# Effect of Tip Mass Length Ratio on Low Amplitude Galloping Piezoelectric Energy Harvesting

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling

_{∞}. The two piezoelectric plates plotted in Figure 1 are connected in parallel with opposite polarity to an electrical impedance R to harvest higher galloping energy. The free-body diagram of the tip mass which is exposed to an incoming flow in the z-direction with a magnitude of V

_{∞}is depicted in the top view of the problem for the D-shape cylindrical mass in Figure 2. The beam mounted bluff body experiences galloping in the x-y plane in the ±y direction when V

_{∞}is greater than the onset galloping velocity. The material properties of the system (cantilever beam, air, and the tip body) are presented in Table 1. As the system is exposed to the wind, it starts to vibrate (in the y-direction) to reach a stable oscillation. The strain produced in the base beam is converted to an electric charge in piezoelectric sheets.

#### 2.1. Beam Modeling

_{i}(t) is the ith mode shape; and q

_{i}(t) is the time-dependent part of the beam displacement, which is also mentioned in the literature as the mode coordinate. Each of the displacement functions is named a mode, and the shape of the displacement curve is named the mode shape. The Euler–Bernoulli beam model is composed of the second derivate of the internal moment; piezoelectric coupling term; and internal damping of the structure (strain rate), where the galloping aerodynamic moment and force are applied by the Dirac delta function at the end of the beam and the viscous air damping coefficient is neglected. The exact mode shapes of the structure are obtained by Euler–Bernoulli beam theory, including forcing, damping, and piezoelectric coupling terms:

^{−1}·s

^{−1}), ρ is the density of material, A is the cross-sectional area, E

_{p}is the piezoelectric material’s Young′s modulus, d

_{31}is the strain coefficient of piezoelectric, w

_{p}is the width of the piezoelectric layer, t

_{p}is the piezoelectric layer thickness, L

_{1}and L

_{2}are respectively the starting and ending points of the piezoelectric material sheets on the beam, δ is the delta Dirac function, V is the produced voltage at the electrodes, M

_{tip}is the effective moment applied at the end of the beam by the tip mass, and F

_{tip}is the effective force applied at the end of the beam by the tip mass. The left-hand side (LHS) of Equation (2) is the simplification of the time-dependent linear theory of elasticity, known as engineer′s beam theory (a special case of Timoshenko beam theory or classical beam theory), which offers a tool for calculating the small deflection characteristics of beams under lateral external loads. The first term in the LHS characterizes the inertial effect derived from kinetic energy while the second one in the LHS signifies the effective stiffness derived from potential energy due to internal forces. The right-hand side of Equation (2) has units of force per length composed of two sources of a distributed load of piezoelectric sheet and tip mass effect at the point of tip mass-beam connection. The piezoelectric load considered here is the ideal case which will be discussed more in Section 2.5, and the tip mass effects are retrieved from the first and second integrations of Equation (2) concerning the longitudinal direction which leads to shear force in the beam and to bending moment in the beam.

_{i}is the natural frequency of the ith mode. The left-hand side of Equation (3) presents the second derivate of the bending moment with respect to the longitudinal position, while the right-hand side of Equation (3) presents the inertial terms. The Rayleigh method offers the same method for computation of the fundamental frequency of the system. As the displacement is not unique and depends on the frequency, the time-dependent part is given by the following:

_{a}is the air density, L

_{r}is the length of the tip body, D is the width of the tip body, V

_{∞}is the wind velocity, L is the length of the beam, t

_{b}is the beam material layer thickness, dot is used to present the derivative with respect to time, and prime is used for the first derivate versus location. The values of q (actually constants as there is one for each mode i), in general, are complex and are found by the initial conditions on the displacements and velocity of the beam. As the characteristic time scale of the flow motion is less than the characteristic time scale of the oscillations, the quasi-steady hypothesis is used to evaluate the galloping aerodynamic terms in the above equations.

