Equations of Motion for Flapping Flight

Gebert, Glenn; Gallmeier, Phillip; Evers, Johnny

doi:10.2514/6.2002-4872

Cited by 24 publications

(25 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Equations of motion for flapping flight have been developed and presented by Gebert et al [8]. However, due to some errors, their equations cannot be used as they are.…”

Section: Equations Of Motion and Fluid Dynamics Equationsmentioning

confidence: 99%

See 1 more Smart Citation

Dynamic flight stability of hovering insects

2007

View full text Add to dashboard Cite

The equations of motion of an insect with flapping wings are derived and then simplified to that of a flying body using the "rigid body" assumption. On the basis of the simplified equations of motion, the longitudinal dynamic flight stability of four insects (hoverfly, cranefly, dronefly and hawkmoth) in hovering flight is studied (the mass of the insects ranging from 11 to 1,648 mg and wingbeat frequency from 26 to 157 Hz). The method of computational fluid dynamics is used to compute the aerodynamic derivatives and the techniques of eigenvalue and eigenvector analysis are used to solve the equations of motion. The validity of the "rigid body" assumption is tested and how differences in size and wing kinematics influence the applicability of the "rigid body" assumption is investigated. The primary findings are: (1) For insects considered in the present study and those with relatively high wingbeat frequency (hoverfly, drone fly and bumblebee), the "rigid body" assumption is reasonable, and for those with relatively low wingbeat frequency (cranefly and howkmoth), the applicability of the "rigid body" assumption is questionable. (2) The same three natural modes of motion as those reported recently for a bumblebee are identified, i.e., one unstable oscillatory mode, one stable fast subsidence mode and one stable slow subsidence mode. (3) Approximate analytical expressions of the eigenvalues, which give physical insight into the genesis of the natural modes of motion, are derived. The expressions identify the speed derivative M u (pitching moment produced by unit horizontal speed) as the primary source of the unstable oscillatory mode and the stable fast subsidence mode and Z w (vertical force produced by unit vertical speed) as the primary source of the stable slow subsidence mode.

show abstract

“…Equations of motion for flapping flight have been developed and presented by Gebert et al [8]. However, due to some errors, their equations cannot be used as they are.…”

Section: Equations Of Motion and Fluid Dynamics Equationsmentioning

confidence: 99%

“…(8) and (9) into Eqs. (6) and (7) and after some manipulation, we have (assuming the insect with two wings):…”

Section: Simplification Of the Equations Of Motion By "Rigid Body" Asmentioning

confidence: 99%

Dynamic flight stability of hovering insects

2007

View full text Add to dashboard Cite

show abstract

“…It follows that the rigid body approximation is only likely to be valid in flapping flight if the pitch rate damping is sufficiently high that the flapping cycle does not normally excite the short period mode. Equations of motion have recently been derived that include three rotational degrees of freedom for each of two flapping wings (Gebert et al, 2002), so the theoretical framework now exists for future experimental work to take account of the wing motions.…”

Section: The Rigid Body Approximation May Be Inappropriatementioning

confidence: 99%

Dynamic flight stability in the desert locustSchistocerca gregaria

Taylor

Thomas

2003

Journal of Experimental Biology

265

274

View full text Add to dashboard Cite

SUMMARYHere we provide the first formal quantitative analysis of dynamic stability in a flying animal. By measuring the longitudinal static stability derivatives and mass distribution of desert locusts Schistocerca gregaria, we find that their static stability and static control responses are insufficient to provide asymptotic longitudinal dynamic stability unless they are sensitive to pitch attitude (measured with respect to an inertial or earth-fixed frame)as well as aerodynamic incidence (measured relative to the direction of flight). We find no evidence for a `constant-lift reaction', previously supposed to keep lift production constant over a range of body angles, and show that such a reaction would be inconsequential because locusts can potentially correct for pitch disturbances within a single wingbeat. The static stability derivatives identify three natural longitudinal modes of motion: one stable subsidence mode, one unstable divergence mode, and one stable oscillatory mode (which is present with or without pitch attitude control). The latter is identified with the short period mode of aircraft, and shown to consist of rapid pitch oscillations with negligible changes in forward speed. The frequency of the short period mode (approx. 10 Hz) is only half the wingbeat frequency (approx. 22 Hz), so the mode would become coupled with the flapping cycle without adequate damping. Pitch rate damping is shown to be highly effective for this purpose — especially at the small scales associated with insect flight — and may be essential in stabilising locust flight. Although having a short period mode frequency close to the wingbeat frequency risks coupling, it is essential for control inputs made at the level of a single wingbeat to be effective. This is identified as a general constraint on flight control in flying animals.

show abstract

“…Although the DelFly Nimble does not resemble an conventional aircraft, previous studies have shown that these EOM can describe the motion of some flapping-wing flyers [28,29,30,31]. In addition, we do not know the dynamics of the DelFly Nimble very well yet therefore, the use of standard aircraft EOM is a logical first step.…”

Section: Longitudinal Grey-box Model Identification Of the Delfly Nimmentioning

confidence: 98%

Longitudinal Grey-Box Model Identification of a Tailless Flapping-Wing MAV Based on Free-Flight Data

Nijboer

Armanini

Karásek

et al. 2020

AIAA Scitech 2020 Forum

View full text Add to dashboard Cite

List of Symbols Actuator output of closed-loop automatic feedback system Control deflection. Measured dihedral deflection [deg] Reference signal of closed-loop automatic feedback system Error signal of closed-loop automatic feedback system Controller output signal of closed-loop automatic feedback system Output system Measured output system , , Position [m] , , , Attitude quaternions Roll rate Pitch rate Yaw ratė Roll acceleratioṅ Pitch acceleratioṅ Yaw acceleration Linear acceleration in x-direction Linear acceleration in y-direction Linear acceleration in z-direction Pitch attitude angle set-point Flapping frequency [Hz] Pitch angle referencė Pitch rate reference Measured pitch anglė Measured pitch rate Rate feedback gain Attitude feedback gaiṅ Derivative of x at time t Measured aerodynamic force in x-direction [N] Measured aerodynamic force in y-direction [N] Measured aerodynamic force in z-direction [N] Measured aerodynamic moment around the x-axis [Nm] Measured aerodynamic moment around the y-axis [Nm] Measured aerodynamic moment around the z-axis [Nm] Mass [kg] Gravitational acceleration [m/s2] Velocity along the body x-axis [m/s] Velocity along the body y-axis [m/s] Velocity along the body z-axis [m/s] Acceleration along the body x-axis [m/s2] Acceleration along the body y-axis [m/s2] Acceleration along the body z-axis [m/s2] Pitch angle Trimmed pitch angle Roll angle Yaw angle Angle of attack Moments of inertia of a

show abstract

Equations of Motion for Flapping Flight

Cited by 24 publications

References 5 publications

Dynamic flight stability of hovering insects

Dynamic flight stability of hovering insects

Dynamic flight stability in the desert locustSchistocerca gregaria

Longitudinal Grey-Box Model Identification of a Tailless Flapping-Wing MAV Based on Free-Flight Data

Contact Info

Product

Resources

About