A structured approach to design-for-frequency problems using the Cayley-Hamilton theorem

Abstract

An inverse eigenvalue problem approach to system design is considered. The Cayley-Hamilton theorem is developed for the general case involving the generalized eigenvalue vibration problem. Since many solutions exist for a desired frequency spectrum, a discussion of the required design information and suggestions for including structural constraints are given. An algorithm for solving the inverse eigenvalue design problem using the generalized Cayley-Hamilton theorem is proposed. A method for solving partially described systems is also specified. The Cayley-Hamilton theorem algorithm is shown to be a good design tool for solving inverse eigenvalue problems of mechanical and structural systems.

A structured method for the design-for-frequency of a brace-soundboard system using a scalloped brace

“…This greatly reduces the computational effort involved. Once again, constraints similar to those provided by (11) can be used to create a physically realistic system.…”

“…Eq. (10) is a matrix containing eighty-one ninth order equations, of which only nine are independent, as shown in [11]. If we solve these nine independent equations for the two unknown geometric/structural parameters and seven unknown eigenvalue parameters, solutions to the system are obtained.…”

“…As described in [11], the Cayley-Hamilton theorem can be used to solve general structured inverse eigenvalue problems. The generalized Cayley-Hamilton theorem states that if p(t) is the characteristic polynomial of the generalized eigenvalue problem (K, M), where K and M are square matrices obtained from p(t) = det (K À tM), then substituting (M À1 K) for t in the polynomial gives the zero matrix [15,16].…”

“…Two methods are proposed and demonstrated for solving this problem. The first method, first described in [11], consists of using the generalized Cayley-Hamilton theorem as an algorithm for solving the inverse eigenvalue problem. The second method is referred to as the determinant method and is described below.…”

“…very few methods exist for matrices which have a more general form. One of the goals of this paper is to demonstrate the use of the recently proposed inverse eigenvalue technique using the Cayley-Hamilton theorem [11]. This approach is particularly interesting because it enables the solution of any matrix structure.…”