Emergence of increasingly smaller electromechanical systems with submilli-Watt power consumption led to the development of scalable micropower generators (MPGs) that harness ambient energy to provide electrical power on a very small scale. A flow MPG is one particular type which converts the momentum of an incident flow into electrical output. Traditionally, flow energy is harnessed using rotary-type generators whose performance has been shown to drop as their size decreases. To overcome this issue, oscillating flow MPGs were proposed. Unlike rotary-type generators which rely upon a constant aerodynamic force to produce a deflection or rotation, oscillating flow MPGs take advantage of cross-flow instabilities to provide a periodic forcing which can be used to transform the momentum of the moving fluid into mechanical motion. The mechanical motion is then transformed into electricity using an electromechanical transduction element. The purpose of this review article is to summarize important research carried out during the past decade on flow micropower generation using cross-flow instabilities. The summarized research is categorized according to the different instabilities used to excite mechanical motion: galloping, flutter, vortex shedding, and wake-galloping. Under each category, the fundamental mechanism responsible for the instability is explained, and the basic mathematical equations governing the motion of the generator are presented. The main design parameters affecting the performance of the generator are identified, and the pros and cons of each method are highlighted. Possible directions of future research which could help to improve the efficacy of flow MPGs are also discussed.

# Micropower Generation Using Cross-Flow Instabilities: A Review of the Literature and Its Implications PUBLIC ACCESS

**Mohammed F. Daqaq**

**Amin Bibo**

**Imran Akhtar**

**Ali H. Alhadidi**

**Meghashyam Panyam**

**Benjamin Caldwell**

**Jamie Noel**

^{1}Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received May 10, 2018; final manuscript received January 9, 2019; published online February 13, 2019. Assoc. Editor: Lei Zuo.

*J. Vib. Acoust*141(3), 030801 (Feb 13, 2019) (27 pages) Paper No: VIB-18-1199; doi: 10.1115/1.4042521 History: Received May 10, 2018; Revised January 09, 2019

## Abstract

## Introduction

Energy surrounds us. It radiates from the sun shining down on Earth. It flows in the air that we breath and the water that we drink. It is hidden beneath the ground that we stand on. It is even entrained in the very matter that forms us. Energy permeates the universe that we live in, dynamic and diffuse, constantly moving and transforming, swirling with infinite complexities. The principle issue with the energy faced by mankind has, therefore, never been with the dearth of it, but rather with our ability to tap into its tremendous flow and siphon off a small amount to put to use for the purposes of sustaining human life and civilization. In this sense, a part of our history can be viewed as the story of man using his ingenuity to develop new and ever more complex ways to harness energy from the world around him.

The earliest humans relied on harvesting solar energy in the form of farming to provide food. Soon, they discovered that they could use energy from the environment to power devices as well; wind to propel sailing ships and flowing water to turn mills and grind grain. These primitive methods simply repurposed some energy in the environment as usable mechanical power, without actually converting its form. However, with the advent of the steam engine and the dawn of the industrial revolution, mankind was able to extract mechanical energy from chemical sources through the combustion of carboniferous fuels. This new energy source provided a more reliable, scalable, portable, and power dense form of powering man's creations, and it soon replaced the historic energy sources. Additionally, with the invention of dynamos and alternators, fossil fuels led to the easy production of electrical power, which ushered human civilization into the modern era.

Nevertheless, owing to concerns about global climate changes due to the release of carbon dioxide into the atmosphere by burning carbon-based fuels [1], there has been a renewed interest in harvesting “clean,” renewable energy from existing sources in the environment; e.g., wind, solar, geothermal, tidal, etc. Large utility-scale energy harvesting technologies from these renewable sources, particularly solar and wind, already exist. More recently however, many research studies have focused on using ambient energy to provide electrical power on the milli- and micro-Watt level. Interest in such small power inputs resulted from many technological advances that have led to the development of increasingly smaller electromechanical systems which consume very little power, on the order of *μ*W or *m*W [2]. While the possible applications for such devices are numerous and constantly growing, from wireless sensor networks [3] to biomedical systems [4], a key obstacle in their development and implementation has always been providing such devices with the power they needed to operate, however small it may be. Such devices are not conducive to centralized grid power, as it limits many of the advantages of their small size, such as mobility, implementation in large quantities, and placement in remote and hard-to-access locations.

Traditionally, small-scale electronic devices have relied on battery power, which also suffers from a variety of drawbacks. The energy density of batteries is low, and they have a finite lifespan. Also, depending on the application, they may or may not be easily replaceable. Nonetheless, the lower power threshold of current electronic devices has opened up the possibility of scavenging low-level ambient energy to generate self-sustaining electrical power for the device. Therefore, energy harvesting technology exists as a potential innovation for powering the next generation of human technology in the information age; a world of completely interconnected, and nearly invisible, devices.

To reduce the dependence on battery power, many different types of micropower generation technologies have been developed to harness different ambient energy sources, including solar [5], thermal [6], fluid flow [7], and mechanical [8]. Application of such devices span different areas of technology including, but not limited to, structural- and machine-health monitoring systems [9], smart road networks, sustainable buildings, biomedical implants [10,11], large-scale railway or industrial systems [3,6,12], and vehicle suspensions [13].

The aforementioned examples and the ever-growing applications of micropower generators (MPGs) indicate that they have a great potential, which is only limited by human imagination and our ability to develop technologies to harvest the energy needed to power such innovative devices.

## Flow Micropower Generators

A flow MPG is a particular type of energy harvester which converts the momentum of an incident fluid flow into an electrical output. The working fluid is most commonly air or water due to their availability in nearly any environment, natural, or otherwise. Traditionally, flow energy has been harnessed using rotary-type generators such as the typical horizontal axis wind turbines or the common water mills. Unfortunately, performance of rotary-type generators drops significantly as their size decreases. Mitcheson et al. [14] reported that the power coefficient of the rotary wind generator can drop below 0.1 as the size of the turbine decreases. The reduction in the power coefficient was attributed to the higher viscous drag on the blades at lower *Reynolds* numbers, as well as the increase in thermal losses and electromagnetic interferences which tend to increase as size decreases. In addition to the reduction in the performance, fabrication of rotary-type generators that require a rotor, a stator, and blades is a very complex and expensive process at smaller scales. This makes their actual implementation for compact applications, such as those concerning distributed sensing networks, an astounding task.

