# Analysis of Vibration Control of Nonlinear Beam Using a Time-Delayed PPF Controller

Authors: Yueli Chen; Juhong Ge
Journal: Shock and Vibration
 (2020)

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## Abstract

This paper presents a study on the performance of a positive position feedback (PPF) controller to suppress the vibration of a horizontal beam under vertical excitation. Time delays in the control loop are taken into consideration to study their effects on the controller performance and the stable region. The integral iterative method is conducted to obtain a second-order approximate solution and the corresponding amplitude equations for the considered system. The stability of the steady-state solutions is ascertained using a combination of Floquet theory and Hill’s determinant. The maximum limits of time delays at which the system remains stable have been determined for different values of control parameters. And the effects of various control parameters on the existence of multiple-solution region are investigated. The analysis illustrates that the appearance of time delay and the elimination of controller damping coefficient are the two main factors to enhance the nonlinear characteristics of the controlled system. The points at which the steady-state amplitude of the main system reaches its minimum are studied analytically. The analyses show that the analytical results are in excellent agreement with the numerical simulations.


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## Body

## 1. Introduction

Active vibration control has been used to suppress the undesired vibrations in different systems for many years. Various types of controllers are designed to channel the excess energy from the excited systems to the slave ones. Applications of the positive position feedback controller to reduce a system’s vibrations have been intensively developed in recent years. Shan et al. [1] employed a positive position feedback controller to suppress multi-mode vibrations while slewing the single-link flexible manipulator. Experimental results showed that PPF controller is more effective than the L-type velocity feedback controller while the slewing process was realized. Jun [2] employed an active linear absorber based on PPF control to reduce the high-amplitude vibration of the single-mode of the flexible beam when subjected to primary resonance excitation. It showed that the control scheme possessed a wide suppression bandwidth if the absorber’s frequency was properly tuned. Numerical and experimental researches on four types of controllers including PPF controller applied to nonlinear beam models were presented in [3]. The results showed that PPF and NSC controllers were most effective in suppressing the high-amplitude vibration of flexible composite beam structure. Mitura et al. [4] presented numerical results for the PPF control method applied to strongly nonlinear horizontal and vertical beam models. They demonstrated that the PPF controller was more effective in reducing the vibration of the vertical beam than that of the horizontal beam. El-Ganaini et al. [5] presented an analytical study for positive position feedback controller to suppress the vibration amplitude of a nonlinear dynamical model when the primary resonance and the 1 : 1 internal resonance occurred simultaneously. They found that it was necessary to tune the controller’s natural frequency to the same value of the excitation frequency in the control process. In [6], a new nonlinear modified PPF controller was introduced to suppress the nonlinear vibration at primary resonance. It was demonstrated that the nonlinear modified PPF controller provided a higher level of suppression in the overall frequency domain compared with the conventional PPF controller. El-Sayed and Bauomy [7] employed two PPF controllers to reduce the vertical vibration in the vertical conveyors. It was shown that PPF controllers were very suitable for small natural frequency dynamical systems subjected to primary resonance excitations. Kandil and Eissa [8] overcame the drawback of the PPF controller by coupling additional NSCs to the main system to impose a V-curve at each one of the peaks out of the effective frequency bandwidth. In this control process, the two peaks could be suppressed to acceptable levels. Saeed and Kamel [9] applied a tuned PPF controller to suppress the lateral vibrations of a Jeffcott-rotor system. They concluded that the controller could reduce the vibration amplitudes close to zero at any spinning speed even at large disc eccentricity. In [10], a positive position feedback controller was proposed to suppress the nonlinear vibrations of a horizontally supported Jeffcott-rotor system. They found that the nonlinear PPF controller could eliminate the nonlinear phenomena of the Jeffcott-rotor system.Time delays inherently exist in many active control systems. They may induce complex dynamic behavior such as undesirable bifurcations, quasiperiodic motions, and chaotic behavior, etc. They limit the performance of active control. Therefore, it is necessary to investigate the effect of time delays on the active control process. In [11–13], Saeed et al. proposed three different controllers to suppress the vibrations of nonlinear Jeffcott-rotor systems. They presented the regions at which the system solutions were stable on the τ1−τ2 plane. The methods for selecting the optimal values of time delays were proposed. They found that time delays could not only improve the vibration suppression performance, but also increase the vibration amplitudes and destabilize the controlled system. In [14], a nonlinear time delay saturation-based controller was proposed to suppress the vibrations of a nonlinear beam. They proposed a concept of “vibration suppression region” at which the amplitude-delay’s response curves exhibit stable solution. They concluded that time delays could adjust the effective frequency bandwidth of the saturation controller and avoid the occurrence of the controller overload. El-Ganaini et al. [15] proposed a time-delayed PPF controller to suppress the horizontal vibration of a magnetically levitated body subjected to multiple force excitations. They concluded that the amplitudes of the controlled system did not depend only on a certain delay, but on the sum of two delays. Similar results also appeared in [16]. The authors demonstrated that time margins which indicated the safe region of operation depended on the overall delay of the controlled system. Kandil and El-Ganaini [17] utilized a time-delayed PPF controller to reduce the nonlinear oscillations of the compressor blade system subjected to a primary excitation at 1 : 1 internal resonance. They investigated the effect of time delay on the control of rotating blade vibrations and presented the safe region of operation.In this paper, a positive position feedback controller is utilized to suppress the vibration of a nonlinear horizontal beam under vertical excitation. Time delays in the control loop are taken into consideration in this work. We get an approximate solution by applying the integral iterative method. The stability of the system is investigated by applying a combination of Floquet theory and Hill’s determinant. The safe operation region of time delays is investigated for different values of control parameters. The main factors affecting the nonlinear characteristics of the controlled system are analyzed. In addition, the points at which the steady-state amplitude of the main system reaches its minimum are investigated analytically. Numerical simulations are presented to validate the analytical predictions. Finally, a comparison with the previously published works is included in the end of this paper.


## 2. Mathematical Model

The nonlinear differential equation that describes the vibration of a nonlinear horizontal beam under vertical excitation [4] is given as follows:(1)v¨+2μω1v˙+ω12v+βv3+δvv˙2+v2v¨=ξy0Ω2sinΩt.The model of the horizontal beam and the experimental setup were presented in [16]. By integrating a time-delayed PPF controller to the nonlinear beam, the equation governing the dynamics of the controlled system is suggested as(2)v¨+2μω1v˙+ω12v+βv3+δvv˙2+v2v¨=ξy0Ω2sinΩt+γut−τ1,u¨+2ζω2u˙+ω22u=αvt−τ2,where v denotes the response of the main system (the horizontal beam), u denotes the response of the PPF controller, ω1 is the natural frequency of the main system, μ is the damping ratio of the main system, β is the curvature nonlinearity coefficient, δ denotes the inertia nonlinearity coefficient, ζ is the damping ratio of the controller, ω2 is the natural frequency of the controller, y0 and Ω represent the amplitude and frequency of the support motion, γ denotes the control signal gain, α denotes the feedback signal gain, and τ1,τ2 are time delays.


