## Modelling of an Explosion of an Improvised Explosive Device a Vehicle

**Turza Jozef**^{1,}, **Eliáš Jozef**^{1}, **Kopiláková Beáta**^{1}, **Rakúsová Danka**^{1}

^{1}Faculty of Special Technology, Trenčín University of A. Dubček in Trenčín

### Abstract

An aim of the paper is to define behaviour of a combat vehicle based on an analysis of its movement from a point of a maximum vertical movement, acceleration and forces through development of a mathematical model of a vehicle during an operation of the vehicle in critical situations of a modern battlefield.

### At a glance: Figures

**Keywords:** ** **combat vehicle, explosives, an explosion, a system, mathematical model, a torque bar, a rocker, a damper, spring and rubber buffer, a vehicle hull

*American Journal of Mechanical Engineering*, 2014 2 (7),
pp 209-215.

DOI: 10.12691/ajme-2-7-8

Received September 05, 2014; Revised September 15, 2014; Accepted October 09, 2014

**Copyright**© 2013 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Jozef, Turza, et al. "Modelling of an Explosion of an Improvised Explosive Device a Vehicle."
*American Journal of Mechanical Engineering*2.7 (2014): 209-215.

- Jozef, T. , Jozef, E. , Beáta, K. , & Danka, R. (2014). Modelling of an Explosion of an Improvised Explosive Device a Vehicle.
*American Journal of Mechanical Engineering*,*2*(7), 209-215.

- Jozef, Turza, Eliáš Jozef, Kopiláková Beáta, and Rakúsová Danka. "Modelling of an Explosion of an Improvised Explosive Device a Vehicle."
*American Journal of Mechanical Engineering*2, no. 7 (2014): 209-215.

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### 1. Introduction

In deployment of a mechanized infantry an explosion of a charge under a vehicle may appear. The paper is aiming to make an analysis based on a developed mathematical model, what influence will it have on a vehicle’s dynamics. In the paper there is a simplified mathematical model of behaviour of the vehicle. The model can determine what acceleration (hence power) influence a vehicle and what large vertical movements may occur. An issue of a vertical movement of a vehicle after an explosion of a charge ^{[1, 7]} and in running over a bump is rather complicated; therefore some simplifications have been applied for the first analysis.

### 2. Description of Dynamics of Vehicle Movement in Explosion under a Vehicle

The BMP-2 combat vehicle has been taken for description of dynamics of a vertical movement of the vehicle, depicted with main parts needed for description of its dynamic features in the Figure 1. The traverse wheels with rockers and torque bars are important for a description, where a torque bar is a spring element for a vertical movement ^{[1, 2, 3, 6]}.

**Fig**

**ure**

**1**

**.**

**A view on a BMP-2 combat vehicle**

An explosion of a charge under a vehicle is supposed in a lower part of a hull places under the gravity center of the vehicle.

As an issue of a movement of the vehicle is rather difficult, the following assumptions have been assumed for modelling ^{[5, 7]}:

- an explosion will occur in the middle of the vehicle under its gravity center,

- the gravity center of the vehicle is situated symmetrically in a longitudinal as well as in a transverse direction,

- we consider no longitudinal, nor transverse vibration of the vehicle, we take only vertical movement of the vehicle into consideration,

- pressure during explosion is regularly distributed on a whole lower surface of the vehicle,

- distribution of forces is regular for all loaded wheels,

- a factor of dumpers absorption is considered as different for an engaging and disengaging of a dumper,

- dumpers are constantly connected with contact points,

- terrain under vehicle is considered flat, clearance of a vehicle is the same,

- vertical movement of the vehicle downwards is limited proportionally for four rockers through lamellar and for four rockers through rubber buffers.

**2.1. The Vehicle as a Whole**

Mathematical description of particular important parts of the vehicle is defined in a subsequent text. Particular important elements and groups needed for description of behaviour of the vehicle are described separately with certain simplification. There are independent units, as a hull of the vehicle, rockers with torque bars, dampers, and spring and rubber buffers of the vehicle. A separate chapter is a description of a force action from an explosion of an improvised charge on a vehicle bottom.

