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Dynamic flight stability in the desert locust Schistocerca gregaria

Graham K. Taylor^* and Adrian L. R. Thomas

Department of Zoology, Oxford University, Tinbergen Building, South Parks Road, Oxford, OX1 3PS, UK

View larger version (18K): [in a new window]   Fig. 1. Black box representation of the rigid body equations of motion underpinning the quantitative framework of this analysis. The integral sign represents a bank of single integrators.

View larger version (19K): [in a new window]   Fig. 2. Definition of the state variables u, w, q and . Each of the variables is signed positive in the direction shown. The body axes are centred upon the centre of mass and are aligned so that the x-axis points in the direction of flight at equilibrium. The locust is shown during a nose-up perturbation: q is zero at equilibrium, and w and are defined so as to be zero at equilibrium.

View larger version (34K): [in a new window]   Fig. 3. Schematic representation of the experimental setup. AD, analogue-to-digital.

View larger version (32K): [in a new window]   Fig. 4. Centre of mass measurements. Dorsal, lateral and ventral views of a locust, showing the average position of the centre of mass (x). The line drawings were traced from images generated by placing locust `G' directly onto a flatbed scanner, and are scaled accordingly. Plumb lines passing through the centre of mass are shown as coloured lines, with the colours denoting the locust from which they were obtained (red `R', green `G', blue `B'). The intersection of the plumb lines corresponds to the position of the centre of mass; dotted lines join the most forward and rearward plumb line intersections on the dorsal and ventral views to give an indication of measurement error.

View larger version (18K): [in a new window]   Fig. 5. Pitching moment of inertia (Iyy) measurements. (A) Stacked bar graphs showing the masses of eight transverse sections of the body corresponding to the sections illustrated in (B) below. The different colours distinguish data from the three locusts according to the scheme `R' (red), `G' (green), `B' (blue). The mass of each section is expressed as a proportion of the total frozen body mass and the y-axis is scaled such that the total area of the bar graphs is 100%. (B) Line drawing showing the position of the eight sections. (C) Block model of a locust. The area of each of the blocks corresponds to the projected area of each section in B and the density of the shading corresponds to the mass per unit area of each section. The white crosses mark the positions of the wing roots. The red cross marks the position of the centre of mass. (D) Bar graph illustrating the percentage contribution of each body section to the total pitching moment of inertia. Note that this bar graph mirrors the bar graph in A above, indicating that percentage contributions to the total pitching moment of inertia are inversely correlated with percentage contributions to the total body mass.

View larger version (30K): [in a new window]   Fig. 6. Graphs of relative lift Lr against b for the unpaired (A—C) and paired (D—F) analyses of the angle series data, and against U (G—I) for the speed series data. The usual red/green/blue colour coding of the locusts applies. Horizontal black lines denote equilibrium levels of dimensionless force production. Vertical black lines denote the reference speed Uref and body angle b,ref. In each of A—F, the single data point at 7° represents the mean of the 14 measurements at b,ref. Regression lines are only drawn if the slope of the individual regression was significant and if U or b attained overall significance in the corresponding pooled general linear model. The error bars on the regressions for the unpaired analysis (A—C, G—I) show the 95% confidence interval for the regression mean (±1.96 S.E.M.). The error bars on the regressions for the paired analyses (D—F) show the 95% confidence interval for the mean of the 14 reference measurements (±1.96 S.E.M.), which in this case provides the only independent estimate of the height of the curve above the x-axis. The broken lines represent the 95% confidence interval for the slope of the regression plus the 95% confidence interval for the slope of the corrections for tunnel speed or balance orientation. Note that the combined confidence interval for the slope may include zero even if the P-value for the regression slope itself is significant.

View larger version (33K): [in a new window]   Fig. 7. Graphs of relative thrust—drag Tr against b for the unpaired (A—C) and paired (D—F) analyses of the angle series data, and against U (G—I) for the speed series data. For further explanation, see legend to Fig. 6. Individual regressions of Tr against U2 (G—I) are shown as black dotted lines where the individual regression and the corresponding GLM treating U2 as the covariate were both significant; it is clear that the deviation from linearity is small over the range of speeds used.

View larger version (31K): [in a new window]   Fig. 8. Graphs of relative pitching moment Mr against b for the unpaired (A—C) and paired (D—F) analyses of the angle series data, and against U (G—I) for the speed series data. For further explanation, see legend to Fig. 6. Individual regressions of Mr against U2 (G—I) are shown as black dotted lines where the individual regression and the corresponding GLM treating U2 as the covariate were both significant; it is clear that the deviation from linearity is small over the range of speeds used.

