Study on the feasibility of piezoelectrically actuated flapping flight

DRAGAN CVETICANIN

Department of Mechanical Engineering Vanderbilt University

Box 1592 Station B, Nashville, TN 37235 USA

cveticanin@uns.ns.ac.yu

Abstract: In this paper the properties of a micro air vehicle (MAV) that simulates the hovering of the hummingbirds are considered. Specifically, a flying micro robot that contains flapping and flexible wings moved by a unimorph piezoelectric actuator through a link-bar mechanism be considered. The MAV system is analyzed experimentally and analytically. For the known wing type the optimal actuator that flaps the wing and creates the highest lift force is determined. It is concluded that a single crystal type actuator gives the best power density and reaches almost zero net lift, which is the same as hovering. Key-Words: - micro air vehicle, flapping flight, piezoelectric actuator, optimization

1 Introduction Bird and insect flight has fascinated humans for many centuries. Many investigations are made which consider the biological aspect of flight especially of flapping wing flight [1]-[4]. Unfortunately, these results can’t help much in solving engineering problems of flapping flight. The main problem is the extremely high power supply requirement for flapping flight. At Vanderbilt University in Nashville a micro air vehicle (MAV) with flapping motion is investigated. A piezoelectric ceramic as the actuator for flapping wings is under construction. The goal of this paper is to investigate the possibilities of piezoelectric actuators used for creating flapping flight. In order to do so, first the wings with one degree of freedom motion were designed and then the influence of wing design on performances (average lift force and power) were analyzed. The mathematical model of flapping wing and piezoelectric actuator were developed. The two models were connected through the lever arm creating a coupled system. The optimization was performed on the coupled system in order to find the best matching actuator type for a particular wing.

2. Wing design and measurement One of the most important issues for successful development of a micro-air vehicle (MAV) is to create wings that will have highly effective aerodynamic performance. The flapping flight of a hummingbird and its hovering was chosen as the closest case for the robot

we are intended to create. During flapping the hummingbird experiences two motions of the wings: translational and rotational. Since our MAV actuated only one degree of freedom of the wing, i.e. wing flapping (i.e., translation), it was necessary to generate rotation passively through wing design. This was achieved by varying the ratio between bending and torsion natural frequencies of the wing. It was assumed that fundamental wing bending and torsion modes were independent. When the two oscillators are excited by the same input they would respond with different phase relative to the input if the natural frequencies are different, and will exhibit the same phase if the natural frequencies are the same. This theory was used in our wing design. As a result, there were four parameters to control: bending and torsion natural frequencies, flapping amplitude and excitation frequency. According to this, there were two non-dimensional parameters under investigation listed in Table 1.

Non-Dimensional Parameter

Definition Associated Dimensional

Parameter Wing Frequency

Ratio t

b

ωω

γ = Wing Bending

and Torsion Frequencies

Excitation Frequency Ratio

t

e

ωω

η = Excitation Frequency

Table 1.

A set of wings with various wing frequency ratios was constructed. The natural torsion frequency for these wings was 14 Hz. All wings had a rectangular shape with a span of 6 inches and aspect ratio of 2.3, in order to preserve the geometric similarity. The wings were made of carbon-fiber strips and mylar sheets. The mass of the wings was about 2 grams. To quantify the importance of the mentioned non-dimensional parameters γ and η on the performance of flapping flight the experimental investigations are done. The test bed was made to measure the instantaneous thrust force and power usage of different wings. The rig consisted of (Fig.1): a low inertia DC motor to create flapping motion, potentiometer for angle measuring, load cell for lift force and moment measurement, strain gage circuit, control circuit, amplifier, AD/DA converter, d-Space for data acquisition and a high-speed video camera. The input parameters were frequency and amplitude of flapping. The wings were excited on wide range of frequencies and the three dimensional surface plot has been drawn, as shown in Fig.2.

