应用数学和力学

2014 Vol. 35, No. 1

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Symplectic Conservation Integration of Rigid Body Dynamics With Quaternion Parameters

XU Xiao-ming, ZHONG Wan-xie

2014, 35(1): 1-11. doi: 10.3879/j.issn.1000-0887.2014.01.001

Abstract(2082) PDF(2569)

Abstract:
A numerical method was proposed with the quaternion representation of rigid body dynamics. Based on the analytical structural mechanics, the action of differential system was introduced for the time integration of the approximated discrete system and the constraint that the norm of quaternion kept constant at 1 was satisfied strictly at the grid points of integration. As was interpreted in the theory of analytical structural mechanics, the numerical integration was symplectic conservative and the constraint was satisfied approximately in the sense of variation principle. The numerical results of heavy tops are satisfying in precision and efficiency.

Symplectic Runge-Kutta Method for Structural Dynamics

GUO Jing, XING Yu-feng

2014, 35(1): 12-21. doi: 10.3879/j.issn.1000-0887.2014.01.002

Abstract(1426) PDF(1251)

Abstract:
An explicit and efficient implementation of the symplectic implicit Gauss-Legendre Runge-Kutta (RK) method of stage s and order 2s,was presented for solution of the dynamical ordinary differential equation with physical damping and external loads. The analytical explicit spectral radii and single-step phase errors of the implicit Gauss-Legendre RK method were given and compared with those of the explicit classical RK method of stage 4 and order 4. Numerical comparisons through the dynamical solution of a linear multi-degree-of-freedom (MDOF) system and a nonlinear Rayleigh system were made to validate the present study and showed the advantages of the symplectic RK method over the classical RK method with numerical dissipation, especially in aspects of the kinematic properties and long time numerical simulation.

Structure-Preserving Algorithm for Steady-State Solution to the Infinite Dimensional Hamilton System

QIN Yu-yue, DENG Zi-chen, HU Wei-peng

2014, 35(1): 22-28. doi: 10.3879/j.issn.1000-0887.2014.01.003

Abstract(1566) PDF(1320)

Abstract:
Based on Hamilton variational principle and Bridges’ multi-symplectic integration theory, a new structure-preserving algorithm was proposed to simulate the steady-state solution to the complex infinite dimensional Hamilton system. With Zufiria’s Boussinesq-type equations as an example, the high-order partial differential equation describing the steady-state solution to the Zufiria model was rewritten into a symmetric form under the energy flux conservation law and momentum flux conservation law firstly; then the box scheme for the symmetric form was constructed to simulate the steady-state solution to the Zufiria model. The numerical results show that the box scheme can well simulate the steady-state solution to the Zufiria model while properly preserving the momentum flux conservation law.

Research on the Vibration Property of the Beam on Elastic Foundation Based on the PCs Theory

CHEN Qi-yong, HU Shao-wei, ZHANG Zi-ming

2014, 35(1): 29-38. doi: 10.3879/j.issn.1000-0887.2014.01.004

Abstract(1273) PDF(1099)

Abstract:
Axial loads imposed on the structure influence the vibration properties and cause the change of vibration resistance functions. Theory about the band gap properties of phononic crystals (PCs) were used to study the flexural vibration band gaps of an Euler beam on elastic foundation. A flexural vibration model of the infinite periodic PCs Euler beam was established, which was under the actions of axial force and Winkler foundation. A modified transfer matrix (MTM) method was applied to calculate the band structure of the beam. The change tendency of the band structure were estimated on the basis of the band structure. Results show that axial loads influence the band gaps and band frequency ranges. Axial tensile loads elevate the band gap frequencies, but the base band gaps remain unchanged; axial compressive loads lower the band gap frequencies, and the base band gap frequencies drop when the amplitudes of the compressive loads increase. Meanwhile, the Euler beam model was numerically simulated, and the results were matched with the analytical ones. Through adjusting the magnitude of the axial loads, different band frequency ranges and effects of vibration reduction could be achieved.

Nonlinear Galloping of Iced Quad-Bundle Conductors With Internal Resonances

YAN Bo, LIU Xiao-hui, ZHAO Li, ZHOU Lin-shu

2014, 35(1): 39-49. doi: 10.3879/j.issn.1000-0887.2014.01.005

Abstract(1404) PDF(1050)

Abstract:
Galloping of iced quad-bundle conductors with internal resonance was numerically investigated by means of nonlinear finite element method. Based on the numerical simulation of galloping processes of the typical iced quad-bundle conductor lines in both the steady and the stochastic wind fields, galloping characteristics of the iced conductor line with the ratio of the frequency of the first symmetric in-plane mode to the frequency of the out-of-plane mode being 2∶1 were discussed. It is shown that energy continuously interchanges between the vertical motion and transverse horizontal motion as internal resonance exists, which is apparently different from those of the line without internal resonance. The result provides a theoretical basis for the understanding of the coupling mechanism of galloping.

On Piecewise Nonlinear Dynamic Characteristics of a New-Type Quasi-Zero-Stiffness Vibration Isolator With Cam-Roller-Spring Mechanism

WANG Xin-long, ZHOU Jia-xi, XU Dao-lin

2014, 35(1): 50-62. doi: 10.3879/j.issn.1000-0887.2014.01.006

Abstract(1947) PDF(1120)

Abstract:
A piecewise nonlinear dynamic model for a new-type quasi-zero-stiffness (QZS) vibration isolation system with cam-roller-spring mechanism was developed. The dynamic characteristics of the QZS system under harmonic base motion were analyzed theoretically with the averaging method, and the primary resonance responses obtained. The analytical solution was verified by the numerical solution to the equation of motion, which confirmed the efficiency of the averaging method to solve the piecewise nonlinear problem. Further, the effects of exciting displacement amplitude and damping on the system responses were discussed. The displacement transmissibility was used to evaluate the performance of vibration isolation. The results show that the exciting displacement amplitude and damping notably influence the system response. When the exciting amplitude is relatively small, the QZS system outperforms the corresponding linear system obviously. The QZS system’s performance goes down as the exciting amplitude increases, but stands no worse than that of the corresponding linear system. This feature of the new-type QZS vibration isolation system is superior to that of the traditional QZS ones.