**2.2. Vehicle Hull**

With respect to the assumptions we are taking into consideration the forces are acting symmetrically only in a cross and longitudinal sections. The action of forces acting on the vehicle hull in a cross and longitudinal sections are illustrated in the Figure 2. There are forces from placing a torque bar *F*_{A}, from the weight of the vehicle *G*_{v}, a force from a vehicle acceleration *F*_{mv}, forces from dampers *F*_{tl} from the first and the sixth pair of wheels, forces from a spring buffer *F*_{pr1} in the first and the sixth pair of wheels, forces from a spring buffer *F*_{pr2} in the second and the fifth pair of wheels having the number of rockers with wheels *n*_{k}=12.

**Fig**

**ure**

**2**Action of forces on a vehicle hull; a) a cross section, b) a longitudinal section

From a balance of forces in a vertical direction the following relation applies.

(1) |

Gravitational force of the vehicle having an *m*_{v} weight is defined as

(2) |

and an accelerating force

(3) |

The other forces and relations will be defined in the following text.

**2.3. A Rocker with a Traverse Wheel and a Torque Bar**

We suppose that a road has a flat surface. Kinematics of a movement and main parameters of a traverse wheel rocker with a torque bar is illustrated in Figure 3. We can simplify it for purposes of a dynamic analysis with a certain approximation as in Figure 4. An action of forces on a rocker in a vertical plane is illustrated in Figure 5. Let us simplify and choose a simulated rocker that the forces of a dumper, the forces of a spring and rubber buffer having a size with 4/*n*_{k} ratio, are acting.

Then in a vertical plane there are acting forces on a simulated rocker as by Figure 5. Particularly there are* *defined the* **M*_{t} momentum of a torque bar and a *M*_{I} momentum from a rotation of a rocker. An *F*_{c} force from a road at *a*_{1}_{ }distance implies an additional bending moment on a rocker and therefore a deformation as well. With regard to a massiveness of a rocker we are not taking its deformation into consideration for our purposes. We do not consider any vertical roughness of a road. The description is made for a group of a simulated rocker of a traverse Wheel with a torque bar. Through an implementation of a course of a vertical movement due to a road roughness, a dynamic behaviour of a driving vehicle could be determined.

**Fig**

**ure**

**3**

**.**Kinematics of a movement of a rocker, a damper, a rubber and a spring buffer

**Fig**

**ure**

**4**

**.**

**Forces acting on a rocker in an explosion of a charge in a simplified way**

**Fig**

**ure**

**5**

**.**Forces acting on a simulated rocker with a wheel in a vertical plane

In Figure 5 there are illustrated all stresses applied to the rocker in a vertical plane. The principle of a movement of a rocker, damper, a rubber and a spring buffer is delineated there. The movement will be explored only on the only one rocker. The number of rockers is *n*=12, number of spring buffers is four and number of rubber buffers is four. The weight of a vehicle per a rocker is reduced in 1/*n*_{k} ratio, therefore the number of buffers is also considered in 1:4 ratio. An uplift of a rocker (vehicle) will be considered approximately

(4) |

The *M*_{t} torque of a torque bar, having an* **l*_{t} length of a torque bar, the *G*_{t} modulus of elasticity in shear and a *J*_{t}* *polar square moment of a torque bar cross-section having a *d*_{t} diameter causes twisting of a rocker by an angle

(5) |

where a polar square moment of a torque bar cross-section is defined from a relation

(6) |

Stiffness of a rocker torque bar is defined using an equation (5)

(7) |

There are applied similar relations between a twisting φ and a shift of a vehicle body in large and small deformations

(8) |

We suppose balanced loading of all rockers of traverse wheels with torque bars. A reduced *I*_{r} moment of inertia for a rotational movement of a rocker with a Wheel is calculated based on a relation

(9) |

where *I*_{r} is a mass moment of inertia of a rocker arm o a traverse wheel and a torque bar. For a vertical movement of an A point (Figure 5) the equation of a balance in a vertical direction is applied

