GA based optimization of flextensional piezoelectric actuator for rotorcraft active vibration...

Proceeding of 5th International Mechanical Engineering Forum 2012 June 20th 2012 – June 22nd 2012, Prague, Czech Republic

ISBN 978 – 80 – 213 – 2291 – 2 838

GA based optimization of flextensional piezoelectric actuator for rotorcraft active

vibration control

S. Shevtsov1, I. Zhilyaev1, M. Flek2 1Department of Aviation Technology, South Center of Russian Academy, Rostov on Don,

Russia 2Rostvertol Helicopters, Rostov on Don, Russia

Abstract

This article concern a structural optimization problem for a flextensional piezoelectric

actuator which consist of the high power piezoelectric stack and polymeric composite shell

intended for amplification of the stroke. This actuator drive an active flap hosted on the

helicopter rotor blade to change its aerodynamic properties at rotor rotation. As the flap

deflection properly synchronized with the azimuthal angle of rotor blade, such active flap

control devices allows to significantly reduce the noise and vibration level of helicopter. The

principal drawback of the piezoelectric transducers is a very small stroke at relatively high

operating force. In order to supply the required stroke some amplification means are used.

The considered flextensional transducer allow to obtain a needed stroke amplification, but

because of aerodynamic forces and torque act on the flap, the initial actuator’s stroke

significantly reduced. The main aim of this article is the structural optimization of the actuator

design to simultaneously provide sufficient stroke and stiffness, allowing counteract external

loads. To do this, we parametrize the shape of the amplificator’s shell by the rational Bezier

curves, which parameters (coordinates and weights of the control points) are changed

iteratively by genetic algorithm according to the fitness function value calculated by the finite

element model of the transducer with varied geometry.

Key words: helicopter rotor blade, vibration reduction, active control, piezoelectric actuator,

structural optimization, genetic algorithm, finite element modelling

INTRODUCTION

Reducing high frequency dynamic structural loads that increase the wear of components by

minimizing the corresponding disturbances, is an important aim for aerospace designers.

Besides, the modern aircraft must meet the strong requirements to limits for emitted noise.


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The helicopter vibrations are mainly excited by periodic forces generated by the main rotor

and transferred through the rotor head and the gearbox into the fuselage. Most intensive

vibrations are generated at forward flight and because of the blade-vortex interaction (BVI).

At the forward flight with the airspeed V the chordwise velocity U of a blade element is

given by

sinVrU , (1)

where is the blade azimutal position, is a rotor angular velocity, and r is the blade

element distance from the rotor axis. On the retreating side the component of rotational

velocity and horizontal velocity are subtractive and a region of reversed flow will exist. Since

the flow in the reversed flow region is from trailing edge to leading edge it has the effect of

pushing the rotor around the mast thereby extracting power from the airflow. BVI is an

example of an intense vertical velocity field with high velocity gradients. While passing an

airfoil at a predetermined distance, a convecting vortex of positive circulation produces a

downwash velocity while upstream of the blade (airfoil), and this changes to an upwash as it

move downstream. This situation leads to a rapidly and continuosly changing angle of attack,

resulting in highly unsteady aerodynamic loads (KESSLER 2011a).

Different approaches have been explored to achieve a reduction of the helicopter vibration.

Passive means, either located in the rotating or fixed system, aim at exclude the transfer of the

exciting forces into the fuselage by especially tuned absorbers. Research activities of last 40

years showed the high potential of active rotor control with a higher harmonic blade pitch

variation for vibration reduction (KESSLER 2011a). In this case a blade pitch oscillation with

harmonics of the rotor rotational frequency is superimposed to the collective and cyclic blade

pitch (see. Fig. 1, a). The Higher Harmonics Control (HHC) provide that the control signal

was fed into actuators located beneath the swashplate. The promising theoretical and

experimental results led to the development of the Individual Blade Control (IBC) system

