Truss structure dynamics under oscillatory and transient conditions

Truss structures form the skeleton of a wide variety of systems, from simple bridges to complex metamaterials. As these constructions are often exposed to a variety of forces, a proper understanding of how they respond to arbitrary perturbations is critical for ensuring optimal performance and stability. However, these dynamics are typically derived using lumped models that fail to properly account for the entire range of motion experienced by each bar. In this work, we derive an expansive model that maintains the microstructure fidelity by considering a network of linearly elastic bars connected by free joints and exploring its dynamics in Fourier space. We show that a linear, frequency dependent matrix relation exists between the forces applied to and displacement of the joints. The natural frequencies and modes can be obtained from null space of this matrix at the singular frequencies and are equivalent to those found by traditional lumped matrix methods under the limit of infinite splitting. We explore the solution set generated by this method for a small example truss and use this to transform back into temporal space so as to examine the transient dynamics of the system under finite time forces.