# Magnetoelastic Principal Parametric Resonance of a Rotating Electroconductive Circular Plate

Authors: Zhe Li; Yu-da Hu; Jing Li
Journal: Shock and Vibration
 (2017)

---

## Abstract

Nonlinear principal parametric resonance and stability are investigated for rotating circular plate subjected to parametric excitation resulting from the time-varying speed in the magnetic field. According to the conductive rotating thin circular plate in magnetic field, the magnetoelastic parametric vibration equations of a conductive rotating thin circular plate are deduced by the use of Hamilton principle with the expressions of kinetic energy and strain energy. The axisymmetric parameter vibration differential equation of the variable-velocity rotating circular plate is obtained through the application of Galerkin integral method. Then, the method of multiple scales is applied to derive the nonlinear principal parametric resonance amplitude-frequency equation. The stability and the critical condition of stability of the plate are discussed. The influences of detuning parameter, rotation rate, and magnetic induction intensity are investigated on the principal parametric resonance behavior. The result shows that stable and unstable solutions exist when detuning parameter is negative, and the resonance amplitude can be weakened by changing the magnetic induction intensity.


---

## Body

## 1. Introduction

An annular or discoid rotating plate member application is widely used in aerospace, large generating units, and mechanical engineering. The change of the rotation speed in complicated circumstance which includes electric, magnetic, and force will cause large amplitude vibration and lead to the loss of stability, even damage the rotatable structures. Hence, it is important to analyze the dynamic response of varying parameters in rotatable system. The dynamic response of a nonlinear system is investigated in [1, 2]. The decomposition and study of the equations of motion proposed are beneficial in the analysis of vibration. The nonlinear response of a base-excited slender beam carrying a lumped mass subjected to principal parametric resonance is investigated. The steady-state response and stability of the system by the method of multiple scales are obtained as discussed by Dwivedy and Kar [3]. EL-Bassiouny [4] investigated the principal parametric resonance of a single-degree-of-freedom system with nonlinear two-frequency parametric and self-excitations and analyzed the influences of damping, nonlinear excitation, and self-excitation. Feng et al. [5] studied principal parametric resonances of a slender cantilever beam subjected to random excitation. Nonlinear vibration equation and many valuable conclusions were given. Feng and Hu [6] investigated principal parametric and internal resonance of flexible beams. By giving nonlinear modulation equations of combined vibration, they obtained the solutions and considered the problem of stability. Hashemi et al. [7] analyzed natural frequency and free vibration of rotating plates under different parameters. Hu et al. [8–11] investigated the problems of vibration and resonance of rectangular or circular thin plate in magnetic field. They discussed the influences of parameters on stability of a system. Özhan and Pakdemirli [12] considered the primary parametric resonance of the parametric excitation of nonlinear continuous system. They used the method of multiple scales to find an approximate analytical solution and discussed steady state solutions and their stability. Shahgholi and Khadem [13] studied the primary and parametric resonances of rotating asymmetrical shaft. They showed the influences of principal axes and external damping on the steady state response of the asymmetrical rotating shaft. The parametric vibrations of orthotropic plates with complex forms for different types of boundary conditions are studied with a novel approach in [14]. Awrejcewicz et al. [15] studied the chaotic vibrations of flexible plates of infinite length by the use of fast Fourier transforms and wavelets and illustrated the advantages of wavelet-oriented analysis of nonlinear vibrations of mechanical systems in [16]. Tang and Chen [17] studied the influences of variation speed of axially moving plates on stability. They calculated and analyzed natural frequency and modal under the method of multiple scales. Zhao et al. [18] investigated the stability of principal parametric resonance of stayed-cable. They obtained the nonlinear equations of vibration under axial excitation with the method of multiple scales and presented numerical simulation. Zhou et al. [19] investigated the dynamic behavior of magneto-elastic-plastic interaction and coupling for beam-plates. They showed the influences of systemic parameters on plastic deformation.In this study, the magnetoelastic axisymmetric parametric vibration equations of a conductive rotating thin circular plate are obtained. The problems of principal parametric resonance and stability of the variable-velocity rotating circular plate in transverse magnetic field are investigated, and amplitude-parameter curves are obtained for analyzing the influences of parameters on amplitude.


## 2. The Magnetoelastic Vibration Equations of the Rotating Circular Plate

The applied computational scheme is based on the Kirchhoff hypotheses and is treated as the model of first approximation. It is sufficient for engineering oriented analysis as it has been pointed out in [15]. According to the spinning circular thin plate in the magnetic field (Figure 1), the displacement vectors of each internal point of the circular plate in three directions in cylindrical coordinates are assumed as(1)ur1r,θ,z,t=urr,θ,t+zu1r,θ,t,uθ1r,θ,z,t=uθr,θ,t+zv1r,θ,t,uzr,θ,z,t=wr,θ,t,where ur and uθ denote the displacement in the middle plane, w denotes the transverse displacement, u1=-∂w/∂r and v1=-1/r∂w/∂θ denote angular displacements, r, θ, and z denote radial, circumferential and normal coordinates, respectively, and t denotes the time variable.Figure 1
Sketch of the rotating circular plate.Velocity components of rotating circular plate, derived from (1), are as follows:(2)Vr=dur1dt-Ωuθ1,Vθ=Ωr+ur1+duθ1dt,Vz=dwdt,where d/dt=∂/∂t+Ω∂/∂θ and Ω denotes the rotation speed. The expression of the kinetic energy can be determined as(3)Tk=12ρ∫-h/2h/2∫0R∫02πdur1dt-Ωuθ12+Ωr+ur1+duθ1dt2+dwdt2rdθdrdz.The deformation potential caused by bending deformation can be determined as(4)Uε1=12∫0R∫02πMrκr+Mθκθ+2Mrθκrθrdrdθ.The potential energy of strain in the middle plane of the plate can be determined as(5)Uε2=12∫0R∫02πNrεr+Nθεθ+Nrθγrθrdrdθ,where Mr and Mθ denote bending moments, Mrθ denotes torque, κr and κθ denote curvatures, κrθ denotes torsion, Nr, Nθ, and Nrθ denote inertial forces in the middle plane, and εr, εθ, and γrθ denote strains in the middle plane. It is assumed that there is transversal microdisplacement which is allowed by the displacement boundary condition, namely, virtual displacement δw. Virtual work done by the electromagnetic forces and moments Fr, Fθ, Fz, mr, and mθ can be written as(6)δUF=∫0R∫02πFrδur+Fθδuθ+Fzδw+mrδu1+mθδv1rdrdθ.Applying the Hamilton principle, we get(7)∫t0t1δTk-δUε1-δUε2+δUFdt=0.Substituting (3)–(6) into (7) and considering axisymmetric vibration problem, one can compute the magnetoelastic axisymmetric vibration equation of spinning circular thin plate, which gives(8)∂Nr∂r+Nr-Nθr+Fr=d2urdt2-Ω2r+urρh,(9)1r∂2rMr∂r2-∂Mθ∂r+∂∂rrNr∂w∂r+1r∂rmr∂r+Fz=ρhd2wdt2-ρh3121rd2dt2∂w∂r+d2dt2∂2w∂r2-Ω21r∂w∂r+∂2w∂r2.


## 3. Electromagnetic Force

The circular plate is affected by Lorentz force in the magnetic field. The electromagnetic force per unit volume of plate is(10)ffr,fθ,fz=J×B=fri+fθj+fzk,where J=σ0e+V×B, e denotes electric field intensity, V denotes the velocity vector of each internal point of the thin plate, and σ0 denotes the electric conductance.Integrating (10) in the direction of the thickness, we obtain the expression of the electromagnetic forces and moments:(11)Fr=∫-h/2h/2frdz=σ0eθ+∂w∂tB0r-durdtB0zB0z-σ0ez+durdtB0θ-Ωr+ur+durdtB0rB0θ,Fz=∫-h/2h/2fzdz=σ0Ωr+urB0z-∂w∂tB0θB0θ-σ0eθ+∂w∂t+Ω∂w∂tB0r-durdtB0zB0r,mr=∫-h/2h/2frzdz=σ0h312B0z2∂2w∂t∂r+σ0h312B0θ2∂2w∂t∂r-Ωr∂w∂r-σ0h312B0rB0θΩ∂w∂r.


## 4. Principal Parametric Resonance of a Rotating Circular Plate in Magnetic Field

According to the spinning circular plate in transverse magnetic field (0,0,B0z) and substituting the expressions of bending moment, internal force, electromagnetic force, and electromagnetic torque into (9), the axisymmetric oscillation differential equation of the spinning circular plate in transverse magnetic field is obtained as(12)-DM∂4w∂r4+2r∂3w∂r3-1r2∂2w∂r2+1r3∂w∂r+12rDN∂w∂r3+32DN∂w∂r2∂2w∂r2+σ0h312B0z21r∂2w∂r∂t+∂3w∂r2∂t=ρh∂2w∂t2-ρh3121r∂3w∂r∂t2+∂4w∂r2∂t2-Ω21r∂w∂r+∂2w∂r2,where DM=Eh3/121-μ2 denotes the flexural rigidity, E denotes Young’s modulus, DN=Eh/1-μ2 denotes the tensile rigidity, and μ denotes the Poisson coefficient.We assume that an approximation to the solution ofw(r,t) can be expressed as(13)wr,θ,t=WrTtcos⁡nθ=∑n=0rn·∑v=0,1,2,…CvrR2vTntcos⁡nθ,where R denotes the radius of the plate. The coefficients denote C0=1, C1=-2, C2=1 for fixed supported conditions and C0=1, C1=-(6+2μ)/(5+μ), C2=(1+μ)/(5+μ) for simply supported conditions and we let n=1.Substituting (13) into (12) and using Galerkin method, the principal parametric resonance differential equation is obtained (n=1,2):(14)A1nT¨0t+B1nT¨1t+A2nT˙0t+B2nT˙1t+A3n+A4nΩ2T0t-B3n+B4nΩ2T1t+A5nT03t+B5nT13t+A6nT02tT1t+B6nT0tT12t=0,where(15)A1n=∫0R∫02πρh-ρh33C1R2+ρhC1R2-4ρh33C2R4r2+ρhC2R4r4Wcosn-1⁡θrndθdr,B1n=∫0R∫02πρh-2ρh33C1R2r+ρhC1R2-2ρh3C2R4r3+ρhC2R4r5Wcosn⁡θrndθdr,A2n=-∫0R∫02πσ0h312B0z24C1R2+16C2R4r2Wcosn-1⁡θrndθdrB2n=-∫0R∫02πσ0h312B0z28C1R2r+24C2R4r3Wcosn⁡θrndθdrA3n=∫0R∫02π64DMC2R4Wcosn-1⁡θrndθdrB3n=∫0R∫02πDM1r3+192C2R4rWcosn⁡θrndθdrA4n=∫0R∫02πρh3124C1R2+16C2R4Wcosn-1⁡θrndθdrB4n=∫0R∫02πρh3128C1R2r+24C2R4r3Wcosn⁡θrndθdrA5n=-∫0R∫02πDN12r2C1R2r+4C2R4r33+322C1R2r+4C2R4r322C1R2+12C2R4r2Wcosn-1⁡θrndθdrB5n=-∫0R∫02πDN12r1+3C1R2r2+5C2R4r43cos3θ+321+3C1R2r2+5C2R4r46C1R2r+20C2R4r3cos3θ-12r21+3C1R2r2+5C2R4r42r+C1R2r3+C2R4r5·cos3θ+2r2r+C1R2r3+C2R4r51+3C1R2r2+5C2R4r42sin2θcosθ-12r3r+C1R2r3+C2R4r52.1+3C1R2r2+5C2R4r4sin2θcosθ+32r4r+C1R2r3+C2R4r53sin2θcosθ-12r26C1R2r+20C2R4r31+3C1R2r2+5C2R4r42·sin2θcosθWcosnθrndθdrA6n=-∫0R∫02πDN32r2C1R2r+4C2R4r321+3C1R2r2+5C2R4r4cosθ+322C1R2r+4C2R4r32·6C1R2r+20C2R4r3+22C1R2r+4C2R4r31+3C1R2r2+5C2R4r42C1R2+12C2R4r2cosθ-12r2r+C1R2r3+C2R4r52C1R2r+4C2R4r32·cosθWcosn-1θrndθdrB6n=-∫0R∫02πDN32r2C1R2r+4C2R4r321+3C1R2r2+5C2R4r42cos2θ+2r22C1R2r+4C2R4r3·r+C1R2r3+C2R4r51+3C1R2r2+5C2R4r4sin2θ+321+3C1R2r2+5C2R4r422C1R2+12C2R4r2+22C1R2r+4C2R4r36C1R2r+20C2R4r3·1+3C1R2r2+5C2R4r4cos2θ-12r3r+C1R2r3+C2R4r52r+C1R2r3+C2R4r5·sin2θ-1r21+3C1R2r2+5C2R4r4·r+C1R2r3+C2R4r52C1R2r+4C2R4r3cos2θ+12r22C1R2+12C2R4r21+C1R2r2+C2R4r42·sin2θWcosnθrndθdrW=1+C1R2r2+C2R4r4.The time history diagrams of principal parametric resonance aboutT0t and T1t are given in Figure 2 (h=8 mm, R=0.8 m, Ω1=400 r/min, B0z=0.1 T, Ω0=7500 r/min). This figure shows that under the same conditions the first-order amplitude T0t is greater than the second-order amplitude T1t. In order to simplify the calculation, we only analyze first-order amplitude:(16)A1T¨0t+A2T˙0t+A3+A4Ω2T0t-A5T03t=0,where(17)A1=ρh2R2+ρh2R2-ρh36C1+ρh3R2-ρh33C2+ρh6R2-ρh312C12+ρh10R2-ρh36C22+ρh4R2-5ρh318C1C2,A2=-σ0h312B0z22C1+4C2+C12+2C22+103C1C2,A3=DMR2323C22+16C1C2+32C2,A4=ρh3122C1+4C2+C12+2C22+103C1C2,A5=DNR283C14+4C13+20C13C2+2645C12C22+24C12C2+48C1C22+1763C1C23+32C23+1607C24.Figure 2
