Vibrations of Continuous Systems: Axial Vibrations of a Uniform Bar
Equation of Motion
Consider a uniform bar shown in Figure 10.4(a) which has a length , cross–sectional area and density (mass per unit volume). Let measure the axial deformation of the material located at position in the equilibrium position. A FBD/MAD of an infinitesimal element of the beam is shown in Figure 10.4(b). Applying Newton’s Laws we get
For a linear elastic material
where is Young’s modulus and is the axial strain given by
For small deformations we can make the approximation that
The equation of motion then becomes
We recognize 10.19 once again as the wave equation. In this case is the speed of the axial waves traveling along the beam as opposed to the transverse motions for the cable seen previously.
Solution for the Axial Response of the Beam
Following our previous solution procedure for cables, we assume a solution of the form
Upon substitution into the equation of motion and proceeding as before we find that
As a result the total solution for the axial motions becomes
Once again the constants and are determined from the boundary conditions while and are subsequently determined from the initial conditions.
Some common boundary conditions for axial motions of beams are illustrated in Figure 10.5. The results for some common beam configurations are shown in Table 10.2. The procedure to find these results is the same as was followed for the vibrating cable considered earlier.