force during firing, and furthermore it can make use of compressed gas or spring to return the gun tube to its original position. In other words, the recoil is the rearward movement of the gun and connected parts during and after firing. It is caused by the reaction to the forward motion of the projectile and propellant gases. After recoil, the gun and connecting parts return to the in-battery, or firing position. This forward movement is called "counter recoil."
impossible to build a carriage to withstand the load imposed upon it without rupturing, overturning, or moving. To bring the carriage stresses down to a reasonable value and to ensure the stability, a recoil system is interposed between the gun and the carriage. The recoil mechanism absorbs the energy of the recoiling parts over a certain convenient length and returns the gun to battery for further firing. The recoiling parts are the recoil mechanism and carriage that move with the gun in recoil and counter recoil.
Figure 1.2-1 Recoil mechanism [1]
A recoil mechanism usually consists of two components as shown in Figure 1.2-2:
(1) Recoil brake – Normally, a recoil system consists of two stationary pistons attached to the slide of the liquid-filled cylinder in the housing. As the housing moves backward during recoil, the trapped liquid is forced around the piston head through metered orifices, slowing the movement of the housing.
in a pressurized cylinder. As the gun recoils, the piston protrudes further into the cylinder. After the end of the recoil period, the nitrogen gas pressure acting on the piston pushes the housing back into the original position. The piston may be attached to the slide or set in a chamber.
Figure 1.2-2 Recoil and counterrecoil system [2]
Besides, the recoil and counterrecoil system, shown in Figure 1.2-1, comprises two recoil cylinders connected hydraulically to a recuperator, and performs five functions:
(1) To absorbs the gun recoil force in prescribed recoil lengths.
(2) To maintain a nearly constant recoil force throughout the recoil stroke.
(3) To store energy to return the gun to the battery.
(4) To regulate the counterrecoil stroke to keep it within the prescribed rate-of-fire limits.
The recuperator is a hydropneumatic device having separate chambers for hydraulic fluid and nitrogen gas. During the recoil stroke, the energy developed by the recoiling gun is absorbed by the recuperator and the recoil cylinder. The energy absorbed by the recuperator is stored as the compressed nitrogen gas. The energy absorbed by the recoil cylinders is dissipated by resistance to the flow of hydraulic fluid through throttling grooves between the stationary and moving parts of the cylinders.
During counterrecoil, the energy (gas pressure) provides the gun moving from its maximum recoil position to the original position. In this phase of counterrecoil, the recoil cylinders act as stroke regulators and release energy stored by the recuperator at a regulated rate.
To achieve the purpose of the recoil brake, there are many kinds of recoil mechanism, such as hydropneumatic and hydrospring. Moreover, the technical literatures are confidential in the foreign companies. Also, it is not easy to get the related technical data except from papers and patents. For this reason, this study will focus the literature review in US patents and Chinese papers from Mainland China to support the results of simulation. Other available will be explained in the following sections.
1.3 Motivations
The recoil mechanism is mainly used to absorb the recoil force during firing, and return the gun tube to its original position. The device, which plays a decisive role in the artillery weapons, is like a human heart. Because of the invention from the ninth decade of century, the performance of guns was improved, and the shooting velocity was increased greatly. Thereby, the device truly deserves to be studied.
Rheinmetall GmbH (Germany), Giat Industries (France), and Bofors Defence AB (Sweden). This technology relates to the national defense, therefore the developed countries will not ignore it when our country begins to develop it. By comparison, the defense industry in our country is with plagiary, and directly purchases weapons from foreign companies. It shows that the knowledge and techniques are insufficient. Fortunately, there are more and more resources available on the recoil mechanism.
The innovation of the recoil mechanism is not easy because there are so many problems which need to be solved. People usually design a new concept by searching patents, marketing research, or some innovative methods. But there is an important subject that the complete definition of the original problem about the recoil force during firing is required. If the definition is not clear, it is difficult to solve related problems and perform further analysis.
According to the above reasons, the purpose of this research is to find a mathematical model for the recoil force during firing. In addition, it can be implemented by the computer program for simulation. The results which are presented and constructed by this thesis can provide a clear understanding for designing the mechanism or improving the performance of recoil ability in the future.
1.4 Literature Reviews
The system of the recoil mechanism consists of eleven components: recoil piston rod, recoil piston, cushion, dish spring, recoil cylinder, recuperating cylinder, counterrecoil buffer, floating piston, recoil throttling valve, regulator, and regulator valve [3]. The general functions are introduced in Chapter 2. Because the recoil mechanism was developing for a
general classification, hydrospring and hydropneumatic type. The classification of the recoil mechanism is often seen in the national defense industry. Theses are two main types of the recoil mechanism. These two types of the recoil mechanism principally differ from their action components, one is the spring and the other is compressed gas. Generally speaking, recuperator was originally with the spring type [3] [4].