Time history diagram.We assume(18)Ω=Ω0+Ω1cos⁡ω1t,where Ω0 denotes a constant term, Ω1 denotes the constant amplitude of disturbance term (the cosinusoid variation of the angular velocity), and ω1 denotes the frequency of disturbance term. The order of Ω1 is less than the order of Ω0.Substituting (18) into (16), one can obtain(19)T¨0t+η1T˙0t+η2+η3Ω02+12Ω12T0t+2Ω0Ω1η3cos⁡ω1t+Ω122η3cos⁡2ω1tT0t-η4T03t=0,where η1=A2/A1, η2=A3/A1, η3=A4/A1, and η4=A5/A1.For simplification, we letω1t=2τ, qτ=T0(τ)/h and obtain(20)q¨+2ξ~q˙+ωn2q+h~1cos⁡2τ+h~2cos⁡4τq-h~3q3=0,where ξ~=η1/ω1, h~1=8Ω0Ω1η3/ω12, h~2=2Ω12η3/ω12, h~3=4h2η4/ω12, and ωn2=4η2+4η3Ω02+1/2Ω12/ω12.We introduce a small parameterε. Equation (20) can be expressed as(21)q¨+2εξq˙+ωn2q+εh1cos⁡2τ+εh2cos⁡4τq-εh3q3=0,where ξ=ξ~/ε, h1=h~1/ε, h2=h~2/ε, and h3=h~3/ε.We assume(22)ωn=1+εσ,where σ denotes the detuning parameter. By the use of the method of multiple scales [20], the solution of (21) in terms of different time scales can be expressed as(23)qτ,ε=q0T0,T1+εq1T0,T1.Substituting (22)-(23) into (21) and equating the coefficients of the same order of ε in both sides, the following equations are obtained:(24)D02q0+q0=0,(25)D02q1+q1=-2D0D1q0-2ξD0q0-2σq0-h1cos⁡2τ+h2cos⁡4τq0+h3q03,where D0=∂/∂T0, D1=∂/∂T1. The solution of (24) is(26)q0=AT1eiT0+A-T1e-iT0.Substituting (26) into (25), one may obtain(27)D02q1+q1=3h3A2A--2iA′-2iξA-2σAeiT0+h3A3e3iT0-h12Ae3iT0+A-eiT0-h22Ae5iT0+A-e3iT0+cc,where A′=∂A/∂T1 and cc is conjugate complex number in right side.The solvability conditions of (27) express that the coefficients of eiT0 must be vanished. Then (27) has nontrivial solution which gives(28)3h3A2A--2iA′-2iξA-2σA-h12A-=0.To solve (28), AT1 is written in the polar form:(29)A=12aT1eiφT1,where a and φ are real numbers. Substituting (29) into (28) and separating the result into its real and imaginary parts, we obtain(30)a′=-ξa+ah14sin⁡2φ,φ′=σ-3a2h38+h14cos⁡2φ.The frequency-response equation of the principal parametric resonance can be obtained by combining the equations of (30) when a′=0, φ′=0:(31)ξ2+σ-3a2h382=h142.The stabilization of the steady-state solutions correspond to the stabilization of the singular points. As in (30), we let [20](32)a=a0+a1,φ=φ0+φ1,where a1 and φ1 are small compared with a0 and φ0. Substituting (32) into (30) and linearizing the resulting equations in a1 and φ1, we obtain(33)a1′=-ξa1+23h38a03-σa0φ1,φ1′=-3h3a04a1-2ξφ1.According to the stability theory, the solution is stable if the real parts of characteristic values are negative in (33). Due to ξ>0, the real parts of characteristic values are negative if and only if(34)3h32a023h38a02-σ>0.As this time, the steady-state motion of rotating system is stable.


