The Scheme to Determine the Convergence Term of the Galerkin Method for Dynamic Analysis of Sandwich Plates on Nonlinear Foundations

Chen, Hongyan; Ding, Hu; Li, Shaohua

doi:10.1007/s10338-020-00208-6

Cited by 17 publications

(5 citation statements)

References 41 publications

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“…e effectiveness of this method for solving L-shaped composite beams is illustrated [5]. Chen et al used the global modal method to study the U-shaped composite beam structure, obtained the frequency equation of the U-shaped beam structure, and obtained the natural frequency and global mode of the system [6]. Ebrahimi-Mamaghani derived the governing equations of the plane motion of the Z-shaped composite beam structure and the boundary conditions of the system by using the Hamilton's principle and theoretically obtained the natural frequency of the Z-shaped composite beam structure and the modal mode shape of the analytical form [7].…”

Section: Literature Reviewmentioning

confidence: 99%

Application of the Lagrange Equation for Intelligent Sensor Vibration Control for Power Network Monitoring

Cheng

2022

International Transactions on Electrical Energy Systems

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Objective. In order to control the vibration of the beam structure more effectively and improve the safety and availability of the beam structure, an application study of the Lagrange equation for vibration control of smart sensors for power grid monitoring is proposed. The vibration of the beam structure, the displacement of the beam structure under the excitation of seismic acceleration, the response analytical electrical formula, and the displacement response formula of the beam structure under the action of the kinematic force are deduced. The optimal parameters of the beam-TMDI system are given, and the parameter sensitivity analysis is carried out. Then, the control effect of the TMDI system is studied by numerical analysis, and the vibration reduction effect of the TMDI system and the tuned mass damper (TMD) system is compared. Experimental results show that when the mass ratio μ of the TMDI system and the TMD system are both set to a fixed value of 0.005, and the parameter β of the TMDI system is set to 0, namely β = b = 0 , at this time, the TMDI system degenerates into a TMD system. The TMD natural frequency is 14.179 rad/s and the damping ratio is 0.0432 by the DH optimization method, while the TMD natural frequency is 14.1812 rad/s, and the damping ratio is 0.0436 by the augmented Lagrangian optimization algorithm. Conclusion. The vibration displacement response spectrum of a beam structure obtained by the frequency domain method can effectively reflect electricity in the displacement response of a beam structure. The parameters that minimize the vibration response of the beam structure can be accurately obtained by using the augmented Lagrangian parameter optimization method. The sensitivity of the TMDI system is controlled by the inertial device, and the inertial device has a significant impact on its robustness. The vibration reduction performance of the TMDI system is obviously better than in the conventional TMD systems.

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Section: Literature Reviewmentioning

confidence: 99%

Application of the Lagrange Equation for Intelligent Sensor Vibration Control for Power Network Monitoring

Cheng

2022

International Transactions on Electrical Energy Systems

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“…Abdelghany et al [24], Kargarnovin et al [25] and Ansari et al [26] used the same method to calculate the responses of the beam under a harmonic load or to calculate the resonance of the beam. Chen et al [27,28] studied the convergence of Galerkin truncation for the sandwich beam on a nonlinear foundation and a scheme to determine the convergence of this model has been presented. Recently, Ouzizi et al [29] presented a model of the dynamic responses of the beam on a nonlinear frictional viscoelastic foundation with the help of an explicit scheme.…”

Section: Introductionmentioning

confidence: 99%

Calculation of dynamic responses of railway sleepers on a nonlinear foundation

Tran,

Hoang,

Foret

et al. 2023

Nonlinear Dyn

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The safety of a passing train depends on different factors, of which one of the most important is the behavior of the foundation. Therefore, the effects of the non-linearity of ballast on the dynamic responses of the railway track are a key research interest. In this paper, a new model of railway sleepers posed on a nonlinear foundation has been developed. By coupling the finite element method (FEM) of the sleeper with an analytical model of the periodically supported beam model, the dynamic equation of the sleeper is developed. On the other hand, by considering a periodic series of moving loads, this equation can be transformed to a forced nonlinear oscillation. Iteration procedures have been built to calculate the periodic solution. This method has demonstrated a good convergence of results by comparison with the analytical solution in the linear case. The influence of the nonlinear foundation has been investigated by two examples: cubic-nonlinear and bi-linear foundations. The parametric studies demonstrate that numerical results converge with a small number of iterations.

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“…For example, Reference [13] employed third-order truncations to study the nonlinear free and forced vibrations of beams on a viscoelastic foundation. References [16][17][18] carried out the convergence analyses of the Galerkin truncation. Theoretically, adding terms in the Galerkin method can improve accuracy, but it also increases the difficulty of theoretical analysis.…”

Section: Introductionmentioning

confidence: 99%

Validity of Galerkin Method at Beam’s Nonlinear Vibrations of the Single Mode with the Initial Curvature

Zhang,

Huang,

2023

Buildings

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A common strategy for studying the nonlinear vibrations of beams is to discretize the nonlinear partial differential equation into a nonlinear ordinary differential equation or equations through the Galerkin method. Then, the oscillations of beams are explored by solving the ordinary differential equation or equations. However, recent studies have shown that this strategy may lead to erroneous results in some cases. The present paper carried out the following three research studies: (1) We performed Galerkin first-order and second-order truncations to discrete the nonlinear partial differential integral equation that describes the vibrations of a Bernoulli-Euler beam with initial curvatures. (2) The approximate analytical solutions of the discretized ordinary differential equations were obtained through the multiple scales method for the primary resonance. (3) We compared the analytical solutions with those of the finite element method. Based on the results obtained by the two methods, we found that the Galerkin method can accurately estimate the dynamic behaviors of beams without initial curvatures. On the contrary, the Galerkin method underestimates the softening effect of the quadratic nonlinear term that is induced by the initial curvature. This may cause erroneous results when the Galerkin method is used to study the dynamic behaviors of beams with the initial curvatures.

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The Scheme to Determine the Convergence Term of the Galerkin Method for Dynamic Analysis of Sandwich Plates on Nonlinear Foundations

Cited by 17 publications

References 41 publications

Application of the Lagrange Equation for Intelligent Sensor Vibration Control for Power Network Monitoring

Application of the Lagrange Equation for Intelligent Sensor Vibration Control for Power Network Monitoring

Calculation of dynamic responses of railway sleepers on a nonlinear foundation

Validity of Galerkin Method at Beam’s Nonlinear Vibrations of the Single Mode with the Initial Curvature

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