Flutter-Mill: a New Energy-Harvesting Device

Flutter-Mill: a New Energy-Harvesting Device
Liaosha Tang 1,∗ , Michael P. Païdoussis 2 and James D. DeLaurier 1
1 University of Toronto Institute for Aerospace Studies,
Toronto, Ontario, M3H 5T6, Canada
2 Department of Mechanical Engineering, McGill University,
Montréal, Québec, H3A 2K6, Canada
This paper introduces the concept of a new energy-harvesting device, i.e., the
flutter-mill. It has been found that cantilevered flexible plates subjected to axial flow
lose stability by flutter at sufficiently high flow velocity. The flow-induced vibrations of
this kind of fluid-structure interaction system can be utilized to extract energy from the
surrounding fluid flow for generation of electric power. In this paper, the dynamics of
two-dimensional cantilevered flexible plates in axial flow is summarized, with special
attention given to the energy transfer between the plate and the flow. Based on the
theoretical work, key design parameters of the proposed flutter-mill can be determined
and the power-extraction capacity is preliminarily evaluated.
1. Introduction
Cantilevered flexible plates in axial flow lose stability dynamically at sufficiently high
flow velocity; flutter occurs beyond the critical point. An everyday example of this
phenomenon is the waving motions of a flag in the wind. The dynamics of this
fluid-structure interaction system has recently been reviewed by Tang and Païdoussis
[1] in a systematic manner. Obviously, when the flow velocity exceeds the critical point,
energy is continuously pumped into the plate from the surrounding fluid flow,
sustaining the flutter motion; therefore, this kind of self-induced vibrations can be
utilized to extract energy from the fluid flow to do useful work, particularly to generate
electric power.
To utilize flow-induced vibrations for the generation of electric power is not a new
idea. The concept of an oscillating-wing windmill (the so-called wingmill) was first
proposed by Adamko and DeLaurier [2] and Mckinney and DeLaurier [3] to utilize the
flutter motion of a wing subjected to air flow to drive an electrical generator. When a
wingmill is properly designed so as to achieve a phase difference between the
plunging and pitching motions, Ly and Chasteau [4] have demonstrated that the
efficiency is equivalent to that of the vertical axis wind turbine (Darrieus-type) [5]. The
wingmill concept still receives extensive attention [6,7], and a patent [8] was recently
granted.
∗ Corresponding author, post-doctoral fellow, e-mail: ltang2008@gmail.com

This paper
introduces the
conceptual
design of a new
energy-
harvesting
device, named
the flutter-mill.
As shown in Fig.
1, a cantilevered
flexible plate
with embedded
conductors is
placed in an
axial (air) flow
and between a
pair of magnetic
panels. When
flutter takes
place, the
motion of each conductor in the magnetic field generates an electric potential
difference between its upstream and downstream ends. In this paper, the dynamics of
two-dimensional cantilevered flexible plates in axial flow is first summarized; special
attention is given to the energy transfer between the plate and the surrounding fluid
flow. In the light of the dynamics and the accompanying energy transfer of the system,
the key design parameters of the flutter-mill are determined. Moreover, the
power-extraction performance of flutter‐mills, with two typical sets of key design
parameters, is preliminarily evaluated and compared to a real Horizontal Axial Wind
Turbine (HAWT) studied by Burton et al. [9].
2. The fluid-elastic model
A schematic diagram of
a cantilevered flexible
plate in axial flow is
shown in Fig. 2. The
geometrical
characteristics of the
rectangular
Fig. 1: The conceptual design of a flutter-mill: (a) the layout of the
system, and (b) the wiring scheme.
Fig. 2: A cantilevered flexible in axial flow.
homogeneous plate are the length of the flexible section L, width B and thickness h;
and for a thin two-dimensional plate. Normally, there is a rigid
segment of length L0 as part of the clamping arrangement at the upstream end. The

other physical parameters of the system are: the plate material density and
bending stiffness , where E and are, respectively, Young's
modulus and the Poisson ratio of the plate material, the fluid density , and the
undisturbed flow velocity U. As shown in Fig. 2, W and V are, respectively, the
transverse and longitudinal displacements of the plate. and are the
aerodynamic loads acting on the plate in the transverse and longitudinal direction. S is
the distance of a material point on the plate from the origin, measured along the plate
centreline in a coordinate system embedded in the plate. Moreover, material damping
of the Kelvin-Voigt type is considered for the plate, with the loss coefficient denoted by
a.
The equations of motion of the plate can be written in nondimensional form as (see
Ref. [1])
where the overdot and the prime represent and , respectively. The
nondimensional tension in the plate is given by
The nondimensional variables used in Eqs. (1)−(3) are defined by
where and are, respectively, nondimensional and dimensional vibration
frequencies. Moreover, the mass ratio and the reduced flow velocity are
defined by
In Eqs. (1) and (3), and are calculated using the unsteady lumped vortex
model (see Ref. [1]). To do this, the flexible plate is evenly divided into panels,
each of length (note that the plate is assumed to be inextensible). On each
individual panel, say the ith panel, the pressure difference across the plate can
be computed and then decomposed into the lift and the drag . That is
(1)
(2)
(3)
(4)
(5)
(6)

where is the incidence angle of the ith panel. In the second equation above, an
additional drag coefficient is considered to be uniformly distributed over the
whole length of the plate, to account for the viscous drag effects.
3. The dynamics and energy transfer
The dynamics of
two-dimensional
cantilevered
flexible plates in
axial flow has
recently been
investigated by
Tang and
Païdoussis [1] in a
multiple-
parameter space.
When the reduced
flow velocity
is low, the plate
remains in the
stretched-straight
state. When
exceeds the
Fig. 3: (a) The power of the work and (b) the accumulated work done
by the fluid load fL, and the calculation of time-averaged power .
The parameters of the system are: , , ,
and .
critical point , flutter takes place; as is increased further, the flutter amplitude
grows but the flutter mode remains qualitatively the same. The flutter mode of the
system is principally determined by the mass ratio : when is small, the plate
vibrates in a second-beam-mode shape (note that the plate never flutters in a pure
first-beam-mode shape). With increasing , higher-order modes participate in the
dynamics and become increasingly significant. It should be emphasized that points at
different locations along the length of the plate do not oscillate in phase; as some
parts of the plate move upwards, others may move downwards.
It follows from Eqs. (1) and (6) that the nondimensional power of the work
done by the transverse fluid load and the nondimensional accumulated work
on the ith panel, in the sense of per unit length along the spanwise dimension of
the plate, can be calculated from
Therefore, over the whole length of the plate, the nondimensional total power and
the nondimensional accumulated total work , respectively, can be calculated by
(7)
(8)

Note that is the time integral of . When the plate flutters, also oscillates
about zero, as shown in Fig. 3. That is, in a vibration cycle, fL does both positive work
(energy transferred from the fluid flow to the plate) and negative work. Since
oscillates, the plot of versus time is not smooth (see Fig. 3(b)). The oscillating
may be evaluated in terms of the time-averaged power . In this paper, as
illustrated in the inset of Fig. 3(b), one may conveniently use the slope of the
versus time, in terms of the line connecting the local maxima, to calculate .
Moreover, according to the nondimensionalization scheme defined in Eq. (4), the
dimensional time-averaged power can be calculated by
Since points at
different locations
along the length of
the plate do not
oscillate in phase,
and the flutter mode
depends on , it is
important to examine
the energy transfer all
along the plate for
various systems with
different values of .
As shown in Fig. 4
for a specific system
with ,
, the
energy transfer at
various locations is
not the same: energy
is pumped from the
fluid flow into the
plate (positive slope
of the versus
time plot) at points
through
; while it is
transferred from the
plate to the fluid flow
at other locations (i.e.,
points
through ). Note
Fig. 4: The energy exchange along the length of the plate: (a) the
flutter modes, (b) the work done by fL at various locations, (c) a
regional enlargement of (b), and (d) the total work done by fL. The
parameters used are: , , ,
and . panels are used in the numerical simulation.
The energy transfer at 40 locations (every 5 panels) is recorded
and the results at 10 locations (every 20 panels) are presented.
(9)

