Royer Oscillator Matlab

Rotational Velocities Of A-Type Stars ⋆,⋆⋆

II. Measurement of v sin i in the northern hemisphere F. Royer1,2, S. Grenier2, M.-O. Baylac2, A.E. G´omez2, and J. Zorec3 1 Observatoire de Gen`eve, 51 chemin des Maillettes, CH-1290 Sauverny, Switzerland 2 GEPI/CNRS FRE 2459, Observatoire de Paris, 5 place Janssen, F-92195 Meudon cedex, France 3 CNRS, Institut d’Astrophysique de Paris, 98 bis boulevard Arago, F-75014 Paris, France

Received / Accepted

Abstract. This work is the second part of the set of measurements of v sin i for A-type stars, begun by Royer et al. (2002). Spectra of 249 B8 to F2-type stars brighter than V = 7 have been collected at Observatoire de Haute-Provence (OHP). Fourier transforms of several line profiles in the range 4200–4600 ˚A are used to derive v sin i from the frequency of the first zero. Statistical analysis of the sample indicates that measurement error mainly depends on v sin i and this relative error of the rotational velocity is found to be about 5 % on average.

The systematic shift with respect to standard values from Slettebak et al. (1975), previously found in the first paper, is here confirmed. Comparisons with data from the literature agree with our findings: v sin i values from Slettebak et al. are underestimated and the relation between both scales follows a linear law v sin inew = 1.03 v sin iold + 7.7.

Finally, these data are combined with those from the previous paper (Royer et al. 2002), together with the cat- alogue of Abt & Morrell (1995). The resulting sample includes some 2150 stars with homogenized rotational velocities.

Key words. techniques: spectroscopic – stars: early-type; rotation

1. Introduction

This paper is a continuation of the rotational velocity study of A-type stars, initiated in Royer et al. (2002, here- after Paper I). The main goals and motivations are de- scribed in the previous paper. The sample of A-type stars described and analyzed in this work is the counterpart of the one in Paper I, in the northern hemisphere.

In Short, It Is Intended To Produce A Homogeneous

sample of measurements of projected rotational velocities (v sin i) for the spectral interval of A-type stars, and this without using any preset calibration.

This article is structured in a way identical to the precedent, except for an additional section (Sect. 5) where data from this paper, the previous one and the catalogue of Abt & Morrell (1995) are gathered, and the total sample is discussed in statistical terms.

Send Offprint Requests To: Fr´Ed´Eric Royer

⋆Based on observations made at Observatoire de Haute

Provence (Cnrs), France

⋆⋆Tables 5 and 9 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.125.5)

2. Observational Data

Spectra were obtained in the northern hemisphere with the AUR´ELIE spectrograph (Gillet et al. 1994) associ- ated with the 1.52 m telescope at Observatoire de Haute- Provence (OHP), in order to acquire complementary data to HIPPARCOS observations (Grenier & Burnage 1995).

The initial programme gathers early-type stars for which v sin i measurement is needed. More than 820 spec- tra have been collected for 249 early-type stars from January 1991 to May 1994. As shown in Fig. 1, B9 to A2- type stars represent the major part of the sample (70 %).

Most of the stars are on the main sequence and only about one fourth are classified as more evolved than the lumi- nosity class III-IV.

These northern stars are brighter than the magnitude V = 7. Nevertheless, three stars are fainter than this limit

And Do Not Belong To The Hipparcos Catalogue (Esa

1997). Derivation of their magnitude from TYCHO obser- vations turned out to be: HD 23643 V = 7.79, HD 73576 V = 7.65 and HD 73763 V = 7.80. These additional stars are special targets known to be δ Scuti stars.

2

F. Royer et al.: Rotational velocities of A-type stars. II. Fig. 1. Distribution of the spectral type for the 249 pro- gramme stars.

– Λ1: 4110–4310 ˚A. This domain was initially chosen to detect chemically peculiar stars by measuring equiva- lent widths of the lines Si ii 4128, Si ii 4131, Sr ii 4216, and Sc ii 4247.

– Λ2: 4226–4432 ˚A or 4270–4475 ˚A. Centered around Hγ, this spectral range allows the determination of the effective temperature.

– Λ3: 4390–4600 ˚A, or 4400–4610 ˚A. This domain is ded- icated to v sin i determination, although all the three ranges are used to measure the rotational velocity.

