Lecture01-Prof.Ju

Lecture01-Prof.Ju - CEE M237A / MAE M269A Lecture 1: Free...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
CEE M237A / MAE M269A Lecture 1: Free Vibration Analysis of SDOF Equation of Motion Professor J. Woody Ju
Background image of page 1

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

View Full DocumentRight Arrow Icon
SDOF Systems y Components of the basic dynamic system (SDOF) 2 Free body diagram for the mass m
Background image of page 2
E.O.M (Linear system) y Method 1: Direct Equilibrium by D’Alembert Principle 3 where ( ) ( ) ( ) ( ) , damping constant () D I D S S I ft m v t c v t c f f tf tk v t mv t cv t kv t p t t p t ++ = == = ⇒∴ + + = = ±± ± ± Inertia force: Damping force: Spring force:
Background image of page 3

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

View Full DocumentRight Arrow Icon
E.O.M (Linear system) (Cont’d) y Method 2: Principle of V.W. (V.D.): 4 Consider a virtual displacement v δ same direction ("negative", because the direction of force is opposite to negative work) () 0 IDS v f vfvf v p t v δδ −− + = ±²²²³²²²´
Background image of page 4
E.O.M (Linear system) (Cont’d) y Method 3: Hamilton’s Principle (Variational) 5 2 2 negative work 1 2 1 2 () nc Tm v m v v Vk v k v v Wp t v c v v δ δδ ⎛⎞ == ⎜⎟ ⎝⎠ =− ±± ± ± ²³´ Kinetic energy: Elastic energy: External work:
Background image of page 5

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

View Full DocumentRight Arrow Icon
E.O.M (Linear system) (Cont’d) y Hamilton’s Principle (Cont’d) 6 [] 22 2 1 1 1 1 2 0( s t a 12 m (integration by parts) ( 0 at and in Hamilton's Principle () ) 0 tt t t t t t mv v cv v kv v p t v dt mv cv kv p mv vdt mv v mv vdt v t δ δδ =− = −−+ = −− ⇒∴ −+ ∫∫ ±± ± ± ± ± ± N 2 1 arbitrary e E.O.M.) 0 t t vd t = ²³³³´³³³µ
Background image of page 6
Influence of Gravity 7 θ v k c f(t) m Consider the inclined SDOF dynamic system at an angle θ :
Background image of page 7

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

View Full DocumentRight Arrow Icon
Influence of Gravity (Cont’d) 8 ± cz ±± mz m gs i n θ f ( ) sin ( ) sin ( ) ( ) sin Define ( , Equilibrium e ) qn.: sin ( ) (Note: ) u tm g c z k z m z mz cz kz f t mg mu cu ku k f t mg mg k mu cu ku f z t zu δ +− = ⇒+ + = + ++ = + ⇒∴ + =− = = = + = ± ± ± ± ± ± u = relative displacement; z = total displacement; δ = static equil. deformation Therefore, the gravity has no effect on the E.O.M. if the relative displ. u is used .
Background image of page 8
Influence of Support Excitation 9 N N N () tg relative displ. total displ. support displ.
Background image of page 9

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

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 30

Lecture01-Prof.Ju - CEE M237A / MAE M269A Lecture 1: Free...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online