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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Beam’s multiple-contact mode, characterized by multiple and discrete contact regions, non-uniform stoppers’ heights, irregular contact sequence, seesaw-like effect, indirect interaction between different stoppers, and complex coupling relationship between loads and deformation is studied. A novel analysis method and a novel high speed calculation model are developed for multiple-contact mode under mechanical load and electrostatic load, without limitations on stopper height and distribution, providing the beam has stepped or curved shape. Accurate values of deflection, contact load, contact region and so on are obtained directly, with a subsequent validation by CoventorWare. A new concept design of high-g threshold microaccelerometer based on multiple-contact mode is presented, featuring multiple acceleration thresholds of one sensitive component and consequently small sensor size.

Beam’s multiple-contact mode has been used in micromachined RF switches [

The measurement of high-g acceleration has been widely studied [

The modeling is complex. The deformation interacts with contact load and contact region, and consequently there exists a complicated coupling relationship between contact region, contact load, other mechanical loads, electrostatic load and deformation, as shown in

Compared with traditional contact mode, multiple-contact mode is more complex because the contact region is discrete and stoppers’ heights are non-uniform. However, prior models of multiple-contact RF switches aren’t universal, because the multiple-contact is only in a zipper-like way. In this special multiple-contact mode, fixed stoppers contact a movable cantilever sequentially according to the order of stoppers’ locations along the cantilever’s length direction, and the stopper maintains contact with the cantilever once the contact has happened [

Besides the complicated coupling relationship and discrete contact region, in this article our multiple-contact analysis also takes into account the following facts: first, contact load influences beam’s deformation and is a crucial factor to ensure electrodes’ low contact resistance [

Analytical methods have been used in the analysis of traditional contact mode where the contact region is continuous and the movable structure is size-uniform, usually based on some simplification assumptions [

As the movable electrode obtained by microfabrication process is very thin, usually under 0.1 μm, its height is negligible for the modeling. As the gap between the two electrodes of each fixed electrode-couple is very small and beam’s stiffness in the width direction is very high, the two electrodes are regarded as one stopper.

Take the cantilever for example. As shown in _{b}_{i-1}_{i}_{0} = 0 and _{n}_{b}_{i}_{i}_{i}_{0} = 0. Then the resultant contact load for the whole cantilever is _{n}_{i}_{i}_{−1}.

Based on material or plate mechanics [_{i}_{i}_{i}_{i}

Assume _{i}_{0} is vacuum permittivity. _{die}_{r}_{i} is the initial distance between the movable electrode at beam’s _{i}_{i}_{i}_{q}_{i}_{q}_{n}_{i}_{i}_{b}_{i}_{b}_{b}_{b}

Substituting (5) into (1) and solving the resulting equation, we obtain the following expression for the cantilever’s deflection:

In _{i}_{−1} and _{n}_{i}_{i}_{i}_{i}

The cantilever’s slope is:

The _{i}

At a cantilever’s fixed end, the deflection and slope both are zero, therefore:

At a cantilever’s free end, the moment is zero, therefore:

If a cantilever’s _{i}

If a cantilever’s

It should be mentioned that when the fixed electrode is curved, the above equations also are applicable only by regarding _{i}_{i}

Based on Equations (

Variables _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

Each stopper’s top,

The beam’s deformation calculation process is as follows:

Select one of the possible combinations of stopper contact points.

Calculate

Calculate

Solve

Calculate the deflection by substituting values of _{i}_{i}_{i}_{i}

Judge whether the deflection is convergent under the current assumption of stoppers’ contact points. If not, update the electrostatic load according to the beam’s new calculated deflection, and then repeat steps (c–f). If yes, go to step (g).

Judge whether each uncontacted segment of the beam has a calculated deflection smaller than the segment’s initial gap to the stopper or substrate and whether each contact load is a push load but not a pull load. If not, select another possible combination of stopper contact points, and then repeat steps (b–g). If yes, the calculation ends.

