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The dynamic, three-dimensional shape of flapping insect wings may influence many aspects of flight performance. Insect wing deformations during flight are largely passive, and are controlled primarily by the architecture and material properties of the wing. Although many details of wing structure are well understood, the distribution of flexural stiffness in insect wings and its effects on wing bending are unknown. In this study, we developed a method of estimating spatial variation in flexural stiffness in both the spanwise and chordwise direction of insect wings. We measured displacement along the wing in response to a point force, and modeled flexural stiffness variation as a simple mathematical function capable of approximating this measured displacement. We used this method to estimate flexural stiffness variation in the hawkmoth Manduca sexta, and the dragonfly Aeshna multicolor. In both species, flexural stiffness declines sharply from the wing base to the tip, and from the leading edge to the trailing edge; this variation can be approximated by an exponential decline. The wings of M. sexta also display dorsal/ventral asymmetry in flexural stiffness and significant differences between males and females. Finite element models based on M. sexta forewings demonstrate that the measured spatial variation in flexural stiffness preserves rigidity in proximal regions of the wing,while transferring bending to the edges, where aerodynamic force production is most sensitive to subtle changes in shape.

During flapping flight, insect wings accelerate masses of air, generating the forces necessary to support the insect's weight and to perform complex maneuvers. At the same time, these ultralight airfoils (generally only 0.5-5%of body mass; Ellington,1984b) must withstand the forces imposed upon them by the surrounding air, as well as the inertial forces caused by accelerating and decelerating their own mass up to several hundred times per second.

Insect wings perform these roles extremely successfully, despite the fact that they are largely passive structures, with no muscular control past the wing base (Wootton, 1992). Although they are strengthened by a network of tubular veins, the wings of many species deform noticeably during flight, especially during slow flight and hovering (Willmott and Ellington,1997a). These dynamic changes in the three-dimensional shape of wings could potentially affect many aspects of force production, yet few models of insect flight have successfully incorporated passive wing flexibility or examined the effects of flexibility on force production.

In the past two decades, models of insect flight, both mathematical and physical, have contributed enormously to our understanding of the basic mechanisms of force production, despite assuming that insect wings are rigid structures [e.g. mathematical models of Ellington(1984a), Lighthill(1973), Sane and Dickinson(2002), Savage et al. (1978),Smith et al. (1996) and Wilkin and Williams (1993), and physical models of Bennett(1966, 1970), Dickinson et al.(1999), Ellington et al.(1996) and Spedding and Maxworthy (1985)]. Some aspects of wing flexibility have been mimicked in models by altering the relative positions of wing regions(Liu et al., 1998; Vest and Katz, 1996) or by modeling deformations as harmonic waves(Combes and Daniel, 2001; Daniel, 1987; Wu, 1971). These approaches provide unique insights into the mechanisms of force generation during flight,but often neglect one or more critical components of wing deflection (e.g. spanwise bending, chordwise bending or torsion), which can have large effects on aerodynamic force production (Batchelor,1967). Models of insect flight that incorporate passive wing flexibility (in which shape changes are driven by forces imposed upon the wing rather than being specified in advance) are exceedingly rare (e.g. Smith, 1996).

One difficulty in modeling passive wing flexibility is that forces applied at the wing base lead to bending and twisting that are influenced not only by overall stiffness and gross anatomical features (e.g. flexion or fold lines),but also by the distribution of flexibility throughout the wing(Wootton, 1999). Extremely detailed finite element models of insect wings (incorporating vein configuration, three-dimensional relief, and variations in vein and membrane characteristics) could potentially reproduce the distribution of flexibility in real wings by accounting for the precise structural and material properties of the wing. Unfortunately, precise data about the local properties of insect wings are often unavailable and finite element models must be constructed using simplifying assumptions. For example, many finite element models assume that the material stiffness of the wing (Young's modulus, E) is equivalent to that of insect chitin and constant throughout the wing(Smith, 1996; Kesel et al., 1998); however,recent measurements reveal that E can vary widely within a wing(Smith et al., 2000) and that other proteins, such as resilin, occupy key positions (e.g. wing vein joints)in insect wings (Gorb, 1999; Haas et al.,2000a,b). Measuring spatial variation in these material properties, as well as the details of vein and membrane structure, is a time-consuming process that would need to be repeated for each new species studied.

Although these detailed approaches can provide important information about functional wing morphology in pivotal, well-studied species (e.g. Herbert et al., 2000), a more general approach to wing stiffness measurements and modeling could facilitate comparative studies and attempts to incorporate passive flexibility into models of force production. Rather than measuring (and modeling) geometric and material properties separately, the overall bending response of the wing, or flexural stiffness (EI, the product of material stiffness Eand second moment of area I) can be determined. Performing measurements of EI averaged over the whole wing is relatively straightforward (see Combes and Daniel,2003a), but measuring spatial variation in flexural stiffness throughout a wing is more challenging. Steppan(2000) approached this issue by measuring average flexural stiffness over increasingly larger sections of dried butterfly wings, and Wootton et al.(2000) measured average flexural stiffness in three isolated sections of locust hindwing. However, no studies to date have demonstrated a method of measuring the spatial variation in local flexural stiffness of intact insect wings.

In this study, we developed a method of approximating spatial variation in flexural stiffness along two axes of the wing (in the spanwise and chordwise direction). We measured flexural stiffness variation in two insects, the hawkmoth Manduca sexta and the dragonfly Aeshna multicolor. These insects have wings of similar size and are both agile fliers, capable of hovering as well as fast, forward flight. However, they are very distantly related and display large differences in wing shape and venation pattern (see Combes and Daniel, 2003a) that may underlie differences in the distribution of flexural stiffness in their wings.

To determine the spatial pattern of flexural stiffness in wings, we first developed a method to measure displacement (due to a point force) along the wing in the spanwise and chordwise directions. Next, we proposed various alternatives for how flexural stiffness might vary along the wing (represented by simple mathematical functions) and predicted the patterns of displacement that a loaded wing with these theoretical stiffness distributions would display. We then found the flexural stiffness distribution that produced a pattern of displacement most similar to the pattern measured in a real wing. We used this method to estimate the flexural stiffness distribution of the wing in response to forces applied from both the dorsal and the ventral side,as dorsal/ventral asymmetry has been noted in previous studies(Ennos, 1988; Steppan, 2000; Wootton, 1993; Wootton et al., 2000).

We also created two different finite element models of an insect wing,based on the planform geometry of a Manduca sexta forewing. Rather than realistically reproducing the structure and behavior of Manducawings, these generalized models provide a method of assessing the consequences of flexural stiffness distributions to wing bending. We attempted to capture major elements of wing structure in the models (e.g. vein position and flexural stiffness distribution) while avoiding detailed structural features,such as variation in wing thickness and three-dimensional relief. We used these models to examine how different spatial patterns of EIdetermine bending patterns in wings subjected to both static and dynamic loads.

### Flexural stiffness measurements

Insect handling and wing preparation were performed as in Combes and Daniel(2003a). Calculations of EI distribution require both the applied force and the displacement along the length of the wing. We measured displacement of wings in the spanwise direction along a line running from the wing base to the tip, and in the chordwise direction along a line running from the leading edge to the trailing edge (approximately midway between the base and tip). We attached a fresh wing at either the wing base or leading edge, and applied a point force to the wing with a pin attached to a force beam (see Combes and Daniel, 2003a),using a small drop of cyanoacrylate glue to prevent the pin from slipping off the wing tip or trailing edge. We illuminated the wing with sheets of laser light; these sheets appear as lines on the wing, and can be adjusted so that one line illuminates a strip of the wing from the base to the tip(Fig. 1A) or from the leading to the trailing edge (intersecting the pin). When a force is applied to the wing, the laser lines follow the surface of the wing, shifting their position in proportion to the displacement of the wing(Fig. 1B). We used a Coolpix 900 digital camera (Nikon, Tokyo, Japan) angled approximately 45° from the surface of the wing to photograph unloaded and loaded wings, as well as a calibration object containing a series of steps of known height.

Fig. 1.

Unloaded (A) and loaded (B) Manduca sexta wing illuminated with laser lines. A Matlab program was used to find the center of a laser line running from the base of the wing to the tip before and after applying a point force at the tip (yellow and red lines, respectively). The change in position of the laser line y(x) was then used to find displacement of the wing δ(x).

Fig. 1.

Unloaded (A) and loaded (B) Manduca sexta wing illuminated with laser lines. A Matlab program was used to find the center of a laser line running from the base of the wing to the tip before and after applying a point force at the tip (yellow and red lines, respectively). The change in position of the laser line y(x) was then used to find displacement of the wing δ(x).

We measured displacement in response to loads of varying magnitude,returning to the unloaded position to photograph the wing before applying each new load. We checked the repeatability of displacement measurements on the wings of a hawkmoth (in both the spanwise and chordwise direction) by applying five different loads to the dorsal surface of the wing and repeating each load three times (applying loads in random order).

For all other hawkmoth and dragonfly wings, we measured displacement on the dorsal surface of the wing at four different loads, gently removed the glued pin from the edge of the wing (without tearing the wing membrane), flipped the slide over, and measured displacement on the ventral side of the wing at four different loads. Dorsal stiffness in this study describes the stiffness of the wing in response to loading on the dorsal surface (resulting in a dorsally convex surface; see Steppan,2000), which is equivalent to ventral flexion in other studies(Ennos and Wootton, 1989; Wootton, 1981, 1992).

After completing the flexural stiffness measurements, we photographed the wings on a white background from above and measured wing area, span and maximum chord length in NIH Image. For Manduca sexta (whose wing morphology may depend on gender; Willmott and Ellington, 1997b), we also calculated aspect ratio(span2/area) of the forewings and wing loading (body mass divided by the area of both forewings).

