AEROELASTIC AND PERFORMANCE BASELINE ANALYSIS … · AEROELASTIC AND PERFORMANCE BASELINE ANALYSIS OF PIEZO- ... Aeroelasticity and Flapping Mechanism, ... The characteristic

First International Symposium on Flutter and its Application (ISFA)

1

AEROELASTIC AND PERFORMANCE BASELINE ANALYSIS OF PIEZO-AEROELASTIC WING SECTION FOR ENERGY HARVESTER

Harijono Djojodihardjo1 Key issues: Aeroelasticity and Flapping Mechanism, Bio-Inspired, Flapping Piezoelectric Energy Harvester, Flow Energy Harvesting, Oscillatory Foil.

ABSTRACT. Piezo-aeroelastic energy harvesters convert airflow induced vibrations into electrical energy, while the availability and affordability of piezoelectric transducers offer a class of flapping foil energy harvesters mostly in micro- to milliwatts scale which need to be tuned to match the characteristic frequencies. The present work presents a brief review of aeroelastic instability of a generic typical wing section due to the free stream flow field which is utilized as an oscillating foil energy converter. For propaedeutic analysis a generic piezo-aeroelastic cantilevered beam is defined and treated as a typical section. The basic governing equation of this generic structure is treated as a three degrees of freedom electro-dynamic system, with the first two-degree-of freedom comprising the standard binary aeroelastic system with additional relevant terms to represent the influence of a piezoelectric embedded element on the cantilevered wing. Following the philosophical approach of binary aeroelastic system, the problem is mathematically formulated and solved for the range of solutions that can be obtained depending on the prevailing physical properties of the system, focusing on the stability characteristics of the generic system. The characteristic of the unsteady aerodynamics of the oscillating system associated with favorable energy harvesting capabilities are assessed. I. INTRODUCTION AND BRIEF REVIEW The progress of energy harvesting technology and self-powered systems that has been prompted by the development of low power electronics offers autonomous systems that require minimum maintenance that are essential for their deployment in previously inaccessible locations (Zhu, 2011). Energy Harvesting can be defined as the process of capturing energy, which are relatively minute, from the naturally occurring (“Green”) energy sources, and then accumulating them and storing them for later use. Autonomous system and smart material technology that are now widely utilized take advantage of the thermal energy that can be converted to electrical energy by the Seebeck-Peltier effect, or mechanical energy extraction in piezoelectric system (first discovered by Jacques and Pierre Curie). These make extracting energy from mechanical energy an attractive approach for powering electronic systems, as in piezoelectric system. The piezoelectric effect was discovered by the bothers Pierre Curie and Jacques Curie in 1880, while the converse effect was later discovered by Gabriel Lippmann in 1881 through the mathematical aspect of the theory. These behaviors were labelled the piezoelectric effect and the inverse piezoelectric effect, respectively (Ledoux, 2011). New approaches have been developed using first-principles computations, based on fundamental physics. First-principles theory laid the framework for a basic understanding of the origins of piezoelectric behavior and properties. The principle of the composite energy harvesting system relies on the fact that the material, such as Polyvinylidene fluoride (PVDF) has natural piezoelectric properties. The conversion of mechanical

1 The Institute for the Advancement of Aerospace Science and Technology, Jakarta, Indonesia 15419; email: [email protected]; [email protected]

First International Symposium on Flutter and its Application, 2016 505

1The Institute for the Advancement of Aerospace Science and Technology, Jakarta, Indonesia 15419;email: [email protected]; [email protected]

This document is provided by JAXA.


2

energy electricity by the piezoelectric PVDF is characterized by the coupling coefficient k, which can be as high as 15%.

Figure 1: Schematic estimation of flow energy-to-electrical energy conversion using vibrating wing section and piezoelectric

effects A generic model for energy-to-electrical energy conversion using vibrating wing section and piezoelectric effects for energy harvesters is illustrated in Fig. 1. The conversion of flow energy to aeroelastic vibrational energy consists of two parts, in conformity to the concept of aeroelasticity, as illustrated in Fig 2, consisting of the mechanical vibration part (a) and the aerodynamic part (b).

Figure 2: Generic model of piezo-aeoelastic vibration energy harvester comprising the mechanical-structural (a), aerodynamic (b) and piezoelectric (c) parts. The damping consists of two parts, the mechanical structural damping incorporated in part (a) and the aerodynamic damping incorporated in the aerodynamic forcing of part (b), as shown in Fig.2. The piezoelectric element then establishes the piezo-aeoelastic vibration energy harvester, which harvests the flow energy to be delivered to the electrical loads (Zhu, 2011) II. MECHANICS OF PIEZOELECTRIC PATCHED CANTILEVERED BEAM AS ENERGY HARVESTER For purposes of propaedeutic analysis a generic piezo-aeroelastic cantilevered beam is defined and treated as a binary aeroelastic cantilevered wing section as the base line, with piezoelectric element embedded at the wing root, as further simplified implementation of the generic piezo-aeoelastic vibration energy harvester. The schematic of the piezoaeroelastic energy harvesting system from airflow excitation is depicted in Fig. 3 as the baseline of the present analysis. Aeroelastic vibrations of the cantilever wing bending (plunge or heaving motion, represented by bending angular deflection) strain the piezoelectric patches dynamically and produces the electrical output. Therefore, if piezoelectric coupling is added to the plunge DOF of the typical section as schematically depicted in Figs.2 and 3, the resultant force on the piezoelectric element will result in a resultant voltage.


3

Figure 3: Generic model of linear piezoaeroelastic energy harvester

The governing linear aeroelastic equations to be utilized in the piezo-aeroelastic energy harvesting system can be derived from the binary aeroelastic system as given in the classical literature (for example, Bisplinghoff, Ashley and Halfman, 1955; Wright and Cooper, 2011; Djojodihardjo and Yee, 2009; Djojodihardjo, 2015):

e h hm m h mbx d h K h L (1)

hmbx h I d K M (2) where m is the airfoil mass per length (in the span direction), me is the fixture mass (connecting the airfoil to the plunge springs) per length, M is the aerodynamic moment, L is the aerodynamic lift, and the over-dot represents differentiation with respect to time (t). Incorporating piezoelectric element in the model, the following set of piezo-aeroelastic model is obtained (similar to that obtained by Marqui and Erturk, 2011):

e h hm m h mbx d h K h V Ll

(3)

hmbx h I d K M (4)

0piezol

VC V hR

(5)

Here the fixture mass (me) is defined for the case when the system slightly deviates from the ideal typical section depicted in Figure 5 due to the masses of the shaft, spring mass, and other attachments in real experiments (Sousa et al., 2011), while it is zero in the ideal representation of Fig. 3, which is assumed here for the baseline solution. After incorporating all these factors, further simplification, modification, and redefinition of the relevant and prevailing piezo-aeroelastic parameters, equations (3-5) can be reduced to:

0Lk

I k I qSC DVs

(6)

2 0LI s I k qSebC (7)

1 0piezol

k D C V VR (8)

which will be utilized as the baseline for the solution procedure developed subsequently. All the symbols are given in the Nomenclature. It should also be noted that for the present case the structural and aerodynamic damping have been ignored; further analysis which incorporrate these ignored terms will folow after the baseline case is established. III. SYNTHESIS OF BASELINE SOLUTION PROCEDURE

The problem posed by the set of equations (6) to (8) will be formulated in the following sequence of problems, which is considered to follow an appropriate rationale

JAXA Special Publication　JAXA-SP-16-008E506



2

energy electricity by the piezoelectric PVDF is characterized by the coupling coefficient k, which can be as high as 15%.

Figure 1: Schematic estimation of flow energy-to-electrical energy conversion using vibrating wing section and piezoelectric

effects A generic model for energy-to-electrical energy conversion using vibrating wing section and piezoelectric effects for energy harvesters is illustrated in Fig. 1. The conversion of flow energy to aeroelastic vibrational energy consists of two parts, in conformity to the concept of aeroelasticity, as illustrated in Fig 2, consisting of the mechanical vibration part (a) and the aerodynamic part (b).

Figure 2: Generic model of piezo-aeoelastic vibration energy harvester comprising the mechanical-structural (a), aerodynamic (b) and piezoelectric (c) parts. The damping consists of two parts, the mechanical structural damping incorporated in part (a) and the aerodynamic damping incorporated in the aerodynamic forcing of part (b), as shown in Fig.2. The piezoelectric element then establishes the piezo-aeoelastic vibration energy harvester, which harvests the flow energy to be delivered to the electrical loads (Zhu, 2011) II. MECHANICS OF PIEZOELECTRIC PATCHED CANTILEVERED BEAM AS ENERGY HARVESTER For purposes of propaedeutic analysis a generic piezo-aeroelastic cantilevered beam is defined and treated as a binary aeroelastic cantilevered wing section as the base line, with piezoelectric element embedded at the wing root, as further simplified implementation of the generic piezo-aeoelastic vibration energy harvester. The schematic of the piezoaeroelastic energy harvesting system from airflow excitation is depicted in Fig. 3 as the baseline of the present analysis. Aeroelastic vibrations of the cantilever wing bending (plunge or heaving motion, represented by bending angular deflection) strain the piezoelectric patches dynamically and produces the electrical output. Therefore, if piezoelectric coupling is added to the plunge DOF of the typical section as schematically depicted in Figs.2 and 3, the resultant force on the piezoelectric element will result in a resultant voltage.


3

Figure 3: Generic model of linear piezoaeroelastic energy harvester

The governing linear aeroelastic equations to be utilized in the piezo-aeroelastic energy harvesting system can be derived from the binary aeroelastic system as given in the classical literature (for example, Bisplinghoff, Ashley and Halfman, 1955; Wright and Cooper, 2011; Djojodihardjo and Yee, 2009; Djojodihardjo, 2015):

e h hm m h mbx d h K h L (1)

hmbx h I d K M (2) where m is the airfoil mass per length (in the span direction), me is the fixture mass (connecting the airfoil to the plunge springs) per length, M is the aerodynamic moment, L is the aerodynamic lift, and the over-dot represents differentiation with respect to time (t). Incorporating piezoelectric element in the model, the following set of piezo-aeroelastic model is obtained (similar to that obtained by Marqui and Erturk, 2011):

e h hm m h mbx d h K h V Ll

(3)

hmbx h I d K M (4)

0piezol

VC V hR

(5)