that in this case the total work done by fL is positive, as shown in Fig. 4(d).
Fig. 5: When flutter takes place, the positive/negative (P/N) work done by the fluid load fL at
various locations along the length of the plate: (a) , , (b) ,
, (c) , , (d) , , (e) , , and (f)
, . The other parameters of the system are: , and
. Note that, is used for the cases and ; while, for the other
cases , and , is used. The energy transfers at 40 locations
(every 5 panels in the case of and every 10 panels when ) are recorded.
The distribution of positive/negative is also examined for various systems
with different values of , as shown in Fig. 5. It should be mentioned that different
values of are used for individual cases of for the purpose of obtaining flutter
motions (in particular, is so chosen about 10% above the corresponding critical
point ). It can be seen in Fig. 5 that the distribution of positive/negative
depends on . When is large, say or 5, higher modes become important
in the dynamics [1], and the distribution of positive/negative has a complicated
pattern. It should be mentioned that the distribution of the positive/negative also
depends on the value of , as one may see in Figs. 5(b) and (c); however, it has

een shown that this variation normally occurs at locations close to the fixed and free
ends of the plate, where the level of energy transfer is much lower than elsewhere
(refer to Fig. 4(b)).
4. The design of the flutter-mill
4.1. Determination of key design parameters
It has been shown that, at different sections of the plate, the energy transfer may
be from the fluid flow to the plate or
vice versa. Therefore, the conductors
embedded in the flexible plate should
be correspondingly arranged in
several sections, or say phases. Too
many phases of conductor
arrangement lead to difficulties in the
design of the wiring scheme and the
rectifier. To this end, in the light of the
vibration modes of the system with
various values of (see Fig. 12 of
Ref. [1], also Figs. 4
and 5 in the present
paper), a system
with mass ratio
should be
considered, for
which the plate
vibrates in the
second beam mode.
Hence, only two
phases of conductor
arrangement are
necessary. The
present paper
primarily considers
two cases:
and 0.2.
In the conceptual
design, dimensional
parameters allow
one to evaluate the
performance of the
device in a more
Table 1: Dimensional parameters of the
flutter-mills with and 0.2.
Plate 1 Plate 2 Unit
1.226 1.226 kg/m 3
2840 2840 kg/m 3
h 0.5 0.5 mm
L 0.58 0.232 m
B 0.2 0.2 m
0.5 0.2
Fig. 6: The dynamics of the system with and the
time-averaged power of the fluid load fL: (a) the bifurcation
diagram, (b) the flutter frequency, and (c) the time-averaged power
. The other parameters of the system are: ,
and .