The issues of efficiency and cost versus scalability of rotary type flow energy generators can be clearly seen in Fig. 1, which demonstrates that the power density of the wind turbines available in the market increases and their cost decreases as their size increases.

To reduce losses associated with rotary-type generation systems (e.g., micro wind turbines (MWTs)) and the fabrication complexities which could limit their scalability, two things must be carried out. First, relative motion between interfaces must be minimized to reduce friction. Second, the design must be simplified to allow fabrication at a small scale without sacrificing efficiency. This has made oscillating flow MPGs a promising solution. Oscillating flow MPGs employ the natural elasticity of materials to provide a means of translation or rotation devoid of a sliding interface. Instead of relying upon a constant aerodynamic force to produce a deflection or rotation in a certain direction, oscillating flow MPGs take advantage of cross-flow instabilities to provide a periodic forcing which can be used to transform the momentum of the moving fluid into mechanical motion. The four primary types of cross-flow instabilities used for micropower generation are galloping, flutter, vortex shedding, and wake-galloping.

From a cost perspective, these oscillating flow MPGs are expected to have a lower manufacturing cost than MWTs; manufacturing microblades and low friction microbearings is more critical and expensive than manufacturing a micro-oscillator with a bluff body which constitute a flow MPG. Moreover, wind turbine performance is strongly related to aerodynamic characteristics of the blades where flow separation should be avoided as much as possible to ensure maximum efficiency. This makes designing blades at the microscale a challenging task, due to a considerable decrease in the lift to drag ratio which causes flow separation and a significant drop in the aero-elastic power coefficient. Flow MPGs, on the other hand, rely on separated flow, making their design simpler and cost effective at a microscale.

From a performance perspective, the efficiency of MWTs drop significantly at a smaller scale as highlighted earlier. This is further illustrated by evaluating the maximum power densities of three different MWTs operating at the same wind speed of 5 m/s with decreasing rotor sizes: 1 mW/cm^{2} for a rotor diameter of 13.8 cm by Park et al. [15], 0.44 mW/cm^{2} for a rotor diameter of 7.6 cm by Xu et al. [16], and 0.17 mW/cm^{2} for a diameter of 4.2 cm by Rancourt et al. [17]. Comparable performance, although far from optimal, was achieved by the galloping MPG investigated by Sirohi and Mahadik [18] which exhibited a power density of 0.5 mW/cm^{2} for a triangular prism with a length of 25 cm at similar wind speeds. Such findings highlight that oscillating flow MPGs can be a viable alternative for micropower generation.

The purpose of this article is to summarize the important research carried out during the past decade on flow micropower generation using cross-flow instabilities. The summarized research will be categorized according to the different instabilities used to excite mechanical motion: namely galloping, flutter, vortex shedding, and wake-galloping. Under each category, the fundamental mechanism responsible for the instability will be explained and the basic mathematical equations governing the motion of the generator will be developed, and solved, whenever possible. The main design parameters affecting the performance of the harvester will be identified and the pros and cons of each method will be highlighted. Future directions of research which could help to improve the efficacy of flow MPGs will also be discussed.

The rest of the paper is organized as follows: Sec. 3 presents the different types of cross-flow instabilities commonly used for micropower generation and defines the important parameters used throughout the paper. Section 4 presents some important remarks regarding the terminologies used. Section 5 presents the basic fundamentals of galloping-based MPGs, their response, and the different designs proposed in the literature. Section 6 presents the basic fundamentals of wake-galloping-based MPGs, their response, and the most important traditional and nontraditional designs proposed in the literature. Sections 7 and 8 follow the organization of Sec. 6 to present the fundamentals and latest advances in vortex-induced and flutter-based MPGs. Section 9 presents a comparative analysis of the advantages and disadvantages of the different types of MPGs. Finally, Sec. 10 presents possible directions for future work.

## Flow-Induced Vibrations Due to Cross-Flow Instabilities

Nature contains numerous illustrations of systems undergoing oscillations due to flow–structure interactions. Examples include wind-induced motion of tree branches, sway of large buildings under aerodynamic loads, oscillations of underwater pipes, or the motion of reeds in wind-driven musical instruments, etc. The painting of *Giovanni da Modena* which illustrates vortices shedding off the legs of *San Cristoforo* while crossing a stream is probably one of the first documented human realizations of the interaction between a flow and a structure [19].

Regardless of the specific nature of the cross-flow instability inducing the motion of the structure, the interaction can be generally understood by inspecting Fig. 2. The interaction starts when an incident flow of given parameters (speed, viscosity, inertia, and turbulence intensity) exerts pressure forces on an elastic structure. Such forces, which are often decomposed into drag and lift components along the freestream and cross flow directions, cause the structure to oscillate, thus changing its orientation with respect to the flow. Depending on the geometry, elasticity, and damping characteristic of the structure, the change in orientation changes the action of the fluid forces by changing the interfacial boundary layer motion. This, in turn, creates a feedback mechanism which alters the forces acting on the structure resulting in the classical flow-induced vibrations. Under some conditions, this feedback mechanism can generate large-amplitude oscillations, which may be detrimental to the safety and performance of many structural systems. On the other hand, such motions can also be effectively employed to harvest energy as will be described throughout this manuscript.

