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Engineering Honors Thesis

—

Fatigue Analysis of a Gallop-based

Piezoelectric Wind Energy Harvester and

Strategies for Long-Term Performance

Optimization

December 3, 2021

By Andrew Trostle

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Abstract

With great demand for novel forms of renewable energy, there has been substantial interest

in wind energy harvesting based on aeroelastic excitations with the aid of piezoelectric

materials. While most research has been dedicated to optimizing power output [17],

relatively little research has been done on the device’s behavior in response to fatigue,

which may be a key failure mode given its persistent cyclic loading. A Python program was

created to simulate a galloping piezoelectric wind energy harvesting (GPEH) under varying

wind conditions in real time, monitoring performance metrics until fatigue failure. This can

generate Monte Carlo distributions for optimizing GPEH system parameters for durability.

This could then determine if such a form of renewable energy is currently feasible and if so,

provide a database of effective materials for the device’s implementation. In tandem with

this, analytical optimization methods were tested, but remain partially inconclusive,

requiring better dimensional analysis to simplify the myriad of system parameters.

However, initial runs of the code still provided useful data, identifying an optimal

performance state of all those tested, indicating that the tradeoff between performance and

durability is not either-or.

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Table of Contents

1.0 Introduction 4

2.0 GPEH Mechanics and Theoretical Background 4

2.1 GPEH in Previous Literature 4

2.1.1 Aerodynamic Model 5

2.1.2 Structural Model 6

2.1.3 Results 7

2.2 Fatigue Analysis 9

2.2.1 Intro 9

2.2.2 Miner’s Rule 10

2.2.3 Crack Growth Equations 11

2.2.4 Additional Fatigue Effects: Crack-Induced Compliance 12

3.0 Numerical Simulation Setup 13

3.1 Assumptions and Default Model Features 13

3.2 Probabilistic Wind Model 15

3.3 Workflow Algorithm and Code Features 16

3.3.1 General Layout 16

3.3.2 Newmark-Beta Method 19

3.3.3 Adaptive Time-Stepping 20

3.3.4 Fracture Toughness Modeling 20

4.0 Numerical Simulation Results 21

4.1 Qualitative Results 22

4.2 Quantitative Results 23

5.0 GPEH Analysis and Optimization Strategy 27

5.1 Analytical Model for GPEH Performance 27

5.2 Optimization Strategy 29

5.2.1 Effect of Material Choice on Efficiency and Durability 29

5.2.1.1 Piezo-ceramics versus Piezo-polymers 29

5.2.1.2 Performance Optimization of a Piezo-Ceramic-Polymer 30

Composite

5.2.2 Methods of Analytically Correlating Design Parameters and

Performance Variables 32

5.2.2.1 Probability-Based Model 32

5.2.2.2 “Lifetime Efficiency”-Based Model 37

6.0 Conclusion 41

7.0 Bibliography 42

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1.0 Introduction

An emerging design concept in the field of small-scale renewable energy systems is the

Galloping Piezoelectric Wind Energy Harvester (GPEH). This design is based on two

phenomena: the aeroelastic phenomena of gallop, a circumstance in which lift dynamics

result in a net negative damping of the lifting body subjected to a quasi-steady fluid flow,

and the piezoelectric effect, exhibited in certain crystalline materials, where vibratory

oscillations in the material can result in an alternating current. Many researchers have

explored various configurations of the basic GPEH mechanic, resulting in gradual

improvements in system performance, achieving a maximum power of 53 mW [17].

With prior research proving that the performance of such a device can be optimized enough

to power low-power sensors and similar instruments, it can now be determined whether

such a system is feasible for long-term use. Sirohi & Mahadik [17] note visible fatigue

damage to their galloping device after testing, implying rapid degradation which must be

addressed. In this paper, we intend to test the fatigue life of a basic GPEH device subjected

to a realistically varying wind signal through numerical simulation and experimental

testing, as well test the effects of material properties on performance and endurance.

Repeated simulation with varying system parameters will lead to a brute-force optimization

of lifetime energy output for this highly nonlinear system.

2.0 GPEH Mechanics and Theoretical Background

2.1 Previous Literature

This analysis will be largely based on a model presented by Sirohi & Mahadik [17]. They

use a triangular prism tip body configured vertically, simply supported downstream by two

thin aluminum beams with a clamped support. On the clamped end, two strips of

piezoelectric material are attached to the top and bottom of each aluminum beam. The

device covers a total area of 16 x 25 cm. This model was simulated with numerical

integration in Matlab and then verified experimentally in a wind tunnel.

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Figure 2.1.0: GPEH Diagram [17]

2.1.1 Aerodynamic Model [17]

The numerical method begins with the calculation of aerodynamic forces. For a prism of

chord length 𝑏 and prism length 𝐿, a windspeed 𝑉∞ will result in a net vertical force 𝐹𝑦

equal to:

𝐹𝑦 = 𝐿𝑐𝑜𝑠𝛼 + 𝐷𝑠𝑖𝑛𝛼

Where:

𝐿 =1

2𝜌𝑉𝑅

2𝑏𝐿𝐶ℓ(𝛼), 𝐷 =1

2𝜌𝑉𝑅

2𝑏𝐿𝐶𝑑(𝛼)

and

𝛼 = 𝛼0 − tan−1 (�̇�𝑡𝑖𝑝

𝑉∞) + 𝑤𝑡𝑖𝑝

′

𝑉𝑅2 = 𝑉∞

2 + �̇�𝑡𝑖𝑝2

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𝑤 here denotes the beam displacement and 𝛼0 is the default angle of attack. For a triangular

cross-section in this orientation with the oncoming wind gust, 𝛼0 = 60∘. Section lift and

drag coefficients are a function of angle of attack, here based on a lookup table provided in

[1].

Figure 2.1.1.1: Section Lift and Drag Coefficients for an Equilateral Triangle Cross-section

Additionally, Sirohi & Mahadik calculated the Den Hartog Criterion based on the above

coefficients. This criterion determines whether the aerodynamic state will result in a

negative damping, also known as transverse galloping. The criterion is defined as:

𝐻(𝛼) =𝜕𝐶ℓ

𝜕𝛼+ 𝐶𝑑 < 0

For a triangular cross-section the following angle-of-attack ranges are susceptible to

galloping:

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Figure 2.1.1.2: Den Hartog Criterion for an Equilateral Triangle Cross-section [17]

2.1.2 Structural Model

With the aerodynamic model functioning, the structural model can now be constructed. For

this, Sirohi & Mahadik [17] used the Rayleigh-Ritz method, an approximate energy method

where mode shapes for the beam displacement are assumed and the weighted sum of these

modes is solved numerically. Beam deflection can be approximated as:

𝑤(𝑥, 𝑡) = 𝜙1(𝑥)𝛾1(𝑡) + 𝜙2(𝑥)𝛾2(𝑡), 𝑤ℎ𝑒𝑟𝑒 𝜙1(𝑥) =𝑥

𝐿𝑏𝑒𝑎𝑚, 𝜙2(𝑥) = (

𝑥

𝐿𝑏𝑒𝑎𝑚)3

𝜙 represents each shape function and 𝛾 represents the “weight” of each mode on

displacement over time. The mass and stiffness matrices of this system of ordinary

differential equations are of the form:

𝑀𝑖𝑗 = ∫ 𝜌𝐴𝜙𝑖(𝑥)𝜙𝑗(𝑥)𝑑𝑥𝐿𝑏𝑒𝑎𝑚

0

𝐾𝑖𝑗 = ∫ 𝐸𝐼𝜙𝑖′′(𝑥)𝜙𝑗

′′(𝑥)𝑑𝑥𝐿𝑏𝑒𝑎𝑚

0

The paper goes into further detail on including electric displacements of the piezoelectric

material into this system of ordinary differential equations, which includes a coupling

matrix representing the resistance to stress introduced by the induced voltage field.

However, this project will be utilizing the coupled electromechanical model from [4].

The numerical method works by providing the beam tip an initial velocity, calculating the

vertical force generated at each moment in time, applying that load to the differential

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equation, updating the beam displacement, and repeating this step in time steps via a

numerical integration method.

For our code simulation, two mode shapes are used which both satisfy the boundary

conditions:

𝑤(0) = 0, 𝑤′(0) = 0, 𝑤′′(𝐿) = 0

𝜙′′1(𝑥) = (1 −𝑥

𝐿) , 𝜙2

′′(𝑥) = (1 −𝑥

𝐿)2

Note 𝑤′′(0) = 𝜙1′′(0)𝛾1 + 𝜙2

′′(0)𝛾2 = 𝛾1 + 𝛾2. This yields the stiffness and mass matrices:

[𝑀] = 𝜌𝐴𝐿5 [

11

420

173

20160173

20160

13

810

] , [𝐾] = 𝐸𝐼𝐿 [1/3 1/41/4 1/5

]

2.1.3 Results

Sirohi & Mahadik’s research [17] showed a rough correlation between the numerical and

experimental model. Experimentally, the device was able to achieve a maximum power of

53 mW from all four strips. While the numerical model predicted a purely linear

relationship between incident windspeed and the device’s output power, the experiment

demonstrated a sharp dropoff in output power after 12 mph winds. The paper attributes this

mismatch to the wind tunnel’s fans, which may have introduced turbulence or unsteady

flow, violating the quasi-steady flow assumption of the numerical model. However, further

investigation is required into the matter as this could present a maximum performance limit

to gallop-based energy harvesters. The paper also notes that the piezoelectric material had

visible signs of fatigue after the wind tunnel experiment, which may have inhibited proper

performance as well. Regardless, this represents the next critical performance metric for the

device.