direct manner;
accordingly plates
made of aluminium in
axial air flow [10] are
considered. The reason
for considering a
metallic plate is that, in
the prototype, (metal)
conductors are
supposed to be
embedded in the plate,
otherwise made of a
very flexible material;
the properties of this
composite plate would
thus be largely
determined by those of
the conductors. The
parameters of the two
systems with
and 0.2 are,
respectively, listed in
Table 1.
Fig. 7: The dynamics of the system with and the
time-averaged power of the fluid load fL: (a) the bifurcation
diagram, (b) the flutter frequency, and (c) the time-averaged
power . The other parameters of the system are: ,
and .
Some explanation is necessary for the data in Table 1. First, since nondimensional
parameters are used in all numerical simulations, the physical parameters of the
systems listed in Table 1 are actually obtained using a reverse approach: for an
aluminium plate in axial air flow, the physical parameters , , E, and a are
fixed; the thickness of the plate is determined as mm and then the length of
the plate L can be calculated for the case or 0.2. Second, the
nondimensional parameters , and are uniformly used in
all numerical simulations for simplicity; the influence of these parameters on the
dynamics of the system has already been discussed in Ref. [1]. Third, m is
considered in the conceptual design in order to avoid possible difficulties in attaining
sufficient strength of the magnetic field between the two magnetic panels. Note that
the value of B is not used in the investigation of the dynamics (nor in the energy
transfer in terms of per unit length along the spanwise dimension) of a
two-dimensional plate in axial flow.
As shown in Figs. 6 and 7, preliminary calculations of the power extraction from the
fluid flow demonstrate that the proposed flutter-mill is very promising for generation of
electric power. In particular, consider a design with Plate 1 ( ), the device is
compact, with overall dimensions 0.58 m (length) x 0.2 m (width) x 0.58 m (height),
where the height is considered as twice the allowed maximum flutter amplitude (i.e.,
half of the length of the flexible plate). Such a system can be expected to extract 10

Watt power (at m/s or 43.2 km/h) from the wind; and, if only 10% of the
extracted wind energy is ultimately converted to electric power, an output of 1 Watt is
guaranteed. Higher power output can be expected from the system with Plate 2
( ), which has even smaller overall dimensions (0.232 m x 0.2 m x 0.232 m)
but works at higher flow velocities. The power extraction is as high as 1 kW/m at
m/s or 144 km/h approximately.
4.2. The performance of power extraction from a flutter-mill
The performance of the flutter-mill is compared to a real HAWT (Horizontal-Axis
Wind Turbine), specifically the Three-Blade Stall Regulated Turbine studied by Burton
et al. [9], in terms of
measured electric
power output, as shown
in Fig. 8. The HAWT
has a disk area
m 2 and the
data for electric power
output is collected at a
rotational speed of 44
rpm. In order to make a
comparison, one
supposes that the
flutter-mill has the
same wind receiving
area A as the HAWT
and accordingly
calculate the effective
Fig. 8: The performance of the flutter-mill as compared to a real
HAWT, i.e., the Three-blade Stall Regulated Turbine studied by
Burton et al. [9].
width Beff of the plate using the formula , where
is the vibration amplitude at the trailing edge of the plate. Note that the effective width
Beff can be attained by considering an array of flutter-mills, each one having a
designed width of m. Moreover, in the calculation of electric power output, it
is assumed that only 10% of the energy captured by the plate is ultimately converted
to electric power; the other 90% is consumed by the plate for sustaining the flutter
motion (under the action of the induced electromagnetic forces). It can be seen in Fig.
8 that when , the flutter-mill works at the lower part of the wind speed range of
the HAWT; however, the electric power output of the flutter-mill is not as high as that of
the HAWT (about 10%). When it is designed with , the flutter-mill works at
high wind speed beyond the normal working condition of the HAWT, but a very high
electric power output can be expected. Therefore, it is possible to design a flutter-mill
with a mass ratio between and 0.2 to achieve an output capacity comparable
to the real HAWT, and to operate in the same range of wind speed as the HAWT.

5. Concluding remarks
A new energy-harvesting concept of utilizing the flutter motions of cantilevered
flexible plates in axial flow to generate electric power, namely the flutter-mill, is
proposed in this paper. The dynamics of the system and the energy transfer between
the plate and the surrounding fluid flow have been studied. Based on the theoretical
work, the key design parameters are determined. Without a comprehensive
optimization process at the present stage of conceptual design, the performance of
the flutter-mill is preliminarily evaluated and compared to a real conventional wind
turbine. It has been shown that the proposed design of flutter‐mill, with the
distinguishing advantages of simple structure and compact size, is very promising for
generation of electrical power.
Acknowledge
The leading author gratefully acknowledges the support by a Natural Sciences and
Engineering Research Council of Canada (NSERC) post-doctoral fellowship.
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