Besides, the more important part is the forces and procedures of the recoil mechanism during firing. A high gas pressure, which acts on the base of the projectile and accelerates it forward, arises in the combustion chamber of the tube. The same gas pressure also acts on the breechblock of the tube, which is forced to the rear with a gas recoil force, which is also called the breech force. It lets the gun tube accelerate back, and the gun tube will be stopped by various braking components. After the recoil motion is completed, the recuperating mechanism returns the gun tube to its original position. Therefore, at the end of counterrecoil, the gun gets no moving forward, and the recoil part is braked hydraulically at the end of counterrecoil by means of the recoil brake. The period which recoil force acts is from the projectile firing to the mechanism stopping. Therefore, this period is a complete cycle during firing [5] [6].
1.5 Thesis Outlines
In order to find the recoil force situation during firing, the force analysis and the mathematical model will be constructed based on Simulink® in this thesis. Simulink® is one of the packages in Matlab®. Furthermore, the model can help to realize the designs on the available products and literature, especially on the mechanism with some parameters. The brief description and outlines in this thesis contents are given as follows.
recoil mechanism, and comparisons for different structures. Besides, there is a simple description about the procedure during firing.
Chapter 3 introduces the force analysis of the gun during firing, especially mathematical descriptions on the recoil force.
Chapter 4 proposes the dynamic analysis and the dynamic model creation of the recoil mechanism. It can be divided into five parts. With free body analyses and experimental data, the dynamic characters can be obtained from system equations. Using Matlab® Simulink, the dynamic model of system equations can be created into individual modulus.
Chapter 5 constructs a simulation model on Simulink®. The simulation results are analyzed with tendency forecasting to match the physical phenomenon. In addition, the simulation chooses some parameters to find the influence of these parameters on the performance of the recoil mechanism. Target and constraints of the optimization problem are defined according to the design requirements. Finally in this chapter, some improvement designs can be obtained by performing optimization.
Chapter 6 provides the concept of controlling the recoil length by controlling orifice area. When the relation between orifice area and recoil length is known, the recoil mechanism can be stopped at the desired recoil length.
Finally, Chapter 7 contains the conclusions and future works, which could assist some aspects for following works on this study.
CHAPTER 2
PRELIMINARY
2.1 Component of the Recoil Mechanism
The system of the recoil mechanism consists of eleven components: recoil piston rod, recoil piston, cushion, dish spring, recoil cylinder, recuperating cylinder, counterrecoil buffer, floating piston, recoil throttling valve, regulator, and regulator valve. All of these components have their respective different functions, which will be introduced in this section. The general functions are introduced as follows [3]:
1. Recoil piston rod: it is a tensile machine part; one end is connected with recoil piston, and the other with breech ring.
2. Recoil piston: the thickness of the piston is controlled by the space of cushion.
3. Cushion: it can prevent the moveable parts such as piston or piston rod from leakage. In Addition, the cushion presses the moveable surface strongly by fluid pressure and spring.
4. Dish spring: it can provide the pressure for the cushion. The dish spring can supply big load with small deformation in a small space.
5. Recoil cylinder: the inner diameter of the cylinder is determined by the diameter of the piston. Moreover, the tube thickness is determined by the pressure which cushion or fluid generates, and by the yielding stress of the material.
6. Recuperating cylinder: this part can store energy, and makes use of gas pressure to return the gun tube to the original position.
7. Counterrecoil buffer: it can be hydraulic or pneumatic. Besides, the buffer controls the velocity of the recuperating stroke.
8. Floating piston: this part is used to separate the liquid and gas in the recuperator cylinder.
9. Recoil throttling valve: this value can be used to control the hydraulic resistance flowing from recoil cylinder to recuperator cylinder.
10. Regulator: a tool, which is installed in the recuperator cylinder, can adjust pressure during recoil and recuperator. Besides, it has to control the flow of hydraulic fluid in any direction.
11. Regulator valve: a tool, which is installed in liquid end of recuperator cylinder, can adjust the flow of hydraulic fluid during recuperating period.
2.2 The Main Types
Because the recoil mechanism was developing for a long time, there are many available devices using in different kinds of guns. There is a general classification, hydrospring and hydropneumatic type, such as shown in Figure 2.2-1. The classification of the recoil mechanism is often seen in the national defense industry.
These two types of the recoil mechanism principally differ from their action components, one is the spring and the other is compressed gas. Generally speaking, recuperator was originally with the spring type [3] [4].
Figure 2.2-1 Classification of recoil mechanism [3]
2.2.1 Hydrospring
The hydrospring system is shown in Figure 2.2-2. There are a recoil brake, a recuperator, and a counterrecoil buffer. In fact, the device consists of two or three parts to ensure the structure more compact. Sometimes, there is a bigger spring which is wound around the barrel to get a more compact assembly.
The recuperator uses a mechanical spring, and the others use hydraulic systems. Although this type is seldom used, there are still some advantages and disadvantage listed in Table 2.2-1. The design of the device is simple, low cost, rapid adjustment, easy manufacture, and there are fewer problems about the airtight leakage. But the life of the spring may not predictable. The volume is huge, and the component replacement is required frequently.
