Natural frequencies of pre-buckled rods and gridshells

A discrete differential geometry (DDG)-based method is proposed to numerically study the natural frequencies of elastic rods and gridshells in their post-buckling configurations. A fully implicit numerical framework is developed based on Discrete Elastic Rods (DER) algorithm, in order to characterize the mechanical behaviors of an elastic gridshell comprised of multiple rods. When their footprints are constrained along a shrinking trajectory, the slender structures as well as their constructed network would experience a geometrically nonlinear instability and deform into an out-of-plane configuration. By checking the eigenvalues and eigenvectors of the mass and stiffness matrix, the linear vibration near the post-buckling equilibria are characterized through a numerical approach. Exploiting the efficiency and the robustness of the developed discrete method, a systematic parameter sweep is performed to quantify the vibration frequency of pre-buckled gridshells with respect to the number of rods and the pre-compressed distance. It is found that the natural frequency for both pre-deformed rods and gridshells would linearly decrease as the enlargement of compressive distance, even though the geometrically nonlinear deformations have been taken into account. Moreover, the vibration frequency almost linearly rises when the number of rods in a gridshell becomes larger. These findings could provide a fundamental insight in revealing more complex structural dynamics and facilitating the designs of buckling-induced assembly in some man-made systems, e.g., avoidance of the resonance in soft electronics.