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Figure 2.1.3.1: Correlation between Measured and Predicted output power for an incident

wind speed of 10 mph. [17]

Figure 2.3.1.2: GPEH Performance vs Wind Speed and Load Resistance [17]

2.2 Fatigue

2.2.1 Introduction

The next step in evaluating the feasibility of a GPEH system is to understand its

susceptibility to fatigue. While most engineered systems are designed to minimize any

aeroelastic excitations, this device intends to amplify such excitations. Such load cycling is

detrimental to structures, accruing damage in the form of cracks which eventually lead to

failure.

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Understanding the effective fatigue life of these devices gives us an insight into their

effective lifetime capacity for energy harvesting. As the GPEH is cited as a possible

alternative to chemical batteries in situations where the replacement of these batteries is

difficult or undesired, we must verify that such a device can provide more energy than a

standard chemical battery, meaning it would require less frequent maintenance.

The simplest model of fatigue is the S-N curve. This represents the relationship between a

given stress and the number of load cycles of that stress a material can withstand before

failure. These curves are created through extensive testing of materials. These curves tend

to follow a linear-log trend [24], with 1 cycle for stress equal to the ultimate tensile stress,

going all the way to a point known as the endurance limit which corresponds with many

orders of magnitude more cycles . The endurance limit is the minimum stress at which

fatigue failure is achievable. For example, a standard aluminum material has an endurance

limit of around 10 ksi, lasting about 107 cycles [9].

The reason for the limit is based in the yielding stress of materials. As stress increases and

the stress-strain relationship grows increasingly nonlinear, the material will begin to

experience permanent deformations which compound with each cycle. But at low stresses,

where the stress-strain relationship is effectively linear, no such permanent damage occurs.

This theory is supported by these experimental S-N curves, which tend to have higher

endurance limits for brittle materials and lower endurance limits for ductile materials. For

example, lead zirconate titanate, or PZT, has an ultimate tensile strength of 80 MPa and an

endurance limit of 60 MPa [13]. While aluminum is only free from fatigue failure for

stresses below one-seventh its maximum allowable stress, PZT is free from fatigue failure

for stresses below three-fourths its maximum allowable stress.

However, such S-N curves are only applicable for constant-amplitude load cycles. Given

that real-world wind scenarios will have variable wind speeds and thus will induce

variable-amplitude load cycles, we must use a new method of fatigue analysis. The two

most common are Miner’s Rule, which is a very approximate method based on the idea of

cumulative damage, and a variety of Crack Growth Equations, which use differential

equations to model the growth of cracks of an initial size.

2.2.2 Miner’s Rule

Miner’s Rule is based on the idea of cumulative damage, that each load cycle contributes

some fractional amount of “damage” to a parameter D, which starts at zero and finally fails

when 𝐷 = 1. To determine the amount of damage, this method uses S-N curves as

reference points and assumes that this accumulated damage is linear.

For example, if a stress of amplitude 𝑆 is applied for 𝑛 load cycles, and that stress 𝑆 is

correlated with 𝑁 cycles until failure, then the amount of damage accrued is equal to [3]:

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Δ𝐷 =𝑛

𝑁

This method is extremely simple, however this simplicity comes at the cost of theoretical

rigor. As cracks grow, the stress concentration near the propagating tip of the crack grows

exponentially [19], disproving the linear damage assumption of Miner’s Rule. However, no

current alternative method following cumulative damage has been proven to out-predict

Miner’s Rule [23]. Miner’s Rule has stood the test of time through its statistical accuracy.

However, the lack of fidelity in the model beyond being based in experimental data makes

this less preferred for our fatigue analysis.

2.2.3 Crack Growth Equations

This method provides a differential equation of crack growth per load cycle, from an initial

flaw size, based on the geometry of the loading scenario and experimentally derived curve-

fit parameters.

For a crack of length 𝑎 and currently applied stress 𝜎, we can calculate a parameter known

as the stress intensity factor, 𝐾 = 𝑓(𝜎, 𝑎). There are lookup tables for a given boundary

condition at the site of the critical plane, where stress is greatest and/or most persistent.

While a beam undergoing bending has a certain associated formula for stress intensity

factor, the individual flanges of the beam will effectively be under pure tension or

compression. Thus, the crack growth for a center crack of length 2a is calculated from the

basic formula [19]:

𝐾 = 𝜎√𝜋𝑎

Given the loading cycle has a range 𝜎𝑚𝑎𝑥 and 𝜎𝑚𝑖𝑛, there is a corresponding range 𝐾𝑚𝑎𝑥

and 𝐾𝑚𝑖𝑛. The crack growth equation relies on Δ𝐾 = 𝐾𝑚𝑎𝑥 − 𝐾𝑚𝑖𝑛.

Every crack growth equation correlates Δ𝐾 with the per-cycle crack growth 𝑑𝑎

𝑑𝑁 through

varying curve fits. The simplest is the Paris-Erdogan Equation [15]:

𝑑𝑎

𝑑𝑁= 𝐶(Δ𝐾)𝑚

Where 𝐶 and 𝑚 are material constants. However, this model disobeys the concept of a

material endurance limit, as a load cycle with Δ𝐾 = 0, in other words, a load cycle of

amplitude zero, will still contribute a crack growth of 𝐶.

Experimentally derived curves between Δ𝐾 and 𝑑𝑎

𝑑𝑁 typically feature three key regions [20].

The first region, at low Δ𝐾 values, in which 𝑑𝑎

𝑑𝑁 asymptotically approaches zero at the Δ𝐾-

value corresponding to the fatigue threshold, an analog for the endurance limit. The second

region, at medium Δ𝐾 values, features power law crack growth, quantified by the Paris-

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Erdogan formula. The third region, at high Δ𝐾 values, where 𝑑𝑎

𝑑𝑁 asymptotically approaches

infinity at a Δ𝐾 value known as the fracture toughness. While more involved formulae can

accurately account for all three of these regions and the transition between them, the more

involved curve-fit parameters are not so widely studied. Additionally, standard materials

such as metals and ceramics tend to have an insignificant fatigue threshold, and thus the

Region II behavior can be assumed all the way to Δ𝐾 = 0. Likewise, the transition from

Region II to Region III is generally very abrupt and thus failure can simply be asserted once

Δ𝐾 = 2𝐾𝐼𝑐. As crack growth is roughly exponential, a steadily growing crack will exceed

fracture toughness very rapidly, and thus failure is already inevitable regardless of curve

modelling.

2.2.4 Additional Fatigue Effects: Crack-Induced Compliance

Galloping problems are best understood as an enforced displacement problem, in which

beams oscillate in their natural frequency modes, to an amplitude determined by their own

stiffness, damping, and aerodynamics. As galloping induces a positive feedback, the tip

displacements will often be much greater than a simple quasi-static aerodynamic force

analysis would indicate.

One result of this model is that it introduces the possibility of stable crack growth. If a

steady load is applied to the device rather than a steady amplitude, the crack will grow due

to a high stress intensity factor, in turn causing an even higher stress intensity factor, and so

on as the crack grows exponentially. With a steady displacement, we observe that crack

growth will drastically reduce the stiffness of the device, decreasing the applied stress as

well as the stress intensity factor, potentially halting crack growth.

The stability of the crack is also dependent on aeroelasticity, as less stiff structures tend to

experience higher loads than those structures which are too stiff to gallop, flutter, etc. Thus,

crack-induced compliance may have the effect of increasing the effective load experienced

by the beam.

To determine the outcome of this multi-faceted problem, we integrate a crack-based

negative stiffness matrix into our Rayleigh-Ritz Model. This is done by adding a negative

strain energy term to represent the strain energy dissipated by the crack. We use the strain

energy release rate, a variable meant to represent the loss of strain energy per unit area

given by [20]:

𝐺 =𝐾2

𝐸, 𝑉𝑐𝑟𝑎𝑐𝑘 = −∫ 2𝑡 𝐺(𝜅, 𝑎𝑏𝑒𝑎𝑚) 𝑑𝑎

2𝑎𝑏𝑒𝑎𝑚

0

− ∫ 2𝑡 𝐺(𝜅, 𝑎𝑝𝑖𝑒𝑧𝑜) 𝑑𝑎2𝑎𝑃𝑍𝑇

0

= −∫2𝑡𝜋𝜎2𝑎

𝐸 𝑑𝑎

2𝑎

0

= −∫ 2𝑡𝜋𝑎𝐸(𝑧𝜅)2 𝑑𝑎2𝑎

0

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Where a is the crack length in either the beam or the piezoelectric strip, 𝑉𝑐𝑟𝑎𝑐𝑘 is the

potential energy stored in the crack, b is the width of the piezoelectric strip and of the

beam, t is the thickness of the ply or flange, kappa is the curvature at the base of the beam,

and z is the distance from the center to the extremity of the cross-section. Note the

multiplication by two since bending fatigue will induce a crack on bottom and top. Since