(see. Fig. 1, b) where the original pitch links located above the swashplate are replaced by

power actuators which change the incidence (or pitch) angle of the blades. Both technologies

work with a higher harmonic excitation of the blade pitch at the blade root, which is a very

energy-consuming, and also can lead to the blades excitation on a first twisting vibration

mode (HÖFINGER and LECONTE 2004). With an active trailing edge (ATE) flap the

excitation is not introduced at the blade root, but localized at a distance 75-90% of the blade

radius (see. Fig. 2). ATE concept can be implemented in the form of flexible or turned trailing

edge flap, and as the locally morphing airfoil (MAUCHER et al. 2007).

https://www.researchgate.net/publication/228659818_Actuator_design_for_the_active_trailing_edge_of_a_helicopter_rotor_blade?el=1_x_8&enrichId=rgreq-82fd8358-fef5-4161-90f5-b8eed4ba0eec&enrichSource=Y292ZXJQYWdlOzI3MzMxODk5NztBUzoyMDUxODY2MjM2NDM2NDhAMTQyNTkzMTcxMTA0MQ==


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(a) (b)

Fig.1 HHC vs IBC concepts (HÖFINGER, LECONTE 2004)

Fig.2 SMART rotor blade with piezoelectric actuator driving the trailing edge flap

The Actively Controlled Flaps are generally 15% of a chord length and most known structures

are driven by the power piezoelectric transducers. Due to the high operating forces and

frequency, smart materials such as piezoelectric actuators are well adapted for this task, but

relatively large displacements require some sort of mechanical amplification of these devices

movement. Because of the very small displacements created by the piezoelectric devices some

design solutions to amplify the stroke of PZT actuators are proposed (MANGEOT et al. 2009,

NOLIAC 2011).

The most important requirements to the piezoelectric actuator for the active flap design are

the following (CEDRAT 2010):

- high force and large displacement in compact sizes;

- high resolution within the micrometre range;

- very short response time below 1 ms;


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- life time greater than 1010 cycles;

- low voltage supply below 150V DC, no backlash and no play;

- low power consumption when static;

- severe environment compatibility (broad temperature and moisture ranges).

Among the known structural concepts of actuators most adoptable properties had the

flextensinal actuator which consist of the high power pezoelectric stack and shell which

supply the stroke amplification (see Fig. 3).

(a) (b)

Fig.3 Piezoelectric amplified stack with steel (a) and polymeric composite (b) shell

These actuators provide a high stiffness and relatively large displacements. On the other hand,

their mass is not optimised and the metallic shell implies a large penalty on the mass. In order

to improve the mass of these flextensional actuators a carbon composite is used instead of

metal for the frame. Fig. 4 which demonstrate the mounting of actuators on the rotor blade

and show the transmission of aerodinamic forces to the actuator through the levers. These

forces deflect the flap in opposite direction to its active deflection. Because of acting

aerodynamic forces a higher flap deflections require larger and therefore heavier actuators,

but additional mass is needed to equilibrate the center of mass of the blade close to active flap

mounting.

Fig.4 Mounting of active flap and driving piezoelectric actuators on the rotor blade

(KLOEPEL 2006)


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In order to realize the active flap on the real-world five-bladed helicopter we need to optimize

the mechanical properties of the considered flextensional actuator. Primarily we investigate

by the finite element method (FEM) the properties of actuator with most widely used elliptic

composite shell. Next we describe the shape of the shell by the rational Bezier curves, which

allow the good flexibility to simultaneously optimize the stiffness and operating stroke. An

optimization problem will be stated with the given performance of piezoelectric material and

the constraints will be imposed on the acting external force as well as the coordinates of the

points controlling the Bezier curve behavior. As the fittness function we accept the actuator’s

stroke at permissible operating voltage on the PZT stack and given external force. Because of

total number of degree of freedom for this problem is very big, we use genetic algorithm

implemented in the soft package MATLAB (Genetic Algorithm Toolbox) and simplified one-

quarter 2D FEM model of actuator worked in Comsol Multiphysics. Finally we demonstrate

the effectiveness of the developed optimization method and a significantly improved

properties of the optimized actuator.

