HYDRO PNEUMATIC MECHANICAL ENERGY ABSORBER ENHANCING PASSIVE SAFETY OF A MOTOR VEHICLE

The purpose of this paper is to describe and analyse a mechanical device designed to enhance the safety of a motor vehicle. The topic is addressed by analysing the method of absorbing kinetic energy during a car collision with an obstacle. The article analyses opportunities to convert motor vehicle’s kinetic energy into another type of energy in the case of collision. For this purpose, various mechanical, hydraulic or pneumatic devices are normally used. Such devices are designed to absorb collision energy and reduce or eliminate its impact on the driver, the passengers or cargo in the motor vehicle. The absorber may be used as an additional element of safety to the passenger and the cargo. The energy absorber described in the present article incorporates hydraulic, pneumatic and mechanical components. The description of the absorber presented here is based on mathematical calculations characterizing mechanical, pneumatic and hydraulic processes in the equipment. The analysis of the developed mechanism employs a special application to calculate major parameters of the motor vehicle and the installed absorber. The article also gives a sensitivity analysis of the effect of the rod length on the decrement of the vehicle's kinetic energy.


INTRODUCTION
The importance of passive security in motor vehicles has been growing since decades. The safety of the passenger and the cargo is one of the key issues in transport security. Safety during a car collision with an obstacle is typically ensured by absorbing excessive kinetic energy. To absorb the kinetic energy during a car collision and to convert it into another type of energy, modern motor vehicles use a range of mechanical, hydraulic and pneumatic devices.
Such energy absorbing devices are designed to absorb collision energy and reduce or eliminate its impact on the driver, the passengers or the cargo in the motor vehicle. At the same time, such devices would eliminate the need to use air bags or might be used as an additional means of safety to passengers or the cargo.
The energy absorber is designed as a hydraulic, pneumatic and, at the same time, mechanical device as all these expose specific characteristics that may positively contribute to the overall construction.
The description of the absorber presented here is based on mathematical calculations characterizing mechanical, pneumatic and hydraulic processes in the equipment.
The analysis of the developed mechanism employs a special application to calculate major parameters of the motor vehicle and the installed absorber: the trajectory, velocity, acceleration and kinetic energy of the motor vehicle. The calculations help to determine the significance and variation 6 M. Bogdevicius, R. Vitkunas of the parameters of individual elements of the absorber. Finally, a research in and an analysis of the sensitivity of the device levers are done.

ANALYSIS OF THE CONDUCTED WORK
To absorb motor vehicle's kinetic energy during a collision, various hydraulic, pneumatic or mechanical solutions may be used [10]. Review and analysis of the conducted research distinguishes a hydraulic system [5, 12 -14], which reduces the impact suffered by the chassis. Also, a pneumatic shock absorbing system [8] and adjustable shock absorbers [9,16] are widely used to reduce impact on the chassis. The chassis may be equipped with various mechanical [11,17,18] or electromechanical safety devices [6,7] that reduce collision forces when a car with a specially equipped chassis is hit by another motor vehicle.
Mechanical devices [1,4,15] may also be used. The development of the construction design has been mostly affected by works by Georg Piontek, Stanislaw Gumula and Premyslaw Lagievka [1][2][3][4]. Their works describe a mechanism that changes the force of the forward motion into rotational inertia of a flywheel, along with the other works [12]. The works also present the principle scheme of the mechanism that changes the linear force and kinetic energy of the collision into the flywheel's rotational kinetic energy and torque. They also describe the design of the mechanism, but they present neither calculations of its individual elements nor works of optimization. Another drawback of the presented mechanism is its comparatively large dimensions.

