The effect of body size on the wing movements of pteropodid bats, with insights into thrust and lift production

Body size governs almost every aspect of animal biology. Many fundamental aspects of ontogeny, physiology, ecology and behavior can be largely predicted with little more information than the mass of an animal. These, and many other important aspects of an organism's life, scale predictably with body mass, according to fundamental form–function relationships (Brown and West, 2000; Dial et al., 2008). The influence of body size on locomotion is no less striking, and biomechanical investigations have revealed that just as body shape changes with size, so too do locomotor kinematics (Biewener, 1983; Biewener, 2005; Heglund and Taylor, 1988).

Part of the reason that animals of different sizes move their bodies differently is that the physical demands of locomotion change with body size. As an extreme example, small flying animals moving through the air must deal with high viscous forces relative to inertial forces, whereas for larger flying animals, viscous forces are much smaller than inertial ones. As a result, a <100 μg wasp uses constant clap and fling wing motions to fly whereas a 10 kg Andean condor travels mostly by simply holding its wings outstretched (Ellington, 1999; McGahan, 1973; Miller and Peskin, 2005). Within the range of body sizes between those extremes, animal flight can differ in subtle but important ways. The mechanics of insect flight differ between fruit flies and hawkmoths, and the way a bird flies also varies from hummingbirds to pigeons to vultures (Combes and Daniel, 2003; Dial and Biewener, 1993; Dickinson and Götz, 1996; Sane, 2003; McGahan, 1973; Warrick et al., 2005). Unlike insects and birds, however, bats have largely been assumed to use similar mechanisms of aerodynamic force production in flight, regardless of size (Bullen and McKenzie, 2002; Hedenström et al., 2007; Norberg and Rayner, 1987), even though bats range in body mass over roughly three orders of magnitude, from the ≤0.002 kg bumblebee bat (Craseonycteris thonglongyai) to >1.2 kg flying foxes (Pteropus spp.) (Hill and Smith, 1981; Kunz and Jones, 2000; Surlykke et al., 1993).

The purpose of this study is to examine the influence of body size on wing kinematics in bats. The body shapes and foraging modes of bats vary widely among families, so we focused on a single family, the Pteropodidae. This family consists of ca. 186 species distributed throughout the paleotropics (Wilson and Reeder, 2005) and is characterized by fruit and nectar-feeding, non-echolocating animals that are ecologically and morphologically similar across a broad range of body sizes. No other chiropteran family possesses so wide a range of body sizes as the pteropodids (Nowak, 1994).

We investigated wing posture, wing kinematics and the patterns with which kinematics change with flight velocity. In this study we also examined the influences of horizontal and vertical accelerations on wing kinematics, as a means of studying how the kinematics of bats reflect aeromechanical force production in those directions.

We recorded the flight kinematics of 27 animals from six species ranging more than 40-fold in body mass, from 0.0278 kg to 1.152 kg (Table 1). Bats of the smallest species, Cynopterus brachyotis, were flown in a 1.4 m×1.2 m×1.2 m (L×W×H) wind tunnel at the Concord Field Station of Harvard University in Bedford MA, USA (Hedrick et al., 2002). Bats belonging to the other five species were flown in a 14.5 m×2.8 m×2.4 m (L×W×H) flight corridor at the Lubee Bat Conservancy in Gainesville FL, USA. Bats were recorded at 1000 Hz using three phase-locked Photron 1024 PCI digital high-speed cameras (Photron USA, Inc., San Diego, CA USA). A fourth phase-locked 1000 Hz camera, a Photron APX, was also used to record corridor flights. All cameras had 1024×1024 pixel resolution.

Before experiments, each bat was anesthetized with isoflurane gas then marked with non-toxic acrylic paint at 17 anatomical landmarks on the fur and skin of the sagittal midline and the left half of the body (Fig. 1A). At least ten flights were recorded from each individual, but only five flights per individual were used for analyses. Those five were selected to sample the broadest range possible of flight velocities for each individual.

The volumes through which bats flew were calibrated using the direct linear transformation (DLT) method (Abdel-Aziz and Karara, 1971), based on a 0.35 m×0.35 m×0.28 m (L×W×H) calibration frame for the wind tunnel trials, and on a series of vertical wires and beads for the corridor experiments, spanning 2.00 m× 1.25 m×2.40 m (L×W×H).

The positions of 17 anatomical markers were digitized in each video frame (Fig. 1). Where a marker was visible in two or more cameras at a given instant, its position could be calculated using the DLT method. In those frames where a marker was not visible to at least two cameras, its position was interpolated based on its three-dimensional positions before and after that interval, using an over-constrained least-squares polynomial fit of its trajectory. For contiguous gaps in the data with rich data at the end points, a third-order polynomial was used, and for gaps with sporadic intermediate points, a sixth-order polynomial was used (Riskin et al., 2008). The interpolations were visually checked for all 17 markers in all 135 trials by superimposing the interpolated three-dimensional path on the original movies. The wing membrane was modeled as a series of 18 flat triangles spanning the xyz coordinates of the markers (Fig. 1A).

