#### 3.1. Equations of Motion

In this section, we investigate the dynamic behavior of the mirror for different excitation voltages. Our mirror is excited by AC and DC combined loads.

${V}_{bias}$ in static simulation is replaced by

${V}_{DC}$ voltage. The governing equation of motion can be obtained using Lagrange’s equation. Lagrange’s equation may be written:

In this equation,

${q}_{i}$ is a generalized coordinate,

V is the potential energy,

T is kinetic energy,

D is the Rayleigh dissipation function that depends on viscous damping and Γ is the torque applied to the system from electrostatic actuation. The rotational motion of the mirror is described by two generalized coordinates of

α and

β. The potential energy of the mirror is given by Equation (

22):

where

${K}_{x}$ and

${K}_{y}$ are the torsional stiffness of serpentine for

α and

β. Furthermore,

${K}_{z}$ is the bending stiffness of the serpentine for vertical motion. The change in height of the center of mass is significantly small, so the potential energy of the mirror plate and gimbal frame can be neglected. The total kinetic energy equation is given as:

where

ρ is the density of mirror material,

${J}_{p}$ and

${J}_{g}$ are the mass moment inertia of the mirror plate and gimbal,

${l}_{m}$ and

${t}_{m}$ are length and thickness of the mirror plate, respectively,

${m}_{p}$ and

${m}_{g}$ are the masses of the plate and gimbal frame, respectively, and

${L}_{gw}$,

${L}_{gl}$,

${L}_{gwi}$,

${L}_{gi}$ and

${t}_{g}$ are the mirror outer width, gimbal outer length, mirror inner width, gimbal inner length and thickness of gimbal frame, respectively. The last parameter for Lagrange Equation (

21) is energy dissipation.

where

${d}_{1}$ and

${d}_{2}$ represent damping coefficients for rotations about

x and

y, respectively. Substituting the relevant terms into Equation (

21), we obtain the equations of motion for

α and

β:

where

$2{\zeta}_{1}{\omega}_{\alpha}={\displaystyle \frac{{d}_{1}}{{J}_{p}+{J}_{g}}}$,

$2{\zeta}_{2}{\omega}_{\beta}={\displaystyle \frac{{d}_{2}}{{J}_{p}}}$,

$2{\zeta}_{2}{\omega}_{\mathrm{z}}={\displaystyle \frac{{d}_{2}}{{m}_{p}+{m}_{g}}}$ and natural frequencies about the

X-,

Y- and

Z-axis are

${\omega}_{\alpha}^{2}$=

$\frac{{K}_{x}}{{J}_{p}+{J}_{g}}$,

${\omega}_{\beta}^{2}$=

$\frac{{K}_{y}}{{J}_{p}}$,

${\omega}_{\mathrm{z}}^{2}$=

$\frac{{K}_{z}}{{m}_{p}+{m}_{g}}$. The state space equation can be written:

#### 3.2. Dynamic Simulation Results

The transient response and frequency response of the dynamic behavior were simulated using the Runge–Kutta numerical integration method assuming zero initial conditions. Comparing to the experimental data from [

19], we estimated the damping ratios as

$[{\zeta}_{1};\phantom{\rule{2.84526pt}{0ex}}{\zeta}_{2};\phantom{\rule{2.84526pt}{0ex}}{\zeta}_{3}]=[0.0382;\phantom{\rule{2.84526pt}{0ex}}0.0208;\phantom{\rule{2.84526pt}{0ex}}0.0208]$ for

α and

β scanning angles, respectively. The damping ratios about two axes of rotations are different, as the corresponding resonance frequencies are not the same. Furthermore, note that because of the geometrical constraints, the maximum reachable

α angle is around 19 degrees and the

β angle is around 35 degrees, since the gimbal and the mirror plate hit the substrate, respectively. The simulations presented below are within these maximum angle ranges. In

Figure 6a, the frequency responses of the scanning angles at

${V}_{bias}$= 55 V are presented.

Figure 6b demonstrates the frequency response of the vertical displacement at

${V}_{bias}$= 55 V. The numerical simulation for scanning angles shows good accuracy with the experimental results. These figures also show that for both the

α and

β angles, superharmonic resonances at orders of two are observed, as reported in the experimental data. It should be noted that secondary resonances have not been reported for the mirror with sidewall electrodes in the literature. There is a slight difference between experiments and simulation that can be explained from the deviation of nominal dimensions after fabrication. The authors did not have access to the actual device to measure the exact dimensions under an optical profiler to examine the variation of thickness across the mirror plate or the gimbal frame. These variations are responsible for the mismatch of the resonant frequencies.

#### 3.3. Analytical Explanation of Secondary Resonances

Close agreement between experimental results and dynamic simulation confirmed accurate modeling of the electrostatic field for the rotational micro-mirror. At low voltages, superharmonic resonance showed the stiffness nonlinearity in the system. In this section, we examine the effect of increasing forcing and decreasing damping on primary and secondary resonances, then describe the underlying nonlinearities from driven mathematical equations of motion. The primary resonance happens at the frequency close to the natural frequency about each axis of rotation, and secondary resonance, such as superharmonic resonance, appears at a frequency away from the natural frequency when nonlinear stiffness terms are present [

23,

24]. Secondary resonances arise from nonlinear coupling, restoring force that is caused by the electrostatic force on the mirror. The secondary resonances are activated at the high forcing and low damping (low pressure environment).

