A mathematical model for the separation behavior of a split type low-shock separation bolt

Acta Astronautica 164 (2019) 393–406

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A mathematical model for the separation behavior of a split type low-shock separation bolt

T

Dae-Hyun Hwanga, Jae-Hung Hana,∗, Juho Leeb, YeungJo Leeb, Dongjin Kimc a

Department of Aerospace Engineering, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, Republic of Korea The 4th R&D Institute, Agency for Defense Development, P.O.Box 35, Yuseong-gu, Daejeon, Republic of Korea c Energetic Materials & Pyrotechnics Department, Hanwha Corporation, Oesam-ro 8beon-gil, Yuseong-gu, Daejeon, Republic of Korea b

A R T I C LE I N FO

A B S T R A C T

Keywords: Pyrotechnics Pyroshock Separation bolt Mathematical model Low-shock separation mechanism

Pressure cartridge type separation devices which are widely used in fairing, stage separations of space launch systems, and many other aerospace fields generate much lower pyroshock and produce no high-speed debris compared to frangible explosive separation devices. However, because the operation is completed in a few milliseconds and the releasing mechanism is complicated, their separation behavior is difficult to experimentally identify. This paper presents a mathematical model to simulate the separation behaviors for a split-type separation bolt, one of the pressure cartridge type separation devices. The mathematical model includes a combustion model, buckling resisting model, split behavior model related to static and dynamic friction, O-ring friction model, contact force model, and slip angle model. Each composing models are obtained by mathematical formulation or numerical analysis. An efficient contact model is constructed by using virtual penetration model appropriately for complex contact phenomenon. To validate the established model, separation experiments were performed; the results are then compared with the mathematical model. Present study show that complex mechanical behaviors coupled with combustion of solid propellant charge can be efficiently simulated by the mathematical model.

1. Introduction Since the 1960s, pyrotechnics separation devices have been widely used in various engineering areas where reliable separation is required. Because they have many advantages such as light device weight, low input energy, high reliability, and fast operation time, space launch systems have adopted pyrotechnics separation devices as single shot separators in stage separation and fairing separation, among many others. For example, in Apollo mission, at least 210 pyrotechnic devices were used to perform a myriad of onboard, inflight, timed, and controlled tasks automatically or on command in each of the spacecraft's systems [1]. All devices in space mission should have extremely high reliability, because improper operation of even small parts or components may cause loss of crew, failure to meet a mission objective, or an aborted mission. The pyrotechnics separation devices can be classified into high-explosive type or pressure cartridge type based on their main pyrotechnic output charge. The high-explosive type devices are separated by breakage of the structure of the device itself (e.g. high explosive bolt) and pressure cartridge type devices are separated through a release mechanism actuated by high pressure [2]. The high-explosive

∗

type devices have high reliability and generate high energy with a very simple principle of operation. However, they produce fragments and strong pyroshock, which could damage electrical devices. According to Moening [3], out of approximately 600 launches, 83 shock-related problems occurred with separation devices up to 1985, and over half of them resulted in catastrophic loss of missions. As an effort to reduce the pyroshock from high-explosive type separation devices, numerical analyses for identifying the separation mechanism using hydrocodes have been performed. Lee et al. [4–6] identified a separation mechanism for a ridge-cut explosive bolt that uses PENT and RDX by using ANSYS AUTODYN, and conducted a parametric study on specific geometrical dimensions. Low-shock separation devices also have been developed and used to reduce the pyroshock when the separation event occurs; the pressure cartridge types such as pin pullers, release nuts, and separation bolts generate much lower pyroshock and no fragments compared with highexplosive type separation devices [7–10]. There are various examples using this types of a separation device. They have been used to secure the guided missile inside the launch pad before the guided missile is fired and the devices are released the missile at the moment of launch.

Corresponding author. E-mail address: [email protected] (J.-H. Han).

https://doi.org/10.1016/j.actaastro.2019.07.035 Received 1 May 2018; Received in revised form 21 May 2019; Accepted 16 July 2019 Available online 26 August 2019 0094-5765/ © 2019 IAA. Published by Elsevier Ltd. All rights reserved.