Two thirds of the sample have observations in each of the three ranges. The Λ3 range is particularly aimed at v sin i measurement, and contains the largest number of lines selected for this purpose (twice as much as Λ1 and Λ2). Besides it is the only one which covers the magnesium doublet at 4481 ˚A. This line remains often alone for mea- surement in fast rotators. It is thus significant to note that among the 249 stars of the sample, three were observed only in the Λ1 range, eight only in Λ2 and eleven in both Λ1 and Λ2 only. Overall 22 stars have no observation in Λ3. The reason is that these stars have already known v sin i. Moreover they are effective temperature standard stars or reference stars for chemical abundances.

Some changes in the configuration of the instrument meant the central wavelengths of the Λ2 and Λ3 domains have been slightly modified during mission period.

The entrance of the slit is a 600 µm hole, i.e. 3′′ on the sky, dedicated to the 1.52 m Coud´e telescope. The dispersion of the collected spectra is 8.1 ˚A mm−1 and the resolving power is about 16 000.

The barrette detector is a double linear array TH 7832, made of 2048 photo-diodes. Reduction of the data has been processed using MIDAS1 procedures.

Flat field correction with a tungsten lamp and wave- length calibration with a Th-Ar lamp have been made with classical procedures. Nevertheless, a problem oc- curred when applying the flat-field correction with the tungsten calibration lamp. The division by the W lamp spectrum produced a spurious effect in the resulting spec- trum at a given position in the pixels axis. This effect

1 Midas Is Being Developed And Maintained By Eso

distorts the continuum, as it can be seen on the spectra of Vega (Fig. 2). This problem triggered the decision to change the instrumental configuration and central wave- lengths of the spectral ranges.

3. Measurement Of The Rotational Velocity

The method adopted for v sin i determination is the com- putation of the first zero of Fourier transform (FT) of line profiles (Carroll 1933; Ramella et al. 1989). For fur- ther description of the method applied to our sample, see Paper I. The different observed spectral range induces some changes, which are detailed below.

3.1. Continuum Tracing

The normalization of the spectra was performed using MIDAS: the continuum has been determined visually, passing through noise fluctuations. The procedure is much like the normalization carried out in Paper I, except for a different spectral window. For the ranges Λ1 and Λ2, the influence of the Balmer lines is important, and their wings act as non negligible contributions to the difference between true and pseudo-continuum, over the major part of the spectral domain, as shown in Paper I. On the other hand, the Λ3 range is further from Hγ. In order to quan- tify the alteration of continuum due to Balmer lines wings and blends of spectral lines, a grid of synthetic spectra of different effective temperatures (10 000, 9200, 8500 and 7500 K) and different rotational broadenings, computed from Kurucz’ model atmosphere (Kurucz 1993), is used to calculate the differences between the true continuum and the pseudo-continuum. The pseudo-continuum is rep- resented as the highest points in the spectra. The differ- ences are listed in Table 1, for different spectral 20 ˚A wide sub-ranges. This table is a continuation of the similar one in Paper I, considering the spectral range 4200–4500 ˚A. It is clear that the pseudo-continuum is much closer to the true continuum in Λ3 than in both bluer ranges.

3.2. Set Of Lines

Put end to end, the spectra acquired with AUR´ELIE cover a spectral range of almost 500 ˚A. It includes that observed with ECHELEC in Paper I. The choice of the lines for the determination of the v sin i in Paper I is thus still valid here. Moreover, in addition to this selection, were adopted redder lines in order to benefit from the larger spectral coverage.

The complete list of the 23 lines that are candidate for v sin i determination is given in Table 2. In order to quantify effects of blends in the selected lines for later spectral types, we use the skewness of syn- thetic line profiles, as in Paper I. The same grid of syn- thetic spectra computed using Kurucz’ model (Kurucz F. Royer et al.: Rotational velocities of A-type stars. II.

Malization Area. It Contains Seven

of the selected lines.

– Λ2: Centered Around Hγ, This Range

only contains five selected lines.

Among Which The Doublet Line Mg Ii

4481.

Λ3).

