Experimental and numerical investigations of a new type of propulsion using modular umbrella-like wings

Abstract

This paper describes the experiment and simulation of a new type of propulsion that uses modular umbrella-like wings oscillating symmetrically in counterphase to generate thrust. The principle of the propulsion and movement of the modular umbrella-like wings was first developed, and the mechanism used to implement the movement of the modular wings was subsequently designed. Then a kinematic model of the propulsion mechanism was established for performance analysis. Finally, experiments and simulations were completed to study the kinematic and mechanical performances of the propulsion mechanism for different parameters. The experimental results indicate that modular umbrella-like wings oscillating symmetrically in counterphase can generate a valid driving power. Increasing the frequency can enhance the average aerodynamic force in one cycle. The numerical results indicate that the wings would achieve a relatively high performance at a lower reciprocating speed. We can obtain a bigger aerodynamic force by increasing the extent of the generatrix of umbrella-like wing. These results can provide theoretical and practical guidance on making a medium-sized prototype and optimizing the structure design.

Keywords

Aerodynamics flapping flight flow characteristics prupulsive performance umbrella-like wing

Introduction

Many animals have excellent flight and swimming capabilities that have developed over a long period of natural evolution. People have acquired basic cognition on the theory of propulsion of these animals and have published numerous studies in the past few centuries. With the development of high-speed photography and particle image velocimetry (PIV), researchers have provided insight through detailed observations of the morphology, kinematics, and dynamics of birds.^1–3 The understanding of the physical principles of bird flight has in turn motivated efforts to replicate this performance. Many research teams have developed a series of bionic robots that mimic the efficient propulsion of animals.

Flapping-wing air vehicles are representative researches on the bionic robots. Typical bionic flying robots have been proposed in^4–10: they can combine lifting capacity, hovering, and propulsion by imitating the flight of insects or birds. Flexible wings¹¹ and passively morphing ornithopter wings^12,13 have been developed to increase the net lift over the flapping cycle, and both have a favorable effect. However, the mechanism of how geometrically nonlinear effects and the anisotropy of the structure impact the aerodynamic characteristics of the flapping wing remains unknown,¹⁴ and the efficiency of flapping bionics is very low compared to flying organisms. The propulsion mechanisms are mainly concentrated on the small and micro size due to the limitations of their weights and flapping frequencies.

In addition to bionic flying robots, many other bionic propulsion mechanisms have been successfully designed and developed. For example, K. D. Jones designed a flapping-wing-propelled micro air vehicle. The mounting location of the flapping wings was different from a traditional ornithopter, i.e., they were arranged in a biplane configuration and flapped in counterphase.¹⁵ Special composite materials have been developed to allow robot jellyfish to mimic the bell deformation cycle of jellyfish based on their high locomotive efficiency and relatively simple design.^16–18

Thus, many bionic propulsion mechanisms have been successfully designed and developed. Some of them have shown high efficiency and performance under testing. The excellent construction, function, and performance of living systems are attractive research targets for translating the evolutionary successes of natural species into artificial systems. To this end, our work has focused on designing a new type of propulsion mechanism inspired by flying birds and swimming jellyfish. This type of propulsion uses modular umbrella-like wings to produce thrust by oscillating symmetrically in counterphase. This type of propulsion may be applied to the robotics as a mechanism to generate lift or thrust. Due to the availability of vertical take-off and landings, well-controlled hover, it has the potential to revolutionize our sensing and information gathering capabilities in areas such as environmental monitoring and homeland security.

The purpose of this work has been to put forward a new type of propulsion based on an interest in mechanical flapping-wing flight. This kind of propulsion makes use of modular umbrella-like wings to produce thrust by reciprocating in a large travel at a relatively low frequency. We first introduce the kinematics of modular umbrella-like wings and the philosophy behind them. Then the experimental and numerical methods were used to analyze the aerodynamic performance.

Principle and motion of the new type of propulsion

Avians can support their bodies in the air and achieve forward flight by repeatedly beating their wings. Wing beat kinematics is highly complex and has been described in detail elsewhere.^19–23 Inspired by this ability, we conceive a neoteric propulsion technique in which a pair of flapping wings oscillates symmetrically in counterphase to generate propulsive force. Considering how jellyfish generates a stable thrust for its axisymmetric shape,^24,25 the flapping wing is improved to an umbrella-like wing.