For flow-induced vibrations to occur, sufficient interaction between an elastic structure and a moving fluid must exist. In its simplest form, as shown in Fig. 3, an elastic structure can be modeled as a single degree-of-freedom mass-spring-damping oscillator while the effective properties of the fluid can be captured by its density, velocity field, viscosity, and turbulence intensity. Such interaction is often characterized in terms of nondimensional parameters governing the fluid flow, the structural dynamics, and their interaction to scale the phenomena. For the sake of clarity, we define the parameters relevant to this review report as follows:

*Slenderness ratio, s*=*L/D:*the length,*L*to width,*D*ratio of the body.*Reduced velocity,*$U\u0302=U/(\omega nD)$*:*ratio of the length of the path traveled by the flow per cycle to the width of the structure. Here,*U*is the freestream velocity, and $\omega n=K/M$ is the natural frequency of structural oscillations.*Mass ratio, μ*=*ρA*ratio of the mass of the displaced fluid, $(1/4)\rho AsL$, to the mass of the moving structure,_{s}L/(4M):*M*. Here,*ρ*is the fluid density, and*A*is the effective area of the incident bluff body._{s}*Reynolds number, Re = UD/ν*: is the ratio of inertial forces to viscous forces, where*ν*is the kinematic viscosity of the fluid.*Strouhal number, St = ω*ratio between the body velocity and the freestream velocity._{n}D/U:*Damping ratio, ζ:*ratio of energy dissipated per cycle to the total energy of the structure.*Turbulence intensity,*$TI=u\u0302/U$*:*ratio of the root-mean-square of the fluctuating velocity, $u\u0302$, to the freestream velocity.

There are four classical types of cross-flow-induced instabilities used for micropower generation. In what follows, we provide a brief description of each.

Galloping is a cross-flow-induced instability which can be easily activated in bluff bodies with a long afterbody.^{2} The analysis of galloping is often traced back to the pioneering observations of Den Hartog [20] in the early 19th century who related the oscillations of electric power transmission lines to the asymmetric geometry caused by the formation of ice around the circumference of the wire. He noted that the asymmetry causes a net lift force which results in steady-state periodic oscillation of the transmission lines when the freestream flow velocity exceeds a certain threshold, *U*_{cr}. As shown in Fig. 4, the magnitude of the resulting oscillations increases monotonically with increasing freestream velocity. From the perspective of a nonlinear dynamical system, the galloping instability occurring at *U*_{cr} represents a Hopf bifurcation that manifests itself as periodic oscillations when the critical (threshold) flow velocity, *U*_{cr}, is exceeded.

To better understand the formation of the lift force, consider Fig. 5, which depicts the main characteristics of the flow around a trapezoidal bluff body angled at 20 deg with respect to the flow direction. As the flow approaches the leading edge of the bluff body, its separates and creates two shear layers (dashed lines) on either side of the body. As the two shear layers try to reattach, they trap circulation bubbles (CB) between the shear and the surface of the body. The size of the trapped CBs is determined by the curvature of the shear layer which, in turn, is governed by the angle of attack and the Reynolds number of the flow. The shear layer forming on the upper side of the prism curves slowly toward the body, and, hence traps a larger CB. The shear layer forming on the lower side curves quickly and, therefore, traps a smaller CB. The relative size, location, and flow velocity within these CBs determine the net lift on the body with the maximum lift occurring when the pressure difference between the upper and lower surfaces is maximized. If the lift acts in the same direction as the motion of the body, and the magnitude of that component grows as the body velocity grows, the system is unstable to galloping oscillations. The effect is a velocity-dependent force which acts as an energy pumping mechanism, or as negative damping.

The reader should bear in mind that, although the production of lift requires asymmetry, the asymmetry does not need to be in the geometry of the prism but can also occur due to the angle of incidence between the moving body and the flow. As such, perfectly symmetric bodies like a square are also prone to galloping.

Wake-galloping is another cross-flow instability which has been exploited in the field of micropower generation. Traditional wake-galloping occurs when an initially steady fluid flows over an obstacle that is placed in the upstream of a mechanical oscillator. For a range of Reynolds numbers, the steady flow undergoes symmetry breaking resulting in a von Kármán vortex street shedding from the trailing edge of the obstacle. For circular cylinders in particular, periodic vortex shedding occurs when the Reynolds number is increased beyond Re ≈ 40. The vortex shedding remains periodic with one dominant frequency up to approximately Re ≈ 10^{5} [21]. Beyond this value of the Reynolds number, aperiodic flow patterns can be observed.

For the range of Reynolds numbers where the vortex shedding is periodic, the shed vortices induce a periodic lift on the oscillator placed in the downstream of the bluff body. As shown in Fig. 6, when the vortex shedding frequency is close to the natural frequency of the oscillator *ω _{n}*,

*lock-in*or

*synchronization*occurs. In this region, flow couples to the natural modes of the oscillator resulting in large-amplitude motions often coined as wake-galloping. Synchronization may also occur, but to a lesser extent, when the vibration frequency of the structure is equal to a multiple or submultiple of the shedding frequency. For a more detailed understanding of the phenomenon, the reader is referred to Sec. 6 and the schematics within.

Vortex-induced vibration (VIV) is a cross-flow instability which occurs when a fluid flow that is incident upon a bluff body separates over a substantial portion of its surface. At Reynolds numbers beyond a critical value, the shear layers on either side of the body rolls up into vortices and shed periodically into its wake. Vortex shedding causes periodic pressure fluctuations on the top and bottom surfaces of the body itself, thereby inducing an alternating lift force on the body in a direction normal to the flow. This results in periodic oscillations of the bluff body which can be used for power generation.

The phenomenon of vortex shedding was first observed and reported by the famous French physicist, Henri Bénard^{3} [23,24], and later formalized by Theodore von Kármán, who studied the stability of point vortex configurations and derived the geometric conditions under which the classic vortex street (which was later named after him) is formed [25,26]. The seminal works of Bénard and Kármán spawned several research studies concerned with understanding the mechanics and instabilities associated with vortex shedding around two-dimensional (2D) bluff bodies of varying geometries [27–32].

Similar to wake-galloping, for the range of the Reynolds number where the shedding frequency is close to the natural frequency of the oscillating bluff body, large-amplitude resonant oscillations are activated resulting in a response behavior similar to that shown in Fig. 6. As discussed earlier, this phenomenon is called lock-in or synchronization and the resulting oscillations are referred to as VIV. A typical VIV cycle is shown in Fig. 7 plotted with a time interval of *T*/5. When the frequency of vibration is slightly below the vortex shedding frequency, the vortex is shed from the side opposite to the side experiencing maximum displacement. As the vibration frequency traverses the natural shedding frequency, the vortex is shed from the same side as the maximum displacement. In other words, as the cylinder vibration nears the vortex shedding frequency, the vortical structure and phase are influenced. A 180 deg phase shift occurs as the vibration frequency passes through the natural shedding frequency leading to a hysteretic response.

