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• An implicit algorithm is developed to simulate the dynamic response of predeformed gridshells. • The frequencies for both pre-buckled rods and gridshells show a linear decrease as the enlarge of compressive distance. • The natural frequency would linearly depend on the number of rods in an elastic gridshell. 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. [ABSTRACT FROM AUTHOR]

As a typical mechanical structure, ribbons are characterized with three distinctly different dimensions, i.e., length ≫ width ≫ thickness, which leads to their exclusive behaviors different from the one-dimensional (1D) case of slender rods and the two-dimensional (2D) case of thin plates. In this paper, we report a discrete differential geometry (DDG)-based numerical method to simulate the geometrically nonlinear deformations of extensible ribbons. With a cross section-dependent regularized parameter introduced, the proposed 1D framework allows both in-plane stretching and out-of-plane bending in the ribbon mid-surface, aimed at bridging the gap between the linear Kirchhoff rod theory and the developable Sadowsky ribbon model. Instead of solving the ordinary differential equations (ODEs) directly with the associated boundary conditions, the mechanical object is discretized into a mass–spring system, and its equilibrium configuration is obtained through a dynamic relaxation method. The numerical framework is applied to seven typical occasions based on either experimental or published datasets in order to verify its performance. Quantitative agreements demonstrate the effectiveness and accuracy of the proposed discrete approach for extensible ribbons, which, as a computationally efficient numerical simulator, could provide a pivotal understanding of a batch of slender structures, and further inspire the simulation-guided design of man-made systems. [ABSTRACT FROM AUTHOR]

Based on the geometrically nonlinear Kirchhoff rod theory, the snap-through behaviors of an asymmetrically clamped ribbon under midpoint loadings are explored through a numerical approach. The pre-compressed elastic ribbon would experience supercritical pitchfork bifurcation and transform into multiple stable/unstable patterns when subjected to lateral end translations. With its midpoint pushed through a prescribed path and beyond a threshold, the pre-deformed ribbon will jump to other inverted equilibrium configurations, which is known as snap-through buckling. Instead of the well-established snap-through behaviors in 2D bistable beams, we observe several different snap-through processes for the elastic ribbons in a 3D scenario. Exploiting the efficiency and robustness of our developed discrete model, a systematic parameter sweep is performed to quantify the snap-through processes of elastic ribbons in different boundary conditions and categories, and the results of which could provide a fundamental understanding of structural mechanics and further motivate the designs and applications of engineered systems. [ABSTRACT FROM AUTHOR]