In each hawkmoth, we measured spanwise flexural stiffness on one forewing and chordwise flexural stiffness on the other. We did not measure the flexural stiffness of the hindwings, which are much smaller and overlap with the forewings to varying degrees during flight. In dragonflies (which have independent fore- and hindwings of similar size), we measured spanwise flexural stiffness on one forewing and one hindwing, and chordwise flexural stiffness on the other forewing and hindwing.

We analyzed images of loaded and unloaded wings with a custom Matlab program (developed by A. Trimble) that finds the center of the laser line running from the wing base to the tip or from the leading to the trailing edge(Fig. 1A,B). We used a 2nd order Butterworth filter to remove noise from the line position data, and splined the unloaded and loaded data to an equal number of points for comparison. We found the difference in line position between the two data sets [y(x); Fig. 1B] and converted this difference to actual wing displacement along the wing δ(x) with a factor derived from analysis of the calibration object.

With this displacement data and the applied force, we were able to estimate local flexural stiffness by using a continuous beam equation to approximate EI variation along the wing (see Appendix). This equation defines local flexural stiffness as a function of the local curvature (second spatial derivative of displacement) and the local moment (applied force times distance from the point of force application). To avoid errors caused by differentiating displacement data, we solved this as an inverse problem; we posed several simple mathematical functions that might approximate flexural stiffness variation along the wing (constant, linear, exponential, or 2nd degree polynomial) and calculated the expected pattern of displacement due to the applied force and these possible EIdistributions. We then used a simplex minimization program to find the EI distribution that provided the best fit to the measured displacement along the wing. Finally, we calculated average flexural stiffness(

$$EI$$
) by integrating the equation describing flexural stiffness variation along the wing's length (see Appendix).

To verify the method of determining flexural stiffness distribution, we measured δ(x) and calculated local flexural stiffness of a rectangular glass coverslip, which is made of homogeneous material and has a constant material stiffness (E) and second moment of area(I).

### Comparisons within and between individuals

Because wing displacement in hawkmoth and dragonfly wings does not increase linearly with applied force (see Combes,2002), our estimates of flexural stiffness varied with the relative displacement of the wing (δT/L, tip displacement divided by wing span or trailing edge displacement divided by chord length). To facilitate comparisons between individuals and species, we standardized measurements at a relative displacement of 0.05 for spanwise measurements and 0.08 for chordwise measurements (values that are within the range measured on real wings). For each individual, we used the slope of the relationship between EI variables (

$$EI$$
and coefficients describing the distribution of flexural stiffness) and relative displacement to estimate the value of these variables at a givenδ T/L.

In Manduca sexta, we collected spanwise data from nine males and four females, and chordwise data from ten males and nine females. In Aeshna multicolor, we used only males, but tested both fore and hindwings (which have different morphologies), so we examined stiffness separately in the two sets of wings. We collected spanwise and chordwise data on both fore and hindwings from eight individuals.

Within each group, we tested for differences between the dorsal and ventral side of the wing with a Wilcoxon signed rank test. We also tested for differences between male and female Manduca sexta with a Mann-Whitney U-test.

### Finite element modeling

To investigate how flexural stiffness variation affects wing bending, we created two simplified finite element models of an insect wing (based on the forewing of a male Manduca sexta) with the same geometry, but dramatically different patterns of spatial variation in flexural stiffness(Fig. 2). We used MSC Marc/Mentat 2001 to construct model wings with an accurate planform configuration of veins and membrane, but a simplified three-dimensional geometry (with no camber or three-dimensional relief; see Discussion). We increased the material stiffness of vein elements beyond that of the surrounding membrane to mimic the higher second moment of area of tubular veins and produce the measured spanwise-chordwise anisotropy in flexural stiffness (see Combes and Daniel,2003a).

Fig. 2.

Finite element models based on Manduca sexta wings. (A) Model wing with membrane elements (blue) and vein elements (pink), each of homogeneous material stiffness E. (B) Model wing in which declining material stiffness of membranes and veins results in an exponential decline in flexural stiffness. Each color represents a different value of material stiffness.

Fig. 2.

Finite element models based on Manduca sexta wings. (A) Model wing with membrane elements (blue) and vein elements (pink), each of homogeneous material stiffness E. (B) Model wing in which declining material stiffness of membranes and veins results in an exponential decline in flexural stiffness. Each color represents a different value of material stiffness.

The models were composed of thin shell elements with a density of 1200 kg m-3 (as measured in insect wings; Wainwright et al., 1982) and a thickness of 45 μm. We used a Poisson's ratio of 0.49, as measured in some biological materials (Wainwright et al.,1982); because the Poisson's ratio of insect wings is unknown, we tested the effects of using a Poisson's ratio of 0.3 and found that the difference in model behavior was negligible. The uniform mass distribution of the models may cause dynamic bending in the distal regions to be overestimated, but qualitative differences in wing bending between the two models should not be affected. To determine the minimum number of elements necessary to capture the bending behavior of wings, we performed a sensitivity analysis with models composed of 200, 350, 865 and 2300 total elements, and found that 865 elements were sufficient to ensure asymptotic performance of the model.

We adjusted the material stiffness E of vein and membrane elements in the two models to produce different spatial patterns of flexural stiffness,but the same overall bending performance (so that tip and trailing edge displacement in response to a point load were the same as displacements measured in real wings). In the first model, we assigned all vein elements a single (homogeneous) Young's modulus of E=1.5×108 N m-2 (similar to values measured in locust hindwing; Smith et al., 2000), and all membrane elements a Young's modulus of E=2.1×1012 N m-2 (representing both increased material stiffness and the increased second moment of area of tubular veins; Fig. 2A). In the second model,we applied declining values of material stiffness to the model wing in 12 strips oriented diagonally (Fig. 2B); these strips are perpendicular to most of the wing veins,which decrease in diameter towards the wing edge and thus are likely to decrease in stiffness along this axis. We adjusted the values of material stiffness in these strips to approximate patterns of overall wing flexural stiffness measured in real Manduca wings (E in the model varies from 4.7×107 N m-2 to 4.5×109 N m-2 in membrane elements, and from 1.9×1011 N m-2 to 1.8×1013 N m-2 in vein elements).

To determine the resulting pattern of flexural stiffness variation in the spanwise direction of the model wings, we fixed each wing at its base (with no displacement or rotation) and applied a point force at the tip. For chordwise measurements, we fixed the model at its leading edge (from the base to 2/3 span) and applied a point force at the trailing edge. We recorded the displacement at 22 nodes (junctions between elements) aligned between the point of attachment and the point of force application, and used this information to estimate spanwise and chordwise flexural stiffness distribution with the Matlab simplex minimization program, as in real wings.

We compared static bending performance of the two models by fixing each wing at its base and applying either a point load of 0.003 N at the tip(within the range of loads applied to real wings) or a pressure load of -14.43 Pa to the wing surface (equivalent to the average lift force that the wing would experience during steady flight). We compared dynamic bending by flapping each model wing at a realistic wing beat frequency and stroke amplitude. We applied boundary conditions to the nodes at the wing base so that they could not translate in any direction and could rotate only in the dorsal-ventral direction (around the y-axis, see Fig. 2). We began the simulation with initial conditions of zero displacement and zero velocity at all nodes, and gradually increased the rotation at the wing hinge to a sinusoidal motion with the following function:
$\ {\theta}(t)=(1-\mathrm{exp}^{-\mathrm{t}{/}{\tau}}){\cdot}\mathrm{sin}({\omega}t),$
1
where θ is rotation at the base nodes, t is time in seconds,τ is the time constant and ω is the angular frequency(2πf, where f is the flapping frequency). We flapped the wings at 26 Hz, and found that a time constant of 1/20 s avoids transient artefacts of rapid initial acceleration and allows the wings to reach their full stroke amplitude of 108° after 6.5 flaps. We ran each model for 19200 time steps, simulating 12.5 flaps in 0.48 s.

Because the finite element program does not calculate aerodynamic forces acting on the wing, a damping factor had to be applied to stabilize the models. We compared the motions of the model wing with declining material stiffness to those of a real Manduca wing attached to a motor and rotated in the same way (Combes and Daniel,2003b; Daniel and Combes,2002). We found that a mass damping factor of 10 reduced high frequency vibrations and provided the closest match to motions of the real wing, and applied this damping factor to both model wings.

### Flexural stiffness measurements

The simplex minimization analyses revealed that proposed EIdistributions based on either exponential or 2nd-degree polynomial equations can provide relatively good predictions of measured wing displacement, while constant or linear distributions of EI cannot(see Combes, 2002). However,the proposed exponential EI distributions were more consistent within and between individuals, and effectively predicted displacement in the spanwise and chordwise directions of both Manduca sexta and Aeshna multicolor. Because these more consistent results allow us to make comparisons within and between individuals, we used an exponential equation to approximate flexural stiffness patterns in the species tested.

Our control measurements validated the procedure for estimating local flexural stiffness and revealed that the method is repeatable. Bending tests on a glass coverslip showed that a constant distribution of flexural stiffness provided the best approximation of measured displacement (as expected for a homogeneous beam). In addition, measurements of Manduca flexural stiffness in which each load was repeated several times provided consistent estimates of the EI distribution each time that a given load was applied (see Combes, 2002).