𝑧 =𝑏

2 and 𝜅(0) = 𝜙1

′′(0)𝛾1 + 𝜙2′′(0)𝛾2 = 𝛾1 + 𝛾2:

𝑉𝑐𝑟𝑎𝑐𝑘 = −∫1

2𝐸𝑏2𝑡𝜋𝑎𝑏𝑒𝑎𝑚(𝛾1 + 𝛾2)

2𝑑𝑎𝑏𝑒𝑎𝑚

2𝑎𝑏𝑒𝑎𝑚

0

− ∫1

2𝐸𝑏2𝑡𝜋𝑎𝑃𝑍𝑇(𝛾1 + 𝛾2)

2𝑑𝑎𝑃𝑍𝑇

2𝑎𝑃𝑍𝑇

0

𝑉𝑐𝑟𝑎𝑐𝑘 = −𝐸𝑏2𝑡(𝑎𝑏𝑒𝑎𝑚2 + 𝑎𝑃𝑍𝑇

2 )(𝛾1 + 𝛾2)2

𝜕𝑉𝑐𝑟𝑎𝑐𝑘

𝜕𝛾𝑖= −2𝐸𝑏2𝑡(𝑎𝑏𝑒𝑎𝑚

2 + 𝑎𝑃𝑍𝑇2 )(𝛾1 + 𝛾2) = 0

We can finally put this in matrix form:

[𝑀]{𝛾}̈ + [𝐶]{𝛾}̇ + [𝐾]{𝛾} − [𝐾𝑎]{𝛾} = {𝐹}

Where:

𝐾𝑎,𝑖𝑗 = 2𝐸𝑏2𝑡(𝑎𝑏𝑒𝑎𝑚2 + 𝑎𝑃𝑍𝑇

2 )

As we can see, crack growth quadratically reduces the stiffness of the beam. Plugging in

some cursory values, we can see that the loss in stiffness rapidly goes from trivial to

substantial, long before the crack becomes critical, meaning this will have significant

effects on the outcome of the simulation.

3.0 Numerical Simulation Setup

3.1 Assumptions and Default Model Features

To model the myriad of features affecting GPEH performance and durability, we will

simulate a general GPEH device with custom device parameters in Python. The device will

respond to a synthetic wind signal based on real anemometer data, from which tip

displacement, stress, voltage, and crack length with be calculated until the device fails.

While the code is set up to be endlessly customizable to test a variety of factors, some basic

design factors were arbitrarily chosen ahead of time to keep the code simple.

The first design factor to be chosen was the tip-body cross-section. As it is meant to

respond to wind signals in all directions, a symmetric body which satisfied the Den Hartog

criterion for the most angles of attack is desired. After analyzing section aerodynamic

coefficients for a number of regular polygons, it was determined that equilateral triangles

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have the most galloping “coverage”. Given sufficient windspeed, an equilateral triangle

will gallop if the wind blows normal to one of its sides, plus or minus 40 degrees, resulting

in a coverage of 67% [1].

The next decision would be the GPEH structural geometry. As PZT is the main

piezoelectric material being studied for this project, a cantilevered beam with short strips of

piezoelectric material adhered to all sides is considered standard and optimal for brittle

piezoelectric materials. The cantilevered beam cross-section must be symmetric, unlike the

simple rectangular cross-section used by Sirohi & Mahadik [17] and curved surfaces would

complicate the application of piezoelectric material. Also, a non-hollow cross-section

would make the beam very stiff unless very compliant material is used. Therefore, the

cross-section is decided to be a hollow, square tube. While less important for aerodynamic

analysis, this detail is extremely important to determine the parameters for crack

propagation. For initial simulation run-throughs, the beam was chosen to be made of 7075

Aluminum Alloy and the piezoelectric material was chosen as PZT-5H.

For the probabilistic wind model, a month of 1 Hz anemometer data on the coast of Froya,

Norway was sampled [6].

While the GPEH setup used in [17] was designed to only gallop in response to a single

wind-direction, the multi-directional GPEH used in the simulation would look something

like the below figure.

Cantilever Support Piezoelectric Strips

Triangular Prism

Tip Bodies

Beam

Rigid

Connections

Figure 3.1.1:

GPEH Diagram

Incoming Wind Galloping

Motion

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3.2 Probabilistic Wind Model

An important aspect of generating this synthetic wind signal was capturing both speed and

direction, and additionally simulating persistence, or the tendency for wind gusts to stay

constant for an arbitrary duration. If the wind was modeled with a basic Weibull or

Rayleigh probability distribution [21] which was randomly sampled every second, the

resulting signal would be erratic and noisy. Galloping occurs best under steady wind

conditions as the device gradually ramps up to the steady-state amplitude. Thus, modeling

persistence is key to ensuring the maintenance of galloping motion for extended periods.

For this, the key choice was to use a Markov Chain Model to independently measure wind

speed and direction [14]. Rather than viewing the entire wind history at once, a Markov

Chain generates a state-dependent probability distribution at each time step. For example, if

the windspeed is currently 5 m/s, the Markov Chain Model estimates what the wind speed

will be one second later, which will very likely be in the range of 4.9 to 5.1 m/s rather than

0 or 10 m/s. Assuming the probability for a wind vector to change is dependent only on the

wind vector itself, the distribution is assumed to be stable and sufficient data sampling will

converge to a single Markov Chain Model.

To create the model, a data time series is reviewed two steps at a time to give us the data

pair (𝑥𝑖 , 𝑥𝑖+1). By breaking the data range into bins from 0 to 𝑥𝑚𝑎𝑥, each with size 𝑑𝑥, each

of these data points can be stored into a 2D histogram of size (𝑥𝑚𝑎𝑥

𝑑𝑥 ,

𝑥𝑚𝑎𝑥

𝑑𝑥) known as the

transition matrix. After all the data is stored into the histogram, each row of the matrix

represents a probability distribution.

Figure 3.2.1: Transition Matrix Example [14]

To create a synthetic wind signal, then, an initial data point is assumed, that windspeed is

associated with a row in the matrix, a random sample from that distribution is made, and

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that result becomes the state after one time step. This process is repeated with each new

state to generate a random walk through the Markov space. While each probability

distribution will have a very low variance, the probability that a wind speed will remain

stable vanishes over time due to the Law of Large Numbers. Below is a comparison of the

sample data with the synthetic wind signal.

Original Synthetic

Figure 3.2.2: Original vs Synthetic Model, Windrose

3.3 Workflow Algorithm and Code Features

3.3.1 General Layout

The general flow of the code is given by the flowchart below. First, the provided wind data

is analyzed and stored into a Markov model using the method described in Section 3.2.

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Then, the material properties of all materials involved are loaded in through several

“Material” class constructors. Likewise, the dimensions of the cantilever beam and

piezoelectric strips are provided through another constructor class. A for-loop is used to

simulate and re-simulate the device under the same wind model, material selections, and

cross-section dimensions, but the length of the cantilever beam is varied each time. This is

done as length is the simplest way to vary the overall beam stiffness without majorly

altering the underlying geometry and mechanics. The simulation is fed these inputs and

spits out a results object which is then visualized in post-processing.

The Rayleigh-Ritz ODE for structural dynamics is solved using the Newmark-Beta method

with an adaptive time step algorithm. There is additionally a function which reads the wind

vector as well as the beam tip velocity to calculate aerodynamic loads, which are evaluated

on each time step. Originally, this stored the results in one large time series for plotting

results and checking for normal functioning. Once this was verified, however, the code was

changed to only store three data points at a time. This minimizes data storage requirements

as it is the minimum amount of data required to identify peaks and valleys.

When a peak or valley (a reversal) is detected, it is appended to a deque (“double-ended

queue”) object. This is data structure is a necessary improvement over a standard array,

which requires many redundant steps to resize as new items are added. When a new item is

appended to a deque, it includes a pointer to the previous element in deque, and thus we are

able to traverse the list by following the list of pointers and not wasting time endlessly

copying entries. The deque structure is used extensively to store data variables of unknown

size which continuously pile up until device failure.

When three or more reversals are detected, the rainflow-counting algorithm, a standard

algorithm for reducing strain data into discrete cycles [25], is called upon to pair peaks and

valleys into cycles or half-cycles. When a cycle is detected, the peak and valley data points

are appended to a “cycles” object and the resulting stress and voltage amplitude values are

calculated from that. The additional stress cycle is passed to the crack growth equation and

the values for current crack lengths are increased. This then updates the stiffness matrix

used in the Newmark-Beta solver for the next time step.

This loop continues until a crack length is recorded to exceed the width of the beam.

18

Figure 3.3.1: Full Program Flowchart

Load Material Properties

For 𝐿 = 𝐿1: 𝐿2 Run Wind Model Calculate [K], [M], etc.

𝑦 = 𝑦0

𝑦𝑛+1 = 𝑓(𝑦𝑛 , 𝐹𝑛)

𝐹 = 𝐴𝑒𝑟𝑜 𝑀𝑜𝑑𝑒𝑙

Re-size Δ𝑡 𝐸𝑟𝑟𝑜𝑟 ≥ 𝜖?

Store value.

Reversal Detected?

Store reversal.

Cycle Detected?

Calculate performance

variables. Store.

𝑑𝑎

𝑑𝑁= 𝐶(Δ𝐾)𝑚

Update Stiffness Matrix

𝐾𝑚𝑎𝑥 ≥ 𝐾𝐼𝑐? Rapid Crack Growth Failure?

End Simulation.

No

Yes

Yes

Yes

No

No

No

Yes

Yes No

Write to File

19

3.3.2 Newmark-Beta Method

The Newmark-Beta Method [26] is a numerical integration technique designed to

accurately solve stiff structural dynamics systems. Rather than relying on the first-order

�̇� = 𝐴𝑥 form, the equation of motion is given in the second-order form:

[𝑀]�̈� + [𝐶]�̇� + [𝐾]𝑥 = 𝐹

Our Rayleigh-Ritz method derived previously is incomplete, however, and must include the

system’s electromechanical coupling. This consists of two circuits, each consisting of two

piezoelectric strips connected to a single load resistor:

[𝑀]�̈� + [𝐶]�̇� + [𝐾]𝑥 − 𝜃𝑉 = 𝐹

𝐶𝑝�̇� +𝑉

𝑅+ 𝜃�̇� = 0

Where R is the resistance of the load resistor, V is the output voltage, 𝐶𝑝 is the capacitance

of the piezoelectric material, and 𝜃 is the electromechanical coupling factor (see [Source]

for calculation).

These can be appended to our pre-existing matrices:

𝑀′ = [[𝑀] 4𝑥4 04𝑥2

02𝑥4 02𝑥2] , 𝐶′ = [

[𝐶]4𝑥4 04𝑥2

𝜃𝜙1(𝐿) 𝜃𝜙2(𝐿) 0 0

0 0 𝜃𝜙1(𝐿) 𝜃𝜙2(𝐿)𝐶𝑝𝐼2𝑥2

]

𝐾′ =

[ [𝐾0 − 𝐾𝑎]4𝑥4

−𝜃/2−𝜃/2

00

00

−𝜃/2−𝜃/2

02𝑥4

1

𝑅𝐼2𝑥2 ]

, 𝐹′ = (𝐹4𝑥1

02𝑥1)

The Newmark-Beta Method relies on the Extended Mean Value Theorem to calculate the

next step. After deriving the applied aerodynamic forces from the aerodynamic model [26]:

𝛾𝑛+1 = 𝐴−1𝐵

�̈�𝑛+1 =1

𝛽(Δ𝑡)2(𝛾𝑛+1 − 𝛾𝑛) −

1

𝛽Δ𝑡�̇�𝑛 − (

1

2𝛽− 1) �̈�𝑛

�̇�𝑛+1 = �̇�𝑛 + (Δ𝑡)(1 − 𝛼)�̈�𝑛 + (Δ𝑡)𝛼�̈�𝑛

Where:

20

𝐴 =1

𝛽(Δ𝑡)2[𝑀] +

1

𝛽(Δ𝑡)[𝐶] + [𝐾]

𝐵 = (𝐹) + [𝑀] ⋅ (1

𝛽(Δ𝑡)2𝛾𝑛 +

1

𝛽Δ𝑡�̇�𝑛 + (

1

2𝛽− 1) �̈�𝑛) + [𝐶]

⋅ (𝛼

𝛽(Δ𝑡)𝛾𝑛 − (1 −

𝛼

𝛽) �̇�𝑛 − Δ𝑡 (1 −

𝛼

2𝛽) �̈�𝑛)

And (𝛼, 𝛽) are method parameters. The most common parameters to use are (0.5, 0.25), as

this method is unconditionally stable [26].

3.3.3 Adaptive Time-Stepping

As wind conditions are subject to change and thus the precision of the ODE solver vary, it

is beneficial to have a method of adapting the time step. Borrowing from [22], I

implemented a method for estimating the error for each time step and either increasing or

decreasing the time step accordingly to stay within an upper and lower bound for relative

error. While most ODE methods only decrease time step according to error estimates, that

is inefficient for simulations like this meant to run for long periods of time. When wind is

high, time steps should be short as negative damping causing all kinds of chaos in the

beam. But once the wind dies down, this short time step is unnecessary, and much runtime

can be saved by checking and speeding up our simulation when possible. The adaptation

algorithm is featured in Figure 3.2.1.

Using the formula for error estimation [22]:

|𝜀| = (𝛽 −1

6) (Δ𝑡)2|�̈�𝑛+1 − �̈�𝑛|

The next time-step is calculated according to the Newmark-Beta method and the error is

estimated. If that error lies within the acceptable bounds, that proposed data point is

accepted and the simulation moves forward. Otherwise, the code enters a while loop where

the time-step is continuously decreased or increased and the next data point is continuously

re-evaluated until the resulting error is acceptable.

This method has the possibility of being inefficient when the bounds for error are narrow,

as it would result in repeated re-evaluations that are otherwise unnecessary.

3.3.4 Fracture Toughness Modeling

Critical to determining the point of failure is the modelling of stable or unstable crack

growth. While crack growth is regularly determined by the linear Paris-Erdogan formula,

the case when the maximum stress intensity factor exceeds fracture toughness must be

studied as well, as the system risks immediate unstable crack growth. When the stress in a

21

particular cycle is expected to exceed fracture toughness, the simulation is temporarily

paused and the following flowchart is initiated:

Figure 3.3.2: Rapid Crack Growth Flowchart

First, the crack length is increased by an arbitrarily small amount, the stiffness matrix is

reduced accordingly, and the Newmark-Beta solver re-calculates the current time step with

the new stiffness matrix. This is repeated until either the stress intensity factor falls below

the fracture toughness or the crack length exceeds the critical amount. In the first case of

stable crack growth, the new stress intensity factor is plugged into the Paris-Erdogan

formula and the simulation continues. In the latter case of unstable crack growth, the

simulation ceases.

4.0 Numerical Simulation Results

The following material properties were used across all runs of the simulation [2,10-13,16-

18]:

Beam (7075 Aluminum) Properties

Side Length 1.5 in

Wall Thickness 0.0625 in

Young’s Modulus 68 GPa

Density 2,710 kg/m^3

Air Density 1.23 kg/m^3

Structural Damping Ratio, ζT 0.01

Fatigue (Paris’ Law) C 88.7 Pa-m1/2

Fatigue (Paris’ Law) m 3.14

Fracture Toughness 14 MPa-m1/2

[K(a)]

𝑦𝑛+1 = 𝑓(𝑦𝑛, 𝐾, 𝐹)

𝜅𝑛+1 = 𝑓(𝑦𝑛+1) a

𝑆𝐼𝐹 = 𝑓(𝑎, 𝜅)

𝐾𝑚𝑎𝑥 ≥ 𝐾𝐼𝑐? 𝑎 = 𝑎 + da 𝑑𝑎

𝑑𝑁= 𝐶(Δ𝐾)𝑚

End

Yes No

22

Base Strip (PZT-5H) Properties

Thickness 0.015 in

Young’s Modulus 63 GPa

d33 180 pC/N

Self-Resistance 1E6 Ω

Self-Capacitance 15 pF

Load Resistance 1E3 Ω

Electromechanical Coupling, θ 662.7 μV / (m/s)

Fatigue (Paris’ Law) C 9.27 Pa-m1/2

Fatigue (Paris’ Law) m 11.8

Fracture Toughness 0.9 MPa-m1/2

4.1 Qualitative Results

Running the simulation for varying cantilever beam lengths with intermittent print-outs of

the data as it was being processed was immediately illuminating. Firstly, the amount of

time in-simulation varied dramatically, with a number of runs lasting less than an hour

before the hypothetical device would fail. The element of randomness involved in the

probabilistic wind model evidently introduces a lot of variation in fatigue life, implying

multiple runs would be required for gathering a more robust dataset. Overall, however,

shorter (and therefore stiffer) beams had significantly longer fatigue lives. Some runs of the

simulation stopped abruptly as the amount of data being stored exceeded the system

capacity. This can be fixed in the future by writing all data to an output file as soon as it

gathered, but for now a sufficient amount of data was gathered from the code and enough to

note important trends.

Notably, the critical crack growth behavior (when exceeding fracture toughness) always

was unstable. Rapidly increasing stresses at the base of the beam were always identified in

the minutes prior to failure, implying that aeroelasticity serves to amplify crack compliance

and rapidly fracture the beam. While crack growth was monitored on both the metallic

beam and the piezoceramic strip, the piezoceramic strip was always the site of critical crack

growth, as expected.

Unfortunately, the code tended to run quite slowly. As a data read-out was printed once

every simulated minute, these statements tended to be displayed every one to three seconds.

While a factor of 20 to 60 times is still good, the runs which integrated over 48 (simulated)

hours ended up taking a few hours. Taking this with the fact that repeated runs are required,

this code is evidently not the most efficient way to develop and test optimization strategies.

This long runtime could be explained by low system resources and thus could have been

improved on a more powerful machine.

23

As the code currently stands, it is a single time-integration thread consisting of an

alternating series of computationally expensive processes which must occur linearly. Thus,

little can be done through parallel programming to speed this up. However, running

multiple simulation instances in parallel on a machine which can handle the storage and

processing requirements would absolutely perform faster.

4.2 Quantitative Results

First, two plots were compiled demonstrating the applied windspeed distribution and the

general trends between windspeed, maximum stress, and beam length.