The typical movement in a cycle consists of two phases (Figure 1). Sketches (a-c) show the forward motion of the modular umbrella-like wings. The upper wing is closed and is moving downwards at the beginning of this phase. The lower wing is spreading and is moving upwards. The status of the two umbrella-like wings is shown in Figure 1(a). Each of the upper and lower wings acquires a momentum under the action of the air: the upper umbrella wing has passively opened to its fullest, and the lower umbrella wing has passively closed to some degree after a time, as shown in Figure 1(b). Then, the upper and lower wings are moving toward each other. The distance between the upper and lower wings is smallest at the final moment of this phase, as shown in Figure 1(c).

Figure 1.

Sectional schematic of umbrella-like wings approaching each other in the phase of forward motion (a–c) and backward motion (c–e). Solid arrows denote aerodynamic forces acting on wings; dashed arrows denote wings direction of rotation; feint arrows denote vertically moving direction of wings.

Figure 1(c) to (e) shows a backward motion of the two umbrella-like wings. Each movement direction of the two wings is opposite compared with the forward motion, as shown in Figure 1(c). The upper wing has passively closed, and the lower wing has passively opened to its fullest for a short time, as shown in Figure 1(d). The distance between two umbrella-like wings is longest at the final moment of this phase, as shown in Figure 1(e).

The upper wing opens to its fullest status, and the lower wing closes to some degree in the course from status b to status c. The projection area of the upper wing is significantly larger than for the lower wing, and thus, the upper wing would predominantly produce lift compared with the lower wing, which would produce a small drag. Similarly, the upper wing would produce a small drag and the lower wing would produce a large lift in the course from status d to e status. Thus, a persistent thrust is obtained by the modular umbrella-like wings throughout the whole period.

Modular umbrella-like wings used for propulsion have two remarkable characteristics. The first is that the change in the projection area of the umbrella-like wing (wing stretch and closure) is similar to the deformation of a bird’s wing.¹⁹ The second is that the shape of the umbrella-like wing is analogous to the bell of a jellyfish and would generate a stable thrust for an axisymmetric shape.²⁵

Model description

Propelling machinery configuration

To achieve the function of this propulsion system, the mechanical design of the propelling machinery is projected as shown in Figure 2. The propelling machinery consists of three parts: the driving mechanism, the stroke amplifier, and modular umbrella-like wings.

Figure 2.

CAD model of the propelling machinery.

A bidirectional thread of ball screw is developed as a driving mechanism to achieve the function, which transmits the high-speed rotating movement of the motor to a low-frequency reciprocating motion. Bidirectional thread of ball screw could translate rotational motion to linear motion, but the travel of the driving mechanism should be enlarged through a stroke amplifier. In this paper, a scissor-like element (SLE) of deployable structure is used as a stroke amplifier to zoom the travel.^26–28 The upper and lower wings are then installed on each end of the amplifier to make the umbrella-like wings oscillate symmetrically in counterphase with a large travel.

Structure of the umbrella-like wing

The modular umbrella-like wings consist of two identical cone-shaped bells installed at each end of the stroke amplifier. This pair of wings is actuated by deployment to symmetrically oscillate in counterphase with a large travel. The wing can achieve momentum under the action of the air and open to its fullest when moving downwards, and it can close to some degree due to the same mechanism when moving upward. The status of the umbrella-like wings during their motion is shown in Figure 3.

Figure 3.

Sketch of the umbrella-like wing: (a) the closed state and (b) the outspread state.

All of the ribs rotate around the pin as a function of the aerodynamic force, inertial force, and gravity; thus, the wings passively open or close when the direction of motion changes. As each runner is connected to the rib by the stretcher, the runners constitute a structure of a rod-crank mechanism, wherein the runner can slide up or down along the tube when the status of the wing changes. The angle between the rib and tube ranges from 40° to 70°. To reduce the impact generated by the runner, two compressed springs are fixed to the ends of the runner.

The likeness of the opening umbrella-like wing is a pyramidal face which formed by the generatrix rotating around its axis. The extent of the generatrix is denoted by l_u_,r, the angle between the generatrix and its axis is indicated by θ_u, each of the models and diagram of the umbrella-like wing for the basic parameters as shown in Figure 4.

Figure 4.

Geometries of the umbrella-like wing: (a) model and (b) structural parameters.

