Engineering NotesActive Control of Helicopter Structural

Response Using Piezoelectric Stack

Actuators

Laishou Song∗ and Pinqi Xia†

Nanjing University of Aeronautics and Astronautics,

210016 Nanjing, People’s Republic of China

DOI: 10.2514/1.C031758

I. Introduction

B ECAUSE the rotor of a helicopter operates in a periodic,asymmetric, unsteady aerodynamic environment, the helicopter

fuselage produces a high level of vibration under the strong excitationof the rotor. The effective method for active vibration control of ahelicopter fuselage by using the inertial actuators has been used inhelicopters, but the inertia actuators have a considerable weightpenalty for a better control effect. Meanwhile, the inertia actuatorshave a limited range of working frequency and a lag response tocontrol signal. The piezoelectric stack actuator has a lot of advan-tages, such as light weight, large control force, wide range ofworkingfrequency, and fast response to control signal, and has been used as anactuation element for active control of structural vibration [1,2].Hence, using the piezoelectric stack actuators is a newway to activelycontrol the vibration of a helicopter fuselage.In an active vibration control system, the locations of actuators

have a great influence on the effect of vibration suppression and thepower requirement. Many approaches such as the controllabilityindex [3], energy dissipation index [4],H2 norm index [5], and recentindex composed of a multi-objective [6] have been developed tofind the optimal actuator locations and control parameters. In theinvestigations of the helicopter fuselage, Hanagud and Babu [7]placed the piezoelectric actuator near the selected control locationand investigated the vibration reduction of the helicopter fuselage byusing H∞ control. Singhvi and Vennkatesan [8] addressed thepiezoelectric stack actuator parallel with the supporting structurebetween the gearbox and fuselage for a simplified helicopter model.Heverly et al. [9] investigated the optimal placement of piezoelectricstack actuators in the fuselage by using the simulated annealing algo-rithm. The investigations indicated that the configuration of optimaldistributed actuators was capable of greater vibration suppressionwith less control effort. However, in the existing investigations, thepiezoelectric stack actuator was idealized as a force generator. Inthis case, the characteristic effect of the piezoelectric stack actuatorwas not included in the optimization process.The optimal locations of actuators may have a lot of selections at

many possible positions in an actual structure. The optimal selectionof actuator locations cannot uniquely be determined by using the

conventional optimization techniques based on the gradient-descentmethods. The genetic algorithm as a stochastic search technique hasbeen effectively used to determine the optimal locations. Rao et al.[10] presented a modified binary-coded genetic algorithm to solvethe optimal placement of discrete actuator locations in the frameworkof a zero–one optimization. Liu et al. [5] directly applied the binary-coded genetic algorithm to find optimal locations of actuators andsensors on plate structures. The real-coded genetic algorithm [4] wasused to address the optimal locations of the piezoelectric actuators ata continuous spatial coordinate on a beam. Roy and Chakraborty [6]used the integer-coded genetic algorithm to optimize the placementof actuators and simultaneously real-coded genetic algorithm todetermine the weighted matrices in the linear quadratic control.In this paper, the active control of a helicopter structural response

by using piezoelectric stack actuators has been investigated. In theformulated dynamic model, the piezoelectric stack actuators andfuselage coupled composite structure was decomposed by usingthe substructure synthesis technique based on frequency responsefunctions. The weighted quadratic of controlled accelerations inthe frequency domain was chosen as the optimization index. An im-proved real-coded genetic algorithm was used to solve the optimi-zation problem with discrete location variables and continuousweighted variables and to minimize the acceleration responses of thetargeted locations. The vibration suppression of a simplified elastichelicopter fuselage model was analyzed. The numerical results showthat the method proposed in this paper can effectively solve theoptimal parameters and improve the control performance.

