Multi-Mode Trimming of Imperfect Rings

Modal Parameter Tuning of an Axisymmetric Resonator via Mass Perturbation

“…The relationship between the modal frequencies {ω 1 , ω 2 } and anti-node orientations {ψ 1 , ψ 2 } of the in-plane modes with n = 2 modal diameters in a perturbed ring resonator are reproduced below from [6],…”

“…Furthermore, ω 0 represents the natural frequency of the degenerate modes of the unperturbed perfect ring and α 2 the amplitude ratio of the radial and tangential displacement for modes with n = 2 modal diameters. The notation in [6] is largely retained, and although the n = 2 case is studied exclusively in this paper, the results developed below can be extended to any pair of nominally degenerate modes. The antinode orientation associated with the ω 1 mode is is given by ψ 1 and the analysis in [6] assumes ψ 2 = ψ 1 + 45…”

“…The notation in [6] is largely retained, and although the n = 2 case is studied exclusively in this paper, the results developed below can be extended to any pair of nominally degenerate modes. The antinode orientation associated with the ω 1 mode is is given by ψ 1 and the analysis in [6] assumes ψ 2 = ψ 1 + 45…”

“…The resonator introduced in this paper is a bulk micromachined planar structure that is axisymmetric under discrete rotations so, consequently, the theory most relevant to this resonator are the results developed for modifying the dynamics of rings. For example, [4], [5], [6] address thin rings where it is assumed that the deviation from perfect symmetry is small, so consequently the required perturbations to the ring mass and stiffness to bring pairs of modes to degeneracy are small compared to their nominal values.…”

“…II accepts the deposition of 75 µm diameter precision solder spheres for coarse point mass perturbations and individual droplets of silver particle-loaded ink for fine point mass perturbations. In contrast to [5] and [6] which have no constraints on the location and amount of the deposited/removed mass, the present resonator has a fixed number of locations where mass can be deposited and furthermore this mass is quantized and cannot be deposited in arbitrarily fine increments. The perturbation masses are consistent enough, however, so that their effect on the dynamics of the structure are readily predictable from the modified ring models developed in this paper.…”