The materials of the rib, stretcher, and tube are carbon fibers. The tensile modulus of elasticity and bending modulus of elasticity are 136 GPa and 9 GPa. The material used to construct the skin of the umbrella-like wings is polytetrafluorothylene (PTFE) membrane. The elastic modulus of PTFE membrane is about 1.4 GPa. As the stiffness of the wing is big enough, it is assumed that the wing is rigid without plastic deformation taking placing in the experiment.

Kinematic model of the propelling machinery

The assembly is a mechanism with one degree of freedom from the stowed/folded configuration till the end of the deployment. The bars are considered as rigid rods. Then the relationships of each joint can be obtained from the basic geometry constraint of the deployable structures. Figure 5 shows a sketch of propelling machinery with the deployable structure in the fully deployed configuration shown on the left and the deployable structure in the fully contracted configuration shown on the right.

Figure 5.

A sketch of propelling machinery under fully stretch and constriction.

The two joints of a SLE element in the middle are connected to the joints of the slider and the kickstand of the motor. The joint connected to the slider functions as an active part, while the joint connected to the kickstand of the motor is an immovable part. Each assembly number of the SLE elements below and above the immovable joint is the same to remove the inertial force generated by the rods of the SLE elements.

Oxyz coordinate on the earth is defined as a static frame and O′x′y′z′ coordinate on the drive institution is defined as a moving coordinate system. All nodes of the amplifiers from top to bottom are symbolized by Q, P, N, L, M and the distance is equidistant between two adjacent nodes, the relationships are as follows:

| y M - y L | = | y L - y N | = | y N - y P | = y P - y Q |

(1)

In the phase of forward motion, the driving mechanism spends half time during the period to move the point P from the nearest position P₁ relative to its farthest position P₀. The distance between position P₀ and P₁ would be H_d, the magnification of the transmission mechanism would be k, the absolute velocity of point N on the driving mechanism would be v_N_,_a, the velocity of point P on the driving mechanism relative to point N would be v_P_,_r, the absolute velocity of point M would be v_M_,_a, the absolute velocity of point Q would be v_Q_,_a. Based on the velocity composition law of particle movement, one can adopt the following expressions:

v M, a = v N, a + (- 1) INT (2 t / T) kv P, r / 2

(2)

v Q, a = v N, a - (- 1) INT (2 t / T) kv P, r / 2

(3)

where t is the run time of the driving mechanism, T is the run time of the reciprocating motion in one cycle, INT(x) is the rounding function.

According to the structure of the propelling machinery, the travel of the umbrella-like wing could be calculated as follows:

H u = k \cdot H d / 2

(4)

where H_u is the travel of the wing, and k is the number of SLE elements.

The governing equation for the propelling machinery response under external load of wings is

Ma = F p - Mg

(5)

where M is total mass of the propelling machinery, a represents the accelerated velocity due to propelling machinery, F_p is the thrust of the propelling machinery, g is acceleration due to gravity.

The propelling force generated by upper wing and lower wing are denoted by F_p_,u and F_p_,d, hence

F p = F p, u + F p, d

(6)

Experimental and numerical methods

Experimental setup and data processing method

A test apparatus, as shown in Figure 6, was built to investigate the kinematic and mechanical characteristics of the modular umbrella-like wings. This test measures the instantaneous thrust and inertial forces, the velocity of the upper and lower wings in hovering flight. Different input powers were tested to compare their dynamic performance.

Figure 6.

Experimental setup.

A prototype of the propelling machinery was constructed according to the model, as shown in Figure 2. The test bed consists of a main support, trails, a prototype, a lithium-polymer battery, a receiver, a speed controller, and data acquisition equipment. The sensors used in this paper include speed sensor and force sensor. A high-speed camera was used to visualize the movement of the reciprocating wings.

The main kinematic parameters of this propelling machinery include the velocities of the modular umbrella-like wings and the rotational speed of the motor. To calculate the inertial force of the machinery working in hovering flight, the accelerations of the deployed structures should also be measured. The movement of the object being tested can be affected by the displacement sensor when using a traditional approach. Hence, we first took photos of the propelling machinery using a high-speed camera in the tests. Then, an object tracking method^29,30 was applied to these images to acquire the displacement of the object.