The cubic stiffness nonlinearity causes the subharmonic resonance of order 1/3 and superharmonic resonance of order three. On the other hand, quadratic nonlinearity triggers subharmonic resonance of order 1/2 and superharmonic resonance of order two. Subharmonic resonances are secondary resonances as a result of nonlinear spring forces, which generate large responses at a fraction of the excitation frequency. For instance, when the excitation frequency is $N\Omega $, the system responds at Ω, where Ω is the natural frequency of the system. This means that this system has subharmonic resonances at the order of $1/N$, where N is an integer greater than zero.

Superharmonic resonances are large responses at integer multiples of the excitation frequency. For instance, when the excitation frequency is

$\frac{\Omega}{N}$, the system responds at Ω, where Ω is the natural frequency of the system. In this case, we deduce that the system has superharmonic resonance of order

N.

Figure 7a,b shows the frequency response as the AC and DC voltages are increased, respectively. The figures reveal the appearance of primary resonance (around 420 Hz) and two superharmonic resonances of order two (around 210 Hz) and order three (around 140 Hz), which increase with the increasing the voltage. One can deduce that by increasing the voltage, the primary resonance peak inclines to the left (softening), as in

Figure 7a,b. We expect that the frequency peak could considerably bend to the left with a further increase of voltage. However, because of the physical limitation on the rotation angle (35 degrees), a further increase of the voltage was not meaningful. It is noted that the effect of AC voltage on the superharmonic resonance of order two was more prominent than that of the DC voltage. The results indicate that there are nonlinear stiffness terms in the system with a dominant quadratic nonlinearity, as the superharmonic resonance of order two is the prominent secondary resonance.

To describe the quadratic nonlinearity in the system, which is the cause of dominant secondary superharmonic resonance seen in the system, we scrutinize the mathematical equation of motion of the mirror, looking for nonlinear stiffness terms. We start from the torque about the

Y-axis

$T{y}_{e13t}$ (Equation (

11)):

which can be integrated analytically to yield:

A similar analysis can be applied to other torque equation. In Equation (

29),

${y}_{14}$,

${y}_{13}$,

${x}_{14}$,

${x}_{13}$ are constants, and the only variable is

β. Knowing the Taylor expansion around zero to be:

We write a Taylor series expansion up to order three for

$f\left(\beta \right)$:

For small rotation angles,

$sin\left(\beta \right)\approx 0$,

$cos\left(\beta \right)\approx 1$ and

$\frac{{\epsilon}_{0}\xb7{sin}^{2}(\pi /2-\beta )\xb7{V}_{1}^{2}}{2{(\pi /2-\beta )}^{2}}}\approx C\xb7{V}_{1}^{2$, where

C is a constant value. Using the appropriate terms, odd power terms that vanish in Equations (

32) and (

29) can be written as:

${a}_{0}$ is defined in Equation (

33). Consequently, the equation of motion for

β is obtained as:

where

${r}_{1}$ and

${r}_{2}$ are constants.

The simplified equation of motion reveals the fact that excitation voltage changes the stiffness of the system in linear and nonlinear fashions, which clearly describes softening and superharmonic resonances in the system. The third term in Equation (

35) shows that the dominant stiffness nonlinearity in the system is quadratic, which explains the measured superharmonic resonance of order two, even at small rotation angles, as in

Figure 6a. We can also see the slight softening caused by quadratic nonlinearity in

Figure 7a,b. The quadratic stiffness nonlinearity does not refer to the mechanical structure here, but indicates the electromechanical coupling effect caused by the electrostatic torque (Equation (

34)). As the angles becomes larger, the odd terms in the Taylor series expansion become considerable, meaning that at larger voltages, cubic nonlinearity arises and is responsible for superharmonic resonances of order three in

Figure 7a.

As most of the MEMS mirrors are vacuum packaged, they experience a reduced pressure environment. Damping ratios are lower at reduced pressure values. The effect of the decrease of the damping ratio on the frequency response is analogous to the increase of forcing. In

Figure 8, we examined the effect of a reduced damping ratio,

$0.001$ compared to

$0.0208$ in

Figure 7a,b, on the frequency response. As is observed, the superharmonic resonance of order two is the dominant secondary resonances and becomes more significant at smaller damping ratios (reduced pressure environment).

Evolution of phase portraits at primary and secondary resonances of the

β angle are shown in

Figure 9.

Figure 9a shows how the elliptic trajectory of the primary resonance is converted to multiple enclosed ellipses, as in the case of superharmonic resonance of order two in

Figure 9b.

Figure 9c shows a superharmonic resonance of order three. As can be deduced, the trajectories shrink along the angular position and angular velocity axes as the order of resonance increases. That indicates a higher signal to noise ratio at a lower order of resonance.