Introducing the use of positive and negative inertial functions in asymptotic modelling

The work presented here shows that the natural frequencies of constrained systems may be obtained from asymptotic models of corresponding systems where the constraints are replaced by artificial mass or moment of inertia of very large positive and negative values. This offers a convenient alternative to the current practice of using artificial elastic restraints of large stiffness, a concept introduced by Courant in 1943, to remove a limitation on the choice of admissible functions. Recent publications show that in order to control the error caused by approximating constraints with restraints of large stiffness, it is necessary to use both positive and negative stiffness values. However, the negative stiffness introduces instability near the lower modes of vibration and the magnitude of negative stiffness parameter used must be greater than the highest critical stiffness to ensure bounded results are obtained. The use of positive and negative artificial inertial parameters overcomes this problem as they do not introduce instability near the lower modes, allowing the natural frequencies of constrained systems to be delimited to any desired accuracy.