System and methods for a hollow-disk radial-contour mode resonator structure. The hollow disk reduces the dynamic mass and stiffness of the structure. Since electromechanical coupling C.sub.x/C.sub.o goes as the reciprocal of mass and stiffness, the hollow disk structure has a considerably stronger electromechanical coupling than a solid one at the same frequency, and thus raises C.sub.x/C.sub.o without excessive gap-scaling. Several embodiments of hollow disk resonators are detailed, including asymmetric and symmetric disk configurations.

A dual-mode resonator assembly includes a plurality of electrodes disposed around the resonator and configured to transduce information related to a first mode of operation of the dual-mode resonator assembly and a second mode of operation of the dual-mode resonator assembly. The plurality of electrodes includes electrodes associated with the first mode of operation and electrodes associated with the second mode of operation. The plurality of electrodes are disposed symmetrically and centered to nodes and antinodes of the first mode of operation and/or the second mode of operation. The electrodes are configured to also minimize feedback and noise from the first and second mode of operation.

A resonator element includes three vibrating arms. Each of the three vibrating arms includes an arm extending from a base and a wide section positioned at a tip end portion of the arm. The wide section is wider than the arm. When, of the three vibrating arms, the vibrating arm positioned at a center of an arrangement is provided with the arm having a width denoted by W1 which is defined as a length of the arm in the second direction, and the vibration arms positioned on both sides of the arrangement are provided with the arms each having a width denoted by W2 which is defined as a length of the arms in the second direction, a relationship 1W1/W22 is satisfied. When a length of the three vibrating arms in the first direction is denoted by L and a length of the wide sections in the first direction is denoted by Lh, a relationship Lh/L0.49 is satisfied.

A resonator element includes first to third vibrating arms and weights disposed at respective tip end portions of the first to third vibrating arms. When a weight mass ratio A represented by M1/M2 is plotted on a horizontal axis and an arm width ratio B represented by W1/W2 is plotted on a vertical axis, where M1 is a mass of the weight disposed on the first vibrating arm, M2 is a mass of the weight disposed on each of the second and third vibrating arms, W1 is a width of the first vibrating arm, and W2 is a width of each of the second and third vibrating arms, a point (A, B) is located in a region surrounded by a polygon formed by connecting six points of (A, B)=(2.39, 2), (0.01, 2), (0.23, 1.6), (1.65, 1), (7.15, 1), and (4.02, 1.6) with straight lines.