A separation-based analytical framework for seismic responses of weakly-coupled electrical equipment

https://doi.org/10.1016/j.jsv.2020.115768Get rights and content

Highlights

  • Differences between weakly-coupled systems and general structures were clarified.

  • Dynamic transmission proves to be the nature of the weakly-coupled effect.

  • Local vibration modes and nonproportional damping were used in this framework.

  • Shaking table testing was carried out to validate the separation-based framework.

Abstract

Weakly-coupled electrical equipment, consisting of a series of supporting posts interconnected by flexible connection parts, is commonly seen in power systems. Compared with general structures, weakly-coupled electrical equipment has two main characteristics—(a) the structural integrity of the equipment is weak due to the flexible connection parts and (b) material damping of the supporting posts and connection parts can be very different. Based on these characteristics, global vibration modes and proportional damping should be cautiously used. This restricts applications of common seismic calculation methods for the weakly-coupled equipment. Therefore, the authors developed a separation-based analytical framework for the weakly-coupled electrical equipment, which is presented herein. In this framework, usage of nonproportional damping and local vibration modes of posts and connection parts can be achieved. This framework also facilitated a modeling analysis, which investigate the mechanism of the weakly-coupled effect. It was found that the dynamic transmission of connection parts is the nature of the weakly-coupled effect, and the proportional damping can produce unacceptable errors for the weakly-coupled equipment. To further validate the analytical framework, we conducted shaking table testing on two insulator posts connected by a slidable busbar. The analytical method and conventional mode-superposition method were both employed to calculate the seismic responses of the specimen, and the calculation results were compared with the testing responses. Comparisons suggest the separation-based analytical framework is more applicable and effective for weakly-coupled electrical equipment.

Introduction

Seismic issues affecting lifeline systems continue to draw much attention all over the world [1,2], especially for the power system, which has proven vulnerable in an earthquakes [3]. In recent years, seismic performances of individual electrical components and equipment have been frequently studied [4], [5], [6]. Individual equipment is generally interconnected due to electrical requirements; thus, it is necessary to focus more on the system of the connected equipment. Weakly-coupled electrical equipment, consisting of a series of cantilever supporting posts interconnected by flexible connection parts. The connection parts are generally tubular busbar or flexible conductors, as shown in Fig. 1. During earthquakes, the coupling effect activated by the connection parts can amplify the seismic responses of some supporting posts and damage weakly-coupled electrical equipment. According to post-earthquake investigations [7], [8], [9], [10], [11], destruction of weakly-coupled equipment is not only frequent but can incur great economic loss.

Studies focusing on interconnected electrical equipment have found that some equipment with flexible connections often demonstrate weakly-coupled attributes. Filiatrault and Kremmidas [12] studied seismic induced interactions between equipment components connected by a tubular busbar, with which the dynamic response of one of the components were always amplified. Filiatrault and Stearns [13] studied equipment components connected by flexible conductors, and found that the fundamental frequencies of two connected components were not consistent and had different changing trends under various connection conditions. Mohammadi and Tehrani [14] examined the interaction between three interconnected electrical equipment components and concluded that the coupling effect was prominent when the fundamental frequencies of the connected components were different. Some researchers have also investigated the interior coupling effects of some self-connected electrical equipment, such as the surge arrester [15] and disconnect switch [16,17].

The conception of the ‘weakly-coupled’ and the coupling vibrations of subsystems interconnected by weak connectors can be seen in studies on the active devices of aircrafts, trains, and other vehicles [18,19]. However, the weakly-coupled structure has not yet been defined clearly in structural engineering; furthermore, the knowledge of mechanisms and characteristics of the weakly-coupled effect are also insufficient. The structural form of weakly-coupled equipment is seemingly similar to that of interconnected adjacent buildings, which is a research focus in the field of structural engineering [20], [21], [22]; however, there are some substantial differences between them. First, the connection parts of the weakly-coupled equipment are too flexible to maintain a uniform vibration among different supporting posts. The global vibration modes are not inconspicuous, and coupling vibrations among posts are marked [23,24]. Consequently, local vibration modes of the posts and the dynamic transmission effect of the connection parts play important roles in their seismic responses. However, vibrations of the connected buildings are generally required to be consistent and synchronous, and the weakly-coupled property is always avoided in conceptual design. Therefore, the dynamic transmission of the connection between buildings is generally slight.

Notably, this present study has found the dynamic transmission of the connection parts to be the nature of the weakly-coupled effect, which is the first point of concern. The second point of concern is that supporting posts and flexible connection parts of the weakly-coupled equipment are made from insulating material and metal material, respectively; thus, their damping properties are obviously different, i.e., nonproportional damping [25] (or nonclassical damping [26]) should be considered. Vibration behaviors of adjacent structures interconnected by passive or semi-active dampers or elements with large damping have been comprehensively studied in both analytical [27], [28], [29] and experimental [30] research. However, as connection parts with large damping can suppress their dynamic transmission effect and the weakly-coupled characteristics (explained in Section 3.1 and 3.2). In contrast, the weakly-coupled effect is more conspicuous and can enlarge coupling vibrations in a system with lightly damped connection parts.