View larger version (15K): [in a new window]   Fig. 9. Graph of relative lift Lr against time through the angle series experiments. The usual red/green/blue colour coding of the locusts applies. The horizontal black line denotes the equilibrium level of relative lift production. Coloured lines join the 14 reference measurements at b,ref. The vertical range of the lines gives an indication of how lift production varies through time. The range of scatter about the lines gives an indication of the relative magnitude of variation in lift due to changes in b.

View larger version (15K): [in a new window]   Fig. 10. The four general types of solution to the longitudinal equations of motion. (A) Monotonic subsidence (stable), corresponding to a negative real root. (B) Monotonic divergence (unstable), corresponding to a positive real root. (C) Damped oscillation (stable), corresponding to a complex conjugate pair of roots with negative real parts. (D) Divergent oscillation (unstable), corresponding to a complex conjugate pair of roots with positive real parts.

View larger version (22K): [in a new window]   Fig. 11. Root locus plots showing the effect of reducing the speed derivative Mu to zero for each of the three locusts upon the roots of the longitudinal equations of motion. The plots are in Argand diagram form, i.e. the real part of the root (n) is plotted along the x-axis and the imaginary part of the root () is plotted along the y-axis. Roots to the left of the vertical black line are stable. The filled circles denote the position of the roots of the system matrices defined in Equations 16; the centres of the triangles denote the position of the roots of the system matrices when Mu=0. (A) Locust `R'. (B) Locust `G'. (C) Locust `B'. Note that reducing the value of Mu causes the divergence mode to move towards stability.

View larger version (16K): [in a new window]   Fig. 12. Results of a Monte Carlo simulation in which the six static stability derivatives were allowed to vary as normally distributed variables, according to the parameters set by the regression analyses. The analysis was repeated 5000 times for each locust, recalculating the eigenvalues after each iteration. Each Argand diagram plot contains 20 000 points: 5000 for each of the four roots. (A) Locust `R'. (B) Locust `G'. (C) Locust `B'. The plots show that the complex conjugate roots (represented by the clouds of points for which 0) are stable (i.e. n<0) in 100% of cases for locusts `R' and `G', and are stable in 99.9% of cases for locust `B'. There are two real roots, one positive and one negative, in 100% of cases.

View larger version (23K): [in a new window]   Fig. 13. Root locus plots showing the effect of partitioning the static stability derivative M into components due to pitch attitude (M) and aerodynamic incidence (Mw). Symbols and axes are as in Fig. 11. All additive combinations of M and Mw that are compatible with the measured values of M are shown. Note that the positive real root part of the unstable divergence mode becomes strikingly less positive (i.e. less unstable) as the ratio of Mw to M tends toward zero. (A) Locust `R'. (B) Locust `G'. (C) Locust `B'. In the case of locust `G' the two real roots eventually converge and split into a complex conjugate pair, corresponding to the phugoid mode of aircraft.

View larger version (23K): [in a new window]   Fig. 14. Root locus plots showing the effect of introducing pitch rate damping by increasing the value of the pitch rate derivative Mq from zero. Symbols and axes are as in Fig. 11. (A) Locust `R'. (B) Locust `G'. (C) Locust `B'. Increasing Mq has practically no effect upon the non-oscillatory modes, but causes the short period oscillatory mode to become more heavily damped. Eventually, critical damping is reached and the complex conjugate roots split into two negative real roots. The short period mode becomes non-oscillatory when the ratio Mq:M is 0.035 in locust `R', 0.021 in locust `G', and 0.040 in locust `B'.

View larger version (11K): [in a new window]   Fig. 15. Classical form of the frequency—response curve of an aircraft, illustrating the effect of making a sinusoidal control input to the elevator. The graph plots how the ratio of the output (in this case, pitch attitude ) to input (in this case, elevator angle) varies with the frequency of the control input. The gain is plotted in decibels and the control input frequency is plotted on a logarithmic scale. The graph indicates that the gain peaks (due to resonance effects) when the control input frequency coincides with the natural frequencies of the natural modes of motion. The gain drops off sharply at control input frequencies higher than the natural frequency of the short period mode, which therefore limits the bandwidth of the aircraft's frequency response. In insects, control inputs made at the level of a single wingbeat will not be effective if the wingbeat frequency is much greater than the natural frequency of the short period mode.