This paper deals with the long term behavior of periodically excited mechanical systems consisting of linear components and local nonlinearities. The number of degrees of freedom of the linear components is reduced by applying a component mode synthesis technique. Lyapunov exponents are used to identify the character of the long term behavior of a nonlinear dynamic system, which may be periodic, quasi-periodic or chaotic. Periodic solutions are calculated efficiently by solving a two-point boundary value problem using finite differences. Floquet multipliers are calculated to determine the local stability of these solutions and to identify local bifurcation points. The methods presented are applied to a beam system supported by a one-sided linear spring, which reveals very rich, complex dynamic behavior.

Issue Section:
Research Papers
Topics:
Bifurcation, Boundary-value problems, Degrees of freedom, Nonlinear dynamical systems, Springs, Stability, Structural dynamics
1.
Craig, Jr., R. R., 1985, “A Review of Time-Domain and Frequency-Domain Component Mode Synthesis Methods,” Combined Experimental/Analytical Modeling of Dynamic Structural Systems Using Substructure Synthesis, D. R. Martinez and A. K. Miller, ed., pp. 1–31.
2.
de Kraker, A., Fey, R. H. B., van Campen, D. H., and Langeveld C. J., 1989, “Some Aspects of the Analysis of Systems with Local Nonlinearities,” Nonlinear Dynamics in Engineering Systems, W. Schiehlen, ed., IUTAM Springer-Verlag, pp. 165–172.
3.
DIANA, 1994, DIANA User’s Manual 6.0 edition, TNO Building and Construction Research, Delft, The Netherlands.
4.
Feigenbaum
 
M. J.
,
1983
, “
Universal Behavior in Nonlinear Systems
,”
Physica
, Vol.
7D
, pp.
16
–
39
.
5.
Fey, R. H. B., 1992, “Steady-State Behaviour of Reduced Dynamic Systems with Local Nonlinearities,” Ph.D. Thesis, Eindhoven University of Technology, Eindhoven, The Netherlands.
6.
Fried
 
I.
,
1984
, “
Orthogonal Trajectory Accession to the Nonlinear Equilibrium Curve
,”
Computer Methods in Applied Mechanics and Engineering
, Vol.
47
, pp.
283
–
297
.
7.
Guckenheimer, J., and Holmes, P., 1983, “Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,” Applied Mathematical Sciences, 42, Springer-Verlag.
8.
Natsiavas
 
S.
,
1990
, “
On the Dynamics of Oscillators with Bi-Linear Damping and Stiffness
,”
Int. Journal of Non-Linear Mechanics
, Vol.
25
, pp.
535
–
554
.
9.
Newhouse
 
S.
,
Ruelle
 
D.
, and
Takens
 
F.
,
1978
, “
Occurence of Strange Axiom-a Attractors Near Quasi-Periodic Flow on Tm, m ≤ 3
,”
Commun. Math. Phys.
, Vol.
64
, pp.
35
–
40
.
10.
Parker, T. S., and Chua, L. O., ed., 1989, “Practical Numerical Algorithms for Chaotic Systems,” Springer-Verlag.
11.
Pereyra
 
V.
,
1966
, “
On Improving an Approximate Solution of a Functional Equation by Deferred Corrections
,”
Numerische Mathematik
, Vol.
8
, pp.
376
–
391
.
12.
Pomeau
 
Y.
, and
Manneville
 
P.
,
1980
, “
Intermittent Transition to Turbulence in Dissipative Dynamical Systems
,”
Commun. Math. Phys.
, Vol.
74
, pp.
189
–
197
.
13.
Seydel, R., 1988, From Equilibrium to Chaos, Practical Bifurcation and Stability Analysis, Elsevier Science Publishing Co., Inc.
14.
Shaw
 
J.
,
Shaw
 
S. W.
, and
Haddow
 
A. G.
,
1989
, “
On the Response of the Nonlinear Vibration Absorber
,”
Int. Journal of Non-Linear Mechanics
, Vol.
24
, pp.
281
–
293
.
15.
Shaw
 
S. W.
, and
Holmes
 
P. J.
,
1983
, “
A Periodically Forced Piecewise Linear Oscillator
,”
Journal of Sound and Vibration
, Vol.
90
, pp.
129
–
155
.
16.
Thompson, J. M. T., and Stewart, H. B., 1986, Nonlinear Dynamics and Chaos, John Wiley & Sons Ltd.
17.
van de Vorst, E. L. B., Fey, R. H. B., van Campen, D. H., and de Kraker, A., 1993, “Manifolds of Nonlinear Single-DOF Systems,” Topics in Applied Mechanics, Integration of Theory & Applications in Applied Mechanics, J. F. Dijksman and F. T. M. Nieuwstadt, ed., Kluwer Academic Publishers, pp. 293–303.
This content is only available via PDF.
Copyright © 1996
 by The American Society of Mechanical Engineers
You do not currently have access to this content.