Figure 4.2.1: Windspeed Overall Probability Distribution

24

Figure 4.2.2: Stress, Windspeed, and Beam Length

(Note: Data points in Figure 4.2.2 are only a portion of all data points recorded) At first,

Figure 4.2.2 should raise some alarm as we expect a hyperbolic relation between the two,

with zero galloping below a critical wind-speed, a sharp increase in windspeed, and a

roughly linear relation as windspeed increases. Here, we instead have something either

linear or quadratic. The reason for this “smearing” of data points near the cut-in windspeed

is because, though galloping cannot be sustained below the cut-in speed due to finite

damping, there will still inevitably be wind-induced oscillations. Thus, our code simulation

is accounting for load histories which a simple hyperbolic curve would not. The tendency

for near-zero stresses to be recorded at high wind speeds also demonstrates the moments

when the beam is transitioning from one steady-state amplitude to another.

This figure implies a good correlation between beam length and cut-in speed, with longer

beams featuring an earlier cut-in speed. Additionally, longer beams feature higher stresses

at the same windspeed due to this earlier galloping onset. Therefore, though longer beams

are less stiff, they are able to harvest more energy because of their susceptibility to

aeroelastic excitation. Data at higher windspeeds is much more sparse, and therefore

difficult to comment on. The lack of data is both due to the low probability of occurring and

the high chance of failure. The high data variance could be a result of the increased

stiffness of the equations of motion in high-wind environments creating more simulation

error.

25

The following graphs will compile the history of all relevant performance variables. Rather

than overlaying histograms, which vary significantly in number of data points, or through

normalized probability densities, which tend to clump together, the data is displayed as a

cumulative density function to better delineate individual lines.

Figure 4.2.3: Tip Displacement Distribution

26

Figure 4.2.4: Stress Distribution

27

Figure 4.2.5: Voltage Distribution

The first characteristic which jumps out of these plots is the differences in skew, with the

𝐿 = 5 𝑚 being a clear outlier. This larger than normal than skew is the result of a very short

simulation time (1 hour) and thus more an anomaly of the wind model than the structural or

aerodynamic models.

Looking at the other curves, we see a general trend regarding median and skewness, which

both initially increase with beam length from the cases of 𝐿 = 3 𝑚 to 𝐿 = 4.25 𝑚, but then

decrease (excluding the 𝐿 = 5 𝑚 case). Evidently, this points to some “sweet spot” around

𝐿 = 4 𝑚 where the beam is compliant enough to sufficiently exploit galloping, but not so

compliant that it becomes unable to generate sufficient stresses. As the 𝐿 = 4 𝑚 run lasted

longer than that of any longer beams, it was still able to deliver a higher power output.

Therefore, there are certain sets of system parameters in which both the durability and

performance of materials increases, indicating an optimal design!

This ideal length will be dependent on the material properties and the wind conditions, so

once materials, geometry, and location are chosen for a GPEH device, this code can

determine the ideal length for the cantilevered section, making the design customizable to

regional conditions. However, there is still significant variation in the figures above and

continued runs of the code will be required for more definitive solutions.

5.0 GPEH Analysis and Optimization Strategy

5.1 Analytical Model for GPEH Performance

To better understand the tradeoff between performance and durability in the context of the

GPEH device, it is necessary to develop an analytic model of GPEH performance. For

example, we need to develop a formula for determining the steady-state galloping

amplitude of such a device based on the wind speed and device parameters.

For that, we turn to Bibo & Daqaq [4], who use a Taylor-series based aerodynamic

coefficient model to identify a steady-state galloping response. This method has the user

represent the net transverse force coefficient in terms of a Taylor series [4].

𝐶𝑦 = 𝐶ℓ𝑐𝑜𝑠𝛼 + 𝐶𝑑𝑠𝑖𝑛𝛼, 𝐶�̇� = ∑ 𝐴𝑛 (�̇�

𝑈)

𝑛𝑚

𝑛:𝑜𝑑𝑑

+ ∑ 𝐴𝑛 (�̇�

𝑈)𝑛 �̇�

|�̇�|

𝑚

𝑛:𝑒𝑣𝑒𝑛

, 𝑛 ≥ 1

This method uses a single lumped-mass (i.e. single natural frequency) estimation of the

system and assumes steady flow. All these parameters can be normalized through the

formulae:

𝑎∗ =𝑎𝜇

𝜁𝑇, 𝑈∗ =

�̅�𝜇

𝜁𝑇, �̅� =

𝑈

𝜔𝑛𝐷𝑡, 𝜇 =

𝜌𝐿𝑡𝐷𝑡2

4𝑀

28

Where a is the steady-state amplitude, 𝜔𝑛 is the beam natural frequency, 𝑀 is the

equivalent lumped mass at the tip, 𝜁𝑇 is total damping ratio, and (𝐿𝑡, 𝐷𝑡) are the length and

aerodynamic chord of the tip bodies. The user-provided Taylor series, unique to the cross-

section geometry of the galloping body, can then be manipulated to provide us with the

condition for steady-state equilibrium [4]:

𝑈∗2𝐶𝑎∗

𝑎∗− 1 = 0

Where:

𝐶𝑎∗ = ∑ [4𝐴𝑛

𝜋(

𝑎∗

2𝑈∗)𝑛

∑(−1)

𝑛2−𝑘

𝑛 + 1 − 2𝑘

𝑛/2

𝑘=0

]

𝑛:𝑒𝑣𝑒𝑛

+ ∑ 𝐴𝑛 (𝑎∗

2𝑈∗)𝑛

(𝑛 + 1

1

2(𝑛 + 1)

)

𝑛:𝑜𝑑𝑑

Thus, for a given (normalized) windspeed, the steady-state amplitude of the galloping body

is given as the real root of a polynomial. Performing this for an equilateral triangular cross-

section, we get the following galloping response curve:

Figure 5.1.1: Normalized Windspeed vs Normalized Tip Displacement for an Equilateral

Triangular Tip Body

Which can be approximated by a hyperbola of the form:

𝑈∗2

𝑐1∗2 −

𝑎∗2

𝑐2∗2 = 1

De-normalizing this formula finally gives us a hyperbolic formula for galloping response

where:

29

𝑈2

𝑐12 −

𝑦𝑡𝑖𝑝2

𝑐22 = 1

The conclusions of this method will be tested with the results of the simulation, which

factors in a variety of confounding variables that are not so easily modeled.

5.2 Optimization Strategy

5.2.1 Effect of Material Choice on Efficiency and Durability

5.2.1.1 Piezo-ceramics versus Piezo-polymers

Before analyzing all design parameters together, we’ll focus on the choice of piezoelectric

material and see how different classes of material can greatly improve device durability.

Generally, common piezoelectric materials fall into two categories. The first are ceramics

with high dielectric constants, high stiffnesses, low ductility, and poor fatigue resistance

[10, 13]. These offer great efficiency in converting strain energy to electrical energy, but

their poor compliance requires they are only applied in short, thin plies to prevent brittle

fracture. Additionally, their lack of intrinsic crack growth resistance mechanisms ensures a

finite, and often short, fatigue life. For example, Sirohi & Mahadik [17] noted visible

fatigue damage to the PZT plies in the GPEH device after conducting wind tunnel tests.

Secondly, there are piezoelectric semi-crystalline polymers (often thermoplastics) with low

dielectric constants, low stiffness, high ductility, and exceptional fatigue resistance [18].

These are very poor at converting strain energy to electrical energy, but their substantial

compliance means the devices themselves can be much more compact while still featuring

a lower cut-in windspeed than their ceramic-based counterparts. In particular, the

thermoplastic polyvinylidene di-fluoride (PVDF), has a 𝑑33 value one-sixth that of the most

common piezoceramic, lead zirconate titanate (PZT), but has been shown to feature an

endurance limit at about 50% of its ultimate tensile strength, whereas PZT will eventually

fail at any stress level [10,18].

For this reason, PVDF is a clear champion when designing durable energy harvesters,

especially if a stopping mechanism can be employed to prevent bending stresses exceeding

the endurance limit. Such a setup would result in an infinite service life while still being

able to deliver half of its maximum achievable power.

That being said, there are instance where PVDF is still not a practical material choice, as

the achievable amounts of power can often be orders of magnitude lower. For samples of

PZT and PVDF strained to the same extent, the power output is given by the ratio:

Power𝑃𝑍𝑇

Power𝑃𝑉𝐷𝐹=

𝑑33,𝑃𝑍𝑇𝜎𝑃𝑍𝑇

𝑑33,𝑃𝑉𝐷𝐹𝜎𝑃𝑉𝐷𝐹=

𝑑33,𝑃𝑍𝑇𝐸𝑃𝑍𝑇

𝑑33,𝑃𝑉𝐷𝐹𝐸𝑃𝑉𝐷𝐹

30

As PZT offers a much higher value for both 𝑑33 and 𝐸, PZT could be the clear champion in

applications where durability is not necessary.

5.2.1.2 Performance Optimization of a Piezo-Ceramic-Polymer Composite

While both materials have their own trade-offs when used on their own, we will see that

when used together in a composite piezoelectric structure, they are able to offer higher

efficiency than when used separately.

