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Research Papers

A Generalized Structure-Dependent Semi-Explicit Method for Structural Dynamics

[+] Author and Article Information
Jinze Li

Department of Astronautic
Science and Mechanics,
Harbin Institute of Technology,
No. 92 West Dazhi Street,
Harbin 150001, China
e-mail: pinkie.ljz@gmail.com

Kaiping Yu

Department of Astronautic
Science and Mechanics,
Harbin Institute of Technology,
No. 92 West Dazhi Street,
Harbin 150001, China
e-mail: yukp@hit.edu.cn

Xiangyang Li

Department of Astronautic
Science and Mechanics,
Harbin Institute of Technology,
No. 92 West Dazhi Street,
Harbin 150001, China
e-mail: lixiangyang@hit.edu.cn

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 17, 2018; final manuscript received August 16, 2018; published online September 12, 2018. Assoc. Editor: Javier Cuadrado.

J. Comput. Nonlinear Dynam 13(11), 111008 (Sep 12, 2018) (20 pages) Paper No: CND-18-1216; doi: 10.1115/1.4041239 History: Received May 17, 2018; Revised August 16, 2018

In this paper, a novel generalized structure-dependent semi-explicit method is presented for solving dynamical problems. Some existing algorithms with the same displacement and velocity update formulas are included as the special cases, such as three Chang algorithms. In general, the proposed method is shown to be second-order accurate and unconditionally stable for linear elastic and stiffness softening systems. The comprehensive stability and accuracy analysis, including numerical dispersion, energy dissipation, and the overshoot behavior, are carried out in order to gain insight into the numerical characteristics of the proposed method. Some numerical examples are presented to show the suitable capability and efficiency of the proposed method by comparing with other existing algorithms, including three Chang algorithms and Newmark explicit method (NEM). The unconditional stability and second-order accuracy make the novel methods take a larger time-step, and the explicitness of displacement at each time-step succeeds in avoiding nonlinear iterations for solving nonlinear stiffness systems.

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References

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Figures

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Fig. 1

Variation of stability limit with δi+1 for the CSE-2, CSE-9, LSE-1, and LSE-2 when ξ = 0

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Fig. 2

Variation of stability limit with δi+1 for the CSE-2, CSE-9, LSE-1, and LSE-2 algorithm when ξ = 0.0, 0.1, 0.5

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Fig. 3

Variation of stability limit with δi+1 for the DSE-15, CSE-10, and LSE-γ algorithm when ξ = 0

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Fig. 4

Two principal roots of the CSE-2, CSE-9, LSE-1, LSE-2, and the Newmark explicit method

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Fig. 5

Variations of the relative period errors with Ω0 for different δi+1 when ξ0 = 0

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Fig. 6

Variations of relative period errors with Ω0 for different algorithms when ξ0=0

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Fig. 7

Variations of numerical damping ratios with Ω0 for different δi+1 when ξ0 = 0.1 is considered

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Fig. 8

Absolute displacement error at the first time-step when ξ = 0. The solid line indicates the initial conditions d0 = 0 and v0 = 1 are used while the dashed line presents the initial conditions d0 = 1, v0 = 0.

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Fig. 9

Absolute velocity error at the first time-step when ξ = 0. The solid line indicates the initial conditions d0 = 0 and v0 = 1 is used while the dashed line presents the initial conditions d0 = 1, v0 = 0.

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Fig. 10

Absolute displacement error at the first time-step when ξ = 0.5. The solid line indicates the initial conditions d0 = 0 and v0 = 1 are used while the dashed line presents the initial conditions d0 = 1, v0 = 0.

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Fig. 11

Absolute velocity error at the first time-step when ξ = 0.5. The solid line indicates the initial conditions d0 = 0 and v0 = 1 is used while the dashed line presents the initial conditions d0 = 1, v0 = 0.

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Fig. 12

Convergence of displacement, velocity and acceleration for the proposed algorithms and the CAA method

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Fig. 18

Numerical responses of the nonlinear SDOF stiffness softening system obtained from five different algorithms

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Fig. 19

Numerical responses of the nonlinear SDOF stiffness softening system obtained from the CSE-7

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Fig. 13

Displacement responses of all of algorithms for linear elastic system

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Fig. 14

Displacement responses of all of algorithms for stiffness softening system

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Fig. 15

Histories of the instantaneous degree of nonlinearity for the first and second modes for stiffness softening system

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Fig. 16

Displacement responses of all algorithms for stiffness hardening system

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Fig. 17

Histories of the instantaneous degree of nonlinearity for the first and second modes for stiffness hardening system

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Fig. 21

Numerical solutions of a linear elastic SDOF system obtained from five algorithms with Δt = 0.5T: (a) ξ = 0.00, (b) ξ = 0.05, and (c) ξ = 0.50

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Fig. 20

Numerical solutions of a linear elastic SDOF system obtained from five algorithms with Δt = 0.05T: (a) ξ = 0.00, (b) ξ = 0.05, and (c) ξ = 0.50

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Fig. 22

Model of the n-degree-of-freedom mass-spring system

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Fig. 23

Displacement responses of the nth degree-of-freedom nonlinear mass-spring system when n is chosen to be 100, 200, 500, and 1000

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