Table of Content

01 January 2022, Volume 43 Issue 1

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Articles

On band gaps of nonlocal acoustic lattice metamaterials: a robust strain gradient model

Binying WANG, Jinxing LIU, A. K. SOH, Naigang LIANG

2022, 43(1):  1-20.  doi:10.1007/s10483-021-2795-5

Abstract ( 2138 )   HTML ( 1421328404)   PDF (1827KB) ( 332 )  

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We have proposed an "exact" strain gradient (SG) continuum model to properly predict the dispersive characteristics of diatomic lattice metamaterials with local and nonlocal interactions. The key enhancement is proposing a wavelength-dependent Taylor expansion to obtain a satisfactory accuracy when the wavelength gets close to the lattice spacing. Such a wavelength-dependent Taylor expansion is applied to the displacement field of the diatomic lattice, resulting in a novel SG model. For various kinds of diatomic lattices, the dispersion diagrams given by the proposed SG model always agree well with those given by the discrete model throughout the first Brillouin zone, manifesting the robustness of the present model. Based on this SG model, we have conducted the following discussions. (I) Both mass and stiffness ratios affect the band gap structures of diatomic lattice metamaterials, which is very helpful for the design of metamaterials. (II) The increase in the SG order can enhance the model performance if the modified Taylor expansion is adopted. Without doing so, the higher-order continuum model can suffer from a stronger instability issue and does not necessarily have a better accuracy. The proposed SG continuum model with the eighth-order truncation is found to be enough to capture the dispersion behaviors all over the first Brillouin zone. (III) The effects of the nonlocal interactions are analyzed. The nonlocal interactions reduce the workable range of the well-known long-wave approximation, causing more local extrema in the dispersive diagrams. The present model can serve as a satisfactory continuum theory when the wavelength gets close to the lattice spacing, i.e., when the long-wave approximation is no longer valid. For the convenience of band gap designs, we have also provided the design space from which one can easily obtain the proper mass and stiffness ratios corresponding to a requested band gap width.

Quaternion methods and models of regular celestial mechanics and astrodynamics

Y. N. CHELNOKOV

2022, 43(1):  21-80.  doi:10.1007/s10483-021-2797-9

Abstract ( 1222 )   HTML ( 11)   PDF (416KB) ( 137 )  

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This paper is a review, which focuses on our work, while including an analysis of many works of other researchers in the field of quaternionic regularization. The regular quaternion models of celestial mechanics and astrodynamics in the Kustaanheimo-Stiefel (KS) variables and Euler (Rodrigues-Hamilton) parameters are analyzed. These models are derived by the quaternion methods of mechanics and are based on the differential equations of the perturbed spatial two-body problem and the perturbed spatial central motion of a point particle. This paper also covers some applications of these models. Stiefel and Scheifele are known to have doubted that quaternions and quaternion matrices can be used efficiently to regularize the equations of celestial mechanics. However, the author of this paper and other researchers refuted this point of view and showed that the quaternion approach actually leads to efficient solutions for regularizing the equations of celestial mechanics and astrodynamics.
This paper presents convenient geometric and kinematic interpretations of the KS transformation and the KS bilinear relation proposed by the present author. More general (compared with the KS equations) quaternion regular equations of the perturbed spatial two-body problem in the KS variables are presented. These equations are derived with the assumption that the KS bilinear relation was not satisfied. The main stages of the quaternion theory of regularizing the vector differential equation of the perturbed central motion of a point particle are presented, together with regular equations in the KS variables and Euler parameters, derived by the aforementioned theory. We also present the derivation of regular quaternion equations of the perturbed spatial two-body problem in the Levi-Civita variables and the Euler parameters, developed by the ideal rectangular Hansen coordinates and the orientation quaternion of the ideal coordinate frame.
This paper also gives new results using quaternionic methods in the perturbed spatial restricted three-body problem.

Dynamic symmetry breaking and structure-preserving analysis on the longitudinal wave in an elastic rod with a variable cross-section

Jingjing HU, Mengbo XU, Weipeng HU, Ruisong JIANG, Zichen DENG

2022, 43(1):  81-92.  doi:10.1007/s10483-022-2809-6

Abstract ( 1220 )   HTML ( 7)   PDF (254KB) ( 137 )  

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The longitudinal wave propagating in an elastic rod with a variable crosssection owns wide engineering background, in which the longitudinal wave dissipation determines some important performances of the slender structure. To reproduce the longitudinal wave dissipation effects on an elastic rod with a variable cross-section, a structure-preserving approach is developed based on the dynamic symmetry breaking theory. For the dynamic model controlling the longitudinal wave propagating in the elastic rod with the variable cross-section, the approximate multi-symplectic form is deduced based on the multi-symplectic method, and the expression of the local energy dissipation for the longitudinal wave propagating in the rod is presented, referring to the dynamic symmetry breaking theory. A structure-preserving method focusing on the residual of the multi-symplectic structure and the local energy dissipation of the dynamic model is constructed by using the midpoint difference discrete method. The longitudinal wave propagating in an elastic rod fixed at one end is simulated, and the local/total energy dissipations of the longitudinal wave are investigated by the constructed structure-preserving scheme in two typical cases in detail.

