Modeling vibration response and damping of cables and cabled structures

Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Modeling vibration response and damping of cables and cabled structures Kaitlin S. Spak a,n, Gregory S. Agnes b,1, Daniel J. Inman c,2 a b c

Virginia Polytechnic Institute and State University, Blacksburg, VA 24060, USA Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr, Pasadena, CA 91011, USA University of Michigan, Ann Arbor, MI 48109, USA

a r t i c l e i n f o

abstract

Article history: Received 31 May 2014 Received in revised form 1 October 2014 Accepted 6 October 2014 Handling Editor: W. Lacarbonara

In an effort to model the vibration response of cabled structures, the distributed transfer function method is developed to model cables and a simple cabled structure. The model includes shear effects, tension, and hysteretic damping for modeling of helical stranded cables, and includes a method for modeling cable attachment points using both linear and rotational damping and stiffness. The damped cable model shows agreement with experimental data for four types of stranded cables, and the damped cabled beam model shows agreement with experimental data for the cables attached to a beam structure, as well as improvement over the distributed mass method for cabled structure modeling. & 2014 Elsevier Ltd. All rights reserved.

1. Introduction Cables have a wide variety of uses, from small cables for power and signal transmission to large wire ropes that serve as structural anchors for buildings and bridges. As structural analysis has become more precise, the behavior of cables has been studied more extensively; goals for these prior investigations included attempts to determine the contact forces between individual wires within a cable, determining stresses in the cable as a whole, and analyzing vibration response [1]. More precise cable models allow for more accurate prediction of failure, resonance, and performance. The models presented herein for predicting vibration characteristics are applicable to any cable that could be described with effective beam properties, although the experimental data was acquired from spaceflight cables. By modeling the cable as a beam, the bending stiffness of the cable can be taken into account and the vibration response can be determined without calculating friction forces between individual wires, which is difficult and computationally intensive. This paper presents a cable model and a cabled-beam model developed from beam-like equations of motion in which the cable properties are determined to create effectively homogenous properties for the beam model. The distributed transfer function method is used as the solution method because it is an exact method, is well-suited to the repeating nature of cables attached to structures at multiple points, and can incorporate non-standard boundary conditions such as damped spring attachment points.

n

Correspondence to: Exponent, 149 Commonwealth Ave, Menlo Park, CA 94025, USA. Tel.: þ 1 650 285 9216. E-mail address: [email protected] (K.S. Spak). 1 R&D Engineer, 355L. 2 Department Chair, Aerospace Engineering Department, 3064 FXB.

http://dx.doi.org/10.1016/j.jsv.2014.10.009 0022-460X/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: K.S. Spak, et al., Modeling vibration response and damping of cables and cabled structures, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.009i

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Cables are like beams in that they have a reasonably constant cross-sectional area that is small compared to the length. While electrical cables may seem string-like, their bending stiffness is actually not negligible and must be considered for accurate vibration results. A string model considers only the physical mass properties of the material (density and area), while a beam model includes the material's bending stiffness. A beam model can also account for tension and damping in the cable. Thus, the overall structure of the cable is suited to modeling with a beam model; this has been verified by Castello and Matt [2]. Work from the Air Force Research Laboratory confirmed this and further showed that inclusion of shear effects was necessary for accurate cable modeling [3]. Using a beam model to characterize cable behavior works best for lowamplitude vibration modeling with minimal curvature so that internal friction forces due to wire slippage are minimized, or at least constant [1]. Recall that beams can be modeled as Euler–Bernoulli beams, in which plane sections of the beam remain plane as it deforms, shear beams, which includes the effect of shear deformation, or Timoshenko beams, in which both shear deformation and rotary inertia effects are taken into account. The spaceflight cables used in this research are about 1 m long and range in outer diameter from 7 mm to 22 mm. Although these cables could be categorized as Euler–Bernoulli beams due to the ratio of length to area, cables of this type must be modeled as shear beams [3]. This is due in part to the composite nature of the cable; the viscoelastic jacketing undergoes shear deformation and plane sections in the cable do not remain plane as the cable bends. Previous research showed that the Timoshenko beam model provided negligible additional accuracy for increased complexity, so all sources agree that the shear beam is the most applicable for cable modeling [4]. Until now, cable modeling has relied extensively on testing for results. Cable modeling is particularly difficult because of variations in cable construction and attachment points, leading to varying results for a single cable. Even cables of the same type show a large variation in their natural frequency response, often due to wire slippage and internal friction between the wires. It is unrealistic to aim for a model that can pinpoint the exact natural frequency and damping ratio for any given cable because of the inherent response variation; therefore, a cable model that bounds the span of the natural frequency responses is the goal for this work. A lumped mass or distributed mass approach are the currently used methods to model cables attached to spacecraft; a cable model that incorporates bending stiffness and attachment characteristics would provide more accurate modeling of the cabled structure. The distributed transfer function method (DTFM) used herein was developed by Yang and Tan for 1D elastic Euler– Bernoulli beams [5]. The DTFM was extended for use in laminated beams, but shear and rotary inertia effects were still neglected [6]. To use the distributed transfer function method, the Laplace transform of the equation of motion is used to populate a matrix F(s) that can be manipulated to give information about the dynamic response of the system. The ultimate goal of this research is to create a cable model that can be added to structural models to quantify the changes in a structure's dynamic response due to the addition of cable harnesses. The aim of this paper is to model cables on a simple host structure to find the natural frequencies and dynamic response of the system. 2. Cable model development To develop the cable model, we begin with a cable element undergoing both bending and shear. Unlike an Euler– Bernoulli beam, in which plane sections remain plane, the cable is made up of individual wires that have a viscoelastic insulation layer which undergoes shear, so a shear beam model is more appropriate. The cables are assumed to have an overwrap or outer jacket which keeps the individual wires in the same basic configuration and maintains radial compression on the cable. As the cable bends, the wires move against each other, which alters the stiffness of the cable as a whole and is taken into account through careful calculation of the bending stiffness as a function of curvature and wire geometry, including number and arrangement of wires and lay angle. From the work of Timoshenko [7], the equations for a beam are ∂2 wðx; tÞ ∂Vðx; tÞ þ qðx; tÞ ¼ ∂x ∂t 2

