Micro-beam resonator parametrically excited by electro-thermal Joule's heating and its use as a flow sensor

We study parametric resonance (PR) of double-clamped micro-beams that are electro-thermally actuated by a time-dependent Joule's heating and cooled by a steady air flow. The developed model demonstrates applicability of such device as a bifurcation-based flow velocity sensor. An AC electric current through the beam induces a time-harmonic compressive force that leads to parametric excitation of the structure. Convective cooling due to the air flow affects the location of the parametric transition curves on the driving voltage-frequency plane. The flow velocity can be obtained by measuring the frequency corresponding to the steep amplitude transition of the response. The device is modeled as an Euler-Bernoulli beam with an axial force parameterized by the electric current. The heat transfer problem is solved analytically; the heat flux due to the air flow is calculated using empirical correlations. The behavior of the beam is studied numerically, by means of finite differences, and analytically, using an approximate single degree of freedom Galerkin model, reduced to the Mathieu-Duffing equation. We show that while the PR always emerges at the driving voltage/current below the critical static buckling value, practical realization of the purely electro-thermal parametric excitation is challenging and is highly influenced by the device dimensions and quality factors. We evaluate the parameters required to assure the PR and demonstrate, using the model, feasibility of the suggested flow-sensing approach in the devices of realistic dimensions.