1993), is used. Skewness is defined as γ1 = m3 m−1.5

(1)

for an absorption line centered at wavelength λc and spreading from λ1 to λL, where F(λi) is the normalized flux corresponding to the wavelength λi. Ranges [λ1, λL] are centered around theoretical wavelengths from Table 2 and the width of the window is taken to be 0.35, 0.90 and 1.80 ˚A for rotational broadening 10, 50 and 100 km s−1 respectively (the width around the Mg ii doublet is larger: 1.40, 2.0 and 2.3 ˚A). Table 3 lists the skewness of the lines for each element of the synthetic spectra grid and is a continuation of Table 3 from Paper I for the lines with wavelength longer than 4500 ˚A. These additional lines are rather isolated and free from blends. Major part of the computed γ1 for the hotter spectrum (10 000 K) is far lower than the threshold 0.15 chosen in Paper I to iden- tify occurrence of blends. The only case where a line must be discarded is the blend occurring with Fe ii 4520 and Fe ii 4523 for v sin i ≳100 km s−1. This non-blended be- havior continues on the whole range of temperature, and the candidate lines remain reliable in most cases.

The comparison between the rotational velocity de- rived from the weak lines and the one derived from the magnesium doublet was already approached in Paper I.

It is here of an increased importance since the Mg ii line is not present in all spectra (i.e. Λ1 and Λ2 spec- tral ranges). Figure 3 shows this comparison between ⟨v sin i⟩and v sin iMg ii using AUR´ELIE data. The devi- ation from the one-to-one relation (solid line) in the low velocity part of the diagram is due to the intrinsic width

4

F. Royer et al.: Rotational velocities of A-type stars. II. Table 1. Differences between the true continuum and the highest points in different spectral bands for the set of synthetic spectra in the Λ3 domain. Wavelength indicates the center of the 20 ˚A wide range.

0.0181

Table 2. List of the 23 spectral lines used (when possi- ble) for the v sin i measurement, and the corresponding spectral range(s) to which they belong.

Fe Ii

† Wavelength of both components are indicated for the mag- nesium doublet line. of the doublet. This deviation is simulated by represent- ing the Mg ii doublet as the sum of two identical gaus- sians separated by 0.2 ˚A. The full-width at half max- imum (FWHM) of the simulated doublet line is plot- ted in Fig. 4 versus the FWHM of its single-lined com- ponents. The relation clearly deviates from the one-to- Table 3. Variation of the skewness γ1 of the lines with Teffand v sin i in the synthetic spectra.

0.07

Fig. 3. v sin iMg ii derived from the 4481 Mg ii line versus ⟨v sin i⟩derived from other metallic lines for early A-type stars. The solid line stands for the one-to-one relation.

The dashed line is the least-squares linear fit for ⟨v sin i⟩> 30 km s−1. F. Royer et al.: Rotational velocities of A-type stars. II.

5

Fig. 4.

Behavior:

FWHM of the sum of two gaussian lines (separated with 0.2 ˚A) as a function of the FWHM of the components. one relation for single line FWHM lower than 0.6 ˚A.

Using the rule of thumb from Slettebak et al. (1975,

A] ≈0.025 V Sin I[Km S−1], This

value corresponds to v sin i = 24 km s−1. This limit coin- cides with what is observed in Fig. 3. For higher veloci- ties (⟨v sin i⟩> 30 km s−1), ⟨v sin i⟩becomes larger than

V Sin Img Ii. A Linear Regression Gives:

v sin iMg ii = 0.88 ⟨v sin i⟩+ 2.2.

(2)

The effect is similar to the one found in Paper I, sug- gesting that blends in lines weaker than Mg ii produce an overestimation of the derived v sin i of about 10 %.

The number of measurable lines among the 23 listed in Table 2 varies from one spectrum to another according to the wavelength window, the rotational broadening and the signal-to-noise ratio. The number of measured lines ranges from 1 to 17 lines. The Λ3 range offers a large number of candidate lines. Fig. 5 shows the variation of this number with v sin i (solid line). Rotational broadening starts to make the number of lines decrease beyond about 70 km s−1. Nevertheless additional lines in the spectral do- main redder than 4500 ˚A makes the number of lines larger than in the domain collected with ECHELEC (Paper I; dotted line). Whereas with ECHELEC the number of lines decreases with v sin i from 30 km s−1 to reach only one line (i.e. the Mg ii doublet) at 100 km s−1, the number of lines with AUR´ELIE is much sizeable: seven at 70 km s−1, still four at 100 km s−1 and more than two even beyond 150 km s−1.

