Modeling of the hysteretic dynamic behavior of the Stockbridge messenger cables Francesco Foti1, Luca Martinelli1 1Department of Civil and Environmental Engineering, Politecnico di Milano, Italy E-mail: foti@stru.polimi.it, luca.martinelli@polimi.it

Keywords: Stockbridge dampers, wire ropes, hysteretic behavior. SUMMARY. A simple mechanical model of a Stockbridge damper is proposed. The model is based on a description of the interwire sliding process related to the bending behavior of the messenger cable of the damper. Subsequently, a procedure aimed at identifying the parameters of the model from quasi-static experimental tests is shown.

1 INTRODUCTION

Stockbridge dampers are widely employed to control the transverse vibrations of suspended cables subjected to wind forces. In fact, they are cheap, reliable and easy to install. As it is well documented in [1], since their invention in 1925 by G.H. Stockbridge [2] these devices have been successively modified by different researchers to improve their performances, so that several slightly different typologies are currently available (see [1] for details). However, the most common Stockbridge dampers are composed by a short metallic strand, which is known as messenger cable, parallel to the suspended cable and equipped with two rigid bodies attached at its extremities. These are characterized, in general, by different inertial properties. The connection between the damper and the suspended cable is ensured by means of a rigid clamp.

When the device is actuated through a clamp motion, the two sides of the messenger cables behave basically as cantilevers with lumped masses at their ends. Two principal vibration modes of each cantilever can be easily recognized. They are related respectively to the translation and to the rotation about the centroid of the lumped mass. During the flexural vibrations of the messenger cable, the wires of the strands undergo internal sliding so providing dissipation of energy due to friction. Due to the presence of these dissipative phenomena the dynamic response of the messenger cable is non-linear. In particular, the structural damping and the dynamic stiffness of the messenger cable are both functions of the amplitude of the support motion (hysteretic behavior). Notwithstanding the capital influence in determining the global characteristic of the damping device [1], only a few models address specifically the issue of the characterization of the flexural hysteretic behavior of the messenger cable, e.g.: [3], [4].

Within this context, the aim of this paper is to develop a simple mechanical model, based on a beam-like description of the messenger cable. Starting from a description of the interwire sliding process, we propose an inelastic non-linear cross-sectional law for the cyclic bending behavior of the cross sections based on the classic Bouc-Wen hysteretic model [5]. The parameters of the model are then identified from experimental results available in literature [4].

2 MECHANICAL MODEL OF THE DAMPER

Let us consider the schematic representation of a Stockbridge damper depicted in figure 1. The connection between the damping device and the cable is realized through a rigid clamp, usually in

aluminum alloys. The clamps can be directly cast on the messenger cable or assembled on the cable by compression [1]. In both cases the manufacturing process ensures the continuity between the clamp and the messenger cables, so that relative displacements and rotations at the interface (points C1 and C2 in figure 1) can be considered as negligible. The messenger cables are realized by small-diameter steel strands, with length usually belonging to a range of 30-50 cm. Typical cross-sections are composed of a core wire surrounded by one or two concentric layers of wires. A rigid body is attached at the end of each branch of the messenger cable (points A1 and A2 in figure 1). Several different design options have been proposed for these bodies (see [1] for further details). Here we simply assume that the connection between them and the messenger cables can be modeled as perfectly rigid.

Figure 1: Schematic representation of a Stockbridge damper.

2.1 The equations of motion of the damper

The Stockbridge damper behaves as a “tuned mass vibration absorber” [6]. When the suspended cable undergoes in-plane oscillations, as it is usually assumed to study aeolian vibrations [1], the clamp translates and rotates rigidly, following the motion of the cable. Then, the two sides of the messenger cable behave basically as uncoupled planar cantilevers, with lumped masses at their ends, subject to a prescribed support motion. By neglecting the effect of the distributed inertia of the messenger cable, the dynamic behavior of each arm of the damper can be described in terms of two Lagrangian coordinates, representing the vertical translation and the rotation of the lumped mass. In the following we will denote as xi and φi, respectively: the relative downward vertical displacement with respect to the clamp and the clockwise rotation of the ith mass of the damper.

The equations of motion of the ith lumped mass can be written by imposing the dynamic equilibrium at the interface section, Ai, between the mass and the messenger cable. By denoting as (Fi

inert, Miinert) and (Fi, Mi) the generalized inertia and restoring forces acting on the mass (see

figure 2), at a generic instant of time, t, the following hold:

( ) ( )( ) ( )

0

0

inerti i

inerti i

F t F t

M t M t

− + =

− + = (1)

Figure 2: Dynamic equilibrium of the ith mass of the Stockbridge damper.

