The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

An introduction to heat transfer

Dynamics of structures

Past, present and future of nonlinear system identification in structural dynamics

Nonlinear Oscillations: Exact Solutions and their Approximations

Piezoelectric energy harvesting from vortex-induced vibrations of circular cylinder

The concept of harvesting energy from transverse galloping oscillations of a bluff body with different cross-section geometries is investigated. The energy is harvested by attaching a piezoelectric transducer to the transverse degree of freedom of the body. The power levels that can be generated from these vibrations and the variations of these levels with the load resistance, cross-section geometry, and freestream velocity are determined. A representative model that accounts for the transverse displacement of the bluff body and harvested voltage is presented. The quasi-steady approximation is used to model the aerodynamic loads. A linear analysis is performed to determine the effects of the electrical load resistance and the cross-section geometry on the onset of galloping, which is due to a Hopf bifurcation. The normal form of this bifurcation is derived to determine the type (supercritical or subcritical) of the instability and to characterize the effects of the linear and nonlinear parameters on the level of harvested power near the bifurcation. The results show that the electrical load resistance and the cross-section geometry affect the onset speed of galloping. The results also show that the maximum levels of harvested power are accompanied with minimum transverse displacement amplitudes for all considered (square, D, and triangular) cross-section geometries, which points to the need for performing a coupled analysis of the system.

https://iopscience.iop.org/article/10.1088/0964-1726/22/1/015014/pdf

Piezoelectric energy harvesting from transverse galloping of bluff bodies

This work investigates the influence of structural and aerodynamic nonlinearities on the dynamic behavior of a piezoaeroelastic system. The system is composed of a rigid airfoil supported by nonlinear torsional and flexural springs in the pitch and plunge motions, respectively, with a piezoelectric coupling attached to the plunge degree of freedom. The analysis shows that the effect of the electrical load resistance on the flutter speed is negligible in comparison to the effects of the linear spring coefficients. The effects of aerodynamic nonlinearities and nonlinear plunge and pitch spring coefficients on the system’s stability near the bifurcation are determined from the nonlinear normal form. This is useful to characterize the effects of different parameters on the system’s output and ensure that subcritical or “catastrophic” bifurcation does not take place. Numerical solutions of the coupled equations for two different configurations are then performed to determine the effects of varying the load resistance and the nonlinear spring coefficients on the limit-cycle oscillations (LCO) in the pitch and plunge motions, the voltage output and the harvested power.

Modeling and analysis of piezoaeroelastic energy harvesters

This paper provides a thorough review of the significant work done so far in the area of flight dynamics and control of flapping-wing micro-air-vehicles (MAVs). It provides the background necessary to do research in that area. Furthermore, it raises questions that need to be addressed in the future. The three main blocks constituting the flight dynamic framework of flapping MAVs are reviewed. These blocks are the flapping kinematics, the aerodynamic modeling, and the body dynamics. The design and parametrization of the flapping kinematics necessary to produce high-control authority over the MAV, as well as design of kinematics suitable for different flight conditions, are reviewed. Aerodynamic models used for analysis of flapping flight are discussed. Particular attention is given to the physical aspects captured by these models. The issues and consequences of averaging the dynamics and neglecting the wing inertia are discussed. The dynamic stability analysis of flapping MAVs is usually performed by either averaging, linearization and subsequent analysis or using Floquet theory. Both approaches are discussed. The linear and nonlinear control design techniques for flapping MAVs are also reviewed and discussed.

Flight dynamics and control of flapping-wing MAVs: a review

A model for harvesting energy from galloping oscillations of a bar with an equilateral triangle cross-section attached to two cantilever beams is presented. The energy is harvested by attaching piezoelectric sheets to cantilever beams holding the bar. The derived nonlinear distributed-parameter model is validated with previous experimental results. The quasi-steady approximation is used to model the aerodynamic loads. The power levels that can be generated from these vibrations, and the variations of these levels with the load resistance and wind speed, are determined. Linear analysis is performed to validate the onset of galloping speed with experimental measurements. The effects of the electrical load resistance on the onset of galloping are then investigated. The results show that the electrical load resistance affects the onset speed of galloping. A nonlinear analysis is also performed to determine the effects of the electrical load resistance and the nonlinear torsional spring on the level of the harvested power. The results show that maximum levels of harvested power are accompanied by minimum transverse displacement amplitudes. It is also demonstrated that there is an optimum load resistance that maximizes the level of the harvested power.

https://iopscience.iop.org/article/10.1088/0964-1726/22/2/025016/pdf

Muhammad R. Hajj

Papers

Piezoelectric energy harvesting from vortex-induced vibrations of circular cylinder

Piezoelectric energy harvesting from transverse galloping of bluff bodies

Modeling and analysis of piezoaeroelastic energy harvesters

Flight dynamics and control of flapping-wing MAVs: a review

Modeling and nonlinear analysis of piezoelectric energy harvesting from transverse galloping