EXISTENCE OF NATURAL FREQUENCIES OF SYSTEMS WITH ARTIFICIAL RESTRAINTS AND THEIR CONVERGENCE IN ASYMPTOTIC MODELLING

A major limitation of the Rayleigh–Ritz method for determining the natural frequencies of a system is the need to choose admissible functions that do not violate the geometric constraints of that system (Courant 1943 Bulletin of the American Mathematical Society49, 1–23). Several researchers have attempted to overcome this problem by asymptotically modelling the rigid constraints with artificial (imaginary) restraints of very large stiffness (Courant 1943Bulletin of the American Mathematical Society49 , 1–23; Warburton and Edney 1984 Journal of Sound and Vibration95, 537–552; Gorman 1989 Journal of Applied Mechanics56, 893–899; Kim et al. 1990 Journal of Sound and Vibration143, 379–394; Yuan and Dickinson 1992 Journal of Sound and Vibration153, 203–216; Yuan and Dickinson 1992 Journal of Sound and Vibration159, 39–55; Cheng and Nicolas 1992 Journal of Sound and Vibration155, 231–247; Yuan and Dickinson 1994Computers and Structures53 , 327–334; Lee and Ng 1994 Applied Acoustics42, 151–163; Amabili and Garziera 1999 Journal of Sound and Vibration224, 519–539; Amabili and Garziera 2000 Journal of Fluids and Structures14, 669–690). While the numerical results thus obtained for the systems considered in the literature were in close agreement with exact values for the natural frequencies corresponding to the first few modes, sample calculations show that the error introduced by the asymptotic modelling increases with mode number and therefore to obtain accurate results for higher modes the magnitude of stiffness should also be increased. In any event, the error due to the asymptotic modelling would remain uncertain, except when the correct frequency values are known. However, the use of artificial restraints with negative stiffness, a new concept which was introduced in a recent publication (Ilanko and Dickinson 1999 Journal of Sound and Vibration219, 370–378) paves the way for estimating the error due to asymptotic modelling. This is possible since in this work, the Rayleigh–Ritz frequencies of the constrained system were found to be bracketed by the frequencies of the asymptotic models with positive and negative restraints. However, the use of artificial restraints with negative stiffness has raised some important questions: would a system with a large negative restraint become unstable, and if so what is the guarantee that the frequencies of the asymptotic model would converge to that of the constrained system? This paper is the result of the authorʹs attempt to answer these questions and gives a proof of existence of natural frequencies for systems with artificial restraints (springs) having positive or negative stiffness coefficients, and their convergence towards constrained systems. Based on Rayleighʹs theorem of separation, it has been shown that a vibratory system obtained by the addition of h restraints to an n -degree-of-freedom (d.o.f.) system, where h<n, will have at least (n÷h) natural frequencies and modes and that as the magnitude of the stiffness of the added restraints becomes very large, these (n÷h) natural frequencies will converge to the (n÷h) natural frequencies of a constrained system in which the displacements restrained by the springs are effectively constrained.