Here the fixture mass (me) is defined for the case when the system slightly deviates from the ideal typical section depicted in Figure 5 due to the masses of the shaft, spring mass, and other attachments in real experiments (Sousa et al., 2011), while it is zero in the ideal representation of Fig. 3, which is assumed here for the baseline solution. After incorporating all these factors, further simplification, modification, and redefinition of the relevant and prevailing piezo-aeroelastic parameters, equations (3-5) can be reduced to:

0Lk

I k I qSC DVs

(6)

2 0LI s I k qSebC (7)

1 0piezol

k D C V VR (8)

which will be utilized as the baseline for the solution procedure developed subsequently. All the symbols are given in the Nomenclature. It should also be noted that for the present case the structural and aerodynamic damping have been ignored; further analysis which incorporrate these ignored terms will folow after the baseline case is established. III. SYNTHESIS OF BASELINE SOLUTION PROCEDURE

The problem posed by the set of equations (6) to (8) will be formulated in the following sequence of problems, which is considered to follow an appropriate rationale




4

1. Formulate the equation as stability equation, which may be met by certain behavior of piezoelectric terms.

2. Establish the relationships between the piezo-electric output in terms of the aeroelastic parameters setting and flow parameters, and determine the related setting of the piezo-electric system to meet certain objectives.

Based on the insight gained in the solution of the problem, preliminary design specifications for a piezo-aeroelastic system can be defined. Following the technique introduced by Wright and Cooper (2011), the general form of the baseline aeroelastic equations (1) and (2) are in the classical second-order form for N degrees of freedom system:

20 0q U q U q A B D C E (9)

where A,B,C,D,E are the structural inertia, aerodynamic damping, aerodynamic stiffness, structural damping and structural stiffness matrices respectively, and q are the generalized coordinates (typically modal coordinates). It is important to note that the B, C matrices only apply for the reduced frequency for which they are defined. Equation (16) can be transformed to Eigenvalue formulation of flutter equations by adding an identity equation of the form 0 Iq Iq (10) Upon combining equation (9) with equation (10), the following equation (11) is obtained

2

00 00 0U U

II q qC E B DA q q

(11)

Equation (11) can further be reduced to as

1 2 1

0 00U U

Iq qA C E A B Dq q

or 0 x Qx (12)

Equations (12) are now in first-order form although the Q matrix has the size of 2N × 2N, twice the size of the matrices in the aeroelastic Equations (6) to (8) and (9). For convenience, following the practice in aeroelasticity, the analysis of flutter stability can be obtained by assuming oscillatory solution of the form:

0 ;te i x x (13) where is the real (damping) part and is the imaginary (oscillatory) part. Hence by assuming x = x0eλt equation (12) can be reduced to the form

0 0 I Q x or 0 0 Q I x (14) Two approaches can be taken to solve equations (12) and/ or (14), namely the frequency domain method and time-integration method in the time domain. a. Frequency Domain Method: The Frequency Domain Method is used to solve the eigenvalue problem as defined by equations (12) or (14).The eigenvalue solutions are computed for a range of reasonable flow velocity range, calculated for any airspeed. The appearance of positive real part of the eigenvalues indicates the occurrence of instability or flutter. b. The time-integration method The time-integration method is used to integrate equations (12) or (14) numerically and calculate the time response for a specific air flow velocity using a conveniently chosen numerical integration method, such as the Runge-Kutta method. IV. FUNDAMENTAL SOLUTION PROCEDURE – DECOUPLED LINEAR APPROACH


5

The baseline aeroelastic equation (14) can be decoupled into two sets of equations, in order that each set has similar solution vector. Hence one can write:

00 0 00

0 0 0 0 00

0 10 0

L

Lpiezo

l

kk D

sI I qSCk

I I qSebCk D C V V

R

(15)

After substituting equation (13) into the equation (15), one obtains 1 1

2

1

0 0 00

0 0 00

0

0

0 010 0

L

Lpiezo

l

I I I IqSCI I I IqSebC

k D C V

kk D

sI Ik

I IV

R

(16)

In the simplified analysis, equation (16) is split into two sets of equations. These are 2 Q M Q M Q M Q 0d k A (17)

and M M 0d kV V (18)

The latter equation can be reduced to 1

1piezol

piezol

k DV C k D V

R CR

(20)

A significant loss of information is noted here, since the simplified piezo-aeroelastic instability formulation failed to include any piezoelectric variable. D disappear from the eigenvalue problem, since the second row of the “simplified” piezo-aeroelastic problem does not contain a term incorporating . Nevertheless, for the interest of developing the solution further, the piezo-aeroelastic eigenvalue problem will be treated. The piezo electric variables, however, is determined by the eigenvalues and the flow

dynamic pressure q=U2/2 in 1piezo

l

k DV

CR

The solution procedure then can be formulated as follows: 1. Find the four eigenvalues =p=σ+i of

2

4 2 2 04

Lcs CI I I k k U k (21)

2. For each set of eigen-value, find the harvested voltage

1piezo

l

k DV

CR

(22)




4

1. Formulate the equation as stability equation, which may be met by certain behavior of piezoelectric terms.

2. Establish the relationships between the piezo-electric output in terms of the aeroelastic parameters setting and flow parameters, and determine the related setting of the piezo-electric system to meet certain objectives.

