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Direct laser writing to fabricate capacitively transduced resonating sensor

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Abstract

This article reports on laser technology based fabrication approach to develop a micro-size capacitive gap based transducer useful for a variety of applications. A low-cost prototype has been fabricated via a rapid and advanced laser micro-milling technique to achieve a parallel kerf-width (capacitive gaps) of about 60 µm into a piece of aluminum and a stainless steel each of 1 and 2 mm thickness, respectively, thus leading to a high-aspect ratio (> 33) structure. A device is demonstrated to facilitate actuation via electrostatic means and sense a capacitive change across its electrode. Experiments have been performed with a structure made of aluminum. Results comprising analytical modeling, fabrication, and electrical characterization are presented. An applicability of a device as a two degree-of-freedom resonating mode-localization sensor that employs a weak electrostatic coupling is demonstrated to offer vibration amplitude based sensitivity to a relative change in the stiffness. This sensor is able to resolve a minimum stiffness perturbation (normalized), \(\delta_{{k_{\rm{min} } }} = \frac{\Delta k}{{K_{eff} }}\) of the order of 7.98 × 10−4.

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Acknowledgements

This work was funded by the University of Liege, Belgium.

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Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, University of Liege, 4000, Liege, Belgium

    Vinayak Pachkawade

  2. Microsystem Laboratory, University of Liege, 4102, Seraing, Belgium

    Delphine Cerica, Samuel Dricot & Serguei Stoukatch

  3. Department of Electrical Engineering (ESAT), KULeuven, 3001, Louvain, Belgium

    Michael Kraft

Authors
  1. Vinayak Pachkawade
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  2. Delphine Cerica
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  3. Samuel Dricot
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  4. Serguei Stoukatch
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  5. Michael Kraft
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Contributions

VP designed, coordinated this research and drafted the manuscript. VP and MK conceived the study. DC contributed in the analytical work. SS and SD participated in the research coordination. The authors read and approved the final manuscript.

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Correspondence to Vinayak Pachkawade.

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Appendix

Appendix

When system experiences imbalance into the initial symmetry i.e. ∆k ≠ 0, governing equation of motion for the two-coupled proof masses as shown in Fig. 1b is given as follows:

$$M\ddot{x}_{1} + (c + c_{c} )\dot{x}_{1} + (K_{m} + K_{c} )x_{1} - c\dot{x}_{2} - K_{c} x_{2} = f_{1} (t)$$
(1)
$$M\ddot{x}_{2} + (c + c_{c} )\dot{x}_{2} + (K_{m} + K_{c} + \Delta k)x_{2} - c\dot{x}_{1} - K_{c} x_{1} = f_{2} (t)$$
(2)

By operating the device under vacuum, we can mitigate the impact of (a) damping force of individual proof mass and (b) damping force that occurs between two proof masses. There exists other damping forces, which can be considered. For simplifying the analysis, however, we assume \(c_{1} = c_{2} = c_{c} = 0\). Therefore, Eqs. (1) and (2) are modified as below:

$$M\ddot{x}_{1} + (K_{m} + K_{c} )x_{1} - K_{c} x_{2} = f_{1} (t)$$
(3)
$$M\ddot{x}_{2} + (K_{m} + K_{c} + \Delta k)x_{2} - K_{c} x_{1} = f_{2} (t)$$
(4)

By applying a Laplace transformation to Eqs. (3) and (4), we can obtain the following:

$$H_{11} (s)X_{1} (s) - H_{12} (s)X_{2} (s) = f_{1} (s)$$
(5)
$$H_{22} (s)X_{2} (s) - H_{21} (s)X_{1} (s) = f_{2} (s)$$
(6)

where,

$$H_{11} (s) = s^{2} M + (K_{m} + K_{c} )$$
(7)
$$H_{12} (s) = H_{21} (s) = K_{c}$$
(8)
$$H_{22} (s) = s^{2} M + (K_{m} + K_{c} + \Delta k)$$
(9)

In Eq. (6), we set \(f_{2} (s) = 0\), and derive an expression for \(X_{1} (s)\) and \(X_{2} (s)\) to use these values back in Eq. (5) to obtain an output transfer function, as follows:

$$H_{1} (s) = \frac{{X_{1} (s)}}{{f_{1} (s)}} = \frac{{H_{22} (s)}}{{H_{11} (s)H_{22} (s) - H_{12} (s)H_{21} (s)}}$$
(10)
$$H_{2} (s) = \frac{{X_{2} (s)}}{{f_{1} (s)}} = \frac{{H_{21} (s)}}{{H_{11} (s)H_{22} (s) - H_{12} (s)H_{21} (s)}}$$
(11)