## 5. Numerical Study

For numerical study of nonlinear primary parametric resonances of rotating circular plate with simply supported conditions, the circular plate with the following parameters is used: the mass densityρ=2670 kg/m3, Young’s modulus E=71 GPa, the electric conductance σ0=3.63×107Ω·m-1, and Poisson’s ratio μ=0.34. In the figures, the solid curves denote stable solution and the dotted line denotes unstable solution.Figure3 shows the amplitude-frequency response curves (B0z=0.1⁡ T). In order to make curves clear, the detuning parameters and amplitude values are small. It is seen that the value of amplitude a decreases with the sign of the detuning parameter εσ changing from negative to positive. Resonance phenomenon vanishes when the amplitude a changes to zero. We also notice the resonance region between the stable and unstable solutions increases with the increase of Ω1, Ω0, and radius R and the decrease of thickness h. There is an intersection in Figure 3(a), and different regularities exist in both sides of this intersection. The stable amplitude increases with the parametric Ω0 decreasing at the left of intersection and decreases with the parametric Ω0 decreasing at the right of intersection. According to numerical conversion, we compare the first-order displacement amplitude (Figure 2) which is obtained by numerical calculation with amplitude frequency diagram (Figure 3(b)) which is obtained by amplitude frequency response equation of steady motion with given parameters and find that they are identical. It shows that the rationality of solving the resonance solution with the first order approximate expression of the multiscale method is confirmed.Figure 3
Amplitude-frequency response curves.
(a)
h = 8 mm, R=0.8 m, Ω1=400 r/min (b)
R = 0.8 m, Ω0=7500 r/min, Ω1=400 r/min (c)
R = 0.8 m, Ω0=7500 r/min, h=8 mm (d)
Ω 1 = 400 r/min, Ω0=7500 r/min, h=8 mmFigure4 shows the characteristic curves of relationship between the amplitude and the magnetic induction intensity B0z (h=8 mm, R=1⁡ m, Ω1=200 r/min). This figure shows that system has stable and unstable solutions. When the magnetic induction intensity B0z is small, the change of amplitude is insensitive. From Figure 3(b), we can see that the magnetic induction intensity remains constant or substantially constant with the variation of the detuning parameter when bifurcation emerges. However, saddle-node bifurcation exists and resonance vanishes when the magnetic induction intensity increases to a critical value. It means that we can attenuate the resonance phenomenon of rotating plate by changing the magnetic field intensity while other parameters are constant.Figure 4
The characteristic curves of amplitude-magnetic induction intensity.
(a)
ε σ = - 0.03 (b)
Ω 0 = 7500 r/minFigure5 shows the characteristic curves of relationship between the amplitude a and the amplitude of velocity disturbance term Ω1 with different magnetic induction intensity B0z (h=8 mm, R=1 m). The figure shows that unstable nontrivial solutions exist when the detuning parameter εσ=-0.01, while stable nontrivial solutions exist when εσ=0. The nonresonance region, called zero solution region, expands with the increase of B0z. Meanwhile, we can see that the distance between different curves decreases with the increase of Ω1.Figure 5
The characteristic curves of resonance amplitude-amplitude of velocity disturbance term.
(a)
ε σ = 0, Ω0=7500 r/min (b)
ε σ = 0, Ω0=9000 r/min (c)
ε σ = 0, Ω0=10500 r/min (d)
ε σ = - 0.01, Ω0=7500 r/min (e)
ε σ = - 0.01, Ω0=9000 r/min (f)
ε σ = - 0.01, Ω0=10500 r/minFigure6 represents the characteristic curves of resonance amplitude-amplitude of velocity disturbance term with parameter Ω0 (h=8 mm, R=1 m). It shows that unstable nontrivial solutions also existed when the detuning parameter εσ=-0.01, while it only has stable nontrivial solutions when εσ=0. The distance between different curves increases with the parametric Ω1 increasing. From Figures 4 and 5, we note that the curves show symmetrical distribution because the direction of the time-varying speed is different. Figure 6 is also shows the similar property.Figure 6