Flutter is an aeroelastic terminology, classically used in aeronautical context, e.g., airfoils, turbomachinery blades, bridge decks, etc., for a two degrees-of-freedom (2DOFs) coupled torsion-plunge instability in flexible structures with relatively flat shapes, where the aerodynamic forces integrate with structural oscillations to yield self-sustained vibrations. In flutter, the incident flow produces a lift force and a pitching moment that typically depend nonlinearly on the displacement, rotation, velocity, and angular velocity of the oscillator. Such forces can alter the effective damping and stiffness of the system. At a certain flow speed, the influence of the lift force and moment on the moving body is such that the effective damping associated with the pitching mode approaches zero resulting in a Hopf bifurcation which activates periodic oscillations denoted as flutter. Flutter was first documented during World War I by the British scientist *Lanchester* who was investigating violent antisymmetric oscillations occurring on the fuselage and tail sections of the Handley Page 0/400 Biplane Bomber [33].

To explain the fundamental physics behind flutter, consider the cross section of an aircraft wing that is initially rotated by an angle as shown in Fig. 8. As the lift forces cause the wing to rise, the torsional restoring force brings the wing to zero rotation at $t=(T/4)$. The bending restoring force pushes the wing down, and the rotary inertia forces the airfoil to rotate in a nose-down position at $t=(T/2)$. The rotation of the airfoil in the nose-down direction creates a downward lift force, which pushes the wing further downward until it reaches the horizontal position at $t=(3T/2)$. The bending restoring moment then pushes the wing upward and the flutter cycle is complete at *t *=* T*. When the energy fed to the structure overcomes all damping mechanisms, the flutter cycle is repeated periodically.

It must be noted that, as shown in Fig. 8, typical flutter occurs when the frequency of the pitch and plunge degrees-of-freedom are identical. Such condition can be satisfied even for structures which are designed such that the frequency of the pitch and plunge degrees-of-freedom are not equal in the absence of the aerodynamic loads. Due to its linear dependence on the pitch degree-of-freedom, the aerodynamic load has the ability to change the effective stiffness of the pitch mode such that both frequencies converge at a certain flow speed.

The response of a fluttering oscillator is similar to the galloping response shown in Fig. 4. As such, the terms galloping and flutter are often used interchangeably in the open literature. Here, we restrict the use of flutter to indicate a two degrees-of-freedom aerodynamic instability.

## Important Remarks

Some of the flow-induced vibrations phenomena have similar underlying physics and at times the terminologies are used interchangeably. Before we delve into the details of the various devices and techniques used for flow MPG, we ask the reader to bear in mind that it is sometimes difficult to create distinct boundaries between the different cross-flow instabilities that induce vibrations. A few remarks, in this regard, are in order as described next:

*Galloping versus flutter:*The difference between galloping and flutter is mostly based on the historical usage of the terms [34]. In aerospace applications, flutter is used for coupled pitch-plunge instability of airfoil structures while galloping is more favored in civil engineering applications for one degree-of-freedom instability of bluff bodies. In other words, the term galloping is typically used to denote structural vibrations that can essentially be captured using a single mechanical degree-of-freedom, whereas the term flutter involves two mechanical degrees-of-freedom, namely pitching and plunging.*Galloping versus vortex-induced vibration:*Unlike galloping, the lift forces associated with vortex shedding are not caused by the asymmetry of the circulation bubbles trapped under the shear layer but rather by the pressure fluctuations resulting from the alternating nature of the shed vortices. In general, depending on the upstream flow velocity, the same bluff body can undergo either galloping or VIV. Nakamura and Tomonari [35] demonstrated that bluff bodies that have a longer afterbody tend to be more prone to galloping, while those with a shorter afterbody tend to be more prone to VIV. Nonetheless, VIV usually occurs at lower flow velocities and is followed by galloping oscillations.*Vortex-induced vibration versus wake-galloping:*The difference between wake-galloping and VIV lies in the location of the oscillating body with respect to the shedding vortices. In wake-galloping, the oscillating body is placed in the downstream of an obstacle. As such, the body moves due to its interaction with the vortices shedding from the obstacle. In VIV, on the other hand, the body oscillates due to alternating lift forces resulting from the vortices shedding off its own trailing edge.

## Galloping Micropower Generators

When the bluff body shown in Fig. 9(a) is connected to a restoring element (spring) to counter the lift force, and the bluff body is prone to the galloping instability as described in Sec. 3.1.1, the body undergoes steady-state fixed-amplitude periodic motions when *U* > *U*_{cr} (see Fig. 9(b)). Using an electromechanical transduction element, the resulting periodic oscillations can be used to produce a periodic current, *I*(*t*), in a closed-circuit loop providing a scalable energy transduction mechanism. Below *U*_{cr}, on the other hand, the energy pumping mechanism is not large enough to overcome energy dissipation resulting from the electrical and mechanical damping and the bluff body cannot sustain its steady-state motion.

^{(1)}Minimizing the cut-in flow speed or*U*_{cr}, at which the steady-state oscillations are initiated.^{(2)}Maximizing the electric output as a function of the flow speed.

These two parameters are, in turn, dependent on the flow pattern around the harvester, namely the relative size of the circulation bubbles on the top and bottom surface of the harvester as function of the angle of attack.

To analyze and optimize the cut-in flow speed and output power of galloping MPGs, reduced-order models are typically developed. In that regard, several lumped- and distributed-parameter models have been presented in the literature. A comparison of the different modeling approaches, for bluff bodies suspended by a cantilever beam, has been performed by Zhao et al. [36]. They used a parametric case study to analyze the generator's response obtained from a lumped-parameter single degree-of-freedom model as compared to a single-mode and three-mode representation of the dynamics of the beam. They showed that there are negligible differences between the response of the single-mode and the three-mode reduced-order models, and that overall, the lumped-parameters approximation using an effective mass, damping, and stiffness terms as shown in Fig. 10(a) can predict the behavior accurately considering its simplicity.