Figure 2.2-2 Hydrospring recoil mechanism [3]
Table 2.2-1 List the features of hydrospring system
Advantages Disadvantages
1. Simple design
2. Easy manufacture
3. Low cost
4. Rapid adjustment
5. Fewer airtight leakage
1. Maintenance
2. Big volume
2.2.2 Hydropneumatic
For the hydropneumatic system, there are two different types: independent and dependent, as shown in Figure 2.2-3 and Figure 2.2-4 respectively. The recuperator fills with compressed gases, and the nitrogen gas is usually used because of its inactivity.
The recuperator of the independent type is separated from recoil brake completely. Furthermore, the piston rods both directly connect with a back part. When the gun recoils, hydraulic fluid or oil will flow to the chamber of compressed gas. The fluid will then press on the gas to make the gas pressure rising, and the action will reverse during the recuperating time.
On the other hand, the recuperator of the dependent type is often connected to the recoil brake, but the gas is separated from fluid by the floating piston. Besides, the recoil piston rod links a back part simply. On the way of throttling, the fluid from recoil cylinder would be pressed on recuperating cylinder. Also, the advantages and disadvantages of the hydropneumatic system are listed in Table 2.2-2. The reliability of the hydropneumatic system is higher, the durability is better, the recoil distance is long, and the design is more flexible. However, the device needs specialized technology and the cost is expansive. In addition, the gas pressure will be easily changed by the atmosphere temperature, and affect the recoil velocity and recoil travel. So it needs some compensation. Moreover, the device is hard to keep the high firing frequency, because of the heat generation.
Table 2.2-2 List the features of hydropneumatic system
Advantages Disadvantages
1. High reliability
2. Good durability
3. Flexible application
4. Long recoil distance
5. Easy maintenance
1. Hard manufacturing
2. Low firing frequency
3. High sensitivity on temperature
4. High cost
2.3 Forces and Procedures during Firing
During firing, a high gas pressure, which acts on the base of the projectile and accelerates it forward, arises in the combustion chamber of the tube. The same gas pressure also acts on the breechblock of the tube, which is forced to the rear with a gas recoil force, which is also called the breech force. The magnitude of this force is the same as the projectile accelerating force. This large gas recoil force on the gun tube does not act directly on the cradle of the tube but on the recoil mechanism.
The gun tube, which is accelerated back by the gas recoil force, is brought to stop by various braking components. These are the hydraulic braking force of the recoil mechanism, the force of the recuperator mechanism, and the friction forces among the components. The braking force acts as a forward direction on the recoil part of the gun to retard the recoil.
The braking of the recoil mechanism generates a mass inertial force, which in magnitude is equal to the total braking force, and acts towards the rear of the gun. Its line of
action goes through the center of gravity of the recoiling mechanism of the gun, regardless of where the individual braking force components act on. The recoil braking force knocks the gun backwards, while the baking force is a force which acts on the recoiling part of the gun in forward direction.
After the recoil motion is completed, the recuperating mechanism returns the gun tube to its original position. The required force is provided by mechanical springs or gas cushioning, which are compressed even more beyond their pretensioned state during the recoil of the tube. Therefore, at the end of counterrecoil, the gun gets no moving forward, and the recoil part is braked hydraulically at the end of counterrecoil by means of the recoil brake. The period which recoil force acts is from the projectile firing to the mechanism stopping. Therefore, this period is a complete cycle during firing [5] [6].
CHAPTER 3
FORCE ANALYSIS
3.1 Introduction
All kinds of recoil mechanisms operate according to same basic principles. The apparatus can control forces, through the specific recoil movement. In other words, it makes use of the force to retard the gun tube, and return the gun tube to original position.
When firing, owing to the action of the gas recoil force and the recoil braking force, the load on the gun body often varies from time to time. So, how to solve the force change of the recoil mechanism is a key discussion about this chapter.
Before the force analysis, some assumptions need to be established first. From a physical viewpoint, there is no external force acting on the artillery weapon during firing. The process is the conservation of momentum, and it reflects properties and conditions of the artillery weapon. It also conforms to actual condition of firing. However, the energy conservation principle is difficult to apply because some energy is lost during firing [7].
The resistance force is composed of hydraulic braking force and spring force. Although these two forces act individually, it is treated as a resultant force in the system. Therefore, the overall system can be treated as a unit. Before the problem is defined, some assumptions have to be made as follows:
1. The boundary condition is free release and free recoil.
The analysis focuses on the motion of the recoil mechanism only. It uses systematic view to analyze the force behavior, and isn’t affected by external force of other components.
2. The supporting structure is immovable.
The supporting structure of the gun body is a rigid body. Its quantity of motion is very small. Thus, the motion of the supporting structure is neglected. It means that the analysis only focus on the first recoil effect.
3. The effect on the muzzle brake is ignored.
The recoil force is balanced by the muzzle brake and the recoil mechanism. The forces of these two parts are with a proportional relation, such as thirty percent for the muzzle brake and seventy percent for the recoil mechanism. Therefore, the effect on the muzzle brake could be ignored.
4. The analysis focuses on the bore period during firing.
The bore period means that a projectile moves along the bore of barrel until exiting the muzzle. After a projectile exits the muzzle, the bore pressure would drop to atmospheric pressure gradually. And the influence on recoil force is very small. For this reason, the analysis only focuses on the bore period.