Morphological measurements showed that female Manduca sexta are significantly heavier and have larger wings than males (body mass, P=0.004; wing area, P=0.003; wing span, P=0.002,chord length, P=0.004; wing mass, P=0.012), but aspect ratio and wing loading were not significantly different in the individuals sampled(aspect ratio, P=0.946; wing loading, P=0.262). Although overall flexural stiffness scales with wing size across a broad range of species (see Combes and Daniel,2003a), spanwise

$$EI$$
was not significantly different in male and female Manduca (dorsal, P=0.165; ventral, P=0.308; Fig. 3A,B). In the chordwise direction,
$$EI$$
was higher in females than in males on the dorsal side of the wing (P=0.021), but not on the ventral side(P=0.477). The exponents of the stiffness distribution in the chordwise direction were significantly higher in females than in males(dorsal, P=0.006; ventral, P=0.008), but exponents in the spanwise direction were the same in males and females (dorsal, P=0.165; ventral, P=0.089).

Fig. 3.

Flexural stiffness distribution in wings of Manduca sexta and Aeshna multicolor, and in finite element models of M. sextawings. In each graph, spanwise flexural stiffness is shown above (longer lines) and chordwise flexural stiffness below; dorsal measurements are in black and ventral measurements in gray. Each line within these groups represents the flexural stiffness distribution estimated from wing displacement measurements performed on a different individual. (A) Flexural stiffness distribution of male Manduca sexta forewings (N=9 spanwise, N=10 chordwise), and of finite element models with homogeneous (blue) and exponentially declining (red) vein and membrane material stiffness. (B) Flexural stiffness distribution of female Manduca sexta forewings (N=4 spanwise, N=9 chordwise). (C)Flexural stiffness distribution of male Aeshna multicolor forewings(N=8). (D) Flexural stiffness distribution of male Aeshna multicolor hindwings (N=8).

Fig. 3.

Flexural stiffness distribution in wings of Manduca sexta and Aeshna multicolor, and in finite element models of M. sextawings. In each graph, spanwise flexural stiffness is shown above (longer lines) and chordwise flexural stiffness below; dorsal measurements are in black and ventral measurements in gray. Each line within these groups represents the flexural stiffness distribution estimated from wing displacement measurements performed on a different individual. (A) Flexural stiffness distribution of male Manduca sexta forewings (N=9 spanwise, N=10 chordwise), and of finite element models with homogeneous (blue) and exponentially declining (red) vein and membrane material stiffness. (B) Flexural stiffness distribution of female Manduca sexta forewings (N=4 spanwise, N=9 chordwise). (C)Flexural stiffness distribution of male Aeshna multicolor forewings(N=8). (D) Flexural stiffness distribution of male Aeshna multicolor hindwings (N=8).

The wings of all Manduca sexta displayed dorsal/ventral asymmetry;the average flexural stiffness of their wings was higher in response to forces applied to the dorsal side of the wing, in both the spanwise and chordwise direction (spanwise, P=0.002; chordwise, P<0.001; Fig. 3A,B). In contrast, Aeshna multicolor displayed no dorsal/ventral difference in its hindwings or in the chordwise direction of its forewings (hindwing spanwise, P=0.327; hindwing chordwise, P=0.735; forewing chordwise, P=0.069; Fig. 3C,D). In the spanwise direction, the forewings were stiffer on the dorsal side than on the ventral side (P=0.018), but the difference between dorsal and ventral flexural stiffness was far smaller than that seen in Manduca.

### Finite element modeling

The finite element model with diagonal strips of declining material stiffness accurately reproduced the sharp decline in flexural stiffness measured in male Manduca wings(Fig. 3A, red lines), while the model with homogeneous vein and membrane regions displayed a vastly different pattern of flexural stiffness (Fig. 3A, blue lines).

The effects of these differences in EI distribution were apparent in static tests on the model wings. Although an applied point force resulted in the same tip displacement, most bending in the exponential wing was confined to the outer third of the wing, while the homogeneous wing bent gradually along much of its span (Fig. 4A). When subjected to a pressure load, the homogeneous wing again bent along a large portion of its length, and its maximum displacement was nearly twice as large as the displacement of the exponential wing(Fig. 4B). In the exponential wing, displacement was localized to the tip and trailing edge of the wing, and bending was apparent in both the spanwise and chordwise directions.

Fig. 4.

Results of static bending tests on finite element model (FEM) wings. Wings are fixed at the base, and displacement from original position (black outline)in the z-direction is indicated by the color bar. The FEM wing with homogeneous membrane and vein material stiffness is shown on the left and the wing with exponentially declining material stiffness is shown on the right.(A) Displacement due to a point force of 0.003 N (green arrow) at the wing tip. (B) Displacement due to a normal face load of -14.43 Pa (green arrows),the approximate pressure on a wing during steady flight.

Fig. 4.

Results of static bending tests on finite element model (FEM) wings. Wings are fixed at the base, and displacement from original position (black outline)in the z-direction is indicated by the color bar. The FEM wing with homogeneous membrane and vein material stiffness is shown on the left and the wing with exponentially declining material stiffness is shown on the right.(A) Displacement due to a point force of 0.003 N (green arrow) at the wing tip. (B) Displacement due to a normal face load of -14.43 Pa (green arrows),the approximate pressure on a wing during steady flight.

The effects of an exponential decline in flexural stiffness were further illustrated by dynamic tests on the models. The homogeneous wing showed little chordwise bending and spanwise bending was most pronounced near the base,while the distal portion of the wing remained relatively rigid(Fig. 5, blue wings). In contrast, the exponential wing bent considerably in the chordwise direction,and bending in the spanwise direction was confined mainly to the outer portion of the wing (Fig. 5, red wings). For a movie of the model wings in motion, see http://faculty.washington.edu/danielt/movies.

Fig. 5.

Sequence of images from flapping finite element model wings with homogeneous (blue) or exponentially declining (red) vein and membrane material stiffness. Wings are moving to the right, viewed from their leading edge. Models are shown near the end of the simulation at the same time steps,beginning at 0.4325 s andproceeding in 0.0029 s intervals. For a movie of the model wings in motion, see http://faculty.washington.edu/danielt/movies.

Fig. 5.

Sequence of images from flapping finite element model wings with homogeneous (blue) or exponentially declining (red) vein and membrane material stiffness. Wings are moving to the right, viewed from their leading edge. Models are shown near the end of the simulation at the same time steps,beginning at 0.4325 s andproceeding in 0.0029 s intervals. For a movie of the model wings in motion, see http://faculty.washington.edu/danielt/movies.

Modeling passive wing flexibility requires some knowledge of the spatial distribution of stiffness throughout a wing. Although few studies have attempted to quantify spatial patterns of stiffness, a number of qualitative hypotheses have been put forward concerning the relationship between the structure of insect wings and their regional mechanical behavior. Most wings have relatively stiff, supporting zones near the wing base and leading edge,and relatively deformable areas near the edges of the wing(Wootton, 1981). Wing veins taper in diameter from base to tip(Wootton, 1992), and the presumed reduction in stiffness that this provides could serve many purposes,including reducing the inertial load at wing tips (thus reducing energy expenditure and stress at the wing base) and allowing wing tips to buckle and rebound from collisions without damage(Wootton, 1992). In addition,a wing that is stiff at the base and more flexible at the tip would be well suited to withstand the bending torques (which decline sharply from base to tip) generated by flapping (Ennos,1989).

Our measurements of spatial variation in flexural stiffness agree with these qualitative observations; flexural stiffness declines sharply from wing base to tip and from the leading to the trailing edge in the wings of both Manduca sexta and Aeshna multicolor. Displacement in the spanwise and chordwise directions of these wings can be predicted fairly well when the flexural stiffness distribution is approximated by an exponential equation. The observed similarities in flexural stiffness distribution in these two species (despite large differences in wing shape and venation pattern) suggest that a sharp decline in wing flexural stiffness towards the tip and trailing edge may be a common feature of many insect wings.

Although detailed inter-specific comparisons are not feasible with only two species, this study demonstrates several interesting intra-specific differences, particularly in Manduca sexta. Male and female Manduca sexta appear to differ in the spatial patterns of flexural stiffness in their wings; chordwise flexural stiffness declines far more sharply in male moths than in female moths(Fig. 3A,B), and as a result,the wings of female moths are significantly stiffer in the chordwise direction(dorsal

$$EI$$
is higher). The functional significance of these differences is unclear, but other aspects of Manduca wing morphology (such as the physical coupling of fore- and hindwings) also display sexual dimorphism(Eaton, 1988), which may be related to higher wing loading in females carrying eggs(Willmott and Ellington,1997b). The traits measured in female hawkmoth wings could in some way compensate for this intermittent higher wing loading, for example by influencing the pattern or extent of chordwise wing deflections during flight.

The wings of both male and female Manduca sexta display a large dorsal/ventral difference in average flexural stiffness, and in the exponents of flexural stiffness distribution. Average flexural stiffness is greater in response to forces applied on the dorsal side than on the ventral side (in both the spanwise and chordwise direction). In the spanwise direction, this difference is greatest near the base (Fig. 3A,B). Steppan(2000) found a similar dorsal/ventral asymmetry near the base of butterfly wings, and higher dorsal flexural stiffness has also been measured in the leading edge of locust hindwings (Wootton et al.,2000).

Although measurements of such bending asymmetry are rare, theoretical studies have suggested that insect wings might display dorsal/ventral stiffness asymmetry due to the camber inherent in most wings(Ennos, 1995, 1997; Wootton, 1993). We explored the effects of camber on bending asymmetry in the finite element model of a Manduca wing, adding 4% spanwise and 5% chordwise camber (the maximum values measured in real wings). Adding this degree of camber to the model wings had a relatively small effect on tip and trailing edge displacement (see Combes and Daniel, 2003a), and failed to produce any dorsal/ventral bending asymmetry (tip displacement was identical whether the force was applied to the convex or the concave side; see Combes, 2002).

