**2.1 Analytical models**

**Figure 2** represents the analytical plots of the 2-DoF CR sensor system. In **Figure 2(a)**, mode-frequencies as a function of induced mass disorder is shown. Lower resonant frequency of the out-of-phase mode indicates that a design uses an electrical coupling between the two resonators. With *δ<sup>m</sup>* = 0, two resonant frequencies are closely spaced apart (frequency separation determined by the value of the coupling spring constant, *Kc* used in the system). However, when *δ<sup>m</sup>* 6¼ 0, resonant mode-frequencies veer away from each other as the magnitude of the *δ<sup>m</sup>* is increased. In **Figure 2(b)**, similar trend can be observed, where, amplitude ratio (AR) as a function of induced mass disorder is shown. Slopes of the curves in each graphs determine the sensitivity of mode-frequencies and AR to the normalized mass perturbation into the system. The theoretical sensitivity norms (in the context of the mass perturbations, *Δm*) for the amplitude ratio (AR), eigenstate and the resonant frequency used in CR sensors are expressed as below:

$$
\left(\frac{\varkappa \mathbf{1}}{\varkappa \mathbf{2}}\right)\_{ji} = \left|\frac{r\_n - r\_0}{r\_0}\right| \approx \left|\frac{\Delta m}{2K\_c}\right|\tag{23}
$$

$$\mathbf{a}(\mathbf{x})\_{ji} = \frac{|\mathbf{a}\_n - \mathbf{a}\_0|}{|\mathbf{a}\_0|} \approx \left| \frac{\Delta m}{4K\_c} \right| \tag{24}$$

$$\left( \left. f\_{ji} \right) = \frac{\left| \left. f\_n - f\_0 \right|}{\left| f\_0 \right|} \approx \left| \frac{\Delta m}{2m\_{\text{eff}}} \right| \tag{25}$$

for *j* th resonator (*j* = 1, 2) at *i* th mode of the frequency response (*i* = 1, 2), respectively. For the electrostatic coupling between the two resonators, an effective value of the coupling spring is given by *Kc* ¼ � <sup>Δ</sup>*v*<sup>2</sup> ð Þ *<sup>ε</sup>*0*<sup>A</sup> <sup>g</sup>*<sup>3</sup> , where *Δv* refers the potential difference between the two masses, *g* is capacitive gap, *ε* is permittivity and *A* is the cross sectional area of the parallel-plate capacitor.

**Figure 3** shows a plot of AR variation as a function of mass perturbation. Two different values of coupling spring, *Kc* are used. For lower effective value of the coupling spring, *Kc* = 100 *N/m*, (coupling factor, *κ* = 0.00075), higher changes in the AR can be extracted (relatively higher slope). This aspect shows the tunable characteristic of a sensitivity in a CR biosensor unit. In **Figure 4**, different forms of the

#### **Figure 2.**

*Mode-frequencies and AR veering phenomenon observed in CR mode-localized mass sensors.* δ<sup>m</sup> *is the applied normalized mass perturbation. Slope of the AR curves determined the sensitivity. Lower value of the coupling spring,* Kc *enhances sensitivity to the mass perturbation for the AR output.*

#### **Figure 3.**

*Amplitude ratio (AR) curve veering in 2-DoF mode-localized coupled resonators mass sensors. The lower coupling factor* κ *leads to higher changes in the AR when the normalized perturbation* δ<sup>m</sup> *is applied.*

#### **Figure 4.**

*Different types of the outputs with coupled resonators (CR) sensors. The amplitude ratio (AR) shift shows the highest percentage changes as a function of the mass perturbation,* δm*.*

outputs (as expressed in (23) through (25)) are compared against the values of applied mass perturbations in a 2-DoF CR sensor system. As seen, the AR output offers the highest achievable sensitivity. Therefore, AR sensing is the preferred method of the sensing in CR biosensor system. It can be written that *SRi* > *Sai* ≫ *S fi* . Here, *SRi* , *Saji* and *S fi* denote the *theoretical maximum sensitivity* for AR, amplitude and frequency for *j* th resonator (*j* = 1, 2) at the *i* th mode of the frequency response (*i* = 1, 2), respectively. Another way to understand the operation of a 2-DoF CR biosensor is through its output frequency response. **Figure 5** shows an output response of resonator 1 and 2 in a 2-DoF coupled resonant (CR) mass biosensor system. As two resonators are used in the CR system, two peaks appear in the output response. Initially, assuming a symmetric system (i.e. *M1 = M2 = M*, *K1* = *K2* = *K*), *j th* resonator (*j* = 1, 2) vibrate with equal amplitudes at the *i th* mode (*i* = 1, 2) of the output response. After the mass disorder, *δ<sup>m</sup>* is introduced, it is observed that vibration amplitudes of the *j th* resonator (*j* = 1, 2) at the *i th* mode (*i* = 1, 2) of the output response change. An amplitude shift is denoted by *Δa*. The frequency shift, *Δf* at both the modes is also seen. After the mass perturbation, for the resonator 1, vibration amplitude is seen decreased at the first mode, whereas amplitude is seen increased at the second mode.
