Exact natural frequencies of multi-level elastically connected taut strings and related problems

The dynamics of a family of simple, but extremely useful structural elements is governed by a second order Sturm-Liouville equation. This equation allows for the uniform distribution of mass and stiffness and enables the motion of strings and shear beams, together with the axial and torsional motion of bars to be described exactly. As a result, each member type in this family has been treated exhaustively when considered as a single member or when joined contiguously to others. However, when such members are linked in parallel by uniformly distributed elastic interfaces, their complexity becomes significantly more intractable and it is this class of structures that has led to renewed interest and which forms the basis of the work that follows. Initially, differential equations governing the coupled motion of the system are developed from first principles. These are organised into the form of a generalised linear symmetric eigenvalue problem, from which a family of uncoupled differential operators can be established. These operators define a series of exact substitute systems that together describe the complete motion of the original structure. These equations can then be used in either of two ways. In their most powerful form they can be developed into exact dynamic stiffness matrices that enable all the powerful features of the finite element method to be utilised. This subsequently enables sets of members carrying point masses and subject to point spring supports to be analysed easily. Alternatively, the equations are able to yield an exact relational model that links any uncoupled frequency of an original member to the corresponding set of coupled system frequencies. This approach enables ‘back of the envelope calculations’ to be undertaken quickly and efficiently. The exact mode shapes of the original structure can be recovered in either case. Due to space limitations, only aspects of the first technique are described briefly herein, but both are covered exhaustively elsewhere [1].