#### 2.2. Aerodynamic Modeling

_{∞}) vibrates with the natural frequency of f

_{n}(for a low frequency, it typically is about one Hertz) exposing a fluid vortex impact at the beginning of a period of galloping, the vortex in fluid (with the center velocity of V

_{∞}) should be moved downstream adequately far away (at minimum, ten times the characteristic length of body) until the end of that period. Since, at one period later when the mass body returns to the location of the beginning of galloping, that vortex no longer disturbs the fluid flow around the body (f

_{n}≤ V

_{∞}/10D). Meanwhile, the frequency disturbance to the mean flow generated by vortex shedding frequency (f

_{S}= V

_{∞}/5D) should be, at minimum, two times greater than the oscillation frequency. Bearman et al. [21] have a more conservative limit for the natural frequency of the square prism structure (f

_{n}≤ V

_{∞}/30D). It means that the frequency disturbance to the mean flow generated by the vortex shedding frequency should be, at minimum, six times greater than the natural oscillation frequency of the structure for the quasi-steady supposition to be appropriate.

_{1}and a

_{3}are aerodynamic empirical coefficients, for which the D-shaped and other cylindrical cross sections are found in Table 2. The force terms in Equation (2) are replaced by the terms appearing in Equation (4) by the integration of F

_{y}over the tip mass:

_{r.}The effective wind momentum over tip mass is calculated around the beam free end.

#### 2.3. Piezoelectric Modeling

_{p}L

_{p}) from the electric displacement:

#### 2.4. Approximate Method

_{tip}can be evaluated from Equation (10). The explicit form of the effective mass is found from the following:

_{p}. The third term of Equation (21) presents the tip mass effect and the cross effect of rotation and displacement. The last term presents the effect of rotation (i.e., $K.{E}_{rot}=\frac{9{I}_{tip}}{4{L}^{2}}\frac{{\dot{w}}_{L}^{2}}{2}={I}_{tip}\frac{{\dot{{w}^{\prime}}}_{L}^{2}}{2}$). The election of the dimensionless mode shape (in Equation (17)) will affect the dimensionless derivate of the mode shape (See Equation (18).), which is used to calculate piezoelectric coupling terms (See Equation (3).), and the effective mass and stiffness distribution, which is used to evaluate the natural frequency of the system. Based on those functions which can be plotted as a function of the dimensionless variable (x/L), the calculations of the system can be performed.

#### 2.5. Analytical Solution

_{max}is the maximum deflection of the beam which occurs at the free end, V

_{max}is the maximum produced voltage by the piezoelectric sheets, ω is the angular velocity of the galloping motion, and φ is the phase difference between the tip motion and voltage production.

_{max}and V

_{max}is obtained as follows:

_{L}and Equation (34) by V, integrating the resulting equations concerning time from instant 0 to T/2 (time duration as half the period), and equating the coupling terms between two equations gives the following:

#### 2.6. A Note on Coupling Term

_{31}transversal coupling factor in the driving part). This parameter could appear as a simpler coefficient of the voltage at the first term in the right-hand side of Equation (34). Since the measured output of the piezoelectric sensing device is usually less than the expected ideal value, the above value needs modification as follows:

_{31}is considered a positive value, while here, the negative sign is considered. The coupling factor has affected the performance of the system and is measured from the relation between displacement and produced voltage. The relation between the voltage and tip mass deflection in the open circuit case is proportional to the χ value. It can be found in Equation (38) that the voltage to displacement coefficient of the system at open circuit condition is as follows:

_{ave}is the average distance between the piezoelectric sheet (midpoint of the piezoelectric sheet) from clamped point (${x}_{ave}=\frac{{x}_{e}+{x}_{i}}{2}$), x

_{i}is the minimum distance between the piezoelectric sheet (left of the piezoelectric sheet) from clamped point, and x

_{e}is the maximum distance between the piezoelectric sheet (right of the piezoelectric sheet) from clamped point. Detailed proof of Equation (51) is in Appendix A5. It is noticeable that Equation (50) is a specific case of Equation (51); by substitution of ${x}_{ave}=\frac{{L}_{p}}{2}$ in Equation (51), Equation (50) is retrieved. Another point in Equation (50) is that the configuration of two piezoelectrics cause the same voltage-displacement relation as that of the single piezoelectric case. The reason is that the factor 2 in the nominator of the two moments generated by the piezoelectric sheets is reduced by factor 2 in the denominator of the series of capacitance (twice capacitor value). The conclusion from Equation (50) is the evaluation of the theoretical maximum harvesting power in the energy harvesting of a cantilever with an attached prism under aeroelastic galloping. Replacing the generated voltage in the case of connecting load resistance (See Equation (41).) and maximum displacement of the case of no damping (internal structural damping and the electrical damping) in Equation (50), the maximum voltage in the energy harvesting case is obtained as follows:

## 3. Results and Discussion

#### 3.1. Validation by Experimental Results

_{1}= 7.2 and a

_{3}= −2.1757. Additionally, the fitted results are present in Figure 5. The fitted coefficients predict the onset of galloping at 5.6 mph (2.5 m/s) for R = 0.7 MΩ, and the maximum error is less than 5 Volts.