## 3. Analytical Solutions

In [18], we improved the integral equation method introduced by Schmidt and Tondl [19] by adding time delay terms and rewriting the successive program. We renamed the improved method as the integral iterative method because of more iterative processes. Although these two methods are less known than other methods, they still have many advantages. In [19], the authors showed that the use of small parameters could lead to solutions of every degree of accuracy. The mechanism of the two methods is simple and clear. They are easier to program than many other perturbation methods. In our previous work [20], we found that the accuracy of the integral iterative method is much better than that of the multiple scales method when dealing with the single-degree-of-freedom problem. In this section, the integral iterative method is applied to obtain the second-order approximations and the amplitude equations for the system response. A more detailed introduction of the integral iterative method is given in Appendix.From the previously published works [4, 5, 16, 21], it is concluded that the simultaneous resonance occurs when Ω=ω1 and ω1=ω2. To study the PPF control, two detuning parameters σ1 and σ2 are introduced as follows:(3)Ω=ω1+σ1,ω2=ω1+σ2.We introduce a dimensionless time again by(4)t1=Ωt,τ1∗=Ωτ1,τ2∗=Ωτ2.Here, for simplicity, we still replacet1, τ1∗, τ2∗ by t, τ1, τ2. So, the closed loop system is transformed into the following form:(5)v¨+v=1Ω2−2μω1Ωv˙−σ12−2Ωσ1v−βv3−δΩ2vv˙2+v2v¨+ξy0Ω2sint+γut−τ1,u¨+u=1Ω2−2ζω2Ωu˙+2Ωσ1−σ2u−σ1−σ22u+αvt−τ2.For equation (5), λi=1,i=1,2, the corresponding generalized Green’s functions are(6)Git,σ=1π12−cost−σ+∑j=2∞cosjt−σ1−j2,i=1,2,Git−τk,σ=1π12−cost−τk−σ+∑j=2∞cosjt−τk−σ1−j2,i=1,2,k=1,2.From equations (A.7), (A.8), and (A.9) in Appendix, we get the first approximation in the following form:(7)v1t=r1cost+s1sint,(8)u1t=r2cost+s2sint,(9)v1t−τk=r1cost−τk+s1sint−τk,k=1,2,(10)u1t−τk=r2cost−τk+s2sint−τk,k=1,2.Substituting equations (7)–(10) into equations (A.7), (A.8), and (A.9), we get the second approximations in the following:(11)v2t=1πΩ2∫02π12−cost−σ+∑j=2∞cosjt−σ1−j2−βv13σ−2μω1Ωv˙1σ−σ12−2Ωσ1v1σ−δΩ2v1σv˙12σ+v12σv¨1σ+ξy0Ω2sinσ+γu1σ−τ1dσ+r1cost+s1sint=r1−12δr13−12δr1s12+3β4Ω2r13+3β4Ω2r1s12+σ12Ω2r1−2σ1Ωr1+2μω1s1Ωcost+−δr1316+316δr1s12+βr1332Ω2−3βr1s1232Ω2cos3t+s1−12δs13−12δs1r12+3β4Ω2s13+3β4Ω2s1r12+s1σ12Ω2−2σ1s1Ω−2μω1r1Ω−y0ξsint+−316δr12s1+δs1316+3βr12s132Ω2−βs1332Ω2sin3t−s2γsint−τ1Ω2−r2γcost−τ1Ω2,(12)u2t=1πΩ2∫02π12−cost−σ+∑j=2∞cosjt−σ1−j2−2ζω2Ωu˙1σ+2Ωσ1−σ2u1σ−σ1−σ22u1σ+αv1σ−τ2dσ+r2cost+s2sint=r2+σ12r2Ω2−2r2σ1σ2Ω2+r2σ22Ω2−2r2σ1Ω+2r2σ2Ω+2s2ζω2Ωcost−r1αcost−τ2Ω2+s2+s2σ12Ω2−2s2σ1σ2Ω2+s2σ22Ω2−2s2σ1Ω+2s2σ2Ω−2r2ζω2Ωsint−s1αsint−τ2Ω2.Substituting equations (11) and (12) into the solvability conditions equation (A.5) (in Appendix) yields(13)−δr132−12δr1s12+3βr134Ω2+3βr1s124Ω2+σ12r1Ω2−2σ1r1Ω+2μω1s1Ω−r2γcosΩτ1Ω2+s2γsinΩτ1Ω2=0,(14)12δr12s1+s13δ2+y0ξ−3βr12s14Ω2−3βs134Ω2−s1σ12Ω2+2s1σ1Ω+2r1μω1Ω+s2γcosΩτ1Ω2+r2γsinΩτ1Ω2=0,(15)−r2σ12Ω2+2r2σ1σ2Ω2−r2σ22Ω2+2r2σ1Ω−2r2σ2Ω−2s2ζω2Ω+r1αcosΩτ2Ω2−s1αsinΩτ2Ω2=0,(16)−s2σ12Ω2+2s2σ1σ2Ω2−s2σ22Ω2+2s2σ1Ω−2s2σ2Ω+2r2ζω2Ω+s1αcosΩτ2Ω2+r1αsinΩτ2Ω2=0.For convenience to investigate the dynamics of the whole controlled system, we denote the amplitude of the main systema1=r12+s12 and the amplitude of the controller a2=r22+s22. To simplify the above four amplitude equations (13)–(16), we set r1=a1cosθ1,s1=a1sinθ1,r2=a2cosθ2,s2=a2sinθ2, and the four amplitude equations yield(17)−3a13β−4a1σ12+8a1σ1Ω+2a13δΩ2cosθ1−8a1μΩω1sinθ1+4a2γcosθ2+Ωτ1=0,(18)4y0ξΩ2+8a1μΩω1cosθ1+4a2γsinθ2+Ωτ1+−3a13β−4a1σ12+8a1σ1Ω+2a13δΩ2sinθ1=0,(19)−a2σ12+2a2σ1σ2−a2σ22+2a2σ1Ω−2a2σ2Ωcosθ2+a1αcosθ1+Ωτ2−2a2ζΩω2sinθ2=0,(20)2a2ζΩω2cosθ2+a1αsinθ1+Ωτ2+−a2σ12+2a2σ1σ2−a2σ22+2a2σ2Ω−2a2σ1Ωsinθ2=0.To simplify equations (17)–(20), some simple calculations are made in the following:Equation (17)×cosθ1+ equation (18) ×sinθ1 is(21)4a2γcosθ1−θ2−Ωτ1+4y0ξΩ2sinθ1−3βa13−4σ12a1+8σ1Ωa1+2δΩ2a13=0.Equation (17)×sinθ1 − equation (18) ×cosθ1 is(22)−2a1μΩω1−y0ξΩ2cosθ1+a2γsinθ1−θ2−Ωτ1=0.Equation (19) ×cosθ2+equation (20)×sinθ2 is(23)−a2σ1−σ2σ1−σ2−2Ω+a1αcosθ1−θ2+Ωτ2=0.Equation (19) ×sinθ2 − equation (20) ×cosθ2 is(24)2a2ζω2Ω+a1αsinθ1−θ2+Ωτ2=0.Solving equations (21)–(24) and eliminating cosθ1, sinθ1, cosθ1−θ2, sinθ1−θ2 by means of the relations cos2θ1+sin2θ1=1,cos2θ1−θ2+sin2θ1−θ2=1 yield the amplitude equations for two possible scenarios.The uncontrolled system: in this case, the PPF controller does not activate, i.e.,a2=0; we have the amplitude equation of the excited beam from (21) and (22) in the following:(25)a124σ1σ1−2Ω+a123β−2δΩ22+64μ2Ω2ω12=16y0ξ2Ω4.The controlled system: in this case, the PPF controller activates, i.e.,a2≠0; then, the amplitude equations of the controlled system are obtained from (21)–(24)(26)a12α2=a22σ1−σ22σ2−σ1+2Ω2+4ζ2Ω2ω22,(27)16y03ξ2Ω4a12α2=a12α4σ1σ1−2Ω+a123β2δΩ2−4a22γσ1−σ2σ1−σ2∗2ΩcosΩτ1+τ2−2ζΩω2sinΩτ1+τ22+162a12αμΩω1+a22γ2ζΩω2cosΩτ1+τ2+σ1−σ2σ1−σ2−2ΩsinΩτ1+τ22.Solving equation (26), we obtain that(28)a22=a12α2σ1−σ222Ω−σ1+σ22+4ζ2Ω2ω22.Substituting (28) into (27), we have the amplitude equation only about a1 in the following:(29)m3a16+m2a14+m1a12+m0=0,where(30)m0=−16y02ξ2Ω4σ1−σ22σ2−σ1+2Ω2+4ζ2ω22Ω2,m1=16α2γ2+σ12σ1−σ22σ1−2Ω2σ2−σ1+2Ω2+4μ2Ω2ω12σ1−σ22σ2−σ1+2Ω2+4ζ2Ω2ω22σ12σ1−2Ω2+4μ2Ω2ω12−2αγσ1σ2−σ2σ1−2Ωσ1−σ2−2Ω+4αγΩμω1σ1−σ2σ1−σ2−2Ω−4ζμΩ2ω1ω2cosΩτ1+τ2+ζσ1ω2σ1−2ΩsinΩτ1+τ2,m2=83β−2δΩ2σ1σ1−2Ωσ1−σ22σ2−σ1+2Ω2+4ζ2Ω2ω22−αγσ1−σ2σ1−σ2−2ΩcosΩτ1+τ2+2αγζΩω2sinΩτ1+τ2,m3=3β−2δΩ22σ1−σ22σ2−σ1+2Ω2+4ζ2Ω2ω22.


