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1. Transverse Vibration of Clamped-Pinned-Free Beam with Mass at Free End

   https://doi.org/10.3390/app9152996  

   Hong, Jonathan; Dodson, Jacob; Laflamme, Simon; Downey, Austin (August 2019, Applied Sciences)

   Engineering systems undergoing extreme and harsh environments can often times experience rapid damaging effects. In order to minimize loss of economic investment and human lives, structural health monitoring (SHM) of these high-rate systems is being researched. An experimental testbed has been developed to validate SHM methods in a controllable and repeatable laboratory environment. This study applies the Euler-Bernoulli beam theory to this testbed to develop analytical solutions of the system. The transverse vibration of a clamped-pinned-free beam with a point mass at the free end is discussed in detail. Results are derived for varying pin locations and mass values. Eigenvalue plots of the first five modes are presented along with their respective mode shapes. The theoretical calculations are experimentally validated and discussed. 

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2. Four end solutions of a free boundary problem

   https://doi.org/10.1016/j.aim.2022.108395  

   Du, Zhuoran; Gui, Changfeng; Wang, Kelei (August 2022, Advances in Mathematics)
3. Twinning in rolled AZ31B magnesium alloy under free-end torsion

   https://doi.org/https://doi.org/10.1016/j.msea.2020.140405  

   Carneiro, L.; Culbertson, D.; Yu, Q.; Jiang, Y. (January 2021, Materials science engineering)

   null (Ed.)

   The mechanical response and microstructure evolution in a rolled AZ31B magnesium alloy were experimentally characterized using companion thin-walled tubular specimens under free-end monotonic torsion. The tubular specimens were made with their axes along the normal direction of the rolled magnesium plate. The shear stress-shear strain response shows a subtle sigmodal shape that is composed of four distinctive stages of strain hardening. Basal slips and tension twinning are operated throughout the shear deformation. Both tension twinning and compressing twinning are favored. Growth and interaction of tension twins with multiple variants lead to formation of twin-twin boundaries (TTBs). The collective hardening effects by twin boundary (TB) and TTB result in a unique rise of the strain hardening rate in Stage II and III. In addition to primary twins, tension-compression double twins and tension-compression-tension tertiary twins with detectable sizes are observed in the tension-twin favorable grains whereas compression-tension double twins are detected in the tension-twin unfavorable grains; all of which become more observable with the increasing shear strain. During Stage IV deformation where TTB formation exhausts, non-basal prismatic slips become more significant and are responsible for the progressive decrease in strain hardening rate in this stage. Swift effect, which is commonly observed in textured materials, is evidenced under free-end torsion. The origin of Swift effect is confirmed to be dislocation slips at a shear strain less than 5% but is predominantly due to tension twinning at a larger plastic strain. 

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4. Differentiable MPC for End-to-end Planning and Control

   Amos, B.; Rodriguez, I.; Sacks, J.; Boots, B.; Kolter, Z. (January 2018, Advances in neural information processing systems)
5. End-to-End Optimization and Learning for Multiagent Ensembles

   Kotary, James; Di Vito, Vincenzo; Fioretto, Ferdinando (November 2022, arXivorg)