Magnetoelastic Principal Parametric Resonance of a Rotating Electroconductive Circular Plate

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.


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-offreedom 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][9][10][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 stayedcable.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.

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 where   and   denote the displacement in the middle plane,  denotes the transverse displacement,  1 = −/ and V 1 = −(1/)(/) denote angular displacements, , , and  denote radial, circumferential and normal coordinates, respectively, and  denotes the time variable.Velocity components of rotating circular plate, derived from (1), are as follows: where / = / + Ω(/) and Ω denotes the rotation speed.The expression of the kinetic energy can be determined as The deformation potential caused by bending deformation can be determined as The potential energy of strain in the middle plane of the plate can be determined as where Applying the Hamilton principle, we get 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

Electromagnetic Force
The circular plate is affected by Lorentz force in the magnetic field.The electromagnetic force per unit volume of plate is where J =  0 (e + V × B), e denotes electric field intensity, V denotes the velocity vector of each internal point of the thin plate, and  0 denotes the electric conductance.Integrating (10) in the direction of the thickness, we obtain the expression of the electromagnetic forces and moments:

Principal Parametric Resonance of a Rotating Circular Plate in Magnetic Field
According to the spinning circular plate in transverse magnetic field (0, 0,  0 ) 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 where   = ℎ 3 /12(1 −  2 ) denotes the flexural rigidity,  denotes Young's modulus,   = ℎ/(1 −  2 ) denotes the tensile rigidity, and  denotes the Poisson coefficient.
Substituting ( 22)-( 23) into (21) and equating the coefficients of the same order of  in both sides, the following equations are obtained: where  0 = / 0 ,  1 = / 1 .The solution of (24) is Substituting ( 26) into (25), one may obtain where   = / 1 and cc is conjugate complex number in right side.The solvability conditions of (27) express that the coefficients of  i 0 must be vanished.Then (27) has nontrivial solution which gives To solve (28), ( 1 ) is written in the polar form:  where  and  are real numbers.Substituting (29) into (28) and separating the result into its real and imaginary parts, we obtain The frequency-response equation of the principal parametric resonance can be obtained by combining the equations of (30) when   = 0,   = 0: The stabilization of the steady-state solutions correspond to the stabilization of the singular points.As in (30), we let [20]  =  0 +  1 , where  1 and  1 are small compared with  0 and  0 .Substituting (32) into (30) and linearizing the resulting equations in  1 and  1 , we obtain According to the stability theory, the solution is stable if the real parts of characteristic values are negative in (33).Due to  > 0, the real parts of characteristic values are negative if and only if As this time, the steady-state motion of rotating system is stable.

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 density  = 2670 kg/m 3 , Young's modulus  = 71 GPa, the electric conductance  0 = 3.63×10 7 (Ω⋅m) −1 , and Poisson's ratio  = 0.34.In the figures, the solid curves denote stable solution and the dotted line denotes unstable solution.
Figure 3 shows the amplitude-frequency response curves ( 0 = 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  decreases with the sign of the detuning parameter  changing from negative to positive.Resonance phenomenon vanishes when the amplitude  changes to zero.We also notice the resonance region between the stable and unstable solutions increases with the increase of Ω 1 , Ω 0 , and radius  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 Ω 0 decreasing at the left of intersection and decreases with the parametric Ω 0 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 4 shows the characteristic curves of relationship between the amplitude and the magnetic induction intensity  0 (ℎ = 8 mm,  = 1 m, Ω 1 = 200 r/min).This figure shows that system has stable and unstable solutions.When the magnetic induction intensity  0 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 5 shows the characteristic curves of relationship between the amplitude  and the amplitude of velocity disturbance term Ω 1 with different magnetic induction intensity  0 (ℎ = 8 mm,  = 1 m).The figure shows that unstable nontrivial solutions exist when the detuning parameter  = −0.01,while stable nontrivial solutions exist when  = 0.The nonresonance region, called zero solution region, expands with the increase of  0 .Meanwhile, we can see that the distance between different curves decreases with the increase of Ω 1 .Figure 6 represents the characteristic curves of resonance amplitude-amplitude of velocity disturbance term with parameter Ω 0 (ℎ = 8 mm,  = 1 m).It shows that unstable nontrivial solutions also existed when the detuning parameter  = −0.01,while it only has stable nontrivial solutions when  = 0.The distance between different curves increases with the parametric Ω 1 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 7 shows the characteristic curves of resonance amplitude-amplitude of velocity disturbance term with different detuning parameter  (Ω 0 = 7500 r/min, ℎ = 8 mm,  = 1 m).From Figures 6(a 6(d) show that the stable solution exists when the detuning parameter  is positive, and the zero solution region broadens with the increase of detuning parameter .These properties have no transformation with the magnetic induction intensity varying.
Figure 8 is the phase trajectory chary with different detuning parameters.The trajectory movement direction is the direction of arrow ( 0 = 0.1 T, Ω 0 = 7500 r/min, Ω 1 = 400 r/min,  = 0.8 m, ℎ = 8 mm).There is only one stable solution when the detuning parameter  is close to zero, and the unstable solution occurs with the increase of  in negative direction.There only exists a stable focus  1 in Figure 8(a) whose stable solution is   1 = 0.008, which is written as  1 (  1 = 0.008); there is only one stable focus  1 (  1 = 0.018) and a saddle point  2 (  2 = 0.016) in Figure 8  a saddle point  2 (  2 = 0.019) in Figure 8(c).These stable solutions coincide with the amplitude characteristic curves in Figure 2.
Figure 9 is phase diagrams with different parameters.We get that the amplitude of the circular plate increases with the increase of the detuning parameter , magnetic induction intensity  0 , and the parametric Ω 1 and decreases with the increase of parametric Ω 0 .The solution coincides with the amplitude characteristic curves in Figure 2. The rotating system represents single period condition.The singular point is (0, 0).

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 with Ω 1 = 0; (c) the magnetic induction intensity can be chosen or adjusted so that the resonance amplitude becomes zero.

Figure 1 :
Figure 1: Sketch of the rotating circular plate.

Figure 7 :
Figure 7: The characteristic curves of resonance amplitude-amplitude of velocity disturbance term.

Figures 6 (
Figures 6(d), 6(e), and 6(f) represent the intersection of the stable and unstable solutions occurring on the axis of abscissa.Figures6(a) and 6(d) show that the stable solution exists when the detuning parameter  is positive, and the zero solution region broadens with the increase of detuning parameter .These properties have no transformation with the magnetic induction intensity varying.Figure8is the phase trajectory chary with different detuning parameters.The trajectory movement direction is the (b); there exist a stable focus  1 (  1 = 0.021) and
and   denote bending moments,   denotes torque,   and   denote curvatures,   denotes torsion,   ,   , and   denote inertial forces in the middle plane, and   ,   , and   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 .Virtual work done by the electromagnetic forces and moments   ,   ,   ,   , and   can be written as )