This paper investigates the dynamic instability of a functionally graded porous arch reinforced with uniformly distributed graphene platelets (GPLs) under the combined action of a static force and a dynamic uniform pressure in the radial direction. The relationship between the elastic modulus and mass density of the material is determined by the closed-cell cellular solids under Gaussian Random Field scheme. The governing equation is derived based on classical Euler-Bernoulli theory. Galerkin approach is used to derive the Mathieu-Hill equation from which the dynamic unstable region is obtained using Bolotin method. A comprehensive parametric study is conducted to examine the effects of GPL weight fraction and dimensions, porosity distribution, pore size, static force, and arch geometry and size on the dynamic stability characteristics of the arch. Numerical results show that the porous arch's resistance against dynamic instability can be considerably improved by using symmetrically non-uniform porosity distribution and the addition of a small amount of GPLs.

Dynamic instability of functionally graded porous arches reinforced by graphene platelets

Bending vibration of isolated structures has always been neglected when the vibration isolation was studied. Isolated structures have usually been treated as discrete systems. In this study, dynamics of a slightly curved beam supported by quasi-zero-stiffness systems are firstly presented. In order to achieve quasi-zero-stiffness, a nonlinear isolation system is implemented via three linear springs. A nonlinear dynamic model of the slightly curved beam with nonlinear isolations is established. It includes square nonlinearity, cubic nonlinearity, and nonlinear boundaries. Then, the mode functions and the frequencies of the curved beam with elastic boundaries are derived. The schemes of the finite difference method (FDM) and the Galerkin truncation method (GTM) are, respectively, proposed to obtain nonlinear responses of the curved beam with nonlinear boundaries. Numerical results demonstrate that both the GTM and the FDM yield accurate solutions for the nonlinear dynamics of curved structures with nonsimple boundaries. The multi-mode resonance characteristics of the curved beam affect the vibration isolation efficiency. The quasi-zero-stiffness isolators reduce the transmissibility of modal resonances and provide a promising future for isolating the bending vibration of the flexible structure. However, the initial curvature significantly increases the resonant frequency of the flexible structure, and thus the frequency range of the effective vibration isolation is narrower. Furthermore, the quadratic nonlinear terms in the curved beam make the dynamic phenomenon more complicated. Therefore, it is more challenging and necessary to investigate the isolation of the bending vibration of the initial curved structure.

Nonlinear vibration of a slightly curved beam with quasi-zero-stiffness isolators

This paper studies the $$(3+1)$$
 -dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation by implementing the Hirota bilinear method. As a consequence, the Hirota bilinear method is successfully employed and acquired a type of the lump solution and five types of interaction solutions in terms of a new merge of positive quadratic functions, trigonometric functions and hyperbolic functions. All solutions have been verified back into its corresponding equation by Maple. We depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3D and 2D illustrations. Finally, we believe that the executed method is robust and efficient than other methods and the obtained solutions are trustworthy in the applied sciences.

Lump solution and its interaction to (3+1)-D potential-YTSF equation

Under the 3:1 internal resonance condition, the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied. The viscoelastic behaviors of the traveling beam are described by the standard linear solid model, and the material time derivative is adopted in the viscoelastic constitutive relation. The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes. For the first time, the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam. The undetermined coefficient method is used to approximately establish the real modal functions. The approximate analytical results are confirmed by the Galerkin truncation. Numerical examples are presented to highlight the effects of the viscoelastic behaviors on the steady-state periodic responses. To illustrate the effect of the internal resonance, the energy transfer between the internal resonance modes and the saturation-like phenomena in the steady-state responses is presented.

