Numerical analysis of dynamic response of vehicle–bridge coupled system on long-span continuous girder bridge

To systematically study the vehicle–bridge coupled dynamic response and its change rule with different parameters, a vehicle model with seven degrees of freedom was built and the total potential energy of vehicle space vibration system was deduced. Considering the stimulation of road roughness, the dynamic response equation of vehicle–bridge coupled system was established in accordance with the elastic system principle of total potential energy with stationary value and the ‘‘set-in-right-position' rule. On the basis of the self-compiled Fortran program and bridge engineering, the dynamic response of longspan continuous girder bridge under vehicle load was studied. This study also included the calculation of vehicle impact coefficient, evaluation of vibration comfort, and analysis of dynamic response parameters.Results show the impact coefficient changes with lane number and is larger than the value calculated by the ‘‘general code for design of highway bridges and culverts(China)'. The Dieckmann index of bridge vibration is also related to lane number, and the vibration comfort evaluation is good in normal conditions.The relevant conclusions from parametric analyses have practical significance to dynamic design and daily operation of long-span continuous girder bridges in expressways. Safety and comfort are expected to improve significantly with further control of the vibration of vehicle–bridge system.     

Interaction of subway LIM vehicle with ballasted track in polygonal wheel wear development

This paper develops a coupled dynamics model for a linear induction motor (LIM) vehicle and a subway track to investigate the influence of polygonal wheels of the vehicle on the dynamic behavior of the system. In the model, the vehicle is modeled as a multi-body system with 35 degrees of freedom. A Timoshenko beam is used to model the rails which are discretely supported by sleepers. The sleepers are modeled as rigid bodies with their vertical, lateral, and rolling motions being considered. In order to simulate the vehicle running along the track, a moving sleeper support model is introduced to simulate the excitation by the discrete sleeper supporters, in which the sleepers are assumed to move backward at a constant speed that is the same as the train speed. The Hertzian contact theory and the Shen- Hedrick-Elkins’ model are utilized to deal with the normal dynamic forces and the tangential forces between wheels and rails, respectively. In order to better characterize the linear metro system (LMS), Euler beam theory based on modal superposition method is used to model LIM and RP. The vertical electric magnetic force and the lateral restoring force between the LIM and RP are also taken into consideration. The former has gap-varying nonlinear characteristics, whilst the latter is considered as a constant restoring force of 1 kN. The numerical analysis considers the effect of the excitation due to polygonal wheels on the dynamic behavior of the system at different wear stages, in which the used data regarding the polygonal wear on the wheel tread are directly measured at the subway site.     

Wide spectrum solar energy harvesting through an integrated photovoltaic and thermoelectric system

This paper proposes a power system concept that integrates photovoltaic （PV） and thermoelectric （TE） technologies to harvest solar energy from a wide spectral range. By introduction of the ＇spectrum beam splitting＇ technique, short wavelength solar radiation is converted directly into electricity in the PV cells, while the long wavelength segment of the spectrum is used to produce moderate to high temperature thermal energy, which then generates electricity in the TE device. To overcome the intermittent nature of solar radiation, the system is also coupled to a thermal energy storage unit. A systematic analysis of the integrated system is carried out, encompassing the system configuration, material properties, thermal management, and energy storage aspects. We have also attempted to optimize the integrated system. The results indicate that the system configuration and optimization are the most important factors for high overall efficiency.     

Partition method for impact dynamics of flexible multibody systems based on contact constraint

The impact dynamics of a flexible multibody system is investigated. By using a partition method, the system is divided into two parts, the local impact region and the region away from the impact. The two parts are connected by specific boundary conditions, and the system after partition is equivalent to the original system. According to the rigid-flexible coupling dynamic theory of multibody system, system＇s rigid-flexible coupling dynamic equations without impact are derived. A local impulse method for establishing the initial impact conditions is proposed. It satisfies the compatibility con- ditions for contact constraints and the actual physical situation of the impact process of flexible bodies. Based on the contact constraint method, system＇s impact dynamic equa- tions are derived in a differential-algebraic form. The contact/separation criterion and the algorithm are given. An impact dynamic simulation is given. The results show that system＇s dynamic behaviors including the energy, the deformations, the displacements, and the impact force during the impact process change dramatically. The impact makes great effects on the global dynamics of the system during and after impact.     

ANALYSIS OF NONLINEAR DYNAMIC STABILITY OF LIQUID-CONVEYING PIPES

Nonlinearly dynamic stability of flexible liquid-conveying pipe in fluid structure interaction was analyzed by using modal disassembling technique . The effects of Poisson . Junction and Friction couplings in the wave-flowing-vibration system on the pipe dynamic stability were included in the analytical model constituted by four nonlinear differential equations . An analyzing example of cantilevered pipe was done to illustrate the dynamic stability characteristics of the pipe in the full coupling mechanisms , and the phase curves related to the first four modal motions were drawn . The results show that the dynamic stable characteristics of the pipe are very complicated in the complete coupling mechanisms, and the kinds of the singularity points corresponding to the various modal motions are different.     

