tailieunhanh - MODELLING OF MECHANICAL SYSTEM VOLUME 2 Episode 3

Tham khảo tài liệu 'modelling of mechanical system volume 2 episode 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 58 Structural elements hold the dilatational P waves are reflected without mode conversion. Again it is stressed that sliding supports would be difficult to manufacture. Incidentally if the solid is replaced by a non-viscous fluid . a medium in which the Poisson effect and the shear stresses do not exist the present calculation takes on its full practical interest. In a fluid the dilatation waves are called acoustic waves and the natural modes defined by the formulas are the acoustic modes of a rectangular enclosure provided with rigid and motionless walls. Shear plane modes of vibration Starting from the plane shear waves it is immediately verifiable that shear stresses ơxy GdY dx and ơxz GdZ dx are the only nonzero stress components. They both must vanish at the faces x 0 and x Lx which are assumed to be free. Whence the boundary conditions dY dx 0 dY dZ dTxLx 0 dxữ dZ -p- 0 dx L Lx From it follows that the partial differential equation to be solved is GA Y x J Z x k M2p0 Y x Zxk 0 Applying the relation in Appendix to calculate the Laplacian of the displacement vector it is readily shown that the vector equation can be split into two identical and uncoupled equations of the same type as those already encountered in the last examples d2Y Ũ 2 d2Z Ũ 2 dx2 Cs Y 0 dx2 Cs Z 0 Cs Hence the waves oscillating in the Oy direction are found to be identical to those oscillating in the Oz direction. Such a result could have been anticipated since the transverse sections parallel to the Oyz plane are not deformed and oscillate as a rigid body in the Oy and in the Oz directions independently. As a consequence the corresponding natural modes of vibration can be split into two families which are identical to each other except that they vibrate into two orthogonal directions. The common modal quantities are T . y nnx nncs pn x cosl - I n j - n 0 1 2 . Lx Lx By using the principle of superposition the .

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