Lecture14-Prof.Ju

Lecture14-Prof.Ju - CEE M237A / MAE M269A Lecture 14:...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
CEE M237A / MAE M269A Lecture 14: Hamilton’s Principle on Dynamic Structural Systems II 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
Potential Energy of External Forces y Potential energy of external forces relates energy that can be recovered from the forces and/or the loads acting on a given structural component or system y It is defined as the negative of the volume integral of the products of applied forces and their corresponding displacements y These potential energy expressions are for the same structural members, for which their kinetic and strain energies were given in the previous two sections 2
Background image of page 2
Potential Energy of External Forces (Cont’d) y For an Axial Force Member: y For a Bernoulli Euler Beam: 3 () ( ) ( ) 0 1 ,, , L Ex x k k k k VF x t u x t F t u x t x x d x δ = ⎧⎫ =− + ⎨⎬ ⎩⎭ ( ) ( ) 0 1 , , , L E zz k k k k x t w x t F t w x t x x wxt mxt d x x = + +
Background image of page 3

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

View Full DocumentRight Arrow Icon
Potential Energy of External Forces (Cont’d) y For a Circular Cylindrical Torsion Member: 4 () ( ) ( ) 0 1 ,, , L Ex x k k k k Vt x t x t T t x t x x d x ΘΘ δ = ⎧⎫ =− + ⎨⎬ ⎩⎭
Background image of page 4
Potential Energy Due to Axial Shortening y Nonlinear strain-displacement relation for the axial strain in two dimensions: y For small axial strains: 5 () 22 ,, , 1 , 2 xx ux t t wx t xt xx x ε ⎛⎞ ∂∂ =+ + ⎜⎟ ⎝⎠ ( ) 2 t t x x <<
Background image of page 5

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

View Full DocumentRight Arrow Icon
Potential Energy Due to Axial Shortening (Cont’d) y If the axial force is significant or large deflections are experienced by the beam 6 () ( ) 2 ,, 1 , 2 xx ux t wx t xt xx ε ⎛⎞ ∂∂ =+ ⎜⎟ ⎝⎠
Background image of page 6
Potential Energy Due to Axial Shortening (Cont’d) y For an inextensional deformation: 7 () ( ) 2 2 2 0 2 0 ,, 1 ,0 2 1 2 , 1 2 , 2 xx L L E ux t wx t xt xx t t wxt Ld x x P VP L d x x ε Δ ⎛⎞ ∂∂ =+ = ⎜⎟ ⎝⎠ ⇒= =
Background image of page 7

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

View Full DocumentRight Arrow Icon
Equations of Motion via Hamilton’s Principle y Hamilton’s principle incorporates all of the essential equations for a given problem in a single variational statement: y A familiarity with the rudiments of the calculus of variations is needed to execute the steps of the variational procedure 8 2 1 ; 0 t t TV d t =− = δ LL
Background image of page 8
Image of page 9
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 05/18/2011 for the course MAE 269A taught by Professor Ju during the Spring '11 term at UCLA.

Page1 / 25

Lecture14-Prof.Ju - CEE M237A / MAE M269A Lecture 14:...

This preview shows document pages 1 - 9. Sign up to view the full document.

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