One-dimensional Mechanical System

• Longitudinal Waves in a Bar
  • Uniform cross-section
  • Loaded with a uniform stress on its end faces
  • Dimensions shown in figure:

• Material properties of the bar:
  •  Young's modulus E
  • Density r

• Longitudinal (extensional or compressional) velocity:

• Approximation of longitudinal velocity by considering lateral displacements:

where:

           ...   wave number

              ...   Poisson ratio

     ...   moment of inertia of the cross-section

• Model corresponding to the simple lossless structure of the preceding figure:

Impedances in this circut are defined as: