Free Vibrations of Structural Hyperbolic Paraboloid Networks

In the context of dynamic analysis, the membrane theory of prestressed networks remains largely undeveloped, except for special cases intended for specific applications. These works are based on models in which transverse motions are decoupled from the remaining components, with the analysis being limited to the transverse motion alone [1]. The validity of this decoupled system is limited by the smallness of the deviation of the prestressed configuration from the plane, which in turn restricts the applicability of the model. In this chapter no such restrictions are imposed. Within the context of continuum mechanics, elastic cable networks may be regarded as elastic membranes, consisting of two families of perfectly elastic fibers. The fibers are assumed to be continuously distributed and tied together at their points of intersection to prevent slipping. Resistance of the network to shear distortions is neglected in this chapter. A general nonlinear theory for prestressed hyperbolic paraboloid cable networks was recently developed in [2]. The derived system is fully coupled, with variable coefficients whose precise form depends on the fiber response functions and the underlying equilibrium deformation. In this chapter a system of three coupled equations is derived in scalar form, which is more convenient to work with for specific examples. Modal analysis is then performed for a one-parameter family hyperbolic paraboloids with nonlinearly elastic fibers for three particular cases—networks over elliptic, rectangular, and circular domains, which are of interest to structural engineers and architects.