The lumped-parameter model used to describe the dynamics of a galloping MPG consists of a single degree-of-freedom mechanical oscillator coupled to a first-order energy harvesting circuit. As shown in Figs. 10(b) and 10(c), the energy harvesting circuit is an *RC* circuit for piezoelectric transduction and *RL* circuit for electromagnetic transduction. The equations governing the motion can be expressed in the following form:

Here, the dot represents a derivative with respect to time, *t*; *M* represents the effective mass of the bluff body and the supporting structure; while *K*, *C*, *θ*, and *F _{a}* are, respectively, the linear stiffness, damping, electromechanical coupling coefficient, and lift force acting on the prism. For piezoelectric coupling,

*C*is the capacitance of the piezoelectric element,

_{p}*r*is the output voltage, and

*R*is the parallel equivalent of the piezoelectric resistance and the load resistance. For electromagnetic coupling,

_{l}*L*is the inductance of the harvesting coil,

*r*is the current, and

*R*is the series equivalent of the load and coil resistance.

_{l}The lift force *F _{a}* is typically approximated using the quasi-steady approximation, which states that dynamics of a galloping body can be well-approximated by using the same aerodynamic forces as those acting on a static body positioned at an equivalent angle of attack. Such assumption is valid so long that the motion of the body is sufficiently slow when compared to the rate at which the wake is swept downstream.

Loosely speaking, if the fluid velocity is fast with respect to the body motion, the flow impacting the body will not be influenced by its own wake at any point in its oscillation, removing any feedback from the body motion. Some debate has been held over the best threshold value of wind velocity *U* to give quasi-steady flow. Paidoussis [37] documented the development of an approximate threshold, with Fung in 1955 giving a threshold *U _{qs}* = 10

*f*, where

_{n}D*f*is the frequency of body oscillation in Hertz and

_{n}*D*is the characteristic body width in meters [38]. Blevins agreed, for different reasons, in 1977 [39], but as a result of further work later increased the threshold value to

*U*= 20

_{qs}*f*in a 1990 revision [40]. A more involved estimation of $Uqs=((4fnD)/St)$ was suggested by Bearman et al. [41], taking into consideration the

_{n}D*Strouhal*number of the body. This threshold appropriately aims to move the incident velocity far from a region where the vortex shedding would interact with the galloping behavior.

Beyond a simple threshold relationship between wind and body velocity, the second requirement is that the shape of the flow be fundamentally similar to that of the stationary case. When bluff bodies gallop in pure translation, one can simply select the body as a fixed reference frame, and it is apparent that, provided the first criterion is met, the flow is indistinguishable from a steady scenario.

Using the quasi-steady assumption, the force acting on the body can be expressed via

where *C _{a}*(

*α*) is the transverse aerodynamic coefficient that is a function of the angle of attack

It can be clearly deduced from Eq. (3) that the magnitude of galloping oscillations depends on the shape of the *C _{a}* curve (Fig. 11). The aerodynamic forces continue to channel energy into the generator only as long as the sign of the oscillator's velocity, $x\u02d9$, is the same as that of

*C*(quadrants I and III). When the amplitude of oscillation becomes large and $x\u02d9$ is pushed outside quadrants I and II, the aerodynamic force acts against the direction of motion, forcing an upper limit on the response amplitude.

_{a}It turns out that the peak in the *C _{a}* curve corresponds to the smallest angle of attack at which the shear layer reattaches to the prism. Beyond the reattachment angle, the lift force decreases sharply. Therefore, increasing the reattachment angle is essential to increase the magnitude of steady-state galloping oscillations and therewith the performance of the generator.

Curves of *C _{a}* are usually obtained empirically from normal aerodynamic force measurements on a static bluff body at different angles of attack [42]. Figure 12 depicts the

*C*curves for four bluff profiles all having the same frontal width,

_{a}*D*, facing the flow: a square, a trapezoid with a 0.75

*D*rear face (trapezoid 1), a trapezoid with a 0.5

*D*rear face (trapezoid 2) and a triangle. The figure demonstrates that the triangle has the highest reattachment angle, followed by trapezoid 2, followed by trapezoid 1, then the square. Typically, larger reattachment angles correspond to larger oscillations, hence better power generation capabilities. However, for the case of the triangle and trapezoid 2, the

*C*curve takes negative values for angle of attacks roughly below 10 deg. This implies that such prisms will not gallop for small angles of attacks, requiring a very large initial condition to set them into motion.

_{a}Under the quasi-steady assumption, the *C _{a}* can be expanded using an

*m*th-order polynomial approximation in $(x\u02d9/U)$ as [42]

Here, the coefficients *A _{n}* take different values for different geometries and aspect ratios of the bluff body.

Different studies have investigated the sensitivity of the generator's response to variations in the aerodynamic force representation. In most studies, *C _{a}*(

*α*) in Eq. (3) was expanded in a power series of $tan(\alpha )$ using curve fitting. Meseguer et al. [43] investigated the influence of the fitting process, number of experimental points used, and the order of polynomial approximation of the aerodynamic force, on the efficiency of the calculations. They showed that, higher-order terms must be retained in the series expansion to reproduce the real behavior and predict the actual efficiency of the MPG. Javed et al. [44] also demonstrated that there exist significant differences in the prediction of the generator's response which result from utilizing third-, fifth-, seventh-, and ninth-order polynomial approximations of the same aerodynamic force. Javed et al. [45] employed shooting method to stable and unstable solutions of the system while considering transverse displacement of a triangular cylinder. The Floquet multipliers thus computed can be used to characterize the response's type of the harvester.

To understand the response behavior of galloping MPGs, we obtain an analytical solution of Eqs. (1) and (2) subject to Eq. (5). To this end, the equations of motion are first nondimensionalized by introducing the following dimensionless quantities in addition to the quantities already defined in Sec. 3:

where $x\xaf$ and $r\xaf$ represent, respectively, the dimensionless transverse displacement and electric quantity, *κ* is the dimensionless electro-mechanical coupling, and τ is the mechanical to electrical time-constant ratio.

The natural frequency, *ω _{n}*, of the harvester at short-circuit conditions is used to introduce the nondimensional time $t\xaf=\omega nt$, whereas the mechanical damping ratio

*ζ*is defined using

_{m}*C*=

*2*

*ζ*.

