In this section, we propose the new design of a directly driven flapping wing system with extension spring. The basic working principles are also briefly explained. The dynamics of the flapping system with extension spring and its natural frequency are derived.

2.1. Design of the Flapping Wing System with Extension SpringFigure 1 is the top view schematic of the dual flapping wing-driving system with extension springs. One driving system consists of a DC motor, a pinion gear, a spur gear, a wing, and an extension spring. The motor is fixed to the body frame, and the pinion gear is fixed to the motor to transmit the rotational force of the motor to the spur gear. The wing is fixed to the spur gear. An extension spring is connected between the wing part and the frame. The extension spring can be mounted any place between the body frame and the wing or the spur gear, so long as it tenses the spring, and puts the wing at zero position at rest. The angle between the extension spring and the wing is set to about 150 degrees to have space so that the spring is a little extended at the zero position, which helps the spring work in the same way as a linear spring, as will be explained later.Figure 1b shows how the driving system works. As the wing starts to rotate in the clockwise direction from the stationary state, that is, the zero point, and performs the down stroke, the extension spring attached to the wing part is stretched by the wing rotation, and elastic energy is accumulated in the extension spring.

When the wing reaches the top of the flap, the rotation direction of the wing is reversed. At this point, the spring is extended to maximum length, and the elastic energy is also maximally stored. When the direction of flapping is reversed and the wing begins to rotate counterclockwise, the elastic energy stored in the spring is released to accelerate the wing to the zero position. As the wing crosses the zero point, a new cycle of energy storing and discharging is repeated.

The frame of the model is manufactured by processing a carbon plate of 1 mm thickness, and to increase the stiffness of the flying robot, a U-shaped carbon plate of 1 mm width is vertically inserted. Two DC motors of 7 mm diameter and 20 mm length are used for driving. The frames of the two motors also work as stiff holding columns for the body of the robot. The weight of a motor is 3.5 gf, and the stall torque is 2.47 N·mm. A metal gear with module 0.3 and 7 teeth is used for the pinion gear, while a plastic gear with module 0.3 and 81 teeth is used for the spur gear. The wings used are the same as those of the KUBeetle [11]. The wingspan is 75 mm, while the wing area is 18.8 cm2.Figure 2 shows the shape of the wings used. The wing membrane is made of thin polyethylene terephthalate, and the thickness is 20 μm. The wing spar is made of 1 mm thin carbon tube. 2.2. Dynamics of the Flapping Wing SystemThe flapping wing system can be modeled as a rotational spring–mass-damper system, as in Equation (1):

τ=ITθ¨+DTθ˙+KTθ

(1)

where, τ is the torque applied, IT is the moment of inertia of the whole system, DT is the damping coefficient of the total system, and KT is the effective spring coefficient.The equation of motion according to motor driving is expressed as Equation (2):

τ=(IM+IP) θ¨M+(DM+DP)θ˙M+τe

(2)

where, θM is the rotational angle of motor, IM and IP are the moment of inertia of the motor and the pinion gear, respectively, and DM and DP are the damping coefficient of the motor and the pinion gear, respectively. The τe is the external torque from the spur gear.Equation (3) can express the equation of motion according to the rotation of the spur gear:

τS=(IS+IW) θ¨S+(DS+DW)θ˙s+krθS

(3)

where, θs is the rotational angle of spur gear, IS and IW are the moment of the inertia of the spur gear and the wing, respectively, DS and DW are the linearly approximated damping coefficient of the spur gear and the wing, respectively, and kr is the rotational spring constant of the extension spring.The relation between the external torque applied by the motor and the reactive torque of the spur gear is expressed as: where, the number of teeth of the pinion gear and that of the spur gear is nP and nS, respectively. The relation between the rotational angle of the motor and that of the spur gear is expressed as:Using the abbreviated notation n=(nSnP), Equation (1) of the total flapping wing system can be described as:

τ=(n(IM+IP)+1n(IS+IW))θ¨S+(n(DM+DP)+1n(DS+DW))θ˙s+(krn)θS

(6)

Denoting the total moment of inertia (n(IM+IP)+1n(IS+IW)) as IT, the total damping coefficient (n(DM+DP)+1n(DS+DW)) as DT, and the effective total spring constant (krn) as KT, the natural frequency of the entire system ωn is expressed as KTIT, and the damped natural frequency ωd is given as ωn1−ζ2=KTIT×1−ζ2.

The rotational elastic constant can be derived from the torque generated from the extension spring. Figure 3 shows the motion of extension spring when the spur gear and wing are rotated from (a) to (b). Referring to Figure 4, when d1, d2, r, the rotation angle θs of the spur gear, and the rotation angle α of the extension spring are given, the torque by the extended spring is obtained as:

τes=ke(d2−d1)×cos(π2−(θS+α))×r

(7)

where, ke is the linear elastic constant of the extension spring, d1 is the spring length at zero position without extension, d2 is the spring length when it is extended according to the rotation of the spur gear, and r is the distance between the center of the spur gear, and the point where the spring is fixed on the spur gear. The ke, d1, and r are fixed constants, and when θs is given, d2 and α are determined. The angle of the spur gear and the rotation angle of the wing are the same. Thus, the torque in Equation (7) is the function of the wing rotation angle θW.The torque enforced by the extension spring to the spur gear with the wing rotation angle θW is shown as the blue line in Figure 5. Here the ke is 0.796 N/mm, the d1 is 13.4 mm and the r is 5 mm.The slope of the graph in Figure 5 is the rotational elastic constant kr. When the wing rotation angle is small, the torque generated by the extension spring to the spur gear is minor.By stretching the extension spring at the zero position by moving away the fixing point to the body frame, tension can be provided, even when the wing is at zero position. In this case, the spring torque at low wing angle range is increased, compared with the toque increase in higher angle range, and thus the rotational elastic constant is linearized. Denoting the increased distance offset as d¯, Equation (7) is modified into Equation (8). The red line in Figure 5 shows that when the offset distance d¯ is given, the effective torque with wing rotation becomes more linear. Here, the d¯ is fixed as 2.4 mm.

τes=ke(d2+d¯−d1)×cos(π2−(θs+α))×r

(8)

With the given d¯ and using root mean square method, the rotational elastic constant kr is obtained as 0.01954 N·mm/rad, and the correlation factor is 0.9965. The number of teeth of pinion gear and spur gear are 7 and 81, respectively. The moment of inertia of all of the rotating components, which is computed by CATIA simulation program, is 7.53 × 10−8 g·m2. Therefore, the natural frequency of the whole flapping system is around 22 Hz, as calculated from Equation (6) with the given parameters ke and n.