Fig.1. Schematic of the experimental set-up. The 3-D plot shows that the highest generated thrust increases as γ increases from zero to one. Once reaching the wing frequency ratio of one the created lift remains almost constant for any γ larger than one. These results indicate that the generated thrust will be essentially the same for any wing whose natural bending frequency is greater or equal to its natural torsion frequency, and that all wings resonate at lower excitation frequencies than their natural undamped torsion frequencies. The efficiency of wings was analyzed by plotting the curves of lift force to power ratio with respect to excitation frequency. Wings with frequency ratio of 1

and 2 are presented in Fig.3. According to this graph, both exhibit the same efficiency characteristics. The final part of experiment addressed the question of how the flapping amplitude influenced the thrust

Fig.2. Average lift force for a variety of wings and

excitation frequencies.

created by a wing. Fig.4 is representative data for the wings tested. The generated thrust increases with amplitude but the resonant frequency decreases.

0

2

4

6

8

10

12

14

16

0 0.5 1 1.5

Excitation Frequency Ratio

Lif

t to

Po

wer

Rat

io [

g/W

]

wing frequency ratioof 1

wing frequency ratioof 2

Fig.3. Wing flapping thrust to power ratio.

The frequency of resonance drops as a result of large flapping amplitude, which creates large air resistance. This air resistance is, in fact, a damper that dissipates energy. The increase in damping is resulting in shifting the peak to the lower excitation frequencies. As the resistance of air increase, the force on the wing increase, such that the created lift force is bigger. From the analysis of the experimental results the following four conclusions are derived: 1) For the constant natural torsion frequency the average lift created on the wing is roughly constant for all wings with wing frequency ratio greater or equal to one. Since

this is the case the wings with γ=1 were used later for the modeling. 2) The average lift created on the wing is directly proportional to the air resistance, which is a damper. The more energy leaves the system the more lift is created. This is evidenced by the fact that the wing resonance (i.e., frequency of maximum lift) decreases as the flapping amplitude (and lift) increase.

0

2

4

6

8

10

12

0 0.5 1 1.5

Excitation Frequency Ratio

Lif

t [g

]

amplitude of 20degrees

amplitude of 30degrees

amplitude of 40degrees

Fig.4. Thrust generated at various flapping amplitudes.

3) The average lift to power ratio for excitation frequency ratio of one is approximately 8 g/W. This means that the power required for creating lift force of one gram is 125 mW.

3. Modeling of flapping wing The flapping wing is modeled as a lumped parameter system with one degree of freedom. The motion of the system is the rotation of the wing about the wing root. The effect of the lift is modeled as the damping. This effect is presumably relative to the tip motion, so the air resistance damper fits between tip motion and ground. An equivalent mass moment of inertia, spring and two dampers, shown in Fig.5, present the first mode of the system. To prove the correctness of the model the analytically obtained values where compared with experimental one. An experimental wing with equal bending and torsion natural frequencies of 20 Hz was fabricated. A set of experiments was performed for various flapping amplitudes in order to get data of amplitude-moment transfer function for different input frequencies. These data were compared to analytical, as shown in Fig.6 (shown only for amplitude of 10 degrees, since it is similar for 20 and 30 degrees). The crosses represent

experimental data. The analytical curve is matching the points for the best chosen model parameter values. The

Fig.5. Lumped-parameter model of the wing.

values for this particular wing were: mass moment of inertia of JL=5x10-5 kgm2, spring constant of kL=0.8 N/m, damping ratio of ζB=0.1 (B=1.26x10-3 Ns/m), and ζb=0.06θL (bL=7.6x10-4 θL).

Fig.6. Angle to moment transfer function for 10 degrees

amplitude.

4. Modeling of piezoelectric actuator The selected actuator has the piezoelectric unimorph structure which is chosen according to the maximum produced displacement. This structure consists of piezoelectric ceramic layer and thin steel strip, as shown in Fig.7. For the physical model of the piezoelectrically actuated unimorph beam, the following was assumed: - the transverse vibrations of the thin two-layered Bernoulli-Euler beam were investigated - the axial stress in ceramic was produced from the applied electric field - the rotation inertia was neglected - the lengths and width of the layers were equal

- the bonding agent between the two layers was thin and inertialess

Fig.7. Piezoelectric unimorph structure.