MATERIALS AND METHODS

As polymeric composite material for the shell is less dense, we have investigated only this

case. We adopted glass fiber epoxy polymeric composite with the longitudinal Young

modulus 10103 Pa and density 1850 kg/m3. Technology of the shell manufacturing includes

the winding of unidirectional glass-fiber tape on a mandrel with appropriate shape, then

curing in autoclave until the complet solidification. The size of the finished shell should

provide a prestress for the assembled piezoelectric stack for a double-side acting actuator.

Stack was constructed of a multilayer piezoelectric ceramic PZT-5H, the thickness of each

layer, polarized in thickness, was 0.5 mm. Driving electric potential was taken as 400 V.

Geometry of a typical FEM model is presented on a Fig. 5.

Fig.5 Geometry of actuator’s FEM model with elliptic shell

and varying ratio of the ellipse semiaxes


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In our numerical experiments we used shell geometry with the thickness 8 mm and 12 mm;

major semi-axis was always 100 mm, whereas the small semi-axis taking values 40, 50, 60,

70 mm.

The FEM static analysis was performed as follows. After applying of the driving potential the

stack lengthened, deforming the shell and causing it to compress in the vertical direction.

Then to the executive surfaces of the shell (small quadrilateral planes) the gradually

increasing tensile force was applied. When the structure is loading the values of operating

stroke, applied reaction force and deformation of piezoelectric stack were monitored. As soon

as the executive surfaces displacement returns to zero, the blocking force is recorded.

Then to the executive surfaces of the shell (small quadrilateral planes) the gradually

increasing tensile force was applied. When loading the values of operating stroke, applied

reaction force and deformation of piezoelectric stack were measured. Some result of

simulations are presented on a Fig. 6.

Fig. 6 Dependencies of the actuator’s stroke (a), amplification ratio (b), reaction force applied

to the PZT stack (c), and compressive deformation of the stack under this reaction force (d) on

the external force for the different dimensional parameters of composite shell

One can see from the graphs (Fig. 6, a, d) that the total compliance of actuator is determined

by the shell, not PZT stack that has a greater stiffness. Indeed, total displacement of actuator


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(40-100-8) at the loading force 2 kN consist of 0.23 mm, whereas deformation of PZT stack

is equal to 0.01 mm only. As might be expected more thick shell has more stiffness, but less

free stroke.

Dependence of the actuator’s parameters on the ratio of the elliptic shell axes is also essential.

The more shallow shells provide greater amplification, but less ability to counteract external

loads. This fact is illustrated on a Fig. 7. where dependencies of the amplification ratio and

blocking force for the different shell dimensions are presented.

Fig.7 Amplification ratio and blocking force of the flextensional actuator

as the functions of the different axes ratio for the elliptic composite shell

with thickness 8 mm (solid lines) and 12 mm (dotted lines)

These preliminary results justify the need and opportunity to optimize the shell’s geometry.

This optimization can be performed by varying the shape of the generatrix and thickness

distribution of the shell. As the objective function is advisable to choose the operating stroke

oph at the given external load actF . To describe the shape of the generatrix with the necessary

flexibility we use the rational Bezier curves because of these lines are inherent for the used

FEM software Comsol Multiphysics. According to the adopted assumptions the one-quarter

of the symmetrical shell can be presented as two branches, each of which consists of three 3rd

order Bezier curves (see Fig. 8). The rational Bezier curve of nth order defined by the 1n

control points iP is described by the equation

n

i

ini

n

i

iini uBwPuBwuR

00

, (2)

where uBin are the Bernstein polynomials, defined as


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inii

n uuini

nuB

)1(

!!

!, (3)

niwi ...1,0, are the weights of the control points, and u is parameter runs through the

values from 0 to 1.

