Ehsan Omidi·S.Nima Mahmoodi

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RESEARCH PAPER

Nonlinear integral resonant controller for vibration reduction in nonlinear systems

Ehsan Omidi1·S.Nima Mahmoodi1

A new nonlinear integral resonant controller（NIRC）is introduced in this paper to suppress vibration in nonlinear oscillatory smart structures.The NIRC consists of a first-order resonant integrator that provides additional damping in a closed-loop system response to reduce highamplitude nonlinear vibration around the fundamental resonance frequency.The method of multiple scales is used to obtain an approximate solution for the closed-loop system. Then closed-loop system stability is investigated using the resulting modulation equation.Finally，the effectsofdifferent controlsystemparameters are illustrated and an approximate solution response is verified via numericalsimulation results. The advantages and disadvantages ofthe proposed controller are presented and extensively discussed in the results.The controlled system via the NIRC shows no high-amplitude peaks in the neighboring frequencies of the resonant mode，unlike conventional second-order compensation methods. This makes the NIRC controlled system robust to excitation frequency variations.

Active vibration control·Cantilever·High

amplitude oscillation·Method ofmultiple scales·Nonlinear vibration·Piezoelectric actuator·Smart structure

? S.Nima Mahmoodi nmahmoodi@eng.ua.edu

1Nonlinear Intelligent Structures Laboratory，Department of Mechanical Engineering，The University of Alabama，Tuscaloosa，AL 35487-0276，USA

1　Introduction

Flexible structures are susceptible to linear and nonlinear vibrations，which are undesirable in most cases，and their suppression is of the utmost importance.A system under vibration can be enhanced by actuation/sensing elements and a controlunit，so thatithas a smartstructure thatcan actively rejectvibrations.The performance ofthe resulting smartflexible structure is highly affected by the controller designed for the system，which works as a bridge between sensors and actuators.Essentially，the designed controller must consider the vibrational behavior of the system，the available actuators/sensors，and possible types of excitation disturbance. Piezoelectric actuators and sensors have been investigated in various studies and successfully applied to various systems［1，2］.These elements are the most frequently used sources forapplying actuation powerto flexible systemsorproviding vibration position feedback.

Various controllers have been designed for the purpose of vibration control in smart structures.Linear controllers can be used suppress low-amplitude linear vibrations；some approaches are described and investigated in Refs.［3-7］. However，proportionate controllersmustbe designed fornonlinear oscillatory smart structures that are compatible with the nonlinear frequency responses of these systems.Additionally，controlled system response and behavior cannot be obtained using typical linear methods for nonlinear systems. In many cases where exact solutions are not readily available，approximate methods are used to provide a solution for these systems，such as the method of multiple scales［8，9］.The problem of nonlinear vibration and approximate solutions for these systems have been addressed in various studies［10-12］.Different linear and nonlinear controllers have also been designed for such systems:cubic velocityfeedback［13］，fuzzy sliding mode control［14］，nonlinear positive position feedback（PPF）-based methods［15-17］，a nonlinear energy sink approach［18］，and delayed feedback control［19］.The nonlinear dynamic response and control of fiber-metal laminated plates were studied in Ref.［20］，and the active vibration isolation problem fora microelectromechanicalsystem（MEMS）device is addressed in Ref.［21］.

Many ofthe previously proposed controllers fornonlinear vibration suppression of the primary resonant frequency are second-order compensators［15-17］.Although these compensators can effectively suppress the resonant frequency at its exact value，vibration amplitude in the neighboring frequency regions may even exceed the initial resonant amplitude.Because the excitation disturbance may not necessarily remain at the resonant frequency，vibrations in neighboring frequencies become a challenge，too.Additionally，these controllers have relatively complicated structures，which makes the solution preparation and controller implementation difficult.

To avoid the complexity of second-order compensators and their high-amplitude peaks in the neighboring frequencies of resonance，a new nonlinear integral resonant controller（NIRC）is introduced in this paper.In the NIRC，a first-order nonlinear vibration compensator fed by a positive linear and a negative nonlinear position term is implemented to target the primary resonant frequency.A solution of the closed-loop system is obtained using the method of multiple scales in a multilayer fashion.A modulation equation is obtained and a stability analysis performed.The solution of the approximate method is then verified using numerical simulation results，and the influences of different control parameters on closed-loop system response are extensively investigated.The structure ofthiscontrollermakes itsuitable for nonlinear vibratory structures，such as beams and plates，in addition to MEMS systems and devices.

2　Dynamics of the nonlinear system and NIRC

High-amplitude oscillations are one of the primary causes of nonlinearity in responses of flexible structures.Cantilevers are prone to these vibrations more than structures with other boundary conditions because of their free end.Of the differentapproaches to applying the controlmomenton a structure to suppress these vibrations，one should use a collocated patch of piezoelectric actuators and sensors.The primary resonant mode for the model of a nonlinear vibrating system can be obtained using the geometrical deflections of the structure.This has been fully addressed in Ref.［15］，and the governing equation is obtained as

Fig.1 Time-domain block diagram representation of closed-loop system

where u（t）is the time-dependent variable of the main system；overdot denotes differentiation with respect to t；Fc（t）is the control input；Fd=f cos（Ωt），where f is the amplitude and Ω the frequency of the external excitation；μ and ωmare the damping ratio and resonant frequency of the main system，respectively；α and β are curvature and inertia nonlinearity coefficients，respectively.Note that positive and negative values can be assigned to β based on the system characteristics.

The NIRC consists of a first-order resonant integrator，with a combination of a positive linear and a negative nonlinear vibration displacement amplitude input.The NIRC is expressed as

where v（t）is the variable of NIRC，ωNis the first-orderintegrator’s frequency，andλandδ are controllerinputgains.The control law is defined as Fc（t）=τv（t）for τ＞0.Figure 1 shows a time-domain block diagram of a closed-loop system consisting of a main system and the NIRC；the figure also shows how different elements are connected to one another. The overalldamping ofthe systemincreaseswhen the control loop of the system is closed in accordance with the defined law.As a result，we would expect to see a significant reduction in the vibration amplitude in the frequency domain.To verify the expectations and provide a more detailed analysis of the system response，an approximate frequency domain solution is obtained in the following section.

3　Controlled system response using method of multiple scales

The method of multiple scales is applied to the closedloop dynamics of a system to provide a uniform nonlinear approximate solution near the fundamental resonant mode［9］.Initially，two time scales are considered，T0=t and T1=εt，and the corresponding time derivatives are

where Dn=?/?Tn，andεisa bookkeeping parameter.Equation（3）is substituted into the equations of the main system and controller，which yields

The variables of Eqs.（4）and（5）are expanded using

The order of Eq.（7）is chosen to be one order higher than the main systemto keep the first-orderdynamicsofthe controller at the same pace with the second-order nonlinear system model and to have all the necessary variables appear in the correct equations.The main system and controller parameters need to be scaled as follows:Separation in orders of ε yields three layers that are required for the analysis

The solution of the homogeneous ordinary differential equation（ODE）of Eq.（8）is assumed to be in the form

where A（T1）is a complex-valued function and cc is the complex conjugate.The solution considered for Eq.（12）is substituted into Eq.（10），and the resulting ODE is solved. The solution is then obtained as

where the overbar denotes the complex conjugate function，and the variable C（T1）will be determined in subsequent stages of the solution.Equations（12）and（13）are substituted into Eq.（9），which yields

Equation（14）can be solved by neglecting the secular terms that will subsequently be set equal to zero.The result is expressed as

The solutions expressed by Eqs.（13）and（15）are used to obtain the solution of Eq.（11）.The substitution yields Eq.（11）

The solution of Eq.（16）is expressed in the form

where the coefficients V1to V14are presented in the appendix.The secular terms in Eq.（16）should be set equal to zero，which also facilitates calculation of the term C（T1）. The obtained equation is

The solution for the ODE of Eq.（18）is obtained as

where cvisa constant.The nextobjective isto obtain the modulation equation.To this end，the secular terms in Eq.（14）are set equal to zero，

where σfis a small detuning parameter，defined using the detuning equation Ω=ωm+εσf.It is necessary to have this variable defined owing to the fact that the excitation frequency is close to the primary frequency of the main system. The solution of Eq.（20）is expressed in polar form using

Equation（21）is substituted into Eq.（20）for real and imaginary parts to be separated.This yields

A variable transformation is applied to Eqs.（22）and（23）by considering ?（t）=σft-θ（t），and the scaled parameters are restored to their original form.This yields

4　Steady-state response and stability analysis

To obtain the amplitude-frequency steady-state response of the controller system via the NIRC，steady-state conditions are considered as being˙a=˙?=0，which yields

Equations（26）and（27）are squared and the two sides are summed together，which yields the final modulation equation:

Next，a stability analysis is performed.Linearization around the equilibrium point is considered using Eqs.（24） and（25）and considering the variable vectorThe Jacobian matrix is then expressed by

The Routh-Hurwitz stability criterion is used here to examine the stability of the closed-loop system controlled by the NIRC.The characteristic equation of the matrix is obtained first using

for γ as an eigenvalue variable of the Jacobian matrix.Equation（30）is expressed in the form

where the aiare the coefficients of the characteristic equation obtained from Eqs.（29）and（30）.In orderforthe system to be stable，the real parts of all eigenvalues must be negative，which is satisfied by having ai＞0 for i=0，1，2.To constructa stable closed-loop controlsystem，the considered gainsmustbe verified according to the obtained stability condition.Controller gain variables complying with the stability condition guarantee a stable system.

5　NIRC controlled system results and discussions

Finally，in this section，the performance of the proposed NIRC method is illustrated and discussed.The obtained amplitude-frequency equation is used to graphically show the effects ofdifferentcontrolsystemvariables，and the solution is subsequently verified using the numerical simulation result，followed by additional numerical results of system performance.

In this section，the main variables of the closed-loop system will be as follows:ωm=12，μ=0.005，α=20，β=-10，ωN=12，and Ω=12.Gain values are noted for each graph separately.To choose stable gains，the equations presented in Sect.4 are used.It should be noted that stability is dependent on the control gains，amplitude，and frequency shift.Therefore，for every case based on the bandwidth of the amplitude and phase，the gain conditions are calculated. For all the cases presented in this section，the bandwidths of λ and τ gains are very large since the amplitude and phase are reasonably small.Thus the system is stable for almost all values ofgains.Forthe firstresult，the responses ofthe openand closed-loop systems are illustrated.Figure 2 shows the steady-state vibration amplitude versus changes in excitation frequency and amplitude.Two separate surfaces are depicted in Fig.2，showing the uncontrolled and NIRCcontrolled system response for controller gain values of λ=τ=15.A jump phenomenon is observed when the vibration amplitude exceeds f=0.22 in the uncontrolled system，and，owing to the selected numerical values for the nonlinear system，the graph bends toward the negative side of the frequency axis.According to the results，the NIRC is able to suppress the vibration amplitude in the neighborhood of the primary resonant frequency.Unlike conventional methods，such as positive position feedback（PPF）and nonlinear modified positive position feedback（NMPPF），discussed in Ref.［22］，the NIRC controlled system shows no high-amplitude peak on eitherside ofthe fundamentalmode in the frequency domain.

Fig.2 NIRC controlled and uncontrolled system responses

To better analyze the suppression performance of the NIRC，F(xiàn)ig.3 isextracted from the depicted surfaces in Fig.2. Figure 3 shows the vibration amplitude and frequency of peak values for variations in the excitation amplitude，f. According to the obtained results，higher suppression levels are achieved for lower excitation amplitudes，as the suppression level reaches as high as 87.6%for f=0.5 and 62.7% for f=4.Deviation from the resonant frequency，σf，tends to grow much slower in the closed-loop NIRC system；σfincreases by 5.3 times from f=0.5 to f=4 in the uncontrolled mode，whereas it increases in the NIRC controlled system by just 2.2 times.

Next，the effects of two control variables on system response are examined.Figure 4 shows the vibration amplitude for changes in controller gain λ（for τ=3 and f=2）and integrator frequency ωN（for λ=τ=14 and f=2）versus changes in the excitation frequency.According to Fig.4a，vibration suppression improves exponentially as this gain value moves away from zero in both the positive and negative directions.Note that the result obtained for positive gain values is slightly betterthan when using negative values. Figure 4b shows that a higher suppression level is obtained for ωN=0.This graph also shows that the peak in the suppressed graph ison the negative side ofthe frequency axis for positive values of integrator frequency，and vice versa.This is because the sign of the integrator frequency，in addition to its amplitude，changes the magnitude of the second term onthe left-hand side of Eq.（28）.However，the inclination of the graph is to the negative direction in both cases.