Non-Stationary Response of a Stochastic System With Fractional Derivative Damping Under Gaussian White-Noise Excitation

LI Wei, ZHAO Jun-feng, LI Rui-hong, Natasa Trisovic

2014, 35(1): 63-70. doi: 10.3879/j.issn.1000-0887.2014.01.007

Abstract(1664) PDF(995)

Abstract:
Non-stationary response of a nonlinear stochastically dynamical system with fractional derivative damping under Gaussian white-noise excitation was investigated. Based on the equivalent linearization method, the original nonlinear system was converted to a linear system with respect to the vibration amplitude and phase, then the stochastic averaging method was applied to obtain the FPK equation, in which the fractional derivative was approximated by a periodic function. The approximate non-stationary response of the FPK equation was derived with Galerkin method. Numerical results verify the efficiency and correction of the proposed method.

Dynamic Responses of the Submerged Floating Tunnel Under Combined Internal Wave and Sea Current Effect

TIAN Xue-fei, DONG Man-sheng, PANG Huan-ping, LIN Zhi, NIU Zhong-rong

2014, 35(1): 71-80. doi: 10.3879/j.issn.1000-0887.2014.01.008

Abstract(1589) PDF(1139)

Abstract:
The stratified ocean internal wave flow field was established based on the potential flow function theory. The joint force by internal wave and sea current was considered with the Morison method to establish a nonlinear vibration mathematical-physical model of tunnel-fluid interaction. The corresponding vibration differential equations were solved numerically with the Galerkin method to study the multimodal dynamic response behaviors of the submerged floating tunnel under combined internal wave and sea current effect. According to the calculation and analysis of a practical example on the proposed tunnel, the first order mode contributed most of the displacement response.The effect of internal wave on the the joint force by internal wave and sea current could not be ignored, it resulted in a substantial increase of the response amplitudes of the first and third order modes and made the responses strongly nonlinear. The research results provide helpful discussions for the structural load analysis of submerged floating tunnels and for the study on environmental loads upon ocean engineering structures in a complex environment.

Free Vibration Analysis of Elastic Foundation Euler Beams With Different Discontinuities Based on Generalized Functions

CHEN Xiao-chao, MAO Qi-bo, XUE Xiao-li

2014, 35(1): 81-91. doi: 10.3879/j.issn.1000-0887.2014.01.009

Abstract(1535) PDF(872)

Abstract:
The general governing differential equations for the vibration of elastic foundation Euler-Bernoulli beams with different discontinuities subject to axial forces were established based on generalized functions. For each discontinuity at a given location, a basic modal displacement function (Dirac delta function) starting at that location was introduced. The differential equations were then solved by means of Laplace transformation. Unlike the classical vibration solutions to problems of beams with discontinuities, the generalized solution was in a single unified expression for the whole beam. Due to unification of the modal function and degeneration of the compatibility conditions, solution of the eigenvalues was greatly simplified. Finally, the free vibration problems of (a) an elastic foundation beam with multiple masses and corresponding rotary inertias, and (b) an elastic foundation beam with multiple cracks under axial force, were solved with the proposed method. Results show that the present method is accurate and effecient for free vibration analysis of beams with different discontinuities.

Lagrangian High-Order Staggered Conservative Gasdynamics Scheme on Unstructured Meshes

GE Quan-wen

2014, 35(1): 92-101. doi: 10.3879/j.issn.1000-0887.2014.01.010

Abstract(1367) PDF(1098)

Abstract:
A Lagrangian high-order staggered conservative gasdynamics scheme on unstructured meshes was presented.The high-order piecewise pressure on the cell arising from the present-moment subcell density and present-moment subcell acoustic speed was used to construct the high-order subcell force with the MUSCL method.The time discretization of the spatial fluxes was performed by means of the Taylor expansions of the spatial fluxes centered in time. Thereupon the Lagrangian high-order staggered conservative gasdynamics scheme was established. Several numerical tests were presented to demonstrate the robustness and accuracy of the new scheme.

A Dynamic Bayesian Network Model for Structural Time-Dependent Reliability Analysis of Resistance Deterioration

SUN Hong-bin, WU Zi-yan, LIU Shu-kui

2014, 35(1): 102-110. doi: 10.3879/j.issn.1000-0887.2014.01.011

Abstract(1470) PDF(1097)

Abstract:
A dynamic Bayesian network (DBN) model was proposed for timedependent reliability analysis of structures in deterioration. The structural resistance deterioration was modeled as a gamma process while the loads as random variables. The stochastic deterioration process was discretized in time domain as deterioration models. A DBN was established and comprised of the reliability model, deterioration model and observation model. Node elimination algorithm and discretization were applied to modify the DBN into a network with only discrete variables. Exact inferences with the DBN were presented to estimate the 3 structural states at present (filtering), in the future (prediction) and in the past (smoothing), respectively. The structural timedependent reliability was updated with the reestimated deterioration model when measurements were available. The proposed model was validated through the timedependent reliability analysis of a onebay example frame in resistance deterioration.