Functionally graded materials(FGMs)are a novel class of composite materials that have attracted significant attention in the field of engineering due to their unique mechanical properties.This study aims to explore th...Functionally graded materials(FGMs)are a novel class of composite materials that have attracted significant attention in the field of engineering due to their unique mechanical properties.This study aims to explore the dynamic behaviors of an FGM stepped beam with different boundary conditions based on an efficient solving method.Under the assumptions of the Euler-Bernoulli beam theory,the governing differential equations of an individual FGM beam are derived with Hamilton’s principle and decoupled via the separation-of-variable approach.Then,the free and forced vibrations of the FGM stepped beam are solved with the transfer matrix method(TMM).Two models,i.e.,a three-level FGM stepped beam and a five-level FGM stepped beam,are considered,and their natural frequencies and mode shapes are presented.To demonstrate the validity of the method in this paper,the simulation results by ABAQUS are also given.On this basis,the detailed parametric analyses on the frequencies and dynamic responses of the three-level FGM stepped beam are carried out.The results show the accuracy and efficiency of the TMM.展开更多
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 re...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.展开更多
A dynamic model for an inclined carbon ?ber reinforced polymer(CFRP)cable is established, and the linear and nonlinear dynamic behaviors are investigated in detail. The partial differential equations for both the in-p...A dynamic model for an inclined carbon ?ber reinforced polymer(CFRP)cable is established, and the linear and nonlinear dynamic behaviors are investigated in detail. The partial differential equations for both the in-plane and out-of-plane dynamics of the inclined CFRP cable are obtained by Hamilton's principle. The linear eigenvalues are explored theoretically. Then, the ordinary differential equations for analyzing the dynamic behaviors are obtained by the Galerkin integral and dimensionless treatments.The steady-state solutions of the nonlinear equations are obtained by the multiple scale method(MSM) and the Newton-Raphson method. The frequency-and force-response curves are used to investigate the dynamic behaviors of the inclined CFRP cable under simultaneous internal(between the lowest in-plane and out-of-plane modes) and external resonances, i.e., the primary resonances induced by the excitations of the in-plane mode,the out-of-plane mode, and both the in-plane mode and the out-of-plane mode, respectively. The effects of the key parameters, e.g., Young's modulus, the excitation amplitude,and the frequency on the dynamic behaviors, are discussed in detail. Some interesting phenomena and results are observed and concluded.展开更多
In previous research on the nonlinear dynamics of cable-stayed bridges,boundary conditions were not properly modeled in the modeling.In order to obtain the nonlinear dynamics of cable-stayed bridges more accurately,a ...In previous research on the nonlinear dynamics of cable-stayed bridges,boundary conditions were not properly modeled in the modeling.In order to obtain the nonlinear dynamics of cable-stayed bridges more accurately,a double-cable-stayed shallow-arch model with elastic supports at both ends and the initial configuration of bridge deck included in the modeling is developed in this study.The in-plane eigenvalue problems of the model are solved by dividing the shallow arch(SA)into three partitions according to the number of cables and the piecewise functions are taken as trial functions of the SA.Then,the in-plane one-toone-to-one internal resonance among the global mode and the local modes(two cables’modes)is investigated when external primary resonance occurs.The ordinary differential equations(ODEs)are obtained by Galerkin’s method and solved by the method of multiple time scales.The stable equilibrium solutions of modulation equations are obtained by using the NewtonRaphson method.In addition,the frequency-/force-response curves under different vertical stiffness are provided to study the nonlinear dynamic behaviors of the elastically supported model.To validate the theoretical analyses,the Runge-Kutta method is applied to obtain the numerical solutions.Finally,some interesting conclusions are drawn.展开更多
The present study aims to investigate the dynamic behaviors and energy transfer between components of a cable stayed beam structure subjected to a concentrated load,in which the primary resonance of the beam and one-t...The present study aims to investigate the dynamic behaviors and energy transfer between components of a cable stayed beam structure subjected to a concentrated load,in which the primary resonance of the beam and one-to-two internal resonance between modes of the beam and the cable occur.Galerkin discretization and multiple time scales method are applied to derive the modulation equations of the system governing the amplitude and phase.Two sags of span ratios are defined to modulate the internal resonance.Frequency response,amplitude response,phase diagram,Poincare map,and time history curves are calculated and used to investigate the modal resonance dynamics.The results demonstrate that the beam and the cable have two resonant peaks in frequency responses,where the beam always shows hardening spring property and the cable may present hardening and softening spring properties affected by sag to span ratio.The system is prone to complex dynamic behavior with the small amplitude excitation in the primary resonance region.展开更多
基金the National Natural Science Foundation of China(Nos.12302007,12372006,and 12202109)the Specific Research Project of Guangxi for Research Bases and Talents(No.AD23026051)。
文摘Functionally graded materials(FGMs)are a novel class of composite materials that have attracted significant attention in the field of engineering due to their unique mechanical properties.This study aims to explore the dynamic behaviors of an FGM stepped beam with different boundary conditions based on an efficient solving method.Under the assumptions of the Euler-Bernoulli beam theory,the governing differential equations of an individual FGM beam are derived with Hamilton’s principle and decoupled via the separation-of-variable approach.Then,the free and forced vibrations of the FGM stepped beam are solved with the transfer matrix method(TMM).Two models,i.e.,a three-level FGM stepped beam and a five-level FGM stepped beam,are considered,and their natural frequencies and mode shapes are presented.To demonstrate the validity of the method in this paper,the simulation results by ABAQUS are also given.On this basis,the detailed parametric analyses on the frequencies and dynamic responses of the three-level FGM stepped beam are carried out.The results show the accuracy and efficiency of the TMM.