3.3. Precision

Fig. 5. The average number of measured lines (running average over 30 points) is plotted as a function of the mean ⟨v sin i⟩. Solid lines stands for the spectra collected with AUR´ELIE (Λ3 range) whereas dotted line represents ECHELEC spectra from Paper I.

3.3.1. Effect Of V Sin I

In Fig. 6, the differences between the individual v sin i val- ues from each measured line in each spectrum and the associated mean value for the spectrum are plotted as a function of ⟨v sin i⟩. In the same way the error associated with the v sin i has been estimated in Paper I, a robust estimate of the standard deviation is computed for each bin of 70 points. The resulting points (open grey circles in Fig. 6) are adjusted with a linear least squares fit (dot-

Dashed Line). It Gives:

σv sin i|v sin i = 0.048±0.010⟨v sin i⟩+ 0.14±0.19.

(3)

This fit is carried out using GaussFit (Jefferys et al. 1998a,b), a general program for the solution of least squares and robust estimation problems. The resulting constant of the linear fit has an error bar of the same order than the value itself, and then the formal error is estimated to be 5 % of the v sin i.

The Slope Is Lower With Aur´Elie Data Than With

ECHELEC spectra (Paper I): 4.8±1.0 % against 5.9±0.3 %. This trend can be explained by the average number of lines for the computation of the mean v sin i. In the veloc- ity range from 15 to 180 km s−1, the number of measured lines (Fig. 5) is on average 2.4 times larger with AUR´ELIE

√

2.4 ≈1.5.

3.3.2. Effect Of Spectral Range

As shown in Fig.1, the distribution of spectral types is mainly concentrated towards late-B and early-A stars, so that a variation of the precision as a function of the spec- tral type would not be very significant. On the other hand, as the observed spectral domain is not always the same, this could introduce an effect due to the different sets of

6

F. Royer et al.: Rotational velocities of A-type stars. II. Fig. 6. Differences between individual v sin i and mean over a spectrum ⟨v sin i⟩. Variation of the standard de- viation associated with the measure with the ⟨v sin i⟩is shown by the open circles. A linear least-square fit on these points (dot-dashed line) gives a slope of 0.05.

selected lines, their quantity and their quality in terms of v sin i determination. For each of the three spectral do- mains, the residuals, normalized by σv sin i|v sin i (Eq. 3), are centered around 0 with a dispersion of about 1 taking into account their error bars, as shown in Table 4. This suggests that no effect due to the measurement in one given spectral range is produced on the derived v sin i.

Table 4. Mean of differences between individual v sin i and average ⟨v sin i⟩over a spectrum, normalized by the formal error due to v sin i, are indicated for each spectral range as well as the standard deviations ˆσv sin i|Λ of these means.

4.1. Results

In total, projected rotational velocities were derived for 249 B8 to F2-type stars, 86 of which have no rotational velocities in Abt & Morrell (1995).

The results of the v sin i determinations are presented in Table 5 which contains the following data: column (1) gives the HD number, column (2) gives the HIP num- ber, column (3) displays the spectral type as given in the HIPPARCOS catalogue (ESA 1997), columns (4, 5, 6) give respectively the derived value of v sin i, the associated standard deviation and the corresponding number of mea- sured lines (uncertain v sin i are indicated by a colon), col- umn (7) presents possible remarks about the spectra: SB2 Table 5. (extract) Results of the v sin i measurements.

Only the 15 first stars are listed below. The whole table is available electronically. Description of the columns is detailed in the text.

11

(“SB”) and shell (“SH”) natures are indicated for stars showing such feature in these observed spectra, as well as the reason why v sin i is uncertain – “NO” for no selected lines, “SS” for variation from spectrum to spectrum and “LL” for variation from line to line (see Appendix A).

4.1.1. Sb2 Systems

Nine stars are seen as double-lined spectroscopic binary in the data sample. Depending on the v sin i of each compo- nent, their difference in Doppler shift and their flux ratio, determination of v sin i is impossible in some cases.

Table 6 displays the results for the stars in our sample which exhibit an SB2 nature. Spectral lines are identified by comparing the SB2 spectrum with a single star spec- trum. Projected rotational velocities are given for each component when measurable, as well as the difference in radial velocity ∆Vr computed from a few lines in the spec- trum.

