Dynamics beyond dynamic jam; unfolding the Painlev\'e paradox singularity

This paper analyses in detail the dynamics in a neighbourhood of a G\'enot-Brogliato point, colloquially termed the G-spot, which physically represents so-called dynamic jam in rigid body mechanics with unilateral contact and Coulomb friction. Such singular points arise in planar rigid body problems with slipping point contacts at the intersection between the conditions for onset of lift-off and for the Painlev\'e paradox. The G-spot can be approached in finite time by an open set of initial conditions in a general class of problems. The key question addressed is what happens next. In principle trajectories could, at least instantaneously, lift off, continue in slip, or undergo a so-called impact without collision. Such impacts are non-local in momentum space and depend on properties evaluated away from the G-spot. The results are illustrated on a particular physical example, namely the a frictional impact oscillator first studied by Leine et al. The answer is obtained via an analysis that involves a consistent contact regularisation with a stiffness proportional to $1/\varepsilon^2$. Taking a singular limit as $\varepsilon \to 0$, one finds an inner and an outer asymptotic zone in the neighbourhood of the G-spot. Two distinct cases are found according to whether the contact force becomes infinite or remains finite as the G-spot is approached. In the former case it is argued that there can be no such canards and so an impact without collision must occur. In the latter case, the canard trajectory acts as a dividing surface between trajectories that momentarily lift off and those that do not before taking the impact. The orientation of the initial condition set leading to each eventuality is shown to change each time a certain positive parameter $\beta$ passes through an integer

The comfortable roller coaster -- on the shape of tracks with constant normal force

A particle that moves along a smooth track in a vertical plane is influenced by two forces: gravity and normal force. The force experienced by roller coaster riders is the normal force, so a natural question to ask is: what shape of the track gives a normal force of constant magnitude? Here we solve this problem. It turns out that the solution is related to the Kepler problem; the trajectories in velocity space are conic sections.Comment: 10 pages, 4 figure

Static deformation of heavy spring due to gravity and centrifugal force

The static equilibrium deformation of a heavy spring due to its own weight is calculated for two cases. First for a spring hanging in a constant gravitational field, then for a spring which is at rest in a rotating system where it is stretched by the centrifugal force. Two different models are considered. First a discrete model assuming a finite number of point masses connected by springs of negligible weight. Then the continuum limit of this model. In the second case the differential equation for the deformation is obtained by demanding that the potential energy is minimized. In this way a simple application of the variational calculus is obtained.Comment: 11 pages, 2 figure

Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators

International audienceGrazing bifurcations are local bifurcations that can occur in dynamical models of impacting mechanical systems. The motion resulting from a grazing bifurcation can be complex. In this paper we discuss the creation of periodic orbits associated with grazing bifurcations, and we give sufficient conditions for the existence of a such a family of orbits. We also give a numerical example for an impacting system with one degree of freedom

Verdsettelse av Dot.Com- boblen

Formålet med denne masteroppgaven har vært å undersøke om IT-bransjen var overpriset i januar 2000. Bakgrunnen for valget av dette tidspunktet var at IT-aksjer opplevde et kraftig fall kort tid etter, nærmere bestemt fra 10. mars samme år. Årsakene til dette fallet blir drøftet i innledningen av oppgaven. Basert på nedgangen i verdien til IT-markedet er hypotesen at bransjen på dette tidspunktet var overpriset. For å kunne teste hypotesen verdsettes noen utvalgte IT-selskaper i januar 2000 med hensyn på fremtidig realiserte regnskaper og sammenlignes mot aksjeprisene på samme tid. Teorien bak verdsettelsen presenteres i kapittel 2, herunder drøfting av de ulike verdsettelsesmodellene og avkastningskrav. Teoridelen avsluttes ved at hypotesen blir presentert, denne hypotesen ønsker jeg å teste for å kunne besvare problemstillingen. Metodene og designet som brukes for å teste hypotesen blir drøftet i kapittel 3, i dette kapittelet blir selskapene som skal danne grunnlaget for verdiestimering av IT-bransjen valgt ut og verdsettelsesmodellen blir valgt. I kapittel 4 blir dataene og forutsetningene som danner utgangspunktet for analysen fremstilt. Forutsetningene for beregning av kontantstrøm til totalkapitalen blir presentert og avkastningskravet til totalkapitalen blir beregnet. I kapittel 5 presenteres resultatene av analysen. I dette kapittelet blir verdien av selskapenes aksjeverdi vurdert opp mot verdien av selskapene basert på de realiserte inntjeningene fra 2000-2009. Resultatene av analysen viser at de realiserte inntjeningene indikerte en samlet verdi tilsvarende omtrent 2,66 milliarder kroner for de 9 utvalgte selskapene, mens aksjeprisene til disse selskapene på samme tidspunkt viste en samlet verdi på omtrent 7,37 milliarder kroner. Dette gir en overprising av selskapene på 177 %. For å kartlegge enkelte parametrers betydning for verdsettelsen har jeg foretatt en sensitivitetsanalyse, denne analysen presenteres i kapittel 6

Simulation and stability analysis of impacting systems with complete chattering

International audienceThis paper considers dynamical systems that are derived from mechanical systems with impacts. In particular we will focus on chattering— accumulation of impacts—for which local disconti-nuity mappings will be derived. We will first show how to use these mappings in simulation schemes, and secondly how the mappings are used to calculate the stability of limit cycles with chattering by solving the first variational equations

On normal form calculations in impact oscillators

International audienceNormal form calculations are useful for analysing the dynamics close to bifurcations. However, the application to non-smooth systems is a topic for current research. Here we consider a class of impact oscillators, where we allow systems with several degrees of freedom as well as nonlinear equations of motion. Impact is due to the motion of one body, constrained by a motion limiter. The velocities of the system are assumed to change instantaneously at impact. By de­fining a discontinuity mapping, we show how Poincaré mappings can be obtained as an expansion in a local coordinate. This gives the mapping the desired form, thus making it possible to employ standard techniques. All calculations are algorithmic in spirit, hence computer algebra routines can easily be developed

Sn-Aided Joining of Cast Aluminum and Steel Through a Compound Casting Process

Obtaining a strong bond between aluminum and steel is challenging due to poor wettability between aluminum melt and steel and brittle intermetallic phases forming in the interface. In this research, a novel coating method, namely hot dipping of Sn, has been developed to treat the steel insert surfaces. Results show that without preheating the mold or Sn-coated insert, a thin, crack-free, and continuous metallurgical bonding layer was achieved in the A356 aluminum/steel compound castings. Intermetallic structures forming in the interface have been characterized in detail. The Sn-coating layer completely melted and mixed with the liquid aluminum during the casting process. The reaction layer at the aluminum/steel interface is composed of ternary Al–Fe–Si particles and a thin layer of binary Al5Fe2 phase with thickness less than 1 µm. A small fraction of dispersed Sn-rich particles was observed distributing in the reaction layer and adjacent to eutectic Si particles in the A356 alloy. A sessile drop wetting test showed that Sn-coated steel substrates can be well wetted by aluminum melt. The improved wettability between A356 alloy melt and steel was attributed to the penetration and breaking of the aluminum oxide layer at the surface of the aluminum droplets by liquid Sn.publishedVersio