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Nomenclature rb P n a m˙ gen

ηg ρzpp Ab e

Rzpp N ρ V ηp Tf cp Apis Q˙ loss R γ Fbolt Fp

= wedge angle of the split on cross-section, rad = effective sliding angle of the split wedge, rad = static and dynamic friction coefficient = virtual penetration depth between surfaces of each component, m FN,contact = normal contact force between the bolt and the split, N FO-ring = friction force of the O-ring, N fc = coefficient of friction with seal compression per unit length, N/mm fh = coefficient of friction with fluid pressure per unit area, N/mm2 Lc = circumferential length of the contact portion where the O-ring slips, m Ap = contact portion area of the O-ring, m2 Dout, Din = outer and inner diameter of the O-ring glands, m vpis, xpis = velocity and the displacement of the piston, m/s vbody, xbod y = velocity and the displacement of the body, m/s vbolt, xbolt = velocity and the displacement of the bolt, m/s

θ θeff μs , μk Δ

= burn rate, m/s = pressure in the combustion chamber, Pa = burn rate exponent = burn rate constant, mm/s⋅Pan = mass flow rate of the gas produced from the burning of zirconium potassium perchlorate particles, kg/s = fraction of gases in product = density of the zirconium potassium perchlorate particles, kg/m3 = burning surface area, m2 = burned distance from the surface of the zirconium potassium perchlorate particle, m = radius of the zirconium potassium perchlorate particles, μm = total number of particles = gas density in the combustion chamber, kg/m3 = volume of the chamber, m3 = non-ideal gas correction factor = temperature of the generated gas, K = specific heat at constant pressure, J/kg·K = pressured area of the piston, m2 = heat loss rate of the gas, J/s = the gas constant, J/kg·K = specific heat ratio = total actuating force on the bolt, N = pressure force on the piston, N

Subscripts pis bolt body split zpp

= the = the = the = the = the

piston bolt body split zirconium potassium perchlorate main charge

energy, ignitibility, peak pressure, and burning time [14]. In addition, the combustion of the charge, the expansion of the chamber, and the complex separation mechanism are respectively coupled. The separation characteristics are also influenced by the amount of propellant, the initial volume, volume increase rate, mass of components, etc. These complex phenomena including the chemical and mechanical separating process make it very difficult to measure or observe the behaviors inside the separation devices. Because experimental identification of performance is difficult and complicated, numerical analysis can be a useful method. Wang et al. [15] predicted a shock response of separation nut and identified sensitive parameters for shock response using the hydrocodes, ANSYS AUTODYN. They investigated the effects of the preload level and the amount of explosive on the pyroshock response. The simulation results showed that the lower preload is the better way to reduce the pyroshock. Zhao et al. [16] established a finite element model of pyrotechnic separation nuts and they simulated separation process based on the explicit dynamic codes LS-dyna with Arbitrary Lagrange-Euler (ALE) algorithm to insight the mechanism of separation shock and provide a reference for the design of separation nuts. The separation shock generated in the unlocking process is analyzed, and they showed two sources of shock and influence of pre stress on the separation shock. To identify detailed separation mechanisms and effects of parameters, establishing a proper mathematical model to analyze the separation behavior would be beneficial. There have been a few studies on the mathematical modeling of a pin puller using a pressure cartridge and a simple expansion chamber. Jang et al. [17] developed an analytical model that simulates the behavior of pin pullers through an 0-D ideal gas model by employing a simple combustion model. A correction factor was introduced on the gas generation rate and the heat loss rate was empirically determined. The mathematical results were validated through comparison with experimental results for the motion of the pin shaft and the pressure in the combustion chamber with various amounts of propellants. Paul and Gonthier [18] proposed a quasi-1-D model to explain the unsteady gas dynamic effect of explosively actuated valves.

In the space launch vehicle, low-shock separation devices can be utilized in a payload separation system of satellite. They also have been used to jettison the retro-rocket packages on the unmanned Surveyor lunar lander and the Viking Mariner lander, and are being used in numerous application on the space shuttle such as separation from an orbiter [2]. Cui et al. [11] conducted a dynamic simulation of the typical satellite separation system, a V-segment clamp band, with a highexplosive bolt. They performed a parameter sensitivity analysis of dynamics of satellite separation and revealed the parameter that has the most prominent effect on the dynamic-envelope of clamp band. Lancho et al. [12] designed a new Clamp Ring Separation System (CRSS) to considerably increase the load capability, while maintaining a low release shock level. This Clamp Ring Separation System uses a pyro-bolt and pyro-cutters as release devices. Michaels and Gani [13] investigated a tube-in-tube concept for separation of bodies in space theoretically and experimentally. This separation system is based on generation of high pressure gas by combustion of a solid propellant and restricting the expansion of the gas only by ejecting the two bodies in opposite directions. They developed an interior ballistics model that takes into account solid propellant combustion, heat losses, and gas phase chemical reactions, and compared the results of the model with the experimental results. The use of these low-shock separation devices is increasing in rocket and space launch vehicle systems. These devices have advantages of high fastening strength, environmental resistance, and high separation reliability without generating a high pyroshock or fragments [2]. The high pressure gas is due primarily to the high temperature of combustion, and therefore a pressure cartridge type separation device should be operated very quickly, within a few milliseconds; otherwise the efficiency of the cartridge will be reduced by heat loss [2]. Propellant or pyrotechnic powders are used as the main charge in numerous impulsetype pressure cartridges, which are utilized for the actuation of valves, thrusters, and guillotine devices. Important factors that influence the selection and sizing of the propellant powder for a given application are maximum required 394

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They also conducted a parametric study to verify that the cross-sectional area of the connection port controls the release of explosive energy and the acoustic energy transfer rate. Lee [19] developed a onedimensional gas dynamics model to simulate unsteady effects within a closed-bomb firing and a normally open pyrotechnic valve. He revealed that unsteady flow effects affect the performance of a pyrotechnic device. Gonthier et al. [20] established a mathematical model including conservation equations of a multiphase system for the NASA Standard Initiator (NSI) driven pin puller and predicted performance. Using this model, a sensitivity analysis was also conducted [21]. Han et al. [22] established an analytical model of the pin puller and compared the experimental results with the pin behavior and pressure in the chamber. Using the model, an optimization study for the amount of propellant, combustion constant, and granule size was conducted in order to obtain the maximum pressure and maximum kinetic energy of the pin. Methods to prevent damage caused by the pyroshock have been also studied. Hwang et al. [23] established a mathematical model to predict the separation behavior of a ball-type separation bolt which operate with complex mechanical interaction. The model consists of simultaneous differential equations that include a combustion model and equations of motion involving interactions between various components. The effects of the coefficient of friction and amount of propellant were investigated using the model. They showed that the complex performances of the separation bolt can be evaluated with a computationally efficient model. Woo et al. [24] proposed an mathematical model to predict the complicated coupling behavior of a pyroshockreduced separation nut, which has two variable-volume chambers connected by the vent hole. Using the model, parametric studies are then carried out to investigate the effects of the design parameters on the separation behavior. Most previous numerical models for low-shock separation devices have dealt with simple pin pullers or pyro-actuated valves; a numerical analysis or mathematical modeling of pressure cartridge type separation bolts with complex mechanical contact interactions has not been reported to date. During the separation of pressure cartridge type separation bolts, there are a series of contact interactions, repeating contact and detaching. On top of these complex contact interactions, combustion of propellant charge, and friction and resisting forces should be considered at the same time. In this study, we established a mathematical model simulating the separation behavior of a split-type separation bolt. The model consists of simultaneous differential equations including a combustion model, buckling resisting model, split behavior model related to static and dynamic friction, O-ring friction model, contact force model, and slip angle model. The buckling resisting model was obtained using a FEA (explicit analysis) for one column of a ring tube. The contact force model was established by defining penetration on the contact surfaces. Relative behaviors of a bolt and splits are described as a function and the effective sliding angle of the splits was evaluated. To validate the established model, separating experiments were performed; these experimental results were then compared to the mathematical model results. This paper consists of six sections. Section 2 describes the splittype separation bolt considered in this study. Section 3 describes the operation principle of the bolt and establishments of the mathematical models. Section 4 summarizes the simulation results and Section 5 provides the experimental method and results, as well as a comparison between the numerical and experimental results. Finally, conclusions are given in Section 6.

Fig. 1. 3D model of the split-type separation bolt.

split-type separation bolts, withstand much heavier load because they have a large contact area owing to their circular shape. 2.1. Split-type separation bolt The split type separation bolt considered in this study consists of a total of eight parts, as shown in Fig. 1. The main charge inside the initiator burns to create pressure, a housing and a cap contain internal components, and a body combines with the initiator and restrains four ring splits from spreading out. A piston receives the pressure force generated inside the chamber and transfers it to the bolt. The diameter of the bolt is 1/2 inch. The ring split divided into four pieces restrains the head groove of the bolt and fixes the bolt so that the bolt does not separate before the operation. The bolt is attached to the structure to provide a locking force, and finally a ring tube is buckled when the body moves back by the combustion pressure, thereby reducing the speed and absorbing shock caused by mechanical impact. The split-type separation bolt generates mechanical shocks due to the inter-component interaction during separation and pyroshock, which is much lower than that of the high-explosive bolt from the initiator. Table 1 presents the materials used for each part and their properties. The separation sequence and a force diagram of the split-type separation bolt are presented in Figs. 2 and 3. In the beginning, the pressure inside the chamber is sharply increased after the initiation by combustion of the main charge, ZPP. The pressure of combustion gas creates a pushing force ‘Fp ’ on the body and the piston, the body moves backward with received resisting forces ‘Fbk ’ from the four columns of the ring tube and the piston is still stopped by the restraint. The ring splits are spread out in the radial direction after the restraint on the ring splits is removed, and at the same time, the piston and the bolt move forward by the pressure force and the pre-load ‘Fpre ‘. When the radius of the inner circle of the ring splits is larger than the radius of the bolt head, the separating operation is completed . 2.2. Initiator (pressure cartridge) In this split-type separation bolt, the initiator, called a PC-300 Table 1 Applied materials for each component. Material

Young's modulus (GPa)

Yield Stress (MPa)

Tensile Stress (MPa)

SUS 630 (45 HRC)

202.9

1227.9

1379.3

SUS 303

184.7

517.4

737.2

2. Separation bolt using split mechanism Housing Cap Body Piston Bolt Split Ring tube

In this study, we design and investigate a low-shock separation bolt called a split-type separation bolt. There are two types of low-shock separation bolt according to the shape of the locking component. The first, ball-type separation bolts, have relatively light weight and operate faster, but can be applied only to light load cases due to the high stress concentration at the contact point of the balls. In contrast, the second, 395

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Fig. 2. Sequence of the separation process.

Table 2 Parameters for analytical combustion model [17]. Parameter

Value

a (burn rate constant) na (burn rate exponent) η pa (non-ideal gas correction factor) η ga (fraction of gases in product) γ a (specific heat ratio) Tf (temperature of the generated gas) Rzpp (radius of the ZPP particles) ρ zpp (density of the ZPP particles)

3.767 mm/ s⋅Pan 0.182 0.68 0.43 1.104 4810 K 24 μm 2440 kg / m 3

a

Fig. 3. The actuating and interacting forces in the separation bolt.

Non-dimensional parameter.

Fig. 6. Configuration of the ring tube and columns. Fig. 4. The PC-300 initiator.

Fig. 5. Schematic description of constituent models for the mathematical model of the separation bolt. 396

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Fig. 7. Mesh of the simplified geometry for one column of the ring tube for the buckling force analysis.

Fig. 8. Boundary conditions for the buckling force analysis.

Fig. 9. Compressing sequence of one column from the buckling analysis.

Fig. 10. Buckled and compressed state of the ring tube after real operating test.

shown in Fig. 4, is used for pressure generation purposes to generate operating force, not to ignite the high-explosives. PC-300, designed and produced by Hanwha Inc., is used as a pressure cartridge, and as a main charge, 65 mg of zirconium potassium perchlorate (ZPP) is applied to produce about 2.07 MPa (300 psi) at 10 cm3 [25]. ZPP is the most common primary charge of initiators and is also referred to as the NASA Standard Initiator.

Fig. 11. Compressing force history with respect to displacement of the body from the buckling analysis.

3. Mathematical model of separation behavior The operation of this separator takes place within a few milliseconds and it also has various mechanical interactions in the separation 397

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combustion model, resisting force models, and interaction models between the components. The actuating force by gas pressure is calculated by the combustion model. As shown in Fig. 5, the resisting force model includes a buckling force analysis for the ring tube, friction force model of the O-ring, and friction force model on the body and the splits. The interaction model consists of the surface interaction model and the wedge slip angle model. 3.1. Stage model of separation sequence Fig. 12. Force diagram for the cross section (3D modeling software) of the separation bolt at initial condition.

A mathematical model was established by dividing the main mechanical events into four stages as shown in Fig. 2. First, in Stage 1, the ZPP charge in the PC-300 starts combustion by an electric signal, and the body moves backward and the restraint of the splits is released. Thereafter, in Stage 2, the four splits spread out, and then the bolt is separated. In Stage 3, the piston keeps moving after separation of the bolt, and then collides with the spread splits. Stage 4 is defined as the interval after the piston collision. 3.2. Combustion model The initiator PC-300 uses zirconium – potassium perchlorate (ZPP) as the main explosive charge. This charge consists of numerous minute granules and the binder mixture. The granules have an uneven surface; however, to simplify the combustion model, the combustion equation is constructed assuming that each particle is perfectly spherical [21]. By using 0-D combustion model based on Saint Robert's law, pressure of the gas inside the chamber which essential to obtain the actuating force can be calculated. As is known in many references, Saint Robert's law (also known as Vieille's law) presents that a logarithm of the burn rate of solid propellants is proportional to the logarithm of the pressure as the relationship a ln p versus ln rb [26]. Hence, the burn rate rb is given by rb = aPn, where P is the pressure, n is the pressure exponent, and a is a constant determined by the chemical composition and the initial propellant temperature. In this study, this law was used to simulate the pressure of the combustion gas generated from the ZPP charge. In conclusion, the mass generation rate of the gas from the ZPP particles is estimated as follows [17,22]:

Fig. 13. Force diagram for the cross section (3D modeling software) of the separation bolt during separation.

m˙ gen = ηg ρzpp Ab rb Fig. 14. Friction coefficient of rubbing surface according to compression [30].

(1)

Since the volume of the condensed phase in the combustion products is very small compared to the volume of the same mass of gas product, the volume of the condensate is assumed to be negligible in calculating the pressure of the chamber. If the burned distance is e, the radius of spherical ZPP particles is reduced. Thus, the total burning surface area is calculated using Eq [2].

Ab = 4πN (Rzpp − e )2

(2)

where Rzpp is the initial radius of the ZPP particles and the total number of particles, N, is equal to

N=

mzpp 3 ρzpp (4πRzpp /3)

(3)

The total number of particles is calculated to be about 460,000 by dividing the total ZPP mass by the mass of one granule, which is obtained from the radius of one ZPP particle and the density of the ZPP. The relationship between the burn rate rb , the burned distance e, and the pressure is as follows:

Fig. 15. Friction coefficient of rubbing surface according to pressure [30].

process, and as such it is very difficult to identify the detailed separation behaviors experimentally. Therefore, many design modifications and iterative experiments must be performed during the development process. This problem can be solved by introducing the mathematical model developed in this study. The mathematical model that simulates the separation behavior of a split-type separation bolt consists of a

rb =

de = aP n dt

(4)

The above combustion model can be used to calculate the pressure and density inside the chamber [22]. First, the law of conservation of the mass in a control volume of the combustion chamber is given by the following equation: 398

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Fig. 16. Installation method of an O-ring on: a) a piston, b) a rod.

d (ρVcp T ) dt

= ηp m˙ gen cp Tf − P (Apis vpis + Abody vbody ) − Q˙ loss

(6)

The heat loss rate is empirically assumed to be 120 J/s. By combining Eqs [5,6]. using the ideal gas state equations Eqs. [7,8], the rate of density and pressure are derived as Eqs. [9,10], respectively.

P = ρRT

(7)

R = c v (γ − 1)

(8)

m˙ gen − ρV˙ dρ = dt V

(9)

ηp m˙ gen RγTf − (γ − 1)[P (Apis vpis + Abody vbody ) + Q˙ loss ] − PV˙ dP = dt V (10) The rate of the chamber volume is the sum of space generated by consuming the ZPP particles and advancing of the piston, as given in the following equation:

Fig. 17. Concept of virtual penetration between the bolt and the split.

dV = Ab rb + Ap vp dt

(11)

The experimentally obtained combustion parameters applied to this combustion model are shown in Table 2. 3.3. Resisting force model 3.3.1. Buckling force analysis for the ring tube The buckling is the main separation force in this model and, at the same time, is a very important mechanism for the operation of the ring tube responsible for shock absorption. As shown in Fig. 6, by designing the 0.5C chamfer at the tip, a buckling mode, which is similar to the clamped-free buckling mode, can be expected. The buckling analysis was performed to confirm the resistance force acting on the body when the four columns of the ring tube were buckled. In the analysis, a simplified geometry just about one-column is used to reduce computational cost and time, as shown in Fig. 7. The buckling column and virtual load cell are filled with 0.07 mm and 0.1 mm hexahedron elements, respectively. In the thickness direction of the column, seven elements are stacked. Considering the previous studies [27–29], this element size is considered to satisfy the mesh convergence for the buckling problem. The remaining parts are meshed with 1 mm elements. In Fig. 8, a fixed boundary condition is applied on

Fig. 18. The sigmoid function applied for mathematical contact model.

d (ρV ) = m˙ gen dt

(5)

Also, the energy conservation relation of the internal gas can be written as Eq [6].

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Fig. 19. The separation behaviors of the ring splits and the bolt.

outer surfaces and a tangentially sliding boundary condition is applied on the cross section of the body, which moves to compress the column. Finally, a 5.4 m/s velocity boundary condition is applied on the end of the virtual load cell. This analysis model is calculated up to 1 ms due to the gap distance is 5.4 mm. Compressing force was estimated at the center cross section of the load cell, and the compressed displacement on the buckling contact surface was recorded. The buckling resistance force on the body is calculated by multiplying the average value of the principal stress and the cross sectional area of the load cell. From the analysis results, we could obtain the shape of the buckling process and the stress on the load cell. The buckling shape is shown in Fig. 9, the contact tip first slips due to the angled cut of the tip, and bending occurs after contact with the inner wall of the housing. Afterwards, when it bends and touches the cap nut, buckling with a similar shape to the clamped-pinned boundary condition occurs. The red arrows indicate the location of stress concentration on the column during the compression process. The analytically obtained compressed column shape is in good agreement with that of the experiment; Fig. 10 shows the buckled and compressed shape of the ring tube after the operating test. Fig. 11 shows the results of the calculation of the resistance force for all four columns with respect to the compressive displacement. The resistance force increases until the body displacement reaches 0.3 mm from the initial state, but is low in the process of breaking, as in the second and third sequences in Fig. 9. Fig. 11 shows much higher resistance when the second buckling of the short column occurs from the third sequence. In the subsequent compression process, the flat surface of the column that is lying sideways is compressed, and consequently the resistance increases sharply. The obtained resisting forces in Fig. 11 were applied to the mathematical model as a function of the displacement of the body.