– 110 Tau (HD 35189) is highly suspected to be a spec- troscopic binary due to its spectra taken in the Λ1 domain. As it can be seen in Fig. 7.a, core of the lines is double in most cases, corresponding to a difference in radial velocity of about 37 km s−1. However, this case does not allow the measurement of the projected rotational velocity.

– β Aur (HD 40183) is a well known early A-type eclips- ing binary of Algol type. The derived v sin i corre- sponds well with the values of Nordstr¨om & Johansen (1994) (respectively 33 and 34 km s−1) that indicate

Hjdmin = 2431076.7269 + 3.96004732 E,

F. Royer et al.: Rotational velocities of A-type stars. II.

7

Fig. 7. Part of the spectra are displayed for the six SB2 stars that have been observed only once: a) HD 35189, b) HD 40183, c) HD 42035, d) HD 181470, e) HD 203439, f) HD 203858. Three of them are well separated (b, d, f), allowing measurement of v sin i for both components. The three others (a, c, e) have low differential Doppler shift (≤60 km s−1) which makes all the lines blended. No v sin i has been determined for these objects.

Fig. 8. The three following SB2 stars have been observed twice, in Λ1 (upper panels) and Λ3 (lower panels): a) HD 79763 at HJD 2449025, b) HD 98353 at HJD 2448274, c) HD 119537 at HJD 2449025, d) HD 79763 at HJD 2449365, e) HD 98353 at HJD 2449413, f) HD 119537 at HJD 2449415. SB2 nature of these objects is not detected in Λ1 spectral range, and the derived v sin i is a “combined” broadening. The triple system HD 98353 is observed close to conjunction, and lines remain blended. For HD 79763 (d) and HD 119537 (f), the difference in radial velocity is large enough to measure separately the rotational velocities.

where E is an integer of phases, and the Julian date of observation (HJD 2449413.3301), the phase is equal to 0.4.

– HD 42035 has no indication of any binary status in the literature and it is flagged as a photometrically

Constant Star From Hipparcos Data. Nevertheless

the observed line profiles in its spectrum tend to sug- gest that it is composite. The cross-correlation function (CCF) of the observed spectrum with a synthetic one (Teff= 10 000 K, log g = 4.0, v sin i = 5 km s−1) has

8

F. Royer et al.: Rotational velocities of A-type stars. II. Table 6. Results for stars seen as SB2. Rotational veloc- ities are given for each component when measurable. ∆Vr stands for the difference in radial velocity between the two components. Dash indicates a non possible measurement (either for v sin i or ∆Vr).

∆Vr

Fig.

37

7a.

127

7b.

12:

7c.

–

8a.

67

8d.

44

8b.

64:

8e.

–

8c.

98

8f.

229

7d.

56

7e.

106

7f. been computed. This CCF is not characteristic of a single star, and it is perfectly fitted by the sum of two gaussian components centered at 26 and 34 km s−1 re- spectively and whose FWHM are 30 and 120 km s−1.

This indicates that the system is composed of a low v sin i star and a faster rotator.

– Hd 79763 Is Known As A Sb2 System Whose Orbital

period is P = 15.986 d (Batten et al. 1989). Using the spectrum in Λ3 range (Fig. 8.d), the difference in radial velocity is sufficient to estimate v sin i of both components.

– 55 UMa (HD 98353) is a triple system for which com- ponents are early A-type stars. Using tomographic sep- aration, Liu et al. (1997) estimate the v sin i of each of them: 30 ± 4 km s−1, 45 ± 5 km s−1, 55 ± 5 km s−1.

Knowing The Orbital Parameters: P = 2.5538380 D

and TminRV = 2449602.588 (Horn et al. 1996) for the close pair, our spectra correspond to phases φ = 0.99 (Fig. 8.b, HJD 2448274.6233) and φ = 0.02 (Fig. 8.e,

Hjd 2449413.5500). This Means That Both Observa-

tions were unfortunately made close to opposition, and the difference in radial velocity is not large enough to see separated lines. The measured v sin i corresponds to a blend.

Catalogue (Hoffleit & Jaschek 1982) And Was Not De-

tected as a double-lined system with the spectrum col- lected at ESO, within the context of the southern sam- ple (Grenier et al. 1999; Royer et al. 2002). The rota- tional velocity derived in Paper I is 13 ± 1 km s−1, whereas it equals 23.9 km s−1 in Ramella et al. (1989).

