The discretization error of newmark's method for numerical integration in structural dynamics

1985 ◽  
Vol 13 (1) ◽  
pp. 43-51 ◽  
Author(s):  
J. M. B. Brown
Keyword(s):  
Numerical Integration ◽  
Structural Dynamics ◽  
Discretization Error ◽  
Newmark’S Method

scholarly journals An Algorithm for the Numerical Integration of Perturbed and Damped Second-Order ODE Systems

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2028
Author(s):  
Fernando García-Alonso ◽  
José Antonio Reyes ◽  
Mónica Cortés-Molina
Keyword(s):  
Numerical Integration ◽  
Structural Dynamics ◽  
Truncation Error ◽  
First Integral ◽  
Second Order ◽  
New Method ◽  
Series Method ◽  
Initial Value ◽  
Satellite Problem ◽  
Damped Systems

A new method of numerical integration for a perturbed and damped systems of linear second-order differential equations is presented. This new method, under certain conditions, integrates, without truncation error, the IVPs (initial value problems) of the type: x″(t)+Ax′(t)+Cx(t)=εF(x(t),t), x(0)=x0, x′(0)=x0′, t∈[a,b]=I, which appear in structural dynamics, astrodynamics, and other fields of physics and engineering. In this article, a succession of real functions is constructed with values in the algebra of m×m matrices. Their properties are studied and we express the solution of the proposed IVP through a serial expansion of the same, whose coefficients are calculated by means of recurrences involving the perturbation function. This expression of the solution is used for the construction of the new numerical method. Three problems are solved by means of the new series method; we contrast the results obtained with the exact solution of the problem and with its first integral. In the first problem, a quasi-periodic orbit is integrated; in the second, a problem of structural dynamics associated with an earthquake is studied; in the third, an equatorial satellite problem when the perturbation comes from zonal harmonics J2 is solved. The good behavior of the series method is shown by comparing the results obtained against other integrators.

Numerical Methods in Structural Dynamics

1974 ◽  
Vol 1 (2) ◽  
pp. 179-193 ◽  
Author(s):  
J. L. Humar ◽  
E. W. Wright
Keyword(s):  
Numerical Integration ◽  
Structural Dynamics ◽  
Single Point ◽  
Computation Time ◽  
Point Method ◽  
Multiple Point ◽  
Considerable Saving ◽  
Random Forcing ◽  
Forcing Function ◽  
Predictor Corrector

The dynamic analysis of a structure subjected to a random forcing function from a source such as earthquake, blast, or wind requires the use of a numerical integration technique. The efficiency and accuracy of the technique employed is of great importance for both research and practical design. The more effective methods of numerical integration belong to the category designated as ‘predictor-corrector’ methods. A systematic method is presented for the derivation of single-point and multiple-point predictor-corrector formulae. It is shown that most of the methods of numerical integration presently employed in structural dynamics are single-point predictor-corrector methods. A scheme of iteration is usually employed for the solution of the difference equations obtained by the application of these methods. It is shown that for problems in structural dynamics, it is not necessary to use an iterative scheme; a process of elimination is feasible and also gives considerable economy in computation time. It is further shown that the choice of an appropriate multi-point method for the numerical integration of the equations of motion of an elastic system can lead to a considerable saving in computation time and cost. One such multi-point method is presented, and its truncation error and stability are examined.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

Structural dynamics of P-type ATPase ion pumps

2019 ◽  
Vol 47 (5) ◽  
pp. 1247-1257 ◽  
Author(s):  
Mateusz Dyla ◽  
Sara Basse Hansen ◽  
Poul Nissen ◽  
Magnus Kjaergaard
Keyword(s):  
Structural Dynamics ◽  
Single Molecule ◽  
New Technologies ◽  
Atp Hydrolysis ◽  
Resonance Energy Transfer ◽  
Resonance Energy ◽  
Time Resolved ◽  
Dynamics Simulations ◽  
Types Of Information ◽  
P Type

Abstract P-type ATPases transport ions across biological membranes against concentration gradients and are essential for all cells. They use the energy from ATP hydrolysis to propel large intramolecular movements, which drive vectorial transport of ions. Tight coordination of the motions of the pump is required to couple the two spatially distant processes of ion binding and ATP hydrolysis. Here, we review our current understanding of the structural dynamics of P-type ATPases, focusing primarily on Ca2+ pumps. We integrate different types of information that report on structural dynamics, primarily time-resolved fluorescence experiments including single-molecule Förster resonance energy transfer and molecular dynamics simulations, and interpret them in the framework provided by the numerous crystal structures of sarco/endoplasmic reticulum Ca2+-ATPase. We discuss the challenges in characterizing the dynamics of membrane pumps, and the likely impact of new technologies on the field.

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