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

Effect of polymer coatings on fatigue strength of aluminum alloy 2024 box beams

Previous investigators have shown that polymer coatings raise the fatigue strength of metals tested in air to about the same level as that of uncoated specimens tested in vacuum. The results are given of tests to determine if a polymer coating would improve the fatigue strength of built-up aluminum alloy members simulating aircraft construction. Aluminum alloy 2024-T4 riveted box beams were subjected to constant amplitude fatigue tests in air as well as in salt water fog. The coating did not improve the fatigue strength of beams tested in either environment. This is believed to result from the fact that most failures originated at rivet holes, which were isolated from both the coating and the environment

Two-parameter nonsmooth grazing bifurcations of limit cycles: classification and open problems

This paper proposes a strategy for the classification of codimension-two grazing bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a non-generic way. Several such codimension-one events have recently been identified, causing for example period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincaré map from a neighbourhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the the grazing cycle is itself degenerate (e.g. non-hyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that have discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.

Transitioning to Adulthood in Sweden: Comparing the Priorities of Immigrant Youths with Disabilities and Caregivers, from Middle-Eastern Countries

Background: There is limited knowledge about perceived challenges during the transition to adulthood among immigrant youths who are originally from Arabic-speaking countries but now residing in Sweden. Aim: The aim of this study was to describe self-identified problems encountered by immigrant youths with disabilities from Middle Eastern countries who were living in Sweden during their transition to adulthood and to compare these descriptions to the problems identified by their caregivers.Methods: Seventeen semi-structured interviews using the Canadian Occupational Performance Measure were conducted with 17 immigrant youths with intellectual disabilities aged 13-24 years and 16 caregivers, originally from Middle Eastern countries. The participants’ prioritized problems were categorized using the International Classification of Functioning, Disability and Health-Children & Youth Version, focusing on Activity and Participation.Results: A difference in priorities during transition was found when comparing the youths’ and the caregivers’ views. Most of the youths’ priorities were identified in the chapter “Major Life Area” about basic economic transactions and seeking employment, whilst the caregivers thought their youths’ prioritized “self-care”.Conclusion and implications for practice: Planning the best transition for immigrant youths with disabilities involves enabling them to identify their own preferences and needs while collaborating with caregivers and taking into consideration the cultural norms and traditions of collective caregiving

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

Chatter, sticking and chaotic impacting motion in a two-degree of freedom impact oscillator

We consider the dynamics of a two-degree of freedom impact oscillator subject to a motion limiting constraint. These systems exhibit a range of periodic and nonperiodic impact motions. For a particular set of parameters, we consider the bifurcations which occur between differing regimes of impacting motion and in particular those which occur due to a grazing bifurcation. Unexpected resonant behavior is also observed, due to the complexity of the dynamics. We consider both periodic and chaotic chatter motions and the regions of sticking which exist. Finally we consider the types of chaotic motion that occur within the parameter range. We discuss the possibility in relating successive low velocity impacts, especially with respect to possible low dimensional mappings for such a system