Fig. 8 Representation of the shell’s profile by the 3rd order rational Bezier curves

At optimization process, external dimensions of the shell remain unchanged, so, only shape of

the generatrices is optimized. In order to supply 1C continuity at the connection points

(denoted by ) it is necessary to impose additional constraints on the location of the control

points, which are adjacent to the point connection of two curves. If cP is the coordinates

vector of connection point, lcP is a coordinates vector of a point adjacent to the left-to-point

connection, then coordinates vector of a point adjacent to the right-to-point connection is lcccr PPP 2 . (4)

The Fig. 8 shows the equality of the corresponding edges of the control polygon. The

placement on the vertical (horizontal) lines of points adjacent to the terminal points (denoted

as ) of the generatrices allows to automatically satisfy condition of 1C continuity due to

symmetry of the shell (see Fig. 8). So, we have 21 degree of freedom for the points

coordinates, and 18 degree of freedom for the weights, total number of DoFs is 39.

All control points positions are constrained by the system of inequalities

niYY

niXX

ii

ii

;2;

)1(;1;

)1(

)1(, (5)

by the relationship (4) for connection points by the equalities

outin

n

outinn

outin

outin

bX

YaX

bY

bYX

,1

0,

,1

,00

0;

;0

, (6)


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where the dimensions of the external generatrix outout ba , are fixed, but the dimensions of the

internal generatrix are satisfied by the relationships

iedbb

fixedaainout

inout

var. (7)

All weights also are constrained by the system of inequalities. Each of the three Bezier curves,

which form the generatrix, is described by its own equation (2). But to ensure the smoothness

of generatrix, end points of one curve and the initial point of the connecting curve have the

same weight. So, we have the following system of constraints for weights

niw

w

i ;1;50

10 . (8)

After a number of numerical simulations some additional restrictions are introduced to narrow

the range of the coordinates of control points

1,...1;164372

,

;,25.0,,25.0,

2,

,

1

,,

1

,

niia

aiwhere

aiakXaiak

outin

outini

k

outini

outini

k

outin

. (9)

These constraints allow to agree the density of the control points with the curvature of the

generatrix, and thereby significantly reduce the search area. This is very important for a

problem with large number of DoFs. Similar optimization problems can be successfully

solved only by the genetic algorithm. Therefore we have used the Genetic Algorithm Toolbox

MATLAB which has the advanced optimization means and direct access to the FEM

computation. To reduce the computational cost our FEM model was very simplified. Due to

symmetry of the shell we consider one-quarter of actuator’s structure.

Fig. 9 Geometry and displacements field for optimized FEM model

of the flextensional actuator

(Black arrows indicate the value and orientation of displacements on the boundaries)


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All materials properties and displacement are considered as linear. Quadrilateral FEM mesh

consist of near 500 elements. FEM model work in the Structural Mechanics – Piezoelectric

mode. At each iteration step GA Toolbox rebuild the shell geometry, remesh it, and perform

the static analysis. Calculated value of the actuator stroke is passed to the GA Toolbox, which

changes the value of the design variables according to the constraints. All computation

flowchart is controlled by MATLAB which in turn refers to modules that performs the

algorithm and finite element analysis. These modules are implemented as standard

MATLAB’s m-files. Most important GA settings are: population size – 20, elite count – 4,

crossover – scattered, mutation – adaptive feasible, hybrid function – “fminsearch”.

RESULTS AND DISCUSSION

The developed optimization technique has been successfully applied to the different sizes of

flextensional actuators with polymeric composite shell. All optimized actuators with the

improved parameters have a more complicated shape of the generatrix and a varied thickness

along it. The following table shows comparative data of the actuator APA500L (CEDRAT

2010) and actuator with a similar size, which is optimized by the proposed method.

