Support loss depends mainly on the anchor types and materials. The energy lost from micro resonators into the support structure is summarized by three cases: the first case that acts as semi-infinite elastic medium with effectively infinite thickness, the second case that is treated as a plate with in-plane flexural vibrations, and the last case that can be treated as a plate with out-of-plane flexural vibrations. The first case, which is applicable to NEMS resonators, can be solved by modeling as a semi-infinite elastic medium with loads applied at a single point on surface of the half space [15]. Other cases are applicable to most MEMS resonators. The second case can be solved by modeling as a 2-D problem and using a plane strain method [11]. The last case is a 3-D problem. The energy lost into the surrounding structure consists of the net work done by the MEMS resonator at the attachment point.

The operational modes of RMG are simulated as shown in Figure 1(a,b), which are the two uniform rocking modes, and its main structural parameters are shown in Figure 1(c). When RMG vibrates in its rocking modes, two beams vibrate in line as out-of-plane flexural vibrations, while the other two perpendicular beams vibrate as torsion vibrations at the same time; viz., the four beams are undergoing coupled bending and torsion vibrations.

The dimensions of the attachment point are small compared with the vibration wavelength in the substrate at the natural frequency of RMG. So the support case of an RMG can be treated as a 3-D problem with a finite thickness plate. For the case of a support with finite thickness, we consider a semi-infinite plate with a thickness that need not be the same as that of the vibrating structure itself. Using the plate-edge admittance results [18], we derive analytical expressions of support loss, which also are applicable to a variety of resonators.

The power lost from a RMG flows mainly into the support structure. Support loss for the RMG can be simplified as a model with a beam attached to the rigid support structure at its end. The model and its main structural parameters are shown in Figure 2, where l₁, w and h are the length, width and thickness of the beam model, respectively and h_p is the thickness of its support structure. In an ideal situation, the rocking mass post can be seen as a rigid body without distortion, which is used to transfer kinetic energy without energy loss. When the microgyroscope vibrates at the original position, the total energy is translated into the kinetic energy of the beams and the rocking mass post.

In such cases, the effects of the microgyroscope on its substrate can be modeled as harmonic point forces and moments acting at the attachment point. An assumption is given: the energy propagated into the support structure would not be reflected, viz., all energy that reaches the support is considered lost.

The estimated Q factor is thus a lower bound. The Q factor is the ratio of the vibration energy of RMG to the energy lost; the reciprocal of Q is loss factor δ [19]: (2) δ = 1 Q = Δ U 2 π U = Π ω Uwhere ΔU is the total energy lost per cycle of oscillation, due to all applicable loss mechanisms, ΔU = 2πΠ/ω. U is the total vibration energy of oscillation. Π is the total net power flow out of RMG, and the average power is simply, Π = 1/2 Re (F·V). F is a vector of the point loads, and V is the corresponding vector of the harmonic linear and angular velocities at that point.

The operational modes of a RMG can be considered equivalent to a superposition of the rocking mode and torsional mode of an equivalent single degree of freedom (SDOF) system, which has the rocking mass post and two beams in line. The loads of the attachment points are described in Figure 3, without considering gravitation. There are two loads at the end of each bending beam, viz., the shear force normal to the substrate F_z, and the bending moment about the axis parallel to the substrate edge M_b. There is only one torsional moment about the axis perpendicular to the substrate edge M_t, at the end of each torsional beam. When a RMG vibrates in its operational modes, the four beams vibrate as a coupled bending and torsion. The arbitrary amplitude of the bending angle and the torsional angle at the end of these beams are equal, denoted by θ₀.

Using micro scanners’ vibration mode frequency results [20], these loads scaled by θ₀ and the spring constant of RMG can be expressed, respectively as: (3) M t = GI p θ 0 / l 1 (4) M b = − EI y   ( 4 l 1 + 3 l 2 ) θ 0 / l 1 2 (5) F z = − 6 EI y   ( l 1 + l 2 ) θ 0 / l 1 3 (6) K = 2 GI p l 1 + Ewh 3 l 1 [ 1 2 ( l 2 l 1 ) 2 + l 2 l 1 + 2 3 ]where E is Young’s modulus, G is shear modulus, ρ is density. h is the thickness of the vibratory structure, l₁ is the length of the beams, l₂ is the length of the center support. I_p is the polar moment of inertia of the beams’ cross-section, I_y is the moment of inertia of the beams’ cross-section, K is the spring constant of the rocking vibration.

