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A COMPUTATIONAL MODEL FOR ESTIMATING THE MECHANICS OF HORIZONTAL FLAPPING FLIGHT IN BATS : MODEL DESCRIPTION AND VALIDATION

PHILIP WATTS¹, ERIKA J. MITCHELL¹ and SHARON M. SWARTZ^1,2,*

¹ Department of Ecology and Evolutionary Biology, Brown University, Providence, RI 02912, USA
² Division of Engineering, Brown University, Providence, RI 02912, USA
^* Author for correspondence (e-mail: sharon_swartz{at}brown.edu )

View larger version (33K): [in a new window]   Fig. 1. Plan-view of the ventral surface of the wing of a Pteropus poliocephalus held in a horizontal plane, a position similar to that at the middle of the downstroke. The wing is subdivided into 14 chordwise strips; there are seven segments between the shoulder and the carpus (the plagiopatagium or armwing), three between the carpus and the metacarpophalangeal (MCP III) joint of the third digit (the proximal handwing) and four between the MCP III joint and the wingtip (the distal handwing). The locations of the centers of mass of these strips relative to a reference line connecting the two shoulders (broken line) are indicated by the filled circles. The large circles labeled shoulder (glenohumeral joint), carpus, MCP III and wingtip were used as digitizing markers for collecting kinematic data. For the third wing segment (subscript p), the length of the wing chord, cp, the leading edge position, ep, and the distance from the segmental center of mass to the reference line, dp, are also indicated. The variation in these parameters within the wingbeat cycle is represented in the model by a second subscript, q, that ranges from 1 to 40 with the 40 equal time increments within the complete wingbeat cycle.

View larger version (13K): [in a new window]   Fig. 2. Orientations of and relationships between the inertial and non-inertial coordinate systems; the actual and mean paths of the shoulder are depicted from a lateral view of the bat moving from left to right across the figure. The x axis is perpendicular to the plane of the page and is represented by the filled circles.

View larger version (16K): [in a new window]   Fig. 3. Representative illustration of the complex kinematics of the bat wing after curve-fitting raw data; this view depicts the movements of the carpus (open squares) and wingtip (filled circles) with respect to the shoulder joint during both a downstroke and an upstroke. The lines connecting the shoulder (coordinates 0,0) and the carpus schematically represent armwing length, and those between the carpus and the wingtip, handwing length. Time intervals between data points are equal and show that the wrist, in particular, moves far more rapidly in midstroke than at other times in the wingbeat cycle.

View larger version (20K): [in a new window]   Fig. 4. Elliptical distribution of the magnitude of aerodynamic forces at each wing segment; segment 1 is adjacent to the shoulder and segment 14 includes the wingtip (see Fig. 1).

View larger version (17K): [in a new window]   Fig. 5. Schematic diagram of the rectangular idealization of the armwing membrane. The x and z coordinate axes used to describe membrane deflection mathematically are indicated; the y axis is perpendicular to the plane of the membrane. Fs indicates the point of application and approximate orientation of the skin internal force, ca is the mean armwing chord and ba is the instantaneous shoulder-to-carpus distance.

View larger version (17K): [in a new window]   Fig. 6. Schematic diagram of coordinate systems employed in the model. The origin of the global, inertial coordinate system is located at the mean vertical position of the shoulder joint and travels forward at constant mean velocity (see also Fig. 2); its path is indicated by the horizontal dotted line, and its location at mid-upstroke is given in the center of the figure. The origin of the global, non-inertial coordinate system is located at the shoulder joint and accelerates vertically and horizontally with the bat. It is shown at three points in the wingbeat cycle: late downstroke (left), mid-upstroke (center) and late upstroke (right). For both these coordinate systems, the x axis is directed perpendicular to the plane of the page, directly to the bat's right. At mid-upstroke and mid-downstroke, the global inertial and non-inertial axes coincide (center). The true flight path of the shoulder is indicated by the dashed line. Local x, y, z coordinate systems can be centered at any anatomical point of interest, such as the shoulder, the midshaft of the humerus or radius or the carpus. In this illustration, the axes of a local (x, y, z) coordinate system with its origin at the humeral midshaft are shown by gray heavy dashed lines; once again, the x axis is directed perpendicular to the plane of the page. Local primed (x', y', z') coordinate systems are employed for computations of local stresses, etc; they are centered at the origin of a corresponding (x, y, z) coordinate system, but are rotated such that the x axis is directed along the length of the humerus (or radius) and the y axis is directed perpendicular to the local wing surface. The local (x', y', z') coordinate system at the humeral midshaft is illustrated here, with axes depicted by barred black lines. Late in the upstroke (rightmost illustration), from a lateral view as depicted here, the undersurface of the wing would be interposed between the observer and the x' and y' axes; this is depicted schematically by gray shading.

View larger version (16K): [in a new window]   Fig. 7. Definitions of the orientations and signs of longitudinal and shear stresses with respect to midshaft cross sections of the humerus and radius. 1,2, shear stress; L, longitudinal stress; T, transverse stress.

View larger version (17K): [in a new window]   Fig. 8. Longitudinal and shear stresses computed by the model at the midshaft of the humerus throughout the wingbeat cycle as a function of spanwise elastic modulus of the skin, Exx. (A) Longitudinal stresses on the cranial surface of the bone. (B) Longitudinal stresses on the dorsal surfaces of the bone. (C) Shear stresses on the dorsal surface of the bone.

View larger version (19K): [in a new window]   Fig. 9. Stresses computed by the model throughout the wingbeat assuming the spanwise elastic modulus of the skin Exx to be 0.5 MPa.

View larger version (18K): [in a new window]   Fig. 10. Longitudinal and shear stresses at the humeral midshaft computed by the model for an entire wingbeat wingbeat assuming the spanwise elastic modulus of the skin Exx to be 0.5 MPa; total stress (solid heavy line) is partitioned into components due to each of the individually modeled forces exerted on the wing.

View larger version (22K): [in a new window]   Fig. 11. Longitudinal and shear stresses at the radial midshaft computed by the model for an entire wingbeat wingbeat assuming the spanwise elastic modulus of the skin Exx to be 0.5 MPa; total stress (solid heavy line) is partitioned into components due to each of the individually modeled forces exerted on the wing.

View larger version (18K): [in a new window]   Fig. 12. Vertical and forward components of the global acceleration of the bat as computed by the model.

View larger version (10K): [in a new window]   Fig. 13. Comparison of bone stresses computed by the model, assuming the spanwise elastic modulus of the skin Exx to be 0.5 MPa, with the forces computed from surface strains measured empirically during natural flight (Swartz et al., 1992).

View larger version (10K): [in a new window]   Fig. 14. Comparison of transversely oriented bone stresses computed from the model, assuming the spanwise elastic modulus of the skin Exx to be 0.5 MPa, with the forces calculated from empirically gathered rosette strain data and the published value for the Poisson's ratio of bone.

View larger version (17K): [in a new window]   Fig. 15. Comparison of the vertical oscillations of the bat's center of mass over one wingbeat cycle as computed from the model and as measured directly from wind-tunnel film footage (Carpenter, 1985). Empirical measurements are means ± 2 S.D. (N=18).