Resonance-based schemes for dispersive equations via decorated trees

We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the partial differential equation (PDE) and to approximate with high-order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular stochastic PDEs (SPDEs) with regularity structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations which encode the dominant frequencies. The structure proposed in this article is new and gives a variant of the Butcher–Connes–Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, numerical analysis is taking advantage of these extended structures and provides a new perspective on them.

Impact of Mode-Area Dispersion on Nonlinear Pulse Propagation in Gas-Filled Anti-Resonant Hollow-Core Fiber

We numerically investigate the effect of mode-area dispersion in a tubular-type anti-resonant hollow-core fiber by using a modified generalized nonlinear Schrödinger equation that takes into account the wavelength-dependent mode area in its nonlinear term. The pulse evolution dynamics with and without the effect of mode-area dispersion are compared and analyzed. We show that strong dispersion of the mode area in the proximity of the cladding wall thickness-induced resonances has a significant impact on the soliton pulse propagation, resulting in considerable changes in the conversion efficiencies in nonlinear frequency mixing processes. The differences become more prominent when the pump has higher energy and is nearer to a resonance. Hence, the mode-area dispersion must be accounted for when modeling such a case.

Establishment and analysis of nonlinear frequency response model of planetary gear transmission system

Based on the lumped parameter theory, a nonlinear bending torsion coupling dynamic model of planetary gear transmission system was established by considering the backlash, support clearance, time-varying meshing stiffness, meshing damping, transmission error and external periodic excitation. The model was solved by the Runge–Kutta method, the dynamic response was analyzed by a time domain diagram and phase diagram, and the nonlinear vibration characteristics were studied by the response curve of the speed vibration displacement. The vibration test of the planetary gearbox was carried out to verify the correctness of frequency domain response characteristics. The results show that the vibration response in the planetary gear system changes from a multiple periodic response to a single periodic response with the increase in input speed. Under the action of the backlash, time-varying meshing stiffness and meshing damping, the speed vibration displacement response curves of internal and external meshing pairs appear to form a nonlinear jump phenomenon and have a unilateral impact area, and the system presents nonlinear characteristics. The nonlinear vibration of the system can be effectively suppressed by decreasing the mesh stiffness or increasing the mesh resistance, while the vibration response displacement of the system increases by increasing the external exciting force, and the nonlinear characteristics of the system remain basically unchanged. The backlash is the main factor affecting the nonlinear frequency response of the system, but it can restrain the resonance of the system in a certain range. The spectrum characteristics of the vibration displacement signal of the planetary gearbox at different speeds are similar to the simulation results, which proves the validity of the simulation analysis model and the simulation results. It can provide a theoretical basis for the system vibration and noise reduction and a dynamic structural stability design optimization.

Modeling and Dynamics Analysis of a Dual-Mass Flywheel with the Conformal Contact Action of Friction Damping Ring and Pressure Plate

To study the nonlinear dynamic characteristics of the dual-mass flywheel (DMF) under the conformal contact action between the friction damping ring and primary flywheel pressure plate, the contact action model is established and analyzed based on Winkler model. Through analysis and calculation, the contact deformation, contact pressure at different contact positions, and equivalent torsional contact stiffness are obtained. The nonlinear dynamic analysis model of three-degree-of-freedom (3DOF) which takes the conformal contact into account and two-degree-of-freedom (2DOF) without considering conformal contact is established. The approximate analytical solution of the nonlinear frequency characteristics of the system at steady state is derived. By comparing with the results obtained from numerical method, the theoretical analysis process is proved to be valid. And it is found that the overall amplitude and angular displacement transmissibility of the 3DOF model are smaller than the 2DOF model, especially at resonance frequency. The effects of the friction damping ring moment inertia, stiffness of DMF, and axial friction torque on the frequency characteristics of system and angular displacement transmissibility are analyzed. The forced vibration response analysis of the 3DOF model is conducted, through which the torsional angle variations of the primary flywheel, friction damping ring, and secondary flywheel with time are obtained. The results show that the amplitude of the secondary flywheel is much smaller than that of the primary flywheel, indicating that the DMF has prominent damping performance.

On the use of nonlinear impact oscillators in vibrating electromagnetic based energy harvesters

This work investigates the use of impact-oscillators in nonlinear electro-magnetic cantilever based broad bandwidth frequency energy harvester. The electro-mechanical model includes the dynamical equations of the nonlinear impact-oscillators, the nonlinear frequency resonant cantilever beam equation of the harvester, and the Lorenz magnetic force resulting from the surrounding nonlinear magnetic field. The simulated results of the harvested power show a band of resonant frequencies of the vibrating cantilever beam extended beyond its fundamental natural frequency. Due to the strong nonlinearity of both the impact oscillators’ forces and the magnetic flux, the harvested power of the enhanced system shows wider bands in its respective frequency response. Moreover, the harvested power of the cantilever beam shows that considering two magnets in repulsion generates higher values mainly due to the strong nonlinear transient magnetic flux, resulting into few times more power as compared to the case of considering only one magnet in the system.

Estimation of intrapulse modulation parameters of LPI radar under noisy conditions

Low probability of intercept (LPI) radars utilize specially designed waveforms for intra-pulse modulation and hence LPI radars cannot be easily intercepted by passive receivers. The waveforms include linear frequency modulation, nonlinear frequency modulation, polyphase, and polytime codes. The advantages of LPI radar are wide bandwidth, frequency variability, low power, and the ability to hide their emissions. On the other hand, the main purpose of intercept receiver is to classify and estimate the parameters of the waveforms even when the signals are contaminated with noise. Precise measurement of the parameters will provide necessary information about a threat to the radar so that the electronic attack or electronic warfare support system could take instantaneous counter action against the enemy. In this work, noisy polyphase and polytime coded waveforms are analyzed using cyclostationary (CS) algorithm. To improve the signal quality, the noisy signal is pre-processed using two types of denoising filters. The denoised signal is analyzed using CS techniques and the coefficients of spectral correlation density are computed. With this method, modulation parameters of nine types of waveforms up to −12 dB signal-to-noise ratio with an accuracy of better than 95% are extracted. When compared with literature values, it is found that the results are superior.