Study on pyroshock propagation through plates with joints and washers

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Study on pyroshock propagation through plates with joints and washers

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Juho Lee

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, Dae-Hyun Hwang , Jae-Hung Han

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The 4th R&D Institute, Agency for Defense Development, P.O. Box 35, Yuseong-gu, Daejeon, 34186, Republic of Korea b Department of Aerospace Engineering, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 34141, Republic of Korea

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Article history: Received 21 September 2017 Received in revised form 7 February 2018 Accepted 30 May 2018 Available online xxxx Keywords: Pyroshock Pyrotechnics Flexural waves Hydrocodes ANSYS AUTODYN Shock response spectrum (SRS) Pyroshock experiments Pyroshock attenuation

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a b s t r a c t

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Pyrotechnic release devices are widely used in various missions in order to reliably separate the structural parts of the system. However, the pyroshock induced by an explosive pyrotechnic event can lead to fatal malfunctions of the adjacent electrical components. In order to mitigate these pyroshock issues, an accurate understanding of the pyroshock propagation in structures is essential. In this study, an experimental setup for the pyroshock propagation experiments is developed with the pyroshock excitation using pyrotechnic initiators. The pyroshock propagation analysis environment with a commercial hydrocode is established and verified through comparison between the analysis and the experimental results. The pyroshock propagates through the thin plates in the form of flexural waves (or anti-symmetric Lamb waves). Using the established numerical and experimental techniques, the effects of the pyroshock attenuation by the joints and washers are investigated. The plates connected by joints with different materials and the plates connected by joints with inserted washers made of different materials and in different thickness are considered. The experimental and numerical results are in good agreement: the pyroshock attenuation is highly effective when the joints are made of higher density and stiffness materials and when the thickness of the washers is increased. The primary reason for the pyroshock attenuation due to the joints and washers is the flexural wave reflection at the discontinuities caused by acoustic impedance mismatching. © 2018 Published by Elsevier Masson SAS.

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1. Introduction

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Over the past several decades, pyrotechnic release devices have been used widely in numerous applications including launch vehicles, missiles, fighters, and more; these devices are used to reliably separate the structural parts of a system [1,2]. Pyrotechnic release devices are very attractive compared with other release devices based on non-explosive release actuators (NEAs) due to their merits of high power-to-weight ratio, high reliability, instantaneous operation with simultaneity, long-term storage capability, and inexpensive cost. Even though pyrotechnic release devices have been successfully adopted with high separation reliability, pyroshock can lead to fatal malfunctions of the adjacent electrical components, and this remains a critical concern [3]. Pyroshock, also known as pyrotechnic shock, is defined as the transient response of the structures, components, and systems due to the loading induced by the pyrotechnic devices attached to the structures [4]. Pyroshock is generally divided into three categories: near-field, mid-field, and

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E-mail address: [email protected] (J.-H. Han). https://doi.org/10.1016/j.ast.2018.05.057 1270-9638/© 2018 Published by Elsevier Masson SAS.

far-field pyroshock [3]. Near-field pyroshock occurs next to the pyrotechnic device with a very high frequency and magnitude acceleration. Here, most of the energy is not transferred to the structural response and the stress wave propagation predominantly governs the pyroshock response. Mid- and far-field pyroshock can be observed away from the pyrotechnic source with a relatively low frequency and magnitude acceleration. The energy is concentrated at specific frequencies, unlike near-field pyroshock, due to the structural resonance. Although pyroshock can easily cause failure in electronic devices due to relay chatter, failures of circuit components, and short circuits due to small broken fragments [3], it rarely damages the supporting structures unless the pyrotechnic devices are used to break the structures intentionally. Furthermore, pyroshock propagates in terms of linear elastic waves to the surrounding structures without plastic deformation in most portions of the structures, except when in close proximity to the pyrotechnic devices [5]. In order to resolve these pyroshock issues, many researchers have investigated the characteristics of pyroshock propagation in structures. Traditionally, pyroshock issues have been experimentally investigated: the pyroshock was excited on test structures