The reason for the convergence judgment in step (f) is that the electrostatic load interacts with the beam’s deflection, so this step is necessary only when an electrostatic load is applied. The reason of the contact load direction judgment in step (g) is that there is no limitation on _{i}_{i}_{i}_{−1}(_{1},

In addition, to speed up the calculations, a dynamic meshing can be used,

Pull-in analysis can be realized based on the above algorithm, by an iteration of deflection calculations under different voltages, similarly to what we have reported for a noncontact beam [

The novel contact analysis method was validated by FEM, using CoventorWare, which is a widely employed CAD software suite for MEMS that can realize contact analysis combined with electromechanical coupling analysis. To verify the analysis method’s applicability to different load types and different devices, both electrostatic load and mechanical load were applied in the validation. Accordingly, in this section, different to ^{3}, 169 Gpa and 0.3, respectively. Acceleration ^{2}) is applied to the cantilever, and voltage ^{′}, respectively. The contact loads applied by the 1st, 2nd, 3rd, 4th and 5th stoppers are recorded as _{1}, _{2}, _{3}, _{4} and _{5}, respectively, when they result from the above model, and as _{1}^{′}, _{2}^{′}, _{3}^{′}, _{4}^{′} and _{5}^{′}, respectively, when they are produced by CoventorWare. A zero contact load means no contact. Acceleration, voltage and stoppers’ initial distance to the cantilever are changed in

The novel model’s accuracy is illustrated in

3,000∼4,000 elements in CoventorWare make the calculation result trend to be stable, regardless of any further increase of the number of segments, and each case of

Multiple-contact beam’s complex deformation and the reliability demand on contact load make the design of the novel microaccelerometer need a search for suitable structure parameters values, resulting in a large amount of calculations. The above calculation model’s high accuracy and high speed make it competent to realize the design. One design example is shown in the following.

In the current design, the beam-mass structure as shown in

Structure parameters of the designed case are listed in

The beam-mass structure’s density, Young’s modulus and Poisson’s ratio are 2,500 kg/m^{3}, 169 Gpa and 0.3, respectively. Each fixed electrode-couple’s top area is small, and only a low voltage is needed, so electrostatic load is negligible compared with the high acceleration load, which results in a higher calculation speed because no convergence judgment is needed. The whole of the beam-mass structure is regarded as a cantilever with height step and width step, affording high calculation accuracy. The 1st, 2nd, 3rd, 4th and 5th fixed electrode-couple’s contact loads _{1}, _{2}, _{3}, _{4} and _{5} resulting from the novel model are listed in

A novel analysis method and a novel calculation model are developed for the analysis of beams’ multiple-contact mode. Deflection, contact load and contact region are obtained directly, with subsequent validation by CoventorWare. The contact analysis includes an electromechanical coupling analysis, and consequently pull-in voltage calculation and so on also are realizable. Though the analysis method is accurate, it isn’t complicated, and consequently it’s time-saving and has a good repeatability.

A novel design of a high-g threshold microaccelerometer is developed, characterized by the advantage that each sensitive component works under multiple-contact mode with multiple acceleration thresholds. This design reduces the sensor’s size considerably. In the design, low contact resistance is ensured by making the contact load above a demanded value.

As a universal model for beams’ contact mode, the model developed in this article can be degenerated to calculate the deflection and contact load at traditional low-g threshold microaccelerometers, microswitches, microgrippers and so on. The novel model also is applicable for acceleration threshold adjustments and built-in self-tests in low-g threshold microaccelerometers where an electrostatic load comparable to the acceleration load is applied by a fixed electrode besides the beam or mass [

Coupling relationship.

Segment division of beam.

Deflection curves of case 4 in

Structure parameters for the validation.

Cantilever’s total length | 1,000 |

Length of cantilever’s 1st part | 300 |

Width of cantilever’s 1st part | 20 |

Height of cantilever’s 1st part | 20 |

Length of cantilever’s 2nd part | 500 |

Width of cantilever’s 2nd part | 10 |

Height of cantilever’s 2nd part | 15 |

Length of cantilever’s 3rd part | 200 |

Width of cantilever’s 3rd part | 20 |

Height of cantilever’s 3rd part | 20 |

Length of each stopper | 10 |

1st stopper’s distance to cantilever’s fixed end in the length direction | 190 |

1st stopper’s initial gap to the cantilever | _{1} |

2nd stopper’s distance to cantilever’s fixed end in the length direction | 390 |

2nd stopper’s initial gap to the cantilever | _{2} |

3rd stopper’s distance to cantilever’s fixed end in the length direction | 590 |

3rd stopper’s initial gap to the cantilever | _{3} |

4th stopper’s distance to cantilever’s fixed end in the length direction | 740 |

4th stopper’s initial gap to the cantilever | _{4} |

5th stopper’s distance to cantilever’s fixed end in the length direction | 890 |

5th stopper’s initial gap to the cantilever | _{4} |

Fixed electrode’s initial gap to the cantilever | 5 |

Model’s Validation.

