Assume the width and thickness of a beam to be 10 μm and 50 μm, respectively. The beam is made of silicon with Young’s modulus, 19000 MPa, and 10 mg weight is attached to the lever. Determine the equivalent spring constant and natural frequency of the beam element in a micro-accelerometer.

Question

Accepted Answer

Referring to Fig. 8.12, the moment of inertia of the beam cross section

[latex]I=\frac{bt^3}{12} =\frac{(10 imes 10^{-6} )(50 imes 10^{-6})}{12} =0.1042 imes 10^{-18}[/latex]

The spring constant

[latex]k=\frac{w}{\delta } =\frac{3EI}{l ^3} [/latex]

where δ is the deflection, E is Young’s modulus, and l is the length of the beam.
Hence

[latex]k=\frac{3(190000 imes 10^{-6} )(0.1042 imes 10^{-18})}{(100 imes 10^{-6})^3} =59.39[/latex] N/m

The natural frequency of the cantilever beam

[latex]=\sqrt{\frac{k}{m} } =\sqrt{\frac{59.39}{10^{-5}} } = 2437 rad/sec[/latex]

Question 8.5: Assume the width and thickness of a beam to be 10 μm and 50 ...

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