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and the propagated pyroshock was measured. The use of pyrotechnic devices for pyroshock experiments is generally accompanied by high cost and safety issues. Therefore, numerous pyroshock simulators with mechanical excitations have been proposed [3,6]. However, these simulators cannot provide sufficient pyroshock excitation because they occasionally have high frequency and magnitude. Furthermore, the measurement of pyroshock has been routinely conducted based on several standards [4,5,7]. Commercial accelerometers can be used to measure pyroshock with limited frequency ranges due to the resonance of the sensing parts. In order to measure the very high frequency content of pyroshock, laser Doppler vibrometers (LDVs) can be used. Recently, two numerical analysis methods for pyroshock propagation have been proposed: the finite element method (FEM) and the statistical energy analysis (SEA)-based method; these are commonly used to estimate pyroshock propagation [6]. Even though FEM can analyze the stress wave propagation and the pyroshock propagation, it has extremely high computational costs to analyze the high frequency structural modes accurately [8]. Previous study that analyze the pyroshock propagation in structures and electronic units [9] has demonstrated that the pyroshock level estimation is only accurate in the low frequency range (under 1 kHz). In contrast, the SEA-based method was originally developed for high frequency vibro-acoustic analyses [10]. In order to analyze the transient response of the pyroshock propagation, the SEA method with virtual mode synthesis and simulation (VMSS) can be used [11]. In order to extend the capabilities of the SEA-based method into the mid-frequency range, the energy flow method (EFM) that combines the SEA-based method and the modal information obtained from FEM has also been proposed [10]. However, the SEAbased analysis only provides the spatially averaged pyroshock level of substructures and it cannot predict the pyroshock level at specific locations in the structures. Recently, analytic approaches to solve the pyroshock propagation have also been presented [12,13]; they provide accurate results, but are mathematically complex and only applicable to simple structures. Therefore, a new numerical method that can analyze the pyroshock propagation in complex structures with a wide frequency range is necessary in order to efficiently resolve the pyroshock issues; this study proposes the use of hydrocodes for precise analysis of the pyroshock propagation. Hydrocodes were first developed in the late 1950s to numerically solve aluminum and steel impact problems [14]; they can analyze highly dynamic events that involve shocks through solving the conservation equations with material models [15]. They have been used to analyze diverse impact phenomena: composite materials in low velocity to hypervelocity impacts [16,17], collisions of space reentry vehicles [18,19], hypervelocity impacts on fused silica sheets [20], diverse impacts on ultra-high molecular weight polyethylene (UHMWPE) [21–23], and more. In addition, problems related to blast loads that require very expensive experiments have been investigated using hydrocodes [24–26]. In pyrotechnics related fields, the separation mechanisms of the pyrotechnic release devices have been analyzed using hydrocodes: explosive bolts [27,28], shaped charges [29,30], separation nuts [31] and so on. Recently, the numerical prediction of pyroshock generation from separation nuts and ridge-cut explosive bolts were proposed using hydrocodes [31,32]. In this study, a new numerical analysis method for pyroshock propagation using commercial hydrocodes (ANSYS AUTODYN) is presented. Unlike other conventional methods, the hydrocodesbased method can precisely analyze pyroshock propagation in complex structures with a wide frequency range. In order to verify the numerical method, the pyroshock propagation experiments were designed and prepared with the pyroshock excitation using pyrotechnic initiators. From the experimental and numerical results, the pyroshock propagation characteristics on simple plates

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Fig. 1. Simple plate for pyroshock propagation experiments: (a) schematic and (b) manufactured simple plate and fixtures.

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are identified. Using the established numerical and experimental techniques, the effect of the pyroshock attenuation at the joints and washers connecting the plates were also investigated. Mechanical joints and washers are commonly recognized as pyroshock attenuation points along structures.

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2. Pyroshock propagation experiments with simple plates

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2.1. Experimental setup

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For the pyroshock propagation experiments, a simple plate (1 m × 0.5 m × 5 mm) made from aluminum alloy 6061 was prepared. Three threaded holes for mounting the shock accelerometer and one non-threaded hole for attaching the initiator fixture were prepared as shown in Fig. 1(a). Two heavy stainless steel fixtures as seen in Fig. 1(b) were also prepared in order to clamp the plates. These stainless steel fixtures clamped the simple plate using compressive force only. The left and right sides of the plate were clamped using a 0.5 m × 0.02 m contact area. The pyroshock was excited on the plates using pyrotechnic initiators. The excitations were applied to the point located 0.25 m to the left of the center of the plate. In order to attach the pyrotechnic initiators to the plates, an initiator fixture was designed and manufactured from stainless steel 304 as shown in Fig. 2. The initiator fixture was connected to the plate using a bolt connection. The volume of the cavity inside the initiator fixture was approximately 1 cm3 . In particular, the contact surface of the initiator fixture and plates was rectangular. A rectangular washer was also used to make both contact surfaces rectangular. This experimental setup enabled the assumption that the pyroshock was excited on plates with rectangular surfaces. The pyrotechnic initiators PC 800 fabricated by Hanwha Corporation were used. Here, the PC 800 ini-

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Fig. 2. Pyroshock excitation using a pyrotechnic initiator: (a) schematic of the initiator fixture, (b) manufactured initiator fixture, (c) assembly of initiator fixture and pyrotechnic initiator, and (d) attached pyrotechnic initiator on a simple plate.

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Fig. 3. Pyroshock measurements with (a) installed accelerometers and (b) LDVs to verify accelerometers.

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tiators, which generate 800 psi pressure per 10 cm volume, were filled with 210 mg of zirconium potassium perchlorate (ZPP). The pyroshock data for the simple plate were measured using four shock accelerometers. The measurement points of the accelerometers were the point of pyroshock excitation (i.e. the bolt of initiator fixture), one point located 30 mm to the left of the excitation point, and two points located 150 mm and 350 mm to the right of the excitation point. PCB 350B03 shock accelerometers and a PCB 482C05 sensor signal conditioner were used with a DAQ system: NI PXIe-1071 (embedded controller) and PXIe-6361 (analog input board). The accelerometers were mounted using threaded holes as shown in Fig. 3(a). The initiator fixture bolt also used a threaded hole. A torque wrench was used to install the accelerometers using the mounting torque (30 kg·cm) recommended in the sensor manual [33]. The sampling frequency and sampling time were set to 100 kHz and 10 s, respectively, with manual triggering.

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Fig. 4. Pyroshock measurement setup for the simple plate.

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Fig. 5. Accelerometer measurement results: (a) 0 mm point, (b) 30 mm point, (c) 150 mm point, (d) 350 mm point, (e) 0 mm point up to 2 ms, and (f) 150 mm point up to 2 ms.