Based on the insight gained in the solution of the problem, preliminary design specifications for a piezo-aeroelastic system can be defined. Following the technique introduced by Wright and Cooper (2011), the general form of the baseline aeroelastic equations (1) and (2) are in the classical second-order form for N degrees of freedom system:

20 0q U q U q A B D C E (9)

where A,B,C,D,E are the structural inertia, aerodynamic damping, aerodynamic stiffness, structural damping and structural stiffness matrices respectively, and q are the generalized coordinates (typically modal coordinates). It is important to note that the B, C matrices only apply for the reduced frequency for which they are defined. Equation (16) can be transformed to Eigenvalue formulation of flutter equations by adding an identity equation of the form 0 Iq Iq (10) Upon combining equation (9) with equation (10), the following equation (11) is obtained

2

00 00 0U U

II q qC E B DA q q

(11)

Equation (11) can further be reduced to as

1 2 1

0 00U U

Iq qA C E A B Dq q

or 0 x Qx (12)

Equations (12) are now in first-order form although the Q matrix has the size of 2N × 2N, twice the size of the matrices in the aeroelastic Equations (6) to (8) and (9). For convenience, following the practice in aeroelasticity, the analysis of flutter stability can be obtained by assuming oscillatory solution of the form:

0 ;te i x x (13) where is the real (damping) part and is the imaginary (oscillatory) part. Hence by assuming x = x0eλt equation (12) can be reduced to the form

0 0 I Q x or 0 0 Q I x (14) Two approaches can be taken to solve equations (12) and/ or (14), namely the frequency domain method and time-integration method in the time domain. a. Frequency Domain Method: The Frequency Domain Method is used to solve the eigenvalue problem as defined by equations (12) or (14).The eigenvalue solutions are computed for a range of reasonable flow velocity range, calculated for any airspeed. The appearance of positive real part of the eigenvalues indicates the occurrence of instability or flutter. b. The time-integration method The time-integration method is used to integrate equations (12) or (14) numerically and calculate the time response for a specific air flow velocity using a conveniently chosen numerical integration method, such as the Runge-Kutta method. IV. FUNDAMENTAL SOLUTION PROCEDURE – DECOUPLED LINEAR APPROACH


5

The baseline aeroelastic equation (14) can be decoupled into two sets of equations, in order that each set has similar solution vector. Hence one can write:

00 0 00

0 0 0 0 00

0 10 0

L

Lpiezo

l

kk D

sI I qSCk

I I qSebCk D C V V

R

(15)

After substituting equation (13) into the equation (15), one obtains 1 1

2

1

0 0 00

0 0 00

0

0

0 010 0

L

Lpiezo

l

I I I IqSCI I I IqSebC

k D C V

kk D

sI Ik

I IV

R

(16)

In the simplified analysis, equation (16) is split into two sets of equations. These are 2 Q M Q M Q M Q 0d k A (17)

and M M 0d kV V (18)

The latter equation can be reduced to 1

1piezol

piezol

k DV C k D V

R CR

(20)

A significant loss of information is noted here, since the simplified piezo-aeroelastic instability formulation failed to include any piezoelectric variable. D disappear from the eigenvalue problem, since the second row of the “simplified” piezo-aeroelastic problem does not contain a term incorporating . Nevertheless, for the interest of developing the solution further, the piezo-aeroelastic eigenvalue problem will be treated. The piezo electric variables, however, is determined by the eigenvalues and the flow

dynamic pressure q=U2/2 in 1piezo

l

k DV

CR

The solution procedure then can be formulated as follows: 1. Find the four eigenvalues =p=σ+i of

2

4 2 2 04

Lcs CI I I k k U k (21)

2. For each set of eigen-value, find the harvested voltage

1piezo

l

k DV

CR

(22)




6

which consists of V=Vreal () +iVimag() (21)

The sustained fluctuating values of voltage V will be considered as plausible solutions, dictated by the prevailing aeroelastic properties to be further analysed. V. PRINCIPAL PIEZO-AEROELASTIC STABILITY EQUATION Following the approach similar to the transformation of Equation (10) to Eigenvalue (flutter equations) formulation Equation (20) by adding an identity equation, 0 IQ IQ (22)

where V

Q (23)

to the equations (5) to (6) which have been written in a form similar to equation(9), then one gets

0 0 0 1 0 01 0 0 0 0 0 0 0 0 0 1 00 1 0 0 0 0 0 0 0 0 0 10 0 1 0 0 0

0 0 00 0 0 00 2 0 0 0 00 0 0 0

10 0 0 0 0 0 0 0 0

L

L

piezol

VV kk qSC DI I s

k qSebCI IVV k D C

R

00

(24)

or 0 0 0

0 ' ' 0

I I QQA C B QQ

(25)

where 0

A 00 0 0

I I' I I

0 0 0B 0 0 0

0

piezok D C

C 0 2 010 0

L

L

l

kk qSC D

s' k qSebC

R

and 1 0 00 1 00 0 1

I (26a,b,c,d)

Equation (24) or (25) can be written as a. Eigen value problem in frequency domain:

10 0 00 ' ' 0

I I QQA C B QQ

(27)

or G A'' B'' G 0 or GG M G 0 (28)

where


7

V

V

QG

Q (29)

10 0;

0 ' '

I IA'' B''

A C B (30) GM A'' B'' (31)

For further elaboration, different from the simplified split equations approach, one should again look at the oscillating solution assumption to obtain the solution of the extended matrix solution of Equation (28): Assuming periodic solution of G in the form:

0 0

i tte e G G G (32)

where i t then Equation (28) can be converted to

1

G GG M G G M G or 0 GM (33)

which leads to finding the roots that satisfy 0 Gλ M (34)

or

10 0 0 1 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0

0 0 00 0 0 0 0 0 0 0 00 2 0 0 0 00 0 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0 0 0 0 0 0 0

L

L

piezol

kk qSC DI I s

k qSebCI I

k D CR

0

(35) Then, for each set of set of piezo-aeroelastic parameters and wind velocity, represented by q, there are 6 roots of and consequently 6 each of the correspondingσand values. The voltage V of the piezoelectric elements will be the solution vector. The latter notion will be the subject of further investigation.

b. Time-Integration Problem in the time domain/ state space

0 0

0 ' '

I I QQA C B QQ

10 0

0 ' '

I I QQA C B QQ

(36a)

or G A'' B'' G (36a) or GG M G (36b) VI. RESULTS 6.1. Baseline Aeroelastic Stability Results For the purpose of comparison and assessment, the baseline flutter stability results for a typical section as a binary aeroelastic system representing a cantilever Isogai Wing with low frequency aerodynamics following Done-Zwaan aeroelastic stability analysis




6

which consists of V=Vreal () +iVimag() (21)

The sustained fluctuating values of voltage V will be considered as plausible solutions, dictated by the prevailing aeroelastic properties to be further analysed. V. PRINCIPAL PIEZO-AEROELASTIC STABILITY EQUATION Following the approach similar to the transformation of Equation (10) to Eigenvalue (flutter equations) formulation Equation (20) by adding an identity equation, 0 IQ IQ (22)

where V

Q (23)

to the equations (5) to (6) which have been written in a form similar to equation(9), then one gets

0 0 0 1 0 01 0 0 0 0 0 0 0 0 0 1 00 1 0 0 0 0 0 0 0 0 0 10 0 1 0 0 0

0 0 00 0 0 00 2 0 0 0 00 0 0 0

10 0 0 0 0 0 0 0 0

L

L

piezol

VV kk qSC DI I s

k qSebCI IVV k D C

R

00

(24)

or 0 0 0

0 ' ' 0

I I QQA C B QQ

(25)

where 0

A 00 0 0

I I' I I

0 0 0B 0 0 0

0

piezok D C

C 0 2 010 0

L

L

l

kk qSC D

s' k qSebC

R

and 1 0 00 1 00 0 1

I (26a,b,c,d)

Equation (24) or (25) can be written as a. Eigen value problem in frequency domain:

10 0 00 ' ' 0

I I QQA C B QQ

(27)

or G A'' B'' G 0 or GG M G 0 (28)

where


7

V

V

QG

Q (29)

10 0;

0 ' '

I IA'' B''

A C B (30) GM A'' B'' (31)

For further elaboration, different from the simplified split equations approach, one should again look at the oscillating solution assumption to obtain the solution of the extended matrix solution of Equation (28): Assuming periodic solution of G in the form:

0 0

i tte e G G G (32)

where i t then Equation (28) can be converted to

1

G GG M G G M G or 0 GM (33)

which leads to finding the roots that satisfy 0 Gλ M (34)

or

10 0 0 1 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0

0 0 00 0 0 0 0 0 0 0 00 2 0 0 0 00 0 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0 0 0 0 0 0 0

L

L

piezol

kk qSC DI I s

k qSebCI I

k D CR

0

(35) Then, for each set of set of piezo-aeroelastic parameters and wind velocity, represented by q, there are 6 roots of and consequently 6 each of the correspondingσand values. The voltage V of the piezoelectric elements will be the solution vector. The latter notion will be the subject of further investigation.

b. Time-Integration Problem in the time domain/ state space

0 0

0 ' '

I I QQA C B QQ

10 0

0 ' '

I I QQA C B QQ

(36a)

or G A'' B'' G (36a) or GG M G (36b) VI. RESULTS 6.1. Baseline Aeroelastic Stability Results For the purpose of comparison and assessment, the baseline flutter stability results for a typical section as a binary aeroelastic system representing a cantilever Isogai Wing with low frequency aerodynamics following Done-Zwaan aeroelastic stability analysis




8

are exhibited in Figure 4. The favourable agreement validate the aeroelastic stability analysis algorithm used in developing the extended piezo-aeroelastic stability analysis.

Figure 4: Upper-left: Frequency-Reduced Velocity Diagram for Low Frequency Aerodynamic Model. Upper-right: Damping-Reduced Velocity Diagram for Low Frequency Aerodynamic Model . Lower images: corresponding diagram reproduced from Zwaan (1990) for similar analysis.

The baseline flutter stability results for a typical section as a binary aeroelastic system representing a cantilever wing section utilizing quasi-steady aerodynamics following the aeroelastic stability algorithm. for a typical section as a binary aeroelastic system representing a cantilever wing section utilizing Theodorsen unsteady aerodynamics are reproduced in Fig.5. The data utilized in the present study is presented in Table 1.

