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

In this paper, the one-to-one interaction of a cable-stayed arch structure under the cable’s primary resonance is investigated. Based on the coupling condition at the arch tip, the partial differential equations governing the planar motion of the system are derived using the extended Hamiltonian principle, while with the application of the Galerkin method, these equations are transformed into a set of ordinary equations. Applying the method of multiple scales to these ordinary equations, the first approximated solutions and solvability condition are obtained. The one-to-one interaction between the cable and the arch is investigated under simultaneous internal and external resonances for an actual cable-stayed arch structure. Based on the shooting method and the pseudo-arclength algorithm, the dynamic solutions of the system are obtained, and a period-doubling route to chaos is analyzed. The effects of the cable’s initial tension, inclination angle, the arch’s rise-to-span ratio and intersection angle between the cable and the arch are explored, and the results show that the interaction response mainly depends on specific parameters.

Nonlinear dynamic analysis of cable-stayed arches under primary resonance of cables

For complex cable-stayed structures, dynamic coupling between cables and deck beam is important. In this work, an asymptotically reduced coupled model, consisting of two inclined cables and one deck beam, is established for understanding dynamic interactions inherent with the cable-stayed structures, after confining oneself to the simplified case that the beam’s motion is much weaker than the cables’ and the deck beam/cable mass ratio is a large parameter. More explicitly, through focusing on the cables–deck interfaces, the double cables–deck beam coupling is decomposed into deck-induced and cable-induced weak boundary/interior modulations on each other, which are analytically characterized by three boundary coupling coefficients. Based upon the asymptotic model, two different kinds of coupled dynamics are fully investigated, the first with only one of the two cables excited, and the second with the deck beam excited, leading to the coupled system’s different nonlinear forced responses. The steady solutions’ stabilities are determined and bifurcations, such as saddle–node bifurcation, Hopf bifurcation, and pitchfork bifurcation, are all detected. Special attentions are paid to the dynamic effects caused by the deck beam/cable mass ratio, cable’s inclinations.

Nonlinear vibrations for double inclined cables-deck beam coupled system using asymptotic reductions

The nonlinear dynamic behaviors of a double cable-stayed shallow arch model are investigated under the one-to-one-to-one internal resonance among the lowest modes of cables and the shallow arch and external primary resonance of cables. The in-plane governing equations of the system are obtained when the harmonic excitation is applied to cables. The excitation mechanism due to the angle-variation of cable tension during motion is newly introduced. Galerkin’s method and the multi-scale method are used to obtain ordinary differential equations (ODEs) of the system and their modulation equations, respectively. Frequency- and force-response curves are used to explore dynamic behaviors of the system when harmonic excitations are symmetrically and asymmetrically applied to cables. More importantly, comparisons of frequency-response curves of the system obtained by two types of trial functions, namely, a common sine function and an exact piecewise function, of the shallow arch in Galerkin’s integration are conducted. The analysis shows that the two results have a slight difference; however, they both have sufficient accuracy to solve the proposed dynamic system.

Analysis of in-plane 1:1:1 internal resonance of a double cable-stayed shallow arch model with cables’ external excitations

A triad mode resonance, or three-wave resonance, is typical of dynamical systems with quadratic nonlinearities. Suspended cables are found to be rich in triad mode resonant dynamics. In this paper, modulation equations for cable’s triad resonance are formulated by the multiple scale method. Dynamic conservative quantities, i.e., mode energy and Manley-Rowe relations, are then constructed. Equilibrium/dynamic solutions of the modulation equations are obtained, and full investigations into their stability and bifurcation characteristics are presented. Various bifurcation behaviors are detected in cable’s triad resonant responses, such as saddle-node, Hopf, pitchfork and period-doubling bifurcations. Nonlinear behaviors, like jump and saturation phenomena, are also found in cable’s responses. Based upon the bifurcation analysis, two interesting properties associated with activation of cable’s triad resonance are also proposed, i.e., energy barrier and directional dependence. The first gives the critical amplitude of high-frequency mode to activate cable’s triad resonance, and the second characterizes the degree of difficulty for activating cable’s triad resonance in two opposite directions, i.e., with positive or negative internal detuning parameter.

Houjun Kang

Papers

Nonlinear dynamic analysis of cable-stayed arches under primary resonance of cables

Nonlinear vibrations for double inclined cables-deck beam coupled system using asymptotic reductions

Analysis of in-plane 1:1:1 internal resonance of a double cable-stayed shallow arch model with cables’ external excitations

Triad mode resonant interactions in suspended cables

A novel modeling method for in-plane eigenproblem estimation of the cable-stayed bridges