The energy loss can be found by considering the plate support responding to the loads applied to its edge because the support structure is modeled as a plate. Consider the shear force F_z, the bending moment M_b, and the torsional moment M_t, as shown in Figure 4. For RMG, the force normal to the plate edge, the shear force parallel to the plate, and the bending moment about the axis perpendicular to the plate are not considered. The admittance at the edge of a plate was first formulated in integral form by Eichler. The elements of the matrix Y are given as closed-form integrals [21]. These integrals have recently been solved in closed form [22].

The point mobility matrix Y relates the normal angular velocity Ω_b, the tangential angular velocity Ω_t, and the transverse linear velocity V_z, of the attachment point to these applied loads by the expression: (7) [ Ω b Ω t V z ] = i ω [ θ ϕ w ] = 1 ρ p h p D [ Y ] [ M b M t F z ] = 1 ρ p h p D [ y 11 k 2 0 0 0 y 22 k 2 y 23 k 0 y 32 k y 33 ] [ M b M t F z ]where D is the plate stiffness, D = Eh_p³/12(1 − υ²), h_p is its thickness, k is the free wave numbers in the solid at frequency ω, k= [ω(ρh_p/D)^1/2]^1/2. The corresponding coefficients have been calculated, for υ = 0.3, which were calculated by Su [22]; for υ = 0.28, which were calculated by Chouvion [16], as shown in Table 1. The material in our experiments is n-type (100) single crystal silicon, Poisson ratio υ = 0.28, so these coefficients can be determined.

The resulting expressions for the power radiated into the plate support are: (8) Π F z = 3 ( 1 − υ 2 ) y 33 F z 2 h p 2 E ρ (9) Π M b = 6   ( 1 − υ 2 ) y 11 ω M b 2 Eh p 3 (10) Π M t = 6   ( 1 − υ 2 ) y 22 ω M t 2 Eh p 3 (11) Π M b   and   F z = Π F z + Π M b (12) Π M b   and   M t = Π M b + Π M t (13) Π M t   and   F z = Π M t + Π F z + [ 12 ( 1 − υ 2 ) ] 3 / 4 y 23 ω 1 / 2 M t F z ρ 1 / 4 E 3 / 4 h p 2 / 5

When each load is considered individually, the Equations (8–10) apply for the isolated load conditions, while the Equations (11–13) apply for two coupled load conditions. Equation (11) is different from Equation (13) presented by Judge [15]. The reason is that its velocity expressions are wrong compared with Equation (4) presented by Su [22]. However, the off-diagonal terms of Y result in an additional contribution that the total power is in Equation (13) if both the torsional moment and the shear force are present.

Using Raleigh’s method, the kinetic energy stored and the effective inertia of the bending beams can be expressed, respectively as [20]: (14) U b = KE b = ∫ 0 l 1 1 2 ( ρ wh )   ( d ω ( x ) dt ) 2 dx = 1 2 ρ whl 1   ( 1 105 l 1 2 + 5.5 105 l 1 l 2 + 9.75 105 l 2 2 ) ( d θ dt ) 2 (15) J b = ρ whl 1   ( 1 105 l 1 2 + 5.5 105 l 1 l 2 + 9.75 105 l 2 2 )

Then, the kinetic energy and effective inertia of the torsion beams and rocking mass post are also solved: (16) U t = KE t = ∫ 0 l 1 1 12 ρ wh   ( w 2 + h 2 )   ( x l 1 d θ dt ) 2 dx = 1 36 ρ whl 1   ( w 2 + h 2 )   ( d θ dt ) 2 (17) J t = 1 18 ρ whl 1   ( w 2 + h 2 ) (18) U m = KE m = 1 2 J m   ( d θ dt ) 2 (19) J m = 1 12 M m 1   ( l 2 2 + h 2 ) + 1 12 M m 2   ( 3 4 D 2 + h 1 2 )where θ is the rocking angle of the rocking mass post, θ = θ₀ sin(ωt), dθ/dt = ωθ₀ cos(ωt). M_m1 is the mass of the center support, and M_m2 is the mass of the rocking mass post. The formulae of the vibratory energy are also scaled by θ₀, the arbitrary amplitude of the rocking vibration mode shape.