By restricting the attention to the case of an imposed vertical translation of the clamp, zC, with

zero rotation, and denoting with a dot time derivatives, the generalized inertia forces acting on the mass can be expressed as follows:

( ) ( ) ( )( )inerti i i CF t m x t z t= − ⋅ +&& && (2)

( ) ( ) ( )2inerti Gi i i iM t I m e tϕ= − + ⋅ ⋅ && (3)

Where we have introduced the following quantities related to the damper mass: the mass, mi;

the mass moment of inertia, IGi, evaluated with respect to the centroid Gi; and the eccentricity, ei, of the centroid with respect to the interface section Ai.

Due to the hysteretic bending behavior of the messenger cables, the restoring forces which appear in (1) can be described, in general terms, by means of two non-linear and non-holonomic functions of the lagrangian coordinates xi and φi. With a slight abuse of terminology, we introduce the “functions”: Fi=Fi(xi(t), φi (t), t) and Mi=Mi(xi(t), φi (t), t), which depend also on the past history of the problem. The problem of the characterization of the mechanical behavior of the messenger cable will be further discussed in paragraph 2.2.

In conclusion, the motion of the ith damper mass for an imposed vertical translation of the clamp is governed by the following system of non-linear coupled ordinary differential equations:

( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )( )2

, ,

, , 0

i i i i C

Gi i i i i

m x t F x t t t m z t

I m e t M x t t t

ϕ

ϕ ϕ

⋅ + = − ⋅

+ ⋅ ⋅ + =

&& &&

&&

(4)

Once the motion of each damper mass has been determined by integrating the equations of

motion (4), the total reaction force, R(t); exerted by the clamp on the suspended cable can be evaluated as:

( ) ( ) ( ) ( ) ( )1 2 1 1 2 2C CR t m m m z t m x t m x t= + + ⋅ + ⋅ + ⋅&& &&&& (5)

Where we have denoted as mC the mass of the clamp.

2.2 The mechanical model of the messenger cable

A strand can be viewed as a composite structural element with a peculiar internal structure which directly affects its mechanical response. In the reference configuration of the mechanical problem, the wires of the strand are helically twisted around an initially straight core, which in the case of messenger cables is another wire (“core wire”), and grouped in concentric layers. Neglecting the imperfections related to the manufacturing process, the centerline of a generic wire can be described as a circular helix, with radius R, and lay angle α (see [7] for more details). Typical cross-sections of the messenger cables are made of seven or nineteen wires, arranged as represented in figure 3. Cross-sections of the wires are usually round shaped. In the following we will denote as d, their diameter.

(a) (b)

Figure 3: Typical cross sections of a messenger cable. (a) 7-wire strand; (b) 19-wire strand. When the strand is bent, wires tend to slip relatively one to each other, as a consequence of the

axial force gradient generated along their length [7]. Relative displacements are contrasted by friction forces, which develop at the contact surfaces as a function of the internal geometry of the strand, the material properties of its components and the intra- and inter-layer contact pressures. When the forces which tend to activate the sliding are greater than the friction ones, then a generic wire undergoes relative displacements with respect to the neighbors.

Hence, we can distinguish two limit cases in bending. The first one correspond to a full stick-state, in which friction forces are high enough to prevent any relative sliding among wires all along the strand. In this case, the cross sections of the strand remain plane and their bending stiffness takes the upper bound value, EImax, close to that of a circular rod with the same diameter of the strand. The second limit case, instead, is attained when the friction forces are no longer able to contrast any relative displacement between wires, which as a consequence behave independently. In this case the plane section hypothesis is no longer valid and the cross section bending stiffness takes the lower bound value, EImin.

Several expressions have been proposed to evaluate the bending stiffness bound values, leading to very close results [7]. For an axisymmetric cross section, denoting as: m the number of the layers; nj the number of the wires belonging to jth layer; EAj and EIj respectively the linearly elastic axial and bending stiffness of the cross section of a wire belonging to jth layer; and reserving the subscript ‘0’ to quantities referred to the core of the strand, we can introduce the following expressions (see [7] for further details):

( ) ( )3 2max 0

1 1

cos cos2

m mj

j j j j j jj j

nEI EI n EI R EAα α

= =

= + ⋅ ⋅ + ⋅ ⋅ ⋅∑ ∑ (6)

( )min 01

cosm

j j jj

EI EI n EIα=

= + ⋅ ⋅∑ (7)

Hence, referring to a monotonic bending loading, a typical cross-sectional moment-curvature

relation of the strand is composed by two linear regions, with tangent stiffness EImax and EImin, which are related through a smooth non-linear curve. This non-linear transition is physically related to the evolution of the interwire relative displacement. When the sign of the bending load is reversed, as in the case of cyclic bending loading, the moment-curvature relation shows symmetric hysteretic cycles. The area of these cycles can be related to the power dissipated through the interwire relative displacements (see [7] for a more detailed discussion on the topic).