The spectrum in the Λ3 domain displays evidence of SB2 nature and Fig. 8.f shows perfectly the faint lines around Ti ii 4501, Fe ii 4508 and Ti ii 4515 which are usually well isolated in a single star with such a spec- tral type.

– HD 181470 is a close binary system first detected by speckle observations by Miura et al. (1993) and later

Confirmed By Hartkopf Et Al. (2000). The Measured

separation is ρ = 0.′′13 with a magnitude difference ∆m = 1.6 ± 0.2. In the observed spectrum, the differ- ence in radial velocity is large: 229 km s−1.

– HD 203439 is known as a spectroscopic binary system in Batten et al. (1989). – HD 203858 is a known spectroscopic binary. Abt & Morrell (1995) do not see the two components of the system and give a v sin i which is likely overestimated, 70 km s−1, because of blend due to binarity. In this

Case, The Components Are Well Separated And Each

v sin i is measured.

4.2.1. South Versus North

Fourteen stars are common to both the southern sample from Paper I and the northern one studied here. Matching of both determinations allows us to ensure the homogene- ity of the data or indicate variations intrinsic to the stars otherwise. Results for these objects are listed in Table 7.

Table 7. Comparison of the computed v sin i for the stars in common in the northern and southern samples (N ≡ this work, S ≡Paper I). CFF is a flag indicating the shape of the cross-correlation function carried out by Grenier et al. (1999) using the ECHELEC spectra (0: symmetric and gaussian peak, 4: probable double, 5: suspected dou- ble, 6: probable multiple system).

–

Instrumental characteristics differ from ECHELEC to AUR´ELIE data. First of all, the resolution is higher in the ECHELEC spectra, which induces a narrower instrumen- tal profile and allows the determination of v sin i down to a lower limit. Taking the calibration relation from SCBWP

Deployment In Out-Of-Position Situations

D. Bendjaballah1, A. Bouchoucha1, M. L. Sahli1,2* and J-C. Gelin2

Abstract

Side-impact collisions represent the second greatest cause of fatality in motor vehicle accidents. Side-impact airbags have been installed in recent model year vehicle due to its effectiveness in reducing passengers’ injuries and fatality rates. In meeting these requirements, simulations of folding and deploying airbags are very useful and are widely used. The paper presents a simulation method for the deploying airbags using three materials in different working conditions. Finite element analysis is primarily used to evaluate this concept. In these simulations, the gas flow is described by the conservation laws of mass, momentum, and energy. The numerical results indicate that the FE method in this paper is capable of capturing airbag deploying process accurately.

Keywords: Airbag simulations, Out-of-position, Crash, Modeling, Out-of-position

Background

The passive safety of cars has become a very high prior- ity issue for the automotive industry. Today, there are not only one or two airbags in a car; certain models have ten times more than that. With the increasing usage of airbags, the number of accidents where the airbag itself can cause an injury to the occupant also increases

(Augenstein Et Al. 2003; Gabauer And Gabler 2010;

Audrey et al. 2011). As is well known, safety belts are also now devices designed to provide protection to the users of vehicles during crash events, minimizing the loads necessary to adapt their movement to the move- ment of the car (Freesmeier and Butler 1999; Schmitt et al. 1997). In general, the seat belt is designed to restrain the occupant in the vehicle and prevent the

Occupant From Having Harsh Contacts With Interior

surfaces of the vehicles. The airbag acts to cushion any impact with vehicle structure and has positive internal pressure, which can exert distributed restraining forces over the head and face. As a safety component of auto- mobile, an airbag decreases occupants’ injury likelihood effectively in case of an accident (Ruff et al. 2007). These safety elements can reduce the death rates on the roads, and its protection effects have been widely approved (Crandall et al. 2001; Teru and Ishikawa 2003). With computational tools such as finite element methods designed for dynamic contact problems, crashworthiness simulations can now be used with reliable accuracy to evaluate occupant protection in various collision condi- tions with safety metric/parameters such as acceleration, head injury criteria, intrusion distance, intrusion vel- ocity, and neck forces (neck injury risk or whiplash).

Thus, new types of airbag products are being developed to handle different collision scenarios.