Initial distance between stopper and cantilever | _{1} |
0.5 | 0.5 | 0.5 | 1 | 1 |

_{2} |
0.5 | 0.5 | 0.5 | 1.5 | 1.5 | |

_{3} |
0.5 | 0.5 | 0.5 | 0.5 | 0.5 | |

_{4} |
1 | 1 | 0.5 | 1.5 | 1.5 | |

_{5} |
2 | 2 | 2 | 2 | 2 | |

Applied load | 5 × 10^{5} |
1 × 10^{6} |
5 × 10^{5} |
1 × 10^{6} |
0 | |

50 | 100 | 50 | 100 | 100 | ||

0.79950 | 0.82300 | 0.70994 | 1.1417 | 0.64497 | ||

Deflection | ^{′} (μm) |
0.80079 | 0.82419 | 0.71238 | 1.1426 | 0.64468 |

(^{′}) / ^{′} |
−0.16% | −0.14% | −0.34% | −0.08% | 0.04% | |

_{1} (μN) |
0 | 612.8 | 0 | 529.3 | 0 | |

_{1}^{′} (μN) |
0 | 646.2 | 0 | 557.9 | 0 | |

(_{1} – _{1}^{′}) / _{1}^{′} |
/ | −5.16% | / | −5.13% | / | |

Calculated contact load (Validation value of CoventorWare is signed with superscript apostrophe.) | _{2} (μN) |
624.8 | 1,201.4 | 679.2 | 499.7 | 0 |

_{2}^{′} (μN) |
625.1 | 1,187.1 | 680.8 | 500.7 | 0 | |

(_{2} – _{2}^{′}) / _{2}^{′} |
−0.05% | 1.20% | −0.24% | −0.20% | / | |

_{3} (μN) |
349.5 | 665.4 | 0 | 1,493.8 | 54.2 | |

_{3}^{′} (μN) |
349.4 | 669.6 | 0 | 1,486.2 | 55.4 | |

(_{3} – _{3}^{′}) / _{3}^{′} |
0.03% | −0.63% | / | 0.51% | −2.17% | |

_{4} (μN) |
340.8 | 560.0 | 851.0 | 0 | 0 | |

_{4}^{′} (μN) |
339.1 | 557.3 | 846.0 | 0 | 0 | |

(_{4} – _{4}^{′}) / _{4}^{′} |
0.50% | 0.48% | 0.59% | / | / | |

_{5} (μN) |
904.3 | 1,950.7 | 687.0 | 2,118.0 | 64.3 | |

_{5}^{′} μN) |
904.9 | 1,950.2 | 690.1 | 2,119.2 | 66.2 | |

(_{5} – _{5}^{′}) / _{5}^{′} |
−0.07% | 0.03% | −0.45% | −0.06% | −2.87% |

Structure parameters of the design example.

Beam’s length | 800 |

Beam’s width | 20 |

Beam’s height | 20 |

Mass’ length | 200 |

Mass’ width | 50 |

Mass’ height | 180 |

Length of each fixed electrode-couple | 10 |

1st fixed electrode-couple’s distance to beam’s fixed end in the length direction | 240 |

1st fixed electrode-couple’s initial gap to the beam | 0.3 |

2nd fixed electrode-couple’s distance to beam’s fixed end in the length direction | 315 |

2nd fixed electrode-couple’s initial gap to the beam | 0.6 |

3rd fixed electrode-couple’s distance to beam’s fixed end in the length direction | 365 |

3rd fixed electrode-couple’s initial gap to the beam | 0.9 |

4th fixed electrode-couple’s distance to beam’s fixed end in the length direction | 415 |

4th fixed electrode-couple’s initial gap to the beam | 1.3 |

5th fixed electrode-couple’s distance to beam’s fixed end in the length direction | 455 |

5th fixed electrode-couple’s initial gap to the beam | 1.7 |

Contact load of the design example (validation value from CoventorWare is in brackets).

1,000 | 2,000 | 3,000 | 4,000 | 5,000 | |

_{1} (μN) |
101.5 (100.9) | 117.7 (113.4) | 0 | 0 | 0 |

_{2} (μN) |
0 | 148.3 (150.7) | 128.3 (124.7) | 0 | 0 |

_{3} (μN) |
0 | 0 | 219.2 (221.7) | 167.9 (164.2) | 0 |

_{4} (μN) |
0 | 0 | 0 | 249.9 (252.7) | 198.7 (194.8) |

_{5} (μN) |
0 | 0 | 0 | 0 | 276.3 (279.5) |