II. Mathematical Modeling of CoupledComposite Structure

An actuation element of a piezoelectric stack actuator is shown inFig. 1. The applied mechanical force and electric field to the actuatorare along the axis direction (3-3 direction of piezoceramic material).Theworking frequency of the active control of the helicopter fuselagevibration system (dozens of Hertz) is much smaller than the firstaxial natural frequency of the piezoelectric stack actuator (thousandsof Hertz), and so the actuator inertia can be ignored. The one-dimensional constitutive equation of piezoceramic material isexpressed as [11]

σ3 � cE33ε3 − e33E3 (1)

where σ3 is the stress, ε3 is the strain, E3 is the applied electric field,cE33 is the modulus of elasticity at zero or constant electric field, ande33 is the piezoelectric stress constant. The electric fieldE3 related tothe applied voltage v is given asE3 � v∕t, where t is the thickness ofeach piezoceramic patch layer.From Eq. (1), the relation among the applied force, produced

displacement, and applied voltage of a piezoelectric stack actuatorcan be derived as

fi � −fj � f � Aσ3 � AcE33uj − uiL

− Ae33v

t(2)

where fi, fj, and ui, uj denote the forces and displacements at theends of the piezoelectric stack actuator, respectively; L is the lengthof the piezoelectric stack actuator; and A is the cross section of thepiezoelectric stack actuator. The extensional displacement δ of thepiezoelectric stack actuator can be expressed as

δ � uj − ui �L

AcE33f� Le33

cE33tv (3)

Received 12 December 2011; revision received 15 October 2012; acceptedfor publication 20 October 2012; published online 22 February 2013.Copyright © 2012 by the American Institute of Aeronautics andAstronautics,Inc. All rights reserved. Copies of this paper may be made for personal orinternal use, on condition that the copier pay the $10.00 per-copy fee to theCopyright Clearance Center, Inc., 222RosewoodDrive, Danvers,MA01923;include the code 1542-3868/13 and $10.00 in correspondence with the CCC.

*Ph.D. Candidate, Laboratory of Rotorcraft Aeromechanics.†Professor, Laboratory of Rotorcraft Aeromechanics; xiapq@nuaa.edu.cn

(Corresponding Author).

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A piezoelectric stack actuator can only produce a limited force forvibration control. Hence, a set of piezoelectric stack actuators can beassembled to produce enough force for vibration control. In this case,a matrix equation of extensional displacement is adopted as

δa � HaaFa �HavV (4)

where the vector δa � � δ1 δ2 : : : δn �T is the extensionaldisplacement vector. The matrices

Haa � diag

�LAcE

33

LAcE

33

: : : LAcE

33

�

and

Hav � diag

�Le33cE33t

Le33cE33t

: : : Le33cE33t

�

are the force-flexibility matrix and the generalized voltage-flexibilitymatrix, respectively. The vectors Fa � � f1 f2 : : : fn �T andV � � v1 v2 : : : vn �T are the applied force vector and thecontrol voltage vector, respectively.Ignoring the actuator inertia, the expression of extensional

displacement in the frequency domain has the same form as Eq. (4) asfollows:

δa�ω� � Haa�ω�Fa�ω� �Hav�ω�V�ω� (5)

where δa�ω�, Fa�ω�, and V�ω� are the extensional displacementvector, applied force vector, and control voltage vector in frequencydomain, respectively.Haa�ω� andHav�ω� are the frequency responsefunctions (FRFs) from themechanical force and control voltage to theextension displacement.The dynamic equation of the fuselage substructure can be given in

terms of frequency response functions and is expressed as

�ub�ω�ac�ω�

���Hbb�ω� Hbd�ω�Hcb�ω� Hcd�ω�

��Fb�ω�Fd�ω�

�(6)

where ub�ω� is the vector of relative displacement between two endsof actuator;Fb�ω� is the vector of control forces to the fuselage by thepiezoelectric stack actuators; ac�ω� is the vector of accelerations atthe vibration-controlled points in the fuselage; Fd�ω� is the vector ofexternal forces; and Hbb�ω�, Hbd�ω�, Hcb�ω�, and Hcd�ω� are theFRF matrices related to the control forces and external forces.According to the conditions of force equilibrium, Fa�ω� �

