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Geometry
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
The most important examples of roulettes arise when M is a circle and C is a straight line or a circle, but an interesting additional example is provided by the catenaryy=acosh(x/a) $ y = a {\text{cosh }}(x/a) $ , which arises by rolling the parabola y = x2/(4a) on the x‐axis with pole the focus of the parabola (that is,P=(0,a) $ is, P = (0, a) $ in the initial position). The catenary is the shape taken under the action of gravity by a chain or string of uniform density whose ends are held in the air.
Anchoring systems
Published in White David, Cassidy Mark, Offshore Geotechnical Engineering, 2017
Floating facilities have historically been secured by catenary moorings. A catenary is a mathematical definition of the curve assumed by a perfectly flexible uniform inextensible string when suspended from its ends. A catenary mooring is, therefore, one that takes up a catenary curve between the floating facility and the seabed. A catenary mooring touches down on the seabed in advance of the anchor, such that the angle of the anchor chain at the mudline is close to zero and the anchor is only subjected to horizontal forces. In a catenary mooring, most of the restoring forces are generated by the weight of the mooring line. Catenary moorings derive their compliance from the change in suspended line weight and thus line tension as the grounded line lifts off and is replaced on the seabed due to motion of the floating system. A typical mooring configuration has 100 mm diameter rope, steel wire or polyester, stretching down to heavy chain links that can weigh more than 200 kg each. The chains connect at the seabed to an anchor system. Mooring spreads have at least eight separate lines coming from the floating system and some have up to 16. The Na Kika FPS, located in 2,200 m depth of water, used 3,200 m of wire and 580 m chain for each of 16 separate lines (Figure 7.3). The anchors to which the mooring chains are attached sit 2.5 km away from the FPS.
Catenary Action of Gfrp Sandwich Panels With Cfrp Catcher System
Published in Alphose Zingoni, Insights and Innovations in Structural Engineering, Mechanics and Computation, 2016
Catenary curve basis for modeling assumes the material is so thin it can be modelled by a curve. It is also assumed to be flexible such that any force of tension exerted by the member is parallel to the member. This means the member is in pure tension. The catenary curve itself describes the mathematical curve a hanging flexible wire or chain makes when it’s supported at the ends, acted on by a uniform gravitational force (Weisstein, N.D.). The catenary curve is described by equation 1, where t is time, and a is a parameter that describes how quickly the curve “opens up”() y(x)=a*cosh(x/a)
A disturbance observer based lumped-mass catenary model for active pantograph design and validation
Published in Vehicle System Dynamics, 2023
Huayu Duan, Roger Dixon, Edward Stewart
More than half of railway lines in the EU have been equipped with electrified power supplies [1]. In the electrification systems that use a catenary (i.e., overhead line equipment), the pantograph is a critical component. It is situated on top of the vehicle and extended to connect the train to the catenary in order to transmit energy, as shown in Figure 1. Therefore, it is essential that the pantograph is kept in good condition in order to maintain the correct dynamic interaction and contact to enable current collection. Given this, perhaps it is not surprising that the contact behaviour between pantograph head and catenary contact wire is widely studied using numerical computer simulation. Generally, this coupled system is modelled using three components: pantograph, catenary system, and interaction.
An improved full Fourier series method approaching the stitched catenary in high-speed railway
Published in International Journal of Rail Transportation, 2022
Jiangwen Wang, Guiming Mei, Liantao Lu
The dynamic performance of the pantograph-catenary system caused by the stitch wire is analysed. The pantograph is the mass-spring-damp model [35–38], and the penalty method is adopted for contact [8,39–41]. The parameters are from EN 50,318 [1]. The statistical values of the dynamic contact force and the uplift at support are shown in Figure 15. Basically, the standard deviation (STD) decreases when the length is longer and increases when the tension force is higher, as shown in Figure 15(a). However, due to wave propagation, the variation of the STD is not linear. The maximum and minimum values change slightly with the variation of the stitch wire, as shown in Figure 15(b). The maximum uplift becomes higher if the length and tension values are larger, as shown in Figure 15(c).
A new methodology to study the pantograph–catenary dynamics in curved railway tracks
Published in Vehicle System Dynamics, 2020
Pedro Antunes, Jorge Ambrósio, João Pombo, Alan Facchinetti
The over-head contact line, also known simply as catenary, is composed of a set of suspended cable wires and its supporting elements that run along the railway track and carry the electrical current, which in turn is collected by the pantograph mounted on the top of the railway vehicle. The energy collection is assured by the sliding contact between the pantograph and the catenary contact wire. The interaction contact force developed must fulfil tight operational requirements that ensure that a reliable and efficient energy collection is achieved. Operating the pantograph–catenary interface at a low average contact force increases the susceptibility to contact loss incidents with consequent arcing, which in turn leads to high electro-mechanical wear and the deterioration of the functional conditions of both the catenary and the pantograph. High contact forces, on the other hand, results in mechanical wear of the contact elements increasing the frequency of the maintenance cycles and risk of failure [1]. Certainly, the present need to increase the rail network capacity and its interoperability puts extra demands on these systems, [2,3], for which the energy collection ability remains a limiting factor of the current railway vehicles operational speeds [4,5].