A double-resonator design has been devised for a cloverleaf-shaped silicon microelectromechanical resonator. The double-resonator design provides for an inner, higher-frequency resonator suspended on an outer, lower-frequency resonator. This design concept affords several advantages, as described below.
A typical prior design of a microelectromechanical resonator calls for a solidly mounted substrate. Solid mounting entails (1) poor vibration isolation and (2) high energy losses in the substrate, with consequent decrease of the resonance quality factor (Q). The double-resonator design was inspired by the realization that solid mounting is not necessarily desirable and that if the substrate of a resonator is suspended on thin springs, what is formed is a double-mass resonator that can have a Q greater than that of the original resonator. In addition, the outer resonator helps to isolate the inner resonator from packaging stresses and from vibrations of external origin.
The figure schematically depicts mathematical models of the previous single-resonator design and the present double-resonator design. The schematic diagrams reflect the observation that it is more accurate to model the substrate as a finite mass with damping than to assume that the substrate is so rigidly mounted that it represents an infinite mass. In the single-resonator design, resonator mass M1 is coupled, via a spring of stiffness k1, to a damped substrate mass M2. This model yields close agreement between predicted and measured Q factors.
In the double-resonator design, inner resonator mass M1 is suspended on a spring of stiffness k1 that is attached to an intermediate mass M2, which, in turn, is coupled to damped substrate mass M3 via a spring of stiffness k2. M2 is chosen to be much greater than M1; consequently, the frequency and mode shape of the higher-frequency (M1,k1,M2) resonance does not differ greatly from that of the single-resonator design. M3 is also chosen to be much greater than M1; this choice, in combination with the choice of M2, and with the choice of k1 and k2 to be approximately equal, ensures that the damping on M3 exerts little effect on the Q of the higher-frequency resonance.
Because of the isolation provided by k2, very little of any mounting stress that might be imposed on M3 is coupled into k1. In addition, because of the largeness of M2 relative to M1, very little of any vibration imposed on M3 propagates to M1. Another advantage of the double-resonator design is that M2 can be tailored to exert a slight effect on the resonances (in other words, to tune the vibrating system); it is easier to tune in this way that to tailor k1.
In the prototype double resonator, the substrate of a cloverleaf resonator substructure is suspended by four springs that connect it to an outer frame. The lowest resonance frequency of the cloverleaf is designed to be 6 kHz, while the lowest resonance frequency for vibration isolation is designed to be 200 Hz. It has been predicted that the cloverleaf resonance will have a Q > 104, and that because of damping in the outer frame, the substrate resonance will have Q < 100.
This work was done by Roman Gutierrez, Tony K. Tang, and Kirill Shcheglov of Caltech for NASA's Jet Propulsion Laboratory. For further information, access the Technical Support Package (TSP) free on-line at www.nasatech.com/tsp under the Mechanics category.
In accordance with Public Law 96-517, the contractor has elected to retain title to this invention. Inquiries concerning rights for its commercial use should be addressed to
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