Considering these two issues above, it is worthwhile to specially investigate the weakly-coupled system, to which the local vibration modes and the nonproportional damping are considered. However, solving the coupled equations of motion without global vibration modes and proportional damping is still a troublesome problem in the field of structural dynamics. Most current seismic designs and calculations for electrical equipment [31,32] basically follow the seismic codes for buildings [33], [34], [35] and adopt the mode-superposition based method characterized the usage of global vibration modes and proportional damping matrix (i.e., damping properties of every part of a system is uniform), which can cause systematic errors for weakly-coupled equipment. The step-by-step calculation based on an elaborate numerical model is an optional method, with which the usage of global modes and the proportional damping can be avoided. However, the weakly-coupled electrical equipment is generally sizable and lacks standard configurations; consequently, modeling and calculating with the step-by-step method can be arduous and time consuming. In addition, Kiureghian et al. [36] and Behnamfar et al. [37] both developed quick calculation methods for two supporting posts interconnected by a linear damped spring to consider the nonproportional damping of the posts and the spring. However, the global modes of the system were still employed, and the dynamic transmission effect of the connection part could not be fully considered with only a spring.

In the context of these issues, Section 2 presents a separation-based analytical framework. In this framework, a separation analysis was performed on a coupled system, and the distributed-parameter model was used to build controlling equations of the separated components to fully consider the dynamic transmission effect of connection parts. Furthermore, a frequency-domain method with substantial reduction—down to degrees of freedomwas developed to solve the complexly coupled controlling equations without the usage of local vibration modes and nonproportional damping. An analytical modeling analysis accompanies a discussion on the influences of the dynamic transmission effect and the nonproportional damping to weakly-coupled electrical equipment in Section 3. Structural identifications have been well developed in fields such as isolated structures [38,39] and structural damage detections [40,41]. Studies for adjacent structures with/without control devices, e.g., [42], [43], [44], were mainly focused on their modal parameters. Basili and Angelis [45] developed a general approach to identify adjacent structures with different control devices and refine corresponding numerical models. For weakly-coupled systems, equivalent stiffness and damping parameters of flexible connection parts are always difficult to determine with experimental data. In Section 3, an identification procedure based on the analytical framework has also been proposed. With this procedure, such physical parameters of weakly-coupled systems can be identified from transfer functions of connection parts defined in this paper. Section 4 presents shaking table testing on two insulator posts connected by the busbar to validate the analytical framework and to further the discussion.

Section snippets

Separation strategy of weakly-coupled electrical equipment

Based on the structural characteristics of the weakly-coupled equipment, two assumptions were employed to build the analytical framework as follows:

  • (a)

    Since the connection part is much more flexible than the supporting post, the moment and torque generated by the connection part acting on the top of the supporting post are ignored.

  • (b)

    Because the connection parts are too flexible to transmit transverse forces, the overall torsion-coupled vibration of the equipment can be ignored; thus, the equipment

Dynamic transmission effects of connection parts

The frequency-domain transfer functions of the CP, obtained in Section 2, can be used to investigate the characteristics and the formation mechanism of the weakly-coupled effect. For instance, given that the ith CP is symmetrical and homogeneous with a length of 8.5 m and a weight of 295 kg, it is also hinged at both ends. The damping ratios of the CP, ξc, are constant at 1%. Focusing on the in-plane motion, i.e., m = 1, we can calculate |Hic1(ω)| with a different generalized stiffness factor Si

Experimental validation of the analytical framework

To further validate the analytical framework built in Section 2, shaking table testing on two insulator posts connected by a tubular busbar was selected as a case study. In a previous study, a busbar was modeled by a linear component [36]; however, as the busbar was connected to equipment through various fittings, some equipment had complex mechanisms. More sufficient demonstrations are needed to support the rationality of linear modeling. In the present validation, the posts and the busbar

Conclusions

In this paper, a separation-based analytical framework for weakly-coupled electrical equipment is presented to avoid the usage of the global vibration modes and proportional damping. Analytical modeling analysis and shaking table testing validation were completed, and some key conclusions follows:

  • 1.

    The dynamic transmission effect of the CP is the nature of the weakly-coupled effect; thus, local vibration modes and nonproportional damping should be used to fully consider the dynamic transmission.

CRediT authorship contribution statement

Jiayi Wen: Conceptualization, Methodology, Formal analysis, Writing - original draft, Investigation. Qiang Xie: Resources, Supervision, Validation, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Funding: This work was supported by the National Natural Science Foundation of China [grant No.51878508], and the National Key R&D Program of China [grant No.2018YFC0809400].

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