Abstract
The highly nonlinear phenomenon of fluid–structure interaction is discussed, including examples drawn from nature and early work on aircraft flutter. Recent work on extracting the energy in a fluid stream by piezoelectric elements is reviewed, including some of the underlying physics. Whilst the energy extracted from fluttering elements is low, it is a subject of interest for powering Ultra-Low Power (ULP) devices and systems since this method of energy extraction is thought to offer a quiet alternative to conventional wind turbines. Researchers have investigated the use of thin piezoelectric patches coupled to a geometrically shaped, polymeric membrane (via a revolute hinge) which can amplify the bending, strain and hence power. Such systems respond via flutter induced by resonant bending instability of the system, or by the utilisation of time-varying external pressure gradients formed around the system. Key factors that influence performance are examined, such as critical flutter speed, mass ratio, position of revolute hinge, aspect ratio and type of piezoelectric material. The chapter concludes with a discussion of the practical implications of such systems in the future.
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- 1.
Flutter is a type of FSI, where the solid-body oscillates in the fluid flow, usually with large deformation amplitudes.
- 2.
Namely, the Bernárd–von-Kármán vortex shedding and the Kelvin–Helmholtz instability.
- 3.
One particular catastrophic case of aircraft wing flutter was the Braniff Airlines Flight 542 Lockheed Electra, in 1959. Everyone onboard perished when the airframe failed due to a flutter mode excited by engine propeller whirl. The crash investigation report was released in 1961 [9].
- 4.
The Strouhal number is discussed in more detail in Sect. 6.2.2.
- 5.
ULP here denotes both ULP consumption devices and ULP generation technologies. Two examples of ULP consumption devices would be wireless sensor nodes and LED lighting.
- 6.
The Reynolds number is a dimensionless number, which is the ratio of the fluid inertial forces to the viscous forces. Mathematically, it is defined as \(\text{Re} = \frac{uD} {\nu }\), where u is the flow velocity, d is the characteristic dimension of the immersed body, and ν is the kinematic viscosity of the fluid.
- 7.
A Rankine vortex is one that rotates at a constant angular velocity, ω, and has radius r v . The velocity v for any r < r v is such that v = ωr. For r > r v , v decreases exponentially.
- 8.
Solving the two-dimensional N–S equations over a circular cylinder, for example, has shown gross over-prediction of the lift coefficient—even with significant mesh refinement, see Dong et al. [15].
- 9.
A mode of vibration with zero strain.
- 10.
Measurement Specialties, Inc., LDT1-028K/L type.
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Acknowledgements
This work is funded by an Australian Research Council linkage scheme (grant no. LP100200034) in conjunction with Fabrics & Composites Science & Technology Pty. The authors would also like to thank Mr. Phred Peterson for the smoke-flow imagery.
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List of Symbols
- \( A\)
-
Cross-sectional area
- \( a\)
-
Acceleration
- \( AR\)
-
Aspect ratio
- \( b\)
-
Leaf base length
- \( C\)
-
Capacitance
- \( c\)
-
Compliance or damping
- \( C_{1,2,3,\ldots,8}\)
-
Constants in the general solutions of the space functions,Y1 Y1
- \( D\)
-
Characteristic dimension of bluff body or electric displacement
- \( d\)
-
Piezoelectric coupling coefficient
- \( E\)
-
Electric field strength or elastic modulus\( \left(\frac{1}{c}\right)\)
- \( EI\)
-
Bending stiffness
- \( F\)
-
Input forcing function
- \( f\)
-
Vortex shedding frequency
- \( h\)
-
Beam thickness or leaf height
- \( I\)
-
Moment of inertia
- \( I_{b}\)
-
Moment of inertia of an object rotating about its base axis
- \( k\)
-
Stiffness
- \( L\)
-
Beam length
- \( \mathcal{L}\)
-
Laplacian operator
- \( M\)
-
Bending moment
- \( m\)
-
Mass per unit length of the beam, or mass
- \( N\)
-
Number of discretisation points
- \( P\)
-
Power
- \( P_{\text{ave}}\)
-
Average power
- \( P_{i}\)
-
Instantaneous power in an i th second interval
- \( R_{L}\)
-
Load resistance
- \( R_{L_{\text{opt}}}\)
-
Optimum load resistance
- \( r\)
-
Radius
- \( r_{v}\)
-
Rankine vortex radius
- \( Re\)
-
Reynolds number
- \( R_{i}\)
-
i th analytical natural frequency ratio
- \( R_{i_{\,\text{comp}}}\)
-
i th computational natural frequency ratio
- \( S\)
-
Mechanical strain
- \( s\)
-
Distance from beam leading-edge to upstream bluff body, or Laplacian coordinate
- \( St\)
-
Strouhal number
- \( T\)
-
Mechanical stress
- \( t\)
-
Time, or when superscript, denotes the transpose of a
- \( T_{g}\)
-
Glass transition temperature
- \( T_{i}\,(t)\)
-
i th time function in variable separable method
- \( u\)
-
Fluid stream-wise velocity component
- \( u_{c}\)
-
Critical flutter speed
- \( u_{q}\)
-
Quenching speed
- \( V _{\text{RMS}}\)
-
Root-mean-square voltage
- \( v\)
-
Velocity or cross-stream velocity component
- \( w\)
-
Beam width
- \( x,\, y,\, z\)
-
Cartesian coordinates
- \( x_{i},\,y_{1},\,z_{i}\)
-
i th local coordinates
- \( Y _{i}\,(x)\)
-
i th space function in variable separable method
- \( Y (s),\,Z(s)\)
-
Laplace transform of a function
- \( z\)
-
Distance from beam neutral axis to point of interest
- \( \beta _{i}\)
-
\(\root{4}\of{\rho _{s}A\omega _{i}^{2}/EI} \)
- \( \delta\)
-
Damping ratio
- \( \epsilon\)
-
Electric permittivity
- \( \eta\)
-
Non-dimensional hinge position, x/L
- \( \lambda\)
-
Eigenvalue
- \( \lambda _{i}\)
-
i th mode shape eigenvalue
- \( \mu\)
-
Mass ratio
- \( \nu\)
-
Kinematic viscosity or normalised critical flutter speed
- \( \varphi\)
-
Phase difference
- \( \rho _{ f}\)
-
Fluid density
- \( \rho _{s}\)
-
Beam density
- \( \omega\)
-
Angular frequency or angular velocity
- \( \omega _{n}\)
-
nth natural frequency
- \( i\)
-
Index
- \( \mbox{ Hinge}\)
-
Hinged beam
- \( \mbox{ Uniform}\)
-
Uniform beam
- \( \mbox{ Comp}\)
-
Computational
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McCarthy, J.M., Deivasigamani, A., Watkins, S., John, S.J., Coman, F. (2014). Energy Harvesting from Flows Using Piezoelectric Patches. In: Jazar, R., Dai, L. (eds) Nonlinear Approaches in Engineering Applications 2. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6877-6_6
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