According to the assumptions, the model can be simplified. Then, the equation of motion of the recoil mechanism can be defined. And the analysis of interior ballistics and the recoil motion can combine together so as to find the force conditions of all parts when the projectile exits.
3.2 Equation of Motion
Figure 3.2-1 1-D mode of gun
The equation of motion adopts a mode of single degree of freedom as shown inFigure 3.2-1. According to the Newton’s Second Law, the equation of motion of recoil mechanism is
( ) ( ) sin
r
F=M X =B t −K t +Wr θ
∑
(3.2-1)where
∑
F is the total force on the recoil mechanism, M is the recoil mass, Xr is therecoil acceleration, B t( )is the breech force, K t( ) is the recoil braking force, is the recoil weight force, and
r
W θ is the elevation.
F
∑
consists of a breech force, a hydraulic braking force, a recuperator force, a frictional force, and the component of the weight, shown in Figure 3.2-2. There are somedescriptions about the forces on the recoiling parts during firing:
Figure 3.2-2 Forces on the parts moving in elevation
1. Forces acting parallel to the axis of the bore:
(1) Breech force B ( on the breechblock )
During firing, a high gas pressure arises in the combustion chamber of the tube; it acts on the base of the projectile and accelerates it forward. The same gas pressure also acts on the breechblock of the tube, so it is forced towards the rear as a gas recoil force, also called the breech force. The magnitude is the same as the projectile accelerating force.
acts towards the rear of the gun as a hydraulic braking force. This force can be basically produced by a fluid filled cylinder. Owing to the fact, the cylinder piston coupled to the recoiling masses presses the fluid through a narrow orifice. The magnitude of the force can be controlled by the valve cross-section. In addition, the cross-section and recoil travel have a constant relation.
(3) Recuperator force FR ( on the recuperator piston rod )
The counterrecoil mechanism returns the gun tube to its firing position. The force is often made used of mechanical springs or gas cushioning, which are compressed to store some recoil force during the recoil of the tube. Before firing, the device has to be an initial force in order to resist gravity, and keeps the gun at the original position. Figure 3.2-3is the relation of the recuperator force and recoil travel. The force seems linear when the spring is used. If a nonlinear force is required, the gas cushion can be used.
(a) By spring (b) By gas cushion
(4) Frictional force R ( between pistons and the sliding track )
The frictional force is composed of frictions at the piston rod seals on the recoil brake and recuperator, and the piston seal in the recuperator. This force, called the packing friction R , is a constant. On the other hand, the sliding P
track friction of the gun tube R is also a component of the frictional force. SL
(5) Weight component Wrsinθ ( on the center of gravity of the recoiling parts )
A component of recoil weight is a constant during recoil and recuperating time.
2. Forces acting perpendicular to the axis of the bore:
(1) Guide forces and ( on the tube claws ,and corresponding sliding surfaces of the gun tube )
1
N N2
The relation between the guide forces and sliding track friction force R is SL
1 2
( )
SL
R =μ N + N (3.2-2)
where μ is the frictional coefficient, N and 1 N are absolutes of the 2
guide forces.
(2) Weight force component Wrcosθ (on the center of gravity of the recoiling parts )
A component of recoil weight force is a constant during recoil and recuperating time.
( )
B t is the breech force B. Besides, B t( ) can be indicated by the burning rate of powder as shown in Figure 3.2-4.
Figure 3.2-4 A Functional relation between B t( ) and time
( )
K t is the recoil braking force. This means that during the recoil braking, an inertial force has to be applied at the center of gravity of the recoiling parts in the direction of recoil, where is the braking retardation. This backwards directed inertial force is equal to the recoil braking force
R R
m a
R
a
( )
K t . From the force equilibrium in the direction of the axis of the bore, the recoil braking force can be expressed as:
( ) R
K t =H+F + R (3.2-3)
where H is the hydraulic braking force, FR is the recuperator force, R is the friction force.
3.3 Analysis of Interior Ballistics
Figure 3.3-1 Three periods of recoil motion
The period of recoil motion, as shown in Figure 3.3-1, can divide into three parts [8] [9]:
1. In bore period
The period means that a projectile moves along the bore of the barrel until exiting the muzzle. During this period, the motion of the recoil mass is an accelerative motion.
2. Aftereffect period
This period means that the bore pressure declines to approach the atmosphere pressure, after the projectile leaves the muzzle. During this period, the motion of recoil mass changes from positive to negative acceleration. Besides, the maximum velocity of recoil mass is generated in the period.
There is no bore pressure in this period, that is B= . The recoil mass moves by 0 the inertia force which is remained from the foregoing periods. In addition, the recoil device is retarded by a recoil resistance until the velocity is zero.
There are so many papers to describe the three periods. By the above-mentioned assumption, the analysis here focuses on the in-bore period to get a general theory.
3.3.1 Basic Equations
The basic equations refer to Figure 3.3-2, one listed as follows:
Figure 3.3-2 Moving projectile and barrel
x= + (3.3-1)s u du V dt = (3.3-2) ds v dt = (3.3-3) b B P A = (3.3-4)
where x is the absolute displacement of the projectile, is the absolute displacement of the gun body, is the projectile travel, V is the projectile velocity, is the absolute
s
velocity of the gun body, P is the recoil pressure, B is the gas force at the breech (or breech force) Ab is the bore area, and U0 is the barrel length.