(10) |

a torque equation to A point

(11) |

respectively to B point

(12) |

The buffer forces *F*_{pr1} and *F*_{pr2} start acting only after having finished an *x*_{max} uplift (Figure 3). A mass moment of inertia for *I*_{vr} rocker is computed based on a relation

(13) |

After having substituted real values for simulation we receive

(14) |

**2.4. Damper**

It is a double action hydraulic damper, the manufacturer states from testing with 40 double lifts per minute its mean power 3.25 kN in piston rod disengagement and 13 kN in piston rod engagement at *h*_{t}=0.1 m uplift. As a speed of a piston rod uplift in testing is the same in engagement and disengagement, time of uplift is

(15) |

and a speed of uplift is

(16) |

a coefficient of a damping the piston rod down in disengaging is

(17) |

and a coefficient of a damping the piston rod down in engaging is

(18) |

**Fig**

**ure**

**6**

**.**

**Linear characteristics of a damper for engagement and disengagement of a piston rod**

A force of a damper can be expressed by an equation

(19) |

An uplift of a damper on a rocker will be approximately

(20) |

**2.5. Spring Buffer**

A plate spring has been used in form of an Archimedean spiral in conformity with Figure 7. Basic dimensions for computation were: number of convolutions *n*_{l}=7, final angle of a spiral φ_{p1}=0.1 rad, φ_{p2}=0.1833 round, plate width *h*_{l}=92 mm, plate thickness *b*_{l}=4 mm, maximum compression *x*_{dm}=86.6 mm, radius of a spiral origin *r*_{1}=12 mm, radius of a spiral end *r*_{2}=44 mm, modulus of elasticity of a plate material *E*_{l}=210 GPa.

**Fig**

**ure**

**7**

**.**

**Construction of a plate buffer in a shape of an Archimedean spiral**

A square moment of a cross-section

(21) |

Length of a spiral plate

(22) |

Length of a plate of an Archimedean spiral

(23) |

Stiffness of a plate spring

(24) |

After having finished an uplift of a spring, stiffness of a buffer will be

(25) |

Course of a spring force will be

(26) |

in compression of a plate spring

(27) |

Course of forces of a plate spiral spring and a fixed buffer is illustrated in the Figure 8.

**Fig**

**ure**

**8**

**.**

**Course of forces of a plate spring buffer and a fixed buffer**

**2.6. Rubber Buffer**

**Fig**

**ure**

**9**

**.**

**Construction of a rubber buffer**

Construction of a rubber buffer is illustrated in the Figure 9. A rubber is vulcanically connected on a segment. A segment is firmly attached to a vehicle hull with a bolted connection. The dimensions and data on a rubber in accordance with the manufacturer’s data are *a*_{g1}=60 mm, *a*_{g2}=70 mm, *b*_{g1}=50 mm, *b*_{g2}=60 mm, *h*_{g}=53 mm. A dynamic modulus of elasticity in pressure for a used rubber is considered to be *E*_{g}=10 MPa.

Compression of a rubber buffer is computed through a relation

(28) |

Where *k*_{1}, *k*_{2} are auxiliary constants

(29) |

Through an integration of an equation 28 and using constants from an equation 29 we receive

(30) |

From equation 30 after an adjustment a stiffness of a rubber buffer can be obtained in a following form

(31) |

After having substituted the dimensions and parameters of a rubber buffer into an equation 31 we receive an intensity of stiffness

(32) |

Relation for computation of a force of a rubber buffer will be

(33) |

Course of a force of a rubber buffer is drawn by an equation 33 in the Figure 10.

**Fig**

**ure**

**10**

**.**Course of a force of a rubber buffer

**2.7. Forces from an Explosion on a Vehicle**

An overpressure spreads on a front of a blast wave in an explosion of a charge in spherical surfaces. The time of a rise and a discharge of a blast wave are very short, in order up to 20 ms. Makovička was dealing with modelling of a course of a blast wave in more details ^{[7]} and he defined the most probable course. His model has been used in the next text for our case. A value of an overpressure on a front of a blast wave will be calculated through following relations. A reduced spacing distance from an *R*_{p} explosion epicenter will be calculated through a distance from an epicenter of an* **R* charge explosion in m, a total equivalent mass of a charge *C*_{w} in m/kg^{1/3} from a relation

(34) |

There is a different value of an overpressure (Figure 11) acting a each point P on a bottom of a vehicle having dimensions *a*_{v}, *b*_{v} on an element of a surface *dA=dx*•*dy* at *R* distance. This value will be calculated from a relation

(35) |

where *R*_{0}_{ } is a vertical distance of a lower surface of the vehicle from a road, *x*, *y* are coordinates of the P point in a coordinate system *x*, *y*.

