Lecture08-Prof.Ju

Lecture08-Prof.Ju - CEE M237A / MAE M269A Lecture 8:...

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 8: Vibrations of Undamped Continuous Systems (Strings) 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
Problem Overview and Summary of General Solution Procedure 2 y Description by partial differential equations (PDEs) involving both spatial and time variables y Methods for generating exact solutions to these PDEs are developed in this lecture y The governing PDE of motion of a structural member has the general form: (next slide)
Background image of page 2
Problem Overview and Summary of General Solution Procedure 3 () { } ( ) ( ) ( ) ( ) {} 123 where , -kinematic varibale(s) with as time and , , ,... of order 2 in the -stiffness differential operator spatial variables -mass distributio ,, , n kk k K ux tt x x tM x u x t F x t F t x x xx p K Mx δ += = ⎛⎞ + ±±
Background image of page 3

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

View Full DocumentRight Arrow Icon
Problem Overview and Summary of General Solution Procedure 4 () , -time-dependent distributed force excitation -time-dependent concentrated force excitation at 1,2,. .. , and is t kk k Fx t Ft x x xx k δ == he Dirac delta function y The dependent variables and the orders of the structural operator for various structural members are summarized in the next Table: { } K
Background image of page 4
Problem Overview and Summary of General Solution Procedure 5 Problem Type Order (2p) of K{ } Spatial Dimensions Stiffness Characteristics Inertial Dependent Variable(s) String 2 1D Tension T Transverse Mass Transverse Displ. Rod 2 1D Extensional Rigidity EA Longitudinal Mass Displ. Torsion (Circular Rod) 21 D Torsional Rigidity GJ Rotatory Inertia Angle of Twist Beam 4 Flexual Rigidity EI Shear Beam (Timoshenko) 2 eqns. Each of 2 1D Flexural Rigidity EI and Shear Stiffness k 2 GA Transverse Mass and Rotatory Inertia Transverse Displ. And Bending Rotation Membrane 2 2D Plate 4 2D D Shear Plate (Mindlin- Reissner) 42 D D and Shear 2 And 2 Bending Rotations
Background image of page 5

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

View Full DocumentRight Arrow Icon
Problem Overview and Summary of General Solution Procedure 6 () {} 0 0 , 0; 1,2,. .., at each boundary point where 's are linear, homogeneous differential operator with Boundary Conditions a derivatives of orders 2 1 and lower re . : i i Bu xt i p x x B p == = ( ) ( ) 00 Initial Conditions are ,0 ; : ux u x v x ±
Background image of page 6
Free Vibration Analysis 7 () { } ( ) ( ) {} 0 0 ; 1,2,. .., at each boundary point Boundary value problem BVP The solution provides data that characterize the total r B esponse o .C.: EO f a structure in the ,0 f M :, , 0 orm i Kuxt Mx B uxt ux p x t ix = = = = + ±± of natural frequencies and corresponding mode shapes Spectral decomposition of the governing operator
Background image of page 7

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

View Full DocumentRight Arrow Icon
Free Vibration Analysis (Cont’d) 8 () ( ) ( ) ( ) {} ( ) 2 2 2 The displacement solution is taken as a product of func 0 0 tions of space and time 0 : , ux t X xT Tt K X x M xX x KXx Mx Xx T t M x X t x ω ⇒+ = ⇒= = ⇒∴ + = = = ±±
Background image of page 8
Free Vibration Analysis (Cont’d) 9 () { } () ( ) ( ) ( ) Hence, we obtain simple harmonic motion in time:
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.

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 / 31

Lecture08-Prof.Ju - CEE M237A / MAE M269A Lecture 8:...

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