The nonlinear dynamics of a very flexible body with time-variant length encompasses several industrial applications such as high-speed automatic coiling machines and copy machines. Among others, the significant increase in the vibration that occurs when the material length is shortened with time is known as the spaghetti problem. In this paper, the modeling method and the experimental procedure for the analysis of the spaghetti problem are presented. The change in the state of the forces and displacements at the boundary with a clearance is taken into consideration by modeling the mechanical interactions resulting from the clearance. A flexible beam is modeled using the finite segment method to account for the geometric nonlinearities due to the large rotation. The contact forces at the boundary are modeled using a set of springs and dampers. The numerical results obtained using the proposed modeling method agree well with the results obtained using the experiment. The effect of the transport velocity and the clearance are demonstrated, and the cause of the significant increase in the flexible body vibration is discussed from an energy balance viewpoint.

Keywords:
beams (structures), flexible structures, vibrations, damping
Topics:
Clearances (Engineering), Modeling, Vibration, Dynamic analysis, Damping, Experimental methods, Springs, Dampers, Energy budget (Physics), Rotation, Strips
1.
Carrier
,
G. F.
,
1949
, “
The Spaghetti Problem
,”
Am. Math. Monthly
,
56
, pp.
669
–
672
.
2.
Tabarrok
,
B.
,
Leech
,
C. M.
, and
Kim
,
Y. I.
,
1974
, “
On the Dynamics of an Axially Moving Beam
,”
J. Franklin Inst.
,
297
, pp.
201
–
220
.
3.
Mansfield
,
L.
, and
Simmond
,
J. G.
,
1987
, “
The Reverse Spaghetti Problem: Drooping Motion of an Elastica Issuing From a Horizontal Guide
,”
ASME J. Appl. Mech.
,
54
, pp.
147
–
150
.
4.
Stolte
,
J.
, and
Benson
,
R. C.
,
1992
, “
Dynamic Deflection of Paper Emerging From a Channel
,”
ASME J. Vibr. Acoust.
,
114
, pp.
187
–
193
.
5.
Cherchas
,
D. B.
,
1971
, “
Dynamics of Spin-Stabilized Satellites During Extension of Long Flexible Booms
,”
J. Spacecr. Rockets
,
8
, pp.
802
–
804
.
6.
Banerjee
,
A. K.
, and
Kane
,
T. R.
,
1989
, “
Extrusion of a Beam From a Rotating Base
,”
AIAA J. Guidance, Control, Dynamics
,
12
, pp.
140
–
146
.
7.
Stylianou
,
M.
, and
Tabarrok
,
B.
,
1994
, “
Finite Element Analysis of an Axially Moving Beam, Part I: Time Integration
,”
J. Sound Vib.
,
178
, pp.
433
–
453
.
8.
Vu-Quoc
,
L.
, and
Li
,
S.
,
1995
, “
Dynamics of Sliding Geometrically-Exact Beams: Large Angle Maneuver and Parametric Resonance
,”
Comput. Methods Appl. Mech. Eng.
,
120
, pp.
65
–
118
.
9.
Behdinan
,
K.
,
Stylianou
,
M.
, and
Tabarrok
,
B.
,
1997
, “
Dynamics of Flexible Sliding Beams-Non-Linear Analysis Part I: Formulation
,”
J. Sound Vib.
,
208
, pp.
517
–
539
.
10.
Zhu
,
W. D.
, and
Ni
,
J.
,
2000
, “
Energetics and Stability of Translating Media With an Arbitrarily Varying Length
,”
ASME J. Vibr. Acoust.
,
122
, pp.
295
–
304
.
11.
Yuh
,
J.
, and
Young
,
T.
,
1991
, “
Dynamic Modeling of an Axially Moving Beam in Rotation: Simulation and Experiment
,”
ASME J. Dyn. Syst., Meas., Control
,
113
, pp.
34
–
40
.
12.
Huston, R. L., 1990, Multibody Dynamics, Butterworth-Heinemann.
13.
Shabana, A. A., 2001, Computational Dynamics, 2nd, Wiley, New York.
14.
Khulief
,
Y. A.
, and
Shabana
,
A. A.
,
1987
, “
A Continuous Force Model for the Impact Analysis of Flexible Multibody Systems
,”
Mech. Mach. Theory
,
22
, pp.
213
–
224
.
15.
Wehage
,
R. A.
, and
Haug
,
E. J.
,
1982
, “
Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems
,”
ASME J. Mech. Des.
,
104
, pp.
247
–
255
.
Copyright © 2005
 by ASME
You do not currently have access to this content.