Become Standard Equipment On Most New Passenger

vehicles (Braver and Kyrychenko 2004; Teng et al. 2007; Yoganandan et al. 2007). The airbag cushion is com- posed of a woven fabric which is rapidly inflated during a car crash. The airbag dissipates the passenger’s kinetic energy thereby reducing injury through biaxial stretching of the fabric bag and escaping gas through vents. There- fore, the performance of the airbag is greatly influenced by the mechanical properties of the fabric. Generally, air bags are designed to deploy in a crash that is equivalent to a vehicle crashing into a solid wall at 8 to 14 mph.

Air bags most often deploy when a vehicle collides with another vehicle or with a solid object like a tree. There are various types of airbags: frontal, side-impact, and curtain airbags. In general, the passenger side airbags are usually larger than the driver airbags (see Fig. 1).

Besançon, France

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Bendjaballah et al. International Journal of Mechanical

Doi 10.1186/S40712-016-0070-2

Extensive studies have shown that the airbag deploy- ment in load cases consists of two occupant loading phases: a punch-out effect where the airbag bursts out of its container with the airbag and airbag module cover accelerating towards the occupant and a second loading phase during which the airbag is taking on its deployed shape and volume (membrane-loading effect). Bankdak et al. (2002) developed an experimental airbag test system to study airbag-occupant interactions during close proximity deployment. The results provided insight for simulating the effect of inflation energy and mass flow on target response. Bedard et al. (2002) found that while left-side (driver-side) impacts accounted for only 13.5% of all crashes, the fatality rate among these

Crashes Was 68.3% In Comparison To Front Impact

(48.3%), right-side impact (31.3%), and rear impact (38.4%). These studies underscore the importance of oc- cupant safety during side-impact collisions. In the last years, the current market requested to reduce the time and cost airbag development. In order to achieve this result, virtual simulations play an important role since they allow to minimize the number of experimental tests (Pei et al. 2013; Cao et al. 2014). Several simulation models of airbag were established (Wang et al. 2007). It is feasible to optimize the parameters of airbag deploy- ment using simulation technology. Experimental and numerical studies have quantified injury risks to close- proximity occupants from deploying side airbags. These studies have focused on the prevention of the most ad- verse effects of airbag deployment (Duma et al. 2003).

Other studies have proposed airbag characteristics to minimize particular biomechanical responses (Haland and Pipkorn 1996). In a more recent study, Marklund and Nilsson (2003) compared deformation patterns with experimental data as well as the computational costs associated with three different airbag deployment simu- lation methods; they concluded that the SPH method is relatively inexpensive and produces incremental deform- ation patterns that compare most closely to the experi- mental results. The process of inflation of an airbag is one of the determining factors in saving lives. The duration from the initial impact of the crash to the full inflation of an airbag is about 40 ms, and during this time, the airbag goes from being in a folded state to a fully inflated state, with a high internal pressure. After achieving this state, the airbag begins to deflate, thus providing a nice cushion for the body impacting it.

Ideally, the person in the crash should come into contact with the airbag at this time. In the present study, a large volume passenger side airbag model is developed to handle different collision scenarios. The main aim is evaluate the performance of deploying of passenger side airbag using finite element methods (FEM).

Materials

The tensile specimens were made in different airbags (P: Peugeot, R: Renault, and VW: Volkswagen) with a length of 200 mm long and a width of 40 mm. Table 1 shows the mechanical properties of the airbag.

Tensile Tests

To determine the mechanical properties of the material of airbag used in the test pieces, tensile tests were performed on Lloyd EZ20 universal testing machine in Constantine. These tests were conducted using rect- angular samples. The axial force and axial displacement acquired during a test are converted into stress and the strain in order to be used for the fabric material model.

The continuous recording of the stress-strain data was performed during both the load and unload phases. A minimum of five samples were made in order to check the repeatability of the measurements. All the data was collected by using a PC-based data acquisition system and analyzed by commercial software. The picture frame test device that is made for this study is shown in Fig. 2.

Fig. 1 a Frontal and side airbags. b Oblique view of facet occupant model in sitting posture following airbag deployment (Lim et al. 2014)

0.150

Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

Page 2 Of 9

Figure 3 shows the stress-strain relationship of the airbag sample under axial tensile loads. The results are showing a linear increase in extension with the increas- ing stresses. This is an expected output and it confirms with the theoretical behavior of a sample subjected to tensile stress. The rupture strain values for different airbags (R/P/VW) were 0.322, 0.441, and 0.472, respect- ively. The measured elastic parameters (i.e., Young’s modulus E and initial yield strength) and Poisson’s ratio are summarized in Table 2. The tensile tests of the woven fabrics can show differences on mechanical prop- erties because woven fabrics can resist in-plane shear loads once the yarn lock-up angle has been reached. The differences of material property on material direction can affect the shape of fully deployed bag (see Fig. 3b).