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Filters were not applied during the accelerometer measurements. However, a mechanical filter with a 23-kHz resonant frequency was embedded in the accelerometers in order to protect the sensors from high frequency-high magnitude excitations. The measurement capabilities of these accelerometers were limited to 10 kHz in the frequency domain with a precision of 1 dB [34]. During the data processing, the 10th order Butterworth bandpass filter between 100 Hz and 10,000 Hz was applied. This filter was used to resolve potential issues such as high frequency errors due to the limited frequency range of the accelerometers and the low frequency data drift. In order to verify the measurement results of the accelerometers, three laser Doppler vibrometers (LDVs) were used. The measurement points of the LDVs were one point located 30 mm to the left of the excitation point and two points located 150 mm and 350 mm to the right of the excitation point. As depicted in

Fig. 3(b), the velocities from the LDVs were measured directly at the bottom surface of the accelerometers. The measurement results from the LDVs and accelerometers were identical if the same filter was applied; the accelerometers can be used to measure pyroshock if the desired frequency range is below 10 kHz. The pyroshock measurement experiments were performed three times for the simple plates. The overall experimental setups are shown in Fig. 4. All measurement procedures were established based on NASA standards [4].

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2.2. Experiment results

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In general, pyroshock is described and quantified in terms of the acceleration time history and the shock response spectra (SRS) [4]. The SRS value at each natural frequency is defined as the maximum of the acceleration responses of a mass for a given base

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measurement system using accelerometers limits the measurement frequency range to less than 10 kHz: a bandpass filter was applied due to the mechanical filter being embedded in the sensor. This frequency range or spectral content (1 kHz to 10 kHz) corresponds to the mid-field and far-field pyroshock environments [4,6]. Considering the difference between each experiment, the acceleration at the 0 mm point of experiment 2 was used as the input for the numerical analyses and the accelerations at the other points in experiment 2 were used for comparison with the numerical results.

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3. Numerical analyses of the pyroshock propagation on a simple plate

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Fig. 6. Maximax shock response spectra from the accelerometer measurements.

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input to a single degree of freedom (SDOF) system [5]. The maximax SRS, which the maximum of the acceleration responses are defined as the maximum absolute value of responses, are most commonly utilized for pyroshock analysis [4]. If the modal properties of the hardware are known, the response of the hardware can be computed from the acceleration time history. However, it is difficult to estimate these parameters at high frequencies; the SRS are computed with twelfth octave spacing over a wide frequency range [4,5]. Here, three repeated pyroshock propagation experiments were performed on a simple plate. In Fig. 5, the acceleration measurement results from the accelerometers are summarized: the results at the point of pyroshock excitation (0 mm) and the points located 30 mm, 150 mm, and 350 mm from the excitation point. The results of up to 2 ms at the 0 mm and 150 mm points are also presented with magnification. The maximax SRS calculated from the measurement results are drawn in Fig. 6. It appears that the generated pyroshock has high repeatability in the time domain. Therefore, the current pyroshock excitation technique generates consistent pyroshock, and it is appropriate for investigating pyroshock propagation. In the maximax SRS, the measurement results were similar but the difference increased under 1 kHz. In particular, the results of experiment 1 exhibited significant differences compared with those of experiments 2 and 3. Therefore, the current technique might only be applicable above 1 kHz excitation. At the same time, the current

In this study, the numerical analysis methodology for pyroshock propagation using commercial hydrocodes (ANSYS AUTODYN) is proposed. In a previous study [32], the authors established the pyroshock prediction method for ridge-cut explosive bolts using hydrocodes. This numerical method provides reliable results for pyroshock generation from pyrotechnic devices and pyroshock propagation to surrounding structures in two-dimensional (2D) axisymmetric environments. However, the analysis scheme that considers the detonation of high explosives and Euler–Lagrange interactions requires extremely high computational cost in order to extend it to three-dimensional (3D) non-axisymmetric structures. Therefore, a new numerical approach that focuses on pyroshock propagation in 3D structures through neglecting the detonation of high explosives should be established. First, the simple plate used in the experiments (Section 2) was modeled as follows. The dimensions of the prepared plate were 1 m × 0.5 m × 5 mm. The location of the pyroshock source was modeled as a separate part with dimensions of 14 × 14 × 5 mm. Here, the additional mass modeled using stainless steel 304 that has a similar mass to the initiator, bolt, and initiator fixture is attached at the point of the pyroshock source. Modeling of the simple plate is shown in Fig. 7. Then, the mesh of the simple plate was constructed as illustrated in Fig. 7. High quality meshes are essential for fast computation times and accurate results. In particular, the Courant–Friedrichs–Lewy (CFL) condition [35] was used to determine the time step of each numerical model: the smallest element dimension in the model determined the time increments. Therefore, uniformly mapped meshes with 1 mm hexagonal elements were prepared; the aspect ratio for all mesh elements was 1. In preliminary study, the mesh convergence test was carried out. Among them, 1 mm, 0.5 mm and smaller hexagonal elements pro-

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Fig. 7. Modeling and meshing of the simple plate with the pyroshock source point.

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Fig. 8. Applied boundary conditions: (a) fixed support boundary condition and (b) velocity boundary condition.

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Fig. 9. Applied pyroshock in terms of (a) acceleration and (b) velocity.

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vide almost identical results; 2.5 mm hexagonal elements produce quite deviated results. In order to neglect the connections between the parts, all parts were merged into a single structure. Fixed support boundary conditions were applied at the edge of the plate where the plate was clamped in the experiments as indicated in Fig. 8(a). Pyroshock was applied to the simple plates using a velocity boundary condition at the point of the pyroshock excitation (i.e. the point of the pyroshock source on the plate and additional masses) as shown in Fig. 8(b). The applied pyroshock for the current numerical analyses was obtained from the accelerometer measurement results at the 0 mm point in experiment 2. This acceleration result was integrated into the velocity for the hydrocodes input. The acceleration and velocity for pyroshock input are drawn in Fig. 9. Here, a velocity of up to 2 ms was used as the input; only the pyroshock excitation due to the pyrotechnic initiator was applied and the reverberation from the boundaries was not applied. In order to extract the analysis results, gauge points were assigned to the plate where the accelerations were measured in the experiments. In this study, aluminum alloy 6061, stainless steel 304, and magnesium alloy AZ31B were used for the experiments. Shock E.O.S., which can manage shock wave propagation in the metals, was utilized [27,28]. Shock E.O.S. is a Mie–Gruneisen form of E.O.S. [36] that related with the shock Hugoniot as the reference states, as follows:

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P − PH =

γ v

(e − e H )

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where P is the pressure, e is the specific internal energy, v is the specific volume, γ is the Gruneisen constant, and the subscript H refers to the shock Hugoniot. The shock Hugoniot for each material can be obtained experimentally through drawing the locus of all shocked states. The shock Hugoniot can be defined in several

forms; the relationship between the shock velocity U and the particle velocity u is a commonly used shock Hugoniot [37]. For most metals, the relationship is given linearly as follows:

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U = C 0 + su

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where C 0 and s are empirical parameters estimated from the experiments. As the strength model, the Steinberg–Guinan strength model [38], which is applicable at high strain rates, was used. Unlike the Johnson–Cook strength model used for the separation behavior analysis and pyroshock prediction of ridge-cut explosive bolts [27, 28,32], the Steinberg–Guinan strength model considers the change of not only the yield strength Y but also the shear modulus G with respect to the equivalent plastic strain ε , pressure P , and internal energy (temperature, T ). The constitutive relations are given as follows:



G = G0 1 +



G P G0





n

Y = Y 0 [1 + β ε ]



P

η1/3 

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+

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P

Y0

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G0 P

η1/3

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  T (T − T r )

G

G0

n Y 0 1 + β(ε + εi ) ≤ Y max .

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(3)

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(4)

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with the limitation that



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(5)

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Here, G 0 is the initial shear modulus, Y 0 is the initial yield stress, η is the compression, β and n are the work-hardening parameters, and T r is the reference temperature. The parameters G P , G T , and Y P are derivatives of the shear modulus and yield strength with respect to the pressure and temperature in the reference state. The material properties of AL 6061-T6 and SS 304

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Table 1 Material properties of the aluminum alloy, stainless steel, and magnesium alloy used in the pyroshock propagation analyses.

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Parameter of shock E.O.S.

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(g/cm3 ) (none) (m/s) (none) (K) (J/kg K)

Magnesium alloy

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1.775 1.43 4516 1.256 295.15 999

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Parameter of Steinberg Guinan strength model

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Initial shear modulus G 0 Initial yield stress Y 0 Maximum yield stress Y max Hardening constant β Hardening exponent n Derivative dG/dP G P Derivative dG/dT G T Derivative dY/dP Y P Melting temperature T m

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2.76 × 10 2.9 × 105 6.8 × 105 125 0.1 1.8 −1.7 × 104 0.018908 1220

7.7 × 10 3.4 × 105 2.5 × 106 43 0.35 1.74 −3.504 × 104 0.007684 2380 7

1.65 × 10 1.9 × 105 4.8 × 105 1100 0.12 1.6995 −8.3985 × 103 0.01957 1150 7

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supplied by AUTODYN are used for the aluminum alloy and stainless steel. The material properties of Shock E.O.S. for MAG AZ-31B and the Steinberg–Guinan strength model for MAGNESIUM supplied by AUTODYN are used for the magnesium alloy. The material properties are summarized in Table 1. The failure model and erosion model are unnecessary because mid- and far-field pyroshock does not induce failures on load-supporting structures as observed in the experiments. In AUTODYN, static damping is the only applicable damping model for the static equilibrium solution [39]. This model applies damping to structures through introducing the penalty of velocity at each time step, as follows: n+1/2

x˙

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n−1/2

= (1 − 2π R d )˙x

n

+ (1 − π R d )¨x t

n

(6)

Therefore, static damping R d should be determined carefully through considering the time steps of each numerical model. It is known that following static damping, R d should yield a critical damping of the lowest mode vibration with period T or frequency f .

Rd =

2 t / T 1 + 2π t / T

(7)

In general, the time step t is smaller than the period T ( t  T ). This condition induces the following:

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Rd ≈

2 t T

= 2 t · f

(8)

In this study, no damping and three levels of damping ( f = 10 Hz, 20 Hz, 30 Hz) were considered, and an appropriate level of static damping was determined through comparing the analysis results with the experimental results. Pyroshock propagation analyses were performed up to 10 ms with approximately 100,000 cycles; 30 hours of calculation time was required for each numerical model under parallel computing using one high performance desktop PC. For the numerical results processing, an additional filter was not applied and only down sampling by cubic spline interpolations was performed.

