Scaling_2

Scaling_2 - Scaling 2 6. Natural Frequency Lets go over the...

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Unformatted text preview: Scaling 2 6. Natural Frequency Lets 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 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 Newtons 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, 2 2 = + + (38) For harmonic motion x(t)= x m e i t , where is the circular frequency, and x m is the maximum displacement. Therefore, 2 = + +- (39) If R=0 , then M K o = = (40) Here is defined as the natural circular frequency. The natural frequency is then defined as M K f o 2 1 2 1 = = (41) So, the natural frequency scales as x 2 / 3 2 / 1 , ,- = =...
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This note was uploaded on 02/02/2008 for the course AME 455 taught by Professor Han during the Spring '08 term at USC.

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Scaling_2 - Scaling 2 6. Natural Frequency Lets go over the...

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