#### 3.2. Effects of the Load Resistance and Tip Mass Length Ratio on the Harvester′s Response

_{p}(here is ${\eta}_{piezo}\chi $) is adopted by comparing their Equation (1) with Equation (48), and ${C}_{p}=2{\epsilon}_{33}\frac{{w}_{p}{L}_{p}}{{t}_{p}}$. By observation in Equation (56), it is clear that, if the electrical damping ratio defined in Equation (56) is multiplied by angular velocity, the electrical damping defined in Equation (58) is obtained (C = 2ωζ

_{e}). If harvested power is considered during a period of motion

_{1}value found in this study is considerably higher than the a

_{1}value presented in Table 2. This estimates the onset of galloping velocity much lower than that in the experimental results (See Equation (42)). Also, the a

_{3}coefficient is 1 order to 2 orders of magnitude higher than the a

_{3}presented in Table 2. Even the difference in estimation of onset of galloping velocity ignores the 1–2 order of magnitude difference in a

_{3}and could cause a one order of magnitude difference in final deflection (See Equation (39).) and voltage (See Equation (38).) and two orders of magnitude difference in power results (See Equation (44), and compare Figure 10 with Figure 4 of Reference [17]). The difference between fitted coefficient for Reference [9] (Re ≈ 10

^{4}) and that of Reference [17] (Re ≈ 10

^{5}) could be attributed to the various characteristic lengths and ranges of velocity which lead to various Reynolds numbers. While the Reynolds number for Reference [9] is about 11,400 the Reynolds number for Reference [17] is about 67,260. It shows that, to globalize the results of the aerodynamic coefficient to the problem of beam galloping perpendicular to the wind direction, more experiments are needed.

## 4. Conclusions

- In this study, a simple analytical model is used to encounter the effect of a tip mass which could be used by engineers for design of energy harvesting devices with piezoelectric materials.
- The dimensionless functions for calculation of effective mass, effective stiffness, and electromechanical coupling coefficient were presented.
- The effect of tip mass length on effective mass was developed.
- The onset of galloping for the system was predicted analytically.
- The current aerodynamic coefficient in the literature causes great numerical errors.
- The equivalent damping ratio of the electrical load impedance as a function of load impedance was calculated.
- The effect of tip mass on the analytical solution of the concentrated mass was calculated.
- The effect of tip mass is a decrease in the tip displacement in comparison with the point mass.
- The results are fitted on the analytical solution, and the new aerodynamics coefficients included the effect of tip mass.
- As shown by increases of the length of tip mass for the constant beam and piezoelectric length, the inertia of the system increases while the tip displacement and onset of galloping decrease.

