Theoretical and Experimental Studies of Active Vibration Control for Beams Using Pole Placement Method

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College of Aerospace and Civil Engineering，Harbin Engineering University，Harbin 150001，P.R.China

Abstract: The vibration control in the frequency domain is significant. Therefore，an active vibration control in frequency domain is studied in this paper. It is generally known that piezo-intelligent structures possess satisfactory performances in the area of vibration control，and macro-fiber composites（MFCs） with high sensitivity and deformability are widely applied in engineering. So，this paper uses the MFC patches and designs a control method based on the pole placement method，and the natural frequency of the beam can be artificially designed. MFC patches are bonded on the top and bottom surfaces of the beam structure to act as the actuators and sensors. Then，the finite element method（FEM）is used to formulate the equation of motion，and the pole placement based on the out-put feedback method is used to design the active controller. Finally，the effectiveness of the active control method is verified.

Key words：piezo-intelligent；macro-fiber composite（MFC）；pole placement；closed-loop control；finite element method（FEM）

0 Introduction

Vibration is a natural phenomenon，which is very common in the field of engineering. The struc?ture exposed to vibration in the long term may cause damage of precision instrument even endanger the structural stability. As a result，vibration control has always been an important challenge in engineering field. Ordinarily，vibration control is composed of passive and active vibration control.

With the rapid development of intelligent mate?rials，piezoelectric materials have been widely used in vibration control fields［1-3］because of their unique piezoelectric effects. In the past decades，research?ers have widely investigated the vibration of beams，plates and shells by using piezoelectric materials as sensors and actuators［4-7］. However，the fragile char?acteristics of piezoelectric patches limit its applica?tion in engineering. In order to overcome this weak?ness，the macro-fiber composite（MFC）was devel?oped by NASA Langley Research Center in 1999.Because of the excellent characteristics of flexibility and actuating bending moment，MFC has been widely used in the field of vibration control recent?ly［8-10］. Williams et al.［11-13］carried out a detailed de?scription of the MFC manufacturing process and in?vestigated the nonlinear behavior of tensile，shear and temperature effects of MFC. Zhang et al.［14］studied the structural deformation of isotropic and cross-ply composite laminated thin-walled smart structures bonded with orthotropic MFCs. MFC has an outstanding performance in the field of active con?trol. Lucyna et al.［15］designed a closed-loop system with MFC sensor and actuators and applied to sup?press circular plate vibrations. Gao et al.［16］utilized MFC to investigate a new multiple model switching adaptive control algorithm to implement the real time active vibration suppression tests with a new multiple switching strategy. Chai et al.［17］studied the aerothermoelastic characteristics of composite laminated panels and carried out active flutter con?trol of composite laminated panels with time-depen?dent boundary conditions in supersonic airflow by using MFC materials. Jia et al.［18］measured vibra?tion data from various sources，including aero?space，automotive，engine，bridge and rail applica?tions，confirmed a finite element model and the FE software to simulate the multiphysical process of piezoelectric vibration energy harvesting，and car?ried out the dynamic mechanical and electrical be?haviors of MFC on carbon fiber composite struc?tures.

In general，the active vibration control mainly uses the control method for the structural vibration.Wu et al.［19］utilized PD and fuzzy control algorithms to build the closed-loop feedback control system.Sohn et al.［20］used MFC actuators to suppress the vibration response of the smart hull structure with linear quadratic Gaussian（LQG）control method.Song et al.［21］investigated the optimal active flutter control of supersonic composite laminated panels with distributed piezoelectric actuators/sensors pairs. Comparing two different controllers，numeri?cal simulations show that the optimal locations ob?tained by the genetic algorithm（GA）can increase the critical flutter aerodynamic pressure significant?ly，and the LQG algorithm is more suitable in flut?ter suppression for supersonic structures than the proportional feedback controller. Through the col?lection and arrangement of the literature，the gener?al active control method cannot change the dynamic characteristics of the structure itself. The pole place?ment method possesses such characteristics. It can arbitrarily assign any natural frequency of the struc?ture，and has been widely used in the structural vi?bration control［22-25］. Tehrani et al.［26］carried out the theory and practical application of the receptance method for vibration suppression in structures by multi-input partial pole placement.

However，most researchers commonly consid?ered vibration control in time domain by using piezo?electric ceramics，but very limited literatures have been found to investigate the active control in fre?quency domain. This paper creatively investigates a vibration control method in frequency domain which is the pole placement method combined with piezo?electric effect. MFC patches are bonded on the top and bottom surfaces of the beam structure to act as the actuators and sensors. By using the MFC patch?es and designing the control process through the pole placement method，the natural frequency of the beam can be artificially designed. The effects of ac?tive vibration control for beam using pole placement method are carried out through the theoretical，nu?merical and experimental verifications.

