Stability Analysis of a Free Falling Pararotor

Abstract

The pararotor is a decelerator device based on the autorotation of a rotating wing. When it is dropped, it generates an aerodynamic force parallel to the main motion direction, acting as a decelerating force. In this paper, the rotational motion equations are shown for the vertical flight without any lateral wind component and some simplifying assumptions are introduced to obtain analytic solutions of the motion. First, the equilibrium state is obtained as a function of the main parameters. Then the equilibrium stability is analyzed. The motion stability depends on two nondimensional parameters, which contain geometric, inertia, and aerodynamic characteristics of the device. Based on these two parameters a stability diagram can be defined. Some stability regions with different types of stability trajectories (nodes, spirals, focuses) can be identified for spinning motion around axes close to the major, minor, and intermediate principal axes. It is found that the blades contribute to stability in a case of spin around the intermediate principal inertia axis, which is otherwise unstable. Subsequently, the equations for determining the angles of nutation and spin of the body are obtained, thus defining the orientation of the body for a stationary motion and the parameters on which that position depends.

Pararotor Dynamics: Center of Mass Displacement from the Blade Plane—Analytical Approach

“…With the objective of comparing the case where k 3i is different from zero with the case where k 3i is zero, the same aerodynamic model used in previous work [16] was adopted. The fact of considering the parameter & 31 implies the introduction of a great complexity because the number of terms of system of equations became extremely large.…”

“…The induced velocity v t was not directly included in the model, following similar assumptions of former studies [16]. Other studies [22] indicated that a real low-aspect-ratio rotary wing shows an induced power correction factor K of about 2 or greater.…”

“…The parameters N' e , k e can account for the effects of principal moments of inertia ratios and are able to define stability regions of the system analyzed. These parameters represent a more general case of the parameters presented in [15,16].…”

“…As analyzed in [15,16], it is considered that 0)3 remains constant when the system is perturbed around an equilibrium point. In a smallaspect-ratio blade pararotor, the induced power correction factor K is considerably larger than in large-aspect-ratio rotary-wing decelerators.…”

“…Previous work has been done to model the stability of a pararotor, mainly by Nadal Mora, Sanz, and Piechocki [9][10][11][12][13][14][15][16]. They conducted investigations that present the stability behavior of pararotors whose blades were aligned with the center of mass of the whole device.…”