ρA

∂Mðx; tÞ ¼ V ðx; t Þ ∂x

(1)

(2)

where ρA is the beam's area density, wðx; t Þ is the transverse displacement of the beam, V ðx; t Þis the shear force in the beam, M ðx; t Þ is the bending moment of the beam, and qðx; t Þ is an applied forcing function. These beam equations are used in conjunction with the constitutive equations for the moment and shear force for a beam that experiences bending and shear, which are M ¼ EI

∂ψ ∂x


 ∂w ψ V ¼  κ AG ∂x

(3) (4)

where the moment, shear, transverse displacement and total rotation are all functions of x and t for those and all subsequent equations until the Laplace transform is taken for the DTFM form. From this starting point, we investigate an undamped model with tension and shear effects and then examine various damping mechanisms that can be included. Please cite this article as: K.S. Spak, et al., Modeling vibration response and damping of cables and cabled structures, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.009i

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2.1. Undamped cable model For an undamped shear beam, the moment and shear terms are substituted into Eqs. (1) and (2) and a tension term is added to give
2  ∂2 w ∂ w ∂ψ þ qðx; tÞ (5) ρA 2 ¼ κ AG  2 ∂x ∂x ∂t EI


 ∂2 ψ ∂w ∂w  ¼  κ AG ψ þT ∂x ∂x ∂x2

(6)

which are combined to yield the governing equation of motion for an undamped shear beam in terms of only one variable: ∂2 w ρEI ∂4 w ∂4 w ∂2 w EI ∂2 q  þ EI 4 þ T 2 ¼ q  κ AG ∂x2 ∂x ∂x ∂t 2 κ G ∂x2 ∂t 2

ρA

(7)

where the cable parameters are density ρ, area A, bending stiffness EI, shear modulus G and shear factor κ , and tension in the cable is T. To use the distributed transfer function method, Eq. (7) is transformed into the Laplace domain and rearranged so that the highest derivative is set equal to the other terms. For the undamped case of the shear cable in tension, this results in the equation
 ρA ρ 2 T Q 1 W″ þ  s  Q″ (8) W″″ ¼  s2 W þ EI EI EI κ AG κG This transformed equation of motion is now equivalent to the form ∂ ηðx; sÞ ¼ FðsÞηðx; sÞ þQ ðx; sÞ ∂x

(9)

where ηðx; sÞ is the Laplace transform of the solution vector and F(s) is the fundamental matrix for the system in which system properties are incorporated. For the undamped case with shear effects and tension, the fundamental matrix is 2 3 0 1 0 0 6 0 0 1 07 6 7 6 7 (10) FðsÞUndamped ¼ 6 0 0 0 17 6 7 4 ρA 2 5 ρ 2 T s 0 s  0  EI EI κG This matrix has two non-zero second-order terms in the bottom row; in contrast, the Euler–Bernoulli formulation has only one term for an untensioned beam and the Timoshenko formulation has two terms, but the two terms contain fourth-order and second-order polynomial expressions in s, leading to greater complexity. As previous studies have shown that the additional complexity of the Timoshenko model does not lead to much greater accuracy, but significantly increases computation time, the shear beam model is considered sufficient. Tension is included because slack cables with zero tension have extreme nonlinearities in their behavior, so lightly tensioned cables were tested to reduce these nonlinearities and the model needed to reflect the test reality. The fundamental matrix forms the core of the DTFM method and includes the physical characteristics of the cable as the coefficient values. The solution to the state space equation for the response of the system is Z 1     ηðx; sÞ ¼ G x; ξ; z v ξ; s dξ þ Hðx; sÞγ ðsÞ; x ϵ ð0; 1Þ (11) 0

where 8  1 MðsÞe  FðsÞξ ; ξ o x   < eFðsÞx MðsÞ þ NðsÞeFðsÞ G x; ξ; z ¼   :  eFðsÞx MðsÞ þ NðsÞeFðsÞ  1 NðsÞeFðsÞð1  ξÞ ; ξ 4 x

(12)

and  1 Hðx; sÞ ¼ eFðsÞx MðsÞ þNðsÞeFðsÞ

(13)   where eFðsÞx is the fundamental matrix using the state space matrix F(s) created from the equation of motion, and G x; ξ; z and Hðx; sÞ known as the transfer functions of the subsystem. MðsÞ and NðsÞ are boundary condition matrices for each side determined from the state-space equation MðsÞηð0; sÞ þ NðsÞηð1; sÞ ¼ γðsÞ:

(14)

However, this is only the solution for a single subsystem, and the value of the DTFM for cabled structure research is the ability to connect many subsystems to model a cable attached at many points. Thus, for any number of interconnected Please cite this article as: K.S. Spak, et al., Modeling vibration response and damping of cables and cabled structures, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.009i

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subsystems, displacement and strain vectors are described by   αðx; sÞ ¼ W ðx; sÞ W 0 ðx; sÞ T   ϵðx; sÞ ¼ W″ðx; sÞ W″0 ðx; sÞ T  T and the solution vector is ηðx; sÞ ¼ αT ðx; sÞ ϵT ðx; sÞ . An internal force vector is assumed to be

σðx; sÞ ¼ EðsÞϵðx; sÞ

(15)

(16)

where E(s) is a constitutive matrix, which is the elastic modulus in the case of a beam model. Set   γðsÞ ¼ αð0; sÞ αð1; sÞ T " #  0 0 I2 0 MðsÞ ¼ ; NðsÞ ¼ I2 0 0 0

(17)

where α are nodal displacement vectors at each end of a subsystem and I2 is a 2  2 identity matrix that either links subsystems or incorporates boundary conditions and constraints as described later. Constraints such as springs between subsystems can be incorporated through either the boundary condition matrices or the global stiffness matrix created through subsystem assembly. The subsystems are assumed to have a length of one unit, so the fundamental matrix must be multiplied by the length of the subsystem. Combining all of these equations and partitioning the G and H matrices appropriately yields solutions of Z 1     αðx; sÞ ¼ Gα x; ξ; s f ξ; s dξ þ Hα0 ðx; sÞα0 ðsÞ þ Hα1 ðx; sÞα1 ðsÞ (18) 0

σðx; sÞ ¼ EðsÞ

Z

1 0

    Gϵ x; ξ; s f ξ; s dξ þ Hϵ0 ðx; sÞα0 ðsÞ þ Hϵ1 ðx; sÞα1 ðsÞ

(19)

Once the nodal displacements are known, the response of the system is completely determined. To determine the nodal displacements, the subsystems must be assembled to make a stiffness-like matrix as follows. Assuming the total system has N nodes, located at xk ; k ¼ 1; 2…N where the subsystems are connected or terminated, the displacement vector of the system at location xk is called uk ðsÞ: Multiple subsystems can be connected at each node, as well as pointwise constraints of the form  C k ðsÞuk and external forces. Force balance at the node xk requires Q A ðsÞ þ Q B ðsÞ þ ⋯ þ Q D ðsÞ  Ck ðsÞuk ðsÞ þ pk ðsÞ ¼ 0

(20)

where Q vectors are the vectors of the forces applied at node k by the subsystems and pk is the vector of nodal external forces. The force Q A ðsÞ for subsystem A is Q A ðsÞ ¼ R A σA ðxk ; sÞ

(21)

where RA is a coordinate transformation matrix for that subsystem. The subsystem nodal displacement vectors αA ðxl ; sÞ and αA ðxi ; sÞ are related to global nodal displacements ul ðsÞ and ui ðsÞ by coordinate transformation matrices detailed in [8] so that combining the solution and coordinate transformation equations yields A

Q A ðsÞ ¼ KAll ðsÞul ðsÞ  KAli ðsÞui ðsÞ  f l

(22) N

where the elements of the stiffness matrix include transformation matrices T determined using the transfer function: KAli ðsÞ ¼ RA Hϵ0 ðxl ; sÞTA ;

KAll ðsÞ ¼ RA Hϵ1 ðxl ; sÞSA

N

and S

for each subsystem, and are (23)

However, these equations assume that xi oxl , and must be rederived otherwise. Deriving the equations for reversed subsystems results in different combinations of the elements of the H transfer matrix with the transformation matrices. After the force balance is determined for each node, an equilibrium equation similar to the typical finite element method can be formulated as KðsÞuðsÞ ¼ qðsÞ

(24)

where the stiffness matrix, displacements, and forces are in global terms. This equation can be solved for the displacements of each subsystem, which can then be substituted back into the solution equations to determine the response of each subsystem. To get eigenvalues and the resulting natural frequencies, the external forces are set to zero so that KðsÞuðsÞ ¼ 0

(25)

and the nontrivial solution for uðλk Þ is found by solving det½K ¼ 0 to give the natural frequencies. The displacements are calculated for each natural frequency, and these displacements can be substituted into the solution equation, again assuming applied external forces are 0, to give the mode shape for each subsystem for each natural frequency. Frequency response functions are developed by assuming that a harmonic forcing function is applied at frequency ω, so the nodal Please cite this article as: K.S. Spak, et al., Modeling vibration response and damping of cables and cabled structures, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.009i

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Fig. 1. Equivalent model for cable test set-up.

Fig. 2. Typical cable attachment point with cable tie fastened to TC105 tab.

Please cite this article as: K.S. Spak, et al., Modeling vibration response and damping of cables and cabled structures, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.009i

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displacements are determined as uðjωÞ ¼ K  1 ðjωÞqðjωÞ

(26)

where the forcing function is assumed to be a harmonic input. Again, the displacements are substituted into the solution equation to get the frequency response as a function of jω. Mode shapes for the system are determined by substituting the natural frequency values for s back into the characteristic equation to solve for the displacement values (η) and then putting them into the solution equation

ηðx; sÞ ¼ expðFðsÞxÞηð0; sÞ;

0 rx r L

(27)

to get a function of x for each s value. Further details about the use of the distributed transfer function for multiple subsystems are available in [9]. Once the fundamental matrix is determined, the incorporation of the system constraints are the next step. Fig. 1 shows the model for the cable as a beam, mounted to a test fixture at two points using a common method of attachment in which the cable is cable-tied to a TC105 mounting tab, as shown in Fig. 2. According to a previous study and verified by these authors, accurate representation of a cable-tie connection could not be modeled with pinned or fixed boundary conditions, so a spring model was used for the connection points [10]. For this DTFM model, both translational and rotational spring and damper models can be used to characterize the attachment connection. Experiments were conducted to determine the coefficient inputs for the model. Effect of spring stiffness values for the connection point is investigated in [11]. For the cable model, the attachment points connect the cable to ground, which is modeled with the DTFM as an extremely large mass. The springs and dampers of the attachment point are included in the force balance by adding a constraint matrix to the global stiffness matrix of the form 2 3 Is2 ðcθ s þkθ Þ 0 6 Is2 þ cθ þkθ 7 6 7 Ci ðsÞ ¼ 6 (28) 7 4 ms2 ðcs þkÞ 5 0 2 ms þ cs þ k where linear and rotational stiffness and damping terms are included, and c; cθ ; k; and kθ are the linear damping, rotational damping, linear stiffness, and rotational stiffness coefficients of the attachment point mechanism, which must be determined experimentally. For the cable attachments to ground, both m representing the rigid mass and I representing the mass inertia must be very large to approximate the constraint condition correctly. Since the constraint matrix multiplies the displacement vector, it can be added into the global stiffness matrix when the force balance at each node representing each attachment point is entered as given in Eq. (21). 2.2. Damped cable models The cable model that takes shear effects and tension into account has been introduced and can now be appended to include damping as well. Damping terms can be included as shear forces, or as additional terms acting on derivatives of the beam's response, or even as a variable bending stiffness term [1]. The simplest damping mechanism to include is viscous damping, sometimes known as motion based damping, in which the damping is directly proportional to the beam cross section velocity or its angular speed. Beam models with viscous damping are common, and are included here not as a novel concept, but to present the full range of damping mechanisms and show how they appear in the final DTF model in comparison to the hysteretic model. A viscous damping term is added to Eqs. (6) and (7), and the equations are combined as previously to yield a new, viscously damped equation of motion and fundamental matrix: EI ρ EIcv EIq″ € þ EIw″″ þ Tw″  _ þ cv w _ ¼ q w″ w″ κG κ AG κ AG 2 3 0 1 0 0 6 0 0 1 07 6 7 6 7 FðsÞViscous ¼ 6 0 0 0 17 6 7 4 ρA 2 cv 5 ρ 2 T cv s  s 0 s  þ s 0  EI EI κ AG EI κG

ρAw€ 

(29)

(30)

A more sophisticated type of damping is hysteretic damping, which deals with internal friction and viscoelastic damping by including a time hysteresis term in the stress expression [12]. For hysteretic damping, the initial starting equations are the same, as is the equation for shear, but for a viscoelastic material, stress is Z t σ ðx; t Þ ¼ Eϵðx; t Þ  g ðt  τÞϵðx; τÞdτ (31) 0

So the moment becomes M o ¼ EI ψ ðx; t Þ0 

Z 0

t

g ðt  τÞwðx; τÞ00 dτ

(32)

Please cite this article as: K.S. Spak, et al., Modeling vibration response and damping of cables and cabled structures, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.009i

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Table 1 Average measured values for each cable type.

Number of wires Mass (kg) Length (m) Outer diameter (m) Lay angle (deg) Lay angle (rad)

1  7 Cable

1  19 Cable

1  48 Cable

7  7 Cable

7 0.0708 0.7692 0.0074 19.6 0.3417

19 0.1905 0.7782 0.0127 16.5 0.2873

48 0.4481 0.7744 0.0204 18.4 0.3217

49 0.4944 0.7744 0.0216 17.4 0.3037

Using the form in which hysteresis is applied to w, substitution of Eq. (33) into Eqs. (2) and (3) yields

ρAw€ ðx; t Þ ¼  κ AGψ ðx; tÞ0 þ κ AGwðx; tÞ00 þ qðx; t Þ EI ψ ðx; tÞ00 þ

Z

t 0

g ðt  τÞwðx; τÞ000 dτ  κ AGψ ðx; t Þ þ κ AGwðx; tÞ0 ¼ 0

(33) (34)

The tension term is added, as well as the viscous damping terms:

ρAw€ ðx; t Þ ¼  κ AGψ 0 ðx; t Þ þ κ AGw00 ðx; t Þ þ cv w_ þ qðx; t Þ EI ψ 00 ðx; t Þ þTw0 þ

Z

t 0

g ðt  τÞw000 dτ  κ AGψ ðx; t Þ þ κ AGw0 ðx; t Þ ¼ 0

Eqs. (36) and (37) are combined into a single equation to eliminate the rotation variable: Z t EI ρ EIc EIq00 _ þTw00 ¼ q  ρAw€  w€ 00 þEIw0000 þ v w_ 00  g ðt  τÞw0000 dτ þcv w κ AG κ AG κG 0

(35) (36)

(37)

Again, the Laplace transform is taken and the equation is rearranged to match the form in which the highest derivative is set equal to the other terms:

ρAWs2 

EI ρ EIcv 1 EIQ ″ W″s2 þEIW″″ þ W″s  GðsÞW″″ þ cv Ws þ TW″ ¼ Q  s κ AG κ AG κG

   ρ 2 EIcv EIQ ″  ρAs2 þcv s W þ EI κ G s  κAGs  T W″ þQ  κ AG   W 0000 ¼ EI  1s GðsÞ

This fulfills the requirement for the state space equation and the hysteretically 2 0 1 0 6 0 0 1 6 6 FðsÞHysteretic ¼ 6 0 0 0 6 EI ρ 2 EIcv 4  ρAs2  cv s κ G s  κ AG s  T 0 EI  1GðsÞ EI  1GðsÞ s

damped fundamental matrix is 3 0 7 07 7 17 7 5 0

(38)

(39)

(40)

s

The hysteretic damping term G(s) was included in the form given by [13] for viscoelastic materials: GðsÞ ¼

αD s 2 þ γ D s s2 þ β D s þ δD

(41)

It is clear that the various damping terms add complexity in varying degrees; terms that add an entire term to the damping matrix (i.e. terms multiplying W0 and W000 ) will slow down computation time considerably. Another consideration is the complexity of the symbolic s term; higher orders of s make for more difficult calculation of the determinant in the solution step. This can be avoided by using a numeric solution, but the accuracy of the natural frequency calculation is then dependent on the step size of the numerical method. Different damping mechanisms can be incorporated by including the appropriate damping terms in the equation of motion. An advantage to the DTFM approach is that adding different damping terms only changes the fundamental matrix, but the solution method remains the same. 2.3. Cable parameters The models presented are applicable for any shear beam that may experience damping or tension. However, to use these models for cables specifically, the model inputs ρ, A, EI, and G must be determined to describe the cable. Because the model assumes a homogenous cross section, these parameters must be calculated to be equivalent to the cable as a whole. Cable parameters are generally found by performing dynamic tests and backing out the beam parameters from the dynamic tests. However, since a related goal is the ability to calculate cable frequency bounds a priori, with no dynamic experimental testing, the authors have determined equivalent cable properties based on the geometry and component materials of the Please cite this article as: K.S. Spak, et al., Modeling vibration response and damping of cables and cabled structures, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.009i

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Table 2 Calculated cable properties used as inputs for each cable type for cables in 2 pt cable model. Calculated 1  7 Cable cable property Value for min frequency

Value for max frequency

Value for min frequency

Value for max frequency

Value for min frequency

Value for max frequency

Value for min frequency

Value for max frequency

ρA (kg/m) EI (kg m3/s2) κAG (kg m/s2)

0.092 0.37 2.13  104

0.448 1.09 7.87  104

0.245 1.18 5.78  104

0.997 4.73 2.02  105

0.580 5.1 1.46  105

1.21 2.14 2.27  105

0.640 2.30 1.49  105

0.145 0.34 2.69  104

1  19 Cable

1  48 Cable

7  7 Cable

Table 3 Equation coefficients based on model inputs for each cable type. Equation coefficient

ρA EI ρ κG T EI 1 κAG

1  7 Cable

1  19 Cable

1  48 Cable

7  7 Cable

Value for min frequency

Value for max frequency

Value for min frequency

Value for max frequency

Value for min frequency

Value for max frequency

Value for min frequency

Value for max frequency

0.43

0.25

0.41

0.21

0.21

0.11

0.56

0.28

5.37  10

6

4.32  10

6

5.69  10

6

4.24  10

6

4.94  10

6

3.97  10

6

5.31  10

6

4.29  10  6

13.09

12.03

4.08

3.77

0.94

0.87

2.08

1.93

3.72  10  5

4.69  10  5

1.27  10  5

1.73  10  5

4.95  10  6

6.85  10  6

4.41  10  6

6.71  10  6

Fig. 3. Cabled beam for experimental testing, side and top views.

Fig. 4. Top view of model for cabled beam with excitation point shown by arrow.

cables [14]. Four types of cable were modeled and subsequently tested: single-stranded cables in 1  7, 1  19, and 1  48 configurations, and a multi-stranded cable made up of seven 1  7 strands, known as a 7  7 cable. The mass, length, outer diameter, and lay angle were measured for each of the five experimental samples of each type of cable and averaged. These cable measurements were used for property calculations as described in [14] and are listed in Table 1. The equivalent cable properties were determined through a combination of cable measurement and weighted average of the material properties of the constituent materials making up the cable; since the rule of mixtures was used for the density and the moduli values, and area could be calculated using either overall area or individual wire area, the cable properties span a range. Lower and upper limits based on uncertainty and alternative methods in the measurements yield a range of input values rather than a single value for each property. These upper and lower limits are the maximum and minimum property parameters used in the model, presented in Table 2. All cables were under 4.45 N of tension during testing, so all model trials used T ¼4.45. Table 3 presents the ranges of values for the equation coefficients, showing that similar sized cables may have very different equation coefficients. Please cite this article as: K.S. Spak, et al., Modeling vibration response and damping of cables and cabled structures, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.009i

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3. Cabled beam model development Modeling a cable as a beam allows for the inclusion of bending stiffness, tension, and damping mechanisms as shown in the previous section. To extend the cable model to a cabled structure model, the characteristics of the attachment points must be determined, and both the structure and cable modeled. Fig. 3 shows the top-view of the cabled beam, while Fig. 4 shows the idealized model view. The cable is attached at five points using the same TC105 tabs and cable ties that were used for the cable attached to ground. The cable and beam are each divided into seven subsystems, with nodes at the ends, each attachment point, and the excitation point, for a total of 14 subsystems and 16 nodes. For the cabled beam model, the cable is modeled as the damped shear beam developed previously, and the aluminum beam is modeled as an Euler–Bernoulli beam. Thus, the subsections for the model have different fundamental matrices as determined using the equations of motion as described previously. The cable properties are determined by methods which require only the basic measurements of the cable and knowledge about the wire composition. For the cabled beam model, the cable is connected to additional model subsystems representing the beam, rather than the ground. For this constraint matrix, the nodes of the cable and the nodes of the beam must be included. Thus, the constraint matrix must fulfill the form (

Q cj

"

) ¼

Q cl

Cjj ðsÞ Clj ðsÞ

Cjl ðsÞ Cll ðsÞ

#(

uj ðsÞ

) (42)

ul ðsÞ

where the forces are represented by Q, and j and l are the nodes for the two subsystem nodes connected through the spring attachment. The C matrices are the transfer functions that describe the attachment point with a linear and rotational spring, determined to be " Ci ðsÞ ¼

0

kθ þ cθ s

k þ cs

0

# (43)

The attachment force affects both the beam and the cable, so the force balance for each node connected to an attachment point, whether it is on the beam or the cable, will have components from the attachment constraint matrix. The input values for the cable properties and attachment point coefficients for the cabled beam model are listed in Table 4. The area and density of each cable is the same as calculated for cables in the two-point fixture, but bending stiffness is slightly higher for the cabled beam model due to the cable having less curvature since it is attached at multiple points with shorter span lengths than the two-point fixture. Table 4 Calculated properties used as inputs for each cable type for cabled beam model. 1  7 Cable

Calculated cable value

Area (m2) Density (kg/m3) Bending stiffness (cabled beam) (kg m3/s2) Shear rigidity (Pa) Cxn stiffness value, rotational Cxn stiffness value, linear

A ρ EI

1  48 Cable

7  7 Cable

Value for min freq.

Value for max freq.

Value for min freq.

Value for max freq.

Value for min freq.

Value for max freq.

Value for min freq.

Value for max freq.

4.35  10  5 3323 0.46

3.44  10  5 2677 0.51

1.27  10  4 3528 1.94

9.33  10  5 2624 2.16

3.27  10  4 3049 16.82

2.36  10  4 2456 18.25

3.66  10  4 3295 3.48

2.41  10  4 2654 3.83

2.13  104

7.87  104 2 N/m rad

5.78  104

2.02  105 2 N/m rad

1.46  105

2.27  105 2 N/m rad

1.49  105

κAG 2.69  104 2 N/m rad kθ k

1  19 Cable

1  104 N/m rad

6  104 N/m rad

1  106 N/m rad

6  105 N/m rad

Fig. 5. Four types of cable investigated; 1  7, 1  19 and 1  48 single stranded cables and 7  7 multi-stranded cable. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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4. Experimental testing To gauge the effectiveness of the cable and cabled beam models, experimental trials were run for each case. Five samples were acquired for each of four types of typical spaceflight cable for a total of 20 cable samples. Fig. 5 shows the three types of

Fig. 6. Cable test set-up; cable attached to ground at two points and modal shaker used for excitation.

Fig. 7. Cabled beam test set-up; cable attached to beam with cable ties and beam suspended for free end condition simulation.

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single-stranded cables and one multi-stranded cable that were investigated. In Fig. 5, the blue rings indicate 1  7 cable strands to show the difference between single stranded cables, in which each successive layer is a concentric ring of individual wires around the previous layer, and multi-stranded cables, in which strands make up the layers. All cable samples were made of MIL 27500TG2T14 wires, contra-helically twisted and machine wrapped with Kapton. Cables were excited with a shaker producing white noise at 0.1–0.4g and the response was measured at the driving point with a non-contact laser vibrometer. Fig. 6 shows the cable test set-up. Each cable sample was tested at least 15 times, for a total of 70þ trials per cable type. Due to the flight cables' inherent variation in wire alignment, wire friction, and attachment, the experimental natural frequency results have variation, spanning a range of values for each mode, which is what the model is designed to capture by running both upper and lower bound property values. The range of experimental frequencies for each mode was recorded, as well as the average natural frequency for the first four modes. Experimental testing is discussed in further detail in [15]. For the cabled beam, the cable of each type that showed the least variation from trial to trial was attached to an aluminum beam. Fig. 7 shows the cabled beam test set-up. Full scans of the beam were run with a laser vibrometer to measure the response along the entire beam so that mode shapes could be compared to the model as well as natural frequencies. 5. Results To show the value of the model, the natural frequency and damping results are compared to experimental data. Frequency response functions were visually compared, and mode shapes and natural frequencies were compared directly 20 Experimental FRF Data Hysteretically Damped 1X7 Cable Model Model Frequency Range

Magnitude (dB)

10

0

-10

-20

-30

-40 0

20

40

60

80

100

Frequency (Hz) Fig. 8. Comparison of experimental data and hysteretically damped cable model for the 1  7 cable in two-point fixture configuration. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

0 Experimental FRF Data

-5

Hysteretically Damped Cable Model Model Frequency Range

-10

Magnitude (dB)

-15 -20 -25 -30 -35 -40 -45 -50

0

20

40

60

80

100

Frequency (Hz) Fig. 9. Comparison of experimental data and hysteretically damped cable model for the 1  19 cable in two-point fixture configuration. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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between experimental and model values. As listed in Table 2, the minimum frequency bound occurs when the ρA term is a maximum, the bending stiffness term EI is a minimum, and the shear term depends on the maximum area. The maximum frequency bound occurs when the ρA term is a minimum, the bending stiffness term is a maximum, and the shear term depends on the minimum area. Improved property calculation will result in a narrower range for the property values. The goal is to have the modeled natural frequency range (determined with parameters calculated only from basic cable measurements) just cover the range of frequencies shown by experimental data from the cables. 5.1. Cable model results Figs. 8–11 show the fitted damped cable model FRFs with the experimental scan average FRFs for each cable in the twopoint fixture. The model FRF is based on the cable inputs for the minimum frequency response; the darker blue bars above each frequency show the damped frequency range. Using the cable inputs for maximum frequency response shifts the frequency response to the upper bound, but only the minimum response is shown for clarity. Varying the cable inputs within the cable parameter range and adjusting damping values can result in a closer match between experimental and analytical data, but the intent here is to capture the wide range of cable responses using straightforward bounds for the cable inputs, which is achieved. Results are best for the larger, more beam-like cables. In each of the damped cable model comparisons, the amplitudes for the first three frequencies are adequately matched, with the exception of the 1  19 cable, which had a second frequency that was higher than predicted. Not only are the amplitudes matched, the damped frequency ranges were generally able to span the experimental frequency. The lower frequencies were better matched by the minimum model, while higher frequencies tended toward the maximum value model. Thus, the minimum to maximum model span seems to be a decent method to cover the first 3–5 modes of the cable. If prediction of fewer modes was required, the minimum model could be used with confidence for the first two modes in each case. 0 Experimental FRF Data Hysteretically Damped Cable Model Model Frequency Range

Magnitude (dB)

-10

-20

-30

-40

-50

-60 0

50

100

150

200

Frequency (Hz) Fig. 10. Comparison of experimental data and hysteretically damped cable model for the 1  48 cable in two-point fixture configuration. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

10 Experimental FRF Data Hyst. Damped Cable Model Model Frequency Range

Magnitude (dB)

0 -10 -20 -30 -40 -50 -60 0

20

40

60

80

100

120

140

160

Frequency (Hz) Fig. 11. Comparison of experimental data and hysteretically damped cable model for the 7  7 cable in two-point fixture configuration. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Recalling the form of hysteretic damping from Eq. (41), the gamma term had the greatest effect on matching the second and third mode amplitudes once the first mode amplitude was matched sufficiently with the alpha coefficient. The delta term could shift the model frequencies to some degree without significantly changing the amplitude of the model. A more sophisticated curve fitting program could probably improve the damping parameters determined for the two-point cables, especially if more terms were added to the hysteretic damping form to improve the curve fit over a larger frequency. The GHM method was initially developed for finite element analysis, so it is typical to include a sum of terms of the form in GðsÞ, instead of the single term as used here since the finite element method was not used. Summing more terms in the GHM damping model could improve the damping fit for the cable models. 0.9 0.8

Mode 1, 5.7 Hz

0.7 2, 19.8 Hz Model Modes

0.6 3, 42.6 Hz

0.5

4, 73 Hz

0.4 0.3

5, 110 Hz 0.2 6, 149 Hz

0.1

9 Hz

19 Hz 40 Hz 61 Hz 118 Hz 140 Hz Experimental Modes

Fig. 12. Representative MAC for 1  7 cable in two-point fixture, showing agreement of both mode shapes and natural frequencies.

1X7 Experimental Average DTFM Model Average Major Structure Frequency Ranges Additional Cable-Induced Frequency Ranges

Magnitude (dB)

-20 -40 -60 -80 -100

0

50

100

150

200

250

300

350

400

450

500

Frequency (Hz)

Fig. 13. Damped average cabled beam model compared to experimental data for 1  7 cabled beam, with ranges for the natural frequency values shown as bars.

1X19 Experimental Average

0

DTFM Model Average Major Structure Frequency Ranges Additional Frequency Ranges

Magnitude (dB)

-20 -40 -60 -80 -100

0

50

100

150

200

250

300

350

400

450

500

Frequency (Hz)

Fig. 14. Damped average cabled beam model compared to experimental data for 1  19 cabled beam, with ranges for the natural frequency values shown as bars.

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To ensure that frequencies were being compared correctly, modal assurance criteria (MAC) were run for each cable to compare the mode shapes from the model and the experimental scans. Fig. 12 shows the MAC for the 1  7 cable with the minimum model and experimental mode frequencies listed; it is clear that the mode shapes and frequencies are in good agreement, since the diagonal of the MAC has values close to 1 and the off-diagonal values are close to zero. 5.2. Cabled beam results The model was first compared run for an experimentally tested solid Acetron GP rod to ensure that the model was sound before introducing the uncertainty of the cable parameters. The rod-on-beam model was able to capture the experimentally determined natural bending frequencies of the structure within 10 percent, acceptable considering the simplicity of the model and assumptions about the connection stiffness for the rod. Figs. 13–16 show the results for the cabled beam model. Due to the complexity of the cabled beam transfer functions, only the damped average (middle) value transfer function is shown, with the minimum and maximum frequency range shown with horizontal bars. The average experimental data is shown with a black dashed line, the model average is shown as a solid line, and the range of natural frequency values based on the minimum and maximum cable input parameters are shown as range bars above the respective frequencies. Just as with the cable models, it is important to compare mode shapes as well as natural frequencies to ensure legitimate comparison between model and experimental data since frequencies that appear to be similar in value may actually represent different modes. This is especially important for the cabled beam as the number of frequencies predicted with the minimum and maximum cable parameter values as inputs will not necessarily be the same; the minimum cable parameter values tend to have additional lower modes. Only mode shapes that were clearly matched for both minimum and maximum inputs are shown as additional frequency ranges in Figs. 13–16. Fig. 13 presents the damped average transfer function response for the 1  7 cabled beam. Note the very narrow range for the first bending frequency (around 50 Hz) and the eighth frequency (around 450 Hz). In fact, all of the bending frequencies that occur similar to the bare beam are modeled with a very narrow result; the additional modes that are introduced due to -10 -20

Magnitude (dB)

-30 -40 -50 -60 -70 -80 1X48 Experimental Average DTFM Average Model Major Structure Frequency Ranges Additional Frequency Ranges

-90 -100 -110 0

50

100

150

200

250

300

350

400

450

500

Frequency (Hz)

Fig. 15. Damped average cabled beam model compared to experimental data for 1  48 cabled beam, with ranges for the natural frequency values shown as bars.

-10 -20

Magnitude (dB)

-30 -40 -50 -60 -70 -80 7X7 Experimental Average

-90

DTFM Model Average

-100

Major Structure Frequency Ranges Additional Frequency Ranges

-110 0

50

100

150

200

250

300

350

400

450

500

Frequency (Hz)

Fig. 16. Damped average cabled beam model compared to experimental data for 7  7 cabled beam, with ranges for the natural frequency values shown as bars.

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the addition of the cable on the structure have a much wider range, which corresponds with the uncertainty in the cable parameters. Overall, the model predicts the cabled beam frequencies very well, and manages to capture the additional frequencies due to the addition of the cable to the structure. This is especially useful when compared to the current method of modeling cables as distributed mass. 5.3. Comparison of cabled beam model to distributed mass model To show the value of the DTFM cabled beam model, it must show improvement when compared to existing models. The best current commonly used model for cabled structure vibrations is the distributed mass model; in this model, the mass of the cable is added to the beam as a distributed mass by changing the density of the beam wherever cables are present. Note that this is a better technique than the previously used lumped mass model, where the cable mass was simply added at the center of gravity of the structure. The distributed mass model simply shifts the natural frequencies of the bare structure; the greater the mode number, the larger the frequency decrease due to the additional mass. This method is acceptable for the first one or two modes of a small mass addition, but does not capture the additional modes that arise due to the interaction between the cable and the host structure, fails to predict frequencies accurately at higher modes, and does not capture the difference between the stiff single stranded large cable and the much more flexible multi-stranded large cable that were observed in the experimental trials. To illustrate these points, the distributed mass model is compared to the average DTFM cabled beam model and experimental data in the following figures. Fig. 17 compares the experimental data and both models for the 1  7 cable. For this small cable, the differences in the model frequencies for the first two bending modes are nearly negligible, although the DTFM model does capture additional modes and shows closer agreement for the third and fourth bending mode frequencies. The medium sized 1  19 cable shows similar results that show similar agreement between the models for the first two modes, and improved prediction of higher modes by the DTFM model.

1X7 Experimental Average

-20

Distributed Mass Model

Magnitude (dB)

DTFM Model Average

-40 -60 -80 -100 0

100

200 300 Frequency (Hz)

400

500

Fig. 17. Comparison of distributed mass model and DTFM cabled beam model with experimental data for 1  7 cable.

0 1X48 Experimental Average Distributed Mass Model DTFM Average Model

Magnitude (dB)

-20 -40 -60 -80 -100 0

100

200

300

400

500

Frequency (Hz) Fig. 18. Comparison of distributed mass model and DTFM cabled beam model with experimental data for 1  48 cable.

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0

7X7 Experimental Average Distributed Mass Model DTFM Average Model

Magnitude (dB)

-20 -40 -60 -80 -100

0

100

200

300

400

500

Frequency (Hz) Fig. 19. Comparison of distributed mass model and DTFM cabled beam model with experimental data for 7  7 cable.

