An Improved Structure-Dependent Explicit Method for Structural Dynamics

2008 ◽  
Author(s):  
Shuenn-Yih Chang ◽  
Chiu-Li Huang
Keyword(s):  
Low Frequency ◽  
Conditional Stability ◽  
Explicit Method ◽  
Time Step ◽  
Structural Dynamic ◽  
Linear Elastic ◽  
Acceleration Method ◽  
The Stability ◽  
Stiffness Softening ◽  
Hardening System

An explicit method is presented herein whose coefficients are determined from the initial structural properties of the analyzed system. Thus, it is structure-dependent. This method has a great stability property when compared to the previously published method [6], which is unconditionally stable for linear elastic and any instantaneous stiffness softening systems while it only has conditional stability for an instantaneous stiffness hardening system. The most important improvement of this method is that it has unconditional stability for general instantaneous stiffness hardening systems in addition to linear elastic and instantaneous stiffness softening systems. This implies that a time step may be selected base on accuracy consideration only and the stability problem might be neglected. Hence, many computational efforts can be saved in the step-by-step solution of a general structural dynamic problem, where the response is dominated by the low frequency modes and the high frequency responses are of no great interest, when compared to an explicit method, such as the Newmark explicit method, and an implicit method, such as the constant average acceleration method.

Analysis of Newmark Explicit Integration Method for Nonlinear Systems

2006 ◽  
Vol 22 (4) ◽  
pp. 321-329 ◽  
Author(s):  
S.-Y. Chang ◽  
Y.-C. Huang ◽  
C.-H. Wang
Keyword(s):  
Nonlinear Systems ◽  
Integration Method ◽  
Stability Limit ◽  
Explicit Method ◽  
Initial Stiffness ◽  
Explicit Integration ◽  
Time Step ◽  
Linear Elastic ◽  
Stiffness Softening ◽  
Theoretical Results

AbstractNumerical properties of the Newmark explicit method in the solution of nonlinear systems are explored. It is found that the upper stability limit is no longer equal to 2 for the Newmark explicit method for nonlinear systems. In fact, it is enlarged for stiffness softening and is reduced for stiffness hardening. Furthermore, its relative period error increases with the increase of the step degree of nonlinearity for a given value of the product of the natural frequency and the time step. It is also verified that the viscous damping determined from an initial stiffness is effective to reduce displacement response in the solution of a nonlinear system as that for solving a linear elastic system. All the theoretical results are confirmed with numerical examples.

Accurate Integration of Nonlinear Systems Using Newmark Explicit Method

2009 ◽  
Vol 25 (3) ◽  
pp. 289-297 ◽  
Author(s):  
S.-Y. Chang
Keyword(s):  
Nonlinear System ◽  
Elastic System ◽  
Stability Limit ◽  
Explicit Method ◽  
Analytical Evaluation ◽  
Time Step ◽  
Linear Elastic ◽  
Instantaneous Degree Of Nonlinearity ◽  
Stiffness Softening ◽  
Relative Period

AbstractIn the step-by-step solution of a linear elastic system, an appropriate time step can be selected based on analytical evaluation resultsHowever, there is no way to select an appropriate time step for accurate integration of a nonlinear system. In this study, numerical properties of the Newmark explicit method are analytically evaluated after introducing the instantaneous degree of nonlinearity. It is found that the upper stability limit is equal to 2 only for a linear elastic system. In general, it reduces for instantaneous stiffness hardening and it is enlarged for instantaneous stiffness softening. Furthermore, the absolute relative period error increases with the increase of instantaneous degree of nonlinearity for a given product of the natural frequency and the time step. The rough guidelines for accurate integration of a nonlinear system are also proposed in this paper based on the analytical evaluation results. Analytical evaluation results and the feasibility of the rough guidelines proposed for accurate integration of a nonlinear system are confirmed with numerical examples.

An Amplification Factor to Enhance Stability for Structure-Dependent Integration Method

2012 ◽  
Vol 28 (4) ◽  
pp. 665-676
Author(s):  
S.-Y. Chang
Keyword(s):  
Amplification Factor ◽  
Elastic System ◽  
Explicit Method ◽  
Stability Range ◽  
Unconditionally Stable ◽  
Tangent Stiffness ◽  
Linear Elastic ◽  
The Difference ◽  
Stiffness Softening ◽  
Hardening System

ABSTRACTChang explicit method (2007) has been shown to be unconditionally stable for a linear elastic system and any instantaneous stiffness softening system while it is only conditionally stable for any instantaneous stiffness hardening system. Its coefficients of the difference equation for displacement increment are functions of initial tangent stiffness. Since Chang explicit method is unconditionally stable for a linear elastic system and any instantaneous stiffness softening system, its stability range can be enlarged if the initial tangent stiffness is enlarged by an amplification factor and then this amplified initial tangent stiffness is used to determine the coefficients. The detailed implementation of this scheme for Chang explicit method is presented and the feasibility of this scheme is verified.

