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cubic nonlinearity and surface shock waves in soft tissue-like materials

ULTRASONICS 2025年 145卷 107469页

作者： Alarcon, Hector Galaz, Belfor Espindola, David Univ Autonoma Chile Fac Ingn Dept Fis & Quim Ave Pedro Valdivia 425 Santiago 7500912 Chile Univ Santiago de Chile Dept Fis Estn Cent Ave Victor Jara 3493Casilla 307Correo 2 Santiago 9160000 Chile Univ Ohiggins Inst Ciencias Ingn Ave Libertador Bernardo Ohiggins 611 Rancagua 2841959 Chile

The cubic nonlinearity of shear wave propagation plays a significant role in brain injury biomechanics. However, soft materials, like the brain, also support the propagation of surface waves, which produce a combination of longitudinal and transverse deformation. The order of the nonlinearity of surface waves in soft materials is still unknown. Here, we directly observe nonlinear Scholte waves propagating in an interface formed by an incompressible gelatin tissue-mimicking phantom and a water layer using ultrasound imaging operated as fast as 16667 frames per second. A two-dimensional correlation-based tracking algorithm was utilized to extract movies of the movement produced by the surface wave. Our results show that the initially nearly monochromatic wave becomes progressively distorted with the propagation due to nonlinearity. The distortion of the wave and its frequency spectrum indicate a high content of odd harmonics when compared with even harmonics. Additionally, by fitting our experimental data to a minimalist one-dimensional model based on the wave speed variation as a function of the perturbation amplitude, we found a cubic nonlinear parameter 46 times larger than the quadratic nonlinear parameter. Overall, the wave distortion, the harmonic development, and the dependence of the wave speed with the amplitude prove that cubic nonlinearity is essential to modeling nonlinear Scholte wave propagation.

关键词： Nonlinear Scholte waves Ultrasfast ultrasonography Nonlinear traveling waves cubic nonlinearity Soft matter

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Peak angle limit for quasi-linear operation of resonant MEMS scanning mirror with cubic nonlinearity

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SENSORS AND ACTUATORS A-PHYSICAL 2025年 383卷

作者： Song, Xudong Qiao, Dayong Song, Xiumin Tian, Youliang Liu, Xianrong Northwestern Polytech Univ Key Lab Micro Nano Syst Aerosp Minist Educ Xian 710072 Peoples R China Northwestern Polytech Univ Ningbo Inst Ningbo 315103 Zhejiang Peoples R China Northwestern Polytech Univ Shaanxi Prov Key Lab Micro & Nano Electromech Syst Xian 710072 Peoples R China Zhisensor Technol Co Ltd Xian 710065 Shaanxi Peoples R China Hisense Laser Display Co Ltd Qingdao 266000 Shandong Peoples R China

Resonant microelectromechanical system (MEMS) scanning mirrors play a crucial role in the rapid and broad manipulation of laser beams, attributed to their compact dimensions and low power requirements. However, in the context of nonlinear dynamics, a bifurcation phenomenon arises when the vibration angle surpasses a certain limit, leading to a discontinuous angle-frequency relationship. This discontinuity significantly complicates the control of resonant MEMS scanning mirrors, particularly when they operate at peak angle. Therefore, it is imperative to investigate the onset of bifurcation, specifically the peak angle limit for quasi-linear operation of nonlinear resonant MEMS scanning mirrors, below which the angle-frequency relationship remains continuous. In this study, an example of resonant MEMS scanning mirrors with cubic nonlinearity was utilized to explore the peak angle limit. By solving the dynamic equation, a theoretical expression for the peak angle limit was derived. The influences of the relevant parameters on the peak angle limit are determined. To obtain a larger peak angle limit, it is required to increase the damping coefficient (c) and decrease the absolute value of the cubic spring constant (k3). The MEMS mirror with a set of specific parameters was fabricated and tested to validate the theoretical findings. The results from both theoretical and experimental investigations demonstrated a high degree of concordance, with a peak optical angle limit difference of 0.9 degrees. These findings provide theoretical foundation for enhancing the peak angle limit for quasi-linear operation and improving stability of MEMS mirrors.