The vibration energy and the resonant frequency for the operational modes of RMG are expressed, respectively as: (20) U total = U m + 2 U b + 2 U t = 1 2 J xx ω 2 θ 0 2 (21) ω = k / J xxwhere the total effective inertia of RMG can be solved as J_xx = J_yy = J_m + 2J_b + 2J_t.

Given w ≥ h, I_p = βwh³, β is the function of w/h [23]. The radiated power can be found by substituting Equations (10,11,20,21) into Equation (2), and the support loss factor is found as Equation (22); Given w < h, the support loss factor is found as Equation (23): (22) 1 Q = 3 ( 1 − ν 2 ) J xx y 33 ( l 1 + l 2 ) 2 ( β 1 + ν + η ) 3 / 2 l 1 4 ρ h w l 1 h 2 h p 2 + [ ( 1 − υ 2 ) y 11 ( 4 l 1 + 3 l 2 ) 2 6 ( β 1 + ν + η ) l 1 2 + 6 ( 1 − υ ) y 22 β 2 ( 1 + ν ) ( β 1 + ν + η ) ] w l 1 h 3 h p 3 (23) 1 Q = 3 ( 1 − ν 2 ) J xx y 33 ( l 1 + l 2 ) 2 ( β 1 + ν w 2 h 2 + η ) 3 / 2 l 1 4 h ρ w l 1 h 2 h p 2 + [ ( 1 − υ 2 ) y 11 ( 4 l 1 + 3 l 2 ) 2 6 ( β 1 + ν w 2 h 2 + η ) l 1 2 + 6 ( 1 − υ ) y 22 β 2 ( 1 + ν ) ( β 1 + ν + h 2 w 2 η ) ] h l 1 w 3 h p 3where η = (l₂/l₁)²/2 + l₂/l₁ + 2/3. It can be seen from Equations (22) and (23) that, for h_p is large relative to h, the shear force term F_z dominates the support loss, and the effect of the bending moment M_b and the torsional moment M_t at the attachment point could be neglected.

For ν = 0.28, w = 90, l₁ = 2,200, l₂ = 2,200, h₁ = 5,000, r = 550. Support loss for the rocking modes in RMG is shown in Figure 5. When w ≥ h, four curves are shown relating Q factor to the thickness of the support plate h_p in Figure 5(a), while w < h, three curves are shown in Figure 5(b). Q factor increases with the thickness of the support h_p increasing; furthermore, the thickness of the beams h is thinner, Q factor will increase more obviously. For h = 90 μm, Q factor increases from almost 50 to 1,600 when the thickness h_p increases from 500 to 3,000 μm; while for the 20 μm thick beams, Q factor can reach almost 16,000. Compared with the predicted results in [17], these results are almost an order of magnitude less than those without considering the rocking mass post, viz., the rocking mass post would badly influence Q factor of the microgyroscope. These curves provide a good order-of-magnitude estimate of the support loss for the microgyroscope.

In this section, the sensitivities of support loss to the main structural parameters involved, such as sizes of the rocking mass post, center support, and beams are studied.

A plot of support loss for ν = 0.28, w = 90, h = 60, h_p = 2,000 is shown in Figure 6(a). Four curves show relating Q factor to the length l₁, l₂, h₁ and the radius r. Q factor increases with the length l₁ and l₂ increasing; l₁ dominates the increasing of Q factor, and l₁ must be large enough to achieve high Q factor. Q factor decreases with the length h₁ and the radius r increasing quickly, and r must be small enough to achieve high Q factor and larger natural frequency.

Furthermore, the thickness h and width w of the beams are also studied. For ν = 0.28, h_p = 2,000, l₁ = 2,200, l₂ = 2,220, r = 500, h₁ = 3,000, a plot of support loss relating Q factor to h and w is shown in Figure 6(b). Two curves show that Q factor decreases quickly with the two parameters increasing; h dominates the decreasing of Q factor, and h must be small enough to achieve high Q factor for RMG.

The analytical model derived provides the design guidelines for achieving high Q factor in RMG, which may be summarized as below:

Choice of materials: in order to increase Q_support, the high-strength material is preferred for the rocking mass post and the substrate (for higher vibration energy and lower energy loss), while the low-strength material for the beams.