Similar procedure can be applied to obtain an expression for \(H_{3} (s) = \frac{{X_{1} (s)}}{{f_{2} (s)}}\) and \(H_{4} (s) = \frac{{X_{2} (s)}}{{f_{2} (s)}}\). Using the values of \(H_{11} (s)\), \(H_{12} (s)\)\(H_{21} (s)\) and \(H_{22} (s)\) derived earlier in Eq. (7) through (9), we can obtain

$$H_{1} (s) = \frac{{X_{1} (s)}}{{f_{1} (s)}} = \frac{{s^{2} M + (K_{\alpha } + \Delta k)}}{{s^{4} M^{2} + s^{2} M(2K_{\alpha } + \Delta k) + K_{\alpha }^{2} + K_{\alpha } \Delta k - K_{c}^{2} }}$$
(12)
$$H_{2} (s) = \frac{{X_{2} (s)}}{{f_{1} (s)}} = \frac{{K_{c} }}{{s^{4} M^{2} + s^{2} M(2K_{\alpha } + \Delta k) + K_{\alpha }^{2} + K_{\alpha } \Delta k - K_{c}^{2} }}$$
(13)

respectively, where \(K_{\alpha } = (K_{m} + K_{c} )\). Using \(s = j\omega\), Eqs. (12) and (13) can be modified to attain

$$H_{1} (j\omega ) = \frac{{X_{1} (j\omega )}}{{f_{1} (j\omega )}} = \frac{{ - \omega^{2} M + (K_{\alpha } + \Delta k)}}{{\omega^{4} M^{2} - \omega^{2} M(2K_{\alpha } + \Delta k) + K_{\alpha }^{2} + K_{\alpha } \Delta k - K_{c}^{2} }}$$
(14)
$$H_{2} (j\omega ) = \frac{{X_{2} (j\omega )}}{{f_{1} (j\omega )}} = \frac{{K_{c} }}{{\omega^{4} M^{2} - \omega^{2} M(2K_{\alpha } + \Delta k) + K_{\alpha }^{2} + K_{\alpha } \Delta k - K_{c}^{2} }}$$
(15)

A denominator of Eqs. (14) and (15) is given by

$$\omega^{4} M^{2} - \omega^{2} M(2K_{\alpha } + \Delta k) + K_{\gamma } = 0$$
(16)

where \(K_{\gamma } = K_{\alpha }^{2} + K_{\alpha } \Delta k - K_{c}^{2}\). Equation (16) is called as characteristic equation of this two DoF coupled system. Roots of Eq. (16) provide poles and zeros (resonance and anti-resonance frequencies, respectively):

$$\omega_{ip}^{2} \approx \frac{{M(2K_{\alpha } + \Delta k) + \sqrt {4M^{2} (K_{c}^{2} - K_{\gamma } ) + M^{2} + \Delta k^{2} } }}{{2M^{2} }}$$
(17)
$$\omega_{op}^{2} \approx \frac{{M(2K_{\alpha } + \Delta k) - \sqrt {4M^{2} (K_{c}^{2} - K_{\gamma } ) + M^{2} + \Delta k^{2} } }}{{2M^{2} }}$$
(18)

where, \(\omega_{ip}^{2}\) and \(\omega_{op}^{2}\) are in-phase and out-of-phase natural mode frequencies of the device. With ∆k = 0, Eqs. (17) and (18) take the form \(\omega_{ip}^{2} = \frac{K}{M}\) and \(\omega_{op}^{2} = \frac{{K + 2K_{c} }}{M}\). Dividing Eq. (14) by Eq. (15), we obtain a ratio of amplitudes

$$\frac{{H_{1} (j\omega )}}{{H_{2} (j\omega )}} = \frac{{ - \omega^{2} M + (K_{\alpha } + \Delta k)}}{{K_{c} }}$$
(19)

We can refer Eqs. (17) and (18) to obtain simplified expressions for the pole frequencies given as

\(\omega_{ip}^{2} \approx \frac{{\Delta k + K_{\alpha } + \sqrt {K_{\alpha }^{2} - K_{\gamma } } }}{M}\); \(\omega_{op}^{2} \approx \frac{{\Delta k + K_{\alpha } - \sqrt {K_{\alpha }^{2} - K_{\gamma } } }}{M}\). Substituting these values in place of ω in Eq. (19), we can obtain the expression for mode amplitude ratio as a function of stiffness perturbation, ∆k as follows:

$$\frac{{H_{1} (j\omega_{ip} )}}{{H_{2} (j\omega_{ip} )}} = \frac{{ - \sqrt {K_{\alpha }^{2} - K_{\gamma } } }}{{K_{c} }};\frac{{H_{1} (j\omega_{op} )}}{{H_{2} (j\omega_{op} )}} = \frac{{\sqrt {K_{\alpha }^{2} - K_{\gamma } } }}{{K_{c} }}$$
(20)

With ∆k = 0, Eq. (20) takes the form as

$$\frac{{H_{1} (j\omega_{ip} )}}{{H_{2} (j\omega_{ip} )}} = 1;\frac{{H_{1} (j\omega_{op} )}}{{H_{2} (j\omega_{op} )}} = - 1$$
(21)

thus representing initial balanced condition of a two coupled resonators.

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Pachkawade, V., Cerica, D., Dricot, S. et al. Direct laser writing to fabricate capacitively transduced resonating sensor. Microsyst Technol 26, 547–562 (2020). https://doi.org/10.1007/s00542-019-04549-2

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