For the purposes of this study, we define a body-centered coordinate system (x_b, y_b, z_b) and a global coordinate system (x_g, y_g, z_g) (Gatesy and Baier, 2005). Both are right-handed, and in both, x and y are horizontal and z is vertical (Fig. 1B,C). The global coordinates represent the motion of the bat relative to still air, with positive x_g in the direction of travel, positive y_g to the bat's left and positive z_g pointing vertical up. To obtain global coordinates for corridor trials, we calculated the positions of the 17 markers throughout the trial relative to the calibration frame, then rotated the coordinate system so that the bat's anterior sternum marker (a in Fig. 1A) moved forward in the x_g direction, beginning and ending the wingbeat cycle at the same y_g value. We then subtracted the initial position of the anterior sternum marker from all positions. Thus, every trial began with the anterior sternum marker at (x_g, y_g, z_g)=(0, 0, 0), and ended with the sternum at some positive x_g position with y_g=0. Global coordinates for wind-tunnel trials were calculated analogously but were adjusted to correct for the velocity and direction of the air so that the global coordinates would also represent the motion of the bat's body relative to still air.

The body-centered coordinate system was calculated from the global coordinates by subtracting the position of the anterior sternum marker in each time step from the positions of all markers in that time step. The net effect was that the anterior sternum marker remained at (x_b, y_b, z_b)=(0, 0, 0) throughout the trial and the head faced positive x_b (Fig. 1B).

We restricted our analyses from each flight to a single wingbeat cycle. Where possible (88 of 135 trials), the upper reversal points of the wrist marker in the z_b direction were used to denote the beginning and end of the wingbeat cycle (Fig. 1B). For some flight corridor trials, a complete wingbeat cycle occurred within the calibrated volume, but not spanning those endpoints. In those cases, we used, in order of preference, the lower reversal points of the wrist (38 of 135), the upper reversal points of the wingtip (6 of 135), or the lower reversal points of the wingtip (3 of 135).

To determine how wing shape changed with body mass (M_b), we extracted information about three-dimensional wing conformation from the kinematic recordings. Whereas morphometric measurements are typically taken from specimens with wings extended on a flat surface, our methods capture posture, one of the most important aspects of wing form for understanding flight mechanics (Nudds, 2007). The following parameters were calculated from each trial.

Maximum wingspan (b_max): two times the maximum distance of the wingtip marker from the mid-sagittal plane (y_b=0) in a trial. This always occurred during the downstroke.

Minimum wingspan (b_min): for each time step we found the y_b-value of the wing marker furthest from the y_b=0 plane. The b_minvalue for a trial was the smallest of those values across the entire wingbeat cycle, multiplied by two. This always occurred during the upstroke.

Maximum wing chord (c_max): the greatest two-dimensional distance in a trial between the wrist and the tip of digit V (e and n in Fig. 1A, respectively), using their x_b and z_b position data only, and ignoring y_b.

Maximum wing area (S_max): the left wing was divided into 18 triangular surfaces (Fig. 1A). The areas of those triangles were summed, then multiplied by two, to arrive at total wing area (S). The result is a coarse three-dimensional mesh which approximates the wetted area, not a two-dimensional projection. This was done for each time step over the course of the wingbeat cycle, and S_max was the maximum value of S for a trial. We chose this method because it uses our high-fidelity measurements to arrive at a more detailed index of wing conformation than would a two-dimensional projected area.

Parameters were calculated as follows.

Horizontal velocity (V_horiz): the horizontal distance traveled by the anterior sternum marker over the course of the wingbeat cycle, divided by the duration of the wingbeat cycle.

Vertical velocity (V_vert): the vertical distance traveled by the anterior sternum marker over the course of the wingbeat cycle, divided by the duration of the wingbeat cycle.

Horizontal acceleration (A_horiz): the change in forward velocity between the beginning of the wingbeat cycle and the end of the wingbeat cycle, divided by the duration of the wingbeat cycle. Both estimates of forward velocity were calculated using the slope of a linear fit of the anterior sternum marker's x_g position over time, using a 4 ms window. Since the posture of the bat was roughly the same at the beginning and end of a wingbeat cycle, inertial effects can be neglected, so the change in speed of the sternum markers approximates the change in the speed of the center of mass.

Vertical acceleration (A_vert): calculated analogously to A_horiz, but using z_g position over time instead of x_g.

The following parameters were obtained.

Wingbeat period (T): the time taken to complete the wingbeat cycle.

Downstroke duration (T_down): the duration of the downstroke, based on the motion of the wrist in z_b dimension.

Wing stroke amplitude (φ): the maximum three-dimensional angle between any two positions of the wrist relative to the shoulder within a wingbeat cycle.

Stroke plane angle (β): the trajectory of the wingtip in the x_b–z_b plane was fitted to a linear function using orthogonal regression. The angle between that regression line and horizontal was multiplied by –1 to calculate β, which is negative by convention (Norberg, 1990) (Fig. 1B).

Wing camber at maximum wingspan: in the time step (camera frame) corresponding to b_max, which always occurred at some time during the downstroke, a parasagittal (x_g–z_g) cross section of the wing at the y_g value of the wrist was made. The straight line from the wrist to trailing edge was defined as the chord line. Of the 18 triangular sections of the modeled wing, exactly six always intersected that plane at the time of b_max (Fig. 2A), resulting in seven intersections of a triangle border and the plane. The first term of a sine series was fitted to those seven intersection points to create a curved line between the wrist and the trailing edge of the wing that came as close to those seven points as possible. To calculate wing camber, we divided the maximum distance to the chord line from that curve by the length of the chord line (Fig. 2B). Our estimate of camber is an instantaneous value for a dynamically changing parameter, and although it might not represent the maximum or even average camber over the whole wingbeat cycle, it is a value that can be defined in a clear, unambiguous manner for ease of comparison among wingbeat cycles.