Consider an inverted sandwich lay-up, with the piezo-ceramic in the interior and the piezo-

polymer on the exterior. This design is chosen as it is able to maximize compliance and

ensure the material can easily gallop. A true sandwich layup with the stiffer ceramic plies

on the exterior would be exceptionally stiff and would rarely gallop in normal wind

conditions.

Borrowing from Classical Laminate Theory, when can use the following graphs to

represent strain, stress, and voltage in response to an applied curvature. Here, a filled in

space to the right of the vertical line indicates a positive value while a space to the left

represents a negative value.

Figure 5.2.1.2.1: Bending Diagram

As we can see, the voltage between plies would equalize, and the rate of voltage drop

across each ply is dependent on both the stress experienced by the ply and the

electromechanical coupling coefficient. There is thus a drastic voltage drop across the

piezo-ceramic which slowly increases through the piezo-polymer plies.

Δ𝑉 = [𝑑𝑃𝑉𝐷𝐹𝐸𝑃𝑉𝐷𝐹 (1

2ℎ𝑃𝑍𝑇 + ℎ𝑃𝑉𝐷𝐹) +

1

2𝑑𝑃𝑍𝑇𝐸𝑃𝑍𝑇ℎ𝑃𝑍𝑇] 𝜅 = �̅�𝜅

Where 𝑑 is the electromechanical coupling constant, 𝐸 is the Young’s Modulus, 𝜅 is the

curvature, and ℎ is the ply thickness. Note that “PVDF” and “PZT” will be used as

subscripts to denote any general piezo-polymer or piezo-ceramic, respectively. Here, �̅� is

Strain Stress Voltage + + +

- - -

31

the effective coupling constant between applied curvature and resultant voltage. Thus, �̅� =

𝑑𝑒𝑓𝑓𝐸𝑒𝑓𝑓ℎ𝑒𝑓𝑓.

From basic geometry, we see that ℎ𝑒𝑓𝑓 = 2ℎ𝑃𝑉𝐷𝐹 + ℎ𝑃𝑍𝑇 and from Classical Laminate

Theory, we can determine:

𝐸𝑒𝑓𝑓 =𝐸𝑃𝑍𝑇𝐼𝑃𝑍𝑇 + 2𝐸𝑃𝑉𝐷𝐹𝐼𝑃𝑉𝐷𝐹

𝐼𝑃𝑍𝑇 + 2𝐼𝑃𝑉𝐷𝐹

𝐸𝑒𝑓𝑓 ≈𝐸𝑃𝑍𝑇 (1 +

18𝛼3) + 𝐸𝑃𝑉𝐷𝐹(1 + 8𝛼3)

2 + 8𝛼3 𝑤ℎ𝑒𝑟𝑒 𝛼 =

ℎ𝑃𝑉𝐷𝐹

ℎ𝑃𝑍𝑇

The approximately-equal sign is used as we simplified the calculation of area moment of

inertia for the PVDF plies assumes the PVDF plies are much thicker than the PZT plies, or

𝛼 > 1. We can therefore solve for the equivalent piezoelectric constant of this layup:

𝑑𝑒𝑓𝑓 =�̅�

𝑑𝑒𝑓𝑓𝐸𝑒𝑓𝑓=

(8𝛼2 + 2)𝑑𝑃𝑉𝐷𝐹

𝛽 (1 +1

8𝛼3) + (1 + 8𝛼3)+

(8𝛼3 + 2)/(2𝛼 + 1)

(1 +1

8𝛼3) +1𝛽

(1 + 8𝛼3)𝑑𝑃𝑍𝑇

𝑤ℎ𝑒𝑟𝑒 𝛽 =𝐸𝑃𝑍𝑇

𝐸𝑃𝑉𝐷𝐹

We can lastly normalize this with another variable 𝛾 =𝑑𝑃𝑍𝑇

𝑑𝑃𝑉𝐷𝐹. Assuming large values for

(𝛼, 𝛽, 𝛾):

𝑑𝑒𝑓𝑓 ≈8𝛼3 + 2

8𝛼3 + 𝛽 + 1(𝛽𝛾

2𝛼+ 1) 𝑑𝑃𝑉𝐷𝐹

As (𝛽, 𝛾) are fixed constants, we optimize this design by varying 𝛼. We can find the

optimal layup by setting 𝜕𝑑𝑒𝑓𝑓

𝜕𝛼 to zero and solving for 𝛼 in the numerator.

𝛼4 + (1

4𝛽𝛾 + 1 −

𝛽𝛾

6(𝛽 − 1)) 𝛼3 +

1

4(𝛽𝛾 + 1)𝛼2 −

𝛽𝛾

24(𝛽 − 1)= 0

Assuming 𝛽

𝛽−1≈ 1, 𝛽𝛾 ≫ 4, 𝛽 ≫

2

3 we can fully simplify to:

𝛼4 +1

4𝛽𝛾𝛼3 +

1

4𝛽𝛾𝛼2 −

1

24𝛾 = 0

Quartic formulas are not easily solved analytically, but using a numerical solver with the

imagined ratios below, we arrive at:

𝛽 = 𝛾 = 100, 𝛼𝑜𝑝𝑡 = 3.1, 𝑑𝑚𝑎𝑥 = 1,148.4𝑑𝑃𝑍𝑇 ≈ 11.5𝑑𝑃𝑍𝑇

32

Note that, while the actual voltage drop in the optimal scenario is still lower than that of a

pure PZT layup, but the efficiency is an order of magnitude higher. If we plot 𝑑𝑒𝑓𝑓 for the

example above, we can note an interested behavior:

Figure 5.2.1.2.2: Piezo Coupling vs Ply Thickness Ratio

The effective piezoelectric constant begins at that of PZT for 𝛼 = 0, falls slightly as it is

replaced with polymer, then drastically rises to a maximum value, then finally decaying to

that of PVDF as 𝛼 approaches infinity. We have evidently stumbled upon an ideal regiment

where the load-sharing induced by materials of differing Young’s moduli is perfectly

calibrated with their differing piezoelectric coupling coefficients to most efficiently convert

strain energy into electrical energy.

5.2.2 Methods of Analytically Correlating Design Parameters and Performance

Variables

5.2.2.1 Probability-Based Model

The windspeed applied to the GPEH mechanism can be modeled as following a Weibull

distribution, with model coefficients unique to the specific region the GPEH is placed. If

we use our hyperbolic formula from Section 5.1, we can then model the GPEH tip

displacement with roughly the shape plotted below.

33

Figure 5.2.2.1.1: Example Probability Distribution for Galloping Tip Displacement

Note that the hyperbolic relation [4] further exaggerates the positive skew of the

distribution while the existence of the cut-in windspeed provides a discrete probability mass

that the steady-state tip displacement equals zero. If we assume a constant mode shape

equal to the first eigenmode, then the curvature and stresses at the base of our cantilevered

beam will be linearly correlated with the tip displacement, meaning we can relate stress in

our piezoelectric material in one of the following ways:

𝑈2

𝑈02 −

𝜎2

𝑚12 = 𝑚2

2, 𝜎 = 𝑚1√(𝑈

𝑈0)2

− 𝑚22, 𝑈 = 𝑈0√

𝜎2

𝑚12 + 𝑚2

2

The variable we are seeking to optimize is the total lifetime energy, that is, the average

power rate times the device fatigue life, or: 𝐸𝑛𝑒𝑟𝑔𝑦 = 𝑑33𝜎𝑃𝑍𝑇𝑁(𝜎𝑃𝑍𝑇). At first glance,

there is a trivial maximum to this variable. As power increases linearly with applied stress

level while the number of fatigue-cycles decreases faster-than-linearly as stress increases.

Therefore, the optimal design is one which consistently undergoes practically zero stress,

with a trivial power output but a nearly infinite fatigue life.

Obviously, such a design would be impractical in most applications as whatever payload is

meant to be powered would have some minimal power requirement. Thus, we have to

approach this problem already assuming some average power output. In other words,

𝐸[𝜎] = 𝜎 = 𝑐𝑜𝑛𝑠𝑡. This assumption can be made as the formula for average power output:

𝐸[𝑃] = 𝑑33𝐸[𝜎 ] = ∫ 𝑑33𝜎 𝑓(𝜎 )𝑑𝜎

∞

0

Since the formula for power is linearly dependent on stress, this simplification can be made.

If we wish to calculate fatigue life via Miner’s Rule, however, the relation is non-linear and

thus must be evaluated in the form:

𝐸[𝑁] = ∫ 𝑁(𝜎)𝑓(𝜎)𝑑𝜎∞

0

P

34

Where 𝑁(𝜎) is the expected number of life cycles associated with an arbitrary applied

stress and 𝑓(𝜎) is the probability distribution of stress, loosely based off the Weibull model

of windspeed. What this tells us is that highly skewed stress distributions, where the

applied stress in the piezoelectric material is low for the majority of the time, will have

significantly longer fatigue life, as the average is weighed more heavily by high-cycle

fatigue than low-cycle fatigue. In summary, our goal for the design parameters is to

maximize skewness of the stress distribution.

