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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 Supplementary Material for “Stable hovering of a jellyfish-like flying machine” We present an aerodynamic force model that allows us to estimate the lift, flapping frequency, and orientational stability of a four-winged hovering ornithopter. We show that the values of the drag and lift coefficients in the model can be uniquely determined from experimental measurements of the motor properties, flapping frequency, and lift generated. The experimentally-inferred force coefficients are then used to inform a model which reveals the values of the body moment of inertia and center-of-mass location needed for upright stability. These stability criteria are found to be invariant under isometric scaling of the flyer. Calculation of lift. Consider a single rigid wing of rectangular planform hinged at the top and driven to oscillate back-and-forth through small amplitude. The wing is of length in span (hinge-to-tip) and in chord, and the total wing area is . For a chord-wise blade element located a distance down from the hinge, the quasi-steady lift is given by . For simplicity, we assume that the wing moves at constant speed during each half-stroke, so that where is the tip speed. Thus, the lift on a single wing is . For our four-winged ornithopter, the total lift must balance body weight during hovering: . To assess whether such a model is promising for our ornithopter, we can estimate the value of from the known quantities in the above equation. Here, we use the root-mean-square (rms) value for the wing tip velocity during hovering: with frequency Hz (Fig. 3B in main text) and nominal flapping amplitude . Further, using cm, cm, g, and , we find , a reasonable (order 1) value for the lift coefficient. We note that this formulation is similar in spirit to the aerodynamic models employed in studies of insect flight, such as those in references 7, 8, 20 and 21 from the main text. Such models estimate the instantaneous forces on a flapping wing by the average forces felt by a wing moving at steady speed and angle of attack. Applying a strict quasi-steady approximation to a hinged rigid wing – which always has an instantaneous attack angle of – would predict zero lift generated. Thus, the nonzero lift coefficient in our model fit to the data must ultimately reflect unsteady effects, such as vortex shedding, present in the experiments. Secondly, such models assume independence of the blade elements, an approximation best suited to high aspect ratio wings. The wings of our ornithopter, like those of insects, have rather low aspect ratio ( ). Nonetheless, lift-drag models have been informative in predicting forces and flight dynamics of insects and thus also serve as a basis for understanding the flight of our ornithopter. Balance of motor and aero-inertial torques. We next consider how the flapping frequency results from a balance of the motor torque with the aerodynamic and inertial torques on the wings. Instantaneously, the equation of motion for a single wing is . 1 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 The force provided by the motor must overcome both the aerodynamic drag and the inertial force associated with acceleration of the wing and the added mass of the surrounding air. Similar to the lift calculation, the rms drag force on 4 wings is given by with . To estimate the inertial force, we assume that once the motor accelerates the wing (and air) during some portion of the stroke, it is unable to recover this inertia at a later portion. Thus, by averaging the absolute value of the acceleration , we estimate that the inertial force for 4 wings is . Here, the factor of comes from a spatial average along the span of the wing, and the factor of comes from the time average of the absolute acceleration. The motor torque is equal to the sum of the drag and inertial forces times the crankshaft arm length : . In addition, we know that the output of the motor is given by the torque-speed curves given in Fig. 3A of the main text: with and . Here, at any given voltage , the torque is approximated as linearly decreasing in the frequency. Further, the stall torque and zero-load frequency are taken as linearly increasing in the voltage, with slopes estimated from the motor data taken at . Torque balance requires that , which provides a quadratic equation for the flapping frequency : Parameter values include the crankshaft arm length of , wing mass , and added mass of air . The positive solution to this equation is shown as the dashed red line in Fig. 3B of the main text, where a best-fit value of is determined by matching the measured value of at . Frequency and wing size leading to optimal lift for a given motor. To explore the effect of different wing sizes driven by a given motor, consider isometric scaling of the wing by a scale factor of . That is, the relevant lengths of , , and are scaled as and the mass as . Then, the torque balance equation provides a relationship between and : 74 75 The lift generated is given by 76 77 78 79 80 81 82 where we have substituted in the expression for to get an equation in . Interestingly, this analysis shows that lift as a function of frequency shows a peak at a frequency that is one-third the zero-load speed of the motor, . For our ornithopter operating near hovering (5.5 V), , so this formulation recommends a target frequency of , to be compared to our value of . The wing size that corresponds to this frequency is or , and this optimal lift is predicted be 1.24 times the lift of the present prototype. 