The subharmonic resonance of van der Pol (VDP) oscillator with fractional-order derivative is studied by the averaging method. At first, the first-order approximate solutions are obtained by the averaging method. Then the definitions of equivalent linear damping coefficient (ELDC) and equivalent linear stiffness coefficient (ELSC) for subharmonic resonance are established, and the effects of the fractional-order parameters on the ELDC, the ELSC, and the dynamical characteristics of system are also analysed. Moreover, the amplitude-frequency equation and phase-frequency equation of steady-state solution for subharmonic resonance are established. The corresponding stability condition is presented based on Lyapunov theory, and the existence condition for subharmonic resonance (ECSR) is also obtained. At last, the comparisons of the fractional-order and the traditional integer-order VDP oscillator are fulfilled by the numerical simulation. The effects of the parameters in fractional-order derivative on the steady-state amplitude, the amplitude-frequency curves, and the system stability are also studied.
To protect the environment and to minimize the water pressure on a tunnel lining , the controlled drainage principle is proposed for deep mountain tunnels below high water tables. The mechanical characteristics of tunnel supports and linings with respect to seepage and pressure are studied by using numerical methods and a test mooel based on the tunnel mechanics and seep age mechanics. The results show that the pressure exerted on primary supports is unrelated to surrounding drainage conditions. The pore water pressure on primary supports diverts to tunnel linings when the drainage system moves from the interlace between rock mass and primary supports to the interlace between primary supports and tunnel linings , and the primary supports are wUoaded and the pressure moves outwards to the rock mass. The water pressure on twmel linings can be neglected if the ground water can be discharged smoothly. When a tunnel drainage system malfwlCtions and leads to poor drainage, the flow gradient in the ground de creases, the effective radial stresses decrease, the radial flow of tunnel decreases and the ground deformations decrease.
To solve the classical problem that the Mohr–Coulomb yield criterion overestimates the tensile properties of geotechnical materials, a modified Mohr–Coulomb yield criterion that includes both maximum tensile stress theory and smooth processing was established herein. The modified Mohr–Coulomb constitutive model is developed using the user-defined material subroutine (UMAT) available in finite element software ABAQUS, and the modified Mohr–Coulomb yield criterion is applied to construct a numerical simulation of a shaking table model test. Compared with the measured data from the shaking table test, the accuracies of the classical Mohr–Coulomb yield criterion and the modified Mohr–Coulomb yield criterion are assessed. Compared to the shaking table test, the classical Mohr–Coulomb model has a relatively large average error (−6.98% in peak acceleration values, −8.47% in displacement values, −23.93% in axial forces), while the modified Mohr–Coulomb model has a smaller average error (+2.71% in peak accelerations value, +3.19% in displacements value, +7.56% in axial forces). The results of numerical simulation using the modified Mohr–Coulomb yield criterion are closer to the measured data.
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