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Innovative approach with harmonic modes and finite element modeling for the nonlinear dynamic analysis of a suspension bridge

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Abstract

The nonlinear characteristics of a general multiple degree-of-freedom suspension bridge under harmonic excitation are studied with an innovative approach consisting of the harmonic modes, the finite element modeling and the incremental harmonic balance method. The geometric nonlinearity induced by large-amplitude vibration is considered. The harmonic mode and mean kinetic energy of the system are proposed to describe the nonlinear responses. The harmonic mode can be expressed as a linear combination of different linearized system modes. The resonance responses of the structure are characterized with the participation ratio of these system modes. Its composition under harmonic excitations can easily be described in terms of the different harmonic mode shapes and then the system modes. The proposed method is further applied to analyze, in detail, the properties of the harmonic modes and the evolution of the steady-state nonlinear responses with variations in the excitation frequency.

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Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Abbreviations

a 1 ~ a 12 :

Physical coefficients

α ik :

Modal phase of the kth system mode

a jk,, b jk :

Coefficients of Fourier series

a k :

Scaling factor of the kth system mode

b ik :

Modal amplitude of the kth system mode

A, B :

Node name

A c :

Sectional area of the cable

\({\mathbf{A}}_{j}\) :

Matrix related to coefficients of Fourier series

C :

Damping matrix

d :

Horizontal projection of l

E :

Elastic modulus of the material

E :

Kinetic energy of each DoF of the system

\({\overline{\mathbf{E}}}\) :

Mean kinetic energy of the whole system

F c :

Initial internal axial force

F h :

Horizontal component of the axial force

F v :

Vertical component of the axial force

F c :

Initial internal axial force

F :

Vector of the amplitude and location of external harmonic excitations

h :

Vertical projection of l

k :

Initial stiffness of cable

\(\overline{k}\) :

Stiffness after considering the initial internal force of cable

K:

Component of stiffness matrix

l :

Cable length in static equilibrium state

l 0 :

Original length of the element with no axial force

m :

Mass of node

M :

Matrix of system mass matrices

M c :

Lumped mass matrix of the link element

P ik :

Parameter related to contribution of different system modes in each harmonic mode

q :

Dynamic cement response

q 0.:

Dynamic displacement cable related to ω0

K :

Stiffness displacement response

R :

Error vector

t :

Time

u 1, u 2 :

Horizontal displacements of node

v 1, v 2 :

Vertical displacements of node

V(t):

Vibration reconstructed by system modes

W :

Real vibration shapes

Y :

Vibration shape fitted by system mode shapes

α, β :

Coefficients of Rayleigh damping

\(\Delta ( \cdot )\) :

Interval of physical variable

ω :

Circular frequency of excitation

ω 0 :

Frequency close to ω

τ :

Non-dimensional time

φ i :

Mode shape vector of the ith mode

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Acknowledgements

This study was supported in part by the 111 project of the Ministry of Education and the Bureau of Foreign Experts of China (No.B18062), National Natural Science Foundation of China (52078087, 51808075), Natural Science Foundation of Chongqing, China (cstc2020jcyj-msxmX0773), the Fundamental Reseach Funds for the Central Universities (2020CDJ-LHZZ-018)

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Authors and Affiliations

  1. Key Laboratory of New Technology for Construction of Cities in Mountain Area (Chongqing University), Ministry of Education, Chongqing, 400045, China

    Yi Hui, Kunpeng Guo & Min Liu

  2. School of Civil Engineering, Chongqing University, Chongqing, 400045, China

    Yi Hui, S. S. Law, Kunpeng Guo & Min Liu

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  1. Yi Hui
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Correspondence to Kunpeng Guo.

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Hui, Y., Law, S.S., Guo, K. et al. Innovative approach with harmonic modes and finite element modeling for the nonlinear dynamic analysis of a suspension bridge. Nonlinear Dyn 111, 4221–4236 (2023). https://doi.org/10.1007/s11071-022-08069-z

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