## 4. Stability of Periodic Solutions

### 4.1. Stability of Periodic Solutions of the Uncontrolled System

For the uncontrolled beam, i.e.,r2=0,s2=0,γ=0,α=0, we obtain the second approximation from equation (11) as(31)v2t=r1cost+s1sint.In the above equation (31), we have omitted the higher order terms cos3t,sin3t because of the weak nonlinearities. To study the stability of the periodic solution of the uncontrolled beam, we first perturb the periodic solution v2t in equation (31) by introducing disturbance terms zt. Replacing v2t by v2t+zt in equation (1) and linearizing on zt, we get the linear variational equation in the following:(32)z¨+z=1Ω2−2μω1Ωz˙−σ12−2Ωσ1z−3βv22z−δΩ22v2v˙2z˙+v˙22z−δΩ2v22z¨+2v2v¨2z.Corresponding to the Floquet theory, the solution of equation (32) is written as(33)zt=expρtZt,and equation (32) is transformed into(34)Z¨+Z=12r12+s12δ−122+r12δ+s12δρ2−3r12+s12β2Ω2−σ12−2σ1Ω+2μρΩω1Ω2+12Ω2−4r1s1δρΩ2+s12−r123β−3δΩ2+δρ2Ω2cos2t+1Ω2r12−s12δρΩ2−r1s13β−3δΩ2+δρ2Ω2sin2tZ1+−1Ω2+r12δ+s12δρΩ+2μω1+δ−2r1s1−r12ρ+s12ρcos2t+δr12−s12−2r1s1ρsin2tZ˙−12r12+s12δ+12r12+s12δcos2t+r1s1δsin2tZ¨.The first approximation of equation (34) can be expressed as(35)Zt=a0+a1cost+a2sint.Substituting equation (35) into equation (34) and equating the constant terms and the coefficients of the same harmonic terms, we obtain the following set of linear homogeneous algebraic equations governing the coefficients a0,a1,a2:(36)k1a0=0,k2a1+k3a2=0,k4a1+k5a2=0,where(37)k1=1+ρ2+σ12Ω2−2σ1Ω+2μρω1Ω+12Ω2r12+s123β−δΩ2+δρ2Ω2;k2=14Ω23r12+s123β−2δΩ2+δρ2Ω2+ρ2+σ12Ω2−2σ1Ω+2μρω1Ω;k3=2ρ+2μω1Ω+r12+s12δρ+12Ω2r1s13β−2δΩ2+δρ2Ω2;k4=−2ρ−2μω1Ω−r12+s12δρ+12Ω2r1s13β−2δΩ2+δρ2Ω2;k5=14Ω23s12+r123β−2δΩ2+δρ2Ω2+ρ2+σ12Ω2−2σ1Ω+2μρω1Ω.Sincea0,a1,a2 are not all zero, the following coefficients matrix must be zero; i.e.,(38)k1000k2k30k4k5=0.Expanding this determinant yields(39)f0∗ρ6+f1∗ρ5+f2∗ρ4+f3∗ρ3+f4∗ρ2+f5∗ρ+f6∗=0.This is a nonlinear algebraic equation. In this case, the Routh–Hurwitz criterion can be utilized to study the stability of the periodic solution.


### 4.2. Stability of Periodic Solutions of the Controlled System

To investigate the stability of the periodic solution of the PPF controlled system (2), we need to examine the behavior of the small perturbations from the steady state solutions v2t,u2t in equations (11) and (12). Thus, we assume that(40)vt=v2t+z1t,ut=u2t+z2t,where z1t and z2t are small perturbations. Substituting equation (40) into equation (2) and linearizing on z1t and z2t yield(41)z¨1+z1=1Ω2−2μω1Ωz˙1−σ12−2Ωσ1z1−3βv22z1−δΩ22v2v˙2z˙1+v˙22z1+2v2v¨2z1+v22z¨1+γz2t−τ1,z¨2+z2=1Ω2−2ζω2Ωz˙2+2Ωσ1−σ2z2−σ1−σ22z2+αz1t−τ2.According to Floquet theory, equation (41) admits solutions of the form(42)zit=expρtZit,i=1,2,where Zit is a periodic function with period T=2π, which is equal to the period of vt,ut. Substituting equation (42) into equation (41) yields(43)Z¨1+Z1=12r12+s12δ−3β2Ω2r12+s12−σ12Ω2+2σ1Ω−2μρω1Ω−122+δr12+δs12ρ2+12Ω2−4r1s1δρΩ2+s12−r123β+δρ2−3Ω2cos2t+1Ω2r12−s12δρΩ2−r1s13β+δΩ2ρ2−3sin2tZ1t+γΩ2exp−ρτ1Z2t−τ1+−2+r12δ+s12δρ−2μω1Ω+δ−2r1s1−r12ρ+s12ρcos2t+δr12−s12−2r1s1ρsin2tZ˙1+−12δr12+s12−12δr12−s12cos2t−r1s1δsin2tZ¨1,(44)Z¨2+Z2=αexp−ρτ2Ω2Z1t−τ2+−ρ2−σ12Ω2+2σ1σ2Ω2−σ22Ω2+2σ1Ω−2σ2Ω−2ζρω2ΩZ2−2ρ+2ζω2ΩZ˙2.The first-order approximate solutions of equations (43) and (44) can be expressed as(45)Z1t=p0+p1cost+p2sint,(46)Z2t=p3+p4cost+p5sint.Substituting (45) and (46) into (43) and (44) and equating the coefficients of the same harmonic terms yield a set of linear homogeneous algebraic equations governing the coefficients p0, p1, p2, p3, p4, and p5. Setting the determinant of the coefficient matrix equal to zero, we obtain the so-called Hill’s determinant governing ρ. Expanding this determinant yields(47)b12ρ12+b11ρ11++⋯+b2ρ2+b1ρ+b0+dc0+dc1ρ+dc2ρ2+⋯+dc7ρ7+dc8ρ8cosΩτ1+τ2+ds0+ds1ρ+ds2ρ2+⋯+ds6ρ6+ds7ρ7sinΩτ1+τ2exp−ρΩτ1+τ2+ec4ρ4+ec3ρ3+ec2ρ2+ec1ρ+ec0cosΩτ1+τ2+es3ρ3+es2ρ2+es1ρ+es0sinΩτ1+τ2exp−2ρΩτ1+τ2+c0exp−3ρΩτ1+τ2=0.Equation (47) can be solved numerically. The approximate solutions v2t,u2t in equations (11) and (12) are asymptotically stable if and only if all of the eigenvalues ρ lie in the left half of the complex plane, and they are unstable if at least one eigenvalue lies in the right half-plane. The solution subsequent to the bifurcation depends on the manner in which the eigenvalues cross from the left half-plane to the right [22, 23].