Camber may in fact influence the maximum load each side of the wing can resist before buckling (the buckling load), and the finite element model does not simulate elastically stable buckling. However, our measurements on real wings do not display the strong non-linear dependence on loading that would be expected if elastically stable buckling were occurring during these static tests. In a more dynamic situation (such as flapping flight), where continuous shape changes can alter flexural stiffness, camber may in fact lead to dorsal/ventral differences in wing deformation. However, wing camber does not appear to explain the large differences measured in this study on wings subjected to static point loading.

The structural source of this bending asymmetry remains unclear. Although Manduca wings have none of the gross anatomical features often associated with bending asymmetry (e.g. a ventral flexion line), their veins may contain one-way hinges or other micro-structural features that facilitate asymmetric bending in other insect wings(Wootton, 1981; Wootton, 1992). In addition,the stress-stiffening effect explored by Kesel et al.(1998) in models of dipteran wings could potentially apply to Manduca. When unloaded, the membranes of Manduca wings lie rather loosely between the veins extending to the trailing edge. A point force applied to the dorsal (convex)side of the trailing edge appears to pull the trailing edge veins further apart, which would first remove any slack in the membrane, and then possibly increase the stiffness of the wing as force is applied to the already-taut membrane (the stress-stiffening effect). In contrast, when a point force is applied to the ventral side of the trailing edge, the veins are pushed together and the membranes between them become looser.

Although the finite element models do not reproduce this dorsal/ventral asymmetry, the strikingly different bending patterns of the two models demonstrate that flexural stiffness distribution is critical in determining how insect wings bend. In the model wing with exponentially declining flexural stiffness, both static and dynamic bending are limited to the tip and trailing edge, whereas bending in the homogeneous model occurs along the entire span.

These results suggest that the sharply declining flexural stiffness measured in real wings helps maintain rigidity near the wing base (despite larger bending moments), while localizing bending to the tip and trailing edge, which are regions of particular importance in controlling aerodynamic force production. The trailing edge is a critical control surface in airplane wings (affecting both the magnitude of lift and the lift-to-drag ratio; Anderson, 1991), and passive deformations in this region are likely to play a similarly important role in controlling flight forces in insects. The integration of passive flexibility and spatial patterns of flexural stiffness into models of insect flight will be an important step in determining the functional significance of wing structure and dynamic bending to insect flight performance.

To calculate local flexural stiffness, we treated each wing as a heterogeneous, two-dimensional beam in the spanwise (or chordwise) direction,and used a continuous beam equation to estimate flexural stiffness along this axis. With the beam attached at one side and a point force applied at the other end, flexural stiffness at any point along the beam is the moment divided by the local curvature (or the second spatial derivative of displacement):
$\ EI(x)=\frac{M(x)}{\mathrm{d}^{2}{\delta}(x){/}\mathrm{d}x^{2}},$
A1
where M(x) is the moment and δ(x) the displacement at position x. If we assume that the point force is applied at an angle approximately 90° from the beam, then the moment is the product of the applied force and the distance between the point of force application and position x on the beam:
$\ EI(x)=\frac{F(L-x)}{\mathrm{d}^{2}{\delta}(x){/}\mathrm{d}x^{2}},$
A2
where F is the applied force and L is the total length of the beam.

If tip displacement is extremely large relative to beam length, the beam is bent down so far that its coordinates in the x-dimension while loaded are significantly different from its coordinates while unloaded, invalidating Equation A2. However, Equation A2 is valid for our measurements for the following reasons: (i) all measurements were standardized to a relative displacement (δT/L) of 0.05 in the spanwise direction and 0.08 in the chordwise direction, (ii) original measurements were rarely performed at relative displacements over 0.08 in the spanwise direction and 0.15 in the chordwise direction, and (iii) even in the most extreme cases(where relative displacement exceeded 0.15), the error in coordinates introduced by length changes in the x-dimension was less than 4%.

We solved Equation A2 in reverse, posing a functional distribution for EI(x), calculating the expected displacementδ(x), and using simplex minimization to find the parameters that minimize the difference between predicted and measured wing displacement. We proposed that stiffness in the spanwise or chordwise direction might be constant, or that its spatial pattern might be approximated by a linear,exponential, or 2nd-degree polynomial equation:
$\ EI(x)=\mathrm{d},$
A3
$\ EI(x)=mx+b,$
A4
$\ EI(x)=c{\cdot}\mathrm{exp}^{\mathrm{ax}},$
A5
$\ EI(x)=px^{2}+qx+r.$
A6
Each of the proposed EI distributions (Equations A3-A6) was inserted into the following equation to find the expected displacement along the span or chord given an applied moment [F(L-x)]:
$\ {\delta}(x)={{\int}_{0}^{L}}\left[{{\int}_{0}^{x}}\frac{F(L-x)}{EI(x)}\mathrm{d}x\right]\mathrm{d}x.$
A7
We used a simplex minimization program in Matlab that tests various values of the coefficients (for example, m and b in Equation A4) to find the specific EI distribution (for each equation type) that most successfully predicts wing displacement along the span or chord.
After estimating the spatial distribution of flexural stiffness, we calculated average flexural stiffness
$$EI$$
of the wing by integrating the continuous distribution EI(x):
$\ EI=\frac{1}{(x_{\mathrm{max}}-x_{0})}{{\int}_{x_{0}}^{x_{\mathrm{max}}}}EI(x)\mathrm{d}x,$
A8
where xo is the starting point (near the base or leading edge) and xmax is the end point of data collection.

List of symbols

• a

coefficient in exponential equation (A5)

•
• b

intercept in linear equation (A4)

•
• c

coefficient in exponential equation (A5)

•
• d constant

(Equation A3)

•
• E

material stiffness (Young's modulus)

•
• EI

flexural stiffness

•
• $$EI$$

average flexural stiffness

•
• f

flapping frequency

•
• F

force

•
• I

second moment of area

•
• L

beam length

•
• m

slope in linear equation (A4)

•
• M

moment

•
• p

coefficient in polynomial equation (A6)

•
• q

coefficient in polynomial equation (A6)

•
• r

coefficient in polynomial equation (A6)

•
• t

time

•
• x,y,z

spatial coordinates

•
• xo

starting point of data collection

•
• xmax

end point of data collection

•
• δ(x)

displacement at position

•
• x δT

displacement at tip or trailing edge

•
• δT/L

relative displacement of tip or trailing edge

•
• θ

rotation at wing base

•
• τ

time constant

•
• ω

circular frequency (2πf)

A. Trimble was critical in devising the method of measuring displacement by a laser ranging technique and in developing Matlab code to analyze loaded and unloaded images. M. Tu and E. Goldman generously provided useful comments and Matlab wisdom. J. Henry provided advice on collecting dragonflies and donated specimens to the study. J. Dierberger at MSC Software provided crucial troubleshooting of the FEM models, without which the dynamic simulations would not have been possible. This work was supported by NSF grant F094801 to T. Daniel, the John D. and Catherine T. MacArthur Foundation, an NSF graduate fellowship to S. Combes, and an ARCS fellowship to S. Combes.

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text:
# enigma / doc / context / third / enigma / examples / enigma-example-latex.tex

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 \documentclass[a5paper]{scrartcl} \parindent0pt %······································································% \usepackage {enigma} %%·····································································% %% The first machine will be used for encryption of our plain text. \defineenigma {encryption} \setupenigma {encryption} { other_chars = yes, day_key = B V III II 12 03 01 GI JV KZ WM PU QY AD CN ET FL, rotor_setting = ben, spacing = yes, verbose = 3, } %%·····································································% %% This second machine below will be used to decrypt the string. It is %% initialized with exactly the same settings as the first one. The %% reason for this is that we can’t reuse the “encryption” machine as it %% will already have progressed to some later state after the %% encryption. Applying it on the ciphertext would yield a totally %% different string. Hence the need for another machine. \defineenigma{decryption} \setupenigma{decryption}{ other_chars = yes, day_key = B V III II 12 03 01 GI JV KZ WM PU QY AD CN ET FL, rotor_setting = ben, } %%·····································································% \begin{document} %%·····································································% %% Ciphertext in the PDF. Rely on the addressee to decrypt the document %% externally. \startencryption Never underestimate the amount of money, time, and effort someone will expend to thwart a security system. \stopencryption %%·····································································% %% Input string generated with: %% mtxrun --script t-enigma \ %% --setup="day_key=B V III II 12 03 01 GI JV KZ WM PU QY AD CN ET FL,\ %% rotor_setting=ben,\ %% other_chars=yes"\ %% --text="Never underestimate the amount of money,\ %% time, and effort someone will expend to\ %% thwart a security system." \startdecryption aqsnwyxgqarpuzrdktscbslaqmdhyonywxkwhcdgphvuqsspfjwhlszahygjbzmfpcpbniahvfcuradntepxsfchnn \stopdecryption %%·····································································% \end{document}
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text:
# 6.7 Examples of Lost Work in Engineering Processes

1. Lost work in Adiabatic Throttling: Entropy and Stagnation Pressure Changes

A process we have encountered before is adiabatic throttling of a gas, by a valve or other device as shown in Figure 6.8. The velocity is denoted by . There is no shaft work and no heat transfer and the flow is steady. Under these conditions we can use the first law for a control volume (the Steady Flow Energy Equation) to make a statement about the conditions upstream and downstream of the valve:

where is the stagnation enthalpy, corresponding to a (possibly fictitious) state with zero velocity. The stagnation enthalpy is the same at stations 1 and 2 if , even if the flow processes are not reversible.

For a perfect gas with constant specific heats, and . The relation between the static and stagnation temperatures is:

where is the speed of sound and is the Mach number, . In deriving this result, use has only been made of the first law, the equation of state, the speed of sound, and the definition of the Mach number. Nothing has yet been specified about whether the process of stagnating the fluid is reversible or irreversible.