Angle of attack (α): the angle of the wing chord relative to the incoming flow of air was calculated at the instant of maximum wingspan, in the same time step as wing camber was calculated. The chord line was defined as a straight line between the wrist and the intersection of the trailing edge with the x_g–z_g plane of the wrist (Fig. 2C). The angle of the chord line above horizontal was defined as α₁. The angle velocity vector of the wrist below horizontal in the x_g–z_g plane, was defined as α₂. Angle of attack (α) was calculated as α₁+α₂. Again, this gives an instantaneous recording that does not capture changes over the course of the wingbeat cycle, but facilitates comparison among wingbeat cycles. Note that this estimate does not account for relative air movement resulting from the induced velocity.

For tests of allometry in body shape parameters, calculations of body shape were made for all 135 flights. Because our species sample size for estimation of scaling relationships was six, we used averages for all individuals within each species. For each parameter, the calculated values from the five trials of an individual were pooled to determine an individual median, then the individual medians within a species were pooled to determine the species median. Those six species medians were used in a log–log orthogonal (reduced major axis or RMA) regression against median body mass for each species (LaBarbera, 1989). To account for statistical non-independence of data resulting from the shared phylogenetic history of the six species (Felsenstein, 1985), we repeated that analysis accounting for phylogeny using a generalized least squares (GLS) model. We used a consensus tree of several recent pteropodid phylogenies (Giannini and Simmons, 2005; Jones et al., 2002; O'Brien et al., 2009), with branch lengths scaled using the method of Pagel (Pagel, 1992). GLM analyses with phylogeny were carried out using REGRESSIONv2 (Lavin et al., 2008) in Matlab. The slope of each regression was compared with that expected under isometry using two-tailed t-tests with four degrees of freedom. For ease of comparison with previous studies, we also present the results of RMA analyses that do not account for phylogeny. The expected relationships under isometry are: b_max∝M_b^1/3, b_min∝M_b^1/3, c_max∝M_b^1/3, S_max∝M_b^2/3, Q_s∝M_b^1/3 and AR∝M_b⁰ (Norberg, 1990).

Assuming that C_L does not increase with body mass, and assuming V_wrist is proportional to V_horiz, large animals should fly faster than small ones in order to generate the lift required to fly. Specifically, preferred flight velocity is predicted to scale ∝M_b^1/6 (Pennycuick, 1975).

We expected bats flown in the corridor to fly with speeds that scaled with M_b^1/6, but since bats in the wind tunnel could not choose their own velocities, we did not expect those individuals to conform to that pattern. We thus performed a linear least squares regression of log(V_horiz) versus log(M_b) for the median V_horiz values of individuals flown in the corridor (N=22). A GLS model that accounted for phylogenetic effects was used, and the slope of that regression was compared with 1/6 using a two-tailed t-test with eight degrees of freedom. For ease of comparison with previous studies, we also include the slope estimate based on an ordinary RMA regression.

To test for systematic changes in the values of kinematic parameters with body size, we performed linear least squares regressions of individual median values against log(M_b). Because bats flown in the wind tunnel (C. brachyotis) had different velocity distributions than corridor-flown bats (see Results), we excluded them from kinematic scaling analyses.

To account for phylogenetic effects, we performed all interspecific regression analyses using a GLS model, as described above, with reduced degrees of freedom to account for soft polytomies in the pteropodid tree (Purvis and Garland, 1993; Garland and Díaz-Uriarte, 1999). Parameters included in these analyses were T, T_down, τ, φ, St, β, wing camber, α, α₁, α₂ and C_L.

We expected T and T_down to scale with M_b^1/3 (Norberg, 1990), and all other regressions were expected not to scale with body mass, in other words, were expected to scale ∝M_b⁰.

We used log–log regression for the majority of tests but could not use that method for angles (φ, β, α, α₁ and α₂), since many angles were negative (Smith, 1984). Instead, we used log–linear regression for angles, which assumes that if there is an effect of body size on the angle, it scales ∝M_b¹.

For ease of comparison with previous studies, we repeated linear regressions of kinematic parameters against body mass using more classical RMA methods that do not account for phylogeny.

To determine how wing kinematics change with V_horiz, A_horiz and A_vert, we used multiple regression. This method permits examination of how each of these three variables correlates with changes in wing kinematics, while correcting for the influence of the other two. Each regression had a single kinematic variable as the dependent variable, and V_horiz, A_horiz and A_vert as model effects. To take into account variability among individuals, we also included individual bat as a random effect (Gelman and Hill, 2007). We performed these regressions separately for each species.

We conducted regression analysis for b_max, b_min, T, T_down, τ, φ, β, St, α, α₁, α₂, wing camber and C_L. Thus, thirteen multiple regression analyses were performed per species, for a total of 78. In each, we tested for significance of partial regression coefficients for each of V_horiz, A_horiz and A_vert using two-tailed t-tests. Degrees of freedom for each multiple regression test were equal to N–k, where N is the number of flight trials, and k is the number of estimated parameters. Estimated parameters were V_horiz partial regression slope, A_horiz partial regression slope, A_vert partial regression slope, and one intercept per individual bat. This resulted in nine degrees of freedom for R. aegyptiacus regressions, 13 for those of P. vampyrus, and 17 degrees of freedom for those of all other species.

