Vibrations of a Taut String With Nonlinear Damper

CHAPTER 5

Vibrations of a Taut String with Nonlinear Damper. I: Equivalent Linear Solutions It is important to recognize some limitations in the performance of passive linear dampers for cable vibration suppression. Firstly, the levels of supplemental damping that can be provided are fairly low (typically less than 2% of critical), because in practical situations the distance from end of the cable to the damper attachment point must be quite small.

Secondly, the optimal damping ratio can be achieved in only one mode of

vibration; if the damper is designed for optimal performance in a particular mode, it will be effectively too stiff in higher modes and too compliant in lower modes. In designing a linear damper, it is then necessary to select the mode in which optimal performance is desired, and the damping performance will be suboptimal in other modes. The first limitation on the attainable values of damping is not a problem in many cases, because it has been observed that fairly low levels of damping appear to be sufficient to suppress the problematic vibrations. As discussed in Chapter 2, Irwin (1997) suggested that rain-wind vibrations could be avoided if the Scruton number is greater than 10; the Scruton number is defined as Sc = m /( D2), where m = cable mass per


length, is the modal damping ratio, is the air density, and D is the cable diameter. This


criterion has gained fairly widespread acceptance in design of dampers for cable vibration suppression. As noted by Irwin (1997), satisfying this criterion requires only a small amount of damping for typical stay properties (e.g., 172






0.5%). Consequently, the modest

levels of supplemental damping that can be provided by a passive linear damper are often more than sufficient for vibration suppression. The second limitation of mode-dependence is potentially a more significant concern, because it is not yet clear how to identify in which mode the optimal performance should be achieved. Analysis of the measured data at the Fred Hartman Bridge indicates that rain-wind induced vibrations occur predominantly in a number of the lower modes, but which specific mode(s) will be excited for a given stay can not yet be predicted a priori, and designing the damper for optimal performance in a specific mode may render the stay vulnerable to vibrations in other modes.

For example,

designing a damper for optimal performance in a lower mode to mitigate rain-wind induced vibrations may leave the stay susceptible to other excitation mechanisms, such as vortex-induced vibrations, in the higher modes, while designing the damper for optimal performance in a higher mode may leave it susceptible to deck-induced vibrations in the fundamental mode. Other types of damping devices have been investigated as alternatives to the passive linear damper for cable vibration suppression. Carne (1981) investigated the performance of a friction damper, extending the analytical solution for a linear damper using an equivalent energy dissipation criterion, and discussed the implementation of a friction damper for suppressing vibrations of a guy cable supporting a wind turbine. Friction dampers were also investigated numerically by Kovacs et al. (1999), and friction dampers were implemented for cable vibration suppression on the Uddevalla Bridge (Hjorth-Hansen et al. 2001). Semiactive dampers have been investigated using numerical

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simulation, and they have been shown to achieve moderate reductions in response levels over the passive linear damper under white noise excitation (Johnson et al. 1999). Nonlinear dampers of the power-law type are often used in seismic protection applications (e.g., Makris et al. 1997); this type of damper provides a dissipative force that is proportional to the piston velocity raised to a positive exponent . As discussed by Taylor and Constantinou (1995), fluid dampers with specially designed orifices can be used to obtain this type of nonlinear force-velocity relationship, and a fairly wide range of the damper exponent approximately 0.2 ≤

can be achieved; the practical range for fluidic dampers is

≤ 1.8, and inertial fluid dampers are characterized by =2. It is of

interest to investigate the performance of this type of damper for stay-cable vibration suppression. Unintentional damper nonlinearities can also occur, such as a viscous damper with a friction threshold. In this chapter, equivalent linear solutions are developed for two types of nonlinear dampers: a power-law damper and a viscous damper with a friction threshold. In the case of the linear damper, the eigenvalue problem was formulated by enforcing equilibrium of forces at the damper location. For the power-law damper, a solution is first developed using a similar approach, by minimizing the RMS error in the force equilibrium over one half-cycle of oscillation, assuming that the solution has the same form as in the linear case. A more general “equivalent viscous damper” formulation is then developed by equating the work done in one cycle of oscillation by a general nonlinear damper to that done by a linear damper, assuming nearly sinusoidal oscillations. This “equivalent viscous damper” formulation is then applied to develop an

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alternative approximate solution for the power-law damper and then an approximate solution for a viscous damper with a friction threshold. In this chapter, and in Chapter 6, only the simpler taut-string model of the cable is considered; the influence of bending stiffness on nonlinear damper performance will be a subject of future investigation.

5.1 Power-Law Damper The problem under consideration is depicted in Figure 5.1. A nonlinear, powerlaw damper is attached to a taut cable at an intermediate point, dividing the cable into two segments, where

2

>

1

. The present investigation focuses on vibrations in the first few

modes for relatively small values of

1

/ L , for which the damper-induced frequency

shifts are small. Since the supplemental damping ratios are of direct interest and the relevant excitation mechanisms are not well understood, so that an appropriate forcing function is not apparent, free vibrations are considered rather than forced vibrations. Considering free vibrations also facilitates comparison with previous investigations of linear dampers and allows an approximate analytical solution to be formulated by building on the exact analytical formulation of the eigenvalue problem for the linear damper.

L 1

T x1

m c, β

2

T x2

Figure 5.1: Taut Cable with Nonlinear Power-Law Damper 175

The nonlinearity of the damper is introduced as a constraint at the damper location, while the vibrations of the cable itself are still governed by the linear wave equation. Due to the nonlinearity introduced by the damper, the decaying oscillations of the cable-damper system are not simply harmonic oscillations with exponentially decaying amplitude. Rather, the rate of decay and the frequency of oscillation vary with amplitude. A single-mode equivalent linear solution will be sought by assuming that for oscillations at a given amplitude in a particular mode, the vibrations of the cable can be expressed in a separable form in space and nondimensional time over each segment, as in the linear damper case: y k ( xk , τ ) = Yk ( xk )e λτ

(5.1)

where yk(xk ,t) is the transverse deflection, xk is the coordinate along the cable chord axis in the kth segment, τ = ω o1t is a nondimensional time, where ω o1 = (π L) T m is the undamped fundamental natural frequency of the cable, and

is an amplitude-dependent

complex eigenvalue, from which the frequency and the effective damping ratio


can be

obtained. The equation of motion governing the cable vibrations and the displacement boundary conditions then yield the following expression for the complex mode shapes Yk ( xk ) = Ak sinh(πλxk / L)

(5.2)

in which the complex coefficients A1 and A2 are related by the continuity condition: A1 sinh(πλ

1

/ L) = A2 sinh(πλ

2

/ L)

In the case of a linear damper, permissible values of the complex eigenvalue

(5.3) were

obtained by enforcing equilibrium of forces at the damper location, where the force in the damper Fd is balanced by a force due to the tension in the cable and the discontinuity of

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slope at the damper location FT. The force in the nonlinear power-law damper can be expressed as β

Fd = c v sgn(v)

(5.4)

where v is the velocity of the cable at the damper attachment point. This force-velocity relationship is depicted schematically in Figure 5.2 for =1, >1, 1

Fd

β=1 β