- the interface bond had infinite stiffness and no damping creating a perfect bond. The subscript 1 is attached to the variables associated with the upper layer, and 2 with the lower layer. The cross section areas are

11 bhA = , 22 bhA = , (1) where b is the width of the strip and h1 and h2 are the corresponding thicknesses. The moments of inertia of the layers are

12

31

1

bhI = ,

12

32

2

bhI = . (2)

Axial force was applied in the upper layer. It is a function of the ceramic material characteristics and input voltage

11 AEFa ε= , (3)

where

131 h

Vod=ε , (4)

and E1 is modulus of elasticity, A1 is cross section, Vo is voltage input and d31 is piezoelectric material constant. This generated force elongates the top face causing the whole system to move in the vertical direction, because of stationary lower layer. The horizontal force was moved to the neutral axis creating a moment as a product of force and distance between center of upper layer and neutral axis. Since only vertical motions and forces were of our interest, a system of cantilevered two layered beam with a torque was generated. This model with torque acting on the beam was modified into the equivalent system with vertical force acting at the end of the beam, as shown in Fig.8. The two systems were matched up by equality of the end point displacement. For a cantilevered beam with length L, moment of inertia I, modulus of elasticity E and acting moment Mf, the maximal vertical displacement of the free end of the beam is

Mf

x y F

Fig.8. Equivalent systems.

EI

LMy f

2

2

max = . (5)

The maximal displacement of the end of a cantilevered beam of length L and vertical force F employed at the end point is

EI

FLy

3

3

max = . (6)

The equivalent vertical force is developed equalizing the two displacements

L

MF f

2

3= . (7)

Further on the fixed-free two-layered beam with vertical force at the end of the beam was analyzed. Due to the previous assumptions it is evident that the both layers have the same motion. It means that the two-layered beam can be considered as an only one homogenous beam with rigidity and mass, which are the sum of the rigidity and mass of the both layers, respectively. Using the assumption of the Euler-Bernoulli beam the differential equation of motion is

( ) ( ) ( )txpt

vAA

t

vc

x

vIEIE v ,2

2

2

22114

4

2211 =∂∂

++∂∂

+∂∂

+ ρρ

(8) with following boundary conditions

( ) 0,0 =tv , ( ) 0,0 =∂∂

tx

v, ( ) 0,

2

2

=∂∂

tlx

v, ( ) 0,

3

3

=∂∂

tlx

v.

(9) where v(x,t) is the displacement of the layers mid point in the vertical direction, E1I1 and E2I2 are the rigidity of the layers, ρ1A1 and ρ2A2 are the unit masses of the layers, cv is the external damping parameter. The force employed at end of the beam is periodic function described as

( ) thtxp Ω= cos, , (10)

where h is amplitude and Ω is frequency. Let us assume the solution of (8) in the form of

)()(),( tTxVtxv = . (11) Substituting (11) in (8) it becomes

thxVtTcA

AxVtTIEIEtTxV

v

IV

Ω=++

+++

cos)()(2)

)(()())(()(

22

112211

&

&&

ρ

ρ (12)

The equation (12) is multiplied by the weighting function

( ) )sinh()cosh()sin()cos( kxkxkxkxxV λλ −−+= (13) which is the solution of the differential equation

( ) ( ) 02

2

22114

4

2211 =∂∂

++∂∂

+t

vAA

x

vIEIE ρρ , (14)

where k is the solution of the characteristic equation 1)()cos( −=klchkl , (15)

and for the first mode of oscillations it is

lk

8751.1= (16)

the eigenfrequency is

2211

221122

)(1

AA

IEIElk

l ρρω

++

= . (17)

and the parameter λ is

)cosh()cos(

)sinh()sin(

klkl

klkl

+−

=λ . (18)

Integrating the so obtained equation along the beam length the following second order differential equation is obtained

thktTbtTmtT AAA Ω=++ cos)()()( *&&& (19) where

∫∫

∫∫

=+=

=+=

llIV

A

l

vA

l

A

dxxVhhdxxVxVIEIEk

dxxVcbdxxVAAm

0

*

0

2211

0

2

0

22211

)(,)()()(

)(2,)()( ρρ

(20) The steady state solution that corresponds only to the forced vibrations is

)sin(4)(

)(22222

βγϖ

χ−Ω

Ω+Ω−= ttT , (21)

where

22

2tan

Ω−Ω

=ϖ

γβ ,

A

A

m

b=γ2 ,

A

A

m

k=2ϖ ,

Am

h*

=χ .