Half mass of the combustion charge acts as the acceleration of a projectile, and the other half as the acceleration of a gun body. There are forces on a projectile and a gun body individually, and the force magnitude is equal but in opposite direction [10].
1 1 ( ) ( ) 2 2 c r p W dv dV W W g + dt = g + c W dt (3.3-5)
where Wc is the charge weight force, and Wp is the projectile weight force. Because
1
2Wc is much smaller than Wr, so it can be neglected. Eq.(3.3-5) can be written as
( ) 2 c p r W W dv dV dt W dt + = (3.3-6)
In addition, B and F can be written as follows: P
r W dv B g dt = (3.3-7) p P W dV F g dt = (3.3-8)
where B is the gas force at the breech (or breech force), and F is the gas force at the P
base of the projectile.
3.3.2 Determination of Le Duc Parameters: a and b
The projectile travel and projectile velocity in the bore can be expressed as a hyperbolic function which is also called Le Duc formula, as follows [10]:
au V b u = + (3.3-9) 2 ( ) dV ab du dt = b u+ dt (3.3-10)
By rearrangement of Eq.(3.3-2), Eq.(3.3-7), and Eq.(3.3-10), B can be expressed as Eq.(3.3-11). 2 3 ( 0.5 ) ( ) p c r W W a b W dv B g dt g b u + = = + u (3.3-11) By 2 2 0 d V
dt = , the maximum breech force is generated when b=2u. 2 max 4 ( 0.5 ) 27 p c a W W B bg + = (3.3-12) 2 2 max max 4 ( 0.5 ) 27 p c b b a W W B a P K A A bg b + = = = (3.3-13)
where Pmax is the maximum bore pressure, and
4( 0.5 ) 27 p c b W W K A g + = .
According to the muzzle position, is the barrel length, and is the muzzle velocity of the projectile. Then the values of a and b are attained as Eq.(3.3-14) and Eq.(3.3-15). 0 U V0 0 0 0 aU V U b V − = 0 (3.3-14) 2 2
0 max 0 max 0 0 max
0 ( ) 4 2 U P U P KV U P a KV ± − = (3.3-15)
To simplify Eq.(3.3-14) and Eq.(3.3-15), a and b can be expressed as follows:
0
0( 1
a=V Q+ ) (3.3-17)
where
2 2
0 max 0 max 0 0 max
2 0 ( ) 4 1 2 U P U P KV U P Q KV ± − = − .
3.3.3 Projectile Velocity and Breech Force
By rearranging of Eq.(3.3-9), Eq.(3.3-16), and Eq.(3.3-17), the projectile velocity as function of projectile travel is expressed as Eq.(3.3-18).
0 0 ( 1) V Q u au V b u QU u + = = + + (3.3-18)
From Eq.(3.3-11), Eq.(3.3-16), and Eq.(3.3-17), the breech force is
2 2 0 3 0 ( 0.5 ) ( 1) ( ) p c W W V Q QU B g QU u + + = + 0u (3.3-19)
Because the maximum breech force is generated when b=2u, Bmax is reworded as Eq.(3.3-20). 2 2 0 max 0 4( 0.5 ) ( 1) 27 p c W W V Q B QU g + + = (3.3-20)
By rearrangement of Eq.(3.3-19) and Eq.(3.3-20), the simplified breech force is represented as Eq.(3.3-21). 2 2 max 0 3 0 27 4( ) B Q U u B QU u = + (3.3-21)
From du au V dt b u = = + b u dt du au + = By integration of dt, ln b u t u a a = + + c
When u=U0, is zero. So the time is negative before a projectile leaves a muzzle. And t
0 0 ln U b c U a a = − − 0 0 0 0 0 0 0 0 0 ln ( ) ( ln ) ( 1) U Q U U u U U u b u t a u a V Q + − − = − + = − + (3.3-22)
But Eq.(3.3-22) makes that the time is negative infinite when the beginning position of projectile ( ). For this reason, the time which the projectile exits muzzle can be found by the law of impulse and momentum.
0 u= 0 r b W Bt v g = and 0 0 0.5 p c r W W v V W + =
where is the recoil velocity of the gun body when the projectile exits the muzzle, is projectile transit time, and
0
v tb
0
0 0
U
BU =
∫
Fduand using Eq.(3.3-21)
0 2 max 0 0 3 max 0 27 27 4 ( ) 8 ( U u B B Q U du B QU u Q = = + +
∫
2 1) Q (3.3-23) 2 0 max 8( 0.5 ) ( 1) 27 p c b W W V Q t gB Q + + = (3.3-24)3.4 Determination of Total Resistance to Recoil
The basic principle of total resistance is assumed that the curve between K t( ) and time is a trapezoid as shown in Figure 3.4-1 [6]. Therefore, when the resistance reaches maximum, the value will be supposed as a constant. Therefore, the problem is to find the constant resistance. For this purpose, the section adopts the moment area method [8] [12].
