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

This study considers a novel nonlinear system, namely a cable-stayed beam with a tuned mass damper (cable-beam-TMD model), allowing the description of energy transfer among the beam, cable and TMD. In this system, the vibration of the TMD is involved and one-to-one-to-one internal resonance among the modes of the beam, cable and TMD is investigated when external primary resonance of the beam occurs. Galerkin’s method is utilized to discretize the equations of motion of the beam and cable. In this way, a set of ordinary differential equations are derived, which are solved by the method of multiple time scales. Then the steady-state solutions of the system are obtained by using Newton–Raphson method and continued by pseudo-arclength algorithm. The response curves, time histories and phase portraits are provided to explore the effect of the TMD on the nonlinear behaviors of the model. Meanwhile, a partially coupled system, namely a cable-beam-TMD model ignoring the vibration of the TMD, is also studied. The nonlinear characteristics of the two cases are compared with each other. The results reveal the occurrence of energy transfer among the beam, cable and TMD. 

/pdf/internal-resonance-and-energy-transfer-of-a-cable-stayed-3ucb9kz9.pdf

Houjun Kang

Papers

Internal resonance and energy transfer of a cable-stayed beam with a tuned mass damper

On internal resonance analysis of a double-cable-stayed shallow-arch model with elastic supports at both ends

Nonlinear vibration analysis of a shallow arch coupled with an elastically constrained rigid body

Vibration control of a continuous beam under moving load using a nonlinear energy sink

Vibration suppression of a cable under harmonic excitation by a nonlinear energy sink