1 Theoretical Formulations

1.1 Structural modeling

MFCs are mainly composed of piezoceramic epoxy matrix and electrodes. Although each of these materials is considered to be isotropic by itself（or transversely isotropic in the case of the poled PZT），the resulting composite behaves in an ortho?tropic manner when manufactured into a fiber-rein?forced lamina［27-28］.

There are mainly two types of MFC，i. e.，MFC-d31and MFC-d33. MFC-d31is polarized along the thickness direction and perpendicular to the fiber direction. It is usually used as sensors. On the con?trary，MFC-d33is polarized along the fiber direc?tion. It may have greater driving force and is usually used as actuators.

A cantilevered beam with MFC patches is shown in Fig.1，where the MFC-d31and MFC-d33patches are bonded on the top and bottom surfaces to act as sensor and actuator，respectively.

To formulate the equation of motion of the structural system，the Hamilton’s principle is ap?plied，shown as

wheret1andt2are the integral limits，andT，UandδWthe kinetic energy，strain energy and virtual work done by the external forces，calculated by

where dot denotes the derivative with respect to time；ρbis the density of base beam；ρPthe density of MFC，in which sensor and actuator have the same mass density；F0the external load；xFthe co?ordinates ofF0；εxthe strain of beam and MFC（as?suming that the MFC is fully bonded to the beam，and they can be considered to have the same strain）；Ethe electric field；Dathe electric displace?ments of MFC actuator；σbandσPare the stress of beam and MFC；andVb，VaandVsthe volume of the beam，actuators and sensors，respectively. The in-plane and transverse displacements in the above equations are given out according to the Euler-Ber?noulli beam theory，shown as

wherewindicates the transverse displacement along thexdirection anduthe in-plane displacement.

For the beam structure，it can be regarded as a one-dimensional model. While for MFC-d33，the electric field is only applied to the polarization direc?tion.The constitutive equation can be shown as

whereεx=?z?2w/?x2；Ex=V（t）/hais the electric field in thexdirection，hereV（t）the external ap?plied voltage；ε33the permittivity coefficient of the MFC；Eb，EsandEaare the Young’s modulus of the base beam，MFC-d31and MFC-d33；andd31andd33the piezoelectric coefficients.

According to the FEM，the displacement field can be expressed in terms of interpolation functions and nodal coordinates，shown as

whereN（x）andqe（t）are the column vectors of in?terpolation functions and nodal coordinates. For the thin beam structure，a two-node Euler beam ele?ment is used. The interpolation function and nodal coordinate vector are

whereθi=?wi/?x；leis the length of element；ζ=(x?xl)/lethe normalized coordinate；xlthe coordi?nates of the node on the left end of the element.

Thus，based on the Hamilton principle，the equation of motion can be obtained as

whereMeandKeare the element structural mass and stiffness matrices；Kaeis the electromechanical vector；Vi（t）the input voltage generated by theith MFC-d31actuator；andFethe element external force vector.They are calculated by

whereAb，AsandAaare the cross-section area of base beam，MFC sensor and actuator，respective?ly；δsandδacan be equal to 1 or 0 depending on whether the structure contains the sensor or actuator layers，and

For MFC sensors，according to the Gauss law，the closed-circuit charge of theith MFC sen?sor measured through the electrodes is

wherebsis the width of the MFC sensor. And the sensing voltage is expressed as

Based on the above mass and stiffness matrices and considering the boundary condition，after as?sembling，the global equation of motion can be ob?tained as

whereMandKare the global mass and stiffness ma?trices，andU（t）=［V1（t）V2（t） …Vp（t）］T.

1.2 Pole placement design

To conduct the pole placement control，the equation of motion in modal space should be trans?formed to state space as

whereA∈R2n×2n，B∈R2n×p，C∈Rp×2n.

Based on the output feedback system，the con?trol input voltage is expressed as

whereG0∈Rp×pis the matrix of feedback control gain.Therefore，the closed-loop system becomes

The eigenvalue equation of the closed-loop sys?tem can be placed in following equation

whereλcis the eigenvalue of the closed-loop system.According to the linear transformation，the eigenval?ue equation can be expressed by

where

Sinceλcis the eigenvalue of the closed-loop sys?tem，the following equation can be obtained as

Therefore，the matrixG0should satisfy the above equation，and it can be obtained by

where

whereφmp(λcp) represents thempth column of the matrixφ(λcp)，and correspondingly，the matrix is composed of the corresponding column of the unit matrixIas

1.3 Decoupling method

For uncoupled state space equation， when poles are placed to the desired poles，the rest poles will also move. As a result，the previous stable sys?tem may lose stability. In order to solve this prob?lem，the decoupling of the equation of motion is con?ducted.The following transformation is introduced

where

whereaj+ibj（j=1，2，…，m）arempairs of conju?gate complex eigenvectors of the state matrixA.This state transformation matrix can lead to the fol?lowing relationship

where

whereγandωare the real and imaginary parts of the eigenvalue ofA.