Statistical analyses were performed using custom-scripts in Matlab R2008b (MathWorks Inc., Natick, MA, USA), then verified using JMP IN 8.0 (SAS Institute, Cary NC, USA). For models with random effects we used the expected mean squares (EMS) method. To account for possibly inflated family-wise type I error rate resulting from performing multiple statistical tests on closely related data (Curran-Everett, 2000; Curran-Everett and Benos, 2004), we carried out a positive false discovery rate (pFDR) analysis on P-values from the data in supplementary material Table S2A–C using the qvalue package (Version 1.20) (Storey, 2002) for R (Version 2.10.1) (R Development Core Team, 2009) with a pFDR rate of 5% (allowing 5% of ‘significant’ results to be false). From that analysis we found that a more conservative alpha level for significance is 0.034, rather than 0.05, and used the more conservative value as the criterion for rejection of the null hypothesis (supplementary material Table S2A–C).

A complete table of summary statistics for all 135 trials used in this study is provided in supplementary material Table S1. In the body of this paper, we report phylogenetically corrected regression statistics, but for ease of comparison with previous studies, we include results without phylogenetic correction in tables as well.

The scaling of maximum wingspan to body mass was positively allometric (b_max∝M_b^0.423 compared with M_b^0.333, P=0.045; Table 2, Fig. 3A). Minimum wingspan scaled isometrically to body mass (b_min∝M_b^0.366 compared with M_b^0.333, P=0.509; Table 2, Fig. 3B). Wing chord scaled isometrically with body mass (c_max∝M_b^0.357 compared with M_b^0.333, P=0.457; Table 2, Fig. 3C). Wing area scaled with positive allometry (S_max∝M_b^0.768 compared with M_b^0.666, P=0.047; Table 2, Fig. 3D) and wing loading increased more gradually with body size than would be expected under isometry (Q_s∝M_b^0.233 compared with M_b^0.333, P=0.024; Table 2, Fig. 3E). Since span increased with positive allometry but chord increased isometrically, aspect ratio increased with M_b, with the slope of that regression approaching significant allometry (AR∝M_b^0.072 compared with M_b^0.000, P=0.068; Table 2, Fig. 3F).

The horizontal velocities of bats (4.98±0.09 m s^–1) were much greater than vertical velocities (0.12±0.03 m s^–1), so flight paths were close to horizontal (1.36±0.36 deg above horizontal). The flight speeds of bats of all body sizes overlapped greatly. Bats in the wind tunnel (C. brachyotis) generally flew faster for their size than bats in the corridor did (Fig. 4A). With wind tunnel flights excluded from analysis, the preferred velocities of animals increased with M_b with a slope not significantly different from that expected under isometry (V_horiz∝M_b^0.005 compared with M_b^0.167; P=0.056; Table 3) but much closer to M_b^0.000 (P=0.948). We found considerable variability in A_horiz and A_vert among trials (Fig. 4B,C), and no trial showed zero net acceleration.

The regression slope of wingbeat period to body mass was significantly lower than that expected under isometry (T∝M_b^0.180 compared with M_b^0.333; P=0.039; Table 3, Fig. 5A), but downstroke duration was not significantly different from predicted (T ∝M_b^0.213 compared with M_b^0.333, P=0.140; Table 3, Fig. 5B). Downstroke ratio also did not change significantly with body size (τ∝M_b^0.036 compared with M_b^0.000; P=0.162; Table 3, Fig. 5C).

Wing stroke amplitude at the wrist did not significantly change with log body mass [φ∝(logM_b) · (–6.058); P=0.257; Table 3, Fig. 5D], nor did stroke plane angle [β∝(logM_b)8.974; P=0.107; Fig. 5E]. Strouhal number also did not change significantly with log body mass (St∝M_b^–0.088 compared with M_b^0.000; P=0.532; Table 3, Fig. 5F).

We found that α increased significantly with log body mass [α∝(logM_b)7.738; P=0.014; Table 3, Fig. 5G], and that the overall change occurred as a result of changes in α₁ but not α₂ [α₁∝(logM_b)7.542; P=0.032; Fig. 5H; and α₂∝(logM_b) · (–0.461); P=0.866; Table 3, Fig. 5I]. Wing camber at maximum span did not change significantly with log body mass (wing camber∝M_b^0.070; P=0.714; Fig. 5J). Coefficient of lift increased significantly with body mass (C_L∝M_b^0.170; P=0.042; Table 3, Fig. 5K).

We observed several changes in the kinematics of bats with changes in V_horiz, A_horiz and A_vert. Some changes that were significant in some species were not significant in others, but in almost no cases did different species show opposing trends; where a partial regression slope was significant for multiple species, it almost always had the same sign, positive or negative, for all other species. These are reported fully in supplementary material Table S2, and summarized in Table 4.

With increases in flight velocity, holding the influences of A_horiz and A_vert constant, we observed decreased maximum wingspan, increased wingbeat period, increased downstroke duration, increased downstroke ratio, decreased stroke plane angle, decreased angle of attack, decreased wing camber, decreased Strouhal number, and decreased lift coefficient. We observed no significant change in amplitude, and observed mixed results among species for changes in minimum wingspan (Table 4, supplementary material Table S2A).

With increases in horizontal acceleration, we observed decreases in minimum wingspan and stroke plane angle, and increases in maximum wingspan, amplitude and angle of attack. Strouhal number and lift coefficient also both increased. We saw no significant changes in wingbeat period, downstroke duration, downstroke ratio or wing camber (Table 4, supplementary material Table S2B).

