Position-insensitive mass estimation in cantilever MEMS via Fejér-weighted mode averaging

Abstract Minute added masses must be estimated with high accuracy while monitoring aerosol deposition, thin film growth, and molecular adsorption. Although the total added mass is reportedly estimated from shifts in the mechanical eigenfrequencies, single-mode measurements exhibit sensitivity degradation near the mode nodes and strongly depend on the attachment position. Therefore, in this study, we focus on “mode averaged” estimators that infer the total added mass ratio from relative frequency changes across multiple modes. Averaging over multiple modes reduces the influence of mode shape variations; however, doing so naively requires many readout channels, which increases the circuit complexity and load. As an alternative technique that preserves accuracy with only a few modes, we proposed a new estimator that linearly combined the fractional frequency shifts d_n with Fejér (triangular) weights, thereby smoothing the finite sum oscillations induced by position dependence via a Cesàro average. Finite element simulations of an Euler–Bernoulli cantilever validated the approach. Compared with equal weight averaging, the proposed estimator reduced position induced estimation errors by approximately an order of magnitude or more. Thus, Fejér weighted mode averaging provides an effective, low barrier means for improving robustness in applications that require stable monitoring of the total added mass.