**Fig**

**ure**

**11**

**.**

**Exerting pressure explosion charges under the vehicle**

**Fig**

**ure**

**12**

**.**

**Values of a course of a maximum overpressure**

A total equivalent mass of a charge in kg TNT (trinitrotoluene) is defined from a relation

(36) |

Where *C*_{N} is a weight of a used charge of an explosive in kg, *k*_{NTp} is a pressure TNT equivalent, *k*_{E}* *coefficient of a charge sealing, *k*_{G} coefficient of a geometry of a blast wave spreading in a space (for a detonation in a free space *k*_{G}=1, on a terrain surface *k*_{G}=2), *k*_{B} is a ballistic ratio (ratio of a case weight in kg/ to weight of explosives in kg). A sealing coefficient is defined based on relation

(37) |

A size of an overpressure in MPa on a front of a blast wave for a ground explosion in MPa at a ground explosion in external environment for *R*_{p}≤1 (in our case the term is applied

(38) |

An overpressure in an overpressure phase of a blast wave in Mpa

(39) |

Duration of an overpressure wave in s

(40) |

Duration of an under pressure wave in s

(41) |

Course of an overpressure and an under pressure wave in time by previous relations is illustrated in figure, where *p*_{0} is a pressure of an environment. For different 4 values of a charge weight in kg of TNT a Figure 13 illustrates a maximum value of an overpressure for a vertical distance from a bottom surface of vehicle from a road* **R*_{0}=0.5 m.

**Fig**

**ure**

**13**

**.**

**Course of a maximum overpressure on a blast wave head under a vehicle**

**Fig**

**ure**

**14**

**.**

**Course of maximum stresses in quadrant I for different sections in a cross direction**

*x*=(1÷10)

In a selected I quadrant a bottom of a vehicle is divided through a grid into ten sections in a cross direction of a vehicle (*x*=1÷10) and also into ten sections in a longitudinal direction (*y*=1÷10).

Figure 14 illustrates a course of maximum overpressures in a cross-section in *t*_{p} time and the Figure 15 in a longitudinal direction.

**Fig**

**ure**

**15**

**.**

**Course of maximum overpressures in the I. quadrant for different sections in a longitudinal section**

*y*=(1÷10)

**Fig**

**ure**

**16**

**.**

**Spatial representation of a maximum value of a pressure in the I quadrant in a cross and longitudinal directions**

Based on courses we can note, that if a blast wave leaves in a longitudinal direction, the value of a maximum pressure is lower than if it leaves under a vehicle in a cross direction. It relates a symmetric action of an overpressure on a bottom part of the vehicle, therefore a calculation can be done only for one quadrant, and for example the first one and a resulting force will be multiplied by 4 x. A value of an under pressure is very small, therefore it will be neglected for a next calculation of forces from a charge explosion. A size of maximum acting force acting on a bottom of a vehicle in *t*_{p} time in accordance with Figure 11 can be defined based on a relation

(42) |

After integration and adaptations from the equation 42 we receive

(43) |

In case, we would considered, that under a whole surface of a vehicle there is acting a maximum overpressure, a value of a maximum force in *t*_{p} time would be

(44) |

A course of a force is acting on a bottom of a vehicle in *t*_{p} time only with a maximum value of *F*_{v} in time *t*_{p}/2, it means in a form of an isoscelent pulse (Figure 12, respectively for particular values by a Figure 13). With respect to small dimensions of a vehicle we do not consider any regressive blast wave.

For combat vehicles we need to consider mostly the mass of a used charge up to *C*_{N}=5 kg TNT placed just about a terrain surface, where *k*_{G}=2.