Track dynamic behavior at rail welds at high speed

As a vehicle passing through a track with different weld irregularities, the dynamic performance of track com- ponents is investigated in detail by using a coupled vehi- cle-track model. In the model, the vehicle is modeled as a multi-body system with 35 degrees of freedom, and a Timoshenko beam is used to model the rails which are dis- cretely supported by sleepers. In the track model, the sleepers are modeled as rigid bodies accounting for their vertical, lat- eral and rolling motions and assumed to move backward at a constant speed to simulate the vehicle running along the track at the same speed. In the study of the coupled vehicle and track dynamics, the Hertizian contact theory and the theory proposed by Shen-Hedrick-Elkins are, respectively, used to calculate normal and creep forces between the wheel and the rails. In the calculation of the normal forces, the coefficient of the normal contact stiffness is determined by transient contact condition of the wheel and rail surface. In the calcu- lation of the creepages, the lateral, roll-over motions of the rail and the fact that the relative velocity between the wheel and rail in their common normal direction is equal to zero are simultaneously taken into account. The motion equations of the vehicle and track are solved by means of an explicit integration method, in which the rail weld irregularities are modeled as local track vertical deviations described by some ideal cosine functions. The effects of the train speed, the axle load, the wavelength and depth of the irregularities, and the weld center position in a sleeper span on the wheel-rail impact loading are analyzed. The numerical results obtained are greatly useful in the tolerance design of welded rail pro- file irregularity caused by hand-grinding after rail welding and track maintenances.     

Effect of wheelset flexibility on wheel–rail contact behavior and a specific coupling of wheel–rail contact to flexible wheelset

The fexibility of a train's wheelset can have a large effect on vehicle–track dynamic responses in the medium to high frequency range.To investigate the effects of wheelset bending and axial deformation of the wheel web,a specifi coupling of wheel–rail contact with a fexible wheelset is presented and integrated into a conventional vehicle–track dynamic system model.Both conventional and the proposed dynamic system models are used to carry out numerical analyses on the effects of wheelset bending and axial deformation of the wheel web on wheel–rail rolling contact behaviors.Excitations with various irregularities and speeds were considered.The irregularities included measured track irregularity and harmonic irregularities with two different wavelengths.The speeds ranged from 200 to400km/h.The results show that the proposed model can characterize the effects of fexible wheelset deformation on the wheel–rail rolling contact behavior very well.     

Integration of uniform design and quantum-behaved particle swarm optimization to the robust design for a railway vehicle suspension system under different wheel conicities and wheel rolling radii

This paper proposes a systematic method, inte-grating the uniform design(UD)of experiments and quantum-behaved particle swarm optimization(QPSO),to solve the problem of a robust design for a railway vehicle suspension system. Based on the new nonlinear creep model derived from combining Hertz contact theory, Kalker's linear the-ory and a heuristic nonlinear creep model,the modeling and dynamic analysis of a 24 degree-of-freedom railway vehi-cle system were investigated.The Lyapunov indirect method was used to examine the effects of suspension parameters, wheel conicities and wheel rolling radii on critical hunting speeds.Generally,the critical hunting speeds of a vehicle sys-tem resulting from worn wheels with different wheel rolling radii are lower than those of a vehicle system having origi-nal wheels without different wheel rolling radii.Because of worn wheels, the critical hunting speed of a running rail-way vehicle substantially declines over the long term. For safety reasons,it is necessary to design the suspension sys-tem parameters to increase the robustness of the system and decrease the sensitive of wheel noises.By applying UD and QPSO,the nominal-the-best signal-to-noise ratio of the sys-tem was increased from?48.17 to?34.05 dB.The rate of improvement was 29.31%.This study has demonstrated that the integration of UD and QPSO can successfully reveal the optimal solution of suspension parameters for solving the robust design problem of a railway vehicle suspension sys-tem.     

Stochastic Hopf bifurcation in quasi-integrable-Hamiltonian systems

A new procedure is developed to study the stochastic Hopf bifurcation in quasiintegrable-Hamiltonian systems under the Gaussian white noise excitation. Firstly, the singular boundaries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system‘s energy levels with respect to the stochastic averaging method. Secondly, the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones. Lastly, a quasi-integrable-Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure.Moreover, simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure. It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system‘s parameters. Therefore, one can see that the numerical results are consistent with the theoretical predictions.     