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3.2. Comparison between numerical and experimental results

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(kPa) (kPa) (kPa) (none) (none) (none) (kPa/K) (none) (K)

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Aluminum alloy

Density ρ0 Gruneisen coefficient γ Empirical parameter C 0 Empirical parameter s Reference temperature T r Specific heat C ν

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In this study, pyroshock propagation on a simple plate is numerically analyzed using the pyroshock input measured in experiment 2; the numerical results are compared with the results of experiment 2. Fig. 10 compares the results between experiment

2 and the numerical model with different damping levels. As the distance from the pyroshock source increased, the level of pyroshock or acceleration decreased. If the damping model was not applied, the stress wave would not dissipate and would exist continuously with a distributed manner. However, the experimental results demonstrated that it dissipated gradually (Fig. 5). Therefore, an appropriate level of damping should be applied in the numerical model. Here, three levels of damping (10 Hz, 20 Hz, and 30 Hz) are compared. The effect of damping in the early part of the time domain is comparably low. The effect of damping appears as the time increases: when the stress wave reflects back and reaches the measurement points again. The acceleration results of up to 2 ms (Figs. 10(e) and 10(f)) demonstrate that the current numerical analysis precisely estimates the time of the stress wave arrival. Furthermore, the shape and magnitude are quite similar. The maximax SRS are also affected by the damping level as shown in Fig. 10(h). Through considering the effect of damping in both time and frequency domains, an appropriate level of damping can be determined. As shown in Fig. 11, the static damping with 20 Hz frequency exhibited the best agreement between the numerical and experimental results. Here, the results are similar between 1 kHz and 10 kHz in the maximax SRS. Again, the results exhibited a difference under 1 kHz. However, the maximax SRS from the experimental results already exhibited a significant difference as shown in Fig. 6; thus, the comparison between numerical and experimental results under 1 kHz are insignificant. Here, the value of the static damping R d for the current numerical model is 3.97 × 105 : the frequency f is 20 Hz and the average time step t is 9.93 × 10−8 s. 3.3. Pyroshock propagation on a plate

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Pyroshock is defined as the response of a structure to high frequency and high magnitude stress waves generated by an explosive event. Even though pyroshock can easily cause failures in electric devices, it rarely damages the structure itself [3,4]; failures and plastic deformations were not observed in the pyroshock propagation experiments (Section 2). Therefore, the pyroshock propagates through the load supporting structures in the form of elastic waves. Here, most aerospace applications use lightweight structures with thin thicknesses; therefore, pyroshock will propagate primarily through the thin plate-like structures. In these thin plates, the Lamb wave is the dominant elastic wave [40]. Lamb waves, which are also known as plate waves, propagate through thin plate-like structures guided by the free surfaces. Lamb waves have two modes: symmetric and anti-symmetric. The dominant

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Fig. 10. Comparison of results from experiment 2 and the numerical analyses with different damping: (a) 0 mm point, (b) 30 mm point, (c) 150 mm point, (d) 350 mm point, (e) 30 mm point up to 2 ms, (f) 150 mm point up to 2 ms, (g) 350 mm point up to 2 ms, and (h) maximax shock response spectra.

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Fig. 11. Comparison of results from experiment 2 and the numerical analysis with 20 Hz damping: (a) 0 mm point, (b) 30 mm point, (c) 150 mm point, (d) 350 mm point, and (e) maximax shock response spectra.

displacements of the symmetric modes are radial in-plane; the particle motions of the anti-symmetric modes are dominated in out-of-plane motions. Here, the symmetric Lamb wave propagates in a compressional manner: the thickness bulges and contracts. In contrast, the anti-symmetric Lamb wave propagates in a flexural manner with a constant thickness. Here, the measurements on both sides of the plate using LDVs demonstrate that both sides of the plate move in the same direction when the pyroshock was excited. This is a feature of flexural waves and anti-symmetric Lamb waves in thin plates. This characteristic was also observed in the numerical results. Therefore, the pyroshock propagates in the thin plates in the form of anti-symmetric Lamb waves.

Although the previous section demonstrated the accuracy of the current analysis method for pyroshock propagation, the reliability of the analysis can be confirmed again through comparing the numerical and analytical wavelengths and the wave speed of the anti-symmetric Lamb waves. Here, the wavelength can be calculated using the Rayleigh–Lamb equations [40]. Here, two different thicknesses of aluminum alloy plates are considered: 5 mm and 10 mm. The harmonic (cosine) excitations are applied with different frequencies of 2000 Hz, 5000 Hz, and 8000 Hz. The wavelengths and wave speeds from the numerical analysis and analytic solution are summarized in Table 2. They exhibit good agreement with each other with less than 5% difference; thus, the current numerical method can analyze anti-symmetric

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Table 2 Comparison of the Lamb wave parameters between the theory and numerical analyses. Aluminum plate thickness Excitation frequency Theory Wave speed Wavelength Numerical Wave speed Wavelength Error of wave speed Error of wavelength

5 mm 2000 Hz 315.4 m/s 0.1577 m 308.6 m/s 0.1604 m −2.16% +1.71%

5000 Hz 496.0 m/s 0.0992 m 485.4 m/s 0.1010 m −2.14% +1.81%

8000 Hz 624.0 m/s 0.0780 m 598.8 m/s 0.0786 m −4.04% +0.77%

10 mm 2000 Hz 444.4 m/s 0.2222 m 434.8 m/s 0.2247 m −2.16% +1.13%

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5000 Hz 695.0 m/s 0.1390 m 714.3 m/s 0.1384 m +2.78% −0.43%

8000 Hz 870.4 m/s 0.1088 m 854.7 m/s 0.1052 m −1.80% −3.31%

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Fig. 12. Modeling of the two plates connected by two joints: (a) side view and (b) boundary conditions and assigned gauge points.