A DYNAMIC MODEL OF THE HYDRO PNEUMATIC MECHANICAL ENERGY ABSORBER THAT ENHANCES PASSIVE SAFETY IN MOTOR VEHICLES
A motor vehicle with the mass has a hydro pneumatic mechanical energy absorber that enhances passive safety in the motor vehicle (Fig. 1). The motor vehicle is coupled with a bumper whose mass is . Between the motor vehicle and the bumper, there is a deformable body with the rigidity of and shock absorption . As the motor vehicle hits an obstacle with the rigidity of and shock absorption factor , the part of the device attached to the motor vehicle's bumper begins to move at the velocity . The lever of the device on the left side of the hydraulic cylinder presses the inside liquid, increasing its pressure . The respective piston areas of the hydraulic cylinder on the left and the right sides are and . The hydraulic cylinder absorbs the initial impact. Perforation in the cylinder partly equalizes the pressure values on both sides of the hydraulic cylinder ( and ). The rack bar attached to the lever lies in the cylinder and moves, on a collision, to the right, rotating the gear wheel mechanism at the same time (the radius of the gear wheel, referred to as the first gear wheel, is marked in Figs. 1 and 2). During the collision, the gear wheel moving to the right rotates the mechanism and finally (on reaching the end point on the right) comes out of contact with the gear wheel with the radius , leaving the gear wheels and other elements of the mechanism in rotation and movement.
Another gear wheel mounted on axle , or the second gear wheel, with the radius ( Fig. 1 and 2) is coupled with the first gear wheel and rotates at the same speed. The second gear wheel ( ) is also coupled with the third gear wheel . The gear wheel is driven by the gear wheel (Figs. 2 and 3).
The gear wheel is mounted on axle with a flywheel (or the fourth gear wheel) with the radius . Also, a traveller is mounted on the axle (Fig. 3), which drives a crank and generates pressure in the pneumatic cylinder (the piston area is ). The pneumatic cylinder absorbs kinetic energy of the flywheel (and the entire mechanism).  The centre of gravity of the traveller is at the distance from the point , and the centre of gravity of the crank lies at the distance from the point of connection (Fig. 3). An additional rod with the length , loaded with an additional mass at its end (point ) rotating about the point , is mounted on the crank.  The angular velocity of the gear wheels mounted on axle is , and the starting position of the first gear wheel is ( Fig. 1 and 2). The angular velocity of the gear wheels mounted on axle is ( Fig.1 and 3). The starting point of the traveller is . The starting point of the additional mass rod is (Fig. 3). The angle of the starting point of the crank measured between the crank and the axle is ( Fig. 3). Moments of inertia of the first to fourth gear wheels are given in Fig. 2, with values , respectively. The design and parameters of the mechanism have to be set so as to allow the hydro cylinder, the mechanical assembly of the rack bar and the gear wheels, the traveller and the crank with the additional mass on the rod and the pneumatic cylinder to maximally absorb kinetic energy passed to the motor vehicle at the moment of collision with an obstacle. Also, the velocity of the motor vehicle has to maximally decrease, but the acceleration of the motor vehicle should not exceed the value that can cause risk to the driver and the passengers.

Coordinates and velocities of the mechanism
Coordinates and velocities of the actual points in the absorber are described in Fig. 1˗3. Coordinates of the centre of gravity of the traveller are set as follows: and  12 11 , , , Hydro pneumatic mechanical energy absorber… 9 and their velocities are as follows: and (7) , (8) where is the length of the traveller. The coordinates of the centre of gravity of the crank are as follows: and (9) , 10) and their velocities are as follows: and (11) , (12) where is the distance from the point of connection of the traveller with the crank and the centre of gravity of the crank ; is the angle between the crank and axis ; and the angular velocity of the crank in respect of axis .
Coordinates of the point of the rod attached to the crank at the point are as follows: and (13) , (14) and their velocities are as follows: and (15) , (16) where is the distance between the point of connection of the traveller to the crank and the point of the rod attachment to the crank .
Coordinates of the rod connection point and the rotating end point are determined as follows: and (17) , (18) where is the length of the rod and is the turning angle between the rod and the crank.
The elocity of the rotating end point of the rod attached at the point is determined as follows: ,

Kinetic and potential energy of the absorber
The kinetic energy of the system is equal to the following: where and are the respective masses of the motor vehicle and the bumper; and are the respective accelerations of the motor vehicle and the bumper; , , are the moments of inertia of the gearwheels, whose radii are , and , respectively; is the mass of the traveller; is the mass of the crank; is the mass of the rod attached to the crank at the point C4; and is the velocity of the piston of the pneumatic cylinder.
The potential energy of the system is equal to the following: where is the obstacle's rigidity factor; is the rigidity factor of the motor vehicle and the bumper assembly; is the rigidity factor between the rack bar and the first gear wheel (with the radius ); is the rigidity factor between the second and the third gear wheels; the respective radii are and ; is the Heaviside function; is the initial distance between the bumper of the motor vehicle and the obstacle; and and are the radii of the first and the second gear wheels (the axis of rotation is ).

Generalized forces
Generalized forces of the assembly are the following: is the friction force between the motor vehicle and the bumper; and are the respective accelerations of the motor vehicle and the bumper; is the friction between the hydraulic cylinder and the piston; and is the velocity of the piston.

