EFFECT OF FRICTIONAL FORCES ON FREEFALL LIFEBOAT DESCENT FROM INCLINED RAMP

Abstract

The problem of the movement of a free-fall lifeboat, which is modeled by a homogeneous rod, when descending from an inclined ramp during the first phase of a fall with an increasing angle of inclination (rotation phase), taking into account the influence of frictional forces, is considered.

The differential equations of motion of the FFLB in the polar coordinate system, taking into account the frictional force equal to half of its maximum value, are composed using the Lagrange equations of the second kind. A solvable system of ordinary differential equations is obtained and the corresponding Cauchy problem is formulated.

On the basis of the proposed approach, numerous experiments were carried out to determine the rolling time of the FFLB, the speed of its center of mass, the angles of rotation and the angular velocity of the boat at the moment of separation from the ramp at the values of the friction coefficient equal to 0; 0.1 and 0.2, the values of the ramp inclination angle α=35° and α=25°, the length of the boat is 5, 10, 15 m and the initial velocity of its center of mass is 6, 8, 10 m/s, respectively.

Also, on the basis of the differential equations of rod motion in the Cartesian and polar coordinate systems, an expression for the normal reaction Fn of the ramp and a solving system of differential equations in the Cartesian coordinate system are obtained. The corresponding Cauchy problem is formulated, which is solved numerically using the Runge-Kutt method of the fourth order of accuracy.

Calculations of the dependencies of the value of the relative dynamic normal reaction Fn/m (N/kg) of the rod at the angle of inclination of the ramp α=35°, different values of the length of the rod, the initial velocity of its center of mass on time and the distance of the center of mass to the edge of the support in the process of rolling were carried out, and their graphs were constructed for a rod with a length of 5 meters and an initial velocity of 6 m/s.

It was shown that the work of the frictional forces during the rotation phase does not exceed the work of the frictional force at a normal response value equal to half the value of the normal reaction at the beginning of the rotation phase.

It was found that the effect of friction on the parameters of movement of the FFLB during the rotation phase is insignificant and depends on the angle of inclination of the ramp. With a decrease in the angle of inclination of the ramp, the effect of friction on the angle of rotation and angular velocity of the FFLB increases, and on the linear velocity of its center of mass, on the contrary, decreases.

It is concluded that it is possible to use the proposed approach and numerical experiments to take into account the frictional forces during the rotation phase and select the parameters of the FFLB movement.