Figs. 18 and 19 compare the distributed mass model and DTFM cabled beam model for the 1  48 and 7  7 cables, respectively. Fig. 18 clearly shows the addition of modes around 350 Hz, which is not captured at all by the distributed mass model. The mode at 350 Hz appears to be a single mode due to averaging of the experimental trials, but in reality the cabled beam trials showed multiple modes in this area, showing good agreement with the DTFM average model. In addition, all of the bending modes are predicted more accurately by the DTFM cabled beam model for the 1  48 cable. The real strength of the DTFM model is shown in Fig. 19, the comparison of the models and experimental data for the 7  7 model. Here, the heavy but very flexible cable has several additional modes in the 100–200 Hz range, and all bending frequencies are predicted closely by the DTFM model. While the 1  48 and 7  7 distributed mass models are nearly identical, the DTFM model manages to capture the differences between the cabled beams more clearly and more closely to the experimental data. For larger cables, the distributed mass model deviates from experimental data widely for the major structural modes, up to 16 percent difference, whereas the maximum difference between a range boundary and the experimental data is less than 6 percent for the DTFM cabled beam model. For the smaller cables with less mass, the distributed mass model predicts the frequency shift adequately for the lowest modes, but for higher modes and all modes of larger cables, the frequency shift is not captured as correctly as by the DTFM model. The DTFM model also provides a range of values, which can capture the inherent variability of cables more completely; for clarity, only the average DTFM value was shown in the comparison plots. The distributed mass model does not capture additional modes caused by the addition of the cables and is not as accurate at predicting natural frequencies for cables of varying stiffnesses. Therefore, there is value in the development of the DTFM cabled beam model to correct these inadequacies. 6. Conclusion A method for modeling cabled structures has been developed and experimentally validated. The method extends previous work in modeling combined lumped parameter and distributed parameter systems by including shear, tension, damping, and rotational elements. The method is then applied to several spacecraft quality cables attached to a beam, and validated experimentally. Whether the reader's interest is in cables or in cabled structures, prediction of vibration response depends not only on the cable properties, but the ability to model the attachment points. The presented models are a useful extension of the DTFM, incorporate linear and rotational connection stiffness and damping, and can be applied to any beam-like system for which equivalent homogenous parameters can be determined. Adding shear effects, tension, and damping to the existing Euler–Bernoulli DTFM model adds complexity, but also increases the utility of the model to predict behavior for more materials. Because cables are inherently unpredictable due to the presence of internal friction and wire slippage, experimental data collected from cables shows a range of frequencies for each mode. Thus, having a model to predict a range of frequencies may actually more useful for a wider variety of cable types. The DTFM cabled beam model shows improvement over the distributed mass model both in terms of natural frequency prediction and ability to identify locations of additional modes due to cable attachment. Although the distributed mass model may be useful for small cables, the DTFM cabled beam model can incorporate cable bending stiffness and shear rigidity, and thus provides more information and a more accurate prediction for larger cables and higher modes of all size cables.

Acknowledgments The first author thanks the NASA Space Technology Research Fellowship program for generous support and the San Gabriel Valley AIAA Chapter and Virginia Space Grant Consortium for additional funding. Cables for testing were provided at Please cite this article as: K.S. Spak, et al., Modeling vibration response and damping of cables and cabled structures, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.009i

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cost by Southern California Braiding, Co. The third author gratefully acknowledges the support of AFOSR Grant number FA9550-10-1-0427 monitored by Dr. David Stargel. This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. References [1] K.S. Spak, G.S. Agnes, D.J. Inman, Cable modeling and internal damping deveopments, Applied Mechanics Reviews 65 (2013) 1–18, http://dx.doi.org/ 10.1115/1.4023489. [2] D.A. Castello, C.F.T. Matt, A validation metrics based model calibration applied on stranded cables, Journal of the Brazilian Society of Mechanical Scientists and Engineers 33 (2011) 417–427, http://dx.doi.org/10.1590/S1678-58782011000400005. [3] J.C. Goodding, E.V. Ardelean, V. Babuska, L.M. Robertson, S.A. Lane, Experimental techniques and structural parameters estimation studies of spacecraft cables, Journal of Spacecraft and Rockets 48 (2011) 942–957, http://dx.doi.org/10.2514/1.49346. [4] K.S. Spak, G.S. Agnes, D.J. Inman, Comparison of damping models for space flight cables, Conference Proceedings of the Society for Experimental Mechanics, Garden Grove, CA, February 2013, pp. 183–194. 10.1007/978-1-4614–6555-3_21. [5] B. Yang, C.A. Tan, Transfer functions of one-dimensional distributed parameter systems, Journal of Applied Mechanics 59 (1992) 1009–1014, http://dx. doi.org/10.1115/1.2894015. [6] A.M. Majeed, M. Al-Ajmi, A. Benjeddou, Semi-analytical free-vibration analysis of piezoelectric adaptive beam using the distributed transfer function approach, Structural Control and Health Monitoring 18 (2011) 723–736, http://dx.doi.org/10.1002/stc.467. [7] S. Timoshenko, D.H. Young, W. Weaver, Vibration Problems in Engineering, 4th ed. John Wiley & Sons, New York, 1974. [8] D. Sciulli, Dynamics and Control for Vibration Isolation Design, PhD Thesis, Virginia Tech, 1997. [9] B. Yang, Distributed transfer function analysis of complex distributed parameter systems, Journal of Applied Mechanics 61 (1994) 84–92, http://dx.doi. org/10.1115/1.2901426. [10] D.M. Coombs, J.C. Goodding, V. Babuska, E. Ardelean, L.M. Robertson, S.A. Lane, Dynamic modeling and experimental validation of a cable-loaded panel, Journal of Spacecraft and Rockets. 48 (2011) 958–973, http://dx.doi.org/10.2514/1.51021. [11] J. Choi, D.J. Inman, Spectrally formulated modeling of a cable-harnessed structure, Journal of Sound and Vibration. 333 (2014) 3286–3304, http://dx.doi. org/10.1016/j.jsv.2014.03.020. [12] H.T. Banks, R.H. Fabiano, Y. Wang, D.J. Inman, H. Cudney, Spatial versus time hysteresis in damping mechanisms, Proceedings of the 27th Conference on Decision and Control, Austin, TX, 1988, pp. 1674–1677, 10.1109/CDC.1988.194613. [13] M.I. Friswell, D.J. Inman, M.J. Lam, On the realization of GHM models in viscoelasticity, Journal of Intelligent Material Systems and Structures 8 (1997) 986–993, http://dx.doi.org/10.1177/1045389  9700801106. [14] K.S. Spak, G.S. Agnes, D.J. Inman, Cable parameters for homogenous cable-beam models for space structures, Conference Proceedings of the Society for Experimental Mechanics, Orlando, FL, February 2014, pp. 7–18. 10.1007/978-3-319–04546-7_2. [15] K.S. Spak, G.S. Agnes, D.J. Inman, Toward modeling of cable harnessed structures: cable damping experiments, Proceedings of the 54th AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamics, and Materials Conference, Boston, MA, April 2013, doi: 10.2514/6.2013–1889.

Please cite this article as: K.S. Spak, et al., Modeling vibration response and damping of cables and cabled structures, Journal of Sound and Vibration (2014), http://dx.doi.org/10.1016/j.jsv.2014.10.009i