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Lamb waves (and pyroshock) propagation accurately. Unlike the longitudinal, distortional, and Rayleigh waves, the wave speed of anti-symmetric Lamb waves depends on the thickness of the plate and the frequency: as the frequency and thickness of the plate increase, the wave speed increases. The speed of Lamb waves is slower than other types of elastic waves.

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4. Numerical analysis of pyroshock propagation through joints and washers

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It is known that the mechanical joints and washers in the pyroshock path attenuate the pyroshock propagation. However, the principle and effectiveness of the joints and washers for pyroshock attenuation have not been clearly explained; only some experimental results have been reported [41]. In addition, a numerical approach that can analyze the pyroshock attenuation at the joints and washers has not yet been proposed. Therefore, most engineers predict pyroshock attenuation at various parts of structures based on their experience. In this study, the effectiveness of the joints and washers for the pyroshock attenuation is predicted using the proposed numerical methods.

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4.1. Numerical analysis models

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First, the simple plate modeled in Section 3.1 was divided in two parts: two separate plates (0.49 m × 0.5 m × 5 mm) were connected by two mechanical joints (0.1 m × 0.5 m × 5 mm) as illustrated in Fig. 12. The total length of the connected plate was identical to the simple plate; it was 1 m. The same material, i.e. aluminum alloy 6061, was used for the plates. Here, joints with three different materials (aluminum alloy 6061, stainless steel 304, and magnesium alloy AZ31B) were considered. In order to simplify the numerical models, detailed modeling of the bolts and nuts was not performed. Instead, only the plates and joints were modeled; the connections between them were modeled as bonded. In reality,

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Fig. 13. Modeling of two plates connected by two joints and four washers: (a) side view with different thicknesses of washers and (b) boundary conditions and assigned gauge points.

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the bolt and nut connections apply compression loads on the contact surfaces between the plates and joints. As the clamping force increases, the friction between the plates and joints increases and the contacts might behave as bonded. All other modeling details followed the modeling techniques described in Section 3.1. Next, two separate plates connected by two joints and four washers (0.04 m × 0.5 m× thickness) were modeled as shown in Fig. 13. Here, two different thicknesses (5 mm and 2 mm) were considered. The same material, i.e. aluminum alloy 6061, was used for the plates and joints. The washers with three different materials (aluminum alloy 6061, stainless steel 304, and magnesium alloy AZ31B) were also considered. Again, the modeling of the bolts and nuts was neglected; all parts were considered to be bonded.

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4.2. Numerical results

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In this section, the results between the simple plate, the plates connected by joints, and the plates connected by joints and washers are compared in terms of the acceleration time histories and the maximax SRS as shown in Figs. 14 to 16. First, the effect of the pyroshock attenuation through the joints was investigated. Through comparing the numerical results between the simple plate and the plates connected by joints, it was clearly demonstrated that the pyroshock was attenuated at the joints; a portion of the pyroshock reflected back at the joints. Therefore, the pyroshock level increased in both the time and frequency domains at the 150 mm point located before the joints; the pyroshock level decreased at the 350 mm point located after the joints. The effectiveness of the pyroshock attenuation depended on the joint material: the stainless steel joints were the most effective and the aluminum alloy joints were the next most effective. The pyroshock attenua-

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Fig. 15. Numerical results comparison between the simple plate, the plates connected by aluminum alloy joints, and the plates connected by aluminum alloy joints with 5 mm-thick washers: (a) acceleration time histories at 150 mm, (b) acceleration time histories at 350 mm, (c) maximax SRS at 150 mm point, and (d) maximax SRS at 350 mm.

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Fig. 16. Numerical results comparison between the simple plate, the plates connected by aluminum alloy joints, and the plates connected by aluminum alloy joints with 2 mm-thick washers: (a) acceleration time histories at 150 mm, (b) acceleration time histories at 350 mm, (c) maximax SRS at 150 mm, and (d) maximax SRS at 350 mm.

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tion was effective when the joint material had a higher density and/or stiffness. Next, the pyroshock attenuation due to the washers was investigated. The numerical results among the simple plate, the plates connected by aluminum alloy joints, and the plates connected by aluminum alloy joints and washers with different materials and thicknesses demonstrated that the efficiency of the pyroshock attenuation could be increased through inserting washers between the plates and joints. In particular, the pyroshock attenuation was highly effective when the washers were thicker. For the same thickness washers, the pyroshock attenuation was affected by the washer material: it was effective when the washer material had a higher density and/or stiffness. Again, the pyroshock reflected back at the joints and washers. Therefore, the pyroshock level increased at the point located before the joints and washers.

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In Section 4, the effect of the pyroshock attenuation at the joints and washers was numerically investigated. In order to verify the numerical results, pyroshock propagation experiments were performed using plates that were connected by the joints and washers.

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5.1. Experimental setup

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5. Pyroshock propagation experiments with joints and washers

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For the pyroshock propagation experiments of the connected plates, two separate plates (0.49 m × 0.5 m × 5 mm) made from aluminum alloy 6061 were prepared. To connect these plates, joints (0.1 m × 0.5 m × 5 mm) made from aluminum alloy 6061,

Fig. 17. Schematic of the plates connected by two joints and four washers for the pyroshock propagation experiments.