Noniterative Integration Algorithms with Controllable Numerical Dissipations for Structural Dynamics

2019 ◽  
Vol 16 (07) ◽  
pp. 1850111 ◽  
Author(s):  
Jinze Li ◽  
Kaiping Yu
Keyword(s):  
Structural Dynamics ◽  
Low Frequency ◽  
Conditional Stability ◽  
Linear Elastic ◽  
New Family ◽  
Special Cases ◽  
Frequency Domains ◽  
Stability And Accuracy ◽  
Stiffness Softening ◽  
New Algorithms

A new family of noniterative algorithms with controllable numerical dissipations for structural dynamics is studied. Particularly, this paper provides nine members of the proposed algorithms and two existing methods are included as two special cases. The proposed algorithms achieve unconditional stability and are second-order accurate for linear elastic systems. The explicit expressions of stability conditions for nonlinear stiffness systems are completely presented, which shows that new algorithms possess unconditional and conditional stability for stiffness softening and hardening systems, respectively. A comprehensive stability and accuracy analysis, including numerical energy dissipations and dispersions, are studied in order to gain insight into spectral properties of new algorithms. Due to the existence of the nonzero spurious root, this paper also pays attention to the influence of the spurious root, which shows that the spurious root does not influence numerical accuracy at low-frequency domains. Although the proposed algorithms exhibit the unusual overshoot behaviors in either displacement or velocity, numerical damping ratios in new algorithms can significantly eliminate this overshoot at a few steps. The new dissipative algorithms are appropriate to solve numerical transient responses of the structure. Numerical examples are also presented to demonstrate the analytical results.

An Efficient Computing Explicit Method for Structural Dynamics

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Shuenn-Yih Chang
Keyword(s):  
Structural Dynamics ◽  
Unconditional Stability ◽  
Explicit Methods ◽  
Explicit Method ◽  
Implicit Methods ◽  
Time Step ◽  
Integration Algorithm ◽  
Structural Dynamic ◽  
Linear Elastic ◽  
Linearized Stability

An integration algorithm, which integrates the most important advantage of explicit methods of the explicitness of each time step and that of implicit methods of the possibility of unconditional stability, is presented herein. This algorithm is analytically shown to be unconditionally stable for any linear elastic and nonlinear systems except for the instantaneous stiffness hardening systems with the instantaneous degree of nonlinearity larger than 43 based on a linearized stability analysis. Hence, its stability property is better than the previously published algorithm (Chang, 2007, “Improved Explicit Method for Structural Dynamics,” J. Eng. Mech., 133(7), pp. 748–760), which is only conditionally stable for instantaneous stiffness hardening systems although it also possesses unconditional stability for linear elastic and any instantaneous stiffness softening systems. Due to the explicitness of each time step, the possibility of unconditional stability, and comparable accuracy, the proposed algorithm is very promising for a general structural dynamic problem, where only the low frequency responses are of interest since it consumes much less computational efforts when compared with explicit methods, such as the Newmark explicit method, and implicit methods, such as the constant average acceleration method.

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

2018 ◽  
Vol 13 (11) ◽  
Author(s):  
Jinze Li ◽  
Kaiping Yu ◽  
Xiangyang Li
Keyword(s):  
Second Order ◽  
Numerical Dispersion ◽  
The Novel ◽  
Explicit Method ◽  
Dispersion Energy ◽  
Time Step ◽  
Linear Elastic ◽  
Special Cases ◽  
Stability And Accuracy ◽  
Stiffness Softening

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.

Partitioned Integration Method Based on Newmark’s Scheme for Structural Dynamic Problems

2016 ◽  
Vol 16 (01) ◽  
pp. 1640009 ◽  
Author(s):  
Chuanguo Jia ◽  
Zhou Leng ◽  
Yingmin Li ◽  
Hongliu Xia ◽  
Liping Liu
Keyword(s):  
Integration Method ◽  
Computational Cost ◽  
Low Frequency ◽  
State Variables ◽  
Time Step ◽  
Order Accuracy ◽  
Mass System ◽  
Structural Dynamic ◽  
Time Integrators ◽  
Stability And Accuracy