When we define the stagnation pressure, however, we do it with respect to isentropic deceleration to the zero velocity state. For an isentropic process

The relation between static and stagnation pressures is

The stagnation state is defined by , . In addition, . The static and stagnation states are shown in - coordinates in Figure 6.9.

Stagnation pressure is a key variable in propulsion and power systems. To see why, we examine the relation between stagnation pressure, stagnation temperature, and entropy. The form of the combined first and second law that uses enthalpy is

 (6..8)

This holds for small changes between any thermodynamic states and we can apply it to a situation in which we consider differences between stagnation states, say one state having properties and the other having properties (see Figure 6.10). The corresponding static states are also indicated. Because the entropy is the same at static and stagnation conditions, needs no subscript. Writing (6.8) in terms of stagnation conditions yields

Both sides of the above are perfect differentials and can be integrated as

For a process with , the stagnation enthalpy, and hence the stagnation temperature, is constant. In this situation, the stagnation pressure is related directly to the entropy as,

 (6..9)

Figure 6.11 shows this relation on a - diagram. We have seen that the entropy is related to the loss, or irreversibility. The stagnation pressure plays the role of an indicator of loss if the stagnation temperature is constant. The utility is that it is the stagnation pressure (and temperature) which are directly measured, not the entropy. The throttling process is a representation of flow through inlets, nozzles, stationary turbomachinery blades, and the use of stagnation pressure as a measure of loss is a practice that has widespread application.

Equation (6.9) can be put in several useful approximate forms. First, we note that for aerospace applications we are (hopefully!) concerned with low loss devices, so that the stagnation pressure change is small compared to the inlet level of stagnation pressure,

Expanding the logarithm (using ),

or

Another useful form is obtained by dividing both sides by and taking the limiting forms of the expression for stagnation pressure in the limit of low Mach number ( ). Doing this, we find:

The quantity on the right can be interpreted as the change in the Bernoulli constant'' for incompressible (low Mach number) flow. The quantity on the left is a non-dimensional entropy change parameter, with the term now representing the loss of mechanical energy associated with the change in stagnation pressure.

To summarize:

1. For many applications the stagnation temperature is constant and the change in stagnation pressure is a direct measure of the entropy increase.
2. Stagnation pressure is the quantity that is actually measured so that linking it to entropy (which is not measured) is useful.
3. We can regard the throttling process as a free expansion'' at constant temperature from the initial stagnation pressure to the final stagnation pressure. We thus know that, for the process, the work we need to do to bring the gas back to the initial state is , which is the lost work'' per unit mass.

Muddy Points

Why do we find stagnation enthalpy if the velocity never equals zero in the flow? (MP 6.13)

Why does remain constant for throttling? (MP 6.14)

2. Adiabatic Efficiency of a Propulsion System Component (Turbine)

A schematic of a turbine and the accompanying thermodynamic diagram are given in Figure 6.12. There is a pressure and temperature drop through the turbine and it produces work. There is no heat transfer so the expressions that describe the overall shaft work and the shaft work per unit mass are

 (6..10) (6..11)

If the difference in the kinetic energy at inlet and outlet can be neglected, Equation (6.11) reduces to

The adiabatic efficiency of the turbine is defined as

The performance of the turbine can be represented in an - plane (similar to a - plane for a perfect gas with constant specific heats) as shown in Figure 6.12. From the figure the adiabatic efficiency is

The adiabatic efficiency can therefore be written as

The non-dimensional term represents the departure from isentropic (reversible) processes and hence a loss. The quantity is the enthalpy difference for two states along a constant pressure line (see diagram). From the combined first and second laws, for a constant pressure process, small changes in enthalpy are related to the entropy change by , or approximately,

The adiabatic efficiency can thus be approximated as

The quantity represents a useful figure of merit for fluid machinery inefficiency due to irreversibility.

Muddy Points

How do you tell the difference between shaft work/power and flow work in a turbine, both conceptually and mathematically? (MP 6.15)

3. Isothermal Expansion with Friction

In a more general look at the isothermal expansion, we now drop the restriction to frictionless processes. As seen in Figure 6.13, work is done to overcome friction. If the kinetic energy of the piston is negligible, a balance of forces tells us that

During the expansion, the piston and the walls of the container will heat up because of the friction. The heat will be (eventually) transferred to the atmosphere; all frictional work ends up as heat transferred to the surrounding atmosphere.

The amount of heat transferred to the atmosphere due to the frictional work only is thus,

The entropy change of the atmosphere (considered as a heat reservoir) due to the frictional work is

The engine operates in a cycle and the entropy change for the complete cycle is zero (because entropy is a state variable). Therefore,

The total entropy change is

Suppose we had an ideal reversible engine working between these same two temperatures, which extracted the same amount of heat, , from the high temperature reservoir, and rejected heat of magnitude to the low temperature reservoir. The work done by this reversible engine is

For the reversible engine the total entropy change over a cycle is

Combining the expressions for work and for the entropy changes,

The entropy change for the irreversible cycle can therefore be written as

The difference in work that the two cycles produce is proportional to the entropy that is generated during the cycle:

The second law states that the total entropy generated is greater than zero for an irreversible process, so that the reversible work is greater than the actual work of the irreversible cycle.

An engine effectiveness,'' , can be defined as the ratio of the actual work obtained divided by the work that would have been delivered by a reversible engine operating between the two temperatures and :

The departure from a reversible process is directly reflected in the entropy change and the decrease in engine effectiveness.

Muddy Points

Why does ? (MP 6.17)

In discussing the terms closed system'' and isolated system,'' can you assume that you are discussing a cycle or not? (MP 6.18)

Does a cycle process have to have ? (MP 6.19)

In a real heat engine, with friction and losses, why is still 0 if ? (MP 6.20)

4. Propulsive Power and Entropy Flux

The final example in this section combines a number of ideas presented in this subject and in Unified in the development of a relation between entropy generation and power needed to propel a vehicle.

Figure 6.15 shows an aerodynamic shape (airfoil) moving through the atmosphere at a constant velocity. A coordinate system fixed to the vehicle has been adopted so that we see the airfoil as fixed and the air far away from the airfoil moving at a velocity . Streamlines of the flow have been sketched, as has the velocity distribution at station 0'' far upstream and station d'' far downstream.

The airfoil has a wake, which mixes with the surrounding air and grows in the downstream direction. The extent of the wake is also indicated. Because of the lower velocity in the wake the area between the stream surfaces is larger downstream than upstream.

We use a control volume description and take the control surface to be defined by the two stream surfaces and two planes at station 0 and station d. This is useful in simplifying the analysis because there is no flow across the stream surfaces. The area of the downstream plane control surface is broken into , which is the area outside the wake and , which is the area occupied by wake fluid, i.e., fluid that has suffered viscous losses. The control surface is also taken far enough away from the vehicle so that the static pressure can be considered uniform. For fluid which is not in the wake (no viscous forces), the momentum equation is

Uniform static pressure therefore implies uniform velocity, so that on the velocity is equal to the upstream value, . The downstream velocity profile is actually continuous, as indicated. It is approximated in the analysis as a step change to make the algebra a bit simpler. (The conclusions apply to the more general velocity profile as well and we would just need to use integrals over the wake instead of the algebraic expressions below.)

The equation expressing mass conservation for the control volume is

 (6..12)

The vertical face of the control surface is far downstream of the body. By this station, the wake fluid has had much time to mix and the velocity in the wake is close to the free stream value, . We can thus write,

 (6..13)

(We chose our control surface so the condition was upheld.)

The integral momentum equation (control volume form of the momentum equation) can be used to find the drag on the vehicle:

 (6..14)

There is no pressure contribution in Eq. (6.14) because the static pressure on the control surface is uniform. Using the form given for the wake velocity and expanding the terms in the momentum equation we obtain

 (6..15)

The last term in the right hand side of the momentum equation, , is small by virtue of the choice of control surface and we can neglect it. Doing this and grouping the terms on the right hand side of Eq. (6.15) in a different manner, we have

The terms in the square brackets on both sides of this equation are the continuity equation multiplied by . They thus sum to zero leaving the curly bracketed terms as

 (6..16)

The wake mass flow is . All this flow has a velocity defect (compared to the free stream) of , so that the defect in flux of momentum (the mass flow in the wake times the velocity defect) is, to first order in ,

The combined first and second law gives us a means of relating the entropy and velocity:

The pressure is uniform ( ) at the downstream station. There is no net shaft work or heat transfer to the wake so that the mass flux of stagnation enthalpy is constant. We can also approximate that the condition of constant stagnation enthalpy holds locally on all streamlines. Applying both of these to the combined first and second law yields

For the present situation, ; , so that

 (6..17)

In Equation (6.17) the upstream temperature is used because differences between wake quantities and upstream quantities are small at the downstream control station. The entropy can be related to the drag as

 (6..18)

The quantity is the entropy flux (mass flux times the entropy increase per unit mass; in the general case we would express this by an integral over the locally varying wake velocity and density). The power needed to propel the vehicle is the product of , . From Eq. (6.18), this can be related to the entropy flux in the wake to yield a compact expression for the propulsive power needed in terms of the wake entropy flux:

 (6..19)

This amount of work is dissipated per unit time in connection with sustaining the vehicle motion. Equation (6.19) is another demonstration of the relation between lost work and entropy generation, in this case manifested as power that needs to be supplied because of dissipation in the wake.

Muddy Points

Is it safe to say that entropy is the tendency for a system to go into disorder? (MP 6.21)

Douglas Quattrochi 2006-08-06


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In my quest to build an electric scooter from the ground up, the first step is to build a speed controller for the motor. Given that brushless DC motors (BLDC) are readily available in hobby shops, are high efficiency, and relatively inexpensive, the plan is to build a controller for a BLDC motor.