Fig. 20. Displacements relationship between the bolt and the split.

Fig. 21. Cross section of the contact between the bolt and the split.

3.3.2. Friction force model on the body and the splits There are two cases where friction must be considered. The first is at the initial state. As described in Fig. 12, when the bolt receives preload and pressure force (Fbolt , blue arrow 1), the contact force between the bolt and the splits spreads the four splits (blue arrow 2), and then the spreading force becomes the contact force with the inner surface of the body (blue arrow 3). If the body is pushed to left in Fig. 12, friction force acts on the body at the contact with splits. This can be considered a static case, and therefore the friction force can be expressed as given in Eq [12].

Table 3 The initial values of the separation behavior simulation. Parameter

Initial value

Density in the chamber Pressure in the chamber Chamber volume

1.225 kg/m3 1 atm 0.485 cm3

Table 4 Maximum velocity and passing time comparison.

Mathematical model Experiment 1 Experiment 2 Difference with respect to the average value of exp. Results (%)

Maximum velocity (m/s)

Passing time, Δtbolt (ms)

20.34 20.23 19.10 3.4

0.823 0.972 0.938 13.8

sin θ − μs cos θ ⎞ Ff , body = μs Fbolt ⎜⎛ ⎟ ⎝ cos θ + μs sin θ ⎠

(12)

where Fbolt = Fpre  load + Fpressure . As described in Fig. 13, after the constraint of the splits is removed, the bolt and the splits begin wedge slip behavior with friction. Because movements of the bolt and the splits are coupled and also this is a dynamic mechanism, the behavior can be modeled as a relationship in terms of acceleration, rather than the friction force. In this case, the accelerations of the bolt and the splits are derived from Eqs [13,14]. 400

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Fig. 22. The time history of the combustion gas properties.

Fig. 23. The time history of the behaviors of the components.

abolt =

Fbolt − 4FN , contact (cos θ + μk sin θ) Mbolt

(13)

asplit =

FN , contact [(sin θ − μk cos θ ) − μk (cos θ + μk sin θ )] Mbolt

(14)

shown in Fig. 14. Thus, to utilize this characteristic for any hardness and compression rate, we obtained the slope change according to Shore A hardness using curve-fitting, and multiplied the seal compression percentage as given below:

fc = 0.02893 exp ⎡−⎛ ⎢ ⎝ ⎣

3.3.3. Friction force model of the O-rings Friction force of the O-ring was established by referring to the technical handbook from the manufacturer. Friction force of the O-ring is the sum of the friction with seal compression and friction with fluid pressure [30]. First, the coefficient of friction with seal compression is affected by the compression rate and shore A hardness of the rubber, as

Shore A − 94.6 ⎞2⎤ × Comp (%) [N / mm] 25.83 ⎠⎥ ⎦

(15)

Next, the coefficient of friction with fluid pressure has the following relationship shown in Fig. 15. By curve-fitting, as in Eq. [15], the coefficient of friction was obtained from Eq [16]. 401

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Fig. 24. Configuration of the experiment.

Fig. 25. The experiment setup: (a) Measure the displacement of the body, (b) Detecting the passing time of the bolt.

Fig. 26. The experiment results.

fh =

31.155P + 6.293e7   [N / mm2] P + 3.425e7

In conclusion, the total friction force of the O-ring is the sum of these two friction forces.

(16)

Each friction force can be estimated by multiplying the characteristic length and area by the obtained coefficients of friction. With the intalled configuration of the O-rings shown in Fig. 16, the characteristic length with seal compression is the circumferential length where the Oring slips, defined as follows:

πDout (installed on rod ) Lc = ⎧ ⎨ πD ⎩ in (installed on piston)

FO − ring = fc × Lc + fh × Ap 3.4. Surface interaction model

To simplify every single mechanical event, we established contact models, as shown in Eq. [20], for three interacting surface pairs under conditions of colliding and sliding by applying a virtual spring and a virtual damper, as shown in Fig. 17; three interacting surface pairs are ‘bolt-splits’, ‘housing-splits’, and ‘piston-splits’. By defining the virtual penetration depth as Δ , the force at contact will be proportional to the stiffness multiplied by Δ , and also the damping coefficient multiplied by Δ˙ when two surfaces are in contact due to the compression. If the

(17)

The contact portion area of the O-ring ‘Ap’ with fluid pressure is the area exposed to the fluid, and is given by Eq [18].

