Abstract

In this paper, the block simply supported on a harmonically moving ground is assumed as a system well representing a typical nonsmooth dynamical behavior. The aim of the work is to carry out the existence conditions of asymmetric responses; an analysis that comes first in any stability investigation. By using simple definitions belonging to the symmetry group theory, it is possible to completely clarify the relationships between the various initial conditions that allow simple asymmetric responses, and to develop tools, which will be very useful in the stability analysis of more complex asymmetric responses.

1.
Ageno
,
A.
, and
Sinopoli
,
A.
, 2005, “
Lyapunov’s Exponents for Non-Smooth Dynamics With Impacts: Stability Analysis of the Rocking Block
,”
Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274,
15
(
6
), pp.
2015
2039
.
2.
Müller
,
P. C.
, 1995, “
Calculation of Lyapunov Exponents for Dynamic Systems With Discontinuities
,”
Chaos, Solitons Fractals
0960-0779,
5
(
9
), pp.
1671
1681
.
3.
Mainzer
,
K.
, 2005,
Symmetry and Complexity—The Spirit and Beauty of Nonlinear Science
,
World Scientific
,
Singapore
.
4.
Spanos
,
P. D.
, and
Koh
,
A.
, 1984, “
Rocking of Rigid Blocks Due to Harmonic Shaking
,”
J. Eng. Mech.
0733-9399,
110
(
11
), pp.
1627
1643
.
5.
Hogan
,
S. J.
, 1989, “
On the Dynamics of Rigid-Block Motion Under Harmonic Forcing
,”
Proc. R. Soc. London
0370-1662,
425
, pp.
441
476
.
6.
Hogan
,
S. J.
, 1990, “
The Many Steady State Responses of Rigid Block Under Harmonic Forcing
,”
Earthquake Eng. Struct. Dyn.
0098-8847,
19
, pp.
1057
1071
.
7.
Sinopoli
,
A.
, and
Ageno
,
A.
, 2003, “
Stability Analysis of the Rocking Block: Numerical Investigations on Analytical Stability Boundaries
,”
ASME
Paper No. DETC 2003/VIB-48478.
8.
Ageno
,
A.
, and
Sinopoli
,
A.
, 2005, “
Lyapunov’s Exponents for Non-Smooth Dynamical Systems: Behaviour Identification Across Stability Boundaries With Bifurcations
,”
ASME
Paper No. DETC 2005-84574.
9.
Housner
,
W. G.
, 1963, “
The Behavior of Inverted Pendulum Structures During Earthquakes
,”
Bull. Seismol. Soc. Am.
0037-1106,
53
, pp.
403
417
.
10.
Yim
,
S. C. S.
, and
Lin
,
H.
, 1991, “
Nonlinear Impact and Chaotic Response of Slender Rocking Objects
,”
J. Eng. Mech.
0733-9399,
117
(
9
), pp.
2079
2100
.
11.
Sinopoli
,
A.
, 1997, “
Unilaterality and Dry Friction: A Geometric Formulation for Two-Dimensional Rigid Body Dynamics
,”
Nonlinear Dyn.
0924-090X,
12
(
4
), pp.
343
366
.
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