The existence of the cut-in speed is significant in this. Standard methods of stiffening the

cantilever beam would greatly enhance the skew of the distribution’s right tail, but will also

result in a higher cut-in speed, increasing the discrete probability that the stress level is zero

for a given time. The extent to which changes to cut-in windspeed affect stress mean and

skewness are highly dependent on the Weibull model parameters. Therefore, the only way

to fully track these opposing factors is to develop a formula for the mean and skewness of

the stress distribution based on the wind model and the relation between windspeed and

stress. To transform a random variable 𝑋 into another 𝑌 through the relation 𝑌 = 𝑔(𝑋)

where 𝑋 has the probability function 𝑓(𝑋), we use the Law of the Unconscious Statistician:

𝐸[𝑌] = ∫ 𝑌𝑓(𝑔−1(𝑥))𝑑𝑌∞

−∞

Thus we must write the windspeed in terms of the stress:

𝑈 = √𝑚1𝜎2 + 𝑚2

Note: the (𝑚1, 𝑚2) values here are not the same as those used previously, but both

represent the hyperbolic relation between windspeed and stress. The key distribution

parameters are therefore [21]:

𝑀𝑒𝑎𝑛 = 𝐸[𝜎] = ∫ 𝜎∞

0

𝑓(𝜎)𝑑𝜎

= ∫ 𝜎∞

𝑈0

𝑘

𝜆𝑘(𝑚1𝜎

2 + 𝑚2)12(𝑘−1) exp [−(

√𝑚1𝜎2 + 𝑚2

𝜆)

𝑘

] 𝑑𝑈

𝑆𝑡𝑑. 𝐷𝑒𝑣.= √𝐸[𝜎2] − 𝐸[𝜎]2

𝑆𝑘𝑒𝑤𝑛𝑒𝑠𝑠 =𝐸[𝜎3] − 3(𝑀𝑒𝑎𝑛) ⋅ (𝑆𝑡𝑑. 𝐷𝑒𝑣. )2 − (𝑀𝑒𝑎𝑛)3

(𝑆𝑡𝑑. 𝐷𝑒𝑣)3

Where:

35

𝐸[𝜎2] = ∫ 𝜎2∞

0

𝑓(𝜎)𝑑𝜎 = ∫ 𝜎2∞

𝑈0

𝑘

𝜆𝑘(𝑚1𝜎

2 + 𝑚2)12(𝑘−1) exp [−(

√𝑚1𝜎2 + 𝑚2

𝜆)

𝑘

] 𝑑𝑈

𝐸[𝜎3] = ∫ 𝜎3∞

0

𝑓(𝜎)𝑑𝜎 = ∫ 𝜎3∞

𝑈0

𝑘

𝜆𝑘(𝑚1𝜎

2 + 𝑚2)12(𝑘−1) exp [−(

√𝑚1𝜎2 + 𝑚2

𝜆)

𝑘

] 𝑑𝑈

Where (𝑘, 𝜆) are Weibull model parameters [21], 𝑈0 is cut-in windspeed, and (𝑚1, 𝑚2) are

simplified design parameters, though in a different from those derived previously. To try

and solve these integrals, we will begin by writing a formula for the general nth raw

moment:

𝐸[𝜎𝑛] = ∫ 𝜎𝑛∞

𝑈0

𝑘

𝜆𝑘(𝑚1𝜎

2 + 𝑚2)12(𝑘−1) exp [−(

√𝑚1𝜎2 + 𝑚2

𝜆)

𝑘

] 𝑑𝑈

Making the following substitution:

𝑔(𝜎) =(𝑚1𝜎

2 + 𝑚2)k/2

𝜆𝑘

𝐸[𝜎𝑛] = ∫𝜆

𝑚1𝜎𝑛−1𝑔1/k exp[−𝑔] 𝑑𝑔

∞

𝑔(𝑈0)

Note that we can re-write 𝜎2 in terms of this new variable 𝑔 easily:

𝜎2 =𝜆2

𝑚1𝑔2/k −

𝑚2

𝑚1

𝐸[𝜎𝑛] = ∫𝜆

𝑚1(

𝜆2

𝑚1𝑔2/k −

𝑚2

𝑚1)

12(𝑛−1)

𝑔1/k exp[−𝑔] 𝑑𝑔∞

𝑔(𝑈0)

To solve an integral of this form, we can make use of the upper incomplete gamma function

[8]:

Γ(𝑎, 𝑏) = ∫ 𝑥𝑏−1 exp[−𝑥] 𝑑𝑥∞

𝑏

Since 1

2(𝑛 − 1) is not always an integer, we need to perform a general binomial expansion

which works for non-integer powers:

36

(𝑥 + 𝑦)𝑟 = ∑(𝑟𝑐) 𝑥𝑟−𝑐𝑦𝑐

∞

𝑐=0

Therefore:

𝐸[𝜎𝑛] = ∫𝜆

𝑚1𝑔1/k exp[−𝑔]∑(

1

2(𝑛 − 1)

𝑐)

𝜆(𝑛−1)−𝑐

𝑚1

12(𝑛−1)−𝑐

𝑔1𝑘(𝑛−1)−𝑐 (−

𝑚2

𝑚1)

𝑐∞

𝑐=0

𝑑𝑔∞

𝑔(𝑈0)

𝐸[𝜎𝑛] = ∫ ∑(−1)𝑐 (1

2(𝑛 − 1)

𝑐)

𝜆𝑛−𝑐𝑚2𝑐

𝑚1

12(𝑛+1)

𝑔(𝑛/𝑘−𝑐)

∞

𝑐=0

exp[−𝑔] 𝑑𝑔∞

𝑔(0)

𝐸[𝜎𝑛] = ∑(−1)𝑐 (1

2(𝑛 − 1)

𝑐)

𝜆𝑛−𝑐𝑚2𝑐

𝑚1

12(𝑛+1)

∞

𝑐=0

Γ(1 +𝑛

𝑘− 𝑐, 𝑔(𝑈0))

Note that this reduces to a finite sum for odd values of n, and for 𝐸[𝜎2], we can still

approximate the solution with a finite number of terms.

The next step is to evaluate this formula at 𝑛 = 1,2,3, solve for mean, standard deviation,

and skewness. Finally, we would be left with two equations:

𝐸[𝜎] = 𝑓1(𝐷𝑒𝑠𝑖𝑔𝑛 𝑃𝑎𝑟𝑎𝑚𝑠) = 𝑐𝑜𝑛𝑠𝑡.

𝑆𝑘𝑒𝑤𝑛𝑒𝑠𝑠 = 𝑓2(𝐷𝑒𝑠𝑖𝑔𝑛 𝑃𝑎𝑟𝑎𝑚𝑠)

Where optimizing skewness while keeping mean stress constant leaves us with a relation

between design parameters which optimizes durability while still maintaining an acceptable

power output. The goal of the user is to first assume all material properties are constants,

and that only the dimensions of the beam, piezoelectric plies, etc, are customizable. This

should provide the user with a single optimal design. The process can be repeated with new

material properties to find a new optimal design. The best of these optimal designs is finally

used and constructed. Unfortunately, the formulae here are highly involved and I was

unable to make further progress on this method. However, the method is now fully laid out

and ready to be implemented.

While the formula for skewness is extremely convoluted, keeping the mean stress constant

is relatively simple.

𝐸[𝜎] =𝜆

𝑚1Γ (1 +

1

𝑘, 𝑔(𝑈0)) = 𝑐𝑜𝑛𝑠𝑡.

By parametrizing this and solving for the parametric relation:

37

𝑔0 = 𝑔(𝑈0) =𝑚2

k/2

𝜆𝑘, 𝑔1 =

𝜆

𝑚1, 𝐸[𝜎] = 𝑔1Γ (1 +

1

𝑘, 𝑔0)

𝑑𝐸[𝜎]

𝑑𝑔0=

𝑑𝑔1

𝑑𝑔0Γ (1 +

1

𝑘, 𝑔0) + 𝑔1𝑔0

1/k𝑒−𝑔0 = 0,

𝑑𝑔1

𝑑𝑔0= −

𝑔1𝑔01/k

exp[−𝑔0]

Γ (1 +1𝑘

, 𝑔0)

Separating variables and integrating yields:

𝑔1 = Γ(1 +1

𝑘, 𝑔0) , 𝐸[𝜎] = Γ2 (1 +

1

𝑘, 𝑔0) = 𝑐𝑜𝑛𝑠𝑡.

Therefore:

𝑔0 = 𝑐𝑜𝑛𝑠𝑡, 𝑈0 = 𝑐𝑜𝑛𝑠𝑡, 𝑚2 = 𝑐𝑜𝑛𝑠𝑡.

In short, to keep mean power output constant, we need to keep the cut-in windspeed

constant.