0 10 20 30 40 50-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

U/b.omegaalpha

omeg

a/om

egaa

lpha

Quasi-Steady, omega/omegaalpha vs U/b.omegaalpha

omega1omega2omega3omega4

0 10 20 30 40 50-20

-15

-10

-5

0

5

U/b.omegaalpha

Sig

ma/

Om

egaa

lpha

Quasi-Steady, Sigma/omegaalpha vs U/b.omegaalpha

sigma1sigma2sigma3sigmao4

(a) (b)


9

0 200 400 600 800 10000

1

2

3

4

5

6x 10

4 omega/omegaalpha vs Reduced Velocity (Theodorsen Unsteady Aerodynamic Model)

Reduced Velocity, Ur

omeg

a/om

egaa

lpha


0 200 400 600 800 1000-4

-3

-2

-1

0

1

2x 10

16 sigma/omegaalpha vs Reduced Velocity (Theodorsen Unsteady Aerodynamic Model)


sigm

a/om

egaa

lpha

sigma1sigma2sigma3sigma4

(c) (d)

Figure 5: (a). Frequency-Reduced Velocity Diagram Low Frequency Aerodynamic Model; (b). Damping-Reduced Velocity Diagram Low Frequency Aerodynamic Model;; (c). Frequency-Reduced Velocity Diagram Theodorsen Unsteady Aerodynamic Model; (d) Damping-Reduced Velocity Diagram Theodorsen Unsteady Aerodynamic Model Table 1: Cantilever Wing Section Parameters

Cantilever Wing Section Parameters

Parameter Magnitude Information s 7.5 m, semi-span c 2.0 m xf 0.48c x=0 at the leading edge e (xf-c/4)-1.0 a 2xf/c-1.0 b c/2 m, Half-chord th 0.12c m, wing section thickness I (10 x 2)2 Nm/rad x b/2 m Area x c x th m2, Wing Section cross-sectional area,

approximated by an ellipse

Table 2: Piezoelectric Element and Cantilever Wing Section and Baseline Parameters

Piezoelectric Element and Cantilever Wing Section and Baseline Parameters Item Magnitude Unit Dpiezo 1.0. m/Volt Reference values and will be followed by parametric study Rpiezo 1.0 Ohm Cpiezo 1.0 Coulomb/Volt Beta 1.0 radian the angular bending deflection of the cantilever beam for

initialization, if required

Fig. 6 exhibits the Flutter stability diagram for the pure wing section treated as a binary system, solved using quartic root approach to solve the eigenvalues of equation (15) using steady aerodynamic approximation. The computation of the corresponding solutions using Theodorsen unsteady aerodynamics is in progress.




8

are exhibited in Figure 4. The favourable agreement validate the aeroelastic stability analysis algorithm used in developing the extended piezo-aeroelastic stability analysis.

Figure 4: Upper-left: Frequency-Reduced Velocity Diagram for Low Frequency Aerodynamic Model. Upper-right: Damping-Reduced Velocity Diagram for Low Frequency Aerodynamic Model . Lower images: corresponding diagram reproduced from Zwaan (1990) for similar analysis.

The baseline flutter stability results for a typical section as a binary aeroelastic system representing a cantilever wing section utilizing quasi-steady aerodynamics following the aeroelastic stability algorithm. for a typical section as a binary aeroelastic system representing a cantilever wing section utilizing Theodorsen unsteady aerodynamics are reproduced in Fig.5. The data utilized in the present study is presented in Table 1.

0 10 20 30 40 50-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

U/b.omegaalpha

omeg

a/om

egaa

lpha

Quasi-Steady, omega/omegaalpha vs U/b.omegaalpha


0 10 20 30 40 50-20

-15

-10

-5

0

5

U/b.omegaalpha

Sig

ma/

Om

egaa

lpha

Quasi-Steady, Sigma/omegaalpha vs U/b.omegaalpha

sigma1sigma2sigma3sigmao4

(a) (b)


9

0 200 400 600 800 10000

1

2

3

4

5

6x 10

4 omega/omegaalpha vs Reduced Velocity (Theodorsen Unsteady Aerodynamic Model)


omeg

a/om

egaa

lpha


0 200 400 600 800 1000-4

-3

-2

-1

0

1

2x 10

16 sigma/omegaalpha vs Reduced Velocity (Theodorsen Unsteady Aerodynamic Model)


sigm

a/om

egaa

lpha

sigma1sigma2sigma3sigma4

(c) (d)

Figure 5: (a). Frequency-Reduced Velocity Diagram Low Frequency Aerodynamic Model; (b). Damping-Reduced Velocity Diagram Low Frequency Aerodynamic Model;; (c). Frequency-Reduced Velocity Diagram Theodorsen Unsteady Aerodynamic Model; (d) Damping-Reduced Velocity Diagram Theodorsen Unsteady Aerodynamic Model Table 1: Cantilever Wing Section Parameters

Cantilever Wing Section Parameters

Parameter Magnitude Information s 7.5 m, semi-span c 2.0 m xf 0.48c x=0 at the leading edge e (xf-c/4)-1.0 a 2xf/c-1.0 b c/2 m, Half-chord th 0.12c m, wing section thickness I (10 x 2)2 Nm/rad x b/2 m Area x c x th m2, Wing Section cross-sectional area,

approximated by an ellipse

Table 2: Piezoelectric Element and Cantilever Wing Section and Baseline Parameters

Piezoelectric Element and Cantilever Wing Section and Baseline Parameters Item Magnitude Unit Dpiezo 1.0. m/Volt Reference values and will be followed by parametric study Rpiezo 1.0 Ohm Cpiezo 1.0 Coulomb/Volt Beta 1.0 radian the angular bending deflection of the cantilever beam for

initialization, if required

Fig. 6 exhibits the Flutter stability diagram for the pure wing section treated as a binary system, solved using quartic root approach to solve the eigenvalues of equation (15) using steady aerodynamic approximation. The computation of the corresponding solutions using Theodorsen unsteady aerodynamics is in progress.