Ap = π (Dout 2 − Din 2)/4

(19)

(18) 402

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the real sliding angle, θeff , is smaller than θC . S according to the displacement of the bolt. Note that the 45° spreading out direction of each split (see Fig. 19) and the geometrical contact edge shape of the bolt and the split result in the effective sliding angle θeff being smaller than θC . S . During the split movement, there is no contact at the centric surface of the split, and hence all contact force is concentrated on both end edges. To identify the contact relationship between the bolt and the split, the displacement relationship was obtained, as shown in Fig. 20. By taking the arc tangent on the inverse slope of the curve ‘ ysplit = f (xbolt ) ’ in Fig. 20, the actual sliding angle was obtained as Eq [22]. As a result, θeff is 24.04° at point-1, which is the initial contact state, and then moving to point-2, it gradually increased to 90°, as shown in Fig. 21. −1 ⎛ df1 ⎞ ⎤ θeff = tan−1 ⎡ ⎥ ⎢ dxbolt ⎠ ⎦ ⎣⎝ ⎜

contact is lost, the normal contact force ‘FN , contact ’ should be set to zero. To consider this phenomenon, the sigmoid function, described in Eq. [21], was used instead of the step function, because the solution of the ODE system does not readily converge when applying the ideal step function. Figure 18 shows the sigmoid function used here for the contact simulation. The stiffness and damping coefficients were chosen in a way that there is no fluctuating motion at the moment of each contact in the dynamic behavior simulation.

sigmoid (Δ) =

1 1 + e−100x (mm)

(22)

The basic mechanics about θcs are as follow. As the θcs becomes larger, there arises a problem that the force of pushing the ring split in the radial direction with respect to the axial force acting on the bolt (pre-load and pressure force) increases and frictional force acting on the body increases accordingly. On the other hand, when the angle θcs becomes smaller, the force that the bolt pushes the ring split is decreased, so the frictional resisting force on the body is decreased. However, after the restraint on the ring split is released, the bolt may not be able to separate because the pushing force on the ring split is insufficient compared to the maximum static frictional force.

Fig. 27. Comparison between the mathematical model and the experiment results.

FN , contact = [K Δ+ C Δ˙ × (Δ˙ > 0)] × sigmoid (Δ)

⎟

4. Results from the mathematical model

(20)

This mathematical model calculates the time histories of the density of the combustion gas, the pressure, the volume of the chamber, the burning distance of the ZPP granule, and the mechanical movement of the body, the piston, the bolt, and the splits. The initial values of the density and pressure in the chamber, and the chamber volume are listed in Table 3. Since the components, the body, the piston, the splits and the bolt are initially stationary, the initial displacements and the velocities of these components are set to zero. From the simulation results, the end point of combustion is 0.328 ms, which is almost the same as the start point of the split

(21)

3.5. Wedge slip angle model The contact angle is one of the most important design parameters that affect separation behaviors of the bolt and the splits. The wedge angle of the split, θC . S , is designed as 30° on the cross section; however,

Fig. 28. Gas properties and combustion progress history with an increasing amount of charge. 403

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Fig. 29. Behavior history of the components with an increasing amount of charge.

Fig. 30. Gas properties and combustion progress history with an increasing initial volume.

the components, which include the body, the piston, the bolt, and the split. At Stage 1, the body is accelerated by pressure force, but the other components are still restricted, as described in Fig. 2. Then, in Stage 2, the restraint on ring splits by the body are removed and the splits spread out in the radial direction and, accordingly, the piston and the bolt move in the forward direction simultaneously while maintaining contact with the four splits. During this stage, the body reaches a maximum distance at 0.4 ms and the travel speed rises to 20.3 m/s and then stops due to the collision. At the beginning of Stage 3, the diameter of the inscribed circle of the ring splits becomes equal to the diameter of the bolt head, and thus separation is achieved. The analysis results show that the separation time is 0.849 ms. The body and the piston are accelerated until the piston collides with the splits. Before this collision,

movement. In Stage 1, when the pressure force on the body becomes higher than the static frictional force on the body from the contact with the split, the body moves backwards (in the direction of the initiator) and the chamber volume increases. Fig. 22 shows that the density and the pressure increase and decrease before the end of combustion because increase of the volume lessens the density and the pressure. The maximum peak pressure is 24.3 MPa at 0.144 ms. Dotted lines (Stage 2) in Fig. 22 show sudden changes at 0.45 ms, because at this moment the ring tube is completely crushed, and accordingly the body movement suddenly stopped. From the moment the piston collides with the split, both the body and the piston, which affect the volume change, are stationary, and thus the gas condition inside the chamber is constant. Fig. 23 shows time histories of the displacement and the velocity of 404

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Fig. 31. Behavior history of the components with an increasing initial volume.

