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![Page 1: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/1.jpg)
18-04-23
Challenge the future
DelftUniversity ofTechnology
Stochastic FEM for analyzing static and dynamic pull-in of microsystems
Stephan Hannot, Clemens Verhoosel and Daniel Rixen
![Page 2: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/2.jpg)
2Stochastic FEM for analyzing pull-in
Introduction
Microsystems or Micro-Electro-Mechanical Systems.
Typical dimensions 1~100 micrometers
Microsystems
![Page 3: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/3.jpg)
3Stochastic FEM for analyzing pull-in
Introduction
At these small scales physical forces act different.
For instances electrostatic forces can deform and move things.
Electro-mechanical coupling
![Page 4: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/4.jpg)
4Stochastic FEM for analyzing pull-in
IntroductionPull-in voltage
![Page 5: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/5.jpg)
5Stochastic FEM for analyzing pull-in
IntroductionFinite element model
![Page 6: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/6.jpg)
6Stochastic FEM for analyzing pull-in
Contents
•Stochastic Finite Element Method•Static pull-in• FEM computation• Sensitivities• Stochastic analysis
•Dynamic pull-in• FEM computation • Sensitivities • Stochastic analysis
•Conclusions
![Page 7: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/7.jpg)
7Stochastic FEM for analyzing pull-in
Stochastic FEM
A material property is not fixed, definitely at the
microscale it can be an highly uncertain value.
•For instance in the 1D example• Assume k is random, but normally
distributed. • What happens to the pull-in voltage?
Problem definition
![Page 8: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/8.jpg)
8Stochastic FEM for analyzing pull-in
Stochastic FEM
Generate N different values of k and compute N
pull-in voltages, subsequently determine the
distribution of the pull-in voltages.
•Advantages• Conceptually simple• Very robust
•Disadvantage• Computationally very expensive
Crude Monte Carlo Simulation
![Page 9: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/9.jpg)
9Stochastic FEM for analyzing pull-in
Compute the sensitivities of V with respect to k, and use these to approximate the distribution.
•Advantages• Computationally very cheap
•Disadvantage• Design sensitivities required• Only information about mean and variance
Stochastic FEMPerturbation Stochastic FEM
![Page 10: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/10.jpg)
10Stochastic FEM for analyzing pull-in
Contents
•Stochastic Finite Element Method•Static pull-in• FEM computation• Sensitivities• Stochastic analysis
•Dynamic pull-in• FEM computation • Sensitivities • Stochastic analysis
•Conclusions
![Page 11: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/11.jpg)
11Stochastic FEM for analyzing pull-in
Static pull-inFEM model
![Page 12: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/12.jpg)
12Stochastic FEM for analyzing pull-in
Static pull-in
Pull-in resembles limit pointbuckling, therefore the classic
limitpoint buckling sensitivity can
beused:
Sensitivities
![Page 13: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/13.jpg)
13Stochastic FEM for analyzing pull-in
Static pull-in
The perturbation FEM will be compared with crude Monte
Carlo.
It is assumed that the Young’s modulus of material is distributed normally with the following characteristics:
Stochastic analysis
![Page 14: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/14.jpg)
14Stochastic FEM for analyzing pull-in
Static pull-in
In that case MC gives
Stochastic analysis
![Page 15: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/15.jpg)
15Stochastic FEM for analyzing pull-in
Static pull-in
And perturbation FEM gives:
Which is almost the same.
Stochastic analysis
![Page 16: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/16.jpg)
16Stochastic FEM for analyzing pull-in
Contents
•Stochastic Finite Element Method•Static pull-in• FEM computation• Sensitivities• Stochastic analysis
•Dynamic pull-in• FEM computation • Sensitivities • Stochastic analysis
•Conclusions
![Page 17: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/17.jpg)
17Stochastic FEM for analyzing pull-in
Dynamic pull-inFEM model
Step load of 40 Volt Step load of 41 Volt
![Page 18: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/18.jpg)
18Stochastic FEM for analyzing pull-in
Dynamic pull-inFEM model
The transition is rather sharp
![Page 19: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/19.jpg)
19Stochastic FEM for analyzing pull-in
Dynamic pull-inSensitivities
There is problem, mathematically it is difficult to define pull-in. However there is a work around.
![Page 20: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/20.jpg)
20Stochastic FEM for analyzing pull-in
Dynamic pull-inUncertainty analysis
Monte Carlo Perturbation approach
![Page 21: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/21.jpg)
21Stochastic FEM for analyzing pull-in
Dynamic pull-inReliability analysis
Monte Carlo Perturbation approach
What is the chance that the dynamic pull-in is below a critical value of V=37
Compute critical Ec, V(Ec)=37
What is the chance that E<Ec
![Page 22: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/22.jpg)
22Stochastic FEM for analyzing pull-in
Contents
•Stochastic Finite Element Method•Static pull-in• FEM computation• Sensitivities• Stochastic analysis
•Dynamic pull-in• FEM computation • Sensitivities • Stochastic analysis
•Conclusions
![Page 23: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/23.jpg)
23Stochastic FEM for analyzing pull-in
Conclusions
•Analytical sensitivities of Static and dynamic pull-in were derived.
•These sensitivities are sufficient for performing a Stochastic analysis.
•A more robust definition of dynamic pull-in would be nice for a more robust sensitivity computation.
![Page 24: 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens.](https://reader036.fdocuments.in/reader036/viewer/2022062714/56649d055503460f949d9411/html5/thumbnails/24.jpg)
24Stochastic FEM for analyzing pull-in
Thank you for your attention