This phenomenological behavior can be represented by means of classic hysteretic models, such as the Bouc-Wen one (see for example [5] for a survey of the model). Hence, denoting as M and χ respectively the cross-sectional bending moment and curvature, we introduce the following modified form of the Bouc-Wen model:

( ) ( ) ( ) ( )min max min 0M t EI t EI EI tχ χ η= ⋅ + − ⋅ ⋅ (8)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1

0

11

n n

i t t t t t t tη χ σ χ η η σ χ ηχ

− = ⋅ − ⋅ ⋅ ⋅ + − ⋅ ⋅

& & & & (9)

Variables χ0, σ and n are model parameters. It’s worth noticing that χ0 can be regarded as the

yielding curvature of a bilinear elasto-plastic model with pre- and post-yielding stiffness equal respectively to EImax and EImin. The variables σ and n , instead, control the shape of the non-linear transition curve between the two linear regions of the moment-curvature relation.

Now, the restoring forces exerted by the messenger cable on the damper masses can be evaluated by modeling each branch of the damper as a non-linear cantilever with negligible shear strains and a moment-curvature relation governed by equations (8) and (9). The cross-sectional constitutive law is assumed as constant all along the beam element. This can be viewed as a strong assumption of the model. Indeed, in [4] it has been shown that the behavior of the cross sections in proximity of the clamping device is strongly affected by the boundary conditions.

Focusing on a generic branch of the damper, we observe that the structural scheme is statically determined, hence a flexibility-based approach can lead straightforwardly to a relation between the restoring forces and the Lagrangian coordinates of the problem. In fact, by introducing an arc-length coordinates, 0 ≤ s ≤ li, with origin at the tip of the damper arm (point Ai in figure 1), the bending moment at a generic instant of time can be expressed as:

( ) ( ) ( ), 1 , 0i i i ii

sM s t M t F t l s l

l

= + ⋅ ⋅ − ≤ ≤

(10)

If the bending moment is known, the cross sectional curvature can be easily evaluated, by

numerically solving equations (8) and (9). To this aim, a standard Newton-Raphson solver can be adopted. Once the distribution of curvatures is known, through a standard application of the virtual work we can express the current values of the Lagrangian coordinates as a function of the generalized restoring forces. Formally, we can write the following equations:

( ) ( )( ) ( ) ( )( )0

, 1 , ,il

i i i i i ii

sx F t M t l s F t M t ds

lχ

= ⋅ − ⋅

∫ (11)

( ) ( )( ) ( ) ( )( )0

, , ,il

i i i i iF t M t s F t M t dsϕ χ= ∫ (12)

In practice, the integral (11) and (12) can be evaluated numerically, by means of a Gauss-

Lobatto integration scheme.

3 PARAMETER IDENTIFICATION

The results of two quasi-static experimental tests, aimed at identifying the hysteretic bending behavior of a Stockbridge messenger cable, have been published in [4], [8].

A schematic representation of the tests is depicted in figure 4. In both cases, the clamp of the damper is fixed and prescribed displacements or rotations are imposed at the end section of a branch of the messenger cable. In the first case (experiment (a)), a cyclic displacement, x(t), is imposed, while the rotation of the end section of the messenger cable is free. In the second case (experiment (b)) a rotation is imposed at the end section of the messenger cable, while keeping its vertical displacement equal to zero.

(a) (b)

Figure 4: Quasi static experimental tests. The length of the branch is equal to 30 cm, while the geometric characteristics of the cross

section are summarized in table 1. The theoretical extreme values of the cross sectional bending stiffness can be evaluated through expressions (6) and (7), which give: EImax = 59.9 [kNm2] and EImin = 2.91 [kNm2].

diameter [mm] helix radius [mm] lay angle [deg]

Core 2 - -

Layer 1 2 2 24.69

Layer 2 2 4 21.44

Table 1. Geometry of the cross section of the strand.

The force-displacement curve F=F(x(t)) of the experiment (a) (see figure(5)) has been used to identify the parameters of the moment-curvature constitutive law. The identification procedure will be briefly described in the following.

First of all, we compared the theoretical values of maximum and minimum bending stiffness with those which can be estimated from the experimental results by recalling the linear elastic solution: F = (3EI / l3) x. We observed a very good agreement between the theoretical and the experimental value of the maximum bending stiffness, while the minimum bending stiffness was underestimated, with an error of about 21%. In fact, form the “post-yielding” branch of the hysteresis cycle shown in figure (5) we can estimate the value: EImin,exp =3.52 [kNm2]. This difference can be explained by recalling the effect of the boundary conditions, which on one end prevent interwire displacements (near the clamping device), while on the other end (at the free end) impose negligible values of the curvature, and hence of interwire displacements as well. Moreover, the minimum bending stiffness estimated from the global response of the strand can be conveniently regarded as an “average” value evaluated along the element.

In our numerical simulations, then, we considered the minimum bending stiffness estimated from the experimental results instead of the theoretical one.

The remaining parameters of the model have been identified by minimizing the error, ε, (sum of squared residuals) between the experimental, Fexp, and the numerical curve Fnum, defined as it follows:

( ) ( )( )2

exp num1

N

i ii

F x F xε=

= −∑ (12)

To this end, we adopted the Differential Evolution algorithm [9] available within the open

source optimization toolbox of OCTAVE. The identified parameters are the following: χ0 = 1.132 [1/m], σ = 0.8 and n = 1.0. It’s worth noticing that the curve F=F(x(t)) is slightly affected by the values of the parameters σ and n. As a matter of fact, these parameters can be almost arbitrarily assumed (as long that they respect the limits for the BIBO stability of the Bouc Wen model [5]). A comparison between the experimental curve and the identified solution is shown in figure(4).

The identified parameters have been used also to simulate experiment (b). Experimental and numerical results are compared in figure (6), where the bending moment at the clamp is plotted against imposed rotation at the end.

The discrepancies can be justified by observing that χ0 has been identified from global response parameter as an “average” yielding curvature from the global response of a single load case. Hence, the identified value of χ0 is directly related to the curvature distribution of experiment (a).

In the second experiment, the cable is subject to a different moment distribution. In particular, a much larger length of the cable is under higher moments than in experiment (a), hence a larger number of sections are expected to have reached the full-slip state, leading globally to a lower “average” minimum bending stiffness and a lower “average” yielding curvature.

4 CONCLUSIONS

A preliminary simple mechanical model of a Stockbridge damper has been identified. The model is based on a description of the interwire sliding process related to the bending behavior of the messenger cable, leading to a reformulation of the classic Bouc-Wen hysteretic model [5] to describe the cross sectional moment-curvature law.

The parameters of the model can be easily estimated from quasi-static experimental tests. The numerical predictions of the model have been compared with experimental results available in literature [4],[8].

Figure 5: Experiment (a). Force vs displacement at the free end.

Figure 6: Experiment (b). Bending moment at the clamp vs. imposed rotation at the fixed end.

References [1] EPRI, Transmission line Reference Book: Wind-Induced Conductor Motion, Palo Alto, CA,

USA: Electric Power Research Institute (2006). [2] Stockbridge, G.H., “Overcoming vibrations in transmission cables”, Electrical World, 86(26)

1304-1305 (1925). [3] Pivovarov, I. and Vinogradov, O. G., “One application of the Bouc’s model for non-linear

hysteresis”, J. Sound and Vib., 118(2), 209-216 (1987). [4] Sauter, D. and Hagedorn, P., “On the hysteresis of wire cables in Stockbridge dampers”, Int.J.

of Non-Linear Mech., 37(8), 1453-1459 (2002). [5] Ismail, M., Ikhouane, F. and Rodellar, J., “The hysteresis Bouc-Wen model, a Survey”, Arch

Comp Methods Eng, 16, 161-188 (2009). [6] Den Hartog, J.P., Mechanical Vibrations, IVth Ed., Dover Publications, Inc., New York (1985). [7] Foti, F., A corotational beam element and a refined mechanical model for the nonlinear

dynamic analysis of cables, Doctoral Dissertation, Politecnico di Milano, Milano (2013). [8] Sauter, D., Modeling the dynamic characteristics of slack wire cables in Stockbridge dampers,

Dissertation, Technische Universitat Darmstadt, Darmstadt (2003). [9] Storn, R. and Price, K., “Differential Evolution - A Simple and Efficient Heuristic for Global

Optimization over Continuous Spaces”, J Glob Opt, 11, 341-359 (1997).