MRIG frequency mismatch and quadrature control

The output variables of an MRIG (Microscale Rate Integrating Gyroscope), x, y, satisfy the equations of motion of a two-dimensional oscillator. (Ref. [1]) To examine the role of a mismatch of the two principal frequencies, ω
₁
and ω
₂
, we look at the response to a constant rate input, Ω, for times short compared to the damping time constants. Eqs. (7) of Ref. [1], when specialized to principal-axis coordinates, reduce to x - 2kΩy + ω
²
₂
x = 0 (1) ÿ + 2kΩx + ω
²
₁
y = 0 when the damping and external-force terms are omitted. k is the angular gain factor. The solution for x(t), y(t) in terms of the initial values x(0), y(0), x(0), y(0) is readily obtained using Laplace transforms. To reduce the writing, we introduce the definitions (Eq. (8) of Ref. [1]).