_{m}Mω_{n}Using the nondimensional parameters, Eqs. (1) and (2) can be expressed in the form

Here, the prime denotes a derivative with respect to $t\xaf$. The nondimensional lateral force coefficient becomes

Using the method of multiple scales [46], an approximate analytical solution of Eqs. (6a) and (6b) can be expressed as [7,47]

where the steady-state amplitude, *a*, of the resulting oscillations can be obtained by solving the following response equation:

Here, $\zeta e=\tau \kappa /[2(1+\tau 2)]$ represents the electrical damping component which is a measure of the energy channeled from the mechanical oscillator to the electric load, and *ζ _{T}* =

*ζ*+

_{m}*ζ*represents the total damping acting on the generator. It is interesting to note that

_{e}*ζ*takes its maximum value at

_{e}*τ*= 1 or when $Rl=(1/(Cp\omega n))$ for piezoelectric transduction and when

*R*=

_{l}*Lω*for electromagnetic transduction. These values correspond to an optimal

_{n}*R*obtained by linear impedance matching between the energy source and the load.

_{l}For a cubic polynomial approximation of *C _{a}*, Eq. (9) can be solved analytically for

*a*to obtain the transverse displacement, electric quantity, and harvested power as

where the corresponding dimensionless quantities are given by

Bibo and Daqaq [7,47] validated the analytical solution represented by the previous set of equations against numerical and experimental data. Thus, the above expressions can be safely used to understand the influence of the flow and design parameters on the response and obtain the optimal values to maximize the harvested power.

A typical response curve of a galloping flow MPG is shown in Fig. 13. The figure demonstrates that steady-state oscillations of the harvester are only achievable beyond a certain $U\u0302cr$. A quick inspection of Eq. (10) reveals that $U\u0302cr=\zeta T/(A1\mu )$. Thus, $U\u0302cr$, which represents a supercritical Hopf bifurcation, increases as the total damping, *ζ _{T}*, increases, or as

*A*

_{1}and

*μ*decrease. Note that an increase in

*A*

_{1}represents an increase in the slope of the

*C*curve at

_{a}*α*= 0.

Once the flow speed exceeds the cut-in speed, the steady-state deflection, output voltage, and power all increase monotonically with the flow speed. When inspecting Eq. (10), it is evident that the deflection, output voltage, and power increase by decreasing the cubic coefficient, *A*_{3}, or by decreasing *μ*. The dependence on *ζ _{T}* however is more complex since both

*r*and

*P*depend on

*r*

_{0}and

*R*which both indirectly influence the value of

_{l}*ζ*. Barrero-Gil et al. [48] were among the first to indicate that the best performance of a galloping MPG can be achieved by utilizing prisms with largest

_{T}*A*

_{1}and smallest

*A*

_{3}values by considering isosceles triangle-shaped cross section.

Depending on the ratio of the electromechanical coupling to the mechanical damping ratio, the electric load at which the output power is maximized can take different values. In particular, when $((\kappa /\zeta m))<2((\mu U\u0302/\zeta m)A1\u22121)$, the optimal load, *R*_{opt}, embedded within optimal time constant ratio *τ*_{opt} is given by

In this range, the optimal load takes a constant value which corresponds to that resulting from traditional linear impedance matching. On the other hand, when $((\kappa /\zeta m))>2((\mu U\u0302/\zeta m)A1\u22121)$, the optimal load is given by

Note that in this case, there exist two different values of the optimal load, both providing the same maximum output power. Figure 14 provides more details into the optimization results. It can be clearly seen that there is one optimal load for small values of *κ*/*ζ _{m}*. As

*κ*/

*ζ*is increased beyond the critical value $2((\mu U\u0302/\zeta m)A1\u22121)$, two new optimal loads branch out. These two optimal loads result in the same maximum power level with the only difference that one optimal load results in high voltage/low current while the other results in high current/low voltage.

_{m}By introducing the dimensionless quantity $U\xaf=(U\u0302\mu /\zeta T)$, Eq. (10) becomes only dependent on the aerodynamic constants *A _{n}* characterizing the cross section of the bluff body. All the other parameters are now contained within the parameter, $U\xaf$. This implies that the response of all galloping MPGs having the same aerodynamic constants, i.e. bluff body, can be described by a universal curve in the plane $U\xaf\xd7a$ irrespective of the other design parameters.

Figure 15 depicts the universal curve for a square bluff body and different design parameters of the MPG. It can be clearly seen that the data collapses nicely onto an effectively single universal curve no matter how the configuration is changed as long as the bluff body is kept the same. This universal curve is a valuable tool to directly compare the performance of galloping MPGs of different bluff bodies. For instance, as shown in Fig. 16, since the universal curve associated with the square bluff body has a lower cut-in flow speed and higher *x*/*x*_{0} as compared to the triangular and “D” cross section, then one can correctly surmise that, there must exist a combination of design parameters for which the square bluff body achieves better output power and lower cut-in speeds than the other two bluff bodies. The reader should bear in mind, however, that such conclusion does not imply that, given any arbitrary design parameters, the square profile will always produce more power than the two other profiles; it only implies that when all generators are designed based on the optimal design parameters, the square profile is capable of producing more power than the other two cross section.

Since most piezoelectric MPGs use a cantilever beam as the stiffness element, the bluff body attached to the tip of the beam does not undergo pure translational galloping. Kluger et al. [49] highlighted the importance of adjusting the angle of attack, *α*, to account for the coupled translation and rotation of the bluff body when attached to a cantilever beam. They indicated that bluff body rotation, due to the structural angle of the beam, counteracts the effective angle of attack resulting from pure translation which in turn, limits galloping oscillations. They showed that, the amplitude of a galloping generator which couples rotation with translation is always less than the amplitude of an oscillator which moves in pure translation. In fact, for a bluff body with coupled translational–rotational motion, the oscillation amplitude reaches a horizontal asymptote with increasing flow speed. For a purely translational bluff body, however, the amplitude continues to increase with increasing wind speed.