The velocities of the umbrella-like wings can be calculated by the rotational speed of the motor according to the structure of the propelling machinery. In order to investigate the experimental errors of the velocities of the umbrella-like wings measured by the object tracking method, the rotational speed of the motor was measured by a photoelectric sensor in a non-contact way. The instantaneous values of the velocity measured by each method are identical, and the relative error is below 1%.

As the umbrella wings are reciprocating vertically, the direction of the propulsive force is consistent with the direction of movement. To relieve the constraint on the propelling machinery in the vertical direction, sliding rails were used to fix the propelling machinery on the support. The prototype was fixed on the sliding blocks of the rails. As the sliding block had one degree of freedom in the vertical direction, the prototype could move freely in the vertical direction. Three force transducers were then connected to the bottom of the sliding blocks, as shown in Figure 6. The forces were measured at the beginning of the test to determine the static values. The relationship between these forces is as follows:

F p = F p, t - F p, 0

(7)

where F_p,t corresponds to the forces measured by the transducers after motor rotation, and F_p,0 corresponds to the forces measured by the transducers before motor rotation.

A random static loading measurement in the range up to 100 N shows that the relative error is below 0.3%. Hysteresis, nonrepeatability, and nonlinearity of this sensor are below 0.1%. The natural frequency of this sensor is well above our reciprocating frequencies.

The thrust of the propelling machinery F_p consists of two components, i.e., the inertial force and the aerodynamic force:

F p = F i + F a

(8)

where F_i is the inertial force of the propelling machinery and F_a is the aerodynamic force of the modular umbrella-like wings.

The inertial force of the propelling machinery can be calculated as follows:

F i = - \sum_{i}^{n} m i a i

(9)

where m_i is the mass of each part of the propelling machinery and a_i is the corresponding acceleration.

The aerodynamic force of the umbrella wings can be obtained by solving equations (8) and (9).

Numerical method

Simulations mimicking the up-and-down motion of the modular umbrella-like wings were studied in order to understand the mechanism of unsteady aerodynamics generated by moving the wings. The aerodynamics of the wings and the characteristics of the flow field were obtained. Scaling effects were also studied in order to understand the effect of increasing the characteristic length or speed.

For the three-dimensional problem of flapping wings moving in a viscous fluid, the turbulent flow can be characterized by fluctuating velocity fields. However, it is too computationally expensive to simulate directly in practical engineering calculations. Here, the Reynolds-averaged Navier–Stokes equations are used instead of these instantaneous equations.

The finite volume method is used to discrete the Navier–Stokes equations based on the concept of local average. The first-order implicit unsteady formulation is used, as it is sufficient for most problems. Renormalization group κ-ɛ model is used to deal with moving flows because this turbulence model considers the rotation under mean flow, it is a good way to deal with high strain rate and big bend streamline for rotating flows.³¹ Specifically, the Pressure Implicit with Splitting of Operators (PISO) method is used for pressure–velocity coupling.

An unstructured tetrahedral grid that was generated for the three-dimensional geometry was implemented. For our numerical simulations, the modular umbrella-like wings are moving back and forth in the y-axis direction. The dynamic mesh method was used to achieve the motion of the model.³² Spring-based smoothing and local re-meshing were applied to automatically update the mesh after each time step relative to the motion of umbrella-like wings.

Experimental and numerical results comparison

Here, the velocity of the single wing is denoted by V_s, and the average aerodynamic force of the wing is indicated by F_a,m. The velocity, propulsive force, and aerodynamic force of the single umbrella-like wing over three cycles are shown in Figure 7. Each moment of a, b, c, d and e is corresponding to the moving status of upper wing in Figure 1.

Figure 7.

Velocity, propulsive force, and aerodynamic force curves for l_u,r = 0.60 m, f = 1.16 Hz.

The velocity curve shows that the umbrella-like wing was oscillating between a large travel driven by the driving mechanism and deployable structures. The velocity of the wing was rapidly reduced to zero when the deployable structures were fully contracted and elongated. Each part of the propelling machinery achieved a very large acceleration. The propelling machinery would generate inertial forces in this moment, as shown in the propulsive force curve.

The propelling machinery generated an upward aerodynamic force when the umbrella-like wing was moving downwards (corresponding to the movement of the upper wing from b to c in Figure 1), as shown in the aerodynamic force curve; the average aerodynamic force generated by the wing was 13.78 N in this process. A very small drag force was generated by the wing when it was moving upwards (corresponding to the movement of the upper wing from d to e in Figure 1); the average aerodynamic force generated by the wing was −0.76 N in this process. The drag generated by the wing from d to e was much smaller than the lift generated from b to c

Experimental and numerical results of the average aerodynamic force generated by the single umbrella-like wing are shown in Figure 8. The time of these results is from status b to status c.