Analytical and finite-element study of optimal strain distribution in various beam shapes for energy harvesting applications

Owing to the increasing demand for harvesting energy from environmental vibration for use in self-powered electronic applications, cantilever-based vibration energy harvesting has attracted considerable interest from various parties and has become one of the most common approaches to converting redundant mechanical energy into electrical energy. As the output voltage produced from a piezoelec-tric material depends largely on the geometric shape and the size of the beam, there is a need to model and compare the performance of cantilever beams of differing geometries. This paper presents the study of strain distribution in various shapes of cantilever beams, including a convex and concave edge profile elliptical beam that have not yet been discussed in any prior literature. Both analytical and finite-element models are derived and the resultant strain distributions in the beam are computed based on a MATLAB solver and ANSYS finite-element analysis tools. An optimum geome-try for a vibration-based energy harvesting system is verified. Finally, experimental results comparing the power density for triangular and rectangular piezoelectric beams are also pre-sented to validate the findings of the study, and the claim, as suggested in the literature, is verified.     

Coupling dynamic analysis of a liquid-filled spherical container subject to arbitrary excitation

Using spherical coordinates, the coupling nonlin- ear dynamic system of a liquid-filled spherical tank, which can be excited discretionarily, is deduced by the H-O variational principle, and the viscous damping is introduced via the liquid dissipation function. The kinetic equations of the coupling system are deduced by the relationship between the velocity of liquid particles and the disturbed liquid surface equation. Normal differential equations are obtained through the Galerkin method. An equivalent mechanical model is developed for liquid sloshing in a spherical tank subject to arbitrary excitation. The fixed and slosh masses, as well as the spring and damping constants, are determined in such a way as to satisfy the principle of equivalence. Numerical simulations illustrate the theoretical results in this paper as well.     

Nonlinear Performance of MEMS Vibratory Ring Gyroscope

Micro-electro-mechanical system(MEMS)gyroscopes are an important sort of inertial sensor for identifying parameters of spinning structures,such as the spinning speed and angular deviation,based on the Coriolis effect.In this paper,the nonlinear mechanism of MEMS vibratory ring gyroscopes is analyzed by applying a fully coupled nonlinear model,in which the gyroscopic coupling and geometrically and structurally nonlinear couplings are all taken into account.The coupled differential equations governing the drive and sense motions are established via the Lagrangian equations.Numerical simulation is conducted,and the key nonlinear components and energy transfer behaviors between the drive and sense modes are elucidated.It is revealed that the cubic rigidity nonlinearity is another significant factor leading to the coupling between the drive and sense modes other than the gyroscopic coupling.Perturbation analysis is also carried out by using the method of multiple scales.The nonlinear frequency-amplitude responses of the drive and sense vibrations are obtained,and comprehensive parametric studies are performed.The significant effects of system damping,excitation amplitude,drive amplitude and spinning speed on the responses are discussed,which will facilitate to improve the nonlinear performance and sensitivity of the gyroscope.     

RESPONSE ANALYSIS OF PIEZOELECTRIC SHELLS IN PLANE STRAIN UNDER RANDOM EXCITATIONS

The random response of a piezoelectric thick shell in plane strain state under boundary random excitations is studied and illustrated with a piezoelectric cylindrical shell. The differential equation for electric potential is integrated radially to obtain the electric potential as a function of displacement. The random stress boundary conditions are converted into homogeneous ones by transformation,which yields the electrical and mechanical coupling differential equation for displacement under random excitations. Then this partial differential equation is converted into ordinary differential equations using the Galerkin method and the Legendre polynomials,which represent a random multi-degree-of-freedom system with asymmetric stiffness matrix due to the electrical and mechanical coupling and the transformed boundary conditions. The frequency-response function matrix and response power spectral density matrix of the system are derived based on the theory of random vibration. The mean-square displacement and electric potential of the piezoelectric shell are finally obtained,and the frequency-response characteristics and the electrical and mechanical coupling properties are explored.     

Complex dynamics of a harmonically excited structure coupled with a nonlinear energy sink

Nonlinear behaviors are investigated for a structure coupled with a nonlinear energy sink. The structure is linear and subject to a harmonic excitation, modeled as a forced single-degree-of-freedom oscillator. The nonlinear energy sink is modeled as an oscillator consisting of a mass,a nonlinear spring, and a linear damper. Based on the numerical solutions, global bifurcation diagrams are presented to reveal the coexistence of periodic and chaotic motions for varying nonlinear energy sink mass and stiffness. Chaos is numerically identified via phase trajectories, power spectra,and Poincaré maps. Amplitude-frequency response curves are predicted by the method of harmonic balance for periodic steady-state responses. Their stabilities are analyzed.The Hopf bifurcation and the saddle-node bifurcation are determined. The investigation demonstrates that a nonlinear energy sink may create dynamic complexity.     