The characteristic curves of resonance amplitude-amplitude of velocity disturbance term.
(a)
ε σ = 0, B0z=0 T (b)
ε σ = 0, B0z=0.5 T (c)
ε σ = 0, B0z=1 T (d)
ε σ = - 0.01, B0z=0 T (e)
ε σ = - 0.01, B0z=0.5 T (f)
ε σ = - 0.01, B0z=1 TFigure7 shows the characteristic curves of resonance amplitude-amplitude of velocity disturbance term with different detuning parameter εσ (Ω0=7500 r/min, h=8 mm, R=1⁡ m). From Figures 6(a), 6(b), and 6(c) we note that the unstable nontrivial solution occurs and increases when detuning parameter εσ increases along the negative direction. Figures 6(d), 6(e), and 6(f) represent the intersection of the stable and unstable solutions occurring on the axis of abscissa. Figures 6(a) and 6(d) show that the stable solution exists when the detuning parameter εσ is positive, and the zero solution region broadens with the increase of detuning parameter εσ. These properties have no transformation with the magnetic induction intensity varying.Figure 7
The characteristic curves of resonance amplitude-amplitude of velocity disturbance term.
(a)
B 0 z = 0 T (b)
B 0 z = 0.5 T (c)
B 0 z = 1 T (d)
B 0 z = 0 T (e)
B 0 z = 0.5 T (f)
B 0 z = 1 TFigure8 is the phase trajectory chary with different detuning parameters. The trajectory movement direction is the direction of arrow (B0z=0.1 T, Ω0=7500 r/min, Ω1=400 r/min, R=0.8 m, h=8 mm). There is only one stable solution when the detuning parameter εσ is close to zero, and the unstable solution occurs with the increase of εσ in negative direction. There only exists a stable focus S1 in Figure 8(a) whose stable solution is aS1=0.008, which is written as S1(aS1=0.008); there is only one stable focus S1(aS1=0.018) and a saddle point S2(aS2=0.016) in Figure 8(b); there exist a stable focus S1(aS1=0.021) and a saddle point S2(aS2=0.019) in Figure 8(c). These stable solutions coincide with the amplitude characteristic curves in Figure 2.Figure 8
The phase plane curves.
(a)
ε σ = 0.00003 (b)
ε σ = - 0.0003 (c)
ε σ = - 0.0004Figure9 is phase diagrams with different parameters. We get that the amplitude of the circular plate increases with the increase of the detuning parameter εσ, magnetic induction intensity B0z, and the parametric Ω1 and decreases with the increase of parametric Ω0. The solution coincides with the amplitude characteristic curves in Figure 2. The rotating system represents single period condition. The singular point is (0,0).Figure 9
The phase portrait.
(a)
Ω 0 = 7500 r/min, Ω1=400 r/min, εσ=0.0008 (b)
Ω 1 = 400 r/min, Ω0=7500 r/min, B0z=0.1 T (c)
Ω 1 = 400 r/min, B0z=0.1 T, εσ=0.0008 (d)
Ω 0 = 7500 r/min, B0z=0.1 T, εσ=0.0008


## 6. Conclusions

Nonlinear principal parametric resonance and stability for rotating circular plate subjected to parametric excitation resulting from the time-varying speed in the magnetic field are investigated. The influence of rotating velocity, the detuning parameter, the magnetic induction intensity, thickness, and radius on the principal parametric resonance is studied.(1)
The magnetoelastic parameter vibration equation of rotating electroconductive circular plate in the magnetic field is derived.(2)
According to the numerical examples, the following results can be concluded: (a) the resonance amplitude presents different change regularities with variation of rotating velocity, the detuning parameter, and the magnetic induction intensity; (b) there are unstable and stable solutions and they are symmetrical withΩ1=0; (c) the magnetic induction intensity can be chosen or adjusted so that the resonance amplitude becomes zero.



---

*Source: 5196847-2017-02-14.xml*