ABSTRACT

In the present paper, an inverse problem of the lateral natural vibration of " risers is discussed". After eigenvalues λ, < λ,<… and corresponding normalization constants ρ, ρ, …, rigidity EI(s) and mass M(s) given, the effective pressure distributionI(s) and added displacement-dependent force coefficient B(s) will be determined.

1. INTRODUCTION

Generally, after the original design of a riser, the natural vibration must be check, in order to avoid the large amplitude resonance motion. Mathematically, this results in a self-adjoint eigenvalue problem of a four-order ordinary differential equation (Gardner, 1976), so the natural frequencies ( o i)and corresponding normalization constants (ρ oi):, are known. But, for the optimun design, we would like to change (oi), into ( i);, therefore, the inverse eigenvalue problem must be considered. Somr progresses of the inverse problem for higher order are obtained by Barcilon(1971). Leibenzon(1966). Mokeuna(1971), MCtaughlin(1976, 1978, 1981). A review was given by Mclaughlin (1982). The constructive technique by Mclaughlin(1976) will be extended in this present paper.

2. THE MATHEMATICAL MODEL OF THE INVERSE PROBLEM

The natural vibration of risers can be described By the following, four-order ordinary differential equation. There are many kind of boundary" conditions of risers, here we only discuss the case of the simple support at two ends. The method presenting in this paper can be extended to other boundary conditions.

4.EXISTENCE OF THE SOLUTION OF THE INTEGRAL EQUATION

Next, we prove there exists at most one solution of the integral equation (12). Therefore, h(s, l)=O for o..;;t..;;s (s";;l) from (l7). This implies that (14) only has a zero solution, or there exists at most one solution of (12)" So, if we can find a solution of(12), that must be the solution we seek. Next, we will construct the solution of (12).

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