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In this paper, the mechanical response of a commercial off-the-shelf, uni-axial polysilicon MEMS accelerometer subject to drops is numerically investigated. To speed up the calculations, a simplified physically-based (beams and plate), two degrees of freedom model of the movable parts of the sensor is adopted. The capability and the accuracy of the model are assessed against three-dimensional finite element simulations, and against outcomes of experiments on instrumented samples. It is shown that the reduced order model provides accurate outcomes as for the system dynamics. To also get rather accurate results in terms of stress fields within regions that are prone to fail upon high-

Several attempts to provide efficient, robust and accurate (or, at least, informative) reduced order models (ROMs) for nonlinear systems, with a specific focus on MEMS, have been recently published. Accounting for the nonlinearities arising from the coupled electro-mechanical, or even electro-thermo-mechanical physics governing the system behaviour, methodologies to define Krylov subspaces and therefore reduce the computational costs of the analyses were proposed in [

Proper orthogonal decomposition (POD) methodologies were also adopted to reduce the order of MEMS modelling [

Physically based ROMs of the mechanical behaviour of microsystems were instead presented in [

Focusing on the physical effects of shocks and drops on micro-inertial sensors, it can be shown that the mechanical side of the problem is by far the most prominent one; the electrical side can be instead disregarded. This is basically linked to the high levels of acceleration induced by the shock loading: in [^{5}

Because of the large diffusion of MEMS devices in commercial applications for consumer electronics, the post-impact response of polysilicon micro-accelerometers has received attention in the recent years [

In a series of recent papers [

In [

Let us consider the commercial off-the-shelf, uni-axial MEMS accelerometer depicted in

The uni-axial accelerometer is designed to sense accelerations normal to the substrate with a target sensitivity of about 0.65 V/_{1} + _{2} = 760

Comparison FE results have been obtained with the commercial Abaqus code (Simulia) [_{x}_{y}_{z}_{xy}_{xz}_{yz}_{xz}_{yz}

By assuming the external acceleration field to be upper bounded by 200 kHz, see also [_{t}^{3} (where _{xz}_{x}

Inertial terms arising from the assumed kinematics are:

The system dynamics, accounting also for proportional damping, is therefore governed in the ROM by the following two coupled equations of motion:
^{T}^{T}^{−l}. Accounting for proportional damping, we set ^{−1}

During oscillations, plate corners may get into contact with the die and cap surfaces. Algorithmically, variables Δ_{d}_{c}

As far as the solution of governing

In this Section, the ROM accuracy and performance (in terms of reduction of the computational costs) are assessed against FE simulations and available experimental data. Both the ROM and the FE model are fed by the input acceleration loadings depicted in

Both the loading conditions depicted in

In the comparison FE analyses, according to the ROM, a proportional damping has been considered, with the same quality factor

Let's start by considering the low-

Because of the spectral density of the input in this low-_{P}_{P}_{P}_{P}

When the output voltage evolutions obtained with the two numerical approaches are compared with the experimental data, see

Let's move now to the high-_{P}_{P}

Because of the wild oscillations of springs and plate induced by the high-

There are now three basic issues to discuss. First, we have to show the computational gain obtained by running the ROM instead of the FE analyses, keeping the accuracy aside (in this regard, see the discussions above and to follow). The speedups relevant to the two test cases are reported in ^{−5}

Second, we check how the quality factor _{A}_{A}

Third, we assess the underestimation of peak values of _{P}_{P}_{P}

To speed up the assessment of the reliability of a uni-axial inertial MEMS sensor under shock/drop loading conditions, we have provided and discussed a reduced order model of the movable parts of the sensor itself. We termed this model “physically-based”, since within a purely mechanical framework we accounted for the physics of the actually excited torsional and bending deformation modes of the sensor springs supporting the seismic, massive plate.

We have assessed the capability of the model and its accuracy against experimental data collected in a former laboratory campaign and also against three-dimensional finite element simulations. The second comparison looks necessary to check the accuracy of the model, due to the nonlinear dynamics induced by impacts featuring acceleration peaks much beyond the working range (in our case, ±2

It has been shown that the reduced order model provides rather accurate estimations of sensor dynamics, up to acceleration peaks in the order of 5, 000

Financial support to this work has been provided by MIUR through PRIN08 project ^{2}-

Geometry of the uni-axial MEMS accelerometer, and notation.

First five vibration modes of the uni-axial accelerometer, and relevant resonance frequencies.

Shock tests. Top row: acceleration histories felt by the sensor during the (

Low-_{A}

Low-

Low-

High-_{A}

High-

High-

Reduced order modelling: effect of damping on the (top) low-_{A}

Model of the plate-spring connection region subjected to a torque, as adopted for the calculation of the stress intensity factor at the re-entrant corners.

Computational gain, given by the ratio between the FE CPU time and the ROM CPU time.

Speedup factor | |
---|---|

Low- |
725 |

High- |
53 |

Test-induced maximum values of the principal stress _{P}

FE (MPa) | ROM (MPa) | Corrected ROM (MPa) | |
---|---|---|---|

Low- |
13.23 | 4.22 | 23.21 |

High- |
33.54 | 5.12 | 28.16 |