Applied Mathematics and Mechanics-English Edition最新文献

Explaining the mechanism of the cochlear active phonosensitive amplification has been a major problem in medicine. The basilar membrane (BM) is the key infrastructure. In 1960, Nobel Laureate von Békésy first discovered BM’s traveling wave motion. Since that time, BM’s models only have considered the traveling wave but not the biological activity. Therefore, a new model considering changes of BM’s stiffness in space and time is established based on the immersed boundary method to describe its biological activity. It not only reproduces the results of traveling wave motion but also explains the mechanization on the generation of traveling wave. An important discovery is that changes of BM’s stiffness in space and time will cause the unstable global resonance, which will induce amplification of sounds in cochlea. An important inference is that biological activity shall be included in the application of mechanical principles to the analysis of life, which is the essential difference between biomechanics and general mechanics.

The nonreciprocity of energy transfer is constructed in a nonlinear asymmetric oscillator system that comprises two nonlinear oscillators with different parameters placed between two identical linear oscillators. The slow-flow equation of the system is derived by the complexification-averaging method. The semi-analytical solutions to this equation are obtained by the least squares method, which are compared with the numerical solutions obtained by the Runge-Kutta method. The distribution of the average energy in the system is studied under periodic and chaotic vibration states, and the energy transfer along two opposite directions is compared. The effect of the excitation amplitude on the nonreciprocity of the system producing the periodic responses is analyzed, where a three-stage energy transfer phenomenon is observed. In the first stage, the energy transfer along the two opposite directions is approximately equal, whereas in the second stage, the asymmetric energy transfer is observed. The energy transfer is also asymmetric in the third stage, but the direction is reversed compared with the second stage. Moreover, the excitation amplitude for exciting the bifurcation also shows an asymmetric characteristic. Chaotic vibrations are generated around the resonant frequency, irrespective of which linear oscillator is excited. The excitation threshold of these chaotic vibrations is dependent on the linear oscillator that is being excited. In addition, the difference between the energy transfer in the two opposite directions is used to further analyze the nonreciprocity in the system. The results show that the nonreciprocity significantly depends on the excitation frequency and the excitation amplitude.

An analytical method, called the symplectic mathematical method, is proposed to study the wave propagation in a spring-mass chain with gradient arranged local resonators and nonlinear ground springs. Combined with the linearized perturbation approach, the symplectic transform matrix for a unit cell of the weakly nonlinear graded metamaterial is derived, which only relies on the state vector. The results of the dispersion relation obtained with the symplectic mathematical method agree well with those achieved by the Bloch theory. It is shown that wider and lower frequency bandgaps are formed when the hardening nonlinearity and incident wave intensity increase. Subsequently, the displacement response and transmission performance of nonlinear graded metamaterials with finite length are studied. The dual tunable effects of nonlinearity and gradation on the wave propagation are explored under different excitation frequencies. For small excitation frequencies, the gradient parameter plays a dominant role compared with the nonlinearity. The reason is that the gradient tuning aims at the gradient arrangement of local resonators, which is limited by the critical value of the local resonator mass. In contrast, for larger excitation frequencies, the hardening nonlinearity is dominant and will contribute to the formation of a new bandgap.

In this paper, we investigate the interfacial behavior of a thin one-dimensional (1D) hexagonal quasicrystal (QC) film bonded on an elastic substrate subjected to a mismatch strain due to thermal variation. The contact interface is assumed to be non-slipping, with both perfectly bonded and debonded boundary conditions. The Fourier transform technique is adopted to establish the integral equations in terms of interfacial shear stress, which are solved as a linear algebraic system by approximating the unknown phonon interfacial shear stress via the series expansion of the Chebyshev polynomials. The expressions are explicitly obtained for the phonon interfacial shear stress, internal normal stress, and stress intensity factors (SIFs). Finally, based on numerical calculations, we briefly discuss the effects of the material mismatch, the geometry of the QC film, and the debonded length and location on stresses and SIFs.

We present a Gaussian process (GP) approach, called Gaussian process hydrodynamics (GPH) for approximating the solution to the Euler and Navier-Stokes (NS) equations. Similar to smoothed particle hydrodynamics (SPH), GPH is a Lagrangian particle-based approach that involves the tracking of a finite number of particles transported by a flow. However, these particles do not represent mollified particles of matter but carry discrete/partial information about the continuous flow. Closure is achieved by placing a divergence-free GP prior ξ on the velocity field and conditioning it on the vorticity at the particle locations. Known physics (e.g., the Richardson cascade and velocity increment power laws) is incorporated into the GP prior by using physics-informed additive kernels. This is equivalent to expressing ξ as a sum of independent GPs ξ^l, which we call modes, acting at different scales (each mode ξ^l self-activates to represent the formation of eddies at the corresponding scales). This approach enables a quantitative analysis of the Richardson cascade through the analysis of the activation of these modes, and enables us to analyze coarse-grain turbulence statistically rather than deterministically. Because GPH is formulated by using the vorticity equations, it does not require solving a pressure equation. By enforcing incompressibility and fluid-structure boundary conditions through the selection of a kernel, GPH requires significantly fewer particles than SPH. Because GPH has a natural probabilistic interpretation, the numerical results come with uncertainty estimates, enabling their incorporation into an uncertainty quantification (UQ) pipeline and adding/removing particles (quanta of information) in an adapted manner. The proposed approach is suitable for analysis because it inherits the complexity of state-of-the-art solvers for dense kernel matrices and results in a natural definition of turbulence as information loss. Numerical experiments support the importance of selecting physics-informed kernels and illustrate the major impact of such kernels on the accuracy and stability. Because the proposed approach uses a Bayesian interpretation, it naturally enables data assimilation and predictions and estimations by mixing simulation data and experimental data.

