A generalized solution methodology based on piecewise linear vector fields is proposed for piecewise linear systems with singular regions or asymmetric restoring forces which vary spatially and temporally. In matrix representation for these systems, state variables in each region can be explicitly expressed as a function of the time the orbit spends between two boundaries or the time when the orbit hits the boundary. The time can be determined by the Brent method, and periodic solutions can then be obtained. Analytical solutions are validated on a system with 3-regions of displacement and 2-regions of time, a circumferential vibration of gear meshing system, by using the newly developed numerical method.

1.
Nakai, M., Etoh, H., and Murata, S., 1997, “An Analytical Method for Piecewise Linear Systems,” Proceedings of Asia-Pacific Vibration Conference ’97, Vol. II, pp. 928–933.
2.
Komuro
,
M.
,
1988
, “
Normal Forms of Continuous Piecewise Linear Vector Fields and Chaotic Attractors Part I: Linear Vector Fields with a Section
,”
Japan Journal of Applied Mathematics
,
5
(
2
), pp.
257
304
.
3.
Komuro
,
M.
,
1988
, “
Normal Forms of Continuous Piecewise Linear Vector Fields and Chaotic Attractors Part II: Chaotic Attractors
,”
Japan Journal of Applied Mathematics
,
5
(
3
), pp.
503
549
.
4.
Blankenship
,
G. W.
, and
Kahraman
,
A.
,
1995
, “
Steady State Forced Response of a Mechanical Oscillator With Combined Parametric Excitation and Clearance Type Non-linearity
,”
J. Sound Vib.
,
185
(
5
), pp.
743
765
.
5.
Kahraman
,
A.
, and
Blankenship
,
G. W.
,
1996
, “
Interactions Between Commensurate Parametric and Forcing Excitations in a System with Clearance
,”
J. Sound Vib.
,
194
(
3
), pp.
317
336
.
6.
Padmanabhan
,
C.
, and
Singh
,
R.
,
1996
, “
Analysis of Periodically Forced Nonlinear Hill’s Oscillator with Application to a Geared System
,”
J. Acoust. Soc. Am.
,
99
(
1
), pp.
324
334
.
7.
Shaw
,
S. W.
, and
Holmes
,
P. J.
,
1983
, “
A Periodically Forced Piecewise Linear Oscillator
,”
J. Sound Vib.
,
90
(
1
), pp.
129
155
.
8.
Kuroda
,
M.
,
Nakai
,
M.
,
Hikawa
,
T.
, and
Matsuki
,
Y.
,
1996
, “
Bifurcations of Higher Subharmonics and Chaos in a Forced Vibratory System with an Asymmetric Restoring Force
,”
JSME Int. J., Ser C
,
39
(
4
), pp.
753
766
.
9.
Sato
,
K.
,
Yamamoto
,
S.
, and
Fujishiro
,
S.
,
1988
, “
Dynamical Distinctive Phenomena in a Gear System (Bifurcation Sets of Periodic Solutions and Chaotically Transitional Phenomena)
,”
Trans. Jpn. Soc. Mech. Eng., Ser. C
,
54
(
507
), C, pp.
2735
2740
.
10.
More
,
J. J.
, and
Cosnard
,
M. Y.
,
1979
, “
Numerical Solution of Nonlinear Equations
,”
ACM Trans. Math. Softw.
,
5
(
1
), pp.
64
85
.
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