Using the relations (11), (13) and (21) the vertical displacement of the beam can be calculated from the relations

( )

22222 4)(

)sin()](

)()sin()[cos(,

Ω+Ω−

−Ω−

−−+=

γϖ

βχλ

λt

klsh

klchklkltlv

. (22)

4.1. Experimental verification of the model In order to verify our model of the actuator, it was necessary to compare experimental to analytical results. Several different parameters have been compared. The displacement of the beam end and its resonant frequency were matched up, as well as the blocking force. A number of different actuators were used for the verification. The experiments were made with “Thunder” PZT actuators manufactured by FACE Int. Corp. “Thunder” PZT model

Dimensions [inches]

Ceramic thickness [inches]

Substrate thickness [inches]

6-R 3.0 x 1.8 x 0.022

0.015 0.011

7-R 3.8 x 2.8 x 0.025

0.010 0.0085

8-R 2.5 x 0.5 x 0.019

0.008 0.0065

Table 2. Three different types of PZT are investigated: 6-R, 7-R and 8-R. The dimensions of these actuators are shown in Table 2. The values for maximal displacement of the end of the actuator at the frequency of 1 Hz (taken from the Catalog [5]) were compared to the mathematical results (22). The results are presented in Table 3. The differences between calculated and measured displacements of different PZT do not exceed 10%. The biggest incorrectness is for 8-R and the smallest for 7-R. “Thunder” PZT

model Displacement

from catalog [mm]

Displacement from calculation

[mm]

6-R 3.12 3.3 7-R 7.62 7.5 8-R 1.98 2.1

Table 3.

In Table 4, the resonance frequencies of the actuators are compared with the calculated values (17). It can be seen that the inaccuracy of the model is not bigger that 10%. The final test for the correctness of the mathematical model was to put side by side the experimental and analytical values of blocking force for different actuators. The blocking force was experimentally measured as a force of the actuator acting on the device that stopped the motion of its end. The analytical results were calculated in MATLAB. The blocking force in mathematical model is the value of h* derived in (20). The comparison is presented in Table 5. “Thunder” PZT

model Resonance

frequency from catalog

[Hz]

Resonance frequency from

calculation [Hz]

6-R 60 63.7 7-R 31 32.0 8-R 65 68.5

Table4. “Thunder” PZT

model Blocking force

from catalog

[N]

Blocking force from calculation

[N]

6-R 1.6 1.5 7-R 1.3 1.0 8-R 0.23 0.17

Table 5.

5. Modeling and optimization of coupled PZT actuator and wing model The models of the wing described in 2 and actuator in 4 were coupled through the lever arm, as shown in Fig.9. This system has two degrees of freedom. The generalized coordinates are displacement xA and the angle position of the wing θL. The connection between the piezoelement and of the wing is the lever arm of length l. The displacement xA can be presented as the angle displacement θA

l

xAA ≈θ . (23)

The Lagrange method was used to describe the behavior of the system. The kinetic energy of the system is

22

2

1

2

1LLAAK JxmE θ&& += . (24)

Fig.9. The actuator model coupled with the wing model. The potential energy of the system is

22 )(2

1

2

1LALAA kxk θθ −+=Π . (25)

The dissipation function is 222

2

1)(

2

1

2

1LLALAA BbxbD θθθ &&&& +−+= . (26)

The virtual work of the non-conservative time periodical force is

AxFW δδ = . (27) The differential equations of vibrations are

Fl

k

l

bl

blb

l

klklm

LL

LL

AL

AAL

AAA

=−−

−++++

θθ

θθθ

&

&&& )()(, (28)