Figure 8.

Comparison between the experimental and computational results.

The maximum degree of the umbrella-like wing designed in the experiment is 70°. It is seen that the numerical results for θ_u = 70° agree well with the experimental results between 1.64 m/s and 2.16 m/s. Results for θ_u = 75° coincide with the experimental results between 2.29 m/s and 2.71 m/s. The maximum degree of the wing increased as the velocity increased because of the flexibility of the wings, hence the experimental results also increased.

Results and discussion

Time histories of the velocities and forces

To acquire the mechanism of the unsteady aerodynamics generated by the modular umbrella-like wings, tests for the propelling machinery at different oscillating frequencies were performed. The oscillating frequency is highlighted to discuss the effect on the aerodynamics of the modular umbrella-like wings.

The velocity, propulsive force, and aerodynamic force of the modular umbrella-like wings for l_u,r = 0.60 m and H_u = 0.90 m are shown in Figure 9.

Figure 9.

Velocity, propulsive force, and aerodynamic force curves for f = 1.14 Hz.

The velocity curve shows that the modular umbrella-like wings were oscillating between a large travel driven by the driving mechanism and deployable structures. The velocities of the wings were rapidly reduced to zero when the deployable structures were fully contracted and elongated. The propelling machinery generated an upward aerodynamic force when either the upper or the lower umbrella-like wing was moving downwards (corresponding to the movement of the wings from b to c and d to e in Figure 1), as shown in the aerodynamic force curve.

The propulsive force measured by the force sensors consists of the inertial force and aerodynamic force. The inertial force generated by the stroke amplifier and modular umbrella-like wings can be counteracted when using a perfectly symmetrical structure, i.e., when the propulsion mechanism is assembled as shown in Figure 5. However, the inertial force of the propelling machinery was not balanced out, as indicated by the propulsive force curve. The peak values of the inertial force were only reduced to one third compared with the results of the single umbrella-like wing for the same frequency and travel. The movement of the upper and lower wings oscillating in counterphase was not strictly consistent, as shown in the velocity curves. According to the analysis of the images captured by the high-speed camera, the flexibility and assembly errors associated with the deployment structures were the main reasons for this effect.

The results of the test using the same methods of an increased oscillating frequency are shown in Figures 10 and 11.

Figure 10.

Velocity, propulsive force, and aerodynamic force curves for f = 1.28 Hz.

Figure 11.

Velocity, propulsive force, and aerodynamic force curves for f = 1.32 Hz.

The changing trends of these curves for different oscillating frequencies were the same in a period. The aerodynamic force generated by the modular umbrella-like wings was increased by increasing the oscillating frequency. For instance, the average aerodynamic force in one cycle increased by 16.08% as the frequency was increased from 1.28 Hz to 1.32 Hz.

A fully opened wing occurred in the phases of both forward and backward motion. The outspread wing was moving downwards to generate an upward aerodynamic force. The percentage of this mode was approximately in the range of 41.6–46.3% in one cycle. The value of average aerodynamic force increased by 85.84% compared with the single wing for the same parameters. It is effective to improve the percentage of this mode in one cycle to enhance the average aerodynamic force.

Characteristics of flow field around the umbrella wings

The vertical wake and pressure distribution are closely associated with the aerodynamic characteristics of wings flapping. We further discuss the velocity vectors and pressure distributions to understand aerodynamic performance of the moving umbrella-like wings. Figure 12 shows the velocity vectors and pressure contours of the modular umbrella-like wings in hovering flight for l_u,r = 0.60 m and H_u = 0.90 m. The reciprocating speeds of the wings are 2.40 m/s in this case.

Figure 12.

Pressure distribution and velocity vectors of air around umbrella-like wings in an external flow-field at z = 0 m: (a) Phase from b to c; (b) Phase from d to e.

The velocity vectors and pressure contours during the phase of forward motion are shown in Figure 12(a). It is seen that the umbrella-like wing generated two vortexes at the left and right tips, respectively near the periphery of the upper wing. The lower wing also generated two vortexes at the tips, respectively along the inner edge of the umbrella-like wing. The tips vortexes generated two strong circulatory motions at the wing tips, which are usually associated with two low-pressure zones reflected in pressure variation. The tip vortexes of the upper wing were moving toward to the top of the wing surface, which affected the wing aerodynamics.

