Lecture16-Prof.Ju

# Lecture16-Prof.Ju - CEE M237A MAE M269A Lecture 16...

This preview shows pages 1–8. Sign up to view the full content.

CEE M237A / MAE M269A Lecture 16: Hamilton’s Principle on Dynamic Structural Systems IV: Equation of Motion via Hamilton’s Principle Professor J. Woody Ju

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Example 5–Forced Vibration of a Beam –Column Supporting a Mass 2 () 2 This column is in the presence of a gravity field whose acceleration of gravity is units: / . This structural system is driven by a forcing function with a displacement history of (such as an earth M gL T Wt 1 quake) applied to mass . Determine the EOM and the appropriate BCs. M
Example 5–Forced Vibration of a Beam – Column Supporting a Mass (Cont’d) y The Lagrangian L is: ( w r is the relative displ.) y The first variation of the time integral of L gives 3 ( ) ( ) Note: 0 M Wt δ =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Example 5–Forced Vibration of a Beam – Column Supporting a Mass (Cont’d) y Integration by parts of the variations w.r.t. the velocities shows that: 4
Example 5–Forced Vibration of a Beam – Column Supporting a Mass (Cont’d) y Integration by parts of the variations of the potential energy gives: 5

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Example 5–Forced Vibration of a Beam – Column Supporting a Mass (Cont’d) y Recomposing the total variation renders: 6
Example 5–Forced Vibration of a Beam – Column Supporting a Mass (Cont’d) y Therefore, the Euler Equation becomes: y The Natural and Kinematic Boundary Conditions are: 7 222 2 22 2 0 rr rM ww Aw M g EI AW xx x ρρ ⎛⎞ ∂∂∂ ++ + = ⎜⎟ ∂∂ ⎝⎠ ±±

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 26

Lecture16-Prof.Ju - CEE M237A MAE M269A Lecture 16...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online