The cubic or third-power (TP) nonlinear energy sink (NES) has been proven to be an effective method for vibration suppression, owing to the occurrence of targeted energy transfer (TET). However, TET is unable to be triggered by the low initial energy input, and thus the TP NES would get failed under low-amplitude vibration. To resolve this issue, a new type of NES with fractional nonlinearity, e.g., one-third-power (OTP) nonlinearity, is proposed. The dynamic behaviors of a linear oscillator (LO) with an OTP NES are investigated numerically, and then both the TET feature and the vibration attenuation performance are evaluated. Moreover, an analogy circuit is established, and the circuit simulations are carried out to verify the design concept of the OTP NES. It is found that the threshold for TET of the OTP NES is two orders of magnitude smaller than that of the TP NES. The parametric analysis shows that a heavier mass or a lower stiffness coefficient of the NES is beneficial to the occurrence of TET in the OTP NES system. Additionally, significant energy transfer is usually accompanied with efficient energy dissipation. Consequently, the OTP NES can realize TET under low initial input energy, which should be a promising approach for micro-vibration suppression.

A moving trapezoidal profiled convective-radiative porous longitudinal fin wetted in a single-phase fluid is considered in the current article. The periodic variation in the fin base temperature is taken into account along with the temperature sensitive thermal conductivity and convective heat transfer coefficients. The modeled problem, which is resolved into a non-linear partial differential equation (PDE), is made dimensionless and solved by employing the finite difference method (FDM). The results are displayed through graphs and discussed. The effects of amplitude, frequency of oscillation, wet nature, Peclet number, and other relevant quantities on the distribution of temperature through the fin length and with the dimensionless time are investigated. It is deciphered that the periodic heat transfer gives rise to the wavy nature of the fin thermal profile against time. The analysis is beneficial in the design of fin structures for applications like solar collectors, space/airborne applications, and refrigeration industries.

The work is devoted to the fractional characterization of time-dependent coupled convection-diffusion systems arising in magnetohydrodynamics (MHD) flows. The time derivative is expressed by means of Caputo’s fractional derivative concept, while the model is solved via the full-spectral method (FSM) and the semi-spectral scheme (SSS). The FSM is based on the operational matrices of derivatives constructed by using higher-order orthogonal polynomials and collocation techniques. The SSS is developed by discretizing the time variable, and the space domain is collocated by using equal points. A detailed comparative analysis is made through graphs for various parameters and tables with existing literature. The contour graphs are made to show the behaviors of the velocity and magnetic fields. The proposed methods are reasonably efficient in examining the behavior of convection-diffusion equations arising in MHD flows, and the concept may be extended for variable order models arising in MHD flows.

We present a novel motion control technique for microrobot clusters to exploit the characteristics of the ultrasonic field. The method comprises two steps, i.e., introducing an ultrasonic actuation (UA) linear model for three-dimensional (3D) locomotion and controlling the topological charge (TC) in the ultrasonic vortex for microrobot clustering. Here, the TC is a controllable parameter for the expansion and contraction of the pressure null space inside the vortex. We present a TC control method to cluster sporadically distributed microrobots in a specific workspace. To validate the concept, a UA system composed of 30 ultrasonic transducers with 1 MHz frequency is fabricated, and the characteristics of the generated acoustic pressure field are analyzed through simulations. Subsequently, the performances of the adaptive controller for precise 3D locomotion and the TC control method for clustering are evaluated. Finally, the UA technology, which performs both clustering and locomotion in a complex manner, is validated with a gelatin phantom in an in-vitro environment.

A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains. To realize this method, we design a new wavelet basis function, by which we construct a fifth-order numerical scheme for the approximation of multi-dimensional functions and their multiple integrals defined in complex domains. In the solution of differential equations, various derivatives of the unknown function are denoted as new functions. Then, the integral relations between these functions are applied in terms of wavelet approximation of multiple integrals. Therefore, the original equation with derivatives of various orders can be converted to a system of algebraic equations with discrete nodal values of the highest-order derivative. During the application of the proposed method, boundary conditions can be automatically included in the integration operations, and relevant matrices can be assured to exhibit perfect sparse patterns. As examples, we consider several second-order mathematics problems defined on regular and irregular domains and the fourth-order bending problems of plates with various shapes. By comparing the solutions obtained by the proposed method with the exact solutions, the new multiresolution method is found to have a convergence rate of fifth order. The solution accuracy of this method with only a few hundreds of nodes can be much higher than that of the finite element method (FEM) with tens of thousands of elements. In addition, because the accuracy order for direct approximation of a function using the proposed basis function is also fifth order, we may conclude that the accuracy of the proposed method is almost independent of the equation order and domain complexity.