Figure 3.4-1 Practical shape of total resistance to recoil
travel, can be calculated easily by the integration of Eq.(3.2-1). Before using the moment area method, the mass of gun body and recoil travel are known, and the relation between breech force and time is also known.
The basic principal of the moment area method is conservation of linear momentum. The recoil mass starts at rest and returns to stop after recoil stroke ends. Hence, the resultant force on the recoil mass has to be zero by linear impulse-momentum law. Consequently, the impulse caused by the breech force and recoil mass must be equal to the impulse of total resistance to recoil in the recoil period. Generally speaking, the law can be expressed as “Impulse in = Impulse out”.
The basic motion of equation:
( ) ( ) sin
r r
F=M X =B t −K t +W θ
∑
By integrating one time of above equation,
( ) ( ) ( sin ) t t r a b r M X =
∫
Bτ τd −∫
K τ τd + W θ (3.4-1)tHere B( )τ acts at t =a, and K( )τ acts at t b= . For simplifying representations:
( ) ( ) t a D t =
∫
Bτ τd (3.4-2) and ( ) ( ) t b H t =∫
K τ τd (3.4-3)time . t
By rearrangement of Eq.(3.4-1), Eq.(3.4-2) and Eq.(3.4-3),
( ) ( ) ( sin ) r r M X =D t −H t + W θ t (3.4-4) Integrating Eq.(3.4-4), 2 1 ( ) ( ) ( sin ) 2 t t r a b r M X =
∫
Dτ τd −∫
H τ τd + W θ t (3.4-5)Then, integrate the right side of Eq.(3.4-5) as follows:
[
]
( )[
]
( ) 1 2 ( ) ( ) ( sin ) 2 t t t t r a b r a b dD dH M X D d H d W d d τ τ t τ τ τ τ τ τ τ τ θ τ τ = −∫
− +∫
+ (3.4-6) or 2 1 ( ) ( ) ( ) ( ) ( sin ) 2 t t r r a b M X =tD t −∫
τ τ τB d −tH t +∫
τ τ τK d + W θ t (3.4-7) where ( ) t a B d τ τ τ∫
is the moment of areas of breech force, and ( )t
b
K d
τ τ τ
∫
is the moment of areas of total resistance to recoil. Therefore, Eq.(3.4-7) can be rewritten as:2 ( ) ( ) 1 ( ) ( ) ( ) ( ) ( sin ) 2 ( ) ( ) t t t t a b r t t r a b a b B d K d M X tD t B d tH t K d W t B d K d τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ = −
∫
∫
− +∫
∫
+∫
∫
θ (3.4-8)( ) ( ) ( ) t a t a B d t B d τ τ τ α τ τ =
∫
∫
(3.4-9) ( ) ( ) ( ) t b t b K d t K d τ τ τ β τ τ =∫
∫
(3.4-10)where α( )t which is generated by B( )τ in time t , is the centroid of the areas, and β( )t , which is generated by K( )τ in time t , is the centroid of the areas, shown inFigure 3.4-2. Then, Eq.(3.4-8) can be rewritten as Eq.(3.4-11),
2 1 ( ) ( ) ( ) ( ) ( ) ( ) ( sin ) 2 r r M X =tD t −α t D t −tH t +β t H t + W θ (3.4-11)t or
[
]
[
]
1 2 ( ) ( ) ( ) ( ) ( sin ) 2 t t r r a b M X = −t α t∫
Bτ τd − −t β t∫
K τ τd + W θ t (3.4-12)When recoil stroke finishes, the motion of recoil mass stops. According to the linear impulse-momentum law, some terms can be defined: L is the recoil length, is the time of the recoil stroke,
r
t *
I is the total impulse, *
B
I is the impulse of the breech force, α( )tr
is the distance from a beginning to the centroid of area under the curve of breech force, and ( )tr
β is the distance from a beginning to the centroid of area under the curve of total resistance to recoil.
By using above definitions, Eq.(3.4-1) can be summarized as:
* *
sin
B r r
or ( ) ( ) sin r r t t r r b a K τ τd = Bτ τd +W t
∫
∫
θ (3.4-14)Eq.(3.4-13) and Eq.(3.4-14) show that the area under the curve of K t( ) is equal to the areas under the curves of breech force and recoil weight force as shown in Figure 3.4-2. When the recoil motion finishes, X =L, and t= . Eq.(3.4-12) can be written as: tr
[
]
*[
]
* 1 ( ) ( ) ( sin ) 2 r r r B r r r 2 r M L= t −α t I − t −β t I + W θ t (3.4-15)As a result, Eq.(3.4-14) and Eq.(3.4-15) can be used to decide tr and K t( ).
3.5 Forces Contributing to the Total Resistance to Recoil
In the design calculation of the recoil brake, one must start with the total braking force
K, where this force is made up of several components, as shown in Section 3.2, Eq.(3.2-3). These components of force include the recuperator force FR, the sliding track friction R , SL
the packing friction R , and the hydraulic braking force P H. In general, an effort is made to achieve a constant braking force K along the recoil travel. However, a certain rise of the curve must be provided at the beginning and end of the recoil motion.