Substituting Eq.（32）into Eq.（21），the decou?pled state equation can be obtained as

where=ψ?1BU.

Based on the decoupling state space equation of motion，and the pole placement method given in Eq.（29），the matrix of feedback control gain can be obtained as

2 Numerical Results and Discussion

In the numerical simulations，the cantilever beam is made of steel 304 whose geometrical sizes arel= 0.25 m，b= 0.02 m，hb= 0.001 m，Eb=210 GPa，ρb= 7 850 kg/m3，andνb= 0.3. M2814-P1 is adopted as actuator and M2814-P2 is adopted as sensor. Their parameters are 38 mm×20 mm×0.3 mm，and the actuating size is 28 mm×14 mm×0.3 mm. MFC patches arranged in the up-and-down surfaces are pasted with epoxy resin glue at a dis?tance of 0.09 m away from the fixed end. The rele?vant material properties are given in Table 1.

Table 1 Relevant material property specifications

In the experiment，the mass of the acceleration sensor cannot be ignored. Therefore，the accelera?tion sensor is replaced by an additional mass about 7.85 g. Before the active control，the natural fre?quencies of the structure are verified. The compari?son results are shown in Table 2. It can be seen that the three results agree well with each other，which verifies the correctness of the formulations and ex?periment.

Table 2 Comparison of natural frequencies obtained by different methods Hz

2.1 Closed?loop control

Based on the above verification，the active con?trol is carried out. The schematic diagram is dis?played in Fig.2. It can be seen that the MFC patch?es are bonded on the upper and lower surfaces with the epoxy resin glue. By this way，the structure can effectively realize the mutual conversion between voltage and displacement.

Fig.2 Positions of external excitation, MFC patches and acceleration sensor

Under the closed-loop control，the sensor is re?sponsible for converting the displacement into the in?duced voltage and acting on the actuator with a cer?tain gain. Considering the piezoelectric effect of piezoelectric materials，the stiffness of the beam can be changed，and as a result，the natural frequencies can be changed.

Experimental devices shown in Fig.3 can be di?vided into four parts，i.e.，the control system，data acquisition system，excitation system and boundary condition device. The control system is composed of dSpace MicroLabBox， voltage amplifier and Computer 1. Its main function is to collect the in?duced voltage generated by the MFC sensor and output voltage to MFC actuator. The data acquisi?tion system consists of Computer 2 and data acquisi?tion equipment. The data acquisition equipment is used to collect the displacement signal generated by the acceleration sensor and perform Fourier trans?form on the signal. External excitation includes a vi?bration exciter and a power amplifier. The boundary condition device is displayed in Fig.4.

Fig.3 Experiment apparatuses

Fig.4 Boundary condition

Firstly，the influences of the closed-loop con?trol on the natural frequency are investigated. Based on output feedback，different values of control gains are adopted in the experiment. In this paper，the control gain in the experiment is verified by the theo?retical calculation. The excitation system is set to be sweep frequency in the range of 0—200 Hz.

The results of Figs.5，6 show that the theoreti?cal results agree well with the experimental results，and the natural frequency increases with the increase of the control gain. Contrary to Figs.5，6，F(xiàn)igs.7，8 show a decreased trend of natural frequency of the beam under negative control gain. The theoretical results are also coincident with the experimental re?sults.

Fig.5 Comparisons of the controlled natural frequencies when G=600 and without control

Fig.6 Comparisons of the controlled natural frequencies when G=800 and without control

With the increase of the control gain，the con?trol effects become more obvious. As shown in Figs.9—11，the experimental results still agree well with the theoretical calculations with the increase of the control gain. The results show that the closedloop experiment can realize the change of low-order natural frequency up to 20 Hz. In other words，the vibration characteristics of the structure can be im?proved by closed-loop control method in practical application.

Fig.7 Comparisons of the controlled natural frequencies when G=?600 and without control

Fig.8 Comparisons of the controlled natural frequencies when G=?800 and without control

Fig.9 Comparisons of the controlled natural frequencies when G=?1 000 and without control

Fig.10 Comparisons of the controlled natural frequencies when G=?1 200 and without control

Fig.11 Comparisons of the controlled natural frequencies when G=?1 400 and without control

2.2 Pole placement control

Poles represent the eigenvalues of structure sys?tem. They are always composed of real and imagi?nary components，and can be expressed byλ1，2=?ζω0±ω0（ζ2?1）1/2. The real part reflects the damping characteristics of the structure，and the imaginary part is related to the vibration frequency of the structure. By changing the value of system poles，the effect of changing a certain order of natu?ral frequency can be realized.