RESEARCH IN STABILITY OF PERIODIC MOTION,BIFURCATION AND CHAOS IN A VIBRATORY SYSTEM WITH A CLEARANCE

A two-degrees-of-freedom vibratory system with a clearance or gap is under consideration based on the Poincard map. Stability and local bifurcation of the period-one doubleimpact symmetrical motion of the system are analyzed by using the equation of map. The routes from periodic impact motions to chaos, via pitchfork bifurcation, period-doubling bifurcation and grazing bifurcation, are studied by numerical simulation. Under suitable system parameter conditions, Neimark-Sacker bifurcations associated with periodic impact motion can occur in the two-degrees-of-freedom vibro-impact system.     

Vibration suppression in multi-tool ultrasonic machining to multi-external and parametric excitations

Ultrasonic machining （USM） is of particular interest for the machining of non-conductive, brittle materials such as engineering ceramics. In this paper, a multi-tool technique is used in USM to reduce the vibration in the tool holder and have reasonable amplitude for the tools. This can be done via dynamic absorbers. The coupling of four nonlinear oscillators of the tool holder and tools representing ultrasonic cutting process are investigated. This leads to a four-degree-of-freedom system subjected to multi-external and multi-parametric excitation forces. The aim of this work is to control the tool holder behavior at simultaneous primary, sub-harmonic and internal resonance condition. Multiple scale perturbation method is used to obtain the solution up to the second order approximations. The different resonance cases are reported and studied numerically. The stability of the system is investigated by using both phase-plane and frequency response techniques. The effects of the different parameters of the tools on the system behavior are studied numerically. Comparison with the available published work is reported.     

CHAOTIC MOTIONS AND LIMIT CYCLE FLUTTER OF TWO-DIMENSIONAL WING IN SUPERSONIC FLOW

Based on the piston theory of supersonic flow and the energy method, the flutter motion equations of a two-dimensional wing with cubic stiffness in the pitching direction are established. The aeroelastic system contains both structural and aerodynamic nonlinearities. Hopf bifurcation theory is used to analyze the flutter speed of the system. The effects of system parameters on the flutter speed are studied. The 4th order Runge-Kutta method is used to calculate the stable limit cycle responses and chaotic motions of the aeroelastic system. Results show that the number and the stability of equilibrium points of the system vary with the increase of flow speed. Besides the simple limit cycle response of period 1, there are also period-doubling responses and chaotic motions in the flutter system. The route leading to chaos in the aeroelastic model used here is the period-doubling bifurcation. The chaotic motions in the system occur only when the flow speed is higher than the linear divergent speed and the initial condition is very small. Moreover, the flow speed regions in which the system behaves chaos axe very narrow.     

Achievement of chaotic synchronization trajectories of master–slave manipulators with feedback control strategy

This paper addresses a master-slave synchro- nization strategy for complex dynamic systems based on feedback control. This strategy is applied to 3-DOF pla- nar manipulators in order to obtain synchronization in such complicated as chaotic motions of end-effectors. A chaotic curve is selected from Duffing equation as the trajectory of master end-effector and a piecewise approximation method is proposed to accurately represent this chaotic trajectory of end-effectors. The dynamical equations of master-slave manipulators with synchronization controller are derived, and the Lyapunov stability theory is used to determine the stability of this controlled synchronization system. In numer- ical experiments, the synchronous motions of end-effectors as well as three joint angles and torques of master-slave manipulators are studied under the control of the proposed synchronization strategy. It is found that the positive gain matrix affects the implementation of synchronization con- trol strategy. This synchronization control strategy proves the synchronization＇s feasibility and controllability for com- plicated motions generated by master-slave manipulators.     

Periodic motions of spinning rigid spacecraft under influence of gravitational and magnetic fields

The motion of a magnetized axisymmetric spacecraft about its center of mass in a circular orbit is considered, taking the gravitational and magnetic effects of the central body into account. Equations of motion of the reduced system are transformed to equations of plane motion of a charged particle under the action of electric and magnetic fields. Stationary motions of the system are determined and periodic motions near to them are constructed using the Lyapounoff theorem of the holomorphic integral.     

A NUMERICAL METHOD FOR SIMULATING NONLINEAR FLUID-RIGID STRUCTURE INTERACTION PROBLEMS

A numerical method for simulating nonlinear fluid-rigid structure interaction problems is developed. The structure is assumed to undergo large rigid body motions and the fluid flow is governed by nonlinear, viscous or non-viscous, field equations with nonlinear boundary conditions applied to the free surface and fluid-solid interaction interfaces. An Arbitrary-Lagrangian-Eulerian (ALE) mesh system is used to construct the numerical model. A multi-block numerical scheme of study is adopted allowing for the relative motion between moving overset grids, which are independent of one another. This provides a convenient method to overcome the difficulties in matching fluid meshes with large solid motions. Nonlinear numerical equations describing nonlinear fluid-solid interaction dynamics are derived through a numerical discretization scheme of study. A coupling iteration process is used to solve these numerical equations. Numerical examples are presented to demonstrate applications of the model developed.