### 3. Simulation Model

A simulation model has been made up in a computation model DYNAST based on above mentioned equations for a particular vehicle through impedance networks. A block simulation model is shown in the Figure 17. Stiffness of a torque bar with a rocker using equations 5 and 7 with a length of a torque bar *l*_{t}=1.93 m, its modulus of elasticity in a material shear *G*_{t}=78.5 GPa and a diameter *d*_{t}=38 mm is

(45) |

In balanced stable conditions a φ_{st} deformation of rockers is defined using equations 8 and 11, equations 11 and 12 and further adjustment without taking forces of a damper and buffers into consideration as follows

(46) |

with a corresponding angle φ_{st}=2.202°, respectively a static compression in accordance with equation 8

(47) |

This value will be an initial condition for a further solution in a simulation. The previous values were calculated for a length of a torque bar *l*_{t}=1.93 m, diameter of a torque bar *d*_{t}=38 mm, weight of a vehicle *m*_{v}=12800 kg, gravity acceleration *g*=9.81 m/s^{2}.

**Fig**

**ure**

**17**

**.**

**A block simulation model for an explosion of a charge under a vehicle**

Aiming to form a block simulation modul there were used an Rtl movement resistance observing a damping feature of a damper, an Lt acceleration resistance describing a mass moment of inertia of a rocket and attached masses, Ct capability of a torque bar observing stiffness of a torque bar, a Epr1 source respecting a feature of a spring buffer, a Epr2 observes a feature of a rubber buffer, Lv an acceleration resistance observing a mass of a vehicle, the Epm source observes a mass of a vehicle, Ev source of force from explosion, J1 and E1 represent a transformer of a vehicle movement and a rocker movement, BIv and BIt are integrating blocks to define uplifts of a vehicle *x*_{v} and uplift at the end of a rocker *x*_{t}. Theory in more details is mentioned in ^{[8]}.

### 4. Results of a Simulation

Some results of a simulation of the explosion under a vehicle with a charge of size *C*_{N}=5 kg TNT in accordance with data from a previous text are drawn in the Figure 18.

**Fig**

**ure**

**18**

**.**

**Results of simulation of a charge explosion under a vehicle**

A course* **x*_{v} of a vehicle uplift is drawn in the Figure18, having a *v*_{v} speed of the uplift and an *a*_{v} acceleration in a vertical direction during an explosion of a charge under a centre of a vehicle. From courses it can be seen that a maximum uplift of a vehicle is 388 mm at 10.8 ms, a minimum uplift -115 mm, becoming stabilized after 1.5 s. Speed of uplift is maximum at 5.873 m/s in time 16.5 ms and minimum -1.419 m in time 1.5 s. A maximum value of acceleration is 6542 m/s^{2} in time of 1 ms and minimum -46.92 m/s^{2} in time of 56.5 ms. A value of a maximum acceleration for a charge being considered is significant and it has a great influence on a vehicle crew. The values would be lesser if also deformation and damping features of the vehicle hull would be taken into consideration. At computed accelerations it would be difficult to design the seats of a crew that would eliminate an effect of acceleration.

Size of a maximum force at explosion would be *F*_{vm}= 7.69 MN of a saw-tooth pulse lasting *t*_{max}= 1.66 ms.

From values of courses in the Figure 18 we can note that kinematics, used stiffness of rockers and dampers of the vehicle are designed very well and they provide for a strong aperiodic course without additional vibration after a vehicle incidence.

### 5. Conclusions

A generated simplified mathematical model and results of a simulation have shown that a construction of a chassis is very well designed.

It has been proved that action by force from an explosion of a charge of a size of *C*_{N}= 5 kg TNT is significant, mainly a value of an acceleration of a vertical movement of a vehicle.

Based on verification by a created program we are presenting its improvements also for an explosion of a charge aside from a vehicle centre.

### Acknowledgement

This publication was created in the frame of the project "Alexander Dubček University of Trenčín wants to offer high-quality and modern education", ITMS code 26110230099, based on the Operational Programme Education and funded from the European Social Fund.

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