Since only one pair of MFC actuator and sen?sor is used in the theoretical and experimental stud?ies，only one pair of poles can be controlled.

For the structural system studied in this paper，the open-loop poles areλ1，2= ±2π×15.94i. Limit?ed by experimental and equipment conditions，the pole is going to be controlled toλ1，2= ±2π×15.50i in the experiment. Based on the pole placement con?trol method，the feedback control gain isG=1.314e?3. The theoretical and experimental results are shown in Fig.12. It can be seen that the two re?sults agree well with each other，which indicates the correctness of the theoretical analysis.

Fig.12 Theoretical and experimental results controlled by the pole placement method

Based on the above verification，the pole place?ment method is used to investigates the active con?trol effect on the first natural frequency，and the re?sults are shown in Figs.13—16.

Fig.13 Natural frequencies controlled to 15 Hz with G=?1 572 and without control

Fig.14 Natural frequencies controlled to 10 Hz with G=?2 319 and without control

Fig.15 Natural frequencies controlled to 20 Hz with G=?2 721 and without control

Fig.16 Natural frequencies controlled to 25 Hz with G=?2 565 and without control

2.3 Decoupled pole placement control

When conducting the pole placement control，although the certain natural frequency can be con?trolled，other poles may also be changed at the same time，which can be observed from Table 3.When the first order of natural frequency is con?trolled to 10 Hz，the second and third orders of natu?ral frequencies are also changed. To solve this prob?lem，the control equation is decoupled. It can be seen from Table 3 that after decoupling，only the first natural frequency is changed and the other two frequencies remain unchanged. The other four de?sired natural frequencies are also considered，and the results are shown in Tables 4—6.

Table 3 Comparisons of the natural frequencies con?trolled to 10 Hz Hz

Table 4 Comparisons of the natural frequencies con?trolled to 15 Hz Hz

Table 5 Comparisons of the natural frequencies con?trolled to 25 Hz Hz

Table 6 Comparisons of the natural frequencies con?trolled to 50 Hz Hz

It can be seen from Tables 5 and 6 that serious problems occur in the situation of without decoupled system. Table 5 shows the structural instability problem. And Table 6 shows that although the tar?get natural frequency is obtained，the first natural frequency is replaced by other values. The above ex?amples prove the decoupling method is very effec?tive in the pole placement for the active vibration control.

3 Conclusions

The piezoelectric material is used to conduct the active vibration control. Theoretically，the natu?ral frequency can be changed by adjusting the feed?back control gains. Numerical calculations and ex?periments verified this conclusion. The equation of motion is formulated by the Hamilton principle and the finite element method. In the experiment，the correctness of the closed-loop control is verified in the frequency domain. Moreover，the control effect of the pole placement method is investigated through the theoretical and experimental analyses.Based on the results，the following conclusion can be drawn.

（1）The closed-loop control and pole place?ment method based on the output feedback is effec?tive in practical application.

（2）The core of this active vibration control method is to change the natural frequencies of struc?ture arbitrarily.

（3）The decoupled state equation can achieve the independence of poles from unplaced and placed，which can effectively avoid the instability of system.

AcknowledgementsThe work was supported by the Na?tional Natural Science Foundation of China（Nos.11802069 and 11761131006），the China Postdoctoral Science Founda?tion（No.3236310534），the Heilongjiang Provincial Postdoc?toral Science Foundation（Nos. 002020830603 and LBHTZ2008），and the China Fundamental Research Funds for the Central Universities（No.GK2020260225）.

AuthorsMs. YANG Shaoxuan is currently studying for a master’s degree at Harbin Engineering University. Her re?search interests include piezoelectric materials, metamaterial structures and active vibration control.

Prof. SONG Zhiguang received the Ph.D. degree in General Mechanics and Mechanics Foundational from Harbin Insti?tute of Technology，Harbin，China，in 2014. He is the win?ner of the“Youth Project of Overseas High-Level Talent In?troduction Program”and a Humboldt scholar in Germany.Now he is the deputy director of the Department of Engineer?ing Mechanics of Harbin Engineering University. His main research interests are aerothermoelasticity，structural vibra?tion control，nonlinear dynamics and computational mechan?ics.

Author contributionsProf. SONG Zhiguang designed and discussed the study. Ms. YANG Shaoxuan complied the models，conducted the analysis，interpreted the results and wrote the manuscript. Mr. HU Yu contributed to the experiment analysis and revision of the study. All authors commented on the manuscript draft and approved the submission.

Competing interestsThe authors declare no competing interests.