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Scaling_2 - Scaling 2 6 Natural Frequency Let's go over the...

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Scaling – 2 6. Natural Frequency Let’s go over the cantilever beam analysis in section 5 again, but with a weight load at the end of the cantilever shown in Figure 3. Assume the width of the beam is b and the thickness of the beam is h . Figure 3. A cantilever beam with a concentrated load at the end The maximum deflection at the free end is EI FL y L 3 3 = (33) And the cross-sectional area moment of inertia is I = bh 3 /12 (34) Therefore, y L = MgL 3 3 E ( bh 3 /12) (35) The deflection scales as y L , i y L , R = M i gL i 3 3 E i ( b i h i 3 /12) M R gL R 3 3 E R ( b R h R 3 /12) = s i 3 s i 3 s i 4 = s i 2 (36) And the slope scales as F = Mg y L h b L
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y L , i / L i y L , R / L R = M i gL i 2 3 E i ( b i h i 3 /12) M R gL R 2 3 E R ( b R h R 3 /12) = s i 3 s i 2 s i 4 = s i 1 (37) Now we would like to consider the bending vibration of such cantilever beam. Recall the Newton’s law for free vibration for a damped mass-spring damping system shown in Figure 4. Figure 4 A damped mass-spring system K (N/m) is the spring constant and R (Ns/m) is the damping constant. We have, 0 2 2 = + + Κξ δτ δξ Ρ τ δ ξ δ Μ (38) For harmonic motion x(t)= x m e i ϖ t , where ϖ is the circular frequency, and x m is the maximum displacement. Therefore, 0 2 = + + - τ ι μ τ ι μ τ ι μ ε Κξ ιε Ρξ ε Μξ ϖ ϖ ϖ ϖ ϖ (39) If R=0 , then M K o = = ϖ ϖ (40) Here ϖ ο is defined as the natural circular frequency. The natural frequency is then defined as M K f o π ϖ π 2 1 2 1 = = (41) So, the natural frequency scales as x
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2 / 3 2 / 1 , , - = = ι ι Ρ Ρ ο ι ο σ Μ Μ φ φ (42) It is much larger in micro-scale. Therefore, a MEMS cantilever beam can substitute an
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