Xu et al. [50] extended the analysis of Kluger et al. [49] and derived analytical expressions for the output power and efficiency of the MPG accounting for body rotation. A comparative analysis was performed against a MPG with bluff body in pure translational motion. They showed that, a harvester with pure transverse motion always outperforms an MPG employing a cantilever beam as the restoring element. Furthermore, the response of the harvester with a cantilever beam depends significantly on the beam's length, with the output power decreasing for shorter beams due to induced rotation. Noel [51] also concluded that, when the bluff body rotation is significant, which occurs for shorter beams, the quasi-steady assumption of the aerodynamic force fails to capture the actual behavior of the harvester. He demonstrated that, in such a scenario, a new lift coefficient, *C _{a}*, which depends on both the angle of attack and the flow speed can be used to accurately capture the MPG response.

Most of the studies on galloping MPGs have considered piezoelectric transduction because of their high energy density and scalability [18,52–55]. Few studies, on the other hand, have incorporated an electromagnetic transduction such as Ali et al. [56] and Dai et al. [57]. Nonetheless, from the analysis point of view, the distinction between these two approaches lies in the dimensionless time constant ratio *τ* as explained earlier in Eq. (6b). Thus, as long as a single transduction mechanism is used, the conclusions presented earlier hold.

Javed et al. [58] analyzed the response of a galloping MPG employing a hybrid piezoelectric-electromagnetic transduction. They noted that, a harvester with a single transduction mechanism outperforms the proposed hybrid harvester. They attributed the reduced performance of the hybrid system to a substantial increase in the electric damping which shifts the cut-in flow speed into higher values and reduces the overall harvested power.

Most of the theoretical and experimental performance studies assume a purely resistive electric load. For real applications however, a more complex interface circuit for alternating current (AC)-to-direct current (DC) signal rectification and regulation should be used before the harvested power is channeled to the load. To simplify the performance analysis associated with complex circuits, Tang et al. [59] used an equivalent circuit representation in which the entire aero-electromechanical system is modeled in a circuit simulator. The generator's response is then studied with standard AC and DC interface circuits. They showed that, at the optimal load, coupling the generator to a standard AC interface shifts the cut-in flow speed to relatively higher values when compared to the DC interface. On the other hand, at the higher values of the flow speed, the harvested power with the DC interface is lower than that with the AC interface.

Zhao et al. [60] also used the equivalent circuit model to investigate the feasibility of utilizing the synchronized charge extraction (SCE) interface to improve the performance of galloping MPGs. Comparison with a standard DC interface showed that the output power from SCE is independent of the electric load, hence eliminating the need for impedance matching required in standard DC interface. It was also noted that the displacement amplitude of the harvester with SCE is smaller which enhances the durability and fatigue life of the structure. Zhao et al. [61] derived analytical solutions for the response of galloping MPGs when interfaced with standard AC, DC, and SCE circuits. They validated those solutions against experimental data and equivalent circuit model simulations.

Due to its significant impact on the performance of the MPG, many researchers investigated the influence of the bluff body shape on the performance of the harvester. In one demonstration, Zhao et al. [54] conducted experiments characterizing the influence of the shape of the bluff body on the performance of the MPG. They compared square (40 mm × 40 mm), rectangular (40 mm × 60 mm), rectangular (40 mm × 26.7 mm), and equilateral triangle (40 mm side) shapes. The study also compared square bluff bodies of different sizes and masses. The same group extended the comparative analysis to include the D-section (40 mm diameter) in Ref. [62]. Both studies showed that a square bluff body with larger cross-flow area yields the best output power among all tested sections. Contrary to this result, Ali et al. [56] incorporated finite element models to obtain the lift and drag coefficients for different geometries and assess their susceptibility to galloping. Their theoretical predictions were validated experimentally and demonstrated that the D-section yields maximum power. These contradictory results do not indicate an issue with the work done by either of these authors but rather is due to the nonoptimal design parameters used in the experiments. As described previously, using the universal design curves which eliminates the effect of the design parameters on the harvester's performance, an MPG with a square-sectioned bluff body can always be designed such that it outperforms the one incorporating the “D” section.

In other demonstrations, Abdelkefi et al. [63,64] utilized the aerodynamic expressions of the lift force reported in the literature to generate the power curves as function of the flow speed and electric load for several bluff bodies. They noted that some sections are better suited for energy harvesting at low wind speeds while other sections provide better results for higher wind speeds.

Kluger et al. [49] investigated shape optimization of bluff bodies with the objective of minimizing the cut-in wind speed, increasing amplitude sensitivity to wind, and minimizing or eliminating amplitude hysteresis. The analysis assumed that a continuously changing bluff body shape continuously changes its aerodynamic force coefficients. This assumption was supported by the experimental data as in the case of adjusting a square-section bluff body to a trapezoid, and finally to a triangle by incrementally decreasing the rear-to-front face ratio. Using this analysis, they illustrated that the square section is the most optimal candidate for power generation.

Hu et al. [65] conducted experiments on galloping MPGs with fins attached to the corners of the square bluff body. They showed that, attaching fins to the leading edge with an optimal length of *D*/6 can improve the harvested power by a factor of 1.5.

Rather than looking at the aerodynamic coefficients only to assess performance, Noel et al. [66] related the harvester's performance directly to the flow pattern around the bluff body. They showed that the prism which has the largest reattachment angle of the shear layer is the one which is most suitable for galloping MPGs. Based on this analysis, they proposed attaching a splitter plate to the trailing edge of the bluff body to increase the reattachment angle and showed experimentally that the power output of an MPG incorporating a square prism increases by 70% when a splitter plate is attached and that the output power of an MPG incorporating a trapezoidal prism increases by 50% when a splitter plate of an optimal length is attached. They also showed that the splitter plate reduces the cut-in flow speed of the generator.

In an attempt to provide a practical implementation and extend the operation life of galloping flow MPGs, Ewere et al. [67] conducted experiments on MPGs designed with bump stoppers to limit the steady-state oscillation amplitude and, thereby, reduce the bending stresses in the beam. They showed that this technique provides longer fatigue life of the harvester but causes the output power to drop by 36%.