Fb�ω� � 0 and the displacement compatibility δa�ω� � ub�ω� at theconnecting points between substructures, and by combining Eqs. (5,6), the control force vector Fb�ω� is obtained by

Fb � −�Haa �Hbb�−1HbdFd � �Haa �Hbb�−1HavV (7)

Substituting Eq. (7) into Eq. (6), the accelerations at the vibration-controlled points in fuselage are obtained by

ac � HcdFd � HcvV (8)

where Hcd � Hcd −Hcb�Haa �Hbb�−1Hbd is the FRFmatrix fromthe external forces to the acceleration responses, and Hcv �Hcb�Haa �Hbb�−1Hav is the FRF matrix from the control voltagesto the acceleration responses.During the optimization procedure, once a new set of optimal

locations were produced, the FRF of the composite structure wasobtained simply through an algebraic operation of FRFs from the

external forces and control voltages to the acceleration responses inthe substructures.

III. Parametric Optimization of ActiveVibration Control System

The vibration of a helicopter fuselage has a characteristic of low-frequency, steady, and harmonic vibration, as shown in Fig. 2 for anacceleration response spectrum measured in a helicopter fuselage.The low-frequency and steady components dominate the vibration.The harmonic responses measured in the fuselage are the sum ofvibrations due to the external forces and control input voltages at thedisturbance frequency ωi and expressed as

ac�ωi� � Hcd�ωi�Fd�ωi� � Hcv�ωi�V�ωi� (9)

Considering the first R harmonic components, the responses can beexpressed as

ac�ω� � Hcd�ω�Fd�ω� � Hcv�ω�V�ω� (10)

where ac�ω� � �ac�ω1�T : : : ac�ωR�T �T , Fd�ω� � �Fd�ω1�T: : : Fd�ωR�T �T , V�ω� � �V�ω1�T : : : V�ωR�T �T , Hcd�ω� �diag� Hcd�ω1� : : : Hcd�ωR� �, and Hcv�ω� � diag� Hcv�ω1�: : : Hcv�ωR��.A weighted quadratic objective function [9] composed of the

controlled acceleration responses and control voltages in frequencydomain is given as

J � aHc �ω�Waac�ω� � VH�ω�WvV�ω� (11)

where the superscriptH denotes the conjugate transpose, andWa andWv are theweighted matrices. Substituting Eq. (10) into Eq. (11) andoperating by ∂J

∂V � 0, the complex vector of the optimal controlvoltage is yielded as

Vopt � −�HHcvWaHcv �Wv�−1HH

cvWaHcdFd (12)

Then, the controlled acceleration response is given by

ac � �Hcd − HHcv�HH

cvWaHcv �Wv�−1HHcvWaHcd�Fd (13)

From Eqs. (12, 13), the control voltage and the controlled responseare related to the actuator locations by HH

cd and HHcv and to the

weighted factors by Wa and Wv. To acquire the best vibration-suppression performance under the limitation of the input voltageinto the piezoelectric stack actuators, the optimization in this studycan be expressed as

min ∶J�P;Wa;Wv� � aHc Waac (14)

Fig. 1 Piezoelectric stack element.

Fig. 2 Measured acceleration response spectrum in a fuselage.

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by the constraints

�P ∈ Possible location set

vopt ≤ vmax(15)

whereP is the selected actuator location set, and vmax is themaximumallowable voltage into the piezoelectric stack actuators.