With increases in vertical acceleration, we observed an increase in maximum wingspan, angle of attack, wing camber, Strouhal number and lift coefficient, along with a decrease in wingbeat period. We observed no significant changes in minimum wingspan, downstroke duration, downstroke ratio, wing stroke amplitude, or stroke plane angle (Table 4, supplementary material Table S2C).

Our results provide experimental evidence that for pteropodid bats many aspects of wing kinematics vary with body size, but that the ways kinematics change with velocity and acceleration are relatively consistent across body sizes. Additionally, we found that the scaling relationships for maximum wingspan, maximum wing area and minimum wing loading in pteropodid bats, based on measurements from actual wing form as employed during flight behavior, differ from the scaling relationships measured from outstretched preserved specimens (Table 2). Importantly, we uncovered a significant positive allometry between body mass and wing area during the downstroke, which may help to offset the consequences of higher wing loading that accompany increased body size. Furthermore, large bats had higher coefficients of lift during flight than small bats did. This result highlights the importance of wing posture as a confounding variable for hypotheses about ecological function based solely on the two-dimensional shape of an outstretched wing.

A bat's wing comprises highly compliant skin membranes that interconnect a jointed skeleton capable of many degrees of freedom (Riskin et al., 2008). By its very morphological structure, the area of a bat wing is highly variable throughout every wingbeat cycle. As a result, measurements of wing area for bat specimens can vary substantially compared with those for insects or birds, depending especially on the degree to which the membrane is stretched before preservation or measurement. It has been extremely valuable to make comparisons of wing area among bat species in a two-dimensional perspective, but our high-fidelity measurements of wing shape permitted the use of actual three-dimensional wing conformation in flight.

It has long been known that the bodies of pteropodid bats do not scale isometrically (Norberg, 1981; Norberg, 1990; Norberg and Rayner, 1987), but several scaling relationships that we observed in this study differ from those previously reported based on measurements of dead specimens measured outstretched on a flat surface. We hypothesize that those discrepancies arise because wing posture during the downstroke does not match the posture in which bats are typically held for morphometric measurements, and the difference between in-flight conformation and measurement conformation varies non-randomly with body size. In other words, we hypothesize that the way bats position their wings during downstroke varies with body size. If true, the actual scaling trends in the mid-flight shapes of bat wings would present themselves only when wing measurements were taken during flight. Given that wing measurements are widely used to predict behavior and foraging ecology for bat species (e.g. Bullen and McKenzie, 2001; Kingston et al., 2000), our hypothesis, if true, would have wide-reaching influence on our understanding of bat evolution and ecology.

In their analysis of wing form in bats, Norberg and Rayner (Norberg and Rayner, 1987) and Norberg (Norberg, 1981) found that wingspan in pteropodids scaled with M_b^0.350 and M_b^0.362, respectively, both only slightly higher than the M_b^0.333 expected under isometry. The scaling relationship uncovered in this study (b_max∝M_b^0.423) was around 20% higher than theirs and suggests that large bats extend the wing membrane more fully during the downstroke than small bats do. Similarly, the scaling coefficient of wing loading to body mass from this study, Q_s∝M_b^0.233, was around 27% lower than the coefficients M_b^0.327 and M_b^0.309 reported by Norberg and Rayner (Norberg and Rayner, 1987) and Norberg (Norberg, 1981). Here again, posture is a probable contributor to the discrepancy. The area of a bat's wing changes throughout the wingbeat cycle and depends greatly on the positions of the carpus and elbow and the degree of extension and abduction of the digits. Whereas specimens in previous studies have been measured in a fully outstretched and flattened posture, our methods capture the three-dimensional shape of the wing with the degree of wing extension that is biologically relevant.

Interestingly, our results suggest that large bats extend the wing more fully on the downstroke, but not on the upstroke, since the scaling relationship of b_min to body mass (b_min∝M_b^0.366) did not differ from the relationship expected under isometry (b_min∝M_b^0.333). This makes sense, since the majority of lift production occurs on the downstroke.

Since wing loading increases with body mass, it is hypothesized to limit the body sizes of the largest flying animals (Greenewalt, 1975). Norberg (Norberg, 1981) and Norberg and Rayner (Norberg and Rayner, 1987) found that wing loading for pteropodid bats scaled isometrically with body mass (Q_s∝M_b^0.333), but we found evidence for negative allometry (Q_s∝M_b^0.233). Our result suggests that the largest bats may effectively reduce wing loading, compensating for their large size, by employing different wing postures than small bats. In his study of allometric scaling of bird wings, Nudds commented on the possible influence of body-size-dependent variation in elbow angle among birds on scaling relationships (Nudds, 2007). Whether or not the relationship we report here for pteropodid bats is also true for birds, or indeed even other bats, is an important topic for future study.