Figure 3.5-1 Curve for the braking force [3] [6]
Figure 3.5-1 is the curve for the braking force. For specified curves for the quantities
R S
K =K +RL (3.5-1)
It can be taken from the diagram as a function of the recoil travel. And is the rod pull force.
R R P
K =F +R +H
Based on the experience with standard guns, the frictional force is about 3 to 7% of the total braking force K.
3.5.1
n
The Recuperator Force
The counterrecoil mechanism should return the gun tube, which has recoiled back, to the in-battery position, i.e., the initial position. Pneumatic recuperator is a common recuperator where the gas in the reservoir makes direct contact with fluid used as a transmission and sealing medium, or through a membrane or a floating position not clear.
During the recoil, the gas is polytropically compressed from the initial volume to the final volume. It follows that:
n g g P V =Constant (3.5-2) n i i x x PV =P V (3.5-3) and P V C n C = (3.5-4)
where Pg is the gas pressure, Vg is the gas volume, is polytropic exponent, is the recuperator gas pressure in battery,
n Pi
x
constant pressure specific heat, and constant volume specific heat. In general, the recuperator uses nitrogen, and the polytropic exponent can be taken as . Eq.(3.5-3) can be rewritten as:
V C 1.6 n= ( / )n x i i x P =P V V (3.5-5) and x i x V = − ΔV V (3.5-6)
where Δ is increase in Vx V , and it is decided by the type of the recoil mechanism. For x
example, the independent type shown in Figure 2.2-3:
x cr
V A X
Δ = (3.5-7)
where Acr is the effective area of the recuperating piston, and X is the recoil length. The dependent type shown in Figure 2.2-4:
x i x R R
V V V A X AX
Δ = − = = (3.5-8)
where AR is the effective area of the recuperating cylinder, X is the displacement of R
control rod, and A is the effective area of the recoil piston.
When Pi and Vi are known, and V can be computed by Eq.(3.5-6), Eq.(3.5-7), and x
Eq.(3.5-8), then P will be calculated by Eq.(3.5-5). So, the recuperator force is: x
R R
3.5.2 Frictional Force of Sliding Surfaces
The sliding track friction is generated by the surface friction of rigid bodies at motion. Besides, the magnitude of the force is decided by the frictional coefficient, the recoil length, and the elevation of the gun.
Figure 3.5-2 Sliding track friction on mechanism
Figure 3.5-2 shows the relation between forces and reacting forces of the gun body. In order to calculate the sliding track friction R , the guide forces SL and are solved first. Refer to Eq.(3.2-2) 1 N N2 1 2 ( SL
R =μ N + N ) , the frictional coefficient μ can be decided by the relative sliding materials. An intersection point of N2 and R is a pivot, and it can be SL
used to balance the moment.
1 p ( rsin ) f R r ( rcos ) a
N a+Bd + W θ d =K d + W θ b+F df (3.5-10)
or
d d
where N1, N2 are the guide forces, B is the breech force, is the rod pull force, is the inertia force of the recoiling parts, , , are the distances from the center of gravity to the guide forces, and ,
R
K Fa
a b c
p
d df, dr are the lengths of the force arm. Furthermore,
B and Fa can be gotten from: sin
a r
F = +B W θ−K (3.5-12)
In addition, the sum force on the vertical direction of the motion is zero. In other words, using the force balance that it means N1+N2 =Wrcosθ . When N1<Wrcosθ ,
2 rcos 1
N =W θ−N (3.5-13)
By rearrangement of Eq.(3.2-2), Eq.(3.5-1), Eq.(3.5-12), and Eq.(3.5-13),
1 cos r f r f p r d d b d d d N K W a a a μ θ − − − = + + B (3.5-14) 2 (1 ) cos r f f p r r d d d d b d N W K a a μ θ B a − − − = − − − (3.5-15)
On the other hand, when N1 >Wrcosθ,
2 1 rcos
N =N −W θ (3.5-16)
By rearrangement of Eq.(3.2-2), Eq.(3.5-1), Eq.(3.5-12), and Eq.(3.5-16),
1 cos 2 2 2 r f r f p r r r d d b d d d N K W a d a d a d μ θ μ μ − − − = + + + + + μ r B (3.5-17) 2 ( 1) cos 2 2 2 r f r f p r r r d d b d d d N K W a d a d a d μ θ μ μ − − − = + − + + + + μ r B (3.5-18)
3.5.3 Frictional Resistance of Packings
The packing friction is at the piston rod packings on the recoil brake and recuperator, as well as the piston packings in the recuperator. The packings are used to prevent the leakage of the sliding parts. Besides, the packings resist the sliding parts tightly because of the fluid pressure and the action of spring. As a result of the hydrostatic property of packings, the axial pressure of packing is equal to the radial pressure of packing used to prevent the leakage. Moreover, the radial pressure has to be greater than the maximum pressure of the fluid in order to ensure sealing up.