Fig.3 Vibration amplitudes.a Deviations from resonant frequency of peak values.b Changes in excitation amplitude in uncontrolled and NIRC controlled systems

Fig.4 NIRC controlled system response for changes in excitation frequency（σf）versus.a Controller gain λ.b Integrator frequency ωN

Figure 5 illustrates the effects of changes in the excitation amplitude and variations in λ（for τ=22 and σf=0.5）and ωN（for λ=τ=14 and σf=0.5）on the system response. According to Fig.5a，positive values of λ better reduce the vibration amplitude.The results presented in Fig.5b confirm the conclusion drawn from Fig.4b that the highest suppression level is achieved for ωN=0，and positive values for ωNare more effective.

According to Eq.（28），two gain values of λ and τ are multiplied by each other where present.To investigate the effects ofchanges in both these variables on closed-loop system response，F(xiàn)ig.6 is used（for f=0.2 and σf=0.1）. Based on the surface obtained，both gain values have similar effects，as expected.However，better suppression is achieved when both gains have the same signs.

Next，the effect of the controller input term δ on system response is studied.As shown by Eq.（28），this term do not presentin the obtained approximate solution.This is because this term appears in a much higher-order layer of ε in perturbation expansion.In addition，even if the perturbation order is increased，the term will not appear in the secular terms of Eq.（14）since its frequency would be different from iωm. Hence，the effect of this controller variable is numerically investigated.MATLAB software is implemented to solve the coupled Eqs.（1）and（2）under the positive control law for different values of δ and for other control variables of λ=τ=7，f=0.1.According to the result illustrated in Fig.7，an increase in the selected value of δ provides a higher level of suppression.When the vibration amplitude is reduced to a lower level by increasing δ，the peak of the suppressed curve moves toward the negative σfaxis.Notethatany arbitrary value can be assigned to thiscontrollergain as long as the closed-loop system remains stable.

Fig.5 NIRC controlled system response for changes in excitation amplitude（f）versus.a Controller gain λ.b Integrator frequency ωN

Next，the obtained perturbation solution is verified using the results of numerical simulation.Figure 8 shows the controlled system response for gain values of λ= τ=1，f=0.02，for both the perturbation solution and a numerically simulated system.As shown，both graphs are in close agreement with one another，which confirms that the perturbation solution is in close agreement with a real system response.

Finally，a numerical simulation is employed to obtain phase portraits ofthe systemfordifferentvalues ofexcitation frequency.In all illustrated results of Fig.9，the controller is switched on 45 s after the start of the process，when the system has reached steady-state.The control variables are set at λ=τ=7，δ=15，f=0.8，and nonzero initial conditions are selected.For σf=-1.5，the system is not excited at itsresonance，and the vibration amplitude is significantly lower than the maximum values ofothercases.When the controller is switched on，no significant change is observed.For other excitation frequencies，inner limit cycles are the steady-state controlled values.

Fig.6 Closed-loop system response to changes in λ and τ

Fig.7 Effectofcontrollerinputgain，δ，on suppressed vibration amplitude

Fig.8 Perturbation solution verification using numerical simulation results

Fig.9 Phase portraits of suppression process for different values of excitation frequency

6　Conclusion

In this paper，an NIRC was proposed for nonlinear vibration suppression in flexible structures.The NIRC，with its first-order design，provides additional damping for a closedloop systemin the neighborhood ofthe resonantfrequency.Alinear positive and a negative quadratic term of the vibration position are used asthe inputs ofthe controller，which provide more flexibility in the controller design process.A nonlinear model of a high-amplitude vibrating cantilever beam was considered，and an approximate steady-state solution was obtained via the method of multiple scales.Following an examination of the closed-loop system’s stability，results were illustrated and extensively discussed.Numerical simulation results were used to verify the analytical solution.In addition，the effects ofdifferentcontrolparameters on system response were analyzed.The NIRC has a relatively simple structure and provides a smooth closed-loop response in the neighborhood of the excitation frequency.The NIRC can be considered an effective and applicable candidate for nonlinear vibration controllers for flexible structures.