Systems of ordinary differential equations (ODEs) arising from transient structural dynamics very often exhibit high-frequency/low-frequency and linear/nonlinear behaviors of subsets of the state variables. With this in mind, the paper resorts to the use of different time integrators with different time steps for subsystems, which tailors each method and its time step to the solution behaviors of the corresponding subsystem. In detail, a partitioned integration method is introduced which imposes continuity of velocities at the interface to couple arbitrary Newmark schemes with different time steps in different subdomains. It is proved that the velocity continuity of the method is the primal factor of its reduction to first-order accuracy. To maintain second-order accuracy without increasing drift and computational cost, a novel method with the acceleration continuity is proposed whose velocity constraint is also ensured by means of the projection strategy. Both its stability and accuracy properties are examined through numerical analysis of a Single-degree-of-freedom (DoF) split mass system. Finally, numerical validations are conducted on Single- and Two-DoF split mass systems and a four-DoF nonlinear structure showing the feasibility of the proposed method.

Nonlinear Numerical Prediction of Gas Foil Bearing Stability and Unbalanced Response

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
Sébastien Le Lez ◽  
Mihaï Arghir ◽  
Jean Frêne
Keyword(s):  
Dry Friction ◽  
Stability Limit ◽  
Time Step ◽  
Structural Dynamic ◽  
Friction Forces ◽  
Foil Bearings ◽  
Mass Unbalance ◽  
Foil Bearing ◽  
Highly Nonlinear ◽  
The Stability

One of the main interests of gas foil bearings lies in their superior rotordynamic characteristics compared with conventional bearings. A numerical investigation on the stability limit and on the unbalanced response of foil bearings is presented in this paper. The main difficulty in modeling the dynamic behavior of such bearings comes from the dry friction that occurs within the foil structure. Indeed, dry friction is highly nonlinear and is strongly influenced by the dynamic amplitude of the pressure field. To deal with these nonlinearities, a structural dynamic model has been developed in a previous work. This model considers the entire corrugated foil and the interactions between the bumps by describing the foil bearing structure as a multiple degrees of freedom system. It allows the determination of the dynamic friction forces at the top and at the bottom of the bumps by simple integration of ordinary differential equations. The dynamic displacements of the entire corrugated sheet are then easily obtained at each time step. The coupling between this structural model and a gas bearing prediction code is presented in this paper and allows performing full nonlinear analyses of a complete foil bearing. The bearing stability is the first investigated problem. The results show that the structural deflection enhances the stability of compliant surface bearings compared with rigid ones. Moreover, when friction is introduced, a new level of stability is reached, revealing the importance of this dissipation mechanism. The second investigated problem is the unbalanced response of foil bearings. The shaft trajectories depict a nonlinear jump in the response of both rigid and foil bearings when the value of the unbalance increases. Again, it is evidenced that the foil bearing can support higher mass unbalance before this undesirable step occurs.

scholarly journals A Weighted Residual Parabolic Acceleration Time Integration Method for Problems in Structural Dynamics

2007 ◽  
Vol 7 (3) ◽  
pp. 227-238 ◽  
Author(s):  
S.H. Razavi ◽  
A. Abolmaali ◽  
M. Ghassemieh
Keyword(s):  
Fourth Order ◽  
Initial Conditions ◽  
Time Integration ◽  
Critical Time ◽  
Linear Acceleration ◽  
Time Step ◽  
Time Integration Method ◽  
Acceleration Method ◽  
The Stability ◽  
Critical Time Step

AbstractIn the proposed method, the variation of displacement in each time step is assumed to be a fourth order polynomial in time and its five unknown coefficients are calculated based on: two initial conditions from the previous time step; satisfying the equation of motion at both ends of the time step; and the zero weighted residual within the time step. This method is non-dissipative and its dispersion is considerably less than in other popular methods. The stability of the method shows that the critical time step is more than twice of that for the linear acceleration method and its convergence is of fourth order.

Comparisons of Structure-Dependent Explicit Methods for Time Integration

2015 ◽  
Vol 15 (03) ◽  
pp. 1450055 ◽  
Author(s):  
Shuenn-Yih Chang
Keyword(s):  
Time Integration ◽  
Low Frequency ◽  
Unconditional Stability ◽  
Explicit Methods ◽  
Explicit Method ◽  
Computationally Efficient ◽  
Total Response ◽  
Time Step ◽  
Order Accuracy ◽  
Practical Applications

Chang explicit method (CEM)1,2 and CR explicit method3 (CRM) are two structure-dependent explicit methods that have been successfully developed for structural dynamics. The most important property of both integration methods is that they involve no nonlinear iterations in addition to unconditional stability and second-order accuracy. Thus, they are very computationally efficient for solving inertial problems, where the total response is dominated by low frequency modes. However, an unusual overshooting behavior for CR explicit method is identified herein and thus its practical applications might be largely limited although its velocity computing for each time step is much easier than for the CEM.

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