IV. Improved Genetic Algorithm for Optimization

The actuator mounting positions are discrete variables, and theweighted parameters are continuous variables. The optimizationdescribed by the objective function [Eq. (14)] and constraints[Eq. (15)] is a nonlinear mixed-variable optimization problem. As aparallel, guided, random adaptive search technique, the geneticalgorithm is a computationally effective algorithm for this kind ofoptimization [4,5]. In this investigation, an improved real-codedgenetic algorithm suitable for the mixed-variable optimization hasbeen developed to solve the discrete location variables andcontinuous weighted variables simultaneously.The genetic algorithm works with an encoding of the parameter

set. Figure 3 shows the encoding used in this paper and the impliedphysical parameters. During the genetic process, the actuatorlocations and weighted parameters were represented by real numbersin which the real numbers denoted that the location variables wererounded into integers to identify the actuator positions.To ensure the offspring within bounds of parameters, a modified

Laplace crossover operator [12] was presented in this paper. Aprobability distribution function was used to create two childrensolutions around the parent solutions as

F�x� �

8<:

12exp

�xη

�x ≤ 0

1 − 12exp

�− x

η

�x > 0

(16)

where η is the scale parameter controlling the extent of spread in thechildren solution.For the selected parents x�1�, x�2�, to be operated by crossover, the

uniformly distributed random numbers u; r ∈ �0; 1� are generated,and then the corresponding random parameter β that satisfies theLaplace distribution is generated as

β �

8>>><>>>:−b ln

�α−�u − 1

2

�� 1

2

�r ≤ 1

2

b ln

�α��u − 1

2

�� 1

2

�r > 1

2

(17)

where α− � 1jx�2�−x�1�j min�x�1� − xl; x�2� − xl�, α� � 1

jx�2�−x�1�jmin�xu − x�1�; xu − x�2��, which ensure a zero probability of creatingchildren outside the prescribed range [xl, xu]. The two offspring areobtained as

y�1� � x�1� � βjx�1� − x�2�jy�2� � x�2� � βjx�1� − x�2�j (18)

A mutation operator based on power distribution is used here. Thedensity function is given by

F�x� � xp; 0 ≤ x ≤ 1 (19)

where p is the index of distribution degree. For a small p, lessperturbation in the children solutions is expected, and for a large p,

more diversity is achieved. For a selected parent x to be operated bymutation, the uniform random numbers u; r ∈ �0; 1� are created, andthen the corresponding random parameter s that satisfies the powerdistribution is generated as

s � up (20)

The offspring solution is obtained as

y ��x − s�x − xl� t < rx� s�xu − x� t ≥ r (21)

where t � x−xlxu−x

.During the genetic operation, two or more same locations may be

generated. This situation is permitted when two ormore piezoelectricstack actuators are allowed to be located at the same position.Otherwise, the corresponding fitness of this chromosome is set to be avery large value so that it should be discarded in the selectionprocedure.

V. Active Vibration Control of Helicopter Fuselage

A. Fuselage and Actuators Coupled Model

As a numerical example for helicopter vibration control by usingthe method presented in this paper, a simplified elastic model of ahelicopter fuselage for numerical analysis is shown in Fig. 4. Themodel has length of 12.2 m and weight of 2.8 tons. The structuralparameters of the fuselage model are listed in Table 1. The degrees offreedom at each node contained the vertical displacement and pitchangle. The first five natural frequencies are analyzed as 8.3, 17.3,34.0, 48.7, and 64.7 Hz. The modal damping ratio of fuselage wasassumed as 2%. The rotor exciting force was located at the point O,4.8 m away from the original point 0 of the coordinate.The vibration control system included four piezoelectric stack

actuators as control inputs and six vibration control points as controloutputs. The six selected control points A, B, C, D, E, and F werelocated at 1.2, 1.8, 3.0, 3.6, 4.5, and 7.2 from the original point 0,respectively. It was assumed that there were 20 possible installationlocations of piezoelectric stack actuators. The first installationlocation was located at 1.2 m from the original point, and the intervallength between two installation locations was 0.3 m. The momentproduced by the offset installation of piezoelectric stack actuator wasapplied to the fuselage model. The offset length was determined bythe stiffness equivalent principle [11]. If the calculation result ofoffset length was less than 0.5 m, then the calculated value was takenas the offset length. If the calculation result was larger than 0.5 m,then 0.5mwas selected as the offset length, considering the height ofthe fuselage. For an actual fuselage, the actuator can be located at theintersection of the longeron/stringer and the frames or in parallel withlongeron between frames, which can produce a moment at the

Fig. 3 Encoding of parameters.