An alternative explanation for the difference between our body size regression slopes and those of Norberg and Rayner is that because we sampled fewer species, six in this study compared with >50 by Norberg and Rayner, we simply got different slopes by chance. To assess the influence of sample bias, we performed orthogonal regressions of log(M_b) versus log(b_max) and log(M_b) versus log(S_max) using Norberg and Rayner's data for the closest species with complete measurements to the six used in this experiment: Cynopterus brachyotis (0.0265 kg), Rousettus aegyptiacus (0.140 kg), Pteropus sp. (0.210 kg), Eidolon helvum (0.274 kg), Pteropus sp. (0.347 kg) and Pteropus vampyrus (1.179 kg). Using Norberg and Rayner's data, we obtained regression slopes of b_max∝M_b^0.331±0.05 and S_max∝M_b^0.651±0.109. Both of those are closer to the regression slopes of Norberg and Rayner's entire pteropodid dataset (b_max∝M_b^0.350 and S_max∝M_b^0.715) than to our regression slopes (b_max∝M_b^0.423 and S_max∝M_b^0.768). This supports our hypothesis that the difference between their results and ours is our use of mid-flight wing shape measurements, and not the species sample used. Further investigations of species-specific differences in posture will help elucidate the aerodynamic relevance of those differences.

Our data suggest that certain information about flight kinematics for any pteropodid species can be assumed to be representative for the group, despite the remarkable range in body sizes in the family. For the kinematic parameters we report here, wing kinematics were similar across the bats surveyed in this study despite more than a 40-fold range in body mass among individuals. Downstroke period scaled as predicted under isometric scaling, and several kinematic variables showed no significant change with body size (τ, φ, β, St, α₂, wing camber). However, a few variables did change with increasing body size, namely T, α and C_L.

Although we found that the relationship between V_horiz and M_b did not differ significantly from expected, there was no significant increase in V_horiz with M_b. Thus, without air moving faster across the wings, the higher wing loading values of large bats should have required them to have higher lift coefficients than small bats did, and this is what we observed. The higher C_L values of large bats were most likely achieved by their higher wingbeat frequencies (for their size) and their higher angles of attack, since no other variables changed across body sizes. Importantly, the departure from isometry in S_max did not contribute to this trend, since the C_L equation (Eqn 5) accounts for wing area.

Strouhal number can be employed as a dimensionless descriptor of flight speed for flapping swimmers and fliers. Regardless of body size, when animals fly with Strouhal numbers between 0.2 and 0.4, propulsive efficiency is thought to be particularly high (Taylor et al., 2003). The Strouhal numbers of corridor-flown animals in this study were around 0.3 to 0.6, slightly higher than the predicted range, but they did not differ significantly with size. Strouhal numbers for pteropodid bats were published previously by Taylor et al. (Taylor et al., 2003) based on data from Bullen and McKenzie (Bullen and McKenzie, 2002), who used two corridor-flown pteropodid species with body masses within the range we investigated here (Pteropus poliocephalus: 0.7 kg, and P. scapulatus 0.4 kg). Their values were around 0.42, roughly the middle of our range for corridor-flown bats.

Bats flown in the wind tunnel had lower Strouhal numbers than corridor-flown bats did, probably as a result of their relatively high flight velocities. After all, every species decreased St with increases in flight speed (Table 4). Interestingly, the Strouhal numbers of wind-tunnel-flown bats fell exactly within the expected 0.2 to 0.4 range. This suggests that when bats fly at high speed they do not adjust their amplitude and frequency to maintain consistent Strouhal numbers. We hypothesize that because we used a wind tunnel to collect data for C. brachyotis, we artificially decreased Strouhal number. Conversely, Tobalske et al. (Tobalske et al., 1997) found that the wingbeat frequencies of birds in their study were higher at a given speed in the wind tunnel compared with in the wild, which would increase Strouhal number. Similarly, Liechti and Bruderer found that wind-tunnel flown birds showed higher wingbeat frequencies than those observed in the wild (Liechti and Bruderer, 2002). Clearly a survey of Strouhal number across taxa that takes experimental conditions into account would be informative.

The results from our regression analyses varied across species, but a noteworthy trend emerged: when the partial regression coefficient between some variable and one of the regression effects was significantly different from zero for any species, other species with significant regression slopes almost always showed the same directional trend (Table 4). For example, not all bats had significant changes in wingbeat period with V_horiz, but where a significant trend existed, it was always positive. This consistency is remarkable considering that we investigated 13 variables with respect to each of V_horiz, A_horiz and A_vert for a total of 39 regressions per species, and found non-conflicting results from all six species in every case but one (b_minvs V_horiz). The stability of our results suggests that across a broad range of body sizes, bats modify their kinematics similarly with changes in velocity and acceleration.

Several previous studies of bat flight have considered the kinematic changes that correlate with differences in flight velocity for bats flying at steady speeds (e.g. Aldridge, 1986; Aldridge, 1987; Lindhe Norberg and Winter, 2006). As V_horiz increases, drag increases so that more thrust is required, and air velocity over the wing is increased, enhancing lift. Thus, changes in wing kinematics with increased V_horiz might be related to increased thrust production, to drag reduction, to C_L alleviation, or to some combination of those (Hedrick et al., 2002; Tobalske et al., 2007).

Strouhal number, by definition, is inversely proportional to velocity (Eqn 4), so it is not surprising that Strouhal number decreased with increasing V_horiz. Similarly, the mid-downstroke velocity of the wrist, correlated with flight velocity, appears in the denominator of the equation for coefficient of lift (Eqn 5), and we see the expected trend of decreased C_L with increased V_horiz. In essence, a bat flying quickly has a greater airspeed across the wings than it does flying slowly, and since it weighs the same under those two circumstances, the coefficient of lift must decrease with speed.

