|For some important knots closed-form solutions are presented for the holding forces which are needed to keep a knot in equilibrium for given pulling forces. If the holding forces become zero for finite pulling forces, the knot is self-locking and is called stable. This is only possible when first, the friction coefficient exceeds a critical value and second, when there is additional pressure on some knot segments sandwiched by surrounding knot segments. The number of these segments depends on the topology of the knot and is characteristic for it. The other important parameter is the total curvature of the knot. In this way, the complete frictional contact inside the knot is taken into account. The presented model can explain the available experiments. Accepted for publication in Archive of Applied Mechanics (2024).
|Two simple formulas for climbing fall forces for static and dynamic belays: A new impact force formula for movable belayers is presented and compared with the standard impact force formula.
|In sport climbing a common method of belaying is to use a static rope brake attached to the belayer’s harness, but the belayer can move freely. This paper investigates the dynamics of a climbing fall with such a belayer. The dynamics are nontrivial because of the belayer’s constraint to be always at or above his initial position. An exact solution for a linear elastic rope is presented. Compared to a fix-point belay, one obtains a considerable force reduction on the belay-chain. However, there is a trade-off of a longer stopping distance of both climber and belayer. In order to calculate the stopping distance, friction between rope and the top carabiner has been taken into account. This article is published in Journal of SPORTS ENGINEERING AND TECHNOLOGY (first published online February 15, 2021); DOI: 10.1177/1754337121992616.
Swinging on a long rope
|About the helplessness of a free-hanging climber or the inability to excite a pendulum oscillation on a long rope. The situation of a free-hanging climber is considered who tries to excite a pendulum oscillation on a rope from a rest position. It turns out that this is almost impossible for rope lengths larger than approx. 3 meters. The main reason is the low excitation frequency of longer ropes and the slow increase in the swing amplitude which strongly depends on the rope length.
|For the fracture process of a climbing rope, two mechanisms are responsible: plastic deformation and local damage of the contact zone between rope and anchor. These mechanisms are described by two analytical models represented by nonlinear difference equations. The plastic deformation equation can be linked to a catastrophe-theoretical model. From the equation describing local damage accumulation, the Palmgren-Miner rule can be derived. The used energy-based approach allows the combination of these models and thus the calculation of the number of falls to failure as a function of the ratio of fall energy/energy storage capacity. The behaviour of climbing ropes tested by subsequent UIAA falls can be quantitatively explained by these models. The article is published in the Journal of SPORTS ENGINEERING AND TECHNOLOGY (first published in November 29, 2018).
The physics of a climbing rope
|In this theoretical paper, the force-elongation behavior of a climbing rope in a heavy fall is investigated and compared with experiments. The experiments show that the state-of-the-art viscoelastic description of a climbing rope with time-independent friction is not able to explain the rope tension as a function of time. A proper description has to take into account time-delayed friction, i.e. a transition from a low-friction regime to strong friction near the force maximum which leads to a fast relaxation of the rope into its equilibrium position. Furthermore, a climbing rope has to be described by a nonlinear tension with increased stiffness for large elongations in order to agree with experiments with varying fall masses. Finally, observed second-mode force oscillations are explained by a continuum description of the rope taking into account its mass. This work is published in Journal of SPORTS ENGINEERING AND TECHNOLOGY. Volume 231 Issue 2, June 2017 (first published June 16, 2016); DOI: 10.1177/1754337116651184.
|This work demonstrates the omnipresence of the known impact force formula. Although originally derived only for the straight fall with a linear elastic rope, it applies almost unchanged for many other, more complex models and falls.
|This paper describes the fracture process of a fiber bundle model in a dynamic loading situation. First, a general system of equations for the resulting forces and elongations together with a fracture criterion is established. After presenting several exact solvable cases for different fiber breaking probabilities, a nonlinear model with a mixture of different fiber types is applied to the fracture process of a climbing rope. Good agreement with norm fall experiments is achieved.
|Experiences of an author with the German Wikipedia
|A physical model for walking uphill is introduced. It is based on simple principles like the conservation of energy and a force dependent efficiency coefficient. Excellent agreement with experimental data was achieved.
|The full equations of motion for a fall in a climbing rope are set up and solved when both internal viscous friction and external dry friction between the rope and one anchor point are taken into account. An essential part of the work is to discuss how the belayer can control the fall by adjusting the rope slip in the belay device. The theory can fully explain measurements of the maximum impact force, the force on the belayer and on the anchor point with and without rope control.