The brushless DC motor is a high-efficiency motor which uses permanent magnets on the rotor, and electromagnets on the stator to control the rotation of the rotor. The rotor and stator are only attached via bearings which support the rotor as it spins; the small contact area allows for the high efficiency of the motor.

In order to make the rotor spin in a desired direction, the electromagnets must be activated in a certain order to attract the magnets on the rotor to the electromagnets on the stator. This magnetic attraction is what causes the rotor to spin, and the sequence of steps is called the commutation sequence.

In the animation above. you can see that as the stators are energized in order, the rotor “chases” the magnetic fields. This is the basic commutation sequence for BLDC motors.

In order to maximize torque on the rotor, the field applied by the stator should be at an angle of 90 degrees to the orientation of the rotor. The relationship between torque $\tau$, magnetic moment $m$ of the rotor, and the field supplied by the stator $B$ is

The cross product is at is greatest when the vectors are at right angles to each other. The control algorithm used to try to maintain the 90 degree separation is called field-oriented control (FOC).

### PCB Design

Common battery packs for hobby electronics don’t usually exceed 60V, so the main parts I chose were:

• IRFS7530 MOSFETs for its 60V $V_{DS}$, low $R_{DS}$ for minimal power dissipation, and high continuous and pulsed current capability (240A and 1000A+, respectively)
• DRV8353RS triple half-bridge driver to drive the MOSFETs; it’s an all in one package that can be programmed by a microcontroller via SPI, has a built-in hardware-configurable buck converter that can convert the high battery voltage to a lower voltage to power other stuff like microprocessors, and has current sense amplifiers on all three phases
• STM32F407 microcontroller; it has plenty of compute power and peripherals, and a lot of community support. Documentation is all over the place but the chips are very good in my opinion (disclaimer, I am a STM shareholder).

There are six MOSFETs arranged in three half-bridges, one for each phase of the three-phase BLDC. Each half-bridge has a current sense resistor on the low-side. The max amperage of the IRFS7530 is insanely high, 240A, so it’s likely that temperature increase on the PCB, MOSFETs, or the sense resistors will be the limiting factor in current capacity. Besides possibly adding heatsinks to the MOSFETs and/or PCBs, it is essential that high-amperage traces on the PCB be wide enough to reduce resistance and heating, and all wires soldered to the motor/battery need to be thick enough to handle all the current.

To accomplish that, I put huge copper pours on the top/bottom layers and stitched them together with vias; together with the internal planes, both VDD and GND have three large traces to go through. I also removed thermal reliefs on all vias and pads on the MOSFETs and sense resistors which should help with reducing resistance, but made soldering those components difficult.

I got the boards made through OSHPark, and after soldering a minimal amount of components to a board and debugging it for a while, I finally got LEDs to blink.

More coming soon…

last updated June 29, 2020


text:
# If $A$ is symmetric, why is $x^TA^Ty=(x^TA^Ty)^T$ obvious?

The question has emerged when I read the proof of the following Theorem:

Let $$V$$ be a $$\mathbb{R}$$-vector-space, with $$\dim(V)=n< \infty, C=(c_{1},...,c_{n})$$ a Basis of $$V$$, and $$B$$ a bilinear form. Let $$A$$ be the Gramian matrix with $$a_{ij}=B(c_{i},c_{j})$$ and $$1\leq i,j\leq n$$ . Then the following is true: $$B$$ is symmetric $$\leftrightarrow A$$ is symmetric.

I'm only interessted in the $$\leftarrow$$Direction.

This is the part of the proof that I don't understand completely: $$x,y\in \mathbb{R}^n$$. Assume that $$A$$ is symmetric, then $$A=A^T$$. Then $$x^TAy=x^TA^Ty\overset{?}{=}(x^TA^Ty)^T=y^TAx$$=... the rest of the proof is not important for my question.

I tried to proof the equality $$x^TA^Ty=(x^TA^Ty)^T$$:

$$d:= x^TA^Ty \quad d= x^TA^Ty=x^TAy=(\sum \limits_{i=1}^{n}x_{i}^T a_{i1},...,\sum \limits_{i=1}^{n}x_{i}^T a_{in})\cdot y \\ =\sum \limits_{j=1}^{n}y_{j}\sum \limits_{i=1}^{n}x_{i}^T a_{ij}\overset{y_{j}=y_{j}^T,...}{=} \sum \limits_{j=1}^{n}y_{j}^T\sum \limits_{i=1}^{n}a_{ji}^T x_{i} =\sum \limits_{j=1}^{n}\sum \limits_{i=1}^{n}y_{j}^T a_{ji} x_{i} = \sum \limits_{i=1}^{n}\sum \limits_{j=1}^{n}y_{j}^T a_{ji} x_{i}= \sum \limits_{i=1}^{n}x_{i}\sum \limits_{j=1}^{n}y_{j}^T a_{ji} =(\sum \limits_{j=1}^{n}y_{j}^T a_{j1},...,\sum \limits_{j=1}^{n}y_{j}^T a_{jn})\cdot x = y^TA^Tx=y^TAx=(x^TA^Ty)^T$$

But since they just wrote $$x^TA^Ty=(x^TA^Ty)^T$$in the proof like it would be obvious, I guess there must be a much shorter explanation that makes my proof unnecessary, do you have an idea of such an explanation?

PS: I'm not used to write about math in english, please ask if something doesn't makes sense to you.

• What is the transpose of a $1 \times 1$ matrix? – Rodrigo de Azevedo Mar 8 at 15:56
• $x^TA^Ty$ is just an element of $\Bbb R$... – TonyK Mar 8 at 15:56
• Any 1x1 matrix is it's own transpose. – mathreadler Mar 8 at 16:01

Just note that $$x^T A y$$ is a scalar, or a $$1 \times 1$$ matrix if you will.


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# In the figure shown, what is the value of x?

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In the figure shown, what is the value of x?

(1) The length of line segment QR is equal to the length of line segment RS

(2) The length of line segment ST is equal to the length of line segment TU

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/in-the-figur ... 25923.html
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Re: In the figure shown, what is the value of x? [#permalink]

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Thanks and kudos!
I've seen a similar problem before, but forgot...
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Re: In the figure shown, what is the value of x? [#permalink]

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02 Sep 2009, 06:18
Intially I couldn't see how you got angle 1 + angle 6 to be 90.....but I got to open my eyes to see the bigger picture. Good explanation.
Definitely agree with C.
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Re: In the figure shown, what is the value of x? [#permalink]

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04 Sep 2009, 10:08
IMO C..too...gr8 effort ...clicking the image on web cam...
n thx...i m taking angles as RQS.....RSQ....taking 1,2,3, makes problem more easy..
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Re: In the figure shown, what is the value of x? [#permalink]

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19 Mar 2011, 03:16
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it is C:
(1) <R + <T = 90 in which:

(2) <R = 180 - <RQS - <RSQ = 180 - 2<RSQ (as RQ = RS)
(3) <T = 180 - <UST - <SUT = 180 - 2<UST ( as ST = TU)

(1) + (2) + (3):
180 - 2<RSQ + 180 - 2<UST = 90
=>360 - 2(<RSQ + <UST) = 90
=> 360 - 2 (180 -x) = 90
=> x= 45
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Re: In the figure shown, what is the value of x? [#permalink]

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13 Jun 2017, 23:12

In the figure shown, what is the value of x?

$$x+\angle{QSR}+\angle{UST}=180$$ (straight line =180) and $$\angle{R}+\angle{T}=90$$ (as PRT is right angle)

(1) The length of line segment QR is equal to the length of line segment RS --> triangle QRS is isosceles --> $$\angle{RQS}=\angle{QSR}=\frac{180-\angle{R}}{2}$$ (as $$\angle{RQS}+\angle{QSR}+\angle{R}=180$$ --> $$2*\angle{QSR}+\angle{R}=180$$ --> $$\angle{QSR}=\frac{180-\angle{R}}{2}$$). Not sufficient.

(2) The length of line segment ST is equal to the length of line segment TU --> triangle UST is isosceles --> $$\angle{SUT}=\angle{UST}=\frac{180-\angle{T}}{2}$$. Not sufficient.

(1)+(2) $$x+\angle{QSR}+\angle{UST}=180$$ --> $$x+\frac{180-\angle{R}}{2}+\frac{180-\angle{T}}{2}=180$$ --> $$x+\frac{360-(\angle{R}+\angle{T})}{2}=180$$ --> since $$\angle{R}+\angle{T}=90$$ --> $$x+\frac{360-90}{2}=180$$ --> $$x=45$$. Sufficient.

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### Home > PC > Chapter 6 > Lesson 6.3.2 > Problem6-128

6-128.

The shadow of Kate is cast on a wall by a light on the ground, which is $d$ meters from the wall. Kate is $162$ centimeters tall and is 3 meters from the wall. Express $d$ (in meters) as a function of the height of the shadow, $h$ (in meters).

1. Label the drawing.
2. Convert Kate's height to meters.
3. You have two similar triangles. Write a proportion with $d$ and $h$.
4. Solve for $d$.

$\frac{d-3}{d}=\frac{1.62}{h}$


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at1p {CreditRisk} R Documentation

## Analytically - Tractable First Passage (AT1P) model

### Description

at1p calculates the survival probability Q(τ > t) and default intensity for each maturity according to the structural Analytically - Tractable First Passage model.

### Usage

at1p(V0, H0, B, sigma, r, t)

### Arguments

 V0 firm value at time t = 0 (it is a constant value). H0 value of the safety level at time t = 0. B free positive parameter used for shaping the barrier Ht. sigma a vector of constant stepwise volatility σ_t. r a vector of constant stepwise risk-free rate. t a vector of debt maturity structure (it is a numeric vector).

### Details

In this function the safety level Ht is calculated using the formula:

H(t) = \frac{H0}{V0} * E_0[V_t] * \exp^{- B \int_0^t σ_u du}

The backbone of the default barrier at t is a proportion, controlled by the parameter H0, of the expected value of the company assets at t. H0 may depend on the level of the liabilities, on safety covenants, and in general on the characteristics of the capital structure of the company. Also, depending on the value of the parameter B, it is possible that this backbone is modified by accounting for the volatility of the company assets. For example, if B > 0 corresponds to the interpretation that when volatility increases - which can be independed of credit quality - the barrier is slightly lowered to cut some more slack to the company before going bankrupt. When B = 0 the barrier does not depend on the volatility and the "distance to default" is simply modelled through the barrier parameter H0.

### Value

at1p returns an object of class data.frame containing the firm value, safety level H(t) and the survival probability for each maturity. The last column is the default intensity calculated among each interval Δ t.

### References

Damiano Brigo, Massimo Morini, Andrea Pallavicini (2013) Counterparty Credit Risk, Collateral and Funding. With Pricing Cases for All Asset Classes.

### Examples

mod <- at1p(V0 = 1, H0 = 0.7, B = 0.4, sigma = rep(0.1, 10), r = cdsdata$ED.Zero.Curve, t = cdsdata$Maturity)
mod

plot(cdsdata$Maturity, mod$Ht, type = 'b', xlab = 'Maturity', ylab = 'Safety Level H(t)',
main = 'Safety level for different maturities', ylim = c(min(mod$Ht), 1.5), col = 'red') lines(cdsdata$Maturity, mod$Vt, xlab = 'Maturity', ylab = 'V(t)', main = 'Value of the Firm \n at time t', type = 's') plot(cdsdata$Maturity, mod$Survival, type = 'b', main = 'Survival Probability for different Maturity \n (AT1P model)', xlab = 'Maturity', ylab = 'Survival Probability') matplot(cdsdata$Maturity, mod\$Default.Intensity, type = 'l', xlab = 'Maturity',
ylab = 'Default Intensity')

[Package CreditRisk version 0.1.3 Index]


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# Will a random walk on [0, inf) tend to infinity? [closed]

Consider a random walk on [0, inf) where you start at 0. With probability p = 0.5, you increase by 1. With probability (1-p) = 0.5, you decrease by 1, but not below 0.

As time goes to infinity, will your position tend to infinity? If not, to what finite value does it converge?

Edit: To be a bit more precise, what is the limit of the average position as time goes to infinity?

If your position tends to infinity for p = 0.5, for which other probabilities p is this true? (Clearly p > 0.5 will cause you to tend to infinity, so p < 0.5 is what I'm after)

What is the probability of being at position x after an arbitrary amount of steps?

I made a simple simulation to test the p = 0.5 case, and after 500 million iterations, it seems to tend to infinity, but I'd like a more solid explanation.

Thanks!

-

## closed as too localized by Yemon Choi, fedja, Peter Shor, Nate Eldredge, Pete L. ClarkApr 3 '11 at 15:12

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

Random walks with barrier conditions ought to be covered in probability textbooks. My memory is hazy here, but have you tried Feller Volume I, or maybe even Grimmett and Stirzaker's book. –  Yemon Choi Apr 3 '11 at 8:54
Maybe you should ask this on math.stackexchange.com; there are plenty of people on this website who can give you good answers, but they all seem to be ignoring your question because it's too elementary to belong on this website, and so it turns out that you're not getting that many good answers. To answer one of your questions, $$\lim_{n\rightarrow \infty} \mathrm{Pr}(\mathrm{at\ position\ }x) \cdot \sqrt{n}\rightarrow c$$ for some constant $c$, where $n$ is the number of steps taken. –  Peter Shor Apr 3 '11 at 13:17

The symmetric random walk $(X_k)$ on $\mathbb{Z}$ is recurrent. Therefore, with probability one, you will visit infinitely often $0$. The same is true for $(|X_k|)$, which is more ore less your random walk (the more or less depends on what happens exactly at the origin).

In other words, with probability one, the $\liminf$ is $0$. The $\limsup$ behaviour is the subject of the law of the iterated logarithm.

-

As it has been said:

1. if $p=\frac{1}{2}$ this is more or less the same thing as the absolute value of the standard symmetric random walk on $\mathbb{Z}$, wich is recurrent.
2. if $p > \frac{1}{2}$, the law of large numbers immediately shows you that it tends to $+\infty$.
3. if $p < \frac{1}{2}$, you can even check the detailed balance equations and find the invariant distribution: the Markov chain is positive recurrent.
-

Let me try again. Let $R_n$ be the random variable denoting the longest contiguous run of heads for $n$ independent $p$-biased coin tosses.

It is well-known 1 that $E R_n\sim\log_{1/p}((1-p)n)$ plus small correction terms (the variance is $O(1)$).

This means that $S_n=\sum_{i=1}^n X_i$ grows at least as $\log n$ for any fixed $p$, and proves almost-sure escape to $\infty$.

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Why "This means that $S_n$"... ? –  camomille Apr 3 '11 at 11:33
With high probability, after $n$ steps you'll see $O(\log n)$ contiguous heads, which will advance you $O(\log n)$ steps to the right. –  Aryeh Kontorovich Apr 3 '11 at 11:45
Maybe I do not understand what you mean by "escape to \infty". If you mean that the walker goes arbitrarely far the the right with probability one this is obvious. If you mean that the walker tends to infinity with probability one this is false (see my answer below). Actually, many thinks are known on such a walk, this is really very standard (and maybe this should not be discussed here actually). –  camomille Apr 3 '11 at 11:49
I wasn't being precise. The random walk visits 0 and every other natural number infinitely many times, as has been said here. –  Aryeh Kontorovich Apr 3 '11 at 12:22
All right, but your argument is a sophisticated way to prove that the walker can reach any point (this is just irreducibility). –  camomille Apr 3 '11 at 12:41

For any $p>0$ and any finite $N$, with probability one, eventually your random walk will exceed $N$.

-
True, but that's not quite what I'm after. –  wjomlex Apr 3 '11 at 8:57
But that's not the same as saying one "almost surely tends to +infinity" ... –  Yemon Choi Apr 3 '11 at 8:57
I'll elaborate. You are flipping a biased coin, with probability $p$ of heads. With probability 1, you will at some point observe a sequence of $p^N$ consecutive heads, which will necessarily take you to the right of $N$. –  Aryeh Kontorovich Apr 3 '11 at 8:59
We understand what you mean, it's just that you're answering a different question :) –  wjomlex Apr 3 '11 at 9:00
True, my answer does not imply almost-sure escape to infinity. Must think some more... –  Aryeh Kontorovich Apr 3 '11 at 9:00


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# Where is the world map in MMA from? How to cite it?

GROUPS:
 I use GeoListPlot[] function to generate a beautiful map and write a paper. I cite the link http://www.wolfram.com/mathematica/ in my paper, but the editor ask me to cite in the figure legend properly, according to Wolfram Mathematica's tems of use (http://www.wolfram.com/legal/agreements/wolfram-mathematica.html). Please where is the world map from? Thank you.
 Gianluca Gorni 1 Vote Try something like this: List @@ GeoGraphics[] You will see MetaInformation -> {"Software" -> "Wolfram Language", "TileSources" -> "Wolfram Knowledgebase with data from OpenStreetMap"} Is this what you need?


text:
## anonymous 3 years ago Solve.

1. anonymous

$0 \le \theta <360$ $\sec \theta= \csc \theta$

2. anonymous

$0≤θ<360$ $\theta<360$ 360 integer solutions

3. anonymous

Not helping :/

4. anonymous

45 degrees and 225 degrees sec theta = 1/costheta csc theta = 1/sintheta so 1/costheta = 1/sintheta on the interval 0=<theta<360 because 45 degrees gives sqrt2/2 for both sin and cos and 225 gives both -sqrt2/2 for sin and cos, these are the only two values that satisfy the equation


text:
# zbMATH — the first resource for mathematics

The limiting behavior of sequences of quasiconformal mappings. (English) Zbl 0699.30016
The limiting behaviour of sequences of K-quasiconformal homeomorphisms of the n-sphere $$S^ n$$ is studied using a substitute to the Poincaré extension of Möbius transformations introduced by Tukia. Adapted versions of the limit set and the conical limit set known in the theory of Kleinian groups are shown to play a crucial role. Most of the results hold for families of homeomorphisms of $$S^ n$$ which have the convergence property inroduced by Gehring and Martin.
Reviewer: B.Aebischer

##### MSC:
 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 40A99 Convergence and divergence of infinite limiting processes
Kleinian groups
Full Text:


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# Transfer Function

The transfer function H(ω) (also called the network function) is a useful analytical tool for finding the frequency response of a circuit. In fact, the frequency response of a circuit is the plot of the circuit’s transfer function H(ω) versus ω, with ω varying from ω = 0 to ω = ∞.
A transfer function is the frequency-dependent ratio of a forced function to a forcing function (or of an output to an input). The idea of a transfer function was implicit when we used the concepts of impedance and admittance to relate voltage and current. In general, a linear network Input Output can be represented by the block diagram shown in Fig. 1.
Fig. 1: A block diagram representation of a linear network.
The transfer function H(ω) of a circuit is the frequency-dependent ratio of a phasor output Y(ω) (an element voltage or current) to a phasor input X(ω) (source voltage or current).
Thus,
$$H(ω) ={Y(ω) \over X(ω)}$$
assuming zero initial conditions. Since the input and output can be either voltage or current at any place in the circuit, there are four possible transfer functions:
$$\bbox[10px,border:1px solid grey]{\begin{split} H(ω) &= \text{Voltage gain} ={V_o(ω) \over V_i(ω)} \\ H(ω) &= \text{ Current gain} ={I_o(ω) \over I_i(ω)}\\ H(ω) &= \text{Transfer Impedance} ={V_o(ω) \over I_i(ω)}\\ H(ω) &= \text{Transfer Admittance} ={I_o(ω) \over V_i(ω)} \end{split}} \tag{1}$$
where subscripts i and o denote input and output values. Being a complex quantity, H(ω) has a magnitude H (ω) and a phase φ; that is, $H(ω) = H (ω) \angle φ$.
To obtain the transfer function using Eq. (1), we first obtain the frequency-domain equivalent of the circuit by replacing resistors, inductors, and capacitors with their impedances $R$, $jωL$, and $1/jωC$. We then use any circuit technique(s) to obtain the appropriate quantity in Eq. (1).
We can obtain the frequency response of the circuit by plotting the magnitude and phase of the transfer function as the frequency varies.
Some authors use H( jω) for transfer instead of H(ω), since ω and j are an inseparable pair. The transfer function H(ω) can be expressed in terms of its numerator polynomial N(ω) and denominator polynomial D(ω) as
$$\bbox[10px,border:1px solid grey]{H(ω) ={N(ω) \over D(ω)}} \tag{2}$$
where N(ω) and D(ω) are not necessarily the same expressions for the input and output functions, respectively. The representation of H(ω) in Eq. (2) assumes that common numerator and denominator factors in H(ω) have canceled, reducing the ratio to lowest terms. The roots of N(ω) = 0 are called the zeros of H(ω) and are usually represented as jω = z1, z2, . . . . Similarly, the roots of D(ω) = 0 are the poles of H(ω) and are represented as jω = p1, p2, . . . .
A zero, as a root of the numerator polynomial, is a value that results in a zero value of the function. A pole, as a root of the denominator polynomial, is a value for which the function is infinite.
To avoid complex algebra, it is expedient to replace jω temporarily with s when working with $H(ω)$ and replace s with $jω$ at the end.
Example 1: For the RC circuit in Fig. 2(a), obtain the transfer function $V_o/V_s$ and its frequency response. Let $$v_s = V_m cos ωt$$
Fig. 2: For Example 1: (a) time-domain RC circuit, (b) frequency-domain RC circuit.
Solution: The frequency-domain equivalent of the circuit is in Fig. 2(b). By voltage division, the transfer function is given by $$\mathbf{H}(\omega)=\frac{\mathbf{V}_{o}}{\mathbf{V}_{s}}=\frac{1 / j \omega C}{R+1 / j \omega C}=\frac{1}{1+j \omega R C}$$ we obtain the magnitude and phase of $\mathbf{H}(\omega)$ as $$H=\frac{1}{\sqrt{1+\left(\omega / \omega_{0}\right)^{2}}}$$ and $$\phi=- tan ^{-1} \frac{\omega}{\omega_{0}}$$ where $\omega_{0}=1 / R C$.
To plot $H$ and $\phi$ for $0<\omega<\infty$, we obtain their values at some critical points and then sketch.
At $\omega=0, H=1$ and $\phi=0$.
At $\omega=\infty, H=0$ and $\phi=-90^{\circ}$.
Also, at $\omega=\omega_{0}, H=1 / \sqrt{2}$ and $\phi=-45^{\circ}$.
With these and a few more points as shown in Table 1, we find that the frequency response is as shown in Fig. 3.
Fig. 3: Frequency response of the RC circuit: (a) amplitude response, (b) phase response.
Table 1.


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# Irreducible algebraic curves

Is it true that two distinct irreducible algebraic curves intersect only at points?

I have two irreducible bivariate polynomials and would like to say something about the geometry of their shared zeros.

-

Sure, this is a consequence of Bezout's theorem which also tells you how many points there are if you count them carefully enough to account for complex roots, multiplicities and points at infinity. The theorem requires that the curves don't share a component, but this is guaranteed by their being irreducible (and distinct).

-


text:
A difficulty in understanding a statement in Hungerford algebra.

While explaining that "In the category of sets the Cartesian product $\prod_{i \in I} A_{i}$ is a product of the family $\{A_{i}: i \in I\}$."

After showing this the author gave the following remark :

But I did not understand why if some $A_{j} = \emptyset,$ then the whole $\prod_{i \in I} A_{i} = \emptyset$ and why there can be no function that satisfies what given in the picture, is not there a mistake in the index j? it must be i instead? could anyone explain this for me please?

$j$ is a particular index for which $A_j$ is the empty set.
Consider an element $x$ in the product. $x$ has a component $x_i$ for each $i\in I$. When it comes to the $j^{th}$ component $x_j$, what can it be? There aren't any options in $A_j$, which is empty, so $x$ can't exist and the product must be empty too.
• $x_{j}$ can not be zero in this case? – user426277 Nov 24 '17 at 14:37


text:
23 views

The minimum number of $2$-input $NAND$ gates required to implement a $2$-input $XOR$ gate is

1. $4$
2. $5$
3. $6$
4. $7$

retagged | 23 views


text:
Example 29.38.5. Let $k$ be an infinite field. Let $\lambda _1, \lambda _2, \lambda _3, \ldots$ be pairwise distinct elements of $k^*$. (This is not strictly necessary, and in fact the example works perfectly well even if all $\lambda _ i$ are equal to $1$.) Consider the graded $k$-algebra

$A_ d = k[U, V, Z]/I_ d \quad \text{with} \quad I_ d = (Z^2 - \prod \nolimits _{i = 1}^{2d} (U - \lambda _ i V)).$

with grading given by $\deg (U) = \deg (V) = 1$ and $\deg (Z) = d$. Then $X_ d = \text{Proj}(A_ d)$ has ample invertible sheaf $\mathcal{O}_{X_ d}(1)$. We claim that if $\mathcal{O}_{X_ d}(n)$ is very ample, then $n \geq d$. The reason for this is that $Z$ has degree $d$, and hence $\Gamma (X_ d, \mathcal{O}_{X_ d}(n)) = k[U, V]_ n$ for $n < d$. Details omitted.

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text:
# Implementing command pattern in Python

I did some study on the command pattern but most of its examples were in Java so, there must be some difference in implementation in Python. I implemented it in Python with some minor differences, please let me know if something is not correct.

from abc import ABCMeta
from abc import abstractmethod
import inspect
import os

class Command(object):
"""
Abstract / Interface base class for commands.
"""
__metaclass__ = ABCMeta

@abstractmethod
def execute(self):
pass

@abstractmethod
def undo(self):
pass

class CreateCommand(Command):
"""
Create command implementation.
"""
def __init__(self, name):
self.file_name = name

def execute(self, name):
open(self.file_name, 'w')
print str(self) + ':::Method:::' + inspect.stack()[0][3]

def undo(self):
os.remove(self.file_name)
print str(self) + ':::Method:::' + inspect.stack()[0][3]

class MoveCommand(Command):
"""
Move command implementation.
"""
def __init__(self, src, dest):
self.src = src
self.dest = dest

def execute(self, src, dest):
os.rename(self.src, self.dest)
print str(self) + ':::Method:::' + inspect.stack()[0][3]

def undo(self):
os.rename(self.dest, self.src)
print str(self) + ':::Method:::' + inspect.stack()[0][3]

class Invoker(object):

def __init__(self, command):
self.command = command

def do(self):
self.command.execute()

def undo(self):
self.command.undo()

# Client for the command pattern
if __name__ == '__main__':
create_cmd = CreateCommand('/tmp/foo.txt')
move_cmd = MoveCommand('/tmp/foo.txt', '/tmp/bar.txt')
create_invoker = Invoker(create_cmd)
move_invoker = Invoker(move_cmd)
create_invoker.do()
move_invoker.do()
move_invoker.undo()
create_invoker.undo()

O/P:

<__main__.CreateCommand object at 0xb705130c>:::Method:::execute
<__main__.MoveCommand object at 0xb705134c>:::Method:::execute
<__main__.MoveCommand object at 0xb705134c>:::Method:::undo
<__main__.CreateCommand object at 0xb705130c>:::Method:::undo

The important thing to know about commands in Python (or any language with first-class functions) is that they're usually trivial. Commands with only one operation (execute) are simply functions. There's no need to define classes. Even commands with two operations can be represented as pairs of functions: (do, undo). The whole, heavyweight pattern is almost never used.

• Command parameters such as filenames should be arguments to the constructor, not to execute. The code that calls execute doesn't know what these values should be, so they need to already be in the command.
• Invoker.execute manually dispatches by command name. Instead it should blindly call execute or undo, and let method dispatch find the appropriate method.
• The Invoker class does nothing useful.
• Why bother getting the method name from inspect.stack when the method already knows its own name?
• I might call execute do, for symmetry with undo.
• ABCMeta is unnecessary complexity. If you must use it, it's clearer to write class Command(metaclass=ABCMeta) rather than assigning to __metaclass__.
• "how it will find the respective command?": Commands should be represented as Commands, not as strings. Do something like MoveCommand('foo', 'bar').execute(), not invoke('move', 'foo', 'bar'), nor invoker.do(). The Invoker class does nothing; it should be removed. – Anonymous May 18 '14 at 4:49
• @vivekpoddar One common use for this pattern is to implement undo/redo. For redo to work, the UI that manages the queue must be able to call execute without passing any information. – David Harkness May 18 '14 at 7:00