关键词： Resonant MEMS scanning mirrors Peak angle limit Quasi-linear operation cubic nonlinearity

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Data-driven dimension reduction for high-dimensional random vibration systems with cubic nonlinearity

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NONLINEAR DYNAMICS 2025年第5期113卷 4163-4177页

作者： Tian, Yanping Jin, Xiaoling Zhu, Guangyu Hu, Yanchao Wang, Yong Huang, Zhilong Hangzhou Dianzi Univ Sch Mech Engn Hangzhou 310018 Peoples R China Zhejiang Univ Dept Engn Mech Key Lab Soft Machines & Smart Devices Zhejiang Pro Hangzhou 310027 Peoples R China Hangzhou Zhongpu Technol Co Ltd Hangzhou 310012 Peoples R China

Dimension reduction is one of the never-ending themes in investigating high-dimensional nonlinear random vibration systems. This work addresses this issue and establishes a theory-guided data-driven method to automatically distill a low-dimensional description of a high-dimensional presentation. It consists of four successive steps, just corresponding to four verbs, i.e., linearizing, truncating, constructing and compensating: linearizing a high-dimensional nonlinear system by deleting all nonlinear terms;truncating the insignificant modal shapes and building an approximate transformation between all physical coordinates and the modal coordinates retained;constructing a family of low-dimensional nonlinear systems by extending the low-dimensional linear system of retained modal coordinates;finally, compensating the adverse influences induced by the approximate transformation by picking out the low-dimensional equivalent system which balances the accuracy of prediction and the complexity of modelling. Through a toy example and a ten-degrees-of-freedom example with cubic nonlinearity, the application, efficacy, and tremendous potential of this method are illustrated in detail.

关键词： Dimension reduction High-dimensional system cubic nonlinearity Modal coordinate Theory-guided data science

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Singularities in Finite Time of a Two-Component Peakon System With cubic nonlinearity

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2025年第10期48卷 10400-10405页

作者： Han, Xuanxuan Li, Gui-Dong Guizhou Univ Dept Math Guiyang Guizhou Peoples R China

Blow-up of the two-component peakon system with cubic nonlinearity is significant and fascinating for physicists and mathematicians. In the paper, by exploiting the fine structure of the system and the conservation law H, we obtain nonlinear estimates on the L infinity-norm of the strong solution during its lifespan, both with and without dependence on the L1-norm assumption. Thus, two new blow-up results are derived.

关键词： blow-up conservation law cubic nonlinearity two-component peakon system

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Application of modified extended direct algebraic method to nonlinear fractional diffusion reaction equation with cubic nonlinearity

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BOUNDARY VALUE PROBLEMS 2025年第1期2025卷 1-27页

作者： Bilal, Muhammad Khan, Alamgir Ullah, Ikram Khan, Hasib Alzabut, Jehad Alkhawar, Hisham Mohammad Abdul Wali Khan Univ Dept Math Mardan 23200 Khyber Pakhtunk Pakistan Cent South Univ Sch Math & Stat Changsha 410083 Hunan Peoples R China Prince Sultan Univ Dept Math & Sci POB 66833 Riyadh 11586 Saudi Arabia Shaheed Benazir Bhutto Univ Dept Math Dir Upper Khyber Pakhtunk Pakistan OSTIM Tech Univ Dept Ind Engn TR-06374 Ankara Turkiye Prince Sultan Univ Comp Dept Preparatory Year Program Riyadh 11586 Saudi Arabia

This work uses the modified Extended Direct Algebraic Method (mEDAM) with conformable derivatives to obtain accurate solutions for the diffusion-reaction equation with cubic nonlinearity and the nonlinear fractional generalised density-independent DR problem. We use fractional derivatives in the conformable sense to achieve precise polynomial-form solutions by converting the equations into autonomous two-dimensional plane systems. The technique solves nonlinear ordinary differential equations (ODEs) produced by a fractional transformation to produce travelling wave solutions. Through the visualisation of various solution profiles, as periodic, shock, kink, anti-kink, and soliton waves, numerical simulations offer important insights into the dynamics of nonlinear fractional partial differential equations. This work builds on earlier research and broadens our understanding of complex events.

