Computational Mechanics JASS 2006 Survey of Wave Types and Characteristics Longitudinal Waves (For reminding only)  Pure longitudinal waves  Quasi-longitudinal waves Transverse Waves  Transverse plane waves  Torsional waves Bending Waves  Pure bending waves  Corrected bending waves Presented by: Xiuyu Gao

Computational Mechanics JASS 2006 Longitudinal Waves ( For reminding only )  Pure longitudinal Waves Longitudinal wave motions can occur in solids bodies. Where the direction of wave propagation coincides with the direction of the particle displacements. (Here D represents the longitudinal stiffness of the material) Displacements, deformations, and stresses in longitudinal wave motion

Computational Mechanics JASS 2006 The kinematics of a sound field in terms of the (particle) velocity: Then write the differentiation of with respect to time as: The Newton’s law relation is: Combination of the couplings and yields a wave equation: Here the propagation velocity is given by:

Computational Mechanics JASS 2006  Quasi-longitudinal Waves on Beams

Computational Mechanics JASS 2006

Computational Mechanics JASS 2006 Transverse Waves  Transverse Plane Waves Solids can resist changes in shape because of the shear stresses. Also because of it, transverse plane wave motions can occur in solids bodies. Where the direction of propagation (here taken as x-direction) is perpendicular to the direction of the displacement η (here taken as the y- direction). Displacements, deformations, and stresses in transverse wave motion

Computational Mechanics JASS 2006 If we replace the displacement in the y-direction by the corresponding velocity: Then write the differentiation of with respect to time as: The Newton’s law relation is: Combination of the couplings and yields a wave equation: Here the propagation velocity is given by:

Computational Mechanics JASS 2006 Here we derive the relationship between G and E :

Computational Mechanics JASS 2006  Torsional Waves If a narrow beam is excited by a torsional moment, all points on a cross- section rotate about the axis of the beam (suppose coincides with the x- axis)

Computational Mechanics JASS 2006

Computational Mechanics JASS 2006

Computational Mechanics JASS 2006 Bending waves  Pure bending waves Bending waves are by far the most important for sound radiation because of the rather large lateral deflections associated with them. Bending waves differs largely from both longitudinal waves and transverse waves. It must be represented by 4 field variables. Also the boundary conditions are more complex.

Computational Mechanics JASS 2006

Computational Mechanics JASS 2006

Computational Mechanics JASS 2006

Computational Mechanics JASS 2006

Computational Mechanics JASS 2006  Corrected bending waves The previously discussed pure bending waves is valid only if the bending wavelength is large compared to the dimensions of solids. In order to widen the theory to a more general one, we need 2 corrections: 1) Taking account of the deformations which are caused by shear stresses acting on the cross-section. (Timoshenko beam theory) 2) We need to add the previously omitted rotational inertia term

Computational Mechanics JASS 2006

Computational Mechanics JASS 2006

Computational Mechanics JASS 2006 Thank you!