All species showed a decrease in stroke plane angle with increased flight velocity, as has been reported for other bat species previously (Aldridge, 1987). Although the wake patterns generated by bats are complex (Hedenström et al., 2007; Hubel et al., 2009), simple models such as actuator disk theory can capture some of the relevant connection between kinematics and aerodynamic force production. According to these ideas, a decrease in stroke plane angle should direct the induced velocity of the wing motion more rearward, thereby shifting the contribution of induced velocity towards increased thrust and away from lift generation, to simultaneously overcome increased drag and diminish the lift production of the wings (Pennycuick, 1975). This explanation has been applied to changes in stroke plane angle with speed in previous studies (Norberg, 1990; Pennycuick, 1975), and fits our results as well.

Bats showed a trend toward longer wingbeat cycle with increased velocity, and the downstroke phase of the wingbeat cycle was particularly long. Amplitude, however, did not change with speed. This differs from the trends reported for birds, in which wingbeat period increases with speed because of changes in the duration of upstroke, and amplitude does not change (Hedrick et al., 2002; Hedrick et al., 2003). The lengthening of the downstroke by bats may contribute to the reduction in C_L that we observed by slowing the downward motion of the wings during downstroke. Angle of attack and wing camber also both decreased with increased flight velocity, and those trends would also both reduce lift.

Only C. brachyotis showed a significant decrease in both maximum and minimum wingspan with increases in flight velocity. We hypothesize that this resulted because C. brachyotis were flown in a wind tunnel, and therefore flew at higher velocities relative to their body sizes than did the corridor-flown bats. Since drag increases exponentially with velocity, C. brachyotis probably encountered relatively larger drag forces than did the other bats. We hypothesize that C. brachyotis decreased overall wingspan with increasing flight velocity to reduce drag and that the difference between C. brachyotis and other species reflects this. We hypothesize that reduction of maximum wingspan with flight speed only occurs for bats at high flight speeds. We predict that maximum wingspan would not change with V_horiz below some threshold speed for C. brachyotis, and that other bat species would show a negative correlation between wingspan and flight speed at high speeds.

Over the course of a wingbeat cycle, the orientations of aerodynamic thrust, drag and lift change relative to global axes and can vary along the span of the flapping wing (Norberg, 1976). However, if we consider only the summation of forces over the whole wingbeat cycle, and if we neglect deviations from horizontal flight, we can treat horizontally directed net force as the sum of thrust and drag, and vertically directed net force as lift. This approach simplifies comparisons among individuals and flight conditions, and we employ that simplification here.

The net A_horiz of a flying bat results from the sum of forward thrust and rearward drag over the course of the entire wingbeat cycle, and the relative contributions of each over time cannot be separated in a purely kinematic analysis (Hedrick et al., 2002). Thus, the kinematic correlates of increased A_horiz might increase thrust, decrease drag, or do both. Still other changes might not influence those aerodynamic forces directly but change as a secondary result of kinematic changes that do.

When bats increased A_horiz, their stroke plane angles became more negative, thereby making the path of the wingtip more vertical. We hypothesize that a more vertical wing path shifts the orientation of shed vortices rearward, thereby increasing thrust (Pennycuick, 1975). With increases in A_horiz some species also increased wingspan, which would reduce the magnitude of induced drag slightly and might also increase thrust by sweeping the wings through a greater volume of air.

Strouhal number increased with A_horiz, probably as a secondary result of the decreased stroke plane angle; because stroke plane angle became more vertical while amplitude either increased or did not change, the maximum vertical distance traveled by the wingtip increased, and thus so did Strouhal number.

Angle of attack did not change significantly with A_horiz for most species, but its component angles, α₁ and α₂, did. All species showed decreases in α₂ with increasing A_horiz, probably also because of the decreased stroke plane angle. Since wingbeat frequency did not change with A_horiz, the decrease in stroke plane angle caused the leading edge of the wing to move more vertically during downstroke relative to the oncoming flow, resulting in an increase in α₂ for all species. That increase alone would have increased the angle of attack, but the overall angle of attack did not increase with A_horiz for any species except P. hypomelanus. This occurred because all species except P. hypomelanus compensated for the increased α₂ by holding the wing in a more horizontal plane during downstroke, that is, decreasing α₁. The decrease in α₁ with A_horiz was significant for three species, and nearly significant for two (P<0.07). We hypothesize that as the result of those kinematic changes, bats were able to take advantage of the added thrust that resulted from a decreased stroke plane angle without suffering the drag-inducing effects of an increased angle of attack.

A few species showed trends with increases in A_horiz not seen in other species. Pteropus pumilus increased wing stroke amplitude with A_horiz, P. hypomelanus decreased the downstroke ratio and increased angle of attack, and P. vampyrus increased its lift coefficient. No bats showed significant changes in wingbeat period, downstroke duration, or wing camber.

Vertical acceleration results from increased lift, so the kinematic correlates of A_horiz should cause increased lift, or result from correlations to kinematic changes that do so. Many kinematic parameters, such as T_down, τ, φ and β, did not change significantly with increased lift production, even though aerodynamic theory would suggest that many of those variables could influence lift if modified (Norberg, 1990). Bat species typically increased lift production by flapping their wings more quickly (decreasing T), though the duration of the downstroke was not correlated with lift production for any species. Bats also increased lift by increasing wingspan and wing camber. Angle of attack increased with lift production for most species, but that trend was only significant for C. brachyotis. Not surprisingly, the lift coefficient, computed with A_vert in the numerator (Eqn 5), was positively correlated with A_vert.