Theory of Reverse Bonus Certificates
|In this work, reverse bonus certificates (RBCs) are discussed in detail and illustrated with examples. It is shown that their statistical properties can be mapped onto those of bonus certificates (BCs). These properties would be identical, if the RBC was obtained by reflecting a BC using its logarithmic prices. However, the RBC is constructed from a BC using its linear prices with the help of a reflection point S that is not present in BCs. S determines the character of the RBC, together with the absorbing barrier and the bonus level. S is also mainly responsible for the differences between the RBC and the BC. Both similarities and differences are discussed in detail. We also show how the fair price of a RBC can be calculated. The fair price in RBCs plays an even more important role than in certificates of simpler construction, such as discount certificates and BCs, because our studies show that there are large deviations between fair price and market price in RBCs.
|We present an approach using a utility function for a large class of risk-averse investors who want to determine what fraction of their portfolio they should expose to risk. It is shown that even for long investment horizons the risky asset portion is relatively small, varying only between 0.2 and 0.3 without depending much on the investor’s risk aversion. The utility function used consists of a linear component in the rate of return and a strongly decreasing component for negative rate of returns. The expectation values of the utility function are calculated with the probability distribution of a Geometric Brownian Motion, the common model of a stock market. Common quadratic approximations, i.e. an analysis by the mean value and the variance only, are not able to reproduce the results of this article because of the long-tail properties of the lognormally distributed rates of return. Furthermore, the same utility function is used for the selection of more complicated investments like discount or bonus certificates. Depending on the investor’s risk aversion, his "optimal" parameters like cap value, bonus level, etc. are calculated.
Bayes and GAUs
|Probability statements about future major accidents at nuclear power plants after Fukushima, Chernobyl, Three Mile Island: We estimate the probability for the next Super-GAU (major accident, meltdown) in nuclear power plants (NPPs) using Bayes’ theorem. The result is a simple formula by which one can also explain the different opinions about nuclear power plants.
Difficult Difficulties in Free Climbing
|In the first part of this paper we want to relate the subjective climbing grades to objective difficulties. The objective difficulty is proportional to a climber’s performance, while the grade systems for free climbing, e.g. the UIAA or the French grade systems, are subjective. We use two different approaches: in the first method this relationship is set up by doubling the length of endurance routes, in the second method climbing is seen as a complex task which is made up of several individual tasks in a multiplicative way. Both approaches yield the same result: the objective difficulty increases approximately with the 3rd power of the subjective difficulty. In the second part of this paper we analyze the French overall grade system (IFAS). We show that it can be almost completely explained by a relationship between the maximum and the mandatory difficulty.
Theory of Bonus Certificates
|We discuss the stochastic properties of bonus certificates and their behavior as a function of the two variables, the absorbing barrier and the bonus level. First, general expressions for the probability distributions of return and final prices are derived. Then we give analytic expressions for these distributions with the assumption that the underlying asset follows a geometric random walk. The different investment strategies that are possible with bonus certificates are discussed in detail. The spectrum extends from the limiting case "underlying asset", through bets with varying odds, to an almost risk-free investment. By identical methods we discuss a variant of bonus certificates, the so-called capped bonus certificates.
|For all kinds of climbing situations, the impact force and the maximum elongation of a climbing rope are calculated by taking into account dry friction between the rope and the protection points. The climbing situations can include an arbitrary number and placement of protection points. In addition, we calculate the rope drag that a climber has to overcome in order to move forward.
Theoretical Description of Discount Certificates and their Iteration
|The mathematics and statistical properties of discount certificates are discussed. Based on the return function of a discount certificate and on the assumption that the underlying asset (e.g. a stock market index) behaves like a geometric random walk (Black-Scholes model), we calculate the moments of the return and the probability of loss and maximum return. The exact analytical expressions are easy enough to program in Excel and can thus be used directly by an investor. The dependence of these quantities on two parameters can be reduced to one parameter by determining the fair price of a discount certificate. With the ability to choose the cap, discount certificates are an attractive instrument for various investor profiles (i.e. various risk-return profiles). Furthermore, we discuss the iteration of discount certificates. For caps smaller or equal to the price of the underlying asset the resulting return functions, which for large iteration numbers can be exactly determined, are completely different from the return function of the underlying asset. The investor thus exposes himself to another random process which he can determine to a certain extent himself by the proper choice of the certificate’s cap and its term.
Climbing speed of teams on alpine-style, multi-pitch routes
|We give a rule of thumb for the average climbing speed of two climber teams as a function of the average difficulty of the climb.
Risk Management with Bayesian Networks
|Bayesian networks are a relatively new method for stochastic models which are made up of many random variables. The key to success is their capability to combine current data with previous knowledge and their capability to learn parameters and structures. In a first application a dynamic Bayesian network is used to model a financial market. In the second example we use a Bayesian network for the situation assessment of a risky scenario with political and military applications. In both examples the Bayesian networks are expanded to decision networks by the introduction of utility variables which allow evaluation and optimization of different decisions.