关键词： mEDAM technique Conformable derivatives Solitons Partial differential equations Density-independent DR equation cubic nonlinearity

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On the regularity and symmetry of periodic traveling solutions to weakly dispersive equations with cubic nonlinearity

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2023年第6期46卷 6403-6415页

作者： Pei, Long Sun Yat Sen Univ Sch Math Zhuhai Zhuhai Peoples R China Sun Yat Sen Univ Sch Math Zhuhai Zhuhai Guangdong Peoples R China

We consider the a priori regularity and symmetry of periodic traveling wave solutions to a class of weakly dispersive equations with cubic nonlinearity. We prove that any continuous periodic traveling wave solution of small amplitude is symmetric if it satisfies the reflection criterion proposed recently by Bruell and Pei. Moreover, we confirm that continuous periodic traveling waves with singular crests or troughs, which correspond to large-amplitude waves, are also symmetric if some cancellation structure appears near the singular points or if the wave profile has only a single singular point per period. Included in the results as particular cases are periodic traveling waves with peak, cusp, or stump type singular structures. The argument and results also apply to dispersive equations with more general nonlinearity which is non-smooth.

关键词： cubic nonlinearity dispersive equations periodic traveling waves singularity symmetry

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Transformation of polarisation singularities in nonlinear mixing of light beams with frequency conversion in a medium with cubic nonlinearity

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QUANTUM ELECTRONICS 2022年第3期52卷 247-253页

作者： Grigoriev, K. S. Makarov, V. A. Lomonosov Moscow State Univ Fac Phys Leninskie Gory 1Stroenie 2 Moscow 119991 Russia

We analyse the electric field transverse distributions in light beams with a tripled frequency and with a frequency of 2 omega(1 )- omega(2) which arise in the hulk of an isotropic cubic-nonlinear medium upon the propagation of nonuniformly polarised beams of fundamental radiation with a multimode transverse structure described by a superposition of Laguerre-Gaussian modes. A relation is found between the total topological indices of the circular polarisation singularity points in the transverse sections of the generated signal beams and the corresponding total indices in the beams of fundamental radiation.

关键词： cubic nonlinearity polarisation singularity topological index

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Nonlinear vibration analysis of a generally restrained double-beam structure coupled via an elastic connector of cubic nonlinearity

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NONLINEAR DYNAMICS 2022年第2期109卷 563-588页

作者： Zhao, Yuhao Du, Jingtao Harbin Engn Univ Coll Power & Energy Engn Harbin 150001 Peoples R China

Double-beam structures are frequently encountered in various engineering applications. Their vibration behavior is attracting more and more research attention. The majority of the existing studies are mainly limited to double-beam structures coupled via linear connectors. Additionally, the rotational boundary restraints of double-beam structures are usually neglected, limiting the dynamic analysis of double-beam structures in engineering applications. To study the potential application of the elastic cubic nonlinearity on double-beam structures, the dynamic analysis model of a generally restrained double-beam structure coupled via a flexible connector of cubic nonlinearity is modeled in this study. The Galerkin truncated method (GTM) is employed to solve the nonlinear governing equations of the double-beam structure. Mode functions of beam structures without any nonlinearity and damping are selected as the trail and weight functions. The Galerkin condition is applied to discretize the nonlinear governing equations. Then, residual equations of the double-beam structure are established and arranged into a matrix form. The Runge-Kutta method is utilized to solve the corresponding matrix. The finite difference method (FDM) is applied to verify the correctness and reliability of the current model. Based on the model established, the influence of the coupling nonlinearity on the dynamic behavior of the double-beam structure is studied and discussed. The research found that the variation of nonlinear stiffness, coupling viscous damping, and coupling position can effectively transform vibration states of the double-beam structure. For the determined boundary conditions and structural parameters, a suitable combination of coupling nonlinear stiffness, coupling position, and coupling viscous damping has a beneficial effect on the vibration suppression of the double-beam structure.