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stainless steel 304, and magnesium alloy AZ31B were prepared; two joints were needed from each material. The four sets of washers (0.04 m × 0.5 m× thickness) inserted between the plates and the joints were prepared for each material and thickness; aluminum alloy 6061, stainless steel 304, and magnesium alloy AZ31B washers were made with 5 mm and 2 mm thicknesses. Nonthreaded holes were drilled to assemble the plates, joints, and washers using the bolts. The plates were connected as shown in Figs. 17 and 18. Twenty M8 bolts (stainless steel A2-70) and nuts

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were used; the bolts and nuts were fastened using an impact wrench (GDX 18 V-EC) in order to provide sufficiently large torque (higher than 18.4 N·m). This bolt torque generates approximately 11,500 N clamping force per each bolt; the torque coefficient K is assumed as 0.2. Under high clamping force, it is almost impossible to slide two contacting surfaces. Instead, the contacts may behave as bonded. The total length of the connected plate was identical to that of the simple plate, i.e. 1 m. Four accelerometers were used to measure the pyroshock data, and the measurement points were the same as in the previous experiments as depicted in Fig. 19(b): the point of the pyroshock excitation (bolt of the initiator fixture), the point located 30 mm to the left of the excitation point, and points located 150 mm and 350 mm to the right of the excitation point. Here, the 150 mm point was located before the joints and the 350 mm point was located after the joints. All other measurement details are the same as described in Section 2.1. The pyroshock measurement experiments were performed three times for each structure. A total of nine structures were tested: the plates connected by the three different material joints, the plates connected by the aluminum alloy joints with three different materials and two different thickness washers.

5.2. Measurement results

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In this section, the experiment results for the simple plate, the plates connected by the joints, and the plates connected by the joints and washers are compared. Here, the pyroshock excitations by the initiator exhibited slight variation for each experiment. In order to compensate these variations, a correction factor was determined for each experiment; through multiplying the correction factor with the measurement results equally for all points, the maximum accelerations at the 0 mm point become identical to that of the simple plate results (experiment 2). The concept of the current correction factor is identical to the normalization. First, the experiment results in terms of the maximax SRS for the plates connected by the aluminum alloy, stainless steel, and magnesium alloy joints are shown in Fig. 20. Above 1 kHz, the experiment results for the same structures were very similar in their maximax SRS. In particular, Fig. 20(d) illustrates that the pyroshock excitations between different experiments were very similar after compensation by the correction factor. Unlike the experiment results for the simple plate that exhibit a similar pyroshock level at the 150 mm and 350 mm points (Fig. 6), the experiment re-

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sults for the connected plate demonstrated the effectiveness of the joints for pyroshock attenuation. It is clear that the stainless steel joints provide the best pyroshock attenuation capability and the aluminum alloy joints provide the next best. For easy comparison, the maximax SRS were averaged at each natural frequency for the repeated experiments and are redrawn for each measurement point in Fig. 21. They demonstrate that the pyroshock level increased at the 150 mm point located before the joints and the pyroshock level decreased at the 350 mm point located after the joints; a portion of the pyroshock was reflected back at the joints. Next, the experiment results for the plates connected using aluminum alloy joints and the 5 mm-thick washers and 2 mm-thick washers are presented in Figs. 22 and 23, respectively. These results demonstrate that the pyroshock attenuation efficiency could be increased through the insertion of the washers between the plates and joints. Here, the efficiency of the pyroshock attenuation was primarily affected by the washer thickness rather than the washer material; the pyroshock attenuation was effective when the washers were thicker. Again, the pyroshock reflected back at the joints and washers; the pyroshock level increased more at the point located before the joints and washers.

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Fig. 22. Experiment results comparison between the simple plate, the plates connected by aluminum alloy joints, and the plates connected by aluminum alloy joints with 5 mm-thick washers: (a) maximax SRS at 150 mm and (b) maximax SRS at 350 mm.

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5.3. Discussion

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The effects of the pyroshock attenuation by the joints and washers were numerically and experimentally investigated. Qualitatively, these results agree with each other: the pyroshock attenuation was highly effective when the joints were made from higher density and/or stiffness materials and when the washers were thicker. In order to compare the results quantitatively, the maximax SRS ratios were calculated between the different points for each experiment and numerical model. The ratios of the maximax SRS for each natural frequency between 1 kHz and 10 kHz were calculated and then averaged.

Average of SRS ratio =

1 n

10 kHz  1 kHz

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SRSmodel SRSref

(9)

This quantification is similar to the mean acceleration difference (MAD) [42,43], which is used to evaluate the similarity between two SRS. Finally, the ratio for the repetitive experiment

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Table 3 Comparison of the SRS ratios between the simple plate and the plates connected by joints.