0)( =−−+++ ALALLLLLLL kbBbkJ θθθθθ &&&& , (29) where mA, bA, kA and F are the parameters of actuator model and mL, bL, B and kL are the parameters of the wing model. For the model (28)-(29) the net power delivered from the actuator to the wing can be calculated as the energy dissipation in time interval on the dampers bL and B

**LALALLL bBP ωωωω += , (30)

where

LL θω &= , ALLALA θθθω &&& −== , and the ωL* and ωLA

* are their conjugates. This value is compared with experimental results. 5.1 Experimental verification of the model The experimental set-up consisted of an actuator and wing connected to each other through a lever arm, as

shown in Fig.10. The actuator was clamped on one side and the other side was glued to the lever arm. The resonant frequency of the system and the flapping amplitude of the wing were the parameters to be measured with this experiment. Three different actuators with constant length of lever arm (10 cm) and one wing of natural frequency of ratio 1 were used in experiments.

Fig.10. Experimental set-up for model verification. The dimensions of the actuators are presented in Table 6. “Thunder” PZT model

Dimensions [inches]

Ceramic thickness [inches]

Substrate thickness [inches]

Custom 1 3.8 x 1.4 x 0.026

0.010 0.010

Custom 2 4.8 x 0.8 x 0.020

0.008 0.006

7-R 3.8 x 2.8 x 0.025

0.010 0.0085

Table 6.

“Thunder“ actuator type

Resonant frequency from the experiment

[Hz]

Resonant frequency from the calculation

[Hz] 7-R 14 16

Custom 1 15 17 Custom 2 7 10

Table 7.

“Thunder” actuator type

Flapping amplitude from the experiment

[degrees]

Flapping amplitude from the calculation

[degrees] 7-R 7 5

Custom 1 5 3 Custom 2 5 4

Table 8.

The comparisons of the experimental and analytical results are presented in Table 7 and 8. It can be seen that the resonant frequency calculated from the model has the maximum inaccuracy for the Custom 2 type of the PZT and the maximum error of flapping angle is for Custom 1 PZT. The analytical model is overshooting in the calculations for the natural frequencies and undershooting for flapping amplitudes.

6. Optimization description Before making the real MAV it was necessary to find the best combination of wings, actuator and mechanism. For this purpose the optimization process was used. The procedure of optimization is based on varying different variables to achieve the best performance of the system. In our case there were five parameters to control in the optimization, which are thickness of the ceramic and substrate of the PZT, width and length of the actuator and the length of lever arm of the mechanism. These five parameters were too many to control at one time, so three of them were constrained while two were changed. The optimization with two parameters was faster and it was possible to observe the change in characteristics of the system. The goal was to discover the best arrangement of variables in order to get highest possible lift force and power density. For this purpose a MATLAB file was created. The block diagram of the program is shown in Fig.11. The wing data (JL, kL, B) without damping bL and PZT data (d31, modulus of elasticity and density of materials) were entered first and the wing resonant frequency was calculated. The five changeable parameters with their upper and lower limits were entered. In the next step PZT model parameters were calculated. Resonant frequency had to be calculated but it was not possible without all wing parameters. Since the damping bL in the wing model and flapping amplitude θA were unknown and dependent of each other, a loop with iteration method was designed. The flapping angle was calculated from the coupled model equation (29) and from the wing model where it was described as linearly dependent on damping bL. The two were compared and the iteration process moved the error between them to zero. In this loop the resonant frequency of the wing was used as the resonant frequency of the system for the first passing through the program. In the second step the real natural frequency of the system was obtained as well as the flapping amplitude. The lift force on the wing was found from the experimental look-up data table for particular calculated frequency and amplitude. The power density of the

system was calculated using the equation (30) for power and divided by the PZT mass.