Wind tunnels provide substantial benefits for scientific investigations over the use of animals in free flight, including that the investigator can control the speed at which the animal carries out locomotion, and can collect data for a considerable period of time without need to move equipment alongside a free-moving animal. However, wind tunnels might induce kinematics that would not otherwise occur. For example, Tobalske et al. and Liechti and Bruderer found that birds flown in a wind tunnel had higher wingbeat frequencies than birds of the same species did in the wild (Liechti and Bruderer, 2002; Tobalske et al., 1997). Similarly, we believe that a number of differences between C. brachyotis and the other species used in this study probably resulted from the use of a wind tunnel for that species. Specifically, C. brachyotis flown in the wind tunnel flew at generally higher velocities and lower Strouhal numbers than would be predicted based on the size–velocity relationship uncovered for corridor-flown bats (Fig. 4A, Fig. 5F). This speed difference probably underlies all other kinematic differences between C. brachyotis and the trends we saw in the corridor-flown bats.

If the best fit lines for the five corridor-flown species are used to predict what C. brachyotis would have done if flown in a corridor, we see that wingbeat period was higher than expected (Fig. 5A), α₂ slightly lower than expected (Fig. 5I), and wing camber and coefficient of lift were lower than those of other species (Fig. 5J,K). Interestingly, all of these trends are what would be expected for high-speed flight based on our regression analyses (Table 4). With increasing V_horiz, bats increased wingbeat period and decreased wing camber and coefficient of lift. Also, when bats increased thrust production to increase A_horiz, they increased α₂, and it is likely that for fast flight, such as that of C. brachyotis in a wind-tunnel, the relatively higher drag necessitated increased thrust. Obviously, complete resolution of flight speed and the use of a wind tunnel on wing kinematics requires detailed investigation of individual bats flown in a corridor and wind tunnel over a comparable range of speeds, but our results provide an intriguing view of biases that may be introduced by wind tunnel experiments in studies of aerial locomotion.

Many of the trends reported here are consistent with patterns observed previously for other bats, but the data in this study might not be representative for bats of all species. Most bats are insectivorous, and many require high levels of maneuverability to catch their evasive food in flight. By contrast, all six species in this study are frugivorous bats that fly nightly between their roosts and food resources and this was probably also true of their last common ancestor (Boon and Corlett, 1989; DeFrees and Wilson, 1988; Jones and Kunz, 2000; Kunz and Jones, 2000; Kwiecinski and Griffiths, 1999; Luft et al., 2003; Giannini and Simmons, 2005; Jones et al., 2002; Teeling et al., 2005). To determine whether bats with improved maneuverability show different kinematic trends with speed and acceleration to those reported here is an exciting objective for future investigations.

Finally, no other family of bats approaches the largest body sizes of pteropodids, and there is a tremendous diversity of bats below the 33 g body mass of our smallest individuals. Small bats tend to fly at lower speeds than the bats in this study did (Akins et al., 2007), suggesting that the V_horiz∝M_b^1/6 trend reported here might be more robust among bats with smaller body sizes than we observed here. Recent work has revealed that small bats generate leading-edge vortices while flying at low speeds, much like flapping insects do (Muijres et al., 2008), and parallel experiments have not yet been performed for larger bats. Thus, the aeromechanics of bats may differ across the lower spectrum of bat body sizes. Our results demonstrate remarkable consistency in flight mechanics for medium to large-sized bats, and we look forward to parallel studies of smaller bats, especially from non-pteropodid families, that test the applicability of our results to bats in general.

 
• A_horiz

  net forward acceleration for the wingbeat cycle (m s^–2)

 
• AR

  aspect ratio (dimensionless)

 
• A_vert

  net vertical acceleration for the wingbeat cycle (m s^–2)

 
• b_max

  maximum wingspan (m)

 
• b_min

  minimum wingspan (m)

 
• C_L

  coefficient of lift (dimensionless)

 
• c_max

  maximum wing chord (m)

 
• COM

  center of mass

 
• d.f.

  degrees of freedom

 
• DLT

  direct linear transformation

 
• g

  acceleration of gravity (9.81 m s^–2)

 
• GLM

  generalized linear model

 
• M_b

  body mass (kg)

 
• Q_s

  wing loading (N m^–2)

 
• RMA

  reduced major axis

 
• S

  wing area (m²)

 
• S_max

  maximum wing area (m²)

 
• St

  Strouhal number

 
• T

  wingbeat period (s)

 
• T_down

  downstroke duration (s)

 
• V_horiz

  forward velocity (m s^–1)

 
• V_vert

  vertical velocity (m s^–1)

 
• V_wrist

  velocity of the wrist in the x_g–z_g plane at the time of max wingspan (m s^–1)

 
• x_b

  body-centered x dimension

 
• x_g

  global x dimension

 
• y_b

  body-centered y dimension

 
• y_g

  global y dimension

 
• z_b

  body-centered z dimension

 
• z_g

  global z dimension

 
• α

  angle of attack at mid-downstroke (deg; α=α₁+α₂)

 
• α1

  angle of wing chord to horizontal at mid-downstroke (deg)

 
• α2

  angle of wrist trajectory to oncoming flow at mid-downstroke (deg)

 
• β

  stroke plane angle (deg)

 
• ρ

  density of air (1.204 kg m^–3)

 
• τ

  downstroke ratio (dimensionless)

 
• φ

  stroke amplitude (deg)