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.

Copyright © 2015 by ASME
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Fig. 1

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

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Fig. 2

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

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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.

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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.

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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

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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

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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.

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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.

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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.

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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.

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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).



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