Theoretical Background

Numerical simulations of airbags use very complex and techniques such as an orthotropic model to identify the mechanical behaviors during the airbag inflation and the fluid mechanics (gas flow) to describe the inflator gas flow (pressure gradient) and improve the representation of the pressures within the airbag. To model the airbag as an orthotropic model, three material constants have to be provided. Assuming a plane stress condition, the

Ð1Þ

where σ is the normal stress and τ is the shear stress, the subscript refers to the principal material directions, i.e., the fill and warp directions. Also, ε and γ are the strain components. The material elastic constants Qij are

Ð2Þ

where E1 and E2 are the Young’s modulus in the fill and wrap directions and G12 is the shear modulus of the fabric material. νij is the Poisson ratio of the material.

The gas exerts a pressure load on the airbag causing it to expand. This expansion puts the airbag under tensile stress lowering the expansion rate. In this study, heat conduction and heat transfer is not taken into account.

Fig. 2 A photograph of Lloyd EZ20 universal testing Fig. 3 Stress versus strain using Lloyd EZ20 machine for a three different airbags at 0° and 90° and b VW airbag test specimens at

Different Angles

Table 2 Physical and mechanical properties of the airbag

Page 3 Of 9

In the deployment of an airbag, an inflator supplies high velocity gas into an airbag causing it to expand rapidly. The gas inside the airbag is assumed to be ideal, to be of constant entropy, and to satisfy the equation of state:

Ð3Þ

Here p, ρ, and e are respectively the pressure, density, and specific internal energy, and γ is the ratio of the heat capacities of the gas. The gas flow is described by the conservation laws for mass, momentum, and energy that

Ð4Þ

here, V is a volume, A is the boundary of this volume,

N Is The Normal Vector Along The Surface A, And U

denotes the velocity vector in the volume. Applying Bernoulli’s equation in the case of an ideal gas with

Ð5Þ

Here, the subscript ex denotes quantities at the throat of the tube. Furthermore u, p, and ρ denote the quan- tities inside that part of the tube that is supplying mass.

Materials And Boundary Conditions

The airbag system mainly consists of three parts: the airbag itself, the inflator unit, and the crash sensor or diagnostic unit. Thus, to study the behavior of the airbag using FE simulations, we need to have an FE model of the airbag in the folded position. A FE model of the airbag was used to simulate the test condition as shown in Fig. 5. LS-DYNA® material model FABRIC (MAT_34) is used to simulate the airbag material. It is a variation of the layered orthotropic material model. Additionally, in the LS-DYNA® material model, fabric leakage can be accounted for. However, for this CAB material, the leak- age is almost negligible and therefore no leakage is specified. The mechanical properties can be determined from the physical test. Typical material properties for airbag fabrics are taken as given in Chawla et al. (2004a) (Table 3). These properties are used to simulate inflation process of airbag (see Table 1). The car dashboard is modeled as the rectangular thin plate using a MAT_RI-

Gid Material, And The Degrees Of Freedom Are Con-

strained in all the directions. The similar properties of thermoplastic polymer are assigned for contact purposes. The porosity of the fabric is assumed zero. The nitro- gen gas is taken for inflating the airbag. Properties of nitrogen gas and initial bag conditions are shown in Table 4. The example on which we perform the study is a typical passenger side airbag. The geometric de- tails have been measured from a commercially avail- able airbag. The initial state of the airbag is a closed rectangular whose sides are to be finished to 482 × 635 mm2 and is shown in Fig. 4.

Table 3 Material properties of airbag and rigid plate used in FE

–

Table 4 Initial values used for FE simulation of the swelling of

3.33 × 10−4

Fig. 4 The initial airbag geometry in the form of a rectangular Bendjaballah et al. International Journal of Mechanical and Materials Engineering (2017) 12:12

Related Journal Articles & DOI Links

Selected peer-reviewed publications relevant to 12 Lead ECG Acquisition. Click the DOI to access the full paper (may require institutional access).

Why Choose Us?

Bangalore guidance for robotics, Spectre and autonomous systems projects.