10

(a) (b)

Figure 6: Aeroelastic Stability of the reference wing using Quartic Root approach; (a) Normalized frequency / versus normalized velocity; (b) Normalized damping / versus normalized velocity 6.2. Decoupled Linear Equations Approach for the Binary Aeroelasticity Based Piezo-Aeroelastic System for solving the System Output Voltage The parameters utilized for the cantilevered wing section are those tabulated in Table 1. Table 2 tabulates the reference value of bending angle β of the cantilever wing section and various reference parameters for the piezoelectric element utilized in the piezo-aeroelastic system.

(a) (b)

Figure 7: The resulting frequency and damping of the wing section represented as typical section as a function of the airflow velocity (represented by its normalized values U/b; (a) Variation of the normalized frequency / versus normalized airflow velocity U/b; (b) Variation of the normalized damping / versus normalized airflow velocity U/b. Results are obtained for β =0.1.

(a) (b) (c) (d) Figure 8: The output voltage produced by the piezoelectric element of the piezo-aeroelastic wing sectionsystem represented as typical section as a function of the airflow velocity (represented by its normalized values U/b and various values of β . (a) For β =0.1. (b) For β =0.5.; (c) For β =1.0; (d) For β =5.0. The resulting frequency and damping of the wing section represented as typical section as a function of the airflow velocity (represented by its normalized values U/b are shown in Fig. 7 (a) and (b). The corresponding voltage generated by the piezoelectric element for various values of the cantilever wing section bending angle β as functions of the airflow velocity are shown in the Fig. 8 (a) to (d).


11

VII. DISCUSSIONS AND CONCLUDING REMARKS Following the simplified aeroelastic binary system based piezo-aerolastic energy harvester depicted in Fig. 3, the aeroelastic stability problem associated with the system using a baseline data has been formulated, elaborated and solved using various approaches. In addition, the baseline solutions obtained and presented in the figures are based on quasi-steady aerodynamics. The formulation and computation of the solution using more elaborate Theodorsen aerodynamics are in progress. The formulation of two fundamental solution approaches for the piezo-aerolastic energy harvester system have also been formulated, which incorporate systematic solution procedures using frequency domain binary aeroelastic based stability approach and time integration in the time domain using a conveniently chosen numerical integration method. Based on the simplest approach presented, the outpout voltage produced by the piezoelectric element for a defined piezo-aeroelastic system parameters can be predicted and have been obtained as presented in Figs. 7 and 8. These results can be used as the basis for assessment and further refinement of the method. Judging from the results, a more or less stable and consistent relationship between output voltage produced and airflow energy provided by the airflow on the piezo-aeroelastic binary based system has been indicated, using linearized approach and quasi-steady aerodynamics. However, based on the comparison of Figs.4 and 5, instability regions could be predicted using more refined unsteady aerodynamics. Further development is in progress. References: [1]. Dubin Zhu, Vibration Energy Harvesting: Machinery Vibration, Human Movement and Flow Induced Vibration, U

Southampton, Intechopen, 2011, http://citeseerx.ist.psu.edu/, accessed 20 February 2016. [2]. Antoine Ledoux, Theory of Piezoelectric Materials and Their Applications in Civil Engineering, 2011, Master of

Engineering Thesis, MIT. [3]. Qing Xiao a, Qiang Zhu, A review on flow energy harvesters based on flapping foils, Journal of Fluids and Structures 46

(2014) 174–191 [4]. Djojodihardjo, H., 2015, Analysis and Parametric Study of Aircraft Wings for Aeroelastic Stability and Flutter

Characteristics, submitted for publication [5]. Djojodihardjo, H. and Yee, HH., Parametric Study of the Aeroelastic Stability and Flutter Characteristics Of Generic

Aircraft Wings, Proceedings, Aerotech II, 2007, Kuala Lumpur [6]. Jan R. Wright and Jonathan E. Cooper, Introduction to Aircraft Aeroelasticity and Loads, John Wiley & Sons, 2007 [7]. V C Sousa, M de M Anicezio, C De Marqui Jr and A Erturk, Enhanced aeroelastic energy harvesting by exploiting

combined nonlinearities: theory and experiment , Smart Mater. Struct. 20 (2011) 094007 (8pp) [8]. Bisplinghoff, R.L., Ashley, H. and Halfman, R.L., 1955, Aeroelasticity, Addison & Welesly, 1955. [9]. Carlos De Marqui Jr and Alper Erturk, 2012, Electroaeroelastic analysis of airfoil based wind energy harvesting using

piezoelectric transduction and electromagnetic induction, Journal of Intelligent Material Systems and Structures, 24(7) 846–854, © The Author(s) 2012.

[10]. Bae, J.S., and Inman, D.J., “Aeroelastic Characteristics of Linear and Nonlinear Piezo-aeroelastic Energy Harvester,” Journal of Intelligent Material System and Structures, Vol. 25, Vol. 4, 2014, (4), pp. 401-416.




