Experimental and theoretical investigation of the energy dissipation of a rolling disk during its final stage of motion

This paper is concerned with the dominant dissipation mechanism for a rolling disk in the final stage of its motion. The aim of this paper is to present the various dissipation mechanisms for a rolling disk which are used in the literature in a unified framework. Furthermore, new experiments on the ‘Euler disk' using a high-speed video camera and a novel image analysis technique are presented. The combined experimental/theoretical approach of this paper sheds some more light on the dominant dissipation mechanism on the time-scale of several second

The historical development of classical stability concepts: Lagrange, Poisson and Lyapunov stability

A brief historical overview is given which discusses the development of classical stability concepts, starting in the seventeenth century and finally leading to the concept of Lyapunov stability at the beginning of the twentieth century. The aim of the paper is to find out how various scientists thought about stability and to which extent their work is related to the stability concepts bearing their names, i.e. Lagrange, Poisson and Lyapunov stability. To this end, excerpts of original texts are discussed in detail. Furthermore, the relationship between the various works is addresse

Andronov-Hopf Bifurcations in Planar, Piecewise-Smooth, Continuous Flows

An equilibrium of a planar, piecewise-$C^1$, continuous system of differential equations that crosses a curve of discontinuity of the Jacobian of its vector field can undergo a number of discontinuous or border-crossing bifurcations. Here we prove that if the eigenvalues of the Jacobian limit to $\lambda_L \pm {\rm i} \omega_L$ on one side of the discontinuity and $-\lambda_R \pm {\rm i} \omega_R$ on the other, with $\lambda_L, \lambda_R >0$, and the quantity $\Lambda = \lambda_L / \omega_L -\lambda_R / \omega_R$ is nonzero, then a periodic orbit is created or destroyed as the equilibrium crosses the discontinuity. This bifurcation is analogous to the classical Andronov-Hopf bifurcation, and is supercritical if $\Lambda < 0$ and subcritical if $\Lambda >0$.Comment: laTex, 18 pages, 8 figure

The Euler-Maupertuis principle of least action as variational inequality

International audienceStarting from Hamilton's Principle, the current paper discusses how we can derive the Euler-Maupertuis Principle of Least Action in the context of non-smooth dynamics. This variational principle allows us to directly obtain the space curve $y(x)$ of a point-mass in a potential field $V (x, y)$ without referring to the temporal dynamics. This paper generalises the Euler-Maupertuis Principle of Least Action to systems with impact by formulating the principle as a variational inequality

Hamilton’s Principle as Variational Inequality for Mechanical Systems with Impact

International audienceThe classical form of Hamilton's principle holds for conservative systems with perfect bilateral constraints. Several attempts have been made in literature to generalise Hamilton's principle for mechanical systems with perfect unilateral constraints involving impulsive motion. This has led to a number of different variants of Hamilton's principle, some expressed as variational inequalities. Up to now, the connection between these different principles has been missing. The aim of this paper is to put these different principles of Hamilton in a unified framework by using the concept of weak and strong extrema. The difference between weak and strong variations of the motion is explained in detail. Each type of variation leads to a variant of the principle of Hamilton in the form of a variational inequality. The conclusion of the paper is that each type of variation leads to different necessary and sufficient conditions on the impact law. The principle of Hamilton with strong variations is valid for perfect unilateral constraints with a completely elastic impact law, whereas the weak form of Hamilton's principle only requires perfect unilateral constraints and no condition on the energy

Effects of Uncoupling the North Sea Link : An empirical analysis of risks and premiums in the two day-ahead auctions in bidding area N0 2

In this thesis, we study the impact of the current auction solution on the North Sea Link (NSL), an interconnector between the electricity systems of Great Britain (GB) and Southern Norway (NO2). Due to Brexit, interconnector capacity is allocated through an implicit day-ahead auction, which closes before the regular NO2 (SDAC) auction. Consequently, traders in NO2 can decide to participate in two consecutive auctions. To address this decision problem, we develop a forecast model to predict day-ahead NO2 (SDAC) prices. Using these forecasts, we approximate the premium that the marginal trader requires to take positions in the first auction (NSL). Contrary to initial expectations, we find that traders are, on average, pricing in a positive premium in the NSL auction. This results in a systematic difference between the NO2 and NSL prices. We further examine the drivers behind the premium to determine when and why it arises. Our findings indicate that when traders have less confidence in their NO2 price forecasts, they demand a larger premium for trading with the NSL. On average, the marginal trader in our model requires a 0 . 2 2 /MWh higher premium for each incremental increase in the prediction interval of the point forecast. In an efficient market with risk-neutral traders, such premiums would not exist as the traders would increasingly take advantage of the arbitrage opportunity. Therefore, we argue that traders in the NSL auction are risk-averse. The risk premium comes with a commercial cost for the interconnector owners. We find that Statnett has lost 2 7 million in congestion revenues since the opening of the NSL on the 1st of October 2021 to the 15th of January 2023. This amounts to approximately 10% of the total congestion revenues accumulated during the period. Overall, we find that the implicit auction solution on the NSL leads to a redistribution of socio-economic welfare from the interconnector owners to the traders in the form of a systematic risk premium. We argue that this finding should be acknowledged in future research on the socio-economic welfare gains from the NSL and in future projects considering the use of an implicit auction to connect uncoupled markets.nhhma