Luo A.C.J. (Ed.) Dynamical Systems: Discontinuity, Stochasticity and Time-Delay

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292 T. Karvinen and E. Keskinen

Fig. 24.6 Theoretical axial energy dissipation in strands, [12] reprinted with permission from

Professional Engineering Publishing

24.3.1 Belt–Pulley Systems

There exists no generally accepted friction model between the rope and the pulley.

Two different theories, both physically motivated but based on different belt–pulley

friction assumptions, are developed in the literature. The creep model assumes that

the existence of friction depends completely on the relative motion (creep) between

the belt and the pulley surfaces. Only kinetic friction is considered in the belt–pulley

interaction, the contact arc is divided into two zones: an adhesion zone in which the

belt moves at the same speed as the pulley and the sliding zone in which the belt

slides (creeps) on the pulley surface. Only the sliding zone contributes to moment

transmission through friction forces, whereas the friction remains zero in the adhe-

sion zone. Important concepts in the creep model are the slip angle and the contact

region, which are shown in Fig. 24.9. Another approach in the friction modeling is

the shear model, which addresses the shearing deformation of the belt assuming it

inextensible. The creep model is much more used than the shear model. The shear

model is mathematically more complicated than the creep model. The solutions

presented in the literature are in most cases for the steady-state and usually for a

belt-two-pulley system.

Important quantities in the analysis of belt, wire rope, and pulley systems are

the normal force, the friction force, the maximum moment that can be trans-

24 Dynamics of Wire-Driven Machine Mechanisms: Literature Review 293

Fig. 24.7 Comparison of damping of single-layer strands as function of radius of curvature

between two methods, [12] reprinted with permission from ASCE

mitted and the slip angle. [18] compared different models which were: the full

model taking into account all the effects of inertia including the acceleration due

to stretching, the engineering and alternate solutions which take into account the

centrifugal acceleration but not the tangential acceleration, and the Capstan solution

neglecting the inertia in the momentum equations. It was found that the three ﬁrst

approaches gave very similar results for the normal and frictional forces of a belt

and two pulleys systems, but the Capstan solution neglecting the inertia gave signif-

icantly different results, which is illustrated in Fig. 24.10. Also the slip angles had

notable differences.

Different approaches can be found in the literature such as those using the curved

Euler–Bernoulli beam including the bending stiffness [19], some models are solved

analytically and numerical values are possible to obtain rather easily [18], and some

models are solved numerically [19–21].

24.3.2 Complex Wire Rope Systems

Despite the establishment of the analysis capabilities, very few actual wire rope

system designs are accomplished through their use and there is a lack of uniﬁed

design methodology for wire rope systems based on the analysis. [22] discusses the

294 T. Karvinen and E. Keskinen

Fig. 24.8 Damping against service time, [12] reprinted with permission from ASCE

Fig. 24.9 Normal force in different zones of belt calculated using different models, [18] reprinted

with permission from ASME

design of wire rope systems, but dynamics is not considered and the emphasis is on

the minimization of the rope weight. The models found in the literature are rather

detailed and there have not been attempts to simplify the systems, probably due to

the fact that the system has been a belt-two-pulley system. [23] presented an efﬁcient

24 Dynamics of Wire-Driven Machine Mechanisms: Literature Review 295

Fig. 24.10 Schematic showing slip angle and contact zone, [18] reprinted with permission from

ASME

method for calculating the eigensolutions and the dynamic response of a serpentine

belt drive consisting of several pulleys. The efﬁciency in the solution is due to the

discretization of the belt spans, which can easily be generalized to a system of ar-

bitrary number of pulleys. Hong [24] presented a method for representing complex

cable–pulley mechanism conﬁgurations and motions. In this study only the motion

of pulleys, ropes, and blocks is considered and the cables must be oriented in x or y

direction and the orientation change is not allowed. Verho [25] analyzed a wire rope

296 T. Karvinen and E. Keskinen

Fig. 24.11 Reducing wire rope mass for moment of inertia of pulley, [25] reprinted with permis-

sion from author

Fig. 24.12 Comparison of full (solid line) and simpliﬁed (dashed line) wire rope models, [25]

reprinted with permission from author

mechanism in which the moment of inertia of the ropes was reduced to the pulley

and a single value tension approximation along the entire belt was presented as

D I



i1

C L



(24.3)

for which the notation shown in Fig. 24.11. The simpliﬁed and the full model in-

cluding different tension between different pulleys and without the wire rope mass

reduction are compared in Fig. 24.12 in which the quantity on the y-axis is the

acceleration of a mass to be accelerated with the mechanism.

24 Dynamics of Wire-Driven Machine Mechanisms: Literature Review 297

There also exist FEM approaches for dynamic analysis and other numerical

models in which cable and pulley elements are developed [26]. Kumaniecka [13]de-

veloped a model for vibrations and stability of ropes with the Kelvin–Voigt model

for rope damping. Most system level simulations involve control of elevators and

cranes [27–29]; Characteristic to these studies is the simpliﬁed mechanical model

in which the mass and stiffness of rope are neglected, taken as a constant and damp-

ing constant or zero and the emphasis is on the control model. The rope is often

carrying a point mass. Such studies are, e.g., swinging prevention of the load [30]

and improvement of riding comfort in the elevator [32].

24.4 Conclusions

The state-of-the-art of modeling the mechanical properties of wire ropes and the

dynamics analysis of wire rope systems has been presented in the paper. Plenty

of literature is found on predicting the Young’s modulus and damping of the wire

rope taking into account all the geometrical parameters such as the wire and strand

diameter, the lay angle and lay type, respectively. Several modeling approaches exist

and they seem to produce accurate results, even though coding the equations in the

computer and obtaining the numerical results is not very simple. The literature is

scarce discussing the system level dynamic analysis of wire rope systems. Most

papers are dealing with the very detailed analysis of the belt pulley contact in the

steady-state case. Several papers are found on the control of wire rope systems in

which the mechanical model is often very simpliﬁed. The emphasis in these papers

is usually in the control of the actuator to minimize the vibration of the load, for

instance. We conclude that there is room for signiﬁcant developments in the system

level dynamic analysis of wire rope systems.

References

1. Raoof M, Davies TJ (2003) Simple determination of the axial stiffness for large-diameter in-

dependent wire rope core or ﬁbre core wire ropes. J Strain Anal 38(6):577–586

2. Love AEH (1944) A treatise on the mathematical theory of elasticity. Dover, New York, p 643

3. Costello GA (1990) Theory of wire rope. Springer, New York

4. Ma J, Ge S, Zhang D (2008) Distribution of wire deformation within strands of wire ropes.

J China Univ Min Technol 18:475–478

5. Elata D, Eshkenazy R, Weiss MP (2003) The mechanical behavior of a wire rope with an

independent wire rope core. Int J Solids Struct 41:1157–1172

6. Raoof M, Kraincanic I (1995) Characteristics of ﬁbre-core wire rope. J Strain Anal

30(3):217–226

7. Raoof M, Kraincanic I (1995) Analysis of large diameter steel ropes. J Eng Mech

121(6):667–675

8. Raoof M, Kraincanic I (1995) Simple derivation of the stiffness matrix for axial/torsional

coupling of spiral strands. Comput Struct 55(4):589–600

298 T. Karvinen and E. Keskinen

9. Zhu ZH, Meguid SA (2007) Nonlinear FE-based investigation of ﬂexural damping of slacking

wire cables. Int J Solids Struct 44:5122–5132

10. Raoof M, Huang YP (1991) Upper-bound prediction of cable damping under cyclic bending.

J Eng Mech 117(12):2729–2747

11. Raoof M (1991). The prediction of axial damping in spiral strands. J Strain Anal 26(4):221–229

12. Raoof M (1991) Methods for analyzing large spiral strands. J Strain Anal 26(3):165–174

13. Kumaniecka A, Niziol J (1994) Dynamic stability of a rope with slow variability of the param-

eters. J Sound Vib 178(2):211–226

14. Ridge IML, Zheng J, Chaplin CR (2000) Measurement of cyclic bending strains in steel wire

rope. J Strin Anal 35(6):545–558

15. Urchegui MA, Tato W, G´omez X (2008) Wear evolution in a stranded rope subjected to cyclic

bending. J Mat Eng Perform 17(4):550–560

16. Keskinen E, Montonen J, Launis S, Cotsaftis M (1999) Simulation of wire and chain mecha-

nism in hydraulic driven booms. Proceedings of the IASTED international conference applied

modelling and simulation, Cairns, QLD, 1–3 September 1999

17. Sun G, Kleeberger M, Liu J (2004) Complete dynamic calculation of lattice mobile crane

during hoisting motion. Mech Mach Theory 40:447–466

18. Bechtel SE, Vohra S, Jacob KI, Carlson CD (2000) The stretching and slipping of belts and

ﬁbers on pulleys. J Appl Mech 67:197–206

19. Kong L, Parker RG (2005) Steady mechanics of belt–pulley systems. J Appl Mech 72:25–34

20. Kong L, Parker RG (2005) Microslip friction in ﬂat belt drives. Proc Inst Mech Eng C: J Mech

Eng Sci 219:1097–1106

21. Rubin MB (2000) An exact solution for steady motion of an extensible belt in multipulley belt

drive systems. J Mech Des 122:311–316

22. Velinsky SA (1993) A stress based methodology for the design of wire rope systems. J Mech

Des 68:69–73

23. Parker RG (2004) Efﬁcient eigensolution, dynamic response, and eigensensitivity of serpentine

belt drives. J Sound Vib 270:15–38

24. Hong DW, Cipra RJ (2003) A method for representing the conﬁguration and analyzing the

motion of complex cable–pulley systems. ASME J Mech Des 125:332–341

25. Verho A (2003) Katapulttimekanismin mallinnus ja simulointi (in Finnish). MSc Thesis,

Tampere University of Technology

26. Hashemi SM, Roach A (2006) A dynamic ﬁnite element for vibration analysis of cables and

wire ropes. Asian J Civil Eng (Build Hous), 7(5):487–500

27. Lee H (2003) A new approach for the anti-swing control of overhead cranes with high speed

load hoisting. Int J Control 76(15):1493–1499

28. Otsuki M, Ushijima Y, Yoshida K, Kimura H, Nakagawa T (2006) Application of nonstationary

sliding mode control to suppression of transverse vibration of elevator rope using input device

using gaps. JSME Int J C 49(2):385–394

29. Otsuki M, Yoshida K, Nagakagi S, Nakagawa T, Fujimoto S, Kimura H (2004) Experimental

examination of non-stationary robust vibration control for an elevator rope. Proc Inst Mech

Eng I: J Syst Control Eng 218:531–544

30. Yanai N, Yamamoto M, Mohri A (2001) Feed-back control of crane based on inverse dynamics

calculation. Proceeding of the 2001 IEEE/RSJ, international conference on intelligent robots

and systems, Maui, Hi, October 29–November 3, 2001

31. Raoof M, Davies TJ (2005) Simple determination of the maximum axial and torsional energy

dissipation in large diameter spiral strands. Comput Struct 84:676–689

32. Kang J-K, Sul S-K (2000) Vertical vibration control of elevator using estimated car acceleration

feedback compensation. IEEE Transactions on Industrial Electronics 47(1):91–99

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CESLIC Report SC6, Civil Engineering Department, Imperial College, London

Chapter 25

Dynamics of Wire-Driven Machine Mechanisms,

Part II: Theory and Applications

Erno Keskinen, Timo Karvinen, and Jori Montonen

Abstract A systematic approach, where the mass conservation theorem is applied

to a wire continuum, leads to system equations, in which the dynamics of a com-

plete wire (or chain) and pulley assembly are coupled with the motion of a boom

mechanism. This methodology makes it possible to develop a simulation model of

complete crane and lift systems where the wire and chain wheels communicating

with the wire element can transmit the effect of boom displacements into the

dynamic variations of wire tension.

25.1 Introduction

Wires and chains are widely used machine elements in devices carrying out work

processes where large amplitude displacements and high capacity tools subject to

tensional loads are needed. In boom systems, the wires are typically used in lift-

ing winches whereas the chains can provide a compact solution in telescopic boom

conﬁgurations. The modeling problem of wires and chains rests upon the basic ques-

tion of how to represent the elastic property of their internal structure into a simple

enough small degree of freedom model. Along the line of very fast technology de-

velopment aiming at delivering more precise and more task-efﬁcient tools, compact

library models are preferable in system level simulations of large technical systems

since, today, low cost computational capability brings fast models closer to real-

time applications needed in man-in-the-loop simulations. A typical example is the

use of virtual product models as training simulators to give operators good practice

and familiarity with the boom behavior in critical lifting maneuvers. The better real

dynamics is coded into the computational model behind the screen animation, the

better touch and feeling of real operation may be reached in the maneuvers.

E. Keskinen (



)

Department of Mechanical Engineering, Tampere University of Technology, P.O. Box 589,

33101 Tampere, Finland

e-mail: erno.keskinen@tut.ﬁ

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,

DOI 10.1007/978-1-4419-5754-2

25,

 Springer Science+Business Media, LLC 2010

299

300 E. Keskinen et al.

Wires are consisting of ﬁber groups and chains are consisting of link-pin sections.

Despite the complicated internal structure, their dominating mechanical property

is axial elasticity, which may be taken as their main behavior in modeling. This has

been shown by many authors in papers mentioned in a comprehensive reference list

of an adjunct state-of-art review paper written by corresponding authors Karvinen

and Keskinen [1]. A homogenization process produces an average axial stiffness

property, which deﬁnes the effective elastic modulus of the material carrying the

axial loads. It has been shown [2] that a similar lumped volume model governing

the dynamic pressure variation within a ﬂuid volume in ﬂuid power circuits can

be developed for the tensional state in the homogenized wire continuum also. Two

worked case studies show the feasibility of proposed methodology in linking to-

gether the dynamical model of a loading crane [3] with a hydraulic winch for ﬁrst

one (Fig. 25.1a), and of an elevator boom with a chain-driven extension mechanism

[4] for second one (Fig. 25.1b).

Fig. 25.1 Hydraulic boom systems. Winch equipped loading crane (a), chain-driven elevating

platform (b)

25 Dynamics of Wire-Driven Machine Mechanisms 301

25.2 Wire Mechanics

25.2.1 Wire Topology

Wire, chain, and wheel arrangements represent conﬁgurations where one-parametric

elastic continuum is stretched to form spans between a group of wheels mounted on

a steady or moving reference frame in Fig. 25.2.

One end is winched into or from a drum whereas the other one is connected to the

work object. In technical systems, the wire typically follows the motion of a boom

mechanism, as shown on Fig. 25.3.

Topologically, the winch systems represent open kinematic chains since the work

object attached to wire end may be moved freely inside the working space covered

by the boom.

Fig. 25.2 Topology of a rigidly mounted wire-wheel-winch-object arrangement

Fig. 25.3 Winch system

mounted on a moving boom

mechanism in an open

kinematic chain