## Conflicts of Interest

## Nomenclature

ρ_{a} | Air density (kg·m^{−3}) |

m_{t} | Tip mass (g) |

L_{r} | Length of the tip body (mm) |

D | Width of the tip body (mm) |

V_{∞} | Wind velocity (m/s) |

L | Length of the beam (mm) |

w_{b} | Width of the beam material layer (mm) |

t_{b} | Beam material layer thickness (mm) |

E_{b} | Beam material Young′s modulus (GN·m^{−2}) |

ρ_{b} | Beam material density (kg m^{−3}) |

x_{p} | Start of the piezoelectric sheets (mm) |

L_{p} | Length of the piezoelectric sheets (mm) |

w_{p} | Width of the piezoelectric layer (mm) |

t_{p} | Piezoelectric layer thickness (mm) |

E_{p} | Piezoelectric material Young’s modulus (GN m^{−2}) |

ρ_{p} | Piezoelectric material density (kg m^{−3}) |

d_{31} | Strain coefficient of the piezoelectric layer (pC N^{−1}) |

ε_{33} | Permittivity component at constant strain (nF m^{−1}) |

m_{b} | Beam mass (g) |

m_{p} | Piezoelectric sheet mass (g) |

m_{eff} | Total mass (g) |

k_{b} | Beam stiffness (N/m) |

k_{p} | Piezoelectric sheet stiffness (N/m) |

k_{eff} | Total stiffness (N/m) |

f_{n} | Natural frequency (1/s) |

f_{exp} | Experimental natural frequency (1/s) |

C_{p} | Total capacitance of the piezoelectric layers (nF) |

χ | Coupling factor (mmCm^{−1}·kg^{−1/2}) |

V_{m} | Voltage of the piezoelectric layers (V) |

w | Tip displacement of the beam (m) |

v_{onset} | Onset of the galloping velocity (ms^{−1}) |

## Appendix A

#### Appendix A1. Find the Effective Stiffness of the System

#### Appendix A2. Find the Effective Mass of the System

_{tip}), Equation (23) (for the dimensionless function of mass effect versus location), and Equations (27)–(30) (for mass constants of tip, beam, and piezoelectric sheet), the kinetic energy of the beam is obtained as follows:

#### Appendix A3. Find the Second-Order Governing Equation of Motion of the System

_{tip}) should be considered. F

_{tip}is found from the following:

_{tip}) should be considered. M

_{tip}is found from

_{L}

_{eff}, Equation (34) is retrieved.

#### Appendix A4. Find the Piezoelectric Effect on the Governing Equation of Motion of the System

#### Appendix A5. Find the Relation of Piezoelectric Voltage and Tip Displacement

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**Figure 3.**Dimensionless function for calculating of effective mass, effective stiffness, and electromechanical coupling coefficient.

**Figure 4.**Evaluation of the damping ratio of the system from the measured impulse response of the beam with electrodes in an open circuit [7].

**Figure 5.**Comparison of measured and predicted steady-state voltages as a function of incident wind velocity (R = 0.7 MΩ).

**Figure 6.**Comparison of measured and predicted steady-state voltages as a function of incident wind velocity at open circuit condition.

**Figure 8.**The ratio of the displacement of the distributed tip mass and the displacement of the point tip mass.

**Figure 10.**Variation of the amplitude of the harvested power with the electrical damping C at different wind speeds.

Symbol | Description and Unit | Value |
---|---|---|

ρ_{a} | Air density (kg·m^{−3}) | 1.225 |

m_{t} | Tip mass (g) | 65 |

L_{r} | Length of the tip body (mm) | 235 |

D | Width of the tip body (mm) | 30 |

V_{∞} | Wind velocity (m/s) | 4.02 |

L | Length of the beam (mm) | 90 |

w_{b} | Width of the beam material layer (mm) | 38 |

t_{b} | Beam material layer thickness (mm) | 0.635 |

E_{b} | Beam material Young’s modulus (GN·m^{−2}) | 70 |

ρ_{b} | Beam material density (kg·m^{−3}) | 2700 |

L_{p} | Length of the piezoelectric sheets (mm) | 72.2 |

w_{p} | Width of the piezoelectric layer (mm) | 36.2 |

t_{p} | Piezoelectric layer thickness (mm) | 0.267 |

E_{p} | Piezoelectric material Young’s modulus (GN·m^{−2}) | 62 |

ρ_{p} | Piezoelectric material density (kg·m^{−3}) | 7800 |

d_{31} | Strain coefficient of the piezoelectric layer (pC N^{−1}) | −320 |

ε_{33} | Permittivity component at constant strain (nF·m^{−1}) | 33.6 |

R | Load resistance (MΩ) | 0.7 |

Isosceles 30° | D-section | Isosceles 53° | Square | |
---|---|---|---|---|

a_{1} | 2.9 | 0.79 | 1.9 | 2.3 |

a_{3} | −6.2 | −0.19 | −6.7 | −18 |

**Table 3.**Numerical and experimental results of Sirohi and Mahadik [9].

Symbol | Description and Unit | Value |
---|---|---|

m_{b} | Beam mass (g) | 5.9 |

m_{p} | Piezoelectric sheet mass (g) | 5.4 |

m_{eff} | Total mass (g) | 654 |

k_{b} | Beam stiffness (N/m) | 77.9 |

k_{p} | Piezoelectric sheet stiffness (N/m) | 69.3 |

k_{eff} | Total stiffness (N/m) | 646.3 |

f_{n} | Natural frequency (1/s) | 5 |

f_{exp} | Experimental natural frequency (1/s) | 4.167 |

C_{p} | Total capacitance of the piezoelectric layers (nF) | 658.7 |

χ | Coupling factor (mmCm^{−1} ·kg^{−1/2}) | −12.8 |

${\frac{{V}_{m}}{{w}_{\mathrm{max}}}|}_{R\to \infty}$ | Voltage to tip displacement of the system (Vmm^{−1}) | 0.57 |