Equation for pressure change in the hydraulic and the pneumatic cylinders
The force of the piston in the hydraulic cylinder is described as follows: (28) where and are piston areas on the left and the right sides of the pneumatic cylinder and and are pressure values on the left and the right sides of the piston in the pneumatic cylinder. The pressure change in the first chamber of the hydraulic cylinder is described by the following equation: where is the volumetric modulus of elasticity of the fluid; is the initial volume of the first chamber; is the piston area in the left side of the hydraulic cylinder; is the fluid debit between the first and the second chamber of the hydraulic cylinder; and and are the pressure values in the left and the right sides of the cylinder, respectively.
Pressure change in the second (right) chamber of the hydraulic cylinder is described by the following equation: is the initial volume of the second chamber and is the piston area in the right side of the hydraulic cylinder. , where is the perforation area of the piston of the hydraulic cylinder; is the fluent deficiency rate in the perforation of the piston; is the Reynolds number; and is the density of the fluid. Pressure change in the left chamber of the hydraulic cylinder is described by the following equation: where is the gas adiabatic index; is the gas constant; is the temperature; is the initial volume of the third chamber of the pneumatic cylinder; is the piston area in the left side of the pneumatic cylinder; is the displacement of the piston of the pneumatic cylinder; is the rate of the gas flow from the third chamber to the environment; is the ambient pressure; and is the pressure value on the left side of the pneumatic cylinder.
On the right side of the pneumatic cylinder: Pressure change in the right chamber of the hydraulic cylinder is described by the following equation: is the initial volume of the fourth chamber; is the piston area in the right side of the pneumatic cylinder; is the rate of the gas flow between the third and the fourth chamber; and is the pressure value on the right side of the pneumatic cylinder.
Piston displacement in the hydraulic cylinder is as follows: , (34) where is the initial coordinate of the piston; and are the respective lengths of the crank and the traveller; is the angle between the traveller and its axle ; and is the angle between the crank and its axle .

Equations to describe the movement of the gear wheels in the assembly
The movement of the gear wheels is described by the following equations: where , , , and are the respective moments of inertia of the gear wheels; and are the acceleration values; , , and are the radii of the respective gear wheels; and are the gear wheels coupling rigidity factors; is the distance between the bumper and the obstacle; and are the initial gaps between the gear wheel teeth in the coupling; is the Heaviside function; and and are the decrement indexes.

DEVELOPMENT OF THE APPLICATION
To carry out the exact calculations of the parameters of the hydro pneumatic mechanical energy absorber, based on the formulas described in chapter 4, a special FORTRAN application has been developed.
The calculations include the following parameters: ü masses of the motor vehicle, the bumper, the crank, the traveller, the pneumatic piston and the gear wheels; ü radii of the gear wheels; ü moments of inertia of the gear wheels; ü distances from the points of connection of the traveller, the crank and the additional mass to the corresponding centres of gravity; ü diameters of the pistons and the rods; ü the number and diameters of the holes in the hydraulic piston; ü clearance between the rack bar and the gear wheel, and clearance between gear wheels (the second and the third gear wheels); ü the volumetric modulus of elasticity and density of the fluid.

RESULTS OF MATHEMATICAL CALCULATIONS
For the actual calculations, the following parameters have been chosen: 1000 kg vehicle mass, 20 kg bumper mass and 40 km/h motor vehicle's velocity. The calculations include the following parameters of the hydraulic and pneumatic cylinders: moments of inertia and radii of the gear wheels, weights and lengths of the traveller and the crank. The major parameters are presented in the table below.
Further analysis of the device parameters comprises vehicle and bumper thrust, velocities, and accelerations, turning angles of the gearwheels and the traveller, piston strokes and velocities, and the mechanism inertia. The most important among them are thrust, velocity, acceleration and kinetic energy of the vehicle.
The obtained results are graphically presented in Figs. 4-7.
The analysed device is designed to absorb vehicle collision energy; therefore, the optimal parameters have to be set so as to suppress maximum kinetic energy of the vehicle in the shortest possible time on its collision with an obstacle. Meanwhile, the motor vehicle's displacement should remain minimum and its acceleration value should remain within the allowable range.

SENSITIVITY ANALYSIS OF THE PARAMETERS OF THE HYDRO PNEUMATIC MECHANICAL ENERGY ABSORBER
To correctly select parameters of the assembly, it is necessary to know which of them have the most significant effect on the reduction of the system's kinetic energy. The actual effect of each parameter on the reduction of the system's kinetic energy is determined as follows: , where and are the respective kinetic energy values of the motor vehicle and the bumper and is the nominal value of parameter k.  The present research aims to find out the system's parameters that have the most significant effect on reduction of the system's kinetic energy at different initial velocities of the motor vehicle.
First, the actual lengths of the moving parts have been selected (see Fig. 3). The impact of the length of the moving parts of the device on the reduction of the system's kinetic energy has been analysed at the initial speed values (prior to the collision) of 26, 50, 70, 90, 110 and 130 km/h: 26 km/h as an average motor vehicle's speed in urban locations; 50 km/h as a speed limit in urban locations; 70 km/h as a speed limit on high speed streets; 90 km/h as a speed limit on rural roads; 110 km/h as a speed limit on motorways in winter; and 130km/h as a maximum allowable speed on motorways. Table 2 Absorber's moving parts ranking table (the parts are ranked in accordance to the impact of their length on reduction of the system's kinetic energy) No.