## 4.1. Stability of Periodic Solutions of the Uncontrolled System

For the uncontrolled beam, i.e.,r2=0,s2=0,γ=0,α=0, we obtain the second approximation from equation (11) as(31)v2t=r1cost+s1sint.In the above equation (31), we have omitted the higher order terms cos3t,sin3t because of the weak nonlinearities. To study the stability of the periodic solution of the uncontrolled beam, we first perturb the periodic solution v2t in equation (31) by introducing disturbance terms zt. Replacing v2t by v2t+zt in equation (1) and linearizing on zt, we get the linear variational equation in the following:(32)z¨+z=1Ω2−2μω1Ωz˙−σ12−2Ωσ1z−3βv22z−δΩ22v2v˙2z˙+v˙22z−δΩ2v22z¨+2v2v¨2z.Corresponding to the Floquet theory, the solution of equation (32) is written as(33)zt=expρtZt,and equation (32) is transformed into(34)Z¨+Z=12r12+s12δ−122+r12δ+s12δρ2−3r12+s12β2Ω2−σ12−2σ1Ω+2μρΩω1Ω2+12Ω2−4r1s1δρΩ2+s12−r123β−3δΩ2+δρ2Ω2cos2t+1Ω2r12−s12δρΩ2−r1s13β−3δΩ2+δρ2Ω2sin2tZ1+−1Ω2+r12δ+s12δρΩ+2μω1+δ−2r1s1−r12ρ+s12ρcos2t+δr12−s12−2r1s1ρsin2tZ˙−12r12+s12δ+12r12+s12δcos2t+r1s1δsin2tZ¨.The first approximation of equation (34) can be expressed as(35)Zt=a0+a1cost+a2sint.Substituting equation (35) into equation (34) and equating the constant terms and the coefficients of the same harmonic terms, we obtain the following set of linear homogeneous algebraic equations governing the coefficients a0,a1,a2:(36)k1a0=0,k2a1+k3a2=0,k4a1+k5a2=0,where(37)k1=1+ρ2+σ12Ω2−2σ1Ω+2μρω1Ω+12Ω2r12+s123β−δΩ2+δρ2Ω2;k2=14Ω23r12+s123β−2δΩ2+δρ2Ω2+ρ2+σ12Ω2−2σ1Ω+2μρω1Ω;k3=2ρ+2μω1Ω+r12+s12δρ+12Ω2r1s13β−2δΩ2+δρ2Ω2;k4=−2ρ−2μω1Ω−r12+s12δρ+12Ω2r1s13β−2δΩ2+δρ2Ω2;k5=14Ω23s12+r123β−2δΩ2+δρ2Ω2+ρ2+σ12Ω2−2σ1Ω+2μρω1Ω.Sincea0,a1,a2 are not all zero, the following coefficients matrix must be zero; i.e.,(38)k1000k2k30k4k5=0.Expanding this determinant yields(39)f0∗ρ6+f1∗ρ5+f2∗ρ4+f3∗ρ3+f4∗ρ2+f5∗ρ+f6∗=0.This is a nonlinear algebraic equation. In this case, the Routh–Hurwitz criterion can be utilized to study the stability of the periodic solution.


## 4.2. Stability of Periodic Solutions of the Controlled System

To investigate the stability of the periodic solution of the PPF controlled system (2), we need to examine the behavior of the small perturbations from the steady state solutions v2t,u2t in equations (11) and (12). Thus, we assume that(40)vt=v2t+z1t,ut=u2t+z2t,where z1t and z2t are small perturbations. Substituting equation (40) into equation (2) and linearizing on z1t and z2t yield(41)z¨1+z1=1Ω2−2μω1Ωz˙1−σ12−2Ωσ1z1−3βv22z1−δΩ22v2v˙2z˙1+v˙22z1+2v2v¨2z1+v22z¨1+γz2t−τ1,z¨2+z2=1Ω2−2ζω2Ωz˙2+2Ωσ1−σ2z2−σ1−σ22z2+αz1t−τ2.According to Floquet theory, equation (41) admits solutions of the form(42)zit=expρtZit,i=1,2,where Zit is a periodic function with period T=2π, which is equal to the period of vt,ut. Substituting equation (42) into equation (41) yields(43)Z¨1+Z1=12r12+s12δ−3β2Ω2r12+s12−σ12Ω2+2σ1Ω−2μρω1Ω−122+δr12+δs12ρ2+12Ω2−4r1s1δρΩ2+s12−r123β+δρ2−3Ω2cos2t+1Ω2r12−s12δρΩ2−r1s13β+δΩ2ρ2−3sin2tZ1t+γΩ2exp−ρτ1Z2t−τ1+−2+r12δ+s12δρ−2μω1Ω+δ−2r1s1−r12ρ+s12ρcos2t+δr12−s12−2r1s1ρsin2tZ˙1+−12δr12+s12−12δr12−s12cos2t−r1s1δsin2tZ¨1,(44)Z¨2+Z2=αexp−ρτ2Ω2Z1t−τ2+−ρ2−σ12Ω2+2σ1σ2Ω2−σ22Ω2+2σ1Ω−2σ2Ω−2ζρω2ΩZ2−2ρ+2ζω2ΩZ˙2.The first-order approximate solutions of equations (43) and (44) can be expressed as(45)Z1t=p0+p1cost+p2sint,(46)Z2t=p3+p4cost+p5sint.Substituting (45) and (46) into (43) and (44) and equating the coefficients of the same harmonic terms yield a set of linear homogeneous algebraic equations governing the coefficients p0, p1, p2, p3, p4, and p5. Setting the determinant of the coefficient matrix equal to zero, we obtain the so-called Hill’s determinant governing ρ. Expanding this determinant yields(47)b12ρ12+b11ρ11++⋯+b2ρ2+b1ρ+b0+dc0+dc1ρ+dc2ρ2+⋯+dc7ρ7+dc8ρ8cosΩτ1+τ2+ds0+ds1ρ+ds2ρ2+⋯+ds6ρ6+ds7ρ7sinΩτ1+τ2exp−ρΩτ1+τ2+ec4ρ4+ec3ρ3+ec2ρ2+ec1ρ+ec0cosΩτ1+τ2+es3ρ3+es2ρ2+es1ρ+es0sinΩτ1+τ2exp−2ρΩτ1+τ2+c0exp−3ρΩτ1+τ2=0.Equation (47) can be solved numerically. The approximate solutions v2t,u2t in equations (11) and (12) are asymptotically stable if and only if all of the eigenvalues ρ lie in the left half of the complex plane, and they are unstable if at least one eigenvalue lies in the right half-plane. The solution subsequent to the bifurcation depends on the manner in which the eigenvalues cross from the left half-plane to the right [22, 23].


## 5. Results and Discussions

In this section, the steady-state response of the nonlinear beam and the PPF controller is studied analytically and numerically. The parameters are fixed atξ=0.89663,ω1=3.06,μ=0.05,β=14.41,δ=3.27,y0∈0,0.1,ζ=0.001,α=3,γ=3,τ1=0,τ2=0,σ2=0,σ1=0, unless otherwise specified. These parameters are derived from the previously published works [3–5, 14, 16, 18, 21]. The main results are presented in graphical forms and tables, such that the solid lines stand for stable solutions, the dashed lines stand for unstable solutions, and the red points stand for numerical solutions.