Tab. 1 Comparative data of optimized actuator and prototype actuator APA500L

(calculation results)

Parameters APA500L Optimized

actuator

Height axis (mm) 50.09 52

Length (mm) 124.55 124

Thickness (mm) 20 20

Stiffness (N/μm) 1.19 1.24

Free displacement (μm) 630 600

Blocked force (N) 750 795

Displacement against

the force 350 N (μm) 510 575

Shell mass (g) 28 30

Total mass (g) 131 135


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Comparison of data for both transducers demonstrates several best features of the optimized

actuator. For a free stroke, less a 5% it has a 5% greater stiffness and a 12% greater stroke

under load. This advantage is crucial, since the power actuators are designed to move against

the acting forces. The total computer time engaged by the process of optimization (530

epochs) was about 6 hours on i5 computer (see Fig. 10).

Fig. 10 Diagram of the best fittness convergence during iterations of the genetic algorithm

Note that the production of an optimized design of composite shell does not require high

manufacturing costs compared with the elliptical shell. This argument proves the usefulness

of the developed methodology and optimization tools, and their attractiveness not only for the

aircraft applications but also in many areas of electromechanics, automatics and

nanopositioning.

CONCLUSION

To provide an actively controlled flap installed on a helicopter rotor blades which reduce

vibrations level an optimization method for flextensional piezoelectric actuator has been

developed and tested. This method is based on the representation of the shape of polymeric

composite shell by the rational Bezier curves, whose parameters (coordinates and weights of

the control points) are the design variables for the optimization algorithm. The optimization


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process is controlled by the genetic algorithm MATLAB, which access a finite element model

of device, calculates the objective functional (the stroke of actuator against external load), and

modifies design variables to achieve the optimum. The effectiveness of the proposed method

is examined by comparative analysis of the prototype structure with elliptic shape of the

composite shell.

ACKNOWLEDGEMENTS

The authors wish to acknowledge the support of the presented work by the Russian National

Helicopter Center, and Russian Foundation for the Basic Research (Grant 10-08013300).

REFERENCES

1. Actively Controlled Flap. Available at: http://www.smartrotorsystems.ca

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3. HÖFINGER, M., LECONTE, P.: Development of an active flap rotor model //

Proceedings of the 6th ONERA-DLR Aerospace Symposium, 22-23 june 2004, 14 p.,

Berlin, Germany

4. KESSLER, C.: Active rotor control for helicopters: motivation and survey on higher

harmonic control. CEAS Aeronaut J., 2011, 1:3–22

5. KESSLER C.: Active rotor control for helicopters: individual blade control and

swashplateless rotor designs. CEAS Aeronaut J., 2011, 1: 23–54

6. KLOEPEL, V.: Advanced Reduction of Helicopter Noise & Vibrations (Eurocopter on

EADS). Proceedings of the International Conference Cooperation with the New EU

Member States in Aeronautics Research, 15th May 2006, 14 p., Berlin, Germany

7. MANGEOT, C., ANDERSEN, B., HILDITCH, R.: New actuators for aerospace.

2009. Available at: www.noliac.com

8. MAUCHER, C.K., GROHMANN, B.A., JÄNKER, P., ALTMIKUS, A., JENSEN, F.,

BAIER, H.: Actuator design for the active trailing edge of a helicopter rotor blade. In:

Proceedings of the 33rd ERF, September 2007, 13 р., Kazan, Russia

9. Noliac Standard Products – ver. 1001. 2011. Available at: www.noliac.com

10. VISWAMURTHY, S.R., GANGULI, R.: Using the Complete Authority of Multiple

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11. SINGH, K.: The Servo Flap Controlled Rotor. Available at: www.helis.com





ISBN 978 – 80 – 213 – 2291 – 2 850

Corresponding author:

Prof. Sergey Shevtsov, Dr.Sc., Department of aviation technology, South Center of Russian

Academy, Tchekhov str. 41, Rostov on Don, 344006, Russia, tel: +79034012285, email:

[email protected]

GA based optimization of flextensional piezoelectric actuator for rotorcraft active vibration...

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