O. Barry 1 Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON M5S 3G8, Canada e-mail: oumar.barry@utoronto.ca J. W. Zu Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON M5S 3G8, Canada D. C. D. Oguamanam Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, ON M5B 2K3, Canada Forced Vibration of Overhead Transmission Line: Analytical and Experimental Investigation An analytical model of a single line transmission line carrying a Stockbridge damper is developed based on the Euler–Bernoulli beam theory. The conductor is modeled as an axi- ally loaded beam and the messenger is represented as a beam with a tip mass at each end. Experiments are conducted to validate the proposed model. An explicit expression is pre- sented for the damping ratio of the conductor. Numerical examples show that the proposed model is more accurate than the models found in the literature. Parametric studies indi- cate that the response of the conductor significantly depends on the excitation frequency, the location of the damper, and the damper parameters. [DOI: 10.1115/1.4027578] Keywords: stockbridge damper, Strouhal frequency, messenger, rated tensile strength, overhead transmission line 1 Introduction Aeolian vibration of overhead transmission lines, also referred to as conductors, is a major factor that contributes to power out- ages. This type of vibration is wind-induced and the wind speed ranges between 1 and 7 m/s. The vibration is observed in 3–150 Hz frequency range and has peak-to-peak amplitudes of up to one conductor diameter. Aeolian vibration causes fatigue dam- age of the point of contact between the conductor and the suspen- sion clamp. This can be reduced or eliminated by minimizing the amplitude of vibration near the clamp. It is commonplace to pro- tect conductors from fatigue failure by attaching Stockbridge dampers near the clamps. The effectiveness of this external damp- ing device is dependent on their position on the conductor, their overall characteristics, and the characteristics of the conductor. The study of Aeolian vibration of overhead transmission lines abound in the literature. The most commonly used approach is the energy balance method (EBM) [1–4], where the vibration level is evaluated by determining the balance between the energy imparted to the conductor by the wind and the energy dissipated by the conductor (via conductor self-damping) and the added dampers. Another approach for predicting the response of the con- ductor is based on impedance models [5–7]. The single conductor is usually modeled as a cable, and the Stockbridge damper is rep- resented by a single concentrated force on the conductor. This concentrated force is obtained experimentally. Other methods employed to study the vibration of transmission lines include matrix transfer by Hardy and Noiseux [8], the statis- tical method by Noiseux et al. [9], and the approach of multiphy- sics by Tsui [10]. While the simplicity of the aforementioned methods is a major attraction, their robustness suffers in that the complex nature of the coupled-dynamics is not well reflected. The limitation of the dynamics to only one-way coupling between the conductor and damper, a situation where the dynamics of the damper influence that of the conductor but not the converse, is worthy of further investigation. Gonc¸alves et al. [11] employ experiments and theoretical mod- els to better understand the system from the viewpoint of a vortex-induced motion or vibration phenomenon. An attempt at modeling a two-way coupling scenario was reported in Refs. [12] and [13], where both the conductor and damper were modeled as one unified system in order to account for their two-way coupling. However, the conductor self-damping was ignored and the analy- sis was based on the finite element method (FEM). The finite ele- ment model was rather complicated and computationally intensive. The present study is aimed at addressing these short- comings by including the conductor self-damping and by present- ing an analytical approach that yields exact solutions with minimal complications. Experiments were conducted to determine the conductor self-damping and to validate the analytical model. Parametric studies were performed to investigate the effect of the magnitude and location of the damper on the response. The role of the Strouhal frequency on the vibration response was also examined. 2 Governing Equations A schematic of the conductor with a damper is shown in Fig. 1. A close-up view of the conductor and damper deformation is depicted in Fig. 2. Following Ref. [14], the equations of motion are given as m c € w ci þ E c I c w 0000 ci  Tw 00 ci ¼ 0 (1) m m € w  c1 þ ð1Þ ðiþ1Þ € w 0 c1 L mi þ € w mi   þ E m I m w 0000 mi ¼ 0 (2) where w  m1 ðw  m2 Þ is the transverse displacement of the right-end (left-end) counterweight, m 1 (m 2 ) is the tip mass on the right-hand (left-hand) side; L m1 ðL m2 Þ is the length of the messenger on the right-hand (left-hand) side; m c (m m ) is the mass per unit length of the conductor (messenger); m m1 (m m2 ) is the mass of the messen- ger on the right-hand (left-hand) side; T denotes the conductor tension; and E c I c (E m I m ) is the flexural rigidity of the conductor Fig. 1 Schematic of a single conductor with a Stockbridge damper 1 Corresponding author. Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 9, 2013; final manuscript received April 29, 2014; published online May 22, 2014. Assoc. Editor: Mohammed Daqaq. Journal of Vibration and Acoustics AUGUST 2014, Vol. 136 / 041012-1 Copyright V C 2014 by ASME Downloaded From: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 09/24/2014 Terms of Use: http://asme.org/terms