### 5.1. Time Delay Stability Margin for Different Values of Various Control Parameters

In this subsection, our efforts are focused on determining the maximum limits of time delays at which the system solution remains stable. In [16], the authors named these limits of time delays as “time delay stability margin.” According to equations (27) and (47), we infer that both the amplitudes and stability of the steady state response of equation (2) are affected by the sum of delays τ1+τ2. Therefore, the time delay stability margin depends on the overall delay τ1+τ2. This result is consistent with the previously published work [16]. Figure 1 shows time histories of the main system and the controller for four different cases, i.e., τ1=0,τ2=0.05; τ1=0.01,τ2=0.04; τ1=0.04,τ2=0.01; τ1=0.03,τ2=0.02. We observe that the amplitudes of the main system and the controller are almost the same in four different cases. Figure 2 presents a comparison of time delay τ1-response curves between numerical simulations and approximate solutions. It can be seen that the approximate solutions are in good agreement with numerical simulations.Figure 1
Time history for the controlled system whenσ1=0,σ2=0,ζ=0.001,y0=0.07; 3000≤t≤3100,τ1=0,τ2=0.05; 3100≤t≤3200,τ1=0.01,τ2=0.04; 3200≤t≤3300,τ1=0.04,τ2=0.01; and 3300≤t≤3400,τ1=0.03,τ2=0.02: (a) the main system; (b) the controller.
(a)(b)Figure 2
Comparison of time delayτ1-response curves between numerical simulations and approximate solutions when τ2=0,y0=0.07,α=3,γ=3,ζ=0.001,andσ1=σ2=0: (a) the main system; (b) the controller.
(a)(b)Figure3 shows time delay stability margin under different parameters. Figures 3(a1) and (b1) illustrate that time delay stability margin will hardly change when y0 varies in the range 0,0.1. It can be seen that the steady state response of the whole system is stable when τ1+τ2 locates in the interval 0,0.1. Beyond this range, the whole system becomes unstable. From Figures 3(a2) to (b4), the parameter y0 is fixed at 0.07. Figures 3(a2), (b2) and (a3), (b3) illustrate that increasing the feedback signal gain α and the control signal gain γ can shrink the stable region. From these figures, we can see that the controlled system is always stable in the time delay interval 0,0.08 when the control signal gain γ and the feedback signal gain α vary in the interval 0,5. From Figures 3(a4) and (b4), we observe that increasing the damping coefficient ζ can broaden the stable region of the solution while decreasing ζ shrinks the stable region. Based on these figures, we conclude that the unavoidable total time delay τ1+τ2 in the controlled system should not exceed 0.08.Figure 3
Time delayτ-response curves of the PPF controlled system where τ=τ1+τ2: (a) the main system; (b) the controller.
(a)(b)Figures4 and 5 show time histories of the main system and the controller for different values of time delays τ1,τ2. Figure 4 shows stable behavior for the main system and the controller at τ1=0.03,τ2=0.04. It can be seen that the whole system passes through a transient region into a stable steady-state region. Figure 5 shows a complex unstable motion for the system and the controller when the total time delay τ1+τ2 exceeds the stable region (i.e., τ1=0.1,τ2=0.04).Figure 4
Time history of the controlled system whenτ1=0.03,τ2=0.04,y0=0.07 with the initial condition v0=0.001,u0=0.001,v˙0=0.1,andu˙0=0.01: (a) the main system; (b) the controller.
(a)(b)Figure 5
Time history of the controlled system whenτ1=0.1,τ2=0.04,y0=0.07 with the initial condition v0=0.001,u0=0.001,v˙0=0.1,andu˙0=0.01: (a) the main system; (b) the controller.
(a)(b)


### 5.2. Effects of Control Parameters on the Existence of Multiple-Solution Region

In this subsection, we investigate the effects of various parameters on the existence of multiple-solution region for equation (2). For convenience’s sake, we denote A1=a12 in equation (29) and rewrite equation (29) in the form(48)m3A13+m2A12+m1A1+m0=0.Based on Girolamo Cardano, we give the discriminant of roots in the following:Case 1: if1/427m32m0−9m3m2m1+2m232+3m3m1−m223<0 and m2/m3<0,m1/m3>0,m0/m3<0, then equation (48) has three different positive real roots which means equation (29) has three different amplitudes, and system (2) has tristable state vibration.Case 2: if1/427m32m0−9m3m2m1+2m232+3m3m1−m223=0 and 3m3m1−m22≠0,27m32m0−9m3m2m1+2m23≠0,m2/m3<0,m1/m3>0,m0/m3<0, then equation (48) has three positive real roots, two of which are equal. It means equation (29) has two different amplitudes and system (2) has bistable steady-state vibration. This is the critical case of transition from monostable state to tristable state, which often means the occurrence of saddle node bifurcation, and jumping phenomena occur at these points.Case 3: except for the above two cases, equation (48) has only one positive real root; i.e., equation (29) has one amplitude and system (2) has simple steady-state vibration.Figure6 shows the existence of multiple-solution region in y0−σ1 plane for the uncontrolled system, the controlled system without time delay, and the controlled system with τ1+τ2=0.03. The blank region represents only one solution. The red dashed line boundary stands for two different solutions and the blue region stands for three different solutions. Figure 6(a) illustrates that there are no multiple solutions for the uncontrolled system when y0 varies in the interval 0,0.1. That is to say, although the uncontrolled beam is a nonlinear beam, it does not show the nonlinear characteristics under such parameters. Figure 6(b) shows that the controlled system without time delay begins to have multiple solutions when y0>0.048, and the larger y0 is, the wider the multiple-solution region is. It means that the controlled system makes the linear system nonlinear even when there is no time delay. Figure 6(c) shows that the multiple-solution region of the controlled system with τ1+τ2=0.03 is obviously increased compared with the first two cases. Even if the force amplitude y0 is very small, the controlled system shows obvious non-linear characteristics.Figure 6
The system solution behavior iny0−σ1 plane: (a) the uncontrolled system; (b) the controlled system without time delay; (c) the controlled system with τ=τ1+τ2=0.03.
(a)(b)(c)Figure7 shows the existence of multiple-solution region in τ−σ1 plane for the controlled system with ζ=0.05 and ζ=0.001, respectively. Figure 7(a) illustrates that when the total delay τ1+τ2>0.13, the controlled system with ζ=0.05 has multiple solutions. That is to say, despite the appearance of time delay, the controlled system with ζ=0.05 cannot exhibit the non-linear characteristics when τ1+τ2<0.13. Figure 7(b) shows that the multiple-solution region with ζ=0.001 increases significantly compared with the controlled system with ζ=0.05.Figure 7
The system solution behavior inτ−σ1 plane when y0=0.07: (a) ζ=0.05; (b) ζ=0.001.
(a)(b)Figures8–10 show the effect of control parameters γ, α, and ζ on the existence of multiple-solution region. From Figures 8(a) and 9(a), we observe that increasing γ and α leads to a slight broadening of the multiple-solution region for the controlled system without time delay. Figures 8(b) and 9(b) illustrate that the increase of γ and α leads to a significant broadening of the multiple-solution region for the controlled system with τ1+τ2=0.03. Figure 10(a) shows that the controlled system without time delay has multiple solutions only when the damping coefficient ζ<0.006. When the total time delay τ1+τ2=0.03, the multiple-solution region of the controlled system is significantly larger than that of the former in Figure 10(b). From Figure 10(b), we also observe that even if the total time delay is not zero, the controlled system has multiple solutions only when the damping coefficient is less than 0.016.Figure 8
The system solution behavior inγ−σ1 plane when y0=0.07: (a) the controlled system without time delay; (b) the controlled system with τ=τ1+τ2=0.03.
(a)(b)Figure 9
The system solution behavior inα−σ1 plane when y0=0.07: (a) the controlled system without time delay; (b) the controlled system with τ=τ1+τ2=0.03.
(a)(b)Figure 10
The system solution behavior inζ−σ1 plane when y0=0.07: (a) the controlled system without time delay; (b) the controlled system with τ=τ1+τ2=0.03.
(a)(b)Figure11 shows the frequency response curves for the uncontrolled system. A comparison between the approximations obtained by the integral iterative method and numerical simulations is made to verify the analytical results in Figure 11(a). Figure 11(b) illustrates that the steady-state amplitude of uncontrolled system is a monotonic increasing function with respect to the force amplitude y0. As the force amplitude y0 increases in the range 0,0.1, the curve is bent to the left indicating a soft effect. However, the jump phenomenon does not occur. It means that the nonlinearity does not work when y0∈0,0.1.Figure 11
Response curves of the uncontrolled beam. (a) Comparison between numerical solutions and approximate solutions wheny0=0.07; (b) effect of y0 on the response curves.
(a)(b)Figures12 and 13 study the frequency σ1-response curves of the controlled system with τ1=0.02,τ2=0.01 at ζ=0.05 and ζ=0.001, respectively. In addition, we observe that the numerical results correspond well with the theoretical results for both cases from these figures. Figure 12 shows that the controlled system with τ2=0.02,τ1=0.01,ζ=0.05 is always stable and never has jump phenomenon throughout the range σ1∈−0.5,0.5. Figure 13 illustrates that the controlled system with τ2=0.02,τ1=0.01,ζ=0.001 has three solutions in the range 0.05,0.3. In addition, we observe that the larger two solutions are unstable and the smallest solution is stable in the range 0.05,0.18. Within the range 0.18,0.3, the intermediate solution is unstable and the other two solutions are stable. The jump phenomenon occurs at σ1=0.3. It means that there is bistability in the range σ1∈0.18,0.3. Figures 14 and 15 confirm the coexistence of two stable solutions at τ1=0.02,τ1=0.01,ζ=0.001 with two different initial conditions.Figure 12
Frequencyσ1-response curves for the controlled system with τ1=0.02,τ2=0.01,α=3,γ=3,ζ=0.05,andy0=0.07: (a) the main system; (b) the controller.
(a)(b)Figure 13
Frequencyσ1-response curves for the controlled system with τ1=0.02,τ2=0.01,α=3,γ=3,ζ=0.001,andy0=0.07: (a) the main system; (b) the controller.
(a)(b)Figure 14
Time history for the controlled system whenτ1=0.02,τ2=0.01,σ1=0.28,y0=0.07 with the initial condition v0=0.748,u0=1.27,v˙0=0.1,andu˙0=0.01: (a) the main system; (b) the controller.
(a)(b)Figure 15
Time history for the controlled system whenτ1=0.02,τ2=0.01,σ1=0.28,y0=0.07 with the initial condition v0=0.239,u0=0.408,v˙0=0.1,andu˙0=0.01: (a) the main system; (b) the controller.
(a)(b)In summary, the appearance of time delay and near zero damping coefficientζ are two main factors leading to the nonlinear characteristics of the controlled system. Perhaps, we can adjust different initial values to achieve the effect of vibration reduction in the region where multiple solutions exist.


