Vector Control Induction Motor Matlab

Abstract— The problem of step tracking control with a switching input and without any continuous-valued inputs is considered. The control objective is to reduce the number of switchings to a minimal value. This approach finds interesting applications when switching comprises costs and should be avoided. To solve the problem, a state dependent switching strategy should be designed and the resulting closed loop is indeed a hybrid system. Therefore, first we investigate the conditions on a hybrid system for being the desired solution.

Then, we propose a method for designing the switching strategy such that the closed loop as a hybrid system solves the problem. The proposed method is applied to the induction motor control problem which results in relatively simple and efficient control algorithm. Comparison with the direct torque control for induction motors show that our method has a superior performance in reducing the number of mode switches.

I. Introduction

Development of technologies in the recent decades has led to emergence of new control applications. In a class of such applications which is usually referred to as switched systems, the dynamical model of system has a discrete-valued mode [1, 2, 3]. The discrete mode affects the continuous-valued state variable dynamics. If the discrete mode also depends on the continuous state variables, then we have a hybrid system [4, 5]. Switched and hybrid systems find important applications in many areas such as mechanical systems, process control, automotives, aerial and ground transport systems, power systems, networked control systems and etc. An important problem in this domain is when the discrete mode acts as an input of the system which is referred to as the switching control problem or designed switching [3, 1]. A category of applications for this case are power electronic circuits and drives with electronic switches that have attracted applications of the switched and hybrid systems theories in various forms [6, 7, 8, 9, 10].

However, the discrete mode may have other roles such as being a source of randomness in the system or representing the structure of a complex system . Various control problems have been studied for the cases of switched and hybrid systems that include the problem of tracking a reference output [13, 14, 15, 16].

In this work, we consider the step tracking control problem for switched and hybrid systems using the discrete mode as an input of the system. The main control objective is to track a step function as the desired output. Our second objective is to apply a minimal number of mode switches in order to achieve the tracking objective. This approach to the problem has not been considered previously. In the existing results, either a continuous-valued input is available [13, 14] or the number of switches is not an issue and only special cases of reference outputs are considered [15, 16]. Our problem finds important applications when there is a cost associated with each mode switch which motivates the reduction of the total number of mode switches. An example is power electronic circuits with switching elements in which an amount of energy loss is associated with each switching [8, 7]. To solve the problem, we observe that the closed loop resulting from the switching control is in fact a hybrid dynamical system. Therefore, first we formulate a hybrid system which is able to track a step command. After this step, we design the switching such that the closed loop hybrid system solves the problem. We also provide the solvability conditions. The switching is designed in a way that the time between two successive mode switches is maximized. This does not necessarily result in the minimum number of switchings over time, but the achived number of switchings can be regarded as minimal (suboptimal). However, this approach simplifies the control algorithm and .

Minimal Switch Step Tracking Control of Switched Systems

Babak Tavassoli

reduces its computations significantly, in comparison with optimization based methods such as the finite horizon optimal control approach in . We will apply our method to an induction motor. The result is presented as a control algorithm which is ready for implementation. Then, we make a performance comparison with the direct torque control method (DTC) [17, 18]. This method is widely regarded as the successful control method for induction motors. But, this method suffers from requiring high switching frequencies for reducing the amplitude of the tracking error fluctuations. Simulations are provided that show our method applies a considerably smaller number of switches in comparison with the DTC method.

The organization of the paper is as following. Problem formulation for hybrid and switched systems is performed in section two. Solution of the problem together with the conditions for solvability is provided in section three. The results are applied to the induction motor in section four where a comparison is also made with the DTC method. Conclusions are made at the end.

Notation: In the following, ℝ is the set of real numbers, ℝ+ is the set of non-negative real numbers and ℤ+ is the set of non-negative integers. The Euclidian norm of a vector ξ∈ℝn is denoted by ||ξ|| and its ith element is denoted as [ξ]i. The boundary of a set M in a metric space is denoted by ∂M and its closure is denoted by cl[M]. For a set A, its cardinality is denoted by |A| and the set of all subsets of A is denoted by 2A. We say that a mapping Φ:ℝn×ℝ→ℝn is a transition function for a smooth vector field f:ℝn→ℝn, if the differential equation ∂Φ(ξ,t)/∂t = f(Φ(ξ,t)) with Φ(ξ,0) = ξ is satisfied for every ξ∈ℝn and t∈ℝ.

Ii. Problem Formulation

In this section, the tracking problem using a switching input is formulated after some preliminaries.

A. Switched And Hybrid Systems

A hybrid system has a set of continuous state variables and a discrete state variable that interact with each other while evolving along time. A change of the discrete state is referred to as a jump. Between jumps, the discrete state is constant and the vector of continuous states evolves according to an ordinary differential equation (ODE) which depends on the discrete state. This type of evolution of the state is denoted as a flow. In a general hybrid system, the continuous states may also change value at a jump which is denoted as a reset. However, for simplicity we ignore the reset function and rewrite the definition of hybrid system in as below.