Fig.11. Block diagram of the optimization program. 6.1 Results of optimization The optimization program was performed for the “Thunder“ PZT type actuator coupled with a wing with resonant frequency ratio of 1. The natural frequency of the wing was 21 Hz. The dimensions of the actuator varied in the program are reasonable. The width varied between 2 and 7 cm, length between 2 and 10 cm, thicknesses of the ceramic and substrate between 0.006 and 0.02 inches. The data for net-lift and power density were gathered from the optimization. The net-lift presented the value of extra lift when the mass of the system was achieved. In other words, net-lift was the difference between lift force and mass of the system. In our case, positive or at least zero net-lift would be a success. The results of the optimization were presented as three-dimensional plots. The thickness of the ceramic and substrate and the length of the lever arm were constant. Figure 12 and 13 shows the behavior of the created net-lift and power density with respect to length and width of the actuator. It can be seen that the highest extra lift was produced for smallest dimensions of the PZT. For the same PZT the power density was quite small. It seems that the power increases as the dimensions are bigger but the net-lift decreases. This was not expected since the lift should increase with the power. The lift did increase with dimensions but the total mass of the model increased faster so their difference decreased.

Fig.12. Plot of net-lift for various dimensions of the

actuator.

Fig.13. Plot of power density of the system for various

dimensions.

The maximum power density created in the system is about 26 W/kg. This was not enough comparing to early calculation that the necessary power density for flapping flight should be about 125 W/kg. The system was lacking power density so it was unable to create positive net-lift. Since this kind of PZT did not show good results for flapping flight, we turned to the new generation of high power single crystal PZT-PN. The difference between the single crystal and “Thunder” actuator was in the characteristic of ceramic. The single crystal ceramic has eight times higher value of d31 than the “Thunder” ceramic so it creates more strain. The increase in strain

results in increase of excitation force of the actuator model. Higher force gives higher power.

Fig.14. Plot of net-lift for various dimensions of the

actuator.

Fig.15. Plot of power density of the system for various

dimensions. Optimization was performed for this actuator type similar to the previous one. In Figure 14 and 15 the behavior of the net-lift and power density with respect to actuator dimensions are presented. The plot of power density shows the increase of power with dimensions of the actuator. The net-lift value drops as the length and width of the actuator rise to the length of 6 cm and then suddenly the net-lift increase between length of 7 and 9 cm. The net-lift plot shows the existence of a small region where the lift force is almost

equal to the total mass of the system. For that region the power density is 110 W/kg, which is close to the desired 125 W/kg.

7. Conclusion From the obtained optimization results it can be concluded that single crystal type actuator is much closer to achieve flapping flight than “Thunder” type. The maximum power that “Thunder” achieved was four times less than needed. The single crystal actuator almost reached zero net lift, which would be same as hovering. The highest net-lift created was for single crystal actuator dimensions of 8.5 cm x 6 cm and thickness of crystal of 0.01 inch and substrate of 0.006 inches. The optimization gave us the best matching of the actuator for the specific wing characteristics. Even though we did not quite achieve positive net-lift it might be worth trying experiments involving the single crystal PZT. Until today only 1 cm2 size PZT-PN has been designed, so it is left to the future researchers to experimentally verify the optimization results. References [1] W. Shyy, M. Berg, D. Ljungquist, Flapping and flexible wings for biological and micro air vehicles, Progress in Aerospace Sciences, Vol.35, 1999, pp.455-505. [2] M.H. Dickinson, F.O. Lehman, K.G. Gotz, The active control of wing rotation by Drosophila, The Journal of Experimental Biology, Vol.182, 1993, pp.173-189. [3] M.H. Dickinson, F.O. Lehman, S.P. Sane, Wing rotation and the aerodynamic basis of insect flight, Science, Vol.284, 1999, pp.1954-1960. [4] R.J. Templin The spectrum of animal flight: insects to pterosaurs, Progress in Aerospace Sciences, Vol.36, 2000, pp.393-436. [5] Technical specification for standard models “Thunder” actuators for 1998/99, Face International Corp., 1998. ACKNOWLEDGEMENT: The investigations are done at the Vanderbilt University in Nashville with the leadership of Prof. M. Goldfarb.