In order to find the packing friction R , the radial force of packing on cylinder P is needed to know: o F 1 o R F =A P (3.5-19)
where A1 =πD1 1b is the contact area of packing on cylinder wall, is the radial pressure in packing. In addition, can be rewritten by as follows:
R P R P Pa R p P =K Pa o (3.5-20) a s P =P +P (3.5-21)
where Kp is the pressure factor depending on material, P is the axial pressure produced s
by spring, and is the fluid pressure on packing at any recoil position. And can be known by conditions of recoil motion. According to the maximum pressure of the fluid,
o
P Po
s
P
can be expressed as:
( )
R m p s
then ( 1) s m p P P K ν = − (3.5-23)
Where ν is the leakage factor, and is the maximum fluid pressure. By rearrangement of the above-mentioned, the packing friction can be calculated,
m
P
P o
R =μF (3.5-24)
However, the independent type,
P c r
R =R +R (3.5-25)
and the dependent type,
P c R A r R R R A = + (3.5-26)
where R is the packing friction of the recoil brake, c R is the packing friction of the r
recuperator, AR is the effective area of the recuperating cylinder, and A is the effective area of the recoil piston.
3.5.4
L
Hydraulic Baking Force
A hydraulic braking force can be basically developed in a fluid filled cylinder. Owing to the fact, a piston coupled to the recoiling masses presses the displaced fluid through a narrow orifice at recoil motion. From the above-mentioned, the total braking force, the recuperator force, the sliding track friction, and the packing friction are known. According to K =KR+RS , and KR =FR +RP +H , the hydraulic braking force is:
( R P SL)
3.5.5
o
Effective Area of the Equivalent Orifice
In a recoil stroke, the fluid has to flow through the connecting ports, piston ports, the orifice, the slots in sleeves, and leakage areas. And all areas the fluid flowing can be equal to the effective areas of a single orifice. In order to understand the relation between the effective areas of an equivalent orifice and the hydraulic braking force, some assumptions on the fluid are established first:
1. Uncompressed fluid.
2. Inviscosity fluid.
3. Steady flow.
4. One-dimensional flow
For example, the flow velocity of the dependent type through the recoil orifice is,
( ) ( ) r o o Q =Av x =C a v x (3.5-28) and ( ) 2 ( ) o v x = gh x (3.5-29)
where Qr is the rate of flow, A is the effective area of recoil piston, is the recoil velocity, is the orifice coefficient, is the area of recoil orifice, is the velocity of flow through orifice, is the acceleration of gravity, and is the velocity head at ( ) v x o C ao v xo( ) g h x( ) x position.
( ) b r P x P P Δ = − x (3.5-30) and ( ) ( ) P x ωh x Δ = (3.5-31)
where is the fluid pressure of recoil cylinder, is the gas pressure at any position of recoil, and
b
P Prx
ω is the density of fluid. By rearrangement of Eq.(3.5-29), Eq.(3.5-31),
can be rewritten as:
( ) o v x 2 ( ( ) o g P x v x ω Δ = ) (3.5-32)
Also, the area of recoil orifice can be rewritten by Eq.(3.5-28), and Eq.(3.5-32):
( ) ( ) ( ) 2 ( ) o o o o Av x Av x a C v x C g P x ω = = Δ (3.5-33)
As a result of ΔP x( ) can be gotten by:
( ) ( ) /
P x H x A
Δ = (3.5-34)
where H x( ) is the hydraulic braking force generated by the orifice. Now, the effective area of an equivalent orifice can be defined as:
e o
a =C ao (3.5-35)
Then, by rearrangement of Eq.(3.5-33), and Eq.(3.5-35):
( ) ( ) 2 ( ) 2 ( e ) A a Av x Av x g P x gH x ω ω = = Δ (3.5-36)
Eq.(3.5-36) means that all fluid flows through the orifice. But if some fluid remains on the gap of the control rod, Eq.(3.5-36) will be rewritten as follows:
(1 ) ( ) (1 ) ( ) 2 ( ) 2 ( C C e R R A A ) A a A v x A v x A g P x A gH x ω ω = − = − Δ (3.5-37)
where is the cross-section area of the control rod, and is the effective area of the recuperating cylinder.
C
A AR
3.6 Remarks
1. In this chapter, all forces on the recoil mechanism are mentioned, and their relations are stated clearly.
2. The basic principles and mathematical calculations of each force are introduced. From this, many parameters which will influence the recoil mechanism are known, as shown in Table 3.6-1.
3. In accordance with the theory of each important part, the mathematical model of the recoil mechanism has its completeness and reliability.
4. Before analyzing or designing, it can check the mathematical theory of the model. The checking process ensures that the product can be not only supported by original information, but also designed correctly.
Table 3.6-1 List all force and parameters of the recoil mechanism
Force Name Parameters
Breech force W , P WC, Pmax, V0, U0, Ab
Total braking force K =H+FR+RP+RSL
z Hydraulic braking force A, X , a , , e g ω
z Recuperator force A , R Pi, Vi, X , n
z Packing friction A , 1 Kp, Pm, μ , υ
z Sliding track friction W , r X , θ, μ
![Figure 2.2-4 Hydropneumatic recoil mechanism (dependent type) [3]](https://thumb-ap.123doks.com/thumbv2/9libinfo/8257103.171936/30.892.154.768.567.927/figure-hydropneumatic-recoil-mechanism-dependent-type.webp)
