Inverse mode problem of a horizontal bar structure, a single branch structure of a rod-beam-rod system, is investigated. Sign-oscillatory properties of inherent vibration stiffness matrix of discrete horizontal bar structure are shown mathematically. In addition, qualitative properties of frequencies and modes are introduced. Physical parameters of horizontal bar structure are constructed from two sets of displacement or strain modes and corresponding circular frequencies. Several numerical examples are given. A particular inverse mode problem is discussed, where structure parameters of horizontal bar are symmetrical for geometrical symmetry axis. It shows that formulation of inverse problems is reasonable and solution method is validated.

Inverse mixed problems of a discrete system for a rod are studied, i.e., stiffness and mass matrices of discrete system of a rod are reconstructed from some frequency data and part of mode data. Three inverse mixed problems are formulated. Solving methods for these problems are given. And conditions of existing solutions for these inverse problems are discussed. Numerical examples are also shown. Potential worth for this type of inverse problems is analysed.

Conditions and method for constructing stiffness distribution function of various parameters symmetric simple supported beams with a symmetric or anti-symmetric mode and specified polynomial density distributing function are discussed. It is shown that the constructed stiffness distribution functions are positive functions in case with different density distributing. Two numeral examples are given. And two problems correlated are explained.

Mode inverse problem for a beam with overhangs is discussed.Flexural rigidity and density of beam with overhangs are solved with two groups of displacement modes or two groups of strain modes and corresponding frequency.Necessary and sufficient conditions for unique existence of solution of the problem are discussed.An algorithm is proposed and numerical calculations are carried out.Two examples show that better results can be attained if strain modes instead of displacement mode are employed.