2 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 Linear model of longitudinal flight dynamics. Motivated by the experimental observation that stability seems to result from a coupling of translation and rotation of the ornithopter, we consider a model of the longitudinal flight dynamics. The linearized relationship between the variables of horizontal speed, tilt, and tilt rate can be written as: . The first and third equations are the Newton-Euler equations for the forces and torques that result from perturbations in the dynamical variables . Note that the third equation contains no term proportional to , reflecting the fact that no torques results from a pure tilt of the body since both the weight and lift vectors project through the center-of-mass (COM). Also, the above system does not include the vertical velocity, since by symmetry a pure motion upward leads to no sideways force and no torque. 115 Calculating coefficients in the model. In general, each of the coefficients in the above system can be calculated by considering the force (or torque) that results from a change in only one variable at a time. For example, the term results from the sideways force for a pure tilt . In this case, the lift vector will have a horizontal component, so that and thus , where we have used a small angle approximation and the hovering condition . The other coefficients result from the modification of drag due to body translation or rotation. Recall that the net drag is zero during hovering, cancelling on the two half-strokes. Once the body is set in motion, however, each wing will have a higher airspeed on one half-stroke than the other, resulting in a net drag. Consider the case of an upright wing undergoing translation, corresponding to the term in the dynamical system. For a single wing, the time averaged drag differential on a blade element is given by . Thus, the drag on each portion of the wing is proportional to the product of the imposed flapping speed and the translational speed of the body. Integrating along the wing, we arrive at the drag on a single wing: . When our ornithopter moves to the side, 2 wings contribute to drag, yielding: 116 117 Here, we again make use of the condition 118 119 120 121 122 123 Thus, the coefficient can readily be estimated from the known parameters and as well as the values of and that are determined above. A similar procedure can be used to calculate the other coefficients in the model. The term, for example, captures the torque that results from a pure translation. In this case, the COM location , measured upward from the wing base, is an important parameter. Integrating the drag-based torques on 2 wings yields: 110 111 112 113 114 to simplify the expression. 3 124 125 126 127 Here, the body moment of inertia about the COM is also important, and we have again used the hovering condition to simplify the expression. Similarly, we calculate the final two coefficients to be: 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 Physically, these coefficients correspond to the force and torque due to rotation of the body. Stability criteria. Stability of the longitudinal dynamics about the hovering state can be formally evaluated by considering the eigenvalues of the system. For this analysis, it is convenient to eliminate and to arrive at a third-order differential equation for the tilt : An eigenvalue analysis considers solutions of the form This leads to a polynomial equation for : where λ is a complex eigenvalue. If all roots of this equation have negative real part, then the system exhibits solutions that decay in time and is thus stable. For this third-order system, the Routh-Hurwitz criteria specify four conditions that all must be met for the system to be stable: , , , and . Thus, these conditions determine a region of stability for different values of the moment of inertia and COM location. In Fig. 6C of the text, this region is shown in gray in the space of the dimensionless quantities . Of the above conditions, we find that the first two are not physically relevant, as they simply require that the moment of inertia and COM height be real and positive. The third condition demands that the COM be no higher than one-third up the wing, which corresponds to the upper stability boundary in Fig. 6C. The final condition leads to the lower stability boundary shown in Fig. 6C. Scaling of stability properties. Consider isometric scaling of the ornithopter, with all lengths scaling as , mass as , and moment of inertia as . The ratio is assumed to be unchanged under scaling. We use the hovering condition to arrive at a scaling for frequency and wing speed: and thus and In general, the coefficients in the stability model depend on and and thus change under this isometric scaling. For example, the coefficient . However, the associated stability boundary (the upper boundary in Fig. 6C) is invariant, always demanding that regardless of scale. Likewise, for the second relevant Routh-Hurwitz condition, , both sides scale as and this factor can be algebraically 4 162 163 eliminated from the inequality. Thus, the lower stability boundary of Fig. 6C is also invariant under isometric scaling. 5