Narrow Results

This note follows up on the study in the paper [Zepeda Ramírez et al., 2021], where we investigate the nonlinear stability of equilibrium points for the planar restricted equilateral four-body problem for the specific values of masses $m_1=0.01$ and $0.99$ of the primary, fixed on the horizontal axis. We give an improvement of the previous results.

This paper investigates the nonlinear bifurcation and chaos characteristics of piezoelectric composite lattice sandwich plates at 1:3 internal resonance. The nonlinear vibration partial differential equation is first discretized to become an ordinary differential equation by applying the Galerkin method. The main resonance modulation equations and the parametric resonance for the external excitation frequency that is close to the system’s second-order modal frequency are then obtained by using the multiscale method. The Newton–Raphson method is subsequently applied to obtain the bifurcation diagram of the steady-state equilibrium of modulation equation with varying system parameters. The equilibrium stability is finally analyzed. We confirm the existence of static bifurcation such as saddle-node bifurcation, pitchfork bifurcation, and Hopf dynamic bifurcation. A detailed analysis is also conducted on the complex nonlinear jump phenomenon caused by the presence of multiple nonlinear steady-state solution regions. In the dynamic Hopf bifurcation interval, the fourth-order Runge–Kutta method is used to continue to track the dynamic periodic solution of the modulation equation in Cartesian coordinates. It is found that there are multiple boundary crises and attractor merging crises in the vibration system for piezoelectric composite lattice sandwich plates.

We study here a model of conservative nonlinear conservation law with a flux function with discontinuous coefficients, namely the equation u_t + (k(x)u(1 - u))_x = 0. It is a particular entropy condition on the line of discontinuity of the coefficient k which ensures the uniqueness of the entropy solution. This condition is discussed and justified. On the other hand, we perform a numerical analysis of the problem. Two finite volume schemes, the Godunov scheme and the VFRoe-ncv scheme, are proposed to simulate the conservation law. They are compared with two finite volume methods classically used in an industrial context. Several tests confirm the good behavior of both new schemes, especially through the discontinuity of permeability k (whereas a loss of accuracy may be detected when industrial methods are performed). Moreover, a modified MUSCL method which accounts for stationary states is introduced.

The present paper is devoted to the computation of two-phase flows using the two-fluid approach. The overall model is hyperbolic and has no conservative form. No instantaneous local equilibrium between phases is assumed, which results in a two-velocity two-pressure model. Original closure laws for interfacial velocity and interfacial pressure are proposed. These closures allow to deal with discontinuous solutions such as shock waves and contact discontinuities without ambiguity with the definition of Rankine–Hugoniot jump relations. Each field of the convective system is investigated, providing maximum principle for the volume fraction and the positivity of densities and internal energies are ensured when focusing on the Riemann problem. Two-finite volume methods are presented, based on the Rusanov scheme and on an approximate Godunov scheme. Relaxation terms are taken into account using a fractional step method. Eventually, numerical tests illustrate the ability of both methods to compute two-phase flows.

We consider the Helmholtz transmission problem with one penetrable star-shaped Lipschitz obstacle. Under a natural assumption about the ratio of the wavenumbers, we prove bounds on the solution in terms of the data, with these bounds explicit in all parameters. In particular, the (weighted) $H^1$ norm of the solution is bounded by the $L^2$ norm of the source term, independently of the wavenumber. These bounds then imply the existence of a resonance-free strip beneath the real axis. The main novelty is that the only comparable results currently in the literature are for smooth, convex obstacles with strictly positive curvature, while here we assume only Lipschitz regularity and star-shapedness with respect to a point. Furthermore, our bounds are obtained using identities first introduced by Morawetz (essentially integration by parts), whereas the existing bounds use the much-more sophisticated technology of microlocal analysis and propagation of singularities. We also adapt existing results to show that if the assumption on the wavenumbers is lifted, then no bound with polynomial dependence on the wavenumber is possible.

We propose a new technique to calculate the position and width of weak resonant states in weakly bound systems, using isospectral potentials. A new potential is constructed which is strictly isospectral with the original one but has desirable properties which make the calculation of the resonance simpler and more accurate. The usefulness of the method is demonstrated by calculating the first 1⁺ resonance in ⁶Li, using a three-body cluster model for the latter.

Cutting off the tail of the Woods–Saxon (WS) and generalized WS (GWS) potentials changes the distribution of the poles of the $S$-matrix considerably. Here, we modify the tail of the cut-off WS (CWS) and cut-off generalized WS (CGWS) potentials by attaching Hermite polynomial tails to them beyond the cut. The tails reach the zero value more or less smoothly at the finite ranges of the potential. Reflections of the resonant wave functions can take place at different distances. The starting points of the pole trajectories have been reproduced not only for the real values and the moduli of the starting points, but also for the imaginary parts.