### 5.3. The Location of the Minimum Steady-State Main System Amplitude

The previously published works [5, 7–10, 16] pointed out that the minimum steady-state amplitude of the main system occurred at σ1=σ2 for the PPF controlled system. In this section, our efforts are focused on finding out where the minimum amplitude a1 occurs. From equations (26)–(29), we conclude that the vibration of the main system vanishes when σ1=σ2 and ζ=0. Equation (31) is only an amplitude equation on a1. According to the Theorem of the Existence of Implicit Functions, the amplitude a1 in equation (31) can be considered as a function on Ω. Owing to the relation A1=a12 and a1≥0, the minimum point of a1 is equivalent to that of A1. According to calculus theory, the minimum point of amplitude A1 must satisfy the following equation set:(49)dA1dΩ=−dm3/dΩA13+dm2/dΩA12+dm1/dΩA1+dm0/dΩ3m3A12+2m2A1+m1=0,m3A13+m2A12+m1A1+m0=0,and the following inequality:(50)d2A1dΩ2<0.Based on equations (49) and (50), we give the excitation frequency Ω which minimizes the amplitude of the main system when the natural frequency of the controller ω2 varies for two different cases in Tables 1 and 2. Table 1 shows that the excitation frequency Ω and the controller natural frequency ω2 that minimize the amplitude of the main system always deviate from each other when ζ=0.05,τ1+τ2=0.05. And with the increase of σ2 (i.e. ω2−ω1), the deviation Ω−ω2 becomes bigger and bigger. For example, for σ2=0.5 the minimum steady-state amplitude of the main system occurs at σ1=0.43. Table 2 shows that the excitation frequency Ω and the controller natural frequency ω2 that minimize the amplitude of the main system are almost equal to each other when ζ=0.001,τ1+τ2=0.05. It can be seen that the minimum amplitudes of the main system are almost zero when ζ=0.001. From these tables, we conclude that it is necessary to tune the controller’s natural frequency ω2 to the same value of the excitation frequency Ω only when the controller damping coefficient ζ is close to zero. Figure 16 shows time histories of the primary system and the controller for four different cases, i.e., ζ=0.05,Ω=ω2=3.36; ζ=0.05,Ω=3.31,ω2=3.36; ζ=0.001,Ω=ω2=3.36 and ζ=0.001,Ω=3.31,ω2=3.36. From Figure 16(a), it is observed that the amplitude of the main system when ζ=0.05,Ω=3.31,ω2=3.36 is smaller than that when ζ=0.05,Ω=ω2=3.36. It can be seen that the best condition for vibration reduction in the four cases is that ζ=0.001 and Ω=ω2. From Figure 16, we also observe that decreasing the controller damping coefficient ζ can better suppress the vibration of the main system. However, it can enhance the vibration of the controller.Table 1
Different minimum points of amplitudea1 when ω1=3.06, ζ=0.05, y0=0.07, and τ=τ1+τ2=0.05.
ω2ΩA1a12.562.560877.41×10−40.027222.662.658121.01×10−30.031722.762.75451.352×10−30.036772.862.849911.796×10−30.042372.962.94422.353×10−30.048513.063.037393.0360×10−30.0550993.163.129353.8429×10−30.0619913.263.220174.755×10−30.0689543.363.309985.7294×10−30.0756933.463.399066.7057×10−30.0818823.563.487717.6153×10−30.08727Table 2
Different minimum points of amplitudea1 when ω1=3.06, ζ=0.001, y0=0.07, and τ=τ1+τ2=0.05.
ω2ΩA1a12.562.563.58×10−75.98×10−42.662.664.86×10−76.98×10−42.762.766.53×10−78.08×10−42.862.868.68×10−79.31×10−42.962.959991.14×10−61.07×10−33.063.059991.49×10−61.22×10−33.163.159991.93×10−61.39×10−33.263.259982.47×10−61.57×10−33.363.359973.15×10−61.77×10−33.463.459963.98×10−61.99×10−33.563.559955.00×10−62.24×10−3Figure 16
Time history for the controlled system whenτ1=0.03,τ2=0.02,y0=0.07: 2800≤t≤3000,ζ=0.05,Ω=ω2=3.36; 3000≤t≤3200,ζ=0.05,Ω=3.31,ω2=3.36; 3200≤t≤3400,ζ=0.001,Ω=ω2=3.36; and 3400≤t≤3600,ζ=0.001,Ω=3.31,ω2=3.36: (a) the main system; (b) the controller.
(a)(b)In general, only when the controller damping coefficientζ approaches zero will the minimum main system amplitude occur at σ1=σ2. And in this case, the vibration of the main system almost vanishes.


## 5.1. Time Delay Stability Margin for Different Values of Various Control Parameters

In this subsection, our efforts are focused on determining the maximum limits of time delays at which the system solution remains stable. In [16], the authors named these limits of time delays as “time delay stability margin.” According to equations (27) and (47), we infer that both the amplitudes and stability of the steady state response of equation (2) are affected by the sum of delays τ1+τ2. Therefore, the time delay stability margin depends on the overall delay τ1+τ2. This result is consistent with the previously published work [16]. Figure 1 shows time histories of the main system and the controller for four different cases, i.e., τ1=0,τ2=0.05; τ1=0.01,τ2=0.04; τ1=0.04,τ2=0.01; τ1=0.03,τ2=0.02. We observe that the amplitudes of the main system and the controller are almost the same in four different cases. Figure 2 presents a comparison of time delay τ1-response curves between numerical simulations and approximate solutions. It can be seen that the approximate solutions are in good agreement with numerical simulations.Figure 1
Time history for the controlled system whenσ1=0,σ2=0,ζ=0.001,y0=0.07; 3000≤t≤3100,τ1=0,τ2=0.05; 3100≤t≤3200,τ1=0.01,τ2=0.04; 3200≤t≤3300,τ1=0.04,τ2=0.01; and 3300≤t≤3400,τ1=0.03,τ2=0.02: (a) the main system; (b) the controller.
(a)(b)Figure 2
Comparison of time delayτ1-response curves between numerical simulations and approximate solutions when τ2=0,y0=0.07,α=3,γ=3,ζ=0.001,andσ1=σ2=0: (a) the main system; (b) the controller.
(a)(b)Figure3 shows time delay stability margin under different parameters. Figures 3(a1) and (b1) illustrate that time delay stability margin will hardly change when y0 varies in the range 0,0.1. It can be seen that the steady state response of the whole system is stable when τ1+τ2 locates in the interval 0,0.1. Beyond this range, the whole system becomes unstable. From Figures 3(a2) to (b4), the parameter y0 is fixed at 0.07. Figures 3(a2), (b2) and (a3), (b3) illustrate that increasing the feedback signal gain α and the control signal gain γ can shrink the stable region. From these figures, we can see that the controlled system is always stable in the time delay interval 0,0.08 when the control signal gain γ and the feedback signal gain α vary in the interval 0,5. From Figures 3(a4) and (b4), we observe that increasing the damping coefficient ζ can broaden the stable region of the solution while decreasing ζ shrinks the stable region. Based on these figures, we conclude that the unavoidable total time delay τ1+τ2 in the controlled system should not exceed 0.08.Figure 3
Time delayτ-response curves of the PPF controlled system where τ=τ1+τ2: (a) the main system; (b) the controller.
(a)(b)Figures4 and 5 show time histories of the main system and the controller for different values of time delays τ1,τ2. Figure 4 shows stable behavior for the main system and the controller at τ1=0.03,τ2=0.04. It can be seen that the whole system passes through a transient region into a stable steady-state region. Figure 5 shows a complex unstable motion for the system and the controller when the total time delay τ1+τ2 exceeds the stable region (i.e., τ1=0.1,τ2=0.04).Figure 4
Time history of the controlled system whenτ1=0.03,τ2=0.04,y0=0.07 with the initial condition v0=0.001,u0=0.001,v˙0=0.1,andu˙0=0.01: (a) the main system; (b) the controller.
(a)(b)Figure 5
Time history of the controlled system whenτ1=0.1,τ2=0.04,y0=0.07 with the initial condition v0=0.001,u0=0.001,v˙0=0.1,andu˙0=0.01: (a) the main system; (b) the controller.
(a)(b)


