In that study, the main rotor blade radius, blade chord length, blade twist, blade linear mass density, blade flapping spring stiffness, and angular speed were the passive morphing parameters.
Consequently, the model is objectively complex with a total of 29 equations: 9 fuselage equations, 16 blade flapping and lead-lagging equations, 3 static main-rotor downwash equations, and an additional flight-path angle algebraic equation.
Figure 7 shows the closed-loop responses of the linear velocities of the helicopter (i.e., [[phi].sub.A] : roll angle; [[theta].sub.A]: pitch angle; and [[psi].sub.A] angle) states, longitudinal linear velocity (i.e., u) state, and collective blade flapping angle (i.e., [[beta].sub.0]) state for the 1st closed-loop system (solid black line), 2nd closed-loop system (solid red line), and 3rd closed-loop system (solid blue line).
Moreover, the other outputs (e.g., linear velocity states, blade flapping states) do not show any catastrophic behavior.
Blade flapping is of significant importance in understanding the natural stability of quadrotors .
Blade Flapping. When a rotor translates horizontally through the air, the advancing blade of the rotor has a higher velocity relative to the free-stream and will generate more lift than the retreating blade which sees a lower effective airspeed.
Thus, in the case of blade flapping the rotor disk tilts, the rotor thrust is also inclined with respect to the airframe and imposes a component in the x and y directions of the body-fixed frame.
The total longitudinal moment created by blade flapping [M.sub.bf] is the sum of these two moments: