Abstract

We seek the steady-state motion of a slack two-pulley belt drive with the belt modeled as an elastic, shear-deformable rod. Dynamic effects and gravity induce significant transverse deflections due to the low pre-tension. In analogy to the belt-creep theory, it is assumed that each contact region between the belt and one of the pulleys consists of a single sticking and a single sliding zone. Based on the governing equations of the rod theory, we for the first time derive the corresponding boundary value problem and integrate it numerically. Furthermore, a novel mixed Eulerian–Lagrangian finite element scheme is developed that iteratively seeks the steady-state solution. Finite element solutions are validated against semi-analytic results obtained by numerical integration of the boundary value problem. Parameter studies are conducted to examine solution dependence on the stiffness coefficients and the belt pre-tension.

Issue Section:
Research Papers
Keywords:
computational mechanics, dynamics, elasticity, structures
Topics:
Belts, Finite element analysis, Pulleys, Shear (Mechanics), Boundary-value problems, Steady state, Deflection, Gravity (Force), Stiffness, Dynamics (Mechanics)

References

1.
Rubin
,
M. B.
,
2000
, “
An Exact Solution for Steady Motion of An Extensible Belt in Multipulley Belt Drive Systems
,”
ASME J. Mech. Des.
,
122
(
3
), pp.
311
–
316
. 10.1115/1.1288404
2.
Frendo
,
F.
, and
Bucchi
,
F.
,
2020
, “
‘Brush Model’ for the Analysis of Flat Belt Transmissions in Steady-State Conditions
,”
Mech. Mach. Theory.
,
143
, pp.
1036
–
1053
. 10.1016/j.mechmachtheory.2019.103653
3.
Alciatore
,
D. G.
, and
Traver
,
A. E.
,
1995
, “
Multipulley Belt Drive Mechanics: Creep Theory Vs Shear Theory
,”
ASME J. Mech. Des.
,
117
(
4
), pp.
506
–
511
. 10.1115/1.2826711
4.
Wu
,
Y.
,
Leamy
,
M. J.
, and
Varenberg
,
M.
,
2019
, “
Belt-Drive Mechanics: Friction in the Absence of Sliding
,”
ASME J. Appl. Mech.
,
86
(
10
), p.
101001
. 10.1115/1.4044019
5.
Bechtel
,
S. E.
,
Vohra
,
S.
,
Jacob
,
K. I.
, and
Carlson
,
C. D.
,
1999
, “
The Stretching and Slipping of Belts and Fibers on Pulleys
,”
J. Appl. Mech.
,
67
(
1
), pp.
197
–
206
. 10.1115/1.321164
6.
Kong
,
L.
, and
Parker
,
R. G.
,
2005
, “
Steady Mechanics of Belt-Pulley Systems
,”
ASME J. Appl. Mech.
,
72
(
1
), pp.
25
–
34
. 10.1115/1.1827251
7.
Belyaev
,
A. K.
,
Eliseev
,
V. V.
,
Irschik
,
H.
, and
Oborin
,
E. A.
,
2017
, “
Contact of Two Equal Rigid Pulleys With a Belt Modelled As Cosserat Nonlinear Elastic Rod
,”
Acta Mech.
,
228
(
12
), pp.
4425
–
4434
. 10.1007/s00707-017-1942-0
8.
Leamy
,
M. J.
,
2004
, “
On a Perturbation Method for the Analysis of Unsteady Belt-Drive Operation
,”
ASME J. Appl. Mech.
,
72
(
4
), pp.
570
–
580
. 10.1115/1.1940660
9.
Vetyukov
,
Y.
,
Oborin
,
E.
,
Krommer
,
M.
, and
Eliseev
,
V.
,
2017
, “
Transient Modelling of Flexible Belt Drive Dynamics Using the Equations of a Deformable String With Discontinuities
,”
Math. Comput. Model. Dyn. Syst.
,
23
(
1
), pp.
40
–
54
. 10.1080/13873954.2016.1232281
10.
Oborin
,
E.
,
Vetyukov
,
Y.
, and
Steinbrecher
,
I.
,
2018
, “
Eulerian Description of Non-Stationary Motion of An Idealized Belt-Pulley System With Dry Friction
,”
Int. J. Solids. Struct.
,
147
, pp.
40
–
51
. 10.1016/j.ijsolstr.2018.04.007
11.
Dufva
,
K.
,
Kerkkänen
,
K.
,
Maqueda
,
L. G.
, and
Shabana
,
A. A.
,
2007
, “
Nonlinear Dynamics of Three-dimensional Belt Drives Using the Finite-Element Method
,”
Nonlinear Dyn.
,
48
(
4
), pp.
449
–
466
. 10.1007/s11071-006-9098-9
12.
Leamy
,
M. J.
, and
Wasfy
,
T. M.
,
2002
, “
Transient and Steady-State Dynamic Finite Element Modeling of Belt-Drives
,”
J. Dyn. Syst. Meas. Control.
,
124
(
4
), pp.
575
–
581
. 10.1115/1.1513793
13.
Vetyukov
,
Y.
,
2018
, “
Non-Material Finite Element Modelling of Large Vibrations of Axially Moving Strings and Beams
,”
J. Sound. Vib.
,
414
, pp.
299
–
317
. 10.1016/j.jsv.2017.11.010
14.
Scheidl
,
J.
,
2020
, “
Motion of a Friction Belt Drive At Mixed Kinematic Description
,”
Int. J. Solids. Struct.
,
200-201
, pp.
158
–
169
. 10.1016/j.ijsolstr.2020.05.001
15.
Synka
,
J.
, and
Kainz
,
A.
,
2003
, “
A Novel Mixed Eulerian–Lagrangian Finite-Element Method for Steady-State Hot Rolling Processes
,”
Int. J. Mech. Sci.
,
45
(
12
), pp.
2043
–
2060
. 10.1016/j.ijmecsci.2003.12.008
16.
Vetyukov
,
Y.
,
Gruber
,
P. G.
,
Krommer
,
M.
,
Gerstmayr
,
J.
,
Gafur
,
I.
, and
Winter
,
G.
,
2017
, “
Mixed Eulerian–Lagrangian Description in Materials Processing: Deformation of a Metal Sheet in a Rolling Mill
,”
Int. J. Numer. Methods Eng.
,
109
(
10
), pp.
1371
–
1390
. 10.1002/nme.5314
17.
Longva
,
V.
, and
Sævik
,
S.
,
2015
, “
A Lagrangian–Eulerian Formulation for Reeling Analysis of History-Dependent Multilayered Beams
,”
Comput. Struct.
,
146
, pp.
44
–
58
. 10.1016/j.compstruc.2014.09.002
18.
Escalona
,
J.
,
Orzechowski
,
G.
, and
Mikkola
,
A.
,
2018
, “
Flexible Multibody Modeling of Reeving Systems Including Transverse Vibrations
,”
Multi. Syst. Dyn.
,
44
(
2
), pp.
107
–
133
. 10.1007/s11044-018-9619-6
19.
Green
,
A. E.
, and
Laws
,
N.
,
1966
, “
A General Theory of Rods
,”
Proc. R. Soc. London. Ser. A, Math. Phys. Sci.
,
293
(
1433
), pp.
145
–
155
.
20.
Essenburg
,
F.
,
1975
, “
On the Significance of the Inclusion of the Effect of Transverse Normal Strain in Problems Involving Beams With Surface Constraints
,”
ASME J. Appl. Mech.
,
42
(
1
), pp.
127
–
132
. 10.1115/1.3423502
21.
Naghdi
,
P.
, and
Rubin
,
M.
,
1989
, “
On the Significance of Normal Cross-Sectional Extension in Beam Theory With Application to Contact Problems
,”
Int. J. Solids. Struct.
,
25
(
3
), pp.
249
–
265
. 10.1016/0020-7683(89)90047-4
22.
Nordenholz
,
T.
, and
O’Reilly
,
O.
,
1997
, “
On Steady Motions of An Elastic Rod With Application to Contact Problems
,”
Int. J. Solids. Struct.
,
34
(
9
), pp.
1123
–
1143
. 10.1016/S0020-7683(96)00054-6
23.
Kim
,
D.
,
Leamy
,
M.
, and
Ferri
,
A.
,
2009
, “
Dynamic Modeling of Flat Belt Drives Using the Elastic-Perfectly-Plastic Friction Law
,” Proceedings of the ASME Design Engineering Technical Conference, Volume 4: 7th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B and C of International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, San Diego, CA, ASME, pp.
483
–
491
.
24.
Ghayesh
,
M. H.
,
Kafiabad
,
H. A.
, and
Reid
,
T.
,
2012
, “
Sub- and Super-Critical Nonlinear Dynamics of a Harmonically Excited Axially Moving Beam
,”
Int. J. Solids. Struct.
,
49
(
1
), pp.
227
–
243
. 10.1016/j.ijsolstr.2011.10.007
25.
Steindl
,
A.
,
2017
, “
Oscillations of a Visco-Elastic Belt Drive
,”
PAMM
,
17
(
1
), pp.
381
–
382
. 10.1002/pamm.201710160
26.
Belyaev
,
A. K.
,
Eliseev
,
V. V.
,
Irschik
,
H.
, and
Oborin
,
E. A.
,
2019
, “Dynamics of Contour Motion of Belt Drive by Means of Nonlinear Rod Approach,”
Dynamics and Control of Advanced Structures and Machines: Contributions From the 3rd International Workshop
,
Matveenko
,
V. P.
,
Krommer
,
M.
,
Belyaev
,
A. K.
, and
Irschik
,
H.
, eds.,
Springer International Publishing
,
Perm, Russia
, pp.
21
–
29
.
27.
Simo
,
J.
, and
Laursen
,
T.
,
1992
, “
An Augmented Lagrangian Treatment of Contact Problems Involving Friction
,”
Comput. Struct.
,
42
(
1
), pp.
97
–
116
. 10.1016/0045-7949(92)90540-G
28.
Yastrebov
,
V.
,
2013
,
Numerical Methods in Contact Mechanics
,
John Wiley & Sons Ltd
.,
Hoboken, NJ
.
29.
Vetyukov
,
Y.
,
Oborin
,
E.
,
Scheidl
,
J.
,
Krommer
,
M.
, and
Schmidrathner
,
C.
,
2019
, “
Flexible Belt Hanging on Two Pulleys: Contact Problem At Non-Material Kinematic Description
,”
Int. J. Solids. Struct.
,
168
, pp.
183
–
193
. 10.1016/j.ijsolstr.2019.03.034
30.
Naghdi
,
P.
, and
Rubin
,
M.
,
1984
, “
Constrained Theories of Rods
,”
J. Elast.
,
14
(
4
), pp.
343
–
361
. 10.1007/BF00125605
31.
Scheidl
,
J.
, and
Vetyukov
,
Y.
,
2020
, “Steady Motion of a Belt in Frictional Contact With a Rotating Pulley,”
Dynamics and Control of Advanced Structures and Machines: Contributions From the 4th International Workshop
,
Matveenko
,
V. P.
,
Krommer
,
M.
,
Belyaev
,
A. K.
, and
Irschik
,
H.
, eds.,
Springer International Publishing
,
Linz, Austria
(accepted for publication).
32.
Oborin
,
E.
, and
Vetyukov
,
Y.
,
2019
, “
Steady State Motion of a Shear Deformable Beam in Contact With a Traveling Surface
,”
Acta Mech.
,
230
(
11
), pp.
4021
–
4033
. 10.1007/s00707-019-02476-x
Google Scholar
PubMed
 
33.
Oborin
,
E.
,
2020
, “
Belt–pulley Interaction: Role of the Action Line of Friction Forces
,”
Acta Mech.
,
231
(
9
), pp.
3979
–
3987
. 10.1007/s00707-020-02724-5
Google Scholar
PubMed
 
34.
Antman
,
S. S.
,
1995
,
Nonlinear Problems of Elasticity
(
Applied mathematical sciences
),
Springer
,
New York
.
35.
O’Reilly
,
O. M.
,
2017
,
Modeling Nonlinear Problems in the Mechanics of Strings and Rods: The Role of the Balance Laws
(
Interaction of Mechanics and Mathematics
),
Springer International Publishing
,
Cham
.
36.
Eliseev
,
V.
,
1988
, “
The Non-Linear Dynamics of Elastic Rods
,”
J. Appl. Math. Mech.
,
52
(
4
), pp.
493
–
498
. 10.1016/0021-8928(88)90039-1
37.
Nordenholz
,
T. R.
, and
O’Reilly
,
O. M.
,
1995
, “
On Kinematical Conditions for Steady Motions of Strings and Rods
,”
ASME J. Appl. Mech.
,
62
(
3
), pp.
820
–
822
. 10.1115/1.2897023
38.
Eliseev
,
V.
, and
Vetyukov
,
Y.
,
2012
, “
Effects of Deformation in the Dynamics of Belt Drive
,”
Acta Mech.
,
223
(
8
), pp.
1657
–
1667
. 10.1007/s00707-012-0675-3
39.
Wickert
,
J.
,
1992
, “
Non-linear Vibration of a Traveling Tensioned Beam
,”
Int. J. Non-Linear Mech.
,
27
(
3
), pp.
503
–
517
. 10.1016/0020-7462(92)90016-Z
40.
Bussetta
,
P.
,
Marceau
,
D.
, and
Ponthot
,
J.
,
2012
, “
The Adapted Augmented Lagrangian Method: a New Method for the Resolution of the Mechanical Frictional Contact Problem
,”
Comput. Mech.
,
49
, pp.
259
–
275
. 10.1007/s00466-011-0644-z
Copyright © 2020 by ASME
You do not currently have access to this content.