Cutting of the tail of the Woods–Saxon (WS) potential influences the distribution of the poles of the $S$-matrix considerably. A new attempt is made here to correct the tail of the cut-off WS potential in order to bring the positions of the poles closer to the WS potential without cut. New types of strictly finite range potentials are introduced in which the tail of the WS and the generalized Woods–Saxon (GWS) potentials are modified to damp them at their ends. This damping affects the energies of the bound and narrow resonant states weakly. The resonant pole positions in the new potentials were affected more significantly and the new pole positions became closer to that of the GWS potential without cut. However, the cut-off effect cannot be diminished completely. The new potentials belong to the $C^{\infty }$ function class.

A concrete beam is a basic element for many applications in architectural engineering, and its vibration property plays an important role in safety and life-span. Previous vibration models were based on continuum beam theories, which could not take into account the porous property of the concrete beam. This paper adopts a two-scale fractal vibration model, and its low frequency property is revealed. This new finding can explain the vibration attenuation and vibration absorption of the porous concrete, and the present theory sheds a new light on the concrete design.

A robustness design of fuzzy control via model-based approach is proposed in this paper to overcome the effect of approximation error between nonlinear system and Takagi-Sugeno (T-S) fuzzy model. T-S fuzz model is used to model the resonant and chaotic systems and the parallel distributed compensation (PDC) is employed to determine structures of fuzzy controllers. Linear matrix inequality (LMI) based design problems are utilized to find common definite matrices P and feedback gains K satisfying stability conditions derived in terms of Lyapunov direct method. Finally, the effectiveness and the feasibility of the proposed controller design method is demonstrated through numerical simulations on the chaotic and resonant systems.

In this work, we study a system of coupled modified KdV (mKdV) equations. Multiple soliton solutions and multiple singular soliton solutions are derived by using the Hirota's bilinear method and the Hietarinta approach. The resonance phenomenon is examined.

This study investigates the error which occurs when numerically integrating the equation of motion of a single degree of freedom system excited by a harmonic force near resonance. The Constant Average Acceleration method was considered in particular as it features in many finite element software packages. It was found that a considerable error in the calculated responses occurs in systems with low damping due to the well known phenomenon of period elongation. However, the error is reduced for systems with higher damping and/or when smaller time step is used. With regard to this, recommendations are given as to the time steps required to obtain solutions with a pre-defined level of accuracy.

This paper intends to study the train-induced vibration of a parabolic tied-arch bridge using an analytical approach. The train loads over the bridge are regarded as a sequence of equidistant moving loads with identical weights. The tied-arch bridge considered is modeled as the combination of a parabolic flat-rise arch with two-hinged supports and a simple beam suspended by densely distributed vertical cables connected to the arch rib. Using the normal coordinate transformation method, the coupled equations of motion of the arch rib and suspended beam are converted into a set of uncoupled equations. Then, one can derive closed form solutions for the response of the tied-arch beam to successive moving loads, by which the resonant conditions of higher modes of the suspended beam can be identified. According to the present study, the critical position for the maximum acceleration on the suspended beam depends upon the vibration shape that has been excited. Moreover, the numerical results indicate that the lower the rise of the arch rib, the larger the acceleration response of the main beam suspended by the arch.

The objective of this study is to investigate the resonance and sub-resonance acceleration response of a two-span continuous railway bridge under the passage of moving train loadings. The continuous bridge is modeled as a Bernoulli–Euler beam with uniform span length and the moving train is simulated as a series of equidistant two degrees-of-freedom (2-DOF) mass–spring–damper units. The modal superposition method is adopted to compute the interaction dynamics of the train–bridge system. The numerical analyses indicate that (1) the train-induced resonance of the two-span continuous beam may result in significant amplification of the dynamic response of the train/bridge system; (2) for a two-span continuous beam, the first two resonant speeds may fall in the range of operating speeds of high-speed trains, which can lead to highly amplified vehicle responses; (3) due to the presence of sub-resonant peaks, the maximum acceleration of the two-span continuous beam need not occur at the midpoint of the beam; (4) inclusion of damping of a beam is helpful for reducing the train-induced resonant response on the beam, but the first two resonant peaks of the coupling system remain unchanged.

The well-known vibration model of axially moving beam is considered. Both axial moving speed and axial force are assumed to vary harmonically. The Method of Multiple Time Scales (a perturbation method) is used. The natural vibrations of beam are considered for different values of beam parameters. Resonances are obtained for seven different conditions. Solvability conditions for each resonance case are found analytically. Effects of transport velocity, axial force, rigidity and damping are discussed. Stability analysis are obtained for principal parametric resonances. Stable and unstable regions are obtained regarding velocity and force effects separately and together.

