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Many important engineering structures such as rotating machinery, including turbine bladed disks, gears, flywheels and satellites are comprised of repeated (nominally identical) substructures arranged circumferentially with cyclic symmetry. Due to this unique arrangement, the system matrices and consequently the dynamics of such structures exhibit specific characteristics (Mitra and Epureanu, ASME Appl Mech Rev. https://doi.org/10.1115/1.4043083, 2019; Olson et al. ASME Appl Mech Rev, 66(4):040803, 2014). Extensive scientific study and analysis has been conducted on this topic in recent decades. Of particular interest is the change in dynamic behavior when there are deviations in substructures from their nominal, even to a small extent. Colloquially termed mistuning, such deviations are practically impossible to avoid. They manifest as material or geometric differences due to causes such as manufacturing tolerances, wear and differential operation conditions (Castanier and Pierre, J Propuls Power 22/2:384, 2006). Mistuning can lead to strain energy localization, higher system responses and reduction of the operational life cycle and should therefore be carefully considered in the design and analysis of structures. The current industrial practice is to use Monte Carlo simulations to characterize mistuning effects using randomly generated deviations in substructures of the nominal design (Mitra and Epureanu, ASME Appl Mech Rev. https://doi.org/10.1115/1.4043083, 2019; Castanier and Pierre, J Propuls Power 22/2:384, 2006). Since thousands of dynamic simulations might be required to characterize a single design, full order high fidelity models remain prohibitively expensive. For such tasks, reduced order models (ROMs) are employed instead (Castanier and Pierre, J Propuls Power 22/2:384, 2006; Baek and Epureanu, ASME J Vib Acoust, 139(6):061011, 2017). However, obtaining fast and accurate ROMs for cyclic structures with nonlinearities (Mitra and Epureanu, ASME Appl Mech Rev. https://doi.org/10.1115/1.4043083, 2019; Baek and Epureanu, ASME J Vib Acoust, 139(6):061011, 2017; Zucca and Firrone, J Sound Vib, 333:916–926, 2014) remains a challenging task. This tutorial aims at summarizing and highlighting some of the most relevant techniques that have been proposed to date, with a specific focus on nonlinear ROMs including contact nonlinearities. |