Zhao et al. [68] proposed a two degrees-of-freedom piezoelectric galloping MPG with a cut-out beam design and magnetic interaction as shown in Fig. 17(a). This design improved the efficacy of power generation in the low wind speed range as it enhanced the output power and reduced the cut-in wind speed. At higher wind speeds, however, the conventional single degree-of-freedom MPG was shown to outperform the proposed design. In another study, Zhao and Yang [69] proposed adding a beam stiffener in cantilevered galloping MPGs as shown in Fig. 17(b). The stiffener serves to magnify the effective electromechanical coupling by modifying the fundamental mode shape of the beam such that it leads to a higher strain energy in the piezoelectric layer. It was shown that such addition can improve the output power by as much as ten times.

Bibo et al. [70] investigated improving the performance of galloping MPGs by intentionally introducing a nonlinear stiffness element in the design. They utilized magnets of different arrangements and polarities to produce a softening, hardening, or bistable restoring force, Fig. 17(c). Analysis showed that there exists a unique universal curve that accurately represents each of those cases independent of the individual design parameters. For certain airflow parameters, the universal curves can be used to optimize the restoring force. Based on the optimal results, it was concluded that, for similar design parameters, a bistable MPG outperforms all other configurations as long as the interwell motions are activated. If the motion is limited to a single well, however, the MPG incorporating a softening restoring force provides the best performance.

Following the results of Bibo et al. [70], Zhang et al. [71] incorporated a buckled beam in the design of a MPG to introduce bistability and improve performance. The resulting bistable oscillations were then used to excite a piezoelectric cantilever beam attached at the middle of the buckled beam. They showed that bistability can be used to improve the output power.

Vicente-Ludlam et al. [72] investigated the potential of improving the performance of galloping MPGs by incorporating a dual-mass configuration in which the galloping mass is elastically coupled to another mass resulting in a two degrees-of-freedom system. They showed that, when selecting the design parameters appropriately of the coupled oscillators, the efficiency and range of wind speeds at which the efficiency is kept higher than a given level can be enhanced significantly.

Abdelkefi et al. [73] studied temperature effects on the properties of piezoelectric materials and its impact on the overall performance of the harvester. They evaluated the response for a temperature range, between –20 °C and 40 °C, and showed that changes in the temperature can lead to significant variations in the cut-in wind speed and the harvested power. When the harvester is operated at the optimal resistive load, however, the temperature effects on the output power become negligible and can be ignored.

Daqaq [74] derived expressions for the average harvested power of galloping MPGs as function of wind statistical averages. He used the probability density function of actual wind data to estimate the output power of a galloping MPG placed in different locations in the UK. He also studied the influence of wind direction on the average power and showed that the direction of the prevailing wind is not necessarily the ideal direction for harnessing energy.

Cantilevered plates subject to axial flow have been shown to exhibit responses following the typical normal form of a Hopf bifurcation in which the plate can undergo steady-state oscillations only beyond a critical wind speed. Such responses have been also utilized for micropower generation. In one demonstration, Tang et al. [75] introduced a galloping cantilevered flexible plate in axial flow for the purpose of energy generation. They analyzed the influence of the different vibration modes and segments along the plate on the output power. They also evaluated the performance of the MPG in terms of key design parameters such as fluid-to-plate mass ratio. They presented two simulation case studies and compared results against a horizontal axis wind turbine. They demonstrated that a galloping flexible plate can be designed to achieve high performance with compact size, and it is therefore, a very promising candidate for power generation.

Dunnmon et al. [76] developed an aeroelastic model of a device similar to the MPG proposed by Tang et al. [75]. They used the vortex lattice method to better understand the aeroelastic nonlinearities responsible for the resulting self-excited limit-cycle oscillations above the onset speed. They employed this understanding to draw conclusions crucial to designing generators that take full advantage of the unique dynamic properties of the flapping plate.

Michelin and Doare [77] performed numerical simulations to study the power generation efficiency of a galloping flexible plate as a function of the flow speed and other design parameters. Their analysis demonstrated a critical challenge in the robustness of the piezoelectric flapping plates because fluctuations in the wind speed lead to switching between the different flapping modes. They showed that this switching can drop the output power and efficiency of the generator. Building on the work of Michelin and Doare [77], Piñeirua et al. [78] used numerical simulations to optimize the position, number, and dimension of piezoelectric patches used in flapping plate generators. They presented the optimal configurations as function of the electromechanical coupling and fluid-to-plate mass ratio.

Sirohi and Mahadik [52] were among the first to perform experimental testing to demonstrate the potential of utilizing galloping MPGs for powering sensor nodes in outdoor applications. In one study, they used a D-section bluff body of 23.5 cm length and 2.5 cm width and reported a maximum output power of 1.14 mW at 4.7 m/s wind speed. In another study, they used a triangular cross section bluff body and fabricated a prototype measuring 33 × 16 × 4 cm^{3}. Experiments showed that, at the optimal load, the generator was capable of producing maximum output power of 53 mW at 5.2 m/s wind speed [18].

Jung et al. [79] studied the feasibility of exploiting wind-induced galloping oscillations of stay cables to power a wireless sensor node installed on an in-service cable-stayed bridge. They showed that, under moderate wind conditions, the MPG can generate sufficient energy to power a wireless sensor placed on the vibrating cable.

In another demonstration, Xiang et al. [80] presented an integrated indoor sensing system that fully sustains itself through a piezoelectric galloping MPG which harvests energy from the airflow exiting the HVAC outlets. The system was used to sense and report the airflow speed in microclimate control applications.

Tsujiura et al. [81] developed a 20 mm × 20 mm thin-film bimorph piezoelectric galloping MPG. The generator produced 36.4 *μ*W of average power at 8 m/s wind speed. He et al. [82] also conducted wind tunnel experiments with a micromachined MEMS piezoelectric galloping MPG. The device, which was of arbitrary design and far from optimal, produced 2.27 *μ*W at 16.3 m/s.

Table 1 reports the cut-in wind speed, the maximum power per unit volume of the bluff body, and the maximum power per unit area of the piezoelectric element for several galloping MPGs. The cut-in flow speed of the reported devices ranges between 1.9 and 3.6 m/s, which is ideal for harvesting energy from low wind speeds. The best device produces around 0.54 mW of average power per cm^{2} of the piezoelectric layer at a wind speed of 6 m/s.