rate of ZPP increases due to the increase of the pressure inside the chamber. An increase in pressure means an increase in operating force, which speeds up the behavior of all components, thus leading to a faster separation time. As can be seen in Fig. 29, the change in behavior at 40 mg–80 mg is greater than that at 80–120 mg. Figs. 30 and 31 show the change of gas properties, progress of combustion for a ZPP granule, and separation behaviors of the components in accordance with the increasing amount of charge. As the initial volume increases, the rising rate and the peak of the density and the pressure decrease for the same ZPP mass (65 mg in this case) and the burning rate is also slowed down. The pressure rising rate and the burning rate are decreased, so that the time of the peak pressure generation is delayed. After the constraint on the ring splits by the body is removed, the piston, the ring splits, and the bolt start to separate concurrently. At this point, the pressure values in the three cases are similar, so the behavior of these three components is only shifted with respect to the time axis as the volume increases, so the history curves of velocity and displacement is almost the same.

the splits hit the inner surface of the housing, and stop when the displacement of the split reaches 2.6 mm. At Stage 4, the collided piston stops its movement and the contact with the bolt is lost and the bolt is simply accelerated by the preload. In this analysis, we assumed a zero preload condition, and therefore the bolt moves with constant velocity. 5. Experimental validation for the mathematical model To validate the established mathematical model, separation experiments using the developed split-type separation bolt were conducted. The experiment concept is described in Fig. 24. The housing was installed in a vertical fixture that was mounted on a heavy base. The displacement of the body was measured by a laser displacement sensor (LDS, Keyence LK-G400) and the passing time of the bolt was measured by a beam sensor installed at Δxbolt (=1.48 mm) away from the bolt end. The beam sensor produces on-off outputs depending on whether the laser beam is broken or not. Fig. 25 shows photographs of the experiment setup. Two separation experiments were performed. As shown in Fig. 26, Δtbolt is calculated by measuring the time gap between the starting time of the body and the time of detecting the bolt. In addition, the displacement history and maximum velocity of the body were recorded, and compared with the analysis results using the mathematical model in Fig. 27. In general, the experimental and analytical results are in good agreement; the bouncing behavior of the body in Fig. 27 is considered to be due to the elasticity of the mount structure, the body, and the housing. Table 4 summarizes the maximum velocity of the body and the passing time of the bolt, showing quite small relative differences between the experimental and analytical results.

7. Conclusion We developed a low-shock split-type separation bolt using a separation mechanism applying a pressure cartridge, and a mathematical model that can simulate the separation behaviors of the developed splittype separation bolt is also proposed. Based on the operating sequence and mechanical events, the mathematical model was divided into four stages. Each stage has a different mechanical state, equation of motions, and resisting forces. To establish the equation of motions, the actuating force was derived using a combustion model based on Saint Robert's law. The main resisting forces including buckling force of the four columns on the ring tube, friction forces acting on the body, the ring splits, and the bolt, and friction of the O-rings installed on the body and the piston were considered in the mathematical model. The buckling behavior of the ring tube, which is also helpful for shock absorption, was analyzed by an explicit analysis method using ANSYS AUTODYN. From the explicit analysis results, we obtained the buckling force history with respect to the displacement of compression. The model of the static friction, which interrupts movement of the body, opposite to the pressure force, was derived related to force around the split. The dynamic friction model was derived from the acceleration relationship of the bolt and the split. Another friction force by the O-ring was also modeled. Proper functions for friction coefficients by seal compression

6. Parametric study using mathematical model Parametric studies for the amount of charge (AoC) and the initial volume were performed using established mathematical model. Three cases for each parameter, the amount of charge of 40 mg, 80 mg, and 120 mg, and the initial volume of 0.3 cm3, 0.5 cm3, and 0.7 cm3 were considered. Figs. 28 and 29 show the change of gas properties, progress of combustion for a ZPP granule, and separation behaviors of the components in accordance with the increasing amount of charge. Naturally, the density and pressure increase, the volume growth rate increases as the speed of the body and the piston increases. Also, the combustion 405

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and fluid pressure, respectively, were chosen. By combining these functions considering Shore-A hardness, seal compression, and fluid pressure, we obtained the total friction force due to the O-ring. If we divide stages according to every contact and detaching of interacting surfaces, the number of stages will be much larger and the model will become too inefficient. To realize a concise and efficient mathematical model, the contact forces between interacting components were modeled using a virtual penetration method on the contact surfaces. By assuming virtual penetration and stiffness on the contact surfaces, and simulating the contact condition using the sigmoid function, normal contact forces were calculated without manually considering every attaching and detaching behavior between the components. The simulation results from the established mathematical model were compared with experimental results for validation. An experiment to measure movement of the body and the bolt was conducted by using a laser displacement sensor and a beam sensor. The comparison results show good agreement for the body displacement history and beam sensor passing time. Parametric studies for the amount of charge and the initial volume were conducted using the established mathematical model. We found that the increasing amount of charge cause larger pressure so that combustion and separation become faster, and the increasing initial volume cause time of the peak pressure generation and separation time are delayed. In conclusion, in this study, we established and validated a mathematical model for simulating the behaviors of a split-type separation bolt.

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