Fig. 4 Simplified fuselage model.

Table 1 Structural parameters of the elastic fuselage model

Location (coordinate x), m Stiffness, Pa · m4�×106� Mass, kg∕m0 ∼ 1.2 4.44 75.51.2 ∼ 3.6 11.13 276.53.6 ∼ 6.0 20.60 603.06.0 ∼ 7.2 8.35 151.07.2 ∼ 8.4 5.02 25.08.4 ∼ 9.6 2.78 25.09.6 ∼ 10.8 2.24 20.5

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fuselage cross section. The physical parameters of the piezoelectricstack actuator are listed in Table 2.

B. Vibration Control Based on Parameter Optimization

In the vibration control analysis, the rotor excitation was assumedas a harmonic force with double frequencies 22 and 44 Hz. Theexcitation was used to simulate theNΩ and 2NΩ (N is the number ofthe blade andΩ is the rotational speed of the rotor) force produced bythe rotor. Generally, the components at the two frequencies dominatethe vibration responses of the helicopter fuselage. The force ampli-tude of the 22 Hz excitation was adjusted to match the accelerationresponse level (0.184 g) at the point A to the flight test value. Theforce amplitude of the 44 Hz excitation was 30% of the 22 Hzexcitation. The voltage amplitudewas limited to 90V,whichwas halfof the working voltage range of the piezoelectric stack actuator. Theweighted matrix for acceleration responses was taken as constantmatrix Wa � diag� 1e4I6 1e3I6 � in which I denoted the identitymatrix. Theweightedmatrix for controlwasWv � diag� β1I4 β1I4 �,where β1 and β2 were the weighted parameters to be optimized withthe installation locations so as to keep the control input within theworking range. The optimization results indicated that the bestcontrol effects could be obtained when the installation locations ofthe piezoelectric stack actuators were located at points 8, 11, 13, and15, and the weighted factors β1 and β2 were 0.088 and 0.308,respectively.Figure 5 shows the acceleration response amplitudes of the

targeted control points A, B, C, D, E, and F without piezoelectricstack actuators (PSAs), with PSAs and without control, and withPSAs and with control. It shows that remarkable vibration suppres-sion has been obtained after the control with the optimizationparameters. Comparing with the response amplitudes withoutcontrol, the acceleration response amplitudes of 22 Hz excitation atpositions A, B, C, D, E, and F decreased by 96, 78, 96, 95, 66, and77%, respectively. The real controlled response amplitudes were alllower than 0.015 g. The response amplitudes of 44 Hz excitationdecreased by 45, 87, 70, 69, 93, and 34%, respectively. The controleffect for 44 Hz excitation was not so obvious compared with thecontrol effect for the 22 Hz excitation, because the response ofNΩ is

much larger than the response of 2NΩ and more control effort wasstressed on the response of NΩ. Figure 5 also shows the responseswithout installation of piezoelectric stack actuators. In the case of nocontrol, the responses with and without installation of piezoelectricstack actuators are different, indicating that establishment of thefuselage/PSAs coupled structural model is necessary.In this investigation, the comparison between the fuselage/PSAs

