CA-AA242B-Ch4

CA-AA242B-Ch4 - AA242B: MECHANICAL VIBRATIONS AA242B:...

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Unformatted text preview: AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Dynamics of Continuous Systems These slides are based on the recommended textbook: M. G eradin and D. Rixen, Mechanical Vibrations: Theory and Applications to Structural Dynamics, Second Edition, Wiley, John & Sons, Incorporated, ISBN-13:9780471975465 AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Outline 1 Hamiltons Principle 2 Wave Propagation in a Homogeneous Elastic Medium 3 Free Vibrations of Continuous Systems and Response to External Excitation AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Hamiltons Principle Definitions Elastic body AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Hamiltons Principle Green Strains ds 2 = dx i dx i square of original length ds 2 = d ( x i + u i ) d ( x i + u i ) square of deformed length ds 2- ds 2 = 2 ij dx i dx j where ij = 1 2 u i x j + u j x i + u m x i u m x j is the Green symmetric strain tensor Note that ij rigid body motion AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Hamiltons Principle Green Strains Linear deformation (geometric linearity) the extension strains remain infinitesimal: u i x i 1 the rotations have small amplitudes: u i x j 1 the above assumptions lead to a linear expression of the infinitesimal strain tensor ij = 1 2 u i x j + u j x i consider a ~ ds parallel to ~ x 1 ds 2- ds 2 = ( ds- ds )( ds + ds ) = 2 11 dx 2 1 = 2 11 ds 2 = 11 = ds- ds ds 1 2 1 + ds ds for infinitesimal strains, the above result becomes 11 = ds- ds ds (engineering or Cauchy strain) AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Hamiltons Principle Stress-Strain Relationships Hyperelastic material: the work of the mechanical stresses is stored in the form of an internal energy and thus is recoverable ij = f ( kl ) Strain energy density: to a strain increment d ij in the stress state ij corresponds a strain energy per unit volume dW = ij d ij Z ij AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Hamiltons Principle Stress-Strain Relationships ij is energetically conjugate to the Green strain ij . It is called the second Piola-Kirchhoff stress tensor. It does not represent the true (Cauchy) stresses inside a structure with respect to the initial reference frame. Rather, it describes the stress field in a reference frame attached to the body and therefore subjected to its deformation but is related to the elementary area of the undeformed structure. In other words, the second Piola-Kirchhoff stress tensor relates forces in the reference configuration to areas in the reference configuration....
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This note was uploaded on 06/17/2010 for the course AA 242B taught by Professor Charbelfarhat during the Spring '10 term at Stanford.

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CA-AA242B-Ch4 - AA242B: MECHANICAL VIBRATIONS AA242B:...

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