A Relationship Between Defective Systems and Unit-Rank Modification of Classical Damping

A common assumption within the mathematical modeling of vibrating elastomechanical system is that the damping matrix can be diagonalized by the modal matrix of the undamped model. These damping models are sometimes called “classical” or “proportional.” Moreover it is well known that in case of a repeated eigenvalue of multiplicity m, there may not exist a full sub-basis of m linearly independent eigenvectors. These systems are generally termed “defective.” This technical brief addresses a relation between a unit-rank modification of a classical damping matrix and defective systems. It is demonstrated that if a rank-one modification of the damping matrix leads to a repeated eigenvalue, which is not an eigenvalue of the unmodified system, then the modified system is defective. Therefore defective systems are much more common in mechanical systems with general viscous damping than previously thought, and this conclusion should provide strong motivation for more detailed study of defective systems. [S0739-3717(00)00602-4]

Degenerate and Near Degenerate Materials

The Stroh formalism presented in Sections 5.3 and 5.5 assumes that the 6×6 fundamental elasticity matrix N is simple, i.e., the three pairs of eigenvalues pα are distinct. The eigenvectors ξα (α=l,2,3) are independent of each other, and the general solution (5.3-10) consists of three independent solutions. The formalism remains valid when N is semisimple. In this case there is a repeated eigenvalue, say p2=p1 ,but there exist two independent eigenvectors ξ2 and ξ1 associated with the repeated eigenvalue. The general solution (5.3-10) continues to consist of three independent solutions. Moreover one can always choose ξ2 and ξ1 such that the orthogonality relations (5.5-11) and the subsequent relations (5.5-13)-(5.5- 17) hold. When N is nonsemisimple with p2=p1, there exists only one independent eigenvector associated with the repeated eigenvalue. The general solution (5.3-10) now contains only two independent solutions. The orthogonality relations (5.5-11) do not hold for α,β=l,2 and 4,5, and the relations (5.5-13)-(5.5-17) are not valid. Anisotropic elastic materials with a nonsemisimple N are called degenerate materials. They are degenerate in the mathematical sense, not necessarily in the physical sense. Isotropic materials are a special group of degenerate materials that happen to be degenerate also in the physical sense. There are degenerate anisotropic materials that have no material symmetry planes (Ting, 1994). It should be mentioned that the breakdown of the formalism for degenerate materials is not limited to the Stroh formalism. Other formalisms have the same problem. We have seen in Chapters 8 through 12 that in many applications the arbitrary constant q that appears in the general solution (5.3-10) can be determined analytically using the relations (5.5-13)-(5.5- 17). These solutions are consequently not valid for degenerate materials. Alternate to the algebraic representation of S, H, L in (5.5-17), it is shown in Section 7.6 that one can use an integral representation to determine S, H, L without computing the eigenvalues pα and the eigenvectors ξα. If the final solution is expressed in terms of S, H, and L the solution is valid for degenerate materials.