A User-Friendly Formulation of the Newtonian Dynamics for the Coupled Wing-Body System in Insect Flight

Abstract

We establish a user-friendly matrix formulation of the Newtonian dynamics for the free flight of an insect. In our formulation, we can easily change the number of insect wings, prescribe either the kinematics of each wing relative to the body or the torque on each wing exerted by the body, and allow for the dependence of fluid force and torque on acceleration. The implementation of the formulation is straightforward.

Primary 70E55, 76Z99

The work of the author is supported by NSF grant DMS-0915237.

Appendix

To write the explicit forms of the frame-to-frame transformation matrices, we use the following Tait-Bryan angles (angles of roll, pitch, and yaw) to describe the orientation of one frame relative to another

• ϕ _lb , θ _lb , and ψ _lb : the roll (around the x ^b axis), pitch (around the y ^b axis), and yaw (around the z ^b axis) angles of the body relative to the lab frame, respectively;

• ϕ _bw , θ _bw , and ψ _bw : the roll (around the x ^w axis), pitch (around the y ^w axis), and yaw (around the z ^w axis) angles of a wing relative to the body, respectively.

We then define $\vec{{\alpha }}_{lb} = {[{\phi }_{lb},{\theta }_{lb},{\psi }_{lb}]}^{T}$ and $\vec{{\alpha }}_{bw} = {[{\phi }_{bw},{\theta }_{bw},{\psi }_{bw}]}^{T}$.

Applying the rotations in the order of yaw, pitch, and roll, we have

$$\begin{array}{rcl} {R}_{lb}& =& {R}_{lb}^{roll}{R}_{ lb}^{pitch}{R}_{ lb}^{yaw}, \\ {R}_{bw}& =& {R}_{bw}^{roll}{R}_{ bw}^{pitch}{R}_{ bw}^{yaw}, \\ {R}_{lw}& =& {R}_{bw}{R}_{lb} \end{array}$$

(34)

The lab-to-body transformation matrices are

$$\begin{array}{rcl} & & {R}_{lb}^{roll} = \left (\begin{array}{ccc} 1& 0 & 0\\ 0 & \cos {\phi }_{lb } &\sin {\phi }_{lb} \\ 0& -\sin {\phi }_{lb}&\cos {\phi }_{lb} \end{array} \right ),\quad {R}_{lb}^{pitch} = \left (\begin{array}{ccc} \cos {\theta }_{lb}&0& -\sin {\theta }_{lb}\\ 0 &1 & 0 \\ \sin {\theta }_{lb}&0& \cos {\theta }_{lb} \end{array} \right ), \\ & & {R}_{lb}^{yaw} = \left (\begin{array}{ccc} \cos {\psi }_{lb} &\sin {\psi }_{lb}&0\\ -\sin {\psi }_{ lb}&\cos {\psi }_{lb}&0\\ 0 & 0 &1 \end{array} \right ). \end{array}$$

(35)

We can obtain body-to-wing transformation matrices similarly. Every transformation matrix is orthogonal, so its inverse is its transpose.

The angular velocity $\vec{{\Omega }}_{B}$ in the body frame is

$$\begin{array}{rcl} & & \vec{{\Omega }}_{B}^{b} = {K}_{ B}\dot{\vec{{\alpha }}}_{lb},\end{array}$$

(36)

where

$$\begin{array}{rcl} & & {K}_{B} = \left (\begin{array}{ccc} 1& 0 & -\sin {\theta }_{lb} \\ 0& \cos {\phi }_{lb} &\sin {\phi }_{lb}\cos {\theta }_{lb} \\ 0& -\sin {\phi }_{lb}&\cos {\phi }_{lb}\cos {\theta }_{lb} \end{array} \right ).\end{array}$$

(37)

Similarly the angular velocity $\vec{{\Pi }}_{W}$ in the wing frame can be written as $\vec{{\Pi }}_{W}^{w} = {K}_{W}\dot{\vec{{\alpha }}}_{bw}$. If the Tait-Bryan angles of the wing relative to the body are prescribed, then $\dot{\vec{{\Pi }}}_{W}^{w} =\dot{ {K}}_{W}\dot{\vec{{\alpha }}}_{bw} + {K}_{W}\ddot{\vec{{\alpha }}}_{bw}$ is known.