基金Project supported by the National Natural Science Foundation of China(Nos.11572117,11502076,and 11872176)
文摘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.
基金Project supported by the National Natural Science Foundation of China(Nos.11572117 and 11502076)
文摘A dynamic model for an inclined carbon ?ber reinforced polymer(CFRP)cable is established, and the linear and nonlinear dynamic behaviors are investigated in detail. The partial differential equations for both the in-plane and out-of-plane dynamics of the inclined CFRP cable are obtained by Hamilton's principle. The linear eigenvalues are explored theoretically. Then, the ordinary differential equations for analyzing the dynamic behaviors are obtained by the Galerkin integral and dimensionless treatments.The steady-state solutions of the nonlinear equations are obtained by the multiple scale method(MSM) and the Newton-Raphson method. The frequency-and force-response curves are used to investigate the dynamic behaviors of the inclined CFRP cable under simultaneous internal(between the lowest in-plane and out-of-plane modes) and external resonances, i.e., the primary resonances induced by the excitations of the in-plane mode,the out-of-plane mode, and both the in-plane mode and the out-of-plane mode, respectively. The effects of the key parameters, e.g., Young's modulus, the excitation amplitude,and the frequency on the dynamic behaviors, are discussed in detail. Some interesting phenomena and results are observed and concluded.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.11972151 and 11872176)。
文摘In previous research on the nonlinear dynamics of cable-stayed bridges,boundary conditions were not properly modeled in the modeling.In order to obtain the nonlinear dynamics of cable-stayed bridges more accurately,a double-cable-stayed shallow-arch model with elastic supports at both ends and the initial configuration of bridge deck included in the modeling is developed in this study.The in-plane eigenvalue problems of the model are solved by dividing the shallow arch(SA)into three partitions according to the number of cables and the piecewise functions are taken as trial functions of the SA.Then,the in-plane one-toone-to-one internal resonance among the global mode and the local modes(two cables’modes)is investigated when external primary resonance occurs.The ordinary differential equations(ODEs)are obtained by Galerkin’s method and solved by the method of multiple time scales.The stable equilibrium solutions of modulation equations are obtained by using the NewtonRaphson method.In addition,the frequency-/force-response curves under different vertical stiffness are provided to study the nonlinear dynamic behaviors of the elastically supported model.To validate the theoretical analyses,the Runge-Kutta method is applied to obtain the numerical solutions.Finally,some interesting conclusions are drawn.
基金supported by the National Natural Science Foundation of China(Grant Nos.11972151 and 11872176).
文摘The present study aims to investigate the dynamic behaviors and energy transfer between components of a cable stayed beam structure subjected to a concentrated load,in which the primary resonance of the beam and one-to-two internal resonance between modes of the beam and the cable occur.Galerkin discretization and multiple time scales method are applied to derive the modulation equations of the system governing the amplitude and phase.Two sags of span ratios are defined to modulate the internal resonance.Frequency response,amplitude response,phase diagram,Poincare map,and time history curves are calculated and used to investigate the modal resonance dynamics.The results demonstrate that the beam and the cable have two resonant peaks in frequency responses,where the beam always shows hardening spring property and the cable may present hardening and softening spring properties affected by sag to span ratio.The system is prone to complex dynamic behavior with the small amplitude excitation in the primary resonance region.