The velocity vectors and pressure contours during the phase of backward motion are shown in Figure 12(b). It also generated two vortexes at the tips of the umbrella-like wings. Low-pressure zones were generated corresponding to the location of the tip vortexes. The distribution of the pressure on the upper surface of the bottom wing was smaller than the one on the lower surface, the maximum was 9 Pa and the minimum was −5 Pa. The differential pressure on the wing in the vertical direction resulted in a lift, and the value was 7.76 N. But on the contrary, the distribution of the pressure on the upper surface of the top wing was bigger compared with the one on the lower surface resulted in a drag, and the value was −1.74 N. The drag generated by the bottom wing was much smaller than the lift generated by the top wing. Due to the upright projected area of the bottom wing being much larger than the one of the top wing, the machinery achieved a thrusr as a whole.

Length and speed scaling

To increase the thrust that the propelling machinery provides, one can simply multiply the extent of the generatrix by a scaling factor, SF. Another method to increase the thrust is to increase the reciprocating frequency. Here, frequency scaling is found by scaling the reciprocating speed for a constant travel of the driving mechanism. We wished to know if the performance of the given propulsion could be improved by scaling upwards or downwards, if an optimum size scale and speed existed, a general trend, or no effect on the performance.

A generic variable called performance (Pe) is defined³³

Pe = | \bar{F a, t * v N, a |} / (| \bar{F a, u * v Q, a |} + | \bar{F a, d * v M, a |})

(10)

SF ES = l u, r / 1

(11)

SF SS = v P, r / 1

(12)

where SF_ES is an extent scaling factor, l_u_,r = 1 m is referred as the value of the characteristic length, SF_SS is a speed scaling factor, v_P_,r = 1 m/s is referred as the value of the characteristic speed.

In order to visualize the effect of increasing the characteristic length or speed, a graph of normalized performance is presented in Figure 13. Two different methods of scaling are simulated in this figure. Extent scaling (ES) is found by scaling the extent of generatrix by a scaling factor, but maintaining constant parameters for v_N_,_a = 2 m/s, v_P,r = 1 m/s and H_u = 0.90 m. Speed scaling (SS) is found by scaling the reciprocating velocity of the driving mechanism while maintaining constant parameters for v_N_,_a = 2 m/s, l_u,r = 1 m, and H_u = 0.90 m.

Figure 13.

Simulated effect on normalized performance due to changes in the scale factor.

One can see that the speed scale increased, the performance sharply decreased in the range of 0.6 to 1.0. A relatively high performance would be achieved by keeping SF_SS in a small scale. It is consistent with the goal to develop the propelling machinery at a relatively low frequency. The performance decreased slowly as the extent scale increased, hence, we can obtain a higher thrust by increasing the extent of the generatrix of the umbrella-like wing.

Conclusion

In this work, a new type of propulsion technology with modular umbrella-like wings was proposed and analyzed. A flapping action of this propulsion was introduced. Afterward, a simplified physical model of the propelling machinery was established based on this principle, the movement relationship between driving mechanism and umbrella-like wings was given by mathematical equations. Then experimental and numerical methods were used to analyze the aerodynamic performance. The results demonstrate that modular umbrella-like wings oscillating symmetrically in counterphase can generate considerable thrust.

The results of testing modular umbrella-like wings demonstrate that the aerodynamic force of the wings increased as the reciprocating frequency increased. The outspread wing was moving downwards to generate an upward aerodynamic force. It is effective to improve the percentage of this mode in one cycle to enhance the average aerodynamic force. The comparison between the results of single and modular wings strongly supports this conclusion.

The distributions of velocity vectors and pressure contours in an external flow-field were researched to analyze the mechanism of unsteady aerodynamics. Those results indicate that modular umbrella-like wings could generate a valid driving power under a reciprocating motion. It would achieve a relatively high performance at a lower reciprocating speed. We can obtain a bigger thrust by increasing the extent of the generatrix of the umbrella-like wing.

We will apply this technology to make a medium-sized prototype and compare to a flapping-wing micro air vehicles in future work.

Footnotes

Acknowledgments

This investigation was supported by the National Key Science and Research Item.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article

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