This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 crosssectional 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 massspring damping system shown in Figure 4. Figure 4 A damped massspring 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 , , = =...
View
Full
Document
This note was uploaded on 02/02/2008 for the course AME 455 taught by Professor Han during the Spring '08 term at USC.
 Spring '08
 Han

Click to edit the document details