Non-smooth dynamic modeling and simulation of an unmanned bicycle on a curved pavement

Kaiming ZHANG, Xudong ZHENG, Zhang CHEN, Bin LIANG, Tianshu WANG, Qi WANG

2022, 43(1):  93-112.  doi:10.1007/s10483-022-2811-5

Abstract ( 1306 )   HTML ( 5)   PDF (1642KB) ( 108 )  

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The non-smooth dynamic model of an unmanned bicycle is established to study the contact-separate and stick-slip non-smooth phenomena between wheels and the ground. According to the Carvallo-Whipple configuration, the unmanned bicycle is reduced to four rigid bodies, namely, rear wheel, rear frame, front fork, and front wheel, which are connected by perfect revolute joints. The interaction between each wheel and the ground is simplified as the normal contact force and the friction force at the contact point, and these forces are described by the Hunt-Crossley contact force model and the LuGre friction force model, respectively. According to the characteristics of flat and curved pavements, calculation methods for contact forces and their generalized forces are presented. The dynamics of the system is modeled by the Lagrange equations of the first kind, a numerical solution algorithm of the dynamic equations is presented, and the Baumgarte stabilization method is used to restrict the drift of the constraints. The correctness of the dynamic model and the numerical algorithm is verified in comparison with the previous studies. The feasibility of the proposed model is demonstrated by simulations under different motion states.

Mixed convective flow of a hybrid nanofluid between two parallel inclined plates under wall-slip condition

Hang XU

2022, 43(1):  113-126.  doi:10.1007/s10483-021-2801-6

Abstract ( 1267 )   HTML ( 6)   PDF (372KB) ( 154 )  

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The mixed convection flow of a hybrid nanofluid in an inclined channel with top wall-slip due to wall stripe and constant heat flux conditions is studied. Explicit analytical solutions are given to the flow velocity, temperature, as well as the pressure in non-dimensional forms. The flow regime domain, the velocity and temperature distributions, and the dependence of various physical parameters such as the hybrid nanoparticle volume fractions, the wall-slip, the Grashof number, the Reynolds number, and the inclined angle are analyzed and discussed. It is found that the hybrid nanofluid delays the appearance of flow reversal on both walls and the wall-slip postpones the flow reversal on the top wall.

Magnetohydrodynamic flow past a shrinking vertical sheet in a dusty hybrid nanofluid with thermal radiation

I. WAINI, A. ISHAK, I. POP

2022, 43(1):  127-140.  doi:10.1007/s10483-022-2807-8

Abstract ( 1209 )   HTML ( 4)   PDF (770KB) ( 103 )  

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The magnetohydrodynamic (MHD) mixed convection flow past a shrinking vertical sheet with thermal radiation is considered. Besides, the effects of Cu-Al₂O₃ nanoparticles and dust particles are considered. The similarity variables reduce the governing equations to the similarity equations, which are then solved numerically. The outcome shows that, for the shrinking case, the solutions are not unique. The rate of heat transfer and the friction factor enlarge with increasing the values of the copper nanoparticle volume fraction as well as the magnetic parameter. Meanwhile, the assisting flow and the rise of the thermal radiation reduce these quantities. Two solutions are found, and the boundary layer separation is dependent on the mixed convection parameter.

Parallel finite element computation of incompressible magnetohydrodynamics based on three iterations

Qili TANG, Yunqing HUANG

2022, 43(1):  141-154.  doi:10.1007/s10483-022-2802-7

Abstract ( 1289 )   HTML ( 6)   PDF (6700KB) ( 134 )  

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Based on local algorithms, some parallel finite element (FE) iterative methods for stationary incompressible magnetohydrodynamics (MHD) are presented. These approaches are on account of two-grid skill include two major phases:find the FE solution by solving the nonlinear system on a globally coarse mesh to seize the low frequency component of the solution, and then locally solve linearized residual subproblems by one of three iterations (Stokes-type, Newton, and Oseen-type) on subdomains with fine grid in parallel to approximate the high frequency component. Optimal error estimates with regard to two mesh sizes and iterative steps of the proposed algorithms are given. Some numerical examples are implemented to verify the algorithm.