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1. Analytical Study of Nonlinear Flexural Vibration of a Beam with Geometric, Material and Combined Nonlinearities

   https://doi.org/10.3390/vibration7020025  

   Madhuranthakam, Yoganandh ; Chakrapani, Sunil Kishore ( June 2024 , Vibration)

   This article explores the nonlinear vibration of beams with different types of nonlinearities. The beam vibration was modeled using Hamilton’s principle, and the equation of motion was solved using method of multiple time scales. Three models were developed assuming (a) geometric nonlinearity, (b) material nonlinearity and (c) combined geometric and material nonlinearity. The material nonlinearity also included both third and fourth nonlinear elasticity terms. The frequency response equation of these models were further evaluated quantitatively and qualitatively. The models capture the hardening effect, i.e., increase in resonant frequency as a function of forcing amplitude for geometric nonlinearity, and the softening effect, i.e., decrease in resonant frequency for material nonlinearity. The model is applied on the first three bending modes of the cantilever beam. The effect of the fourth-order material nonlinearity was smaller compared to the third-order term in the first mode, whereas it is significantly larger in second and third mode. The combined nonlinearity models shows a discontinuous frequency shift, which was resolved by utilizing a set of transition assumptions. This results in a smooth transition between the material and geometric zones in amplitude. These parametric models allow us to fine tune the nonlinear response of the system by changing the physical properties such as geometry, linear and nonlinear elastic properties.

    

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2. The effects of nonlinear damping on degenerate parametric amplification

   https://doi.org/10.1007/s11071-020-06090-8  

   Li, Donghao ; Shaw, Steven W. ( December 2020 , Nonlinear Dynamics)

   null (Ed.)

   Abstract This paper considers the dynamic response of a single degree of freedom system with nonlinear stiffness and nonlinear damping that is subjected to both resonant direct excitation and resonant parametric excitation, with a general phase between the two. This generalizes and expands on previous studies of nonlinear effects on parametric amplification, notably by including the effects of nonlinear damping, which is commonly observed in a large variety of systems, including micro- and nano-scale resonators. Using the method of averaging, a thorough parameter study is carried out that describes the effects of the amplitudes and relative phase of the two forms of excitation. The effects of nonlinear damping on the parametric gain are first derived. The transitions among various topological forms of the frequency response curves, which can include isolae, dual peaks, and loops, are determined, and bifurcation analyses in parameter spaces of interest are carried out. In general, these results provide a complete picture of the system response and allow one to select drive conditions of interest that avoid bistability while providing maximum amplitude gain, maximum phase sensitivity, or a flat resonant peak, in systems with nonlinear damping. 

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3. The hawkmoth wingbeat is not at resonance

   https://doi.org/10.1098/rsbl.2022.0063  

   Gau, Jeff ; Wold, Ethan S. ; Lynch, James ; Gravish, Nick ; Sponberg, Simon ( May 2022 , Biology Letters)

   Flying insects have elastic materials within their exoskeletons that could reduce the energetic cost of flight if their wingbeat frequency is matched to a mechanical resonance frequency. Flapping at resonance may be essential across flying insects because of the power demands of small-scale flapping flight. However, building up large-amplitude resonant wingbeats over many wingstrokes may be detrimental for control if the total mechanical energy in the spring-wing system exceeds the per-cycle work capacity of the flight musculature. While the mechanics of the insect flight apparatus can behave as a resonant system, the question of whether insects flap their wings at their resonant frequency remains unanswered. Using previous measurements of body stiffness in the hawkmoth, Manduca sexta , we develop a mechanical model of spring-wing resonance with aerodynamic damping and characterize the hawkmoth's resonant frequency. We find that the hawkmoth's wingbeat frequency is approximately 80% above resonance and remains so when accounting for uncertainty in model parameters. In this regime, hawkmoths may still benefit from elastic energy exchange while enabling control of aerodynamic forces via frequency modulation. We conclude that, while insects use resonant mechanics, tuning wingbeats to a simple resonance peak is not a necessary feature for all centimetre-scale flapping flyers. 

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4. Phase-matched nonlinear second-harmonic generation in plasmonic metasurfaces

   https://doi.org/10.1515/nanoph-2018-0181  

   Shams Mousavi, S. Hamed ; Lemasters, Robert ; Wang, Feng ; Dorche, Ali Eshaghian ; Taheri, Hossein ; Eftekhar, Ali A. ; Harutyunyan, Hayk ; Adibi, Ali ( February 2019 , Nanophotonics)

   Abstract The phase matching between the propagating fundamental and nonlinearly generated waves plays an important role in the efficiency of the nonlinear frequency conversion in macroscopic crystals. However, in nanoscale samples, such as nanoplasmonic structures, the phase-matching condition is often ignored due to the sub-wavelength nature of the materials. Here, we first show that the phase matching of the lattice plasmon modes at the fundamental and second-harmonic frequencies in a plasmonic nanoantenna array can effectively enhance the surface-enhanced second-harmonic generation. Additionally, a significant enhancement of the second-harmonic generation is demonstrated using stationary band-edge lattice plasmon modes with zero phase. 

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5. Convex Optimization for Spring Design of Parallel Elastic Actuators

   Guo, S. ; Gregg, R. ; Bolivar-Nieto, E. ( January 2022 , Proceedings of the American Control Conference)

   Elastic actuation can improve human-robot interaction and energy efficiency for wearable robots. Previous work showed that the energy consumption of series elastic actuators can be a convex function of the series spring compliance. This function is useful to optimally select the series spring compliance that reduces the motor energy consumption. However, series springs have limited influence on the motor torque, which is a major source of the energy losses due to the associated Joule heating. Springs in parallel to the motor can significantly modify the motor torque and therefore reduce Joule heating, but it is unknown how to design springs that globally minimize energy consumption for a given motion of the load. In this work, we introduce the stiffness design of linear and nonlinear parallel elastic actuators via convex optimization. We show that the energy consumption of parallel elastic actuators is a convex function of the spring stiffness and compare the energy savings with that of optimal series elastic actuators. We analyze robustness of the solution in simulation by adding uncertainty of 20% of the RMS load kinematics and kinetics for the ankle, knee, and hip movements for level-ground human walking. When the winding Joule heating losses are dominant with respect to the viscous losses, our optimal PEA designs outperform SEA designs by further reducing the motor energy consumption up to 63%. Comparing to the linear PEA designs, our nonlinear PEA designs further reduced the motor energy consumption up to 31%. From our convex formulation, our global optimal nonlinear parallel elastic actuator designs give two different elongation-torque curves for positive and negative elongation, suggesting a clutching mechanism for the final implementation. In addition, the different torque-elongation profiles for positive and negative elongation for nonlinear parallel elastic actuators can cause sensitivity of the energy consumption to changes in the nominal load trajectory. 

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