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1. Noise influenced response movement in coupled oscillator arrays with multi-stability

   https://doi.org/10.1016/j.jsv.2022.116951  

   Alofi, A.; Acar, G; Balachandran, B. (April 2022, Journal of sound and vibration)

   In this work, the authors explore the influence of noise on the dynamics of coupled nonlinear oscillators. Numerical studies based on the Euler–Maruyama scheme and experimental studies with finite duration noise are undertaken to examine how the response can be moved from one response state to another by using noise addition to a harmonically forced system. In particular, jumps from a high amplitude state of each oscillator to a low amplitude state of each oscillator and the converse are demonstrated along with noise-influenced localizations. These events are found to occur in a region of multi-stability for the system, and the corresponding noise levels are reported. A method for recognizing how much noise is required to induce a change the system dynamics is developed by using the response basins of attraction. The findings of this work have implications for weakly coupled, nonlinear oscillator arrays and the manner in which noise can be used to influence energy localization and system dynamics in these systems. 

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2. Cantilevers attached with bluff bodies: vortex-induced vibrations

   https://doi.org/10.1007/s11071-024-10679-8  

   Wani, Khawar Zamman; Pandey, Manoj; Balachandran, Balakumar (December 2024, Nonlinear Dynamics)

   Vortex-induced vibrations are oscillatory motions experienced by a body interacting with an external flow. These vibrations can be harnessed for energy harvesting purpose. A cantilever beam with a cylinder attached at the free end represents the bluff body oscillator of interest here. Vortex-induced vibrations of two adjacent bluff-body oscillators are studied by varying the transverse spacing between the oscillators. A finite element model of the system is used to numerically study the associated fluid–structure interactions. For the case with two oscillators, the effect of varying the oscillator spacing on the system response is studied. Dynamic mode decomposition is used for extracting coherent spatio-temporal structures in pressure fields. The system spectral response for the single oscillator and coupled oscillators cases are studied to examine the system dynamics. The obtained numerical results for the system dynamics are found to agree with previously reported experimental results in the literature. The present work can form a basis for constructing computational models of fluid coupled bluff-body oscillators and configuring arrays of bluff-body oscillators for energy harvesting. 

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3. Using noise to augment synchronization among oscillators

   https://doi.org/10.1038/s41598-021-83806-9  

   Vaidya, Jaykumar; Bashar, Mohammad Khairul; Shukla, Nikhil (December 2021, Scientific Reports)

   null (Ed.)

   Abstract Noise is expected to play an important role in the dynamics of analog systems such as coupled oscillators which have recently been explored as a hardware platform for application in computing. In this work, we experimentally investigate the effect of noise on the synchronization of relaxation oscillators and their computational properties. Specifically, in contrast to its typically expected adverse effect, we first demonstrate that a common white noise input induces frequency locking among uncoupled oscillators. Experiments show that the minimum noise voltage required to induce frequency locking increases linearly with the amplitude of the oscillator output whereas it decreases with increasing number of oscillators. Further, our work reveals that in a coupled system of oscillators—relevant to solving computational problems such as graph coloring, the injection of white noise helps reduce the minimum required capacitive coupling strength. With the injection of noise, the coupled system demonstrates frequency locking along with the desired phase-based computational properties at 5 × lower coupling strength than that required when no external noise is introduced. Consequently, this can reduce the footprint of the coupling element and the corresponding area-intensive coupling architecture. Our work shows that noise can be utilized as an effective knob to optimize the implementation of coupled oscillator-based computing platforms. 

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4. Nonlinear coupling of closely spaced modes in atomically thin MoS2 nanoelectromechanical resonators

   https://doi.org/10.1038/s41378-024-00844-9  

   Yousuf, S_M_Enamul_Hoque; Shaw, Steven_W; Feng, Philip_X-L (December 2024, Microsystems & Nanoengineering)

   Abstract Nanoelectromechanical systems (NEMS) incorporating atomic or molecular layer van der Waals materials can support multimode resonances and exotic nonlinear dynamics. Here we investigate nonlinear coupling of closely spaced modes in a bilayer (2L) molybdenum disulfide (MoS₂) nanoelectromechanical resonator. We model the response from a drumhead resonator using equations of two resonant modes with a dispersive coupling term to describe the vibration induced frequency shifts that result from the induced change in tension. We employ method of averaging to solve the equations of coupled modes and extract an expression for the nonlinear coupling coefficient (λ) in closed form. Undriven thermomechanical noise spectral measurements are used to calibrate the vibration amplitude of mode 2 (a₂) in the displacement domain. We drive mode 2 near its natural frequency and measure the shifted resonance frequency of mode 1 (f_1s) resulting from the dispersive coupling. Our model yieldsλ = 0.027 ± 0.005 pm⁻² · μs⁻²from thermomechanical noise measurement of mode 1. Our model also captures an anomalous frequency shift of the undriven mode 1 due to nonlinear coupling to the driven mode 2 mediated by large dynamic tension. This study provides a direct means to quantifyingλby measuring the thermomechanical noise in NEMS and will be valuable for understanding nonlinear mode coupling in emerging resonant systems. 

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5. Energy Exchange between Two Sub-systems Coupled with a Nonlinear Elastic Path

   Sen, O.T.; Singh, R. (May 2018, NOVEM 2018 Conference)

   The chief objective of this paper is to explore energy transfer mechanism between the sub-systems that are coupled by a nonlinear elastic path. In the proposed model (via a minimal order, two degree of freedom system), both sub-systems are defined as damped harmonic oscillators with linear springs and dampers. The first sub-system is attached to the ground on one side but connected to the second sub-system on the other side. In addition, linear elastic and dissipative characteristics of both oscillators are assumed to be identical, and a harmonic force excitation is applied only on the mass element of second oscillator. The nonlinear spring (placed in between the two sub-systems) is assumed to exhibit cubic, hardening type nonlinearity. First, the governing equations of the two degree of freedom system with a nonlinear elastic path are obtained. Second, the nonlinear differential equations are solved with a semi-analytical (multi-term harmonic balance) method, and nonlinear frequency responses of the system are calculated for different path coupling cases. As such, the nonlinear path stiffness is gradually increased so that the stiffness ratio of nonlinear element to the linear element is 0.01, 0.05, 0.1, 0.5 and 1.0 while the absolute value of linear spring stiffness is kept intact. In all solutions, it is observed that the frequency response curves at the vicinity of resonant frequencies bend towards higher frequencies as expected due to the hardening effect. However, at moderate or higher levels of path coupling (say 0.1, 0.5 and 1.0), additional branches emerge in the frequency response curves but only at the first resonant frequency. This is due to higher displacement amplitudes at the first resonant frequency as compared to the second one. Even though the oscillators move in-phase around the first natural frequency, high amplitudes increase the contribution of the stored potential energy in the nonlinear spring to the total mechanical energy. The out-of-phase motion around the second natural frequency cannot significantly contribute due to very low motion amplitudes. Finally, the governing equations are numerically solved for the same levels of nonlinearity, and the motion responses of both sub-systems are calculated. Both in-phase and out-of-phase motion responses are successfully shown in numerical solutions, and phase portraits of the system are generated in order to illustrate its nonlinear dynamics. In conclusion, a better understanding of the effect of nonlinear elastic path on two damped harmonic oscillators is gained. 

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