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