Vibroacoustic Simulation

Appendix B Specific Solutions

B.1 Second Moments of Area

The bending wave behavior of beams is determined by the cross-section of the beam. We saw in Equations (3.94)–(3.95) that those quantities are a result of an area integration for the infinitesimally small area times the lever related to the neutral axis. There are some definitions and rules that are mandatory to deal with beam dynamics. Figure B.1 shows a typical cross section from building construction. The I or double T-beam. The second moments of area for the axis through the centroid and thus neutral axes are

Figure B.1 Cross section of I-beam with axes ${\mathit{\text{x}}}^{\prime },{\mathit{\text{z}}}^{\prime }$ coinciding with the centroid. Source: Alexander Peiffer.

 (B.1)

 (B.2)

 (B.3)

Here, the first two terms are called the second moment of area with respect to the $y^{\prime }$ and $z^{\prime }$ axes, respectively. The last term is called the product moment of area. There is a further moment linked to the torsion of the beam, the polar moment of area $J_{xx}$

 (B.4)

The polar moment is related ...