Bifurcation of a Microelectromechanical Nonlinear Coupling System with Delay Feedback

The dynamics of a kind of electromechanical coupling deformable micromirror device torsion micromirror with delay are investigated. Based on the distribution of eigenvalues, we prove that a sequence of Hopf bifurcation occurs at the equilibrium as the delay increases and obtain the critical values of Hopf bifurcation. Explicit algorithms for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived, using the theories of normal form and center manifold.


Introduction
The microelectromechanical systems (MEMS) represent a very important class of systems having applications in all fields.Because of the nonlinearity and the complexity, modeling and dynamics of MEMS have strongly attracted people's attention [1][2][3][4].MEMS often involve the nonlinear coupling of electrostatic and mechanical physical fields in engineering, so the dynamic characters are complicated.For example, DMD (deformable micromirror device) torsion micromirror, which is widely used in optical communication, optical computing, projection display, and high definition television, is a new type of MEMS based spatial light modulator.The principle is to adjust the distribution of light through controlling the signal, that is, to make the torsional displacement using the electrostatic interaction between the poles so that the spatial light can be modulated.Initially, the scholars usually focus on the static characters of DMD torsion micromirror [5][6][7].With the development of microsystem dynamics, there has been some work related to its dynamic characters [8][9][10].
Our paper is organized as follows.In Section 2, the structure of a kind of electromechanical coupling DMD torsion micromirror and its kinetic equation are given.In Section 3, the previous results about the ordinary differential equation (ODE) are listed.In Section 4, the existence and the critical values of Hopf bifurcation are obtained.In Section 5, the normal form method and the center manifold theory are used to analyze the properties of Hopf bifurcation.In Section 6, we summarize our results.

Description of Model
The electromechanical coupling structure of DMD torsion micromirror is shown in Figure 1(a).Using angular displacement  as generalized coordinate (see Figure 1(b)), the kinetic equation of torsion vibration system can be formed as follows: where  is rotary inertia,   is damping,   is torsional stiffness,  is turning angle, and  is the external driving moment of the system.Based on a series of calculation like [8,9] and setting  = / max = (/2)/ 0 = /(2 0 ), (1) can be changed into where  is the voltage imposed on the electrode,  is a dielectric constant of air,  0 is the initial distance between two electrodes, and  max is the maximum turning angle as the edge of micromirror plane meets the plate electrode.Introduce the dimensionless time  =  0  with  0 =   /.Let  =   / 0 and  =  3 /16   3 0 .Then (2) can be rewritten as where  = / max ,  is a damping coefficient,  is an excitation voltage, and  is a coefficient determined by geometric parameters of the micromirror system.Equation ( 3) is thus the dimensionless kinetic equation of a simplified physical model of a kind of electromechanical coupling structure of DMD torsion micromirror.

Hopf Bifurcation of System with Delay
Introducing linear time delay feedback for (4), we have where  ≥ 0 is the time delay and  > 0 is the coefficient of feedback gain.
Then  = ( * , 0) is also an equilibrium point of system (5).Now we begin to consider the stability of the system at the point .The characteristic equation of its corresponding linear system around  is where Refer to the eigenvalue analysis in [11] and when  = 0, (6) has roots  1,2 = (− ± √ 2 − 4( − ))/2.
(1) If () is not satisfied, then all roots of (6) have negative real parts for any  ≥ 0.
The proof is complete.

Using the lemmas above, we have Theorem 8.
Theorem 8. Suppose that  >  is satisfied.

Conclusion
In this paper, we investigate the Hopf bifurcation of a kind of electromechanical coupling DMD torsion micromirror with delay feedback.Using the normal form method for functional differential equations (FDEs) and the center manifold theory in [13], we have obtained the properties of Hopf bifurcation.Because the microscale exists, the pull-in phenomena may be caused between microdevices based on the static electricity effect [14].Electrostatic pull-in is a very important character of MEMS dynamics, and it has significant impact on electrostatic driven microstructure design.For example, the pull-in phenomena should be avoided to prevent leakage of driving mode of energy while designing the microcomb drive [15,16]; we need to use the pull-in character to control the structure of switch for the design of the microswitch [17].Pull-in instability is a common problem for microstructural vibration, and it limits the operating range (deformation and voltage) to reduce the safety and reliability of microsystem.So we hope to avoid the phenomenon whether to utilize it or not in the design.From the dynamic perspective, the pull-in between microdevices is corresponding to generalized instability of nonlinear system; that is, the amplitude of vibration system exceeds a certain threshold, and the escape phenomenon occurs [18].Our results indicate that the time delay feedback can make the equilibrium or the periodic solution be stable; thus the pull-in phenomenon is restrained well.

2 JournalFigure 1 :
Figure 1: The electromechanical coupling structure of DMD torsion micromirror and the simple scheme for computation.