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Primary resonance of traveling viscoelastic beam under internal resonance

Dynamical modeling and multi-pulse chaotic dynamics of cantilevered pipe conveying pulsating fluid in parametric resonance

To better understand the dynamic behaviors of cable-stayed bridges, this study investigates the dynamic behaviors of a cable-stayed shallow arch subjected to two external harmonic excitations using the analytical approach First, dimensionless planar vibration equations of the system are obtained by applying the Hamilton principle, and three ordinary differential equations of the arch and the two cables are obtained by using the Galerkin discretization method Second, modulation equations involving the amplitude and phase of the dynamic response of the system are derived by applying the method of multiple scales Third, three simultaneous resonance cases are considered Finally, parametric study results are illustrated through frequency responses, amplitude responses, phase plane and bifurcation diagrams Chaos phenomenon is also detected and presented To validate the developed analytical solutions, numerical simulations are conducted by applying the Runge–Kutta method to integrate the original ordinary differential equations The results demonstrate that acceptable consistency is reached in the results obtained from the analytical solutions and the Runge–Kutta method in the three simulated cases The obtained results show that the system’s dynamic responses in the three simulated cases exhibit similarities in their frequency and amplitude responses, while some qualitative differences exist in the phase plane portraits (eg, period-1, period-2, period-3 solutions) and their bifurcation diagrams

Theoretical Analysis of Dynamic Behaviors of Cable-Stayed Bridges Excited by Two Harmonic Forces

Cable’s non-planar coupled dynamics under asynchronous out-of-plane support motions is investigated in this paper. The moving boundary difficulty is coped by transforming the small support motions into two resonant boundary modulation terms, and cable’s one-to-one resonant coupled non-planar dynamics is reduced through attacking directly the continuous dynamic equations by the multiple scale method. Two boundary dynamic coefficients are derived, characterizing the boundary modulation effects, which are equal for symmetric out-of-plane modes, while opposite for asymmetric ones. Both cable’s frequency and amplitude (of support deflection) response diagrams, with phase lags between supports, are constructed using numerical continuation algorithms, and the steady-state solutions’ stability and bifurcation properties are determined, based upon which the phase lags’ effects on dynamic responses, or travelling wave effects, are thoroughly investigated, through varying the phase lags.

Cable’s non-planar coupled vibrations under asynchronous out-of-plane support motions: travelling wave effect

Cables’ in-plane nonlinear vibrations under multiple support motions with a phase lag are investigated in this paper, which is a continuation of our previous work (Guo et al. in Arch Appl Mech 1647–1663, 2016). The main results are twofold. Firstly, asymptotically reduced models for the cable’s in-plane nonlinear vibrations, with or without internal resonance, excited by multiple support motions, are established using a boundary modulation approach. Two in-plane boundary dynamic coefficients are derived for characterizing moving supports’ effects, which depend closely on the cable’s initial sag, distinct from the out-of-plane ones Guo et al. 
(2016), and are also found to be equal for symmetric in-plane dynamics while opposite for antisymmetric dynamics. Secondly, cable’s in-plane nonlinear responses due to multiple support motions are calculated and phase lag’s dynamic effects are fully investigated. Two important factors associated with phase lags, i.e., the excitation-reduced and excitation-amplified factors, are both derived analytically, indicating theoretically that the phase lags would weaken the cable’s single-mode symmetric dynamics but amplify the antisymmetric dynamics. Furthermore, through constructing frequency response diagrams, the phase lag is found to change the characteristics of cables’ two-to-one modal resonant dynamics, both qualitatively and quantitatively. All these semi-analytical results obtained from the reduced models are also verified by applying the finite difference method to the cable’s full model, i.e., the continuous partial differential equation.

An investigation into cables’ in-plane dynamics under multiple support motions using a boundary modulation approach

Cable-arch structure is a combined structure, which utilizes flexibility of cable and rigidity of arch. Cable-arch structure has been widely used in bridge engineering. In this work, we proposed a novel mechanical model of cable-arch structure. The out-of-plane buckling and in-plane buckling were studied using the energy method. The formula of critical loads of both the first order out-of-plane buckling and in-plane buckling were derived using Rayleigh-Ritz method. An example is ultimately investigated numerically. The results indicate that the cable can improve considerably the out-of-plane and in-plane stability of arch. Therefore, the research about the stability of cable-arch structure are both valuable not only in theoretical research but also in design of engineering structure.

Houjun Kang

Papers

Theoretical Analysis of Dynamic Behaviors of Cable-Stayed Bridges Excited by Two Harmonic Forces

Cable’s non-planar coupled vibrations under asynchronous out-of-plane support motions: travelling wave effect

An investigation into cables’ in-plane dynamics under multiple support motions using a boundary modulation approach

Stability of Cable-Arch Structure

One-to-one internal resonance of a cable-beam structure subjected to a concentrated load