Abstract: A symplectic eigenspace expansion method was proposed for the dynamic analysis of linear systems with damping and gyroscopic effect. The modal expansion method based on the Rayleigh quotient had been widely used in the dynamics of complex structures. However, the applicability of the modal expansion method was restricted since the gyroscopic effect was not taken into account. Herein, the relation between the Rayleigh quotient eigenvalue problems and the symplectic eigenvalue problems was established firstly. For these two types of problems, it was found that the latter was more general and can be reduced to the former through neglection of the gyroscopic effect. Then, the symplectic eigenspace expansion method was derived to conduct the dynamic analysis of linear damped gyroscopic systems. Finally, effectiveness of the proposed method is shown with the numerical examples of damped systems with and without gyroscopic effect simulated respectively.
Abstract: Previously the circular ring specimen containing an inner single radial crack subjected to diametrically compressive loading had been studied, here it was modified with a pair of parallel flattened facets, which were beneficial to loading, to form a newtype specimen―the holed single cracked flattened Brazilian disc (HSCFBD). In addition, the holed (double) cracked flattened Brazilian disc (HCFBD) was further investigated. The dimensionless stress intensity factors of HSCFBD and HCFBD with different inner-radius-to-outer-radius ratios, different dimensionless crack lengths and different flattened angles, were comprehensively calibrated through finite element simulation, and the curve-fitting formulas of the factor were given. The relative errors between the fitting formulas and the numerical calibration results were within ±1.39%. The effects of geometric factors on the dimensionless stress intensity factor were also analyzed. The results show that the dimensionless stress intensity factor increases with the inner-radius-to-outer-radius ratio, and decreases with the flattened angle. According to the varying law of the stress intensity factor, the specimen parameters of HSCFBD and HCFBD were recommended for testing the mode-I fracture toughness. The preliminary test of HCFBD specimens was carried out for a sand stone, and the comparative test of CCNBD specimens suggested by the International Society for Rock Mechanics was also conducted.
Abstract: The boundary element method (BEM) in time domain was employed for the dynamic analysis of saturated porous media subjected to external forces. Based on Biot’s porodynamic equations, the U-P formulation of Green’s function obtained through decoupling of the fast and slow dilational waves, the transformation of Stokes’state as well as Somigliana’s representation, the discretization forms of the boundary integration equations in time domain were discussed in detail. Specially, with the aid of achievement for a single-phase medium, the singularity in the integration of the BEM for a porous medium was successfully treated in numerical implementation. Finally, in several examples, the response results of the displacements and pore pressures from numerical calculation with dimensionless material parameters were presented. Since the time-domain BEM calculation is hardly found in porodynamics as yet, the proposed method makes a new way for the research of dynamic response of 2phase saturated porous media.
Abstract: The fractional calculus was introduced to describe the damped oscillator in viscoelastic medium and the Caputo-type fractional nonlinear oscillation equations were established. The fractional variational iteration method (FVIM) was modified with a small parameter and the Lagrange multiplier was derived. For the linear fractional oscillation equations, both the homogeneous equations and the sinusoidal force-excited nonhomogeneous equations were analyzed with the FVIM to obtain the approximate analytical solution sequence. The varying curves of the displacement for different values of the fractional order were given in the case of the Bagley-Torvik equation. The relationship between oscillator motion and fractional derivative was also studied according to the extent of memorability for different fractional orders. Compared with the ordinary variational iteration method, the proposed FVIM modified with a small parameter expands the interval of convergence significantly for the solution. In the end, the Van der Pol equation with fractional derivative as an example illustrates the method’s effectiveness and convenience to solve non-linear fractional differential problems.
Abstract: A heat jet approach for atomic simulation at finite temperature in both local space and time was proposed based on the 2-way boundary condition and phonon heat bath, without any dissipation factor and empirical parameter introduced. A subsystem was extracted from a space lattice for analysis of the exact molecular dynamics lest the entire lattice was to be solved numerically. For an extracted linear harmonic chain, the 2-way boundary condition allowed effective incoming waves fully enter the subsystem, and meanwhile, non-thermal motion and thermal fluctuation propagate freely out of the subsystem, to realize dynamic equilibrium of the system energy. During numerical calculation, the 2-way boundary condition worked like a wave diode which let in the positive-going waves while keeping out the negative-going ones. The normal mode of phonon heat bath well described the atomic heat vibration, then it was decoupled into positive-going and negative-going input waves of which the former was used to build the heat source term. For the molecular dynamics simulation of linear harmonic chains, the numerical tests demonstrate effectiveness of the proposed heat jet approach, which makes the chain rapidly reach the expectant temperature, keeps it in a steady state thereafter, and reasonably depicted the additional non-thermal atomic motion at finite temperature.
