A theoretical analysis of the refracted shadows produced bysteady andtime-decaying liquid vortices under uniform illumination from above is givenin this article. An expression for the induced shadow intensity is derivedand found to be a function of the vortex's free surface profile, i.e.,function of the static pressure distribution. The patterns for differentfocusing depth are given and compared with previous visualization resultsfrom the literature. The phenomenon is examined and illustrated as a benchmark case by using both steady and time-decaying algebraic vortex models.However, this study can be extended to check the feasibility of recoveringthe main flow properties by analyzing the luminous image intensity of therefracted patterns. The present analysis is valid only when the swirlvelocity is order of magnitude higher than the meridional flow componentsand the vorticity is concentrated within the core region and of intenseconditions.
When the cell width of the incident detonation wave (IDW) is comparable to or larger than the Mach stem height, self-similarity will fail during IDW reflection from a wedge surface. In this paper, the detonation reflection from wedges is investigated for at the wave dynamic processes occurring in the wave front, including transverse shock motion and detonation cell variations behind the Mach stem. A detailed reaction model is implemented to simulate two-dimensional cellular detonations in stoichiometric mixtures of H2/O2 diluted by Argon. The numerical results show that the transverse waves, which cross the triple point trajectory of Mach reflection, travel along the Mach stem and reflect back from the wedge surface, control the size of the cells in the region swept by the Mach stem. It is the energy carried by these transverse waves that sustain the triple-wave-collision of higher frequency within the over-driven Mach stem. In some cases, local wave dynamic processes and wave structures play a dominant role in determining the pattern of cellular record, leading to the fact that the cellular patterns after the Mach stem exhibit some peculiar modes.
Analytical studies on electromagnetoelastic behaviors arepresented for the functionally graded piezoelectric material (FGPM) solidcylinder and sphere placed in a uniform magnetic field and subjected to theexternal pressure and electric loading. When the mechanical, electric andmagnetic properties of the material obey an identical power law in theradial direction, the exact displacements, stresses, electric potentials andperturbations of magnetic field vector in the FGPM solid cylinder and sphereare obtained by using the infinitesimal theory of electromagnetoelasticity.Numerical examples also show the significant influence of materialinhomogeneity. It is interesting to note that selecting a specific value ofinhomogeneity parameter can optimize the electromagnetoelasticresponses, which will be of particular importance in modern engineeringdesigns.
Finite element simulations are carried out to examine the mechanical behavior of the metallic hollow sphere (MHS) material during their large plastic deformation and to estimate the energy absorbing capacity of these materials under uniaxial compression. A simplified model is proposed from experimental observations to describe the connection between the neighboring spheres, which greatly improves the computation efficiency. The effects of the governing physical and geometrical parameters are evaluated; whilst a special attention is paid to the plateau stress, which is directly related to the energy absorbing capacity. Finally, the empirical functions of the relative material density are proposed for the elastic modulus, yield strength and plateau stress for FCC packing arrangement of hollow spheres, showing a good agreement with the experimental results obtained in our previous study.
An alternative strain energy method is proposed for the prediction of effective elastic properties of orthotropic materials in this paper. The method is implemented in the topology optimization procedure to design cellular solids. A comparative study is made between the strain energy method and the well-known homogenization method. Numerical results show that both methods agree well in the numerical prediction and sensitivity analysis of effective elastic tensor when homogeneous boundary conditions are properly specified. Two dimensional and 3 dimensional microstructures are optimized for maximum stiffness designs by combining the proposed method with the dual optimization algorithm of convex programming. Satisfactory results are obtained for a variety of design cases.
In this paper, nonlinear modeling for flexible multibody system with large deformation is investigated. Absolute nodal coordinates are employed to describe the displacement, and variational motion equations of a flexible body are derived on the basis of geometric nonlinear theory, in which both the shear strain and the transverse normal strain are taken into account. By separating the inner and the boundary nodal coordinates, the motion equations of a flexible multibody system are assembled. The advantage of such formulation is that the constraint equations and the forward recursive equations become linear because the absolute nodal coordinates are used. A spatial double pendulum connected to the ground with a spherical joint is simulated to investigate the dynamic performance of flexible beams with large deformation. Finally, the resultant constant total energy validates the present formulation.
多尺度法是为解决含小参数系统发展起来的应用最广泛的摄动法之一. 在求解高阶近似方程时，多尺度法一般只求特解. 用多尺度法求解van der Pol 方程的三阶解时将出现矛盾. 以van der Pol方程为例，证明了忽略一阶修正量中的一阶谐波项使得混合偏导数不能交换顺序，从而导致了多尺度法的二义性和另一个数学矛盾.在求解一阶修正量时采用含有一阶谐波项的全解，消除了二义性和该矛盾. 该方法所求得的近似解与数值解进行了比较，结果非常吻合，验证了其合理性.