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Table 1.

Kinematic variables for the physical wing model (PM) and virtual wing model (VM) comparisons

Case	Source	Fig.	\({\Delta}{\hat{{\tau}}}_{\mathrm{t}}\)	\({\Delta}{\hat{{\tau}}}_{\mathrm{r}}\)	\({\hat{{\tau}}}_{\mathrm{f}}\)	x̂ _o	α_down	α_up
PM1	1	1	0.12	0.32	-0.08	0.2	-50	70
PM2	1	3	0.12	0.32	-0.08	0.2	-50	50
PM3	1	3	0.12	0.32	0.08	0.2	-50	50
VM1	2	6	0.24	0.32	-0.08	0.2	-50	50
VM2	2	6	0.105	0.32	-0.08	0.2	-50	50
VM3	2	11	0.24	0.32	0	0.2	-50	50
VM4	2	11	0.24	0.32	0.08	0.2	-50	50

Case	Source	Fig.	\({\Delta}{\hat{{\tau}}}_{\mathrm{t}}\)	\({\Delta}{\hat{{\tau}}}_{\mathrm{r}}\)	\({\hat{{\tau}}}_{\mathrm{f}}\)	x̂ _o	α_down	α_up
PM1	1	1	0.12	0.32	-0.08	0.2	-50	70
PM2	1	3	0.12	0.32	-0.08	0.2	-50	50
PM3	1	3	0.12	0.32	0.08	0.2	-50	50
VM1	2	6	0.24	0.32	-0.08	0.2	-50	50
VM2	2	6	0.105	0.32	-0.08	0.2	-50	50
VM3	2	11	0.24	0.32	0	0.2	-50	50
VM4	2	11	0.24	0.32	0.08	0.2	-50	50

Source 1, Dickinson et al.,1999; Source 2, Sun and Tang,2002. Fig. refers to the figure in the source paper from which the force curves were digitized.

In PM1, PM2, VM1 and VM2, wing rotation is `advanced' relative to stroke reversal. In PM3 and VM4, rotation is `delayed'. In VM3, wing rotation is symmetric about stroke reversal.

\({\Delta}{\hat{{\tau}}}_{\mathrm{t}}\)

is the non-dimensional duration of wing translational velocity;

\({\Delta}{\hat{{\tau}}}_{\mathrm{r}}\)

is the non-dimensional duration of wing rotation;

\({\hat{{\tau}}}_{\mathrm{f}}\)

is the non-dimensional rotational timing parameter;x̂_o is the rotational axis;α _down and α_up are the pitch angles of the wing during the translation phase of the down and up strokes,respectively.

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