5.2.2.2 “Lifetime Efficiency”-Based Model

As the probabilistic model results in a difficult solution that is not analytically solvable, it’s

possible that another optimization method could tackle the same issue more simply. This

time, instead of maximizing absolute power output, which is heavily dependent on the scale

of the particular GPEH device and local wind conditions, we could instead try to maximize

a new variable we’ll call “Lifetime Efficiency”, given by:

𝐿. 𝐸. = 𝜂𝑝𝑖𝑒𝑧𝑜𝜂𝑎𝑒𝑟𝑜𝑁

Where:

𝜂𝑝𝑖𝑒𝑧𝑜 =𝑇𝑜𝑡𝑎𝑙 𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 𝐸𝑥𝑡𝑟𝑎𝑐𝑡𝑒𝑑

𝑇𝑜𝑡𝑎𝑙 𝑆𝑡𝑟𝑎𝑖𝑛 𝐸𝑛𝑒𝑟𝑔𝑦

𝜂𝑎𝑒𝑟𝑜 =𝑇𝑜𝑡𝑎𝑙 𝑆𝑡𝑟𝑎𝑖𝑛 𝐸𝑛𝑒𝑟𝑔𝑦 𝑝𝑒𝑟 𝐶𝑦𝑐𝑙𝑒

𝑇𝑜𝑡𝑎𝑙 𝑊𝑖𝑛𝑑 𝐸𝑛𝑒𝑟𝑔𝑦=

∫12𝐸𝐼𝜅(𝑥)2 ⋅

𝜔𝑛

2𝜋12𝜌𝐴𝑡𝑈3

And 𝐴𝑡 is the surface area of the tip body, 𝜅 is the beam curvature, and N is the number of

cycles the device can endure before fatigue failure. In short, this variable represents the

total energy harvested in terms of how many seconds of pure wind power would generate

the equivalent amount of power. When a specific power requirement is not so critical,

designing in terms of energy efficiency can ensure the GPEH device operates cost

effectively even when its output is less substantial.

38

Piezoelectric efficiency is mostly a function of the piezoelectric coupling constants and the

portion of the beam which is made of piezoelectric material, which is not as customizable

as the aeroelastic efficiency, thus most of our efforts will focus on that variable. Using

statics, we can find the displacement across a cantilever beam in response to a tip load in

terms of the tip displacement.

𝑤(𝑥) = 𝑤𝑡𝑖𝑝 (3

2(𝑥

𝐿)2

−1

2(𝑥

𝐿)3

) , 𝜅(𝑥) = 3𝑤𝑡𝑖𝑝 (1 −𝑥

𝐿)

Using our hyperbolic model of galloping displacement:

𝑤𝑡𝑖𝑝 = 𝑐2√(𝑈

𝑐1)2

− 1

Where (𝑐1, 𝑐2) are design parameter variables to be examined later on. Plugging this in:

𝑉 (𝑆𝑡𝑟𝑎𝑖𝑛 𝐸𝑛𝑒𝑟𝑔𝑦) = ∫9

2𝐸𝐼𝑐2

2 [(𝑈

𝑐12) − 1] (1 −

𝑥

𝐿) 𝑑𝑥

𝐿

0

=3

2𝐸𝐼𝐿𝑐2

2 [(𝑈

𝑐22) − 1]

Next, we need to divide this by the period of oscillation:

𝑇 =𝜔𝑛

2𝜋=

𝑐32

2𝜋𝐿2√

𝐸𝐼

𝜌𝐴

Where 𝑐3 is the non-dimensional coefficient unique to the natural frequency of a cantilever

beam.

𝜂𝑎𝑒𝑟𝑜 =3

4𝜋

𝑐22𝑐3

2

𝜌𝑎𝑖𝑟𝐴𝑡

(𝐸𝐼)3/2

𝐿(𝜌𝐴)1/2

1

𝑈3[(

𝑈

𝑐1)2

− 1]

As we can see, the aeroelastic efficiency is the difference between two inverse polynomial

terms, which results in a well-known graph as shown below:

Figure 5.2.2.2.1: Aerodynamic Efficiency vs Windspeed (Sketch)

𝜂

𝑈 𝑈0 √3𝑈0

39

Note the efficiency intercepts zero at the cut-in windspeed, rises sharply to a maximum

efficiency at 𝑈 = √3 𝑐1, then decays back to zero. Therefore, our device is most efficient at

windspeeds just above the cut-in speed, but as power increases about linearly at higher

windspeeds while available wind power increases cubically, our device becomes noticeably

less efficient.

The next step is to expand the condensed (𝑐1, 𝑐2) terms given by:

𝑐12 =

4𝜌𝐴𝜔𝑛𝜁𝑇

𝜌𝑎𝑖𝑟𝐷𝑡𝑘1

2, 𝑐22 =

4𝜁𝑇𝜌𝐴𝜔𝑛

𝜌𝑎𝑖𝑟𝐷𝑡𝑘2

2

Where 𝜁𝑇 is the total damping ratio between all mechanical and electrical components, 𝐷𝑡

is the aerodynamic chord of the system such that 𝐴𝑡 = 𝐿𝐷𝑡, and (𝑘1, 𝑘2) are the original

normalized hyperbolic fit constants unique to the equilateral triangular cross-section as

discussed in Section 5.1. The maximum efficiency is then given by:

𝜂𝑎𝑒𝑟𝑜,𝑚𝑎𝑥 =𝑐3

4𝜋√3

𝑘22

𝑘12

(𝐸𝐼)5/4

(𝜌𝐴)3/4𝐿√𝜌𝑎𝑖𝑟𝐷𝑡𝜁𝑇

From this we can see that short, stiff beams will have the highest maximum efficiency.

This makes sense as stiff structures are able to store more strain energy.

Next, we will analyze the N variable. The ASTM-E379-10 standard [24] fits all S-N curves

to the log-log linear fit given by:

log 𝜎 = −𝑎 log𝑁 + 𝑏, 𝑁 = 10b/a𝜎-1/a

Where maximum stress is here given by:

𝜎 = 𝐸𝑧𝜅(0) = 3𝐸𝑧𝑐2√(𝑈

𝑐1)2

− 1

Where z is the distance from the center of the cross-section to the extremity. Plugging

everything in, we are finally left with:

𝜂𝑎𝑒𝑟𝑜 ⋅ 𝑁 = 10b/a(3𝐸𝑧𝑐2)-1/a

3𝑘22𝑐3

4

𝜋

(𝐸𝐼)2𝜁𝑇

𝐿4(𝜌𝑎𝑖𝑟𝐷𝑡)2[

𝜌𝑎𝑖𝑟𝐷𝑡

4𝜁𝑇(𝜌𝐴)𝜔𝑛𝑘12 𝑈−1 − 𝑈−3]

1−12𝑎

This tells us that the behavior of our lifetime efficiency variable varies depending on the S-

N curve fit. When 𝑎 >1

2, which is to say that the S-N curve decreases sharply and fatigue

life is less sensitive to applied stress, then the efficiency variable dominates and the shape

of the curve takes the form:

40

Figure 5.2.2.2.1: Efficiency versus Windspeed for Fatigue-Insensitive Materials

When 𝑎 <1

2, which is to say that the S-N curve decreases gradually and fatigue life is more

sensitive to the applied stress, then the fatigue life dominates and the shape of the curve

takes the form:

Figure 5.2.2.2.2: Efficiency vs Windspeed for Fatigue-Sensitive Materials

41

Note that, as the windspeed approaches cut-in speed, the efficiency approaches zero, but the

number of cycles approaches infinity faster and thus prevails. The location of the extrema

in both forms is the same, but the a-value determines whether it is the maximum or

minimum efficiency. The materials where 𝑎 <1

2 are clearly the most desired as they

approach infinite lifetime efficiency at high speeds and at low (near cut-in) speeds.

Research on S-N curve fits has shown that aluminum alloys belong to the 𝑎 < 0.5 class

[27]. As we can see, stiffening the structure will increase all values in either graph:

Figure 5.2.2.2.3: Lifetime Efficiency Graph vs Stiffness

Thus, the optimal strategy after choosing a durable material is to design an infinitely stiff

structure with a high cut-in speed but which performs perfectly for windspeeds above cut-

in. Obviously, this is not always a practical option, and structures seeking to optimize

lifetime efficiency should design their structure to have a cut-in windspeed aligned with an

arbitrary cumulative probability. For example, say we were to design a GPEH device such

that it spent 50% of the time galloping, which we would do by setting the cut-in windspeed

to the mean windspeed.

6.0 Conclusion

The various mechanisms of galloping-based piezoelectric wind energy harvesters were

studied, including aerodynamics, structural dynamics, piezoelectricity, and critically,

fatigue mechanics. These phenomena were modeled numerically and analytically, and

strides toward a formula for durability optimization were tried, focusing on both physical

and probabilistic variables. A code simulation was performed for a basic GPEH device with

varying system parameters and provided useful insights to how these correlate with

performance metrics. This code currently is inefficient and requires significant runtime

improvements, but is able to account for a myriad of factors and in that way offers

something over a standard analytical model.

Stiffness

Stiffness

a > 0.5 a < 0.5

42

While this project intended to conclude by testing the derived optimization formulae with

the program results, all optimization methods proved partially inconclusive, requiring

extensive dimensional analysis to better simplify the problem. However, with the next steps

of the derivation process clearly laid out, this research could be continued in a future

project using all the formulae provided in this paper.

7.0 Bibliography

1 Alonso, G & Meseguer, J. (2006). A parametric study of the galloping stability of

two-dimensional triangular cross-section bodies. Journal of Wind Engineering and

Industrial Aerodynamics, 94(4), 241-253. ISSN 0167-6105.

https://doi.org/10.1016/j.jweia.2006.01.009

2 ASM Material Data Sheet. [Webpage]. www.asm.matweb.com

3 Bathias, Claude & Pineau, Andre. (2013). Fatigue of Materials and Structures. Pg.

238. Wiley. ISBN 1118623371.

4 Bibo, Amin & Daqaq, Mohammed. (2015). An analytical framework for the design

and comparative analysis of galloping energy harvesters under quasi-steady

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