Appendix

The coefficients of the time-domain response of Eq.（17）are as follows

1.Seigler，T.，Ghasemi，A.H.，Salehian，A.:Distributed actuation requirements of piezoelectric structures under servoconstraints.J. Intell.Mater.Syst.Struct.22，1227-1238（2011）.doi:10.1177/ 1045389X11411222

2.Seigler，T.，Ghasemi，A.:Specified motion ofpiezoelectrically actuated structures.J.Vib.Acoust.134，021002（2012）

3.Daraji，A.H.，Hale，J.M.:Active vibration reduction by optimally placed sensors and actuators with application to stiffened plates by beams.Smart Mater.Struct.23，115018（2014）

4.Li，S.，Li，J.，Mo，Y.etal.:Composite multi-modalvibration control for a stiffened plate using non-collocated acceleration sensor and piezoelectric actuator.Smart Mater.Struct.23，015006（2014）

5.Omidi，E.，Mahmoodi，S.N.:Multiple mode spatialvibration reduction in flexible beamsusing H2-and H∞-modified positive position feedback.J.Vib.Acoust.137，011004（2015）

6.Omidi，E.，Mahmoodi，S.N.:Hybrid positive feedback control for active vibration attenuation of flexible structures.IEEE/ASME Trans.Mechatron.（2014）.doi:10.1109/TMECH.2014.2354599

7.Omidi，E.，Mahmoodi，S.N.:Consensuspositive position feedback control for vibration attenuation of smart structures.Smart Mater. Struct.24，045016（2015）

8.Marinca，V.，Herisanu，N.:Nonlinear Dynamical Systems in Engineering:Some Approximate Approaches.Springer，New York（2012）

9.Nayfeh，A.H.:Problems in Perturbation.Wiley，New York（1985）

10.Hosseini，S.M.，Shooshtari，A.，Kalhori，H.:Nonlinear-forced vibrations of piezoelectrically actuated viscoelastic cantilevers. Nonlinear Dyn.78，571-583（2014）

11.Lazarus，A.，Thomas，O.，Deü，J.:Finite element reduced order modelsfornonlinearvibrationsofpiezoelectric layered beamswith applications to NEMS.Finite Elem.Anal.Des.49，35-51（2012）

12.Mahmoodi，S.N.，Jalili，N.，Khadem，S.E.:An experimental investigation of nonlinear vibration and frequency response analysis of cantilever viscoelastic beams.J.Sound and Vib.311，1409-1419（2008）

13.Oueini，S.S.，Nayfeh，A.H.:Single-mode control of a cantilever beam under principal parametric excitation.J.Sound Vib.224，33-47（1999）

14.Dai，L.，Sun，L.:Vibration control of a translating beam with an active control strategy on the basis of the fuzzy sliding mode control.In:Proceedings of ASME International Mechanical Engineering Congressand Exposition（IMECE），Anonymous4B（2013）

15.Omidi，E.，Mahmoodi，S.N.:Nonlinear vibration suppression of flexible structures using nonlinear modified positive position feedback approach.Nonlinear Dyn.79，835-849（2015）

16.El-Ganaini，W.，Saeed，N.，Eissa，M.:Positive position feedback（PPF）controller for suppression of nonlinear system vibration. Nonlinear Dyn.72，517-537（2013）

17.Omidi，E.，Mahmoodi，S.N.:Sensitivity analysis of the nonlinear integralpositive position feedback and integralresonantcontrollers on vibration suppression of nonlinear oscillatory systems.Commun.Nonlinear Sci.Numer.Simul.22，149-166（2015）

18.Ahmadabadi，Z.N.，Khadem，S.E.:Nonlinearvibration controlofa cantilever beam by a nonlinear energy sink.Mech.Mach.Theory. 50，134-149（2012）

19.Zhao，Y.，Xu，J.:Effects of delayed feedback control on nonlinear vibration absorber system.J.Sound Vib.308，212-230（2007）

20.Shao，X.，F(xiàn)u，Y.，Chen，Y.:Nonlinear dynamic response and active control of fiber metal laminated plates with piezoelectric actuators and sensors in unsteady temperature field.Smart Mater.Struct.24，055023（2015）

21.Meyer，Y.，Cumunel，G.:Active vibration Isolation with a MEMS device.Effectsofnonlinearitieson controlefficiency.SmartMater. Struct.24，085004（2015）

22.Omidi，E.，Mahmoodi，S.N.:Nonlinear vibration control of flexible structures using nonlinear modified positive position feedback approach.In:Proceedings of ASME Dynamic Systems and Control Conference，San Antonio，TX，USA，Anonymous pp. V003T52A002（2014）

12 December 2015/Revised:31 March 2016/Accepted:21 April 2016/Published online:25 June 2016

?The Chinese Society of Theoretical and Applied Mechanics；Institute of Mechanics，Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2016