Excessive vibrations seriously affected the comfort of residents living on the upper floors of a high-rise shear walled building in Beijing. The ambient vibration tests were conducted to measure the floor acceleration responses, which were found to contain almost periodic signals likely to be excited by vibration sources with frequency of about 1.5Hz. The transverse vibration levels of the building above the 8th floor are not acceptable as revealed by the one-third octave spectra and weighted acceleration levels according to the ‘Standard for Allowable Vibration of Building Engineering’ of China. The modal properties of the building are identified by a Bayesian FFT method, indicating that the resonance between the building and the vibration sources caused the excessive vibrations. For comparison, the vibration test of an adjacent building with the same structural design was also conducted, together with modal analysis by the finite element method. It is found that as the story level increases, different trends of amplification in floor root mean square (RMS) acceleration and mode shape component of the two buildings cause different vibration levels. After tests outside the residence community, the main vibration sources were identified to be the working machines in two stone processing factories a few hundred meters away from the building. The vibration tests with measurements in the building and near the vibration sources with different number of machines in the two factories were also conducted. The results show that the vibration levels of the building can be controlled below the acceptance value by reducing the number of machines.

The paper is aimed at investigating the longitudinal vibration and vibration reduction of a cable-stayed bridge under vehicular loads with emphasis on the longitudinal resonance. To investigate the phenomenon of longitudinal resonant vibration, the equivalent longitudinal excitation for the bridge deck due to moving vertical loads is approximately expressed as longitudinal loads with a sine-wave form. A formula for estimating the longitudinal resonant speed of the cable-stayed bridge is developed. A long-span cable-stayed railway bridge is considered in the case study to calculate the longitudinal response of the bridge under moving loads at different speeds. The numerical results indicate that the longitudinal resonance for the cable-stayed bridge occurs when the speeds of the moving loads approach the resonant speed predicted by the analytical formula. A fluid viscous damper (FVD) is employed to reduce the longitudinal vibration of the bridge under moving loads. The results show that the longitudinal resonant responses of the cable-stayed bridge can be effectively mitigated by the FVD adopted.

This paper is concerned with the lateral and torsional coupled vibration of monosymmetric I-beams under moving loads. To this end, a train is modeled as two subsystems of eccentric wheel loads of constant intervals to account for the front and rear wheels. By assuming the lateral and torsional displacements to be restrained at the two ends of the beam, both the lateral and torsional displacements are approximated by a series of sine functions. The method of variation of constants is adopted to derive the closed-form solution. For the most severe condition when the last wheel load is acting on the beam, both the conditions of resonance and cancellation are identified. Once the condition of cancellation is enforced, the resonance response can always be suppressed, which represents the optimal design for the beam. Since the condition for suppressing the torsional resonance is exactly the same as that for the vertical resonance, this offers a great advantage in the design of monosymmetric I-beams, as no distinction needs to be made between the suppression of vertical or torsional resonance.

This paper investigates the free vibration and dynamic response of functionally graded sandwich beams resting on an elastic foundation under the action of a moving harmonic load. The governing equation of motion of the beam, which includes the effects of shear deformation and rotary inertia based on the Timoshenko beam theory, is derived from Lagrange’s equations. The Ritz and Newmark methods are employed to solve the equation of motion for the free and forced vibration responses of the beam with different boundary conditions. The results are presented in both tabular and graphical forms to show the effects of layer thickness ratios, boundary conditions, length to height ratios, spring constants, etc. on natural frequencies and dynamic deflections of the beam. It was found that increasing the spring constant of the elastic foundation leads to considerable increase in natural frequencies of the beam; while the same is not true for the dynamic deflection. Additionally, very large dynamic deflection occurs for the beam in resonance under the harmonic moving load.

In this paper, the vector form intrinsic finite element (VFIFE) method is presented for analysis the train–bridge systems considering the coach-coupler effect. The bridge is discretized into a group of mass particles linked by massless beam elements and the multi-body coach with suspension systems is simulated as a set of mass particles connected by parallel spring-dashpot units. Then the equation of motion of each mass particle is solved individually and the internal forces induced by pure deformations in the massless beam elements are calculated by a fictitious reverse motion method, in which the structural stiffness matrices need not be updated or factorized. Though the vector-form equations resulting from the VFIFE method cannot be used to compute the structural frequencies by the eigenvalue approach, this study proposes a numerical free vibration test to identify the bridge frequencies for evaluating the bridge damping. Numerical verifications demonstrate that the present VFIFE method performs as accurately as previous numerical ones. The results show that the couplers play an energy-dissipating role in reducing the car bodies’ response due to the bridge-induced resonance, but not in their response due to the train-induced resonance because of the bridge’s intense vibration. Meanwhile, a dual-resonance phenomenon in the train–bridge system occurs when the coach-coupler effect is considered in the vehicle model.