10

(a) (b)

Figure 6: Aeroelastic Stability of the reference wing using Quartic Root approach; (a) Normalized frequency / versus normalized velocity; (b) Normalized damping / versus normalized velocity 6.2. Decoupled Linear Equations Approach for the Binary Aeroelasticity Based Piezo-Aeroelastic System for solving the System Output Voltage The parameters utilized for the cantilevered wing section are those tabulated in Table 1. Table 2 tabulates the reference value of bending angle β of the cantilever wing section and various reference parameters for the piezoelectric element utilized in the piezo-aeroelastic system.

(a) (b)

Figure 7: The resulting frequency and damping of the wing section represented as typical section as a function of the airflow velocity (represented by its normalized values U/b; (a) Variation of the normalized frequency / versus normalized airflow velocity U/b; (b) Variation of the normalized damping / versus normalized airflow velocity U/b. Results are obtained for β =0.1.

(a) (b) (c) (d) Figure 8: The output voltage produced by the piezoelectric element of the piezo-aeroelastic wing sectionsystem represented as typical section as a function of the airflow velocity (represented by its normalized values U/b and various values of β . (a) For β =0.1. (b) For β =0.5.; (c) For β =1.0; (d) For β =5.0. The resulting frequency and damping of the wing section represented as typical section as a function of the airflow velocity (represented by its normalized values U/b are shown in Fig. 7 (a) and (b). The corresponding voltage generated by the piezoelectric element for various values of the cantilever wing section bending angle β as functions of the airflow velocity are shown in the Fig. 8 (a) to (d).


11

VII. DISCUSSIONS AND CONCLUDING REMARKS Following the simplified aeroelastic binary system based piezo-aerolastic energy harvester depicted in Fig. 3, the aeroelastic stability problem associated with the system using a baseline data has been formulated, elaborated and solved using various approaches. In addition, the baseline solutions obtained and presented in the figures are based on quasi-steady aerodynamics. The formulation and computation of the solution using more elaborate Theodorsen aerodynamics are in progress. The formulation of two fundamental solution approaches for the piezo-aerolastic energy harvester system have also been formulated, which incorporate systematic solution procedures using frequency domain binary aeroelastic based stability approach and time integration in the time domain using a conveniently chosen numerical integration method. Based on the simplest approach presented, the outpout voltage produced by the piezoelectric element for a defined piezo-aeroelastic system parameters can be predicted and have been obtained as presented in Figs. 7 and 8. These results can be used as the basis for assessment and further refinement of the method. Judging from the results, a more or less stable and consistent relationship between output voltage produced and airflow energy provided by the airflow on the piezo-aeroelastic binary based system has been indicated, using linearized approach and quasi-steady aerodynamics. However, based on the comparison of Figs.4 and 5, instability regions could be predicted using more refined unsteady aerodynamics. Further development is in progress. References: [1]. Dubin Zhu, Vibration Energy Harvesting: Machinery Vibration, Human Movement and Flow Induced Vibration, U

Southampton, Intechopen, 2011, http://citeseerx.ist.psu.edu/, accessed 20 February 2016. [2]. Antoine Ledoux, Theory of Piezoelectric Materials and Their Applications in Civil Engineering, 2011, Master of

Engineering Thesis, MIT. [3]. Qing Xiao a, Qiang Zhu, A review on flow energy harvesters based on flapping foils, Journal of Fluids and Structures 46

(2014) 174–191 [4]. Djojodihardjo, H., 2015, Analysis and Parametric Study of Aircraft Wings for Aeroelastic Stability and Flutter

Characteristics, submitted for publication [5]. Djojodihardjo, H. and Yee, HH., Parametric Study of the Aeroelastic Stability and Flutter Characteristics Of Generic

Aircraft Wings, Proceedings, Aerotech II, 2007, Kuala Lumpur [6]. Jan R. Wright and Jonathan E. Cooper, Introduction to Aircraft Aeroelasticity and Loads, John Wiley & Sons, 2007 [7]. V C Sousa, M de M Anicezio, C De Marqui Jr and A Erturk, Enhanced aeroelastic energy harvesting by exploiting

combined nonlinearities: theory and experiment , Smart Mater. Struct. 20 (2011) 094007 (8pp) [8]. Bisplinghoff, R.L., Ashley, H. and Halfman, R.L., 1955, Aeroelasticity, Addison & Welesly, 1955. [9]. Carlos De Marqui Jr and Alper Erturk, 2012, Electroaeroelastic analysis of airfoil based wind energy harvesting using

piezoelectric transduction and electromagnetic induction, Journal of Intelligent Material Systems and Structures, 24(7) 846–854, © The Author(s) 2012.

[10]. Bae, J.S., and Inman, D.J., “Aeroelastic Characteristics of Linear and Nonlinear Piezo-aeroelastic Energy Harvester,” Journal of Intelligent Material System and Structures, Vol. 25, Vol. 4, 2014, (4), pp. 401-416.



AEROELASTIC AND PERFORMANCE BASELINE ANALYSIS … · AEROELASTIC AND PERFORMANCE BASELINE ANALYSIS OF PIEZO- ... Aeroelasticity and Flapping Mechanism, ... The characteristic

Documents