Mode count and modal density of structural systems: relationships with boundary conditions

The effects of boundary conditions on the mode count and modal density of one- and two-dimensional structural systems, beams and plates, respectively, are investigated by using the wavenumber integration method. Bending vibrations are examined first for a single beam. In this case it is demonstrated that the average mode count is reduced by between 0 and 1 for each boundary constraint, depending on the type of boundary conditions. For more generalised mass and stiffness constraints a frequency-dependent coefficient, which need not lie between 0 and 1, is obtained. The effects of line constraints on the mode count of two-dimensional systems are similar to the equivalent one-dimensional constraints but they are always frequency dependent. Then the mode count of systems of multiple collinear beams and coplanar plates is studied. It is found that an intermediate constraint has the same effect on the average mode count as the same type of constraint applied at an end of the system. The modal density is largely independent of boundary conditions for one-dimensional systems although there are exceptions, while it is dependent on boundary conditions for two-dimensional systems. The results are compared with those from previously published formulae for natural frequencies and with results from finite element method (FEM) analysis. Inclusion of the effect of the boundary conditions in statistical energy analysis (SEA) estimations will result in improved agreements with both analytical and numerical results.