Definition 1: A Non-Reset Hybrid System (NRHS) is a sextuple H = (X, U, E, {Inx}x∈X, {fx}x∈X,

• A Transition Relation E ⊆ X×U×X ;

• a non-empty set Inx ⊆ ℝn for each x∈X denoted as invariant set of x; • a smooth vector field fx : Inx → ℝn for each x∈X ; • a guard set ∅ ≠ Gu(x,u,x′) ⊆ Inx for each (x,u,x′)∈E.

In the above definition, n is the dimension of the continuous state. The state of the NRHS is (x, ξ) with x∈X and ξ∈ℝn . Remark 2: If ξ belongs to the interior of Inx and it also belongs to Gut for some t = (x,u,x′), then both jump and flow are possible at (x, ξ). This situation is regarded as a form of uncertainty . In this paper we avoid such an uncertainty by assuming that a jump has priority over flow (i.e. when both are possible a jump occurs). Another form of uncertainty is possibility of having two jumps with the same input which is also avoided by the following assumption.

Assumption 3: For every (x,u,x′), (x¯, u¯,x¯′)∈E we have

U = U¯ ∧ X = X¯ ⇒ X′ = X¯′

A property of the Definition 1 (inherited from ) is that the input acts only on jumps without affecting the flows. This property conforms to our objective to control with only switchings. A switched system is an NRHS with Inx = ℝn, Gut = ℝn for every x∈X, t∈E, U = X and E = {(x,u,x′)∈X×U×X : u = x′}. Hence, we have a more compact definition for a switched system as the following.

Definition 4: A switched system is a pair S = (X, {fx}x∈X) composed of a finite set of discrete states X and a set of vector fields fx : ℝn → ℝn for each x∈X. According to U = X when representing a switched system as an NRHS, x acts as an input of the switched system. Hence, in the case of a switched system the discrete state x∈X may be referred to as switching input or mode.

The switched system of Definition 4 is a special case of the NRHS in Definition 1 from a mathematical point of view. However, we can build an NRHS from a switched system by selecting guards and invariant sets to restrict the switchings or the jumps. Hence, a switched system may be regarded as more general than an NRHS from a practical viewpoint. Based on this observation, we add the following definition.

Definition 5: An NRHS H = (X, U, E, {Inx}x∈X, {fx}x∈X, {Gut}t∈E) is a restriction of a switched system S = (X, {fx′}x∈X) if U = X, E = {(x,x′,x′) : x,x′∈X} and fx′= fx for every x∈X. For an NRHS which is a restriction of a switched system we can compactly write the triple (x,x′,x′)∈E as an ordered pair (x,x′).

B. Problem Statement

To define the tracking problem we need to define an output for the system. Since the discrete variable x∈X acts as an input of the switched system, we define the output as a function of the continuous state variables only.

Definition 6: For an NRHS or for a switched system with continuous state vector ξ∈ℝn, the vector y∈ℝm is an output if there exist a mapping h: ℝn →ℝm denoted as the output function such that y = h(ξ) for every ξ∈ℝn.

The output at time t∈ℝ+ denoted by y(t) is required to track a constant desired output yd which belongs to a set Yd ⊆ℝm as in the following. Problem 7: Consider the NRHS in Definition 1 with the set of inputs U, an output whose value at t is

M. At A Jump Instant T′, Select U(T′)∈U

according to the hybrid state of NRHS at t′ and the desired output yd∈Yd such that for every t∈[t′,t″] and 1 ≤ i ≤ m the condition in (1) holds where t″ is either the time instant of the next jump or t″ is infinite if there are no jumps after t′.

(1)

The above conditions requires that each element of the tracking error y(t) − yd is either within the error bound or it moves toward such a region. If at every jump instant it is possible to select the input such that (1) is satisfied, then we say that Problem 7 is solvable. If the Problem 7 is solvable, then there may exist multiple choices of input that satisfy (1). In this case, we can select the input that optimizes a measure. In this work, we try to reduce the number of jumps (or the switchings) as below.

Problem 8: Considering the NRHS in Problem 7, among the multiple choices of u(t′)∈U at a jump instant t′ that result in establishment of (1) until the time instant of the next jump t″, select the one that maximizes t″−t′.

For solvability of the Problem 8 there must be at least one choice of input that can establish (1) for t∈[t′,t″]. Hence, the Problem 8 is solvable if and only if the Problem 7 is solvable. In the case of switched systems, there is an additional degree of freedom of choosing guards and invariant set to achieve the control performance. Therefore, the tracking control problem for switched systems is defined based on the Problem 8 as below.

Problem 9: For a switched system, find a restriction H such that Problem 8 is solvable for H.

Iii. Main Results

Our main objective is to solve the Problem 9 which requires the solution of the Problems 7 and 8 for the NRHS case. In the context of the Definitions 1 and 4, for every x∈X, we denote the transition function of fx by Φx. Also, we hide the dependencies on the desired output yd in (1) for brevity since it is a constant.

A. Solution For The Nrhs Case

If the Problem 7 (or 8) is solvable, obtaining the solution is straightforward. Denoting the NRHS by H, first we define the set of hybrid states that belong at least to one guard set as below. G(H) = {(x, ξ) | x∈X, ξ∈Inx, ∃ T=(x,u,x′)∈E : ξ∈GuT}

(2)

If a flow reaches G(H), then a jump occurs according to the priority of jump over flow as described in Remark 2. We denote the time to next jump when starting to flow from a hybrid state (x,ξ) by θ(x,ξ) which

Can Be Calculated As

θ(x,ξ) = sup{t∈ℝ+ | ∀s∈[0,t) : Φx(ξ,s)∉G(H)}.

(3)

Then, for ξ∈ℝn and the vector of error bounds ε∈ℝ+ m, we denote by Xε(ξ) the set of modes in X that can result in the establishment of (1) at a time instant at which continuous state is ξ as below.

ௗ௧|[Y(T) − Yd]I||T=0 < 0, Y(T) = H(Φx(Ξ,T))}

The time derivative in the above equation can be eliminated by applying the chain rule to obtain

(4)

We are interested in the set of modes that can keep (1) to hold until the next jump which is given by

(5)

We calculate the set of inputs that can cause a jump that result in the establishment of (1) as below.

(6)

At the end of each flow, when G(H) is reached and a jump is about to occur, Uε(x,ξ) must be non-empty such that the establishment of (1) can continue in time. This gives the condition for solvability of the

Problem 7 As

Uε(x,ξ) ≠ ∅ ∀ (x,ξ)∈G(H).

(7)

If the Problem 7 is solvable, then the Problem 8 is also solvable since we should only select the element of Uε(x,ξ) which gives the largest value of θ(x,ξ) in (3). We summarize this part as below.

Proposition 10: For an NRHS denoted by H, at every hybrid state (x,ξ)∈G(H), the solution of Problem 7 is an arbitrary element from Uε(x,ξ) in (6) and the solution of Problem 8 is an element from U*

Ε(X,Ξ) In (8)

with θ(x,ξ) in (3).

B. Solution For The Switched System Case

Considering a switched system S = (X, {fx}x∈X), we should determine guard sets and invariant sets such that the Problem 7 (and 8) is solvable for the restriction of the switched system S. The invariant set Inx for every x∈X should be such that (1) holds in Inx. Therefore, based on the definition of Xε(ξ) in (4), for every x∈X the largest possible invariant set is obtained as Inx = cl[{ξ∈ℝn | x∈Xε(ξ)}].

(9)

We select the invariant sets to be the largest possible one in order to be able to have longer time intervals between jumps in the context of Problem 9. Also, to maximize controllability, we consider all possible jumps such that (1) remains valid and obtain the guard sets as Gu(x,x′) = {ξ∈ℝn | x∉Xε(ξ), x′∈Xε(ξ)}.

(10)

It is mentioned that at a point ξ∈∂Inx we may have x∉Xε(ξ) due to taking the closure in (9). The invariant sets are defined to be closed sets to have intersection with the guards in (10) such that every flow in Inx can be followed by a jump after arriving at ∂Inx.

C(Ξ) = Xε(Ξ) For Every Ξ∈ℝn. In An Nrhs Which A

restriction we have U = X and according to (10), the equation (6) reduces to (11).

(11)

By the definition of Xε(ξ), the guards and invariant sets in (9) and (10) establish (1). More precisely, a flow continues in Inx if x∈Xε(ξ) and a jump occurs if x∉Xε(ξ). Therefore, the possibility of such a jump is the only condition for solvability of Problem 9 which is expressed as below.

Xε(ξ) ≠ ∅ ∀ ξ∈ℝn.

(13)

We can now summarize this part as the following. Proposition 11: For a switched system S, the Problem 9 is solvable if (12) holds. The corresponding restriction of S to an NRHS which solves the problem is obtained by the invariant sets in (9) and the guards in (10). Also, the control input at each jump is selected from U* ε in (13).

Remark 12: The condition (12) for solvability of the tracking problem can be interpreted as having sufficient actuation for the plant (or the switched system) through the available modes or the switching input. It is noticed that the condition (12) also depends on the choice of the output of the system and we may need to define the output suitably in order to solve the tracking problem (see the next part).

Remark 13: We can obtain a condition which is independent of yd and is a sufficient condition for (12). This can simplify the task of determining the solvability of the Problem 9. To do this, we require that only the expression on the second line of (4) must hold for some x∈X. To make it independent of yd we should

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.

Figure: System Model & Simulation Flow for Ansys Airbag Injury Simulation

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).

Figure: System Model & Simulation Flow for Ansys Airbag Injury Simulation

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.

Figure: System Model & Simulation Flow for Ansys Airbag Injury Simulation

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).

Figure: System Model & Simulation Flow for Ansys Airbag Injury Simulation

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.

Figure: System Model & Simulation Flow for Ansys Airbag Injury Simulation

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

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