## 5.2. Effects of Control Parameters on the Existence of Multiple-Solution Region

In this subsection, we investigate the effects of various parameters on the existence of multiple-solution region for equation (2). For convenience’s sake, we denote A1=a12 in equation (29) and rewrite equation (29) in the form(48)m3A13+m2A12+m1A1+m0=0.Based on Girolamo Cardano, we give the discriminant of roots in the following:Case 1: if1/427m32m0−9m3m2m1+2m232+3m3m1−m223<0 and m2/m3<0,m1/m3>0,m0/m3<0, then equation (48) has three different positive real roots which means equation (29) has three different amplitudes, and system (2) has tristable state vibration.Case 2: if1/427m32m0−9m3m2m1+2m232+3m3m1−m223=0 and 3m3m1−m22≠0,27m32m0−9m3m2m1+2m23≠0,m2/m3<0,m1/m3>0,m0/m3<0, then equation (48) has three positive real roots, two of which are equal. It means equation (29) has two different amplitudes and system (2) has bistable steady-state vibration. This is the critical case of transition from monostable state to tristable state, which often means the occurrence of saddle node bifurcation, and jumping phenomena occur at these points.Case 3: except for the above two cases, equation (48) has only one positive real root; i.e., equation (29) has one amplitude and system (2) has simple steady-state vibration.Figure6 shows the existence of multiple-solution region in y0−σ1 plane for the uncontrolled system, the controlled system without time delay, and the controlled system with τ1+τ2=0.03. The blank region represents only one solution. The red dashed line boundary stands for two different solutions and the blue region stands for three different solutions. Figure 6(a) illustrates that there are no multiple solutions for the uncontrolled system when y0 varies in the interval 0,0.1. That is to say, although the uncontrolled beam is a nonlinear beam, it does not show the nonlinear characteristics under such parameters. Figure 6(b) shows that the controlled system without time delay begins to have multiple solutions when y0>0.048, and the larger y0 is, the wider the multiple-solution region is. It means that the controlled system makes the linear system nonlinear even when there is no time delay. Figure 6(c) shows that the multiple-solution region of the controlled system with τ1+τ2=0.03 is obviously increased compared with the first two cases. Even if the force amplitude y0 is very small, the controlled system shows obvious non-linear characteristics.Figure 6
The system solution behavior iny0−σ1 plane: (a) the uncontrolled system; (b) the controlled system without time delay; (c) the controlled system with τ=τ1+τ2=0.03.
(a)(b)(c)Figure7 shows the existence of multiple-solution region in τ−σ1 plane for the controlled system with ζ=0.05 and ζ=0.001, respectively. Figure 7(a) illustrates that when the total delay τ1+τ2>0.13, the controlled system with ζ=0.05 has multiple solutions. That is to say, despite the appearance of time delay, the controlled system with ζ=0.05 cannot exhibit the non-linear characteristics when τ1+τ2<0.13. Figure 7(b) shows that the multiple-solution region with ζ=0.001 increases significantly compared with the controlled system with ζ=0.05.Figure 7
The system solution behavior inτ−σ1 plane when y0=0.07: (a) ζ=0.05; (b) ζ=0.001.
(a)(b)Figures8–10 show the effect of control parameters γ, α, and ζ on the existence of multiple-solution region. From Figures 8(a) and 9(a), we observe that increasing γ and α leads to a slight broadening of the multiple-solution region for the controlled system without time delay. Figures 8(b) and 9(b) illustrate that the increase of γ and α leads to a significant broadening of the multiple-solution region for the controlled system with τ1+τ2=0.03. Figure 10(a) shows that the controlled system without time delay has multiple solutions only when the damping coefficient ζ<0.006. When the total time delay τ1+τ2=0.03, the multiple-solution region of the controlled system is significantly larger than that of the former in Figure 10(b). From Figure 10(b), we also observe that even if the total time delay is not zero, the controlled system has multiple solutions only when the damping coefficient is less than 0.016.Figure 8
The system solution behavior inγ−σ1 plane when y0=0.07: (a) the controlled system without time delay; (b) the controlled system with τ=τ1+τ2=0.03.
(a)(b)Figure 9
The system solution behavior inα−σ1 plane when y0=0.07: (a) the controlled system without time delay; (b) the controlled system with τ=τ1+τ2=0.03.
(a)(b)Figure 10
The system solution behavior inζ−σ1 plane when y0=0.07: (a) the controlled system without time delay; (b) the controlled system with τ=τ1+τ2=0.03.
(a)(b)Figure11 shows the frequency response curves for the uncontrolled system. A comparison between the approximations obtained by the integral iterative method and numerical simulations is made to verify the analytical results in Figure 11(a). Figure 11(b) illustrates that the steady-state amplitude of uncontrolled system is a monotonic increasing function with respect to the force amplitude y0. As the force amplitude y0 increases in the range 0,0.1, the curve is bent to the left indicating a soft effect. However, the jump phenomenon does not occur. It means that the nonlinearity does not work when y0∈0,0.1.Figure 11
Response curves of the uncontrolled beam. (a) Comparison between numerical solutions and approximate solutions wheny0=0.07; (b) effect of y0 on the response curves.
(a)(b)Figures12 and 13 study the frequency σ1-response curves of the controlled system with τ1=0.02,τ2=0.01 at ζ=0.05 and ζ=0.001, respectively. In addition, we observe that the numerical results correspond well with the theoretical results for both cases from these figures. Figure 12 shows that the controlled system with τ2=0.02,τ1=0.01,ζ=0.05 is always stable and never has jump phenomenon throughout the range σ1∈−0.5,0.5. Figure 13 illustrates that the controlled system with τ2=0.02,τ1=0.01,ζ=0.001 has three solutions in the range 0.05,0.3. In addition, we observe that the larger two solutions are unstable and the smallest solution is stable in the range 0.05,0.18. Within the range 0.18,0.3, the intermediate solution is unstable and the other two solutions are stable. The jump phenomenon occurs at σ1=0.3. It means that there is bistability in the range σ1∈0.18,0.3. Figures 14 and 15 confirm the coexistence of two stable solutions at τ1=0.02,τ1=0.01,ζ=0.001 with two different initial conditions.Figure 12
Frequencyσ1-response curves for the controlled system with τ1=0.02,τ2=0.01,α=3,γ=3,ζ=0.05,andy0=0.07: (a) the main system; (b) the controller.
(a)(b)Figure 13
Frequencyσ1-response curves for the controlled system with τ1=0.02,τ2=0.01,α=3,γ=3,ζ=0.001,andy0=0.07: (a) the main system; (b) the controller.
(a)(b)Figure 14
Time history for the controlled system whenτ1=0.02,τ2=0.01,σ1=0.28,y0=0.07 with the initial condition v0=0.748,u0=1.27,v˙0=0.1,andu˙0=0.01: (a) the main system; (b) the controller.
(a)(b)Figure 15
Time history for the controlled system whenτ1=0.02,τ2=0.01,σ1=0.28,y0=0.07 with the initial condition v0=0.239,u0=0.408,v˙0=0.1,andu˙0=0.01: (a) the main system; (b) the controller.
(a)(b)In summary, the appearance of time delay and near zero damping coefficientζ are two main factors leading to the nonlinear characteristics of the controlled system. Perhaps, we can adjust different initial values to achieve the effect of vibration reduction in the region where multiple solutions exist.


