ON ISOSPECTRAL SPRING-MASS SYSTEMS

The paper concerns an in-line system of masses (mi()1)(n) connected to each other and to the end supports by ideal massless springs (k(i))(1)(n+1). Four ways are given for constructing a system which is isospectral to a given one: by using the interchange m(i) --> k(n-i+1)(-1), k(i) --> m(n-i+1)(-1) for a cantilever (k(n+1) = 0); by using the indeterminacy associated with the reduction to standard form; by using one or more shifted LL(T) factorizations and reversals; by using one or more shifted QR factorizations and reversals. It is shown that one may pass from any system to any isospectral system by a reduction to standard form, n - 1 QR factorizations and reversals, and a reversed reduction to standard form.