Abstract: The magneto-elastic coupled vibration theoretical model for axially moving conductive and magnetic beams in magnetic field environment was studied. Based on the Timoshenko beam theory and with the geometric nonlinearity considered, the expressions for the deformation potential energy, kinetic energy, electromagnetic force and the virtual work of mechanical force of the elastic beam in axial motion and lateral bidirectional vibration were gained. Then the Hamilton variational principle was applied to get the nonlinear magneto-elastic coupled vibration equations for the axially moving Timoshenko beam in a magnetic field, and get those for the simplified Euler-Bernoulli beam. Based on the electromagnetic theory and the constitutive relation of the corresponding electromagnetism, the expressions for the electromagnetic force of the current-conducting elastic beam, and for the magnet force and magnet force couple of the magneto-elastic beam based on the magnetic dipole-current loop model, were derived. Through the numerical example, the singularity distribution and stability of the conductive and elastic beam in axial movement were analyzed.
Abstract: The Falkner-Skan flow equation is a strongly nonlinear differential equation, which describes the flow around a wedge. In order to overcome the difficulties originated from the semi-infinite interval and asymptotic boundary condition in this flow problem, transformations were simultaneously conducted for both the independent variable and the correponding function to convert the problem to a 2-point boundary value one within a finite interval. The deduced new-form nonlinear differential equation was subsequently solved with the fixed point method (FPM). The present analytical results obtained with the FPM agree well with the previous referential numerical ones. The accuracy of the present solution is conveniently improved through iteration under the FPM framework, which shows that the FPM makes a promising tool for nonlinear differential equations.
Abstract: The liquid bridge is the main reason for image distortion of an atomic force microscope (AFM) in the atmospheric ambiance, meanwhile the capillary force resulting from the liquid bridge dominates the adhesion force in this condition. Investigation of the liquid bridge is of great importance to understand the imaging mechanisms and the sample properties. Herein, 3 different growth processes were presented and analyzed for the growth mechanisms of the liquid bridge in AFMs, including the squeezing process, the capillary condensation and the motion of thin liquid film. The characteristic equilibrium times of the 3 processes are of great importance to the understanding of the liquid bridge’s growth dynamics. The equilibrium time of the squeezing process depends on the contact mode, that of the capillary condensation is at the μs order of magnitude and that of the liquid film motion varies drastically with different viscosities of the liquid film. The contribution of the corresponding 3 growth mechanisms to liquid bridge volume, capillary force and energy dissipation was comaratively studied in different AFM operation modes.
Abstract: A train of periodic deep water stationary waves with finite amplitudes were investigated analytically with the homotopy analysis method. The vertical distribution of water density was considered as variable in a continuous exponential trend. A new form of partial differential equations were proposed as the auxiliary equations and the new-form solution expressions were obtained in order to match the level boundary condition at the bottom and the hypothetical infinitely rigid condition. The detailed recursive relation of the coefficient in the solution expression was given and the explicit expressions of the permanent stationary periodic internal waves were presented. The convergent series solutions were obtained for the global domain both in vertical and horizontal directions. The relation between the density variable and the internal wave amplitude was revealed.
Abstract: According to the derivative relation of physic variables between the Lagrangian space and Eulerian space, a new method of manufacturing solutions was proposed for the verification of Lagrangian radiation hydrodynamic codes. With this method, the manufactured solutions to the 1D Lagrangian radiation hydrodynamic equations were given to be smooth and differentiable in the whole computional domain, without source term in the mass equation. Then the manufactured solutions were applied to verify the correctness of the 2D Lagrangian radiation hydrodynamic codes. The numerical results show the effectiveness of the presented method.