R | Load resistance (MΩ) | 0.7 |

R_{max} | Optimal resistance (kΩ) | 57 |

v_{onset} | Onset of the galloping velocity in open circuit Condition (ms^{−1}) | 22.79 |

v_{onset} | Onset velocity in load resistance of 0.7 MΩ (ms^{−1}) | 40.95 |

**Table 4.**Geometrical and material properties [17].

Symbol | Description and Unit | Value |
---|---|---|

ρ_{a} | Air density (kg·m^{−3}) | 1.225 |

m_{t} | Tip mass (g) | 31.6 |

L_{r} | Length of the tip body (mm) | 70 |

D | Width of the tip body (mm) | 60 |

V_{∞} | Wind velocity (m/s) | 9–11 |

L | Length of the beam (mm) | 210 |

w_{b} | Width of the beam material layer (mm) | 50 |

t_{b} | Beam material layer thickness (mm) | 1 |

E_{b} | Beam material Young’s modulus (GN·m^{−2}) | 69 |

ρ_{b} | Beam material density (kg·m^{−3}) | 3067 |

x_{p} | Start of the piezoelectric sheets (mm) | 5 |

L_{p} | Length of the piezoelectric sheets (mm) | 30 |

w_{p} | Width of the piezoelectric layer (mm) | 40 |

t_{p} | Piezoelectric layer thickness (mm) | 0.5 |

E_{p} | Piezoelectric material Young’s modulus (GN·m^{−2}) | 61 |

ρ_{p} | Piezoelectric material density (kg·m^{−3}) | 8148 |

d_{31} | Strain coefficient of the piezoelectric layer (pC·N^{−1}) | −310 |

ε_{33} | Permittivity component at constant strain (nF·m^{−1}) | 38.3 |

**Table 5.**Numerical and experimental results of Jamalabadi et al. [17].

Symbol | Description and Unit | Value |
---|---|---|

m_{b} | Beam mass (g) | 32.2 |

m_{p} | Piezoelectric sheet mass (g) | 4.4 |

m_{eff} | Total mass (g) | 57.6 |

k_{b} | Beam stiffness (N/m) | 31.0 |

k_{p} | Piezoelectric sheet stiffness (N/m) | 27.7 |

k_{eff} | Total stiffness (N/m) | 122.35 |

f_{n} | Natural frequency (1/s) | 7.33 |

f_{exp} | Experimental natural frequency (1/s) | 7.5 |

C_{p} | Total capacitance of the piezoelectric layers (nF) | 92 |

$\chi $ | Coupling factor (mmCm^{−1}·kg^{−1/2}) | −9 |

${\frac{{V}_{m}}{{w}_{\mathrm{max}}}|}_{R\to \infty}$ | Voltage to tip displacement of the system (Vmm^{−1}) | 0.467 |

${\frac{{V}_{m}}{{w}_{\mathrm{max}}}|}_{\mathrm{exp}}$ | Voltage to tip displacement of the system (Vmm^{−1}) | 0.444 |

v_{onset} | Onset of the galloping velocity in open circuit condition (ms^{−1}) | 27.8 |

v_{onset} | Onset velocity in load resistance of 0.7 MΩ (ms^{−1}) | 148.8 |

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## Share and Cite

**MDPI and ACS Style**

Abdollahzadeh Jamalabadi, M.Y. Effect of Tip Mass Length Ratio on Low Amplitude Galloping Piezoelectric Energy Harvesting. *Acoustics* **2019**, *1*, 763-793.
https://doi.org/10.3390/acoustics1040045

**AMA Style**

Abdollahzadeh Jamalabadi MY. Effect of Tip Mass Length Ratio on Low Amplitude Galloping Piezoelectric Energy Harvesting. *Acoustics*. 2019; 1(4):763-793.
https://doi.org/10.3390/acoustics1040045

**Chicago/Turabian Style**

Abdollahzadeh Jamalabadi, Mohammad Yaghoub. 2019. "Effect of Tip Mass Length Ratio on Low Amplitude Galloping Piezoelectric Energy Harvesting" *Acoustics* 1, no. 4: 763-793.
https://doi.org/10.3390/acoustics1040045