coupled structuremethod presented in this paper, dented bymethod 1and themethod bymodeling the piezoelectric stack actuator as a forcegenerator [10], dented bymethod 2, was also analyzed. Bymethod 2,the obtained optimal installation locations 7, 10, 13, and 17 weredifferent from the optimal installation locations 8, 11, 13, and 15obtained by method 1. Figure 6 shows the controlled accelerationresponses with the two sets of different optimal locations under thesame control voltage. The controlled acceleration responses bymethod 2 are much higher than the results by method 1 except for theposition E due to 22 Hz excitation and the position D due to 44 Hzexcitation. The coupled structural model can obtain a much bettereffect of activevibration control than the no coupled structuralmodel.To further study the effect of control, the simulations in the time

domainwere conducted based on the optimal parameters. The controlsignals were synthesized by using the adaptive method based on thenegative gradient for avoiding the sharp change of the control inputand calculation of the inverse matrix. From the objective function[Eq. (11)], the following formulation was obtained:

∂J∂V� 2HH

cvWaac � 2WvV (22)

Table 2 Parameters of the

piezoelectric stack actuator

Parameter Value

Free displacement, μm 100Blocked force, N 1.4 × 104

Stiffness, N∕m 8 × 107

Voltage range, V −30 ∼ 150First natural frequency, kHz 8.0Mass, kg 0.35

Fig. 5 Response amplitudes before and after control with optimal

parameters.

Fig. 6 Controlled responses with two optimal configurations.

Fig. 7 Acceleration responses of controlled positions by dual-frequency

excitation.

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Then, the adaptive iteration equation of the control voltage complexvector was given by

V�i� 1� � �I − 2μWv�V�i� − 2μHHcvWaac�i� (23)

where μ is the iteration interval, and i is the iteration order. It wasproved [13] that, when 0 < μ < 1∕λmax, the control voltage vectorconverged to the optimal control voltage as expressed in Eq. (12),where λmax was the maximum characteristic value of the matrixWv � HH

tvWaHHtv. The simulation continued to 20 s.When the system

is running to 5 s, the control systemwas tuned on. Figure 7 shows theacceleration responses of the six selected control positions. It can beseen that the controlled responses reach quickly to the expected levelof vibration suppression. Figure 8 shows the control voltages forpiezoelectric stack actuators at the four locations.

VI. Conclusions

Themethod for active control of a helicopter structural response byusing piezoelectric stack actuators has been established in this paper.The dynamic equations of a coupled helicopter fuselage and piezo-electric stack actuators in the frequency domain were formulated byusing the substructure-synthesis technique. The weighted quadraticfunction of controlled acceleration responses was taken as theobjective function for parameter optimization of the active vibrationcontrol system. The improved real-coding genetic algorithm wasdeveloped to optimize the actuator positions and the controllerparameters.A simplified elastic helicopter fuselage model by double-

frequency excitation was used for numerical analysis of the controlsystem with four control inputs and six response outputs. Thenumerical results show that the method proposed in this paper caneffectively find the best actuator positions and controller parametersas well as obtain the obvious effect of vibration control. The fuselageand piezoelectric stack actuators coupled structural model can

produce more effective optimal results of vibration control thanthe model idealizing the piezoelectric stack actuator as the forcegenerator.

References

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[10] Rao, S. S., Pan, T.-S., and Venkayya, V. B., “Optimal Placement ofActuator in Actively Controlled Structures Using Genetic Algorithms,”AIAA Journal, Vol. 29, No. 6, 1991, pp. 942–943.doi:10.2514/3.10683

[11] Giurgiutiu, V., and Rogers, C. A., “Power and Energy Characteristics ofSolid-State Induced-Strain Actuators for Static and DynamicApplication,” Journal of Intelligent Material Systems and Structures,Vol. 8, No. 9, 1997, pp. 738–750.doi:10.1177/1045389X9700800903

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[13] Haykin, S.,Adaptive Tilter Theory, 4th ed., Prentice–Hall, Upper SaddleRiver, NJ, 2003, pp. 208–213.

Fig. 8 Control voltages for four piezoelectric stack actuators by

dual-frequency excitation.

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