Abstract
Based on the Finite Element Method (FEM) model of a practical silicon beam resonator attached to a square diaphragm used for measuring pressure, this paper presents two location error models which exist in actual fabrication. We calculate, analyze and investigate the relationship between the basic natural frequency of the beam resonator and the measured pressure for two error models by making use of FEM. In order to improve the exchangeability of the sensor, it is necessary to monitor the processing accuracy inx-andy-axes, and the reference angle relative to the ideal location within the positive stress range. It is also necessary to monitor the processing accuracy in thex-axis within the negative stress range, as the beam axial direction is along thex-asis the square diaphragm.
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Abbreviations
- A,H :
-
Half length and thickness of the square diaphragm
- X,Y :
-
Cartesian coordinate of the square diaphragm
- P :
-
Pressure
- A 0,A 0+L :
-
Axial coordinates of the beam in Cartesian coordinate of the square diaphragm
- L, b,h :
-
Length, width and thickness of the beam which is attached to the square diaphragm
- a,a :
-
The local coordinates of the beam in Cartesian coordinate of the beam
- B,D :
-
x-axis andy-axis deviations of the beam relative to its ideal location
- α[deg.]:
-
The angular deviation of the beam relative to its ideal location
- W (x,y) :
-
Displacement of the square diaphragm under the applied pressureP
- Ds :
-
The flexural rigidity of the square diaphragm
- E,ρ,μ:
-
Young modulus, density and poisson ratio of the sensing structure
- W max :
-
Ratio between the maximum normal displacement and the thickness of the square diaphragm
- σ x (x,y),σ y (x,y):
-
Stresses of the square diaphragm
- u (s,z,t)-w (s,t):
-
Axial and normal vibrating displacements of the beam in Cartesian coordinate of the beam
- t :
-
Time
- s,z :
-
Axial and normal coordinates of the beam in Cartesian coordinate of the beam
- σ s 0(s):
-
Initial axial stress of the beam
- S :
-
Integrated length of the beam
- ω[rad/s],w(s) :
-
Natural frequency and its corresponding vibrating shape along the axial direction of the beam
- U :
-
Potential energy of the beam
- T :
-
Kinetic energy of the beam
- U 0 :
-
Initial potential energy of the beam, which is caused byσ s 0(s)
- U T :
-
Total potential energy of the beam
- S j ,S j+1 :
-
The jth and the (j+1)th node of the beam element
- N :
-
Total number of the beam element
- q :
-
Dimentionless variable in domains∈[S j ,S j+1]
- l :
-
Half length of the beam element in domains∈[S j ,S j+1]
- w j (s),w j (q) :
-
Displacement of the beam element in domains[S j ,S j+1]
- Q 2 0 :
-
The second order Hermite interpolation vector [1q q 2 qs q 4 q 5]
- G 2 :
-
The second order Hermite interpolation matrix
- a j :
-
Element nodal displacement vector [w(−1)w(−1)w″ (−1)w(+1)w′(+1)w″ (+1)]T
- U 2 :
-
Potential energy of the beam element in domains∈[S j ,S j+1]
- T 2 :
-
Kinetic energy of the beam element in domains∈[S j ,S j+1]
- U o 3 :
-
Initial potential energy of the beam element in domains∈[S j ,S j+1]
- U T j :
-
Total potential energy of the beam element in domains∈[S j ,S j+1]
- K j :
-
The beam element stiffness matrix
- M j :
-
The beam element mass matrix
- K o j :
-
The beam element initial stiffness matrix
- K T j :
-
The beam element total stiffness matrix
- K :
-
The assembly stiffness matrix
- M :
-
The assembly mass matrix
- a :
-
The assembly nodal displacement vector
- f(P,B,D,α) [Hz]:
-
Basic natural frequency of the beam for pressureP, withx-axis deviationB, y-axis deviationD and the angular deviation α relative to its ideal location
- Δf (B,D,α) [H z ]:
-
Variation of the basic natural frequency of the beam within (0,P), withx-axis deviationB, y-axis deviationD and the angular deviation α relative to its ideal location
- β (B,D,α):
-
Relative variation of the basic natural frequency variation of the beam within (0,P) withx-axis deviationB, y-axis deviationD and the angular deviation α relative to its ideal location
- j :
-
Number of the element
- T :
-
Transpose of matrix
- j :
-
Number of the element
- T :
-
Total
- FEM :
-
Finite element method result
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Fan, S., Liu, G., Lee, M.H. et al. Modeling and simulation of a silicon beam resonator attached to a square diaphragm. KSME International Journal 12, 339–346 (1998). https://doi.org/10.1007/BF02946348
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DOI: https://doi.org/10.1007/BF02946348