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49.55% 51.08% 104.35% 61.21% 42.34% 64.15% 65.51% 31.33% 47.55% 61.06% 38.79% 61.45%

46.33% 46.20% 100.05% 58.37% 41.73% 66.82% 66.69% 31.67% 47.24% 58.58% 39.85% 65.29%

46.91% 43.76% 94.97% 58.51% 42.45% 66.72% 61.31% 29.05% 47.48% 60.73% 37.86% 60.23%

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to to to to to to to to to to to to

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Exp. Average 47.59% 47.01% 99.79% 59.36% 42.17% 65.90% 64.50% 30.68% 47.42% 60.12% 38.83% 62.32%

Numerical results 53.92% 54.98% 106.75% 67.59% 51.17% 74.82% 71.68% 41.53% 57.62% 67.54% 49.78% 72.96%

Attenuation Efficiency

17

Reference

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Al 5 mm washer Mg 2 mm washer

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to to to to to to to to to to to to to to to to to to to to to to to to

2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3

49.55% 51.08% 104.35% 61.06% 38.79% 61.45% 73.13% 24.71% 32.74% 78.67% 24.06% 28.79% 79.52% 25.40% 31.66% 73.98% 33.42% 44.13% 72.94% 31.91% 41.03% 76.03% 33.44% 43.05%

46.33% 46.20% 100.05% 58.58% 39.85% 65.29% 75.49% 25.36% 32.92% 78.33% 23.95% 29.13% 77.20% 26.15% 32.97% 70.99% 33.17% 45.92% 71.97% 30.27% 39.33% 72.35% 30.66% 42.22%

46.91% 43.76% 94.97% 60.73% 37.86% 60.23% 74.97% 24.62% 32.27% 83.85% 25.77% 29.34% 83.83% 27.92% 32.32% 71.01% 33.72% 46.33% 81.03% 32.35% 36.93% 70.76% 29.64% 41.06%

Exp. Average 47.59% 47.01% 99.79% 60.12% 38.83% 62.32% 74.53% 24.90% 32.64% 80.28% 24.59% 29.09% 80.18% 26.49% 32.32% 71.99% 33.44% 45.46% 75.31% 31.51% 39.10% 73.05% 31.25% 42.11%

Numerical results 53.92% 54.98% 106.75% 67.54% 49.78% 72.96% 69.80% 31.75% 45.26% 72.72% 30.69% 41.41% 70.23% 31.59% 44.42% 71.56% 43.58% 60.85% 68.72% 41.67% 60.15% 71.59% 43.22% 60.32%

Attenuation efficiency

80 82 83 84 85 86

Reference

87 88 89

Reference

90 91

3rd

92 93 94

1st

95 96

2nd

97 98 99

6th

100 101

4th

102 103 104

5th

40 41

70

81

19 21

69

73

Table 4 Comparison of the SRS ratios between the simple plate and the plates connected by joints and washers.

18 20

68

71

15 16

67

105 106

results were averaged and compared with the numerical results. The maximax SRS ratios are summarized in Tables 3 and 4. First, the pyroshock levels at point 2 (150 mm) and point 3 (350 mm) for the simple plate were almost identical; this was observed from both numerical and experimental results. If the joints were located between point 2 and point 3, the pyroshock levels exhibited a significant difference; the pyroshock was attenuated at the joints. The ratios were not precisely the same. However, the superiority depending on the joint material was identical; the stainless steel joints exhibited the highest efficiency and the aluminum alloy joints exhibited the next highest. The difference between the aluminum alloy joints and the magnesium alloy joints was not remarkable. Here, the primary reason for the pyroshock attenuation due to the joints was the flexural wave reflection at the discontinuities caused by the acoustic impedance mismatching; the acoustic impedance is defined as the density multiplied by the sound speed (or the longitudinal wave speed). However, this is an imprecise explanation; there is no acoustic impedance mismatch between the aluminum alloy plates and the aluminum alloy joints. Moreover, the flexural (or bending) wave reflection occurred due to the beam impedance difference [44]. However, the mathematical formulation of the beam impedance is complex and the concept of the beam impedance is not widely used. Therefore, the concept of the acoustic impedance is introduced here.

Next, the pyroshock attenuation efficiencies of the washers were compared between the numerical and experimental results; the 5 mm washers exhibited more attenuation than the 2 mm washers regardless of the washer material. However, the pyroshock attenuation efficiencies observed in the experiments were slightly higher than that estimated by numerical analyses. This resulted from the numerical models neglecting the nonlinear contacts between the plates, joints, and washers.

107 108 109 110 111 112 113 114 115

6. Conclusion

116 117

In this study, the pyroshock propagation on plate structures was investigated; the pyroshock propagation analysis environment was established and verified through comparison with the experiment results. The pyroshock propagate through the thin plates in the form of flexural waves (or anti-symmetric Lamb waves). The plates connected by joints with different materials and the plates connected by joints with inserted washers with different materials and thicknesses were considered. The experimental and numerical results were in good agreement: the pyroshock attenuation was highly effective when the joints were made from higher density and/or stiffness materials and when the washers were thicker. The primary reason for the pyroshock attenuation due to the joints and washers was the flexural wave reflection at the discontinuities caused by the acoustic impedance mismatching. The developed pyroshock propagation analysis provides a powerful method for

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Fig. 23. Experiment results comparison between the simple plate, the plates connected by aluminum alloy joints, and the plates connected by aluminum alloy joints with 2 mm-thick washers: (a) maximax SRS at 150 mm and (b) maximax SRS at 350 mm.

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structural design that minimizes pyroshock propagation. Without expensive and repeated experiments, the pyroshock propagation characteristics can be predicted and suitable structures for pyroshock attenuation can be designed.

50 51

Conflict of interest statement

52 53 54

The authors declare that there is no conflict of interest regarding the publication of this manuscript.

55 56

Acknowledgements

57 58 59 60 61

This work was supported by Global Surveillance Research Center (GSRC) program funded by the Defense Acquisition Program Administration (DAPA) and Agency for Defense Development (ADD).

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