关键词： Nonlinear vibration Double-beam structure Elastic connector cubic nonlinearity General boundary restraints

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Nonlinear vibration and stability analysis of a flexible rotor-SFDs system with cubic nonlinearity

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NONLINEAR DYNAMICS 2022年第3期109卷 1441-1461页

作者： Ri, Kwangchol Jong, Yongson Yun, Cholil Kim, Kwangchol Han, Poknam Pyongyang Univ Mech Engn Dept Light Ind Machinery Engn Pyongyang 999093 North Korea Kim Chaek Univ Technol Dept Thermal Engn Pyongyang 999093 North Korea Kim Il Sung Univ Fac Forest Sci Pyongyang 999093 North Korea Acad Sci Inst Mech Engn Pyongyang 999093 North Korea Kim Chaek Univ Technol Coll Ship & Ocean Technol Pyongyang 999093 North Korea

In this paper, the nonlinear vibration phenomena occurring in a flexible rotor supported on squeeze-film dampers (SFDs) are investigated. The complex nonlinearities caused by cubic nonlinearity in the rotor-bearing system and the fluid-film forces of SFDs are considered as sources of nonlinear phenomena in the system. Also, the dynamic response characteristics of the system, which occur due to asymmetric characteristics of parameters used to model SFDs, are analyzed. The unbalance response of these systems is analyzed by the modified incremental harmonic balance method, and stability analysis is performed on the solutions calculated using the Floquet theory. By comparing the results calculated using the proposed method with the results calculated using the Runge-Kutta method, the validity of the method proposed in this paper is verified. The frequency-response characteristics according to the change of all parameters used to model the system are considered, and whirl orbits, time history, and Poincare sections are constructed at specific locations. This work can be effectively applied to the study of the nonlinear vibration phenomena of rotor-bearing systems supported on fluid-film bearings such as SFDs.

关键词： Flexible rotor Squeeze-film damper cubic nonlinearity Incremental harmonic balance method Alternating frequency Time Nonlinear forced vibration

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Two-way collinear mixing of a longitudinal and a transverse plane wave in materials with cubic nonlinearity

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WAVES IN RANDOM AND COMPLEX MEDIA 2022年第5期32卷 2138-2157页

作者： Liu, Xiqiang Wang, Li Gong, Ziwei Wang, Xulong Yang, Jing Liang, Bin Cheng, Jianchun Nanjing Univ Inst Acoust Dept Phys MOECollaborat Innovat Ctr Adv Microstruct Nanjing Peoples R China Nanjing Univ Inst Acoust Dept Phys MOEKey Lab Modern Acoust Nanjing Peoples R China Army Engn Univ PLA Dept Gen Educ Nanjing Peoples R China

The resonance conditions of two-way collinear mixing are derived when a longitudinal wave and a transverse wave are propagating in the opposite directions based on cubic nonlinearity. The analytical solutions for mixing waves are obtained. An interesting result is that the resonant longitudinal wave and the resonant transverse wave can be generated, respectively, when considering different resonance conditions. This is different from the case with quadratic nonlinearity that only resonant transverse wave can be generated. For the convenience of applications, the wave mixing of a time-harmonic longitudinal pulse and a time-harmonic transverse pulse is also investigated. A clear mixing process is presented by discussing the parameters introduced in the derivation. Numerical calculation is conducted on material Al, the hexagonal waveforms of the resonant transverse waves are depicted when considering different cases. The results could provide new opportunities for mixing wave applications in nondestructive evaluation.

关键词： Mixing wave cubic nonlinearity resonance conditions nonlinear media nondestructive evaluation

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