## 5.3. The Location of the Minimum Steady-State Main System Amplitude

The previously published works [5, 7–10, 16] pointed out that the minimum steady-state amplitude of the main system occurred at σ1=σ2 for the PPF controlled system. In this section, our efforts are focused on finding out where the minimum amplitude a1 occurs. From equations (26)–(29), we conclude that the vibration of the main system vanishes when σ1=σ2 and ζ=0. Equation (31) is only an amplitude equation on a1. According to the Theorem of the Existence of Implicit Functions, the amplitude a1 in equation (31) can be considered as a function on Ω. Owing to the relation A1=a12 and a1≥0, the minimum point of a1 is equivalent to that of A1. According to calculus theory, the minimum point of amplitude A1 must satisfy the following equation set:(49)dA1dΩ=−dm3/dΩA13+dm2/dΩA12+dm1/dΩA1+dm0/dΩ3m3A12+2m2A1+m1=0,m3A13+m2A12+m1A1+m0=0,and the following inequality:(50)d2A1dΩ2<0.Based on equations (49) and (50), we give the excitation frequency Ω which minimizes the amplitude of the main system when the natural frequency of the controller ω2 varies for two different cases in Tables 1 and 2. Table 1 shows that the excitation frequency Ω and the controller natural frequency ω2 that minimize the amplitude of the main system always deviate from each other when ζ=0.05,τ1+τ2=0.05. And with the increase of σ2 (i.e. ω2−ω1), the deviation Ω−ω2 becomes bigger and bigger. For example, for σ2=0.5 the minimum steady-state amplitude of the main system occurs at σ1=0.43. Table 2 shows that the excitation frequency Ω and the controller natural frequency ω2 that minimize the amplitude of the main system are almost equal to each other when ζ=0.001,τ1+τ2=0.05. It can be seen that the minimum amplitudes of the main system are almost zero when ζ=0.001. From these tables, we conclude that it is necessary to tune the controller’s natural frequency ω2 to the same value of the excitation frequency Ω only when the controller damping coefficient ζ is close to zero. Figure 16 shows time histories of the primary system and the controller for four different cases, i.e., ζ=0.05,Ω=ω2=3.36; ζ=0.05,Ω=3.31,ω2=3.36; ζ=0.001,Ω=ω2=3.36 and ζ=0.001,Ω=3.31,ω2=3.36. From Figure 16(a), it is observed that the amplitude of the main system when ζ=0.05,Ω=3.31,ω2=3.36 is smaller than that when ζ=0.05,Ω=ω2=3.36. It can be seen that the best condition for vibration reduction in the four cases is that ζ=0.001 and Ω=ω2. From Figure 16, we also observe that decreasing the controller damping coefficient ζ can better suppress the vibration of the main system. However, it can enhance the vibration of the controller.Table 1
Different minimum points of amplitudea1 when ω1=3.06, ζ=0.05, y0=0.07, and τ=τ1+τ2=0.05.
ω2ΩA1a12.562.560877.41×10−40.027222.662.658121.01×10−30.031722.762.75451.352×10−30.036772.862.849911.796×10−30.042372.962.94422.353×10−30.048513.063.037393.0360×10−30.0550993.163.129353.8429×10−30.0619913.263.220174.755×10−30.0689543.363.309985.7294×10−30.0756933.463.399066.7057×10−30.0818823.563.487717.6153×10−30.08727Table 2
Different minimum points of amplitudea1 when ω1=3.06, ζ=0.001, y0=0.07, and τ=τ1+τ2=0.05.
ω2ΩA1a12.562.563.58×10−75.98×10−42.662.664.86×10−76.98×10−42.762.766.53×10−78.08×10−42.862.868.68×10−79.31×10−42.962.959991.14×10−61.07×10−33.063.059991.49×10−61.22×10−33.163.159991.93×10−61.39×10−33.263.259982.47×10−61.57×10−33.363.359973.15×10−61.77×10−33.463.459963.98×10−61.99×10−33.563.559955.00×10−62.24×10−3Figure 16
Time history for the controlled system whenτ1=0.03,τ2=0.02,y0=0.07: 2800≤t≤3000,ζ=0.05,Ω=ω2=3.36; 3000≤t≤3200,ζ=0.05,Ω=3.31,ω2=3.36; 3200≤t≤3400,ζ=0.001,Ω=ω2=3.36; and 3400≤t≤3600,ζ=0.001,Ω=3.31,ω2=3.36: (a) the main system; (b) the controller.
(a)(b)In general, only when the controller damping coefficientζ approaches zero will the minimum main system amplitude occur at σ1=σ2. And in this case, the vibration of the main system almost vanishes.


## 6. Conclusion

In this paper, a time-delayed positive position feedback (PPF) controller has been investigated for the simultaneously resonance case (Ω=ω1,ω1=ω2) of a dynamical system. The integral iterative method is used to derive the second-order approximations and the amplitude equations. Then, the stability of the periodic solution is studied by utilizing a combination of Floquet theory and Hill’s determinant. The analysis revealed the following:(1)
The analytical results obtained by the integral iterative method correspond well with numerical simulations both quantitatively and qualitatively.(2)
The amplitude or stability of the controlled system (including the main system and the controller) does not depend on the individual time delayτ1,τ2, but on their summation τ1+τ2.(3)
The unavoidable total time delayτ1+τ2 in the control loop should not exceed 0.08. Otherwise, the controlled system may destabilize.(4)
Although the uncontrolled beam is nonlinear in form, it does not show nonlinear characteristics wheny0 varies in (0, 0.1).(5)
The appearance of time delay and near zero damping coefficientζ are two main factors to enhance the nonlinear characteristics of the controlled system. We can adjust different initial values to achieve the effect of vibration reduction in the region where multiple solutions exist.(6)
Only when the controller damping coefficientζ approaches zero will the minimum amplitude of the main system occur at Ω=ω2. That is to say, if the controller damping coefficient ζ is very close to zero, device should be designed to trace the excitation frequency and modifies the controller natural frequency adaptively to the same new value of the excitation frequency.


## 7. Comparison with Previously Published Work

In recent years, the vibration control for flexible beam structure has received much attention [2–5, 21]. First of all, the approximate solutions of the considered system in the above works were obtained by applying the multiple time scales perturbation technique and the stability of the steady-state solution is ascertained by using frequency-response equations. In [2], Li employed a PPF controller to reduce the high-amplitude vibration of the single mode of the flexible beam when subjected to primary resonance excitation. The author concluded that the PPF control scheme possessed a wide suppression bandwidth if the absorber’s frequency is properly tuned. Warminski et al. [3] used four different controllers to control the high-amplitude vibration of a flexible, geometrically nonlinear beam and showed that the positive position feedback controller and the nonlinear saturation controller were most effective. However, they only presented an analytical study for the NSC controlled system. Mitura et al. [4] utilized the PPF algorithm for vibration reduction of a horizontal beam near the primary resonance and a vertical beam near principal parametric resonance from a numerical point of view. The results showed that the vibration suppression effect of vertical beam was much better than that of horizontal beam. The PPF control method for the horizontal beam needed further improvement. In [5, 16], the authors investigated positive position feedback controllers for suppression of nonlinear beam vibration. They graphically inferred that the minimum amplitude of the main system occurred when the excitation frequency equaled the controller natural frequency.Based on the PPF controlled system in [4], we consider the effect of time delays on the controller performance in this paper. The approximate solutions are obtained by the integral iterative method and the stability of the steady-state solution is studied by utilizing a combination of Floquet theory and Hill’s determinant. The effects of various control parameters on time delay stability margins and the existence of multiple-solution region are investigated. The location of the minimum steady state main system amplitude is studied analytically. We found that only when the controller damping coefficient approaches zero will the minimum main system amplitude occur when the excitation frequency equaled the controller natural frequency.



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