0
Research Papers

Inconsistent Stability of Newmark's Method in Structural Dynamics Applications

[+] Author and Article Information
Richard Wiebe

Structural Sciences Center,
Air Force Research Laboratory,
WPAFB OH, 45433
e-mail: rwiebe@co.uw.edu

Ilinca Stanciulescu

Rice University,
208 Ryon Laboratory, MS 318,
6100 Main Street, Houston, TX 77005
e-mail: ilinca.s@rice.edu

1Present address: University of Washington, 201 More Hall, Box 352700, Seattle, WA 98195-2700

2Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 27, 2014; final manuscript received August 5, 2014; published online April 2, 2015. Assoc. Editor: Dan Negrut.

J. Comput. Nonlinear Dynam 10(5), 051006 (Sep 01, 2015) (8 pages) Paper No: CND-14-1057; doi: 10.1115/1.4028221 History: Received February 27, 2014; Revised August 05, 2014; Online April 02, 2015

The stability of numerical time integrators, and of the physical systems to which they are applied, are normally studied independently. This conceals a very interesting phenomenon, here termed inconsistent stability, wherein a numerical time marching scheme predicts a stable response about an equilibrium configuration that is, in fact, unstable. In this paper, time integrator parameters leading to possible inconsistent stability are first found analytically for conservative systems (symmetric tangent stiffness matrices), then several structural arches with increasing complexity are used as numerical case studies. The intention of this work is to highlight the potential for this unexpected, and mostly unknown, behavior to researchers studying complex dynamical systems, especially through time marching of finite element models. To allow for direct interpretation of our results, the work is focused on the Newmark time integrator, which is commonly used in structural dynamics.

FIGURES IN THIS ARTICLE
<>
Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

References

Hughes, T. J. R., 2000, The Finite Element Method. Linear Static and Dynamic Finite Element Analysis, Dover Publications, Inc., Mineola, NY.
Addison, P. S., Chan, A. H. C., Ervine, D. A., and Williams, K. J., 1992, “Observations on Numerical Method Dependent Solutions of a Modified Duffing Oscillator,” Commun. Appl. Numer. Methods, 8(8), pp. 519–528. [CrossRef]
Arnol'd, V. I., 1989, Mathematical Methods of Classical Mechanics, Vol. 60, Springer, New York.
Thompson, J. M. T., and Hunt, G. W., 1973, A General Theory of Elastic Stability, John Wiley, London.
Hughes, T. J. R., 1977, “A Note on the Stability of Newmark's Algorithm in Nonlinear Structural Dynamics,” Int. J. Numer. Methods Eng., 11(2), pp. 383–386. [CrossRef]
Kalmar-Nagy, T., and Stanciulescu, I., 2014, “Can Complex Systems Really be Simulated?,” Appl. Math. Comput., 227, pp. 199–211. [CrossRef]
Newmark, N. M., 1959, “A Method of Computation for Structural Dynamics,” Proc. ASCE, 85, pp. 67–94.
Thompson, J. M. T., and Stewart, H. B., 2002, Nonlinear Dynamics and Chaos, John Wiley & Sons Inc, West Sussex, England.
Wiebe, R., 2012, “Nonlinear Dynamics of Discrete and Continuous Mechanical Systems With Snap-Through Instabilities,” Ph.D. thesis, Duke University, Durham, NC.
Strogatz, S. H., 2001, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Perseus Books Group, Cambridge, MA.
Chandra, Y., Wiebe, R., Stanciulescu, I., Virgin, L. N., Spottswood, S. M., and Eason, T. G., 2013, “Characterizing Dynamic Transitions Associated With Snap-Through of Clamped Shallow Arches,” J. Sound Vib., 332(22), pp. 5837–5855. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Stability regions of the continuous (a) and discrete (b) systems

Grahic Jump Location
Fig. 2

Stability domain of discrete solution for unstable physical systems (a) ω02Δt2→-∞ and (b) γ = 0.5

Grahic Jump Location
Fig. 3

An SDOF arch that exhibits multiple stable and unstable equilibria. (a) Schematic, (b) static equilibrium relationship, and (c) potential energy for F = 0, and one example where F ≠ 0.

Grahic Jump Location
Fig. 4

Unforced response of the SDOF system obtained using the Newmark method (β = 0.4 and γ = 0.55) for various time steps (dt = 0.01 s and dt = 0.10 s initiated from opposite sides for clarity). (a) Time series showing inconsistent stability for dt = 1.0 s. (b) Basins of attraction for consistent stability (unshaded), inconsistent stability (dark gray), and nonconverging (light gray) solutions for dt = 1.0 s. F = 0.0, x(0) = ±0.1,x·(0) = 0.0.

Grahic Jump Location
Fig. 5

Forced response of the SDOF system obtained using the Newmark method (β = 0.4 and γ = 0.55) for various time steps (black) with overlayed instantaneous stable (red/dark shaded curve) and unstable (gray/light shaded curve) static equilibria. F = 0.1sin((2π/10)t),x(0) = 0.01,x·(0) = 0.0

Grahic Jump Location
Fig. 6

Unforced response of the SDOF system, now for k/m = 2800, obtained using the Newmark method with standard parameters (β = 0.25 and γ = 0.5) for dt = 1 s, x(0) = 0.1,x·(0) = 0.0

Grahic Jump Location
Fig. 7

A two-degree-of-freedom arch that exhibits multiple stable and unstable equilibria. (a) Schematic, (b) static equilibrium relationship, and (c) stability via natural frequency (squared) of the five equilibria (red empty circles) present under F = 0.05.

Grahic Jump Location
Fig. 8

True response of the 2DOF system under constant force F = 0.05 obtained using the Newmark method (β = 0.4 and γ = 0.55) for dt = 0.1 s. (a) Phase portrait in (θ1,θ2) projection, i.e., angular velocities are not shown. (b) Time series of θ1. The circles denote the positions of the static equilibria.

Grahic Jump Location
Fig. 9

Consistent and inconsistent stability of the 2DOF system under dead load F = 0.05 obtained using the Newmark method (β = 0.4 and γ = 0.55) for various time steps. (a) Consistent stability for dt = 6 s. (b) Inconsistent stability for dt = 8 s. All responses initiated near unstable equilibria.

Grahic Jump Location
Fig. 10

A structural arch that exhibits multiple stable and unstable equilibria. (a) Schematic showing central point loading configuration and material properties. (b) Arch geometry; (solid curves) stable FEA equilibria, (dashed curve) unstable FEA equilibrium, and (blue circle) FEA nodal locations (shown on unloaded configuration). (c) Stable (solid) and unstable (dashed) static equilibrium relationship versus central displacement, d (measured from unloaded configuration as shown), with specific loaded (points 2,3,4) and unloaded (point 1) arch configurations. An FE time series of the free decay from near the unstable equilibrium configuration is shown inset in part (c). Note that configuration 2 is omitted in part (b) as it is nearly indistinguishable from configuration 1.

Grahic Jump Location
Fig. 11

Unforced dynamic response of arch obtained with the Newmark method using various time steps for an at-rest initial condition nearby but not on the unstable equilibrium configuration (point 3 in Fig. 10). (a) Two different Newmark parameter sets yielding consistent stability of configurations 2 and 4. (b) A Newmark parameter set yielding inconsistent stability, i.e., false stabilization of configuration 3. Note that the vertical axis is different for parts (a) and (b).

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In