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Theodore Stanoev

Publications and source records attributed to Theodore Stanoev.

At least 19 recordsLinked to original sources
arXiv2025

On the brachistochrone problem for cycling ascents

VAM ({\it velocit\`a ascensionale media}) is a measurement that quantifies a cyclist's climbing ability. We show that to minimize the time to attain a given height gain\, -- \,which is tantamount to maximizing VAM\, -- \,a cyclist should climb as steep a constant-grade hill as possible. Apart from the power-to-weight ratio, the limit of steepness is imposed by such factors as the efficiency of pedalling, which is related to feasible cadence, maintaining balance, preventing lifting of the front, and skidding of the rear, wheel. In an appendix, we discuss steepness constraints due to pedalling efficiency. The article itself is focused on consequences of the power available to the cyclist, which can be viewed as a necessary condition to examine other aspects of climbing strategy. We show that\, -- \,for given start and end points, and for any fixed average-power constraint\, -- \,the brachistochrone, which is the trajectory of minimum ascent time, is the straight line connecting these points, covered with a constant speed, which along such a line is equivalent to a constant power. This is in contrast to the classical solution of a descent brachistochrone under gravity, which is a cycloid along which the speed is not constant.

physics.pop-ph↗
arXiv2025

On minimizing cyclists' ascent times: Part II

We formulate an optimization of a bicycle ascent time under the constraints of the average, maximum, and minimum powers. In contrast to the first part of this study, we do not restrict the departure to flying starts with an initial speed determined by the model and its optimization. We allow for various initial speeds, from a standstill to a launched start. We accomplish this by generalizing the discontinuous piecewise constant speed model to a continuous piecewise linear speed model. Regardless of the initial speed, steepness or profile of the ascent the optimal strategy tends to a constant ground speed, in agreement with the conclusion of the previous, more restricted, formulation. This new formulation allows us to compare various initial-speed strategies and, hence, has a direct application to competitive cycling. Notably, in timetrials composed of flat and steep sections, it helps one decide whether or not to change bicycle, which requires stopping and restarting, from one that is more appropriate for flats to one that is more appropriate for uphills.

physics.class-ph↗
arXiv2024

On minimizing cyclists' ascent times

We prove that, given an average power, the ascent time is minimized if a cyclist maintains a constant ground speed regardless of the slope. Herein, minimizing the time is equivalent to maximizing -- for a given uphill -- the corresponding mean ascent velocity (VAM: velocit\`a ascensionale media), which is a common training metric. We illustrate the proof with numerical examples, and show that, in general, maintaining a constant instantaneous power results in longer ascent times; both strategies result in the same time if the slope is constant. To remain within the athlete's capacity, we examine the effect of complementing the average-power constraint with a maximum-power constraint. Even with this additional constraint, the ascent time is the shortest with a modified constant-speed -- not constant-power -- strategy; as expected, both strategies result in the same time if the maximum and average powers are equal to one another. Given standard available information -- including level of fitness, quantified by the power output, and ascent profile -- our results allow to formulate reliable and convenient strategies of uphill timetrials.

physics.class-ph↗
arXiv2022

On technical considerations of UCI-regulated velodrome track design

A novel approach to velodrome design for UCI-regulated tracks is presented. The mathematical model uses differential geometry to form a three-dimensional ruled surface. The surface accounts for the safety zone, blue band, and track region, the latter of which is comprised of three types of segments: straight lines, the arcs of circles, and connecting transition curves. Following a first-principles approach, the general expressions are derived from the Frenet-Serret relations, as a function of the banking and curvature profiles, lengths of curve segments, and turn radii of the bends. Given the underdetermined nature of the design problem, particular solutions are obtained using a least-squares minimization of an objective function, within the framework of numerical optimization. Computer renderings of two designs, a symmetric assembly of quadrants as well as an asymmetric one, are presented to demonstrate the versatility of the approach, which may be used to design velodrome tracks of any UCI Category and track geometry specification.

physics.pop-ph↗
arXiv2022

Modelling of a cyclist's power for time trials on a velodrome

We formulate a phenomenological model to study the power applied by a cyclist on a velodrome\, -- \,for individual timetrials\, -- \,taking into account the straights, circular arcs, connecting transition curves and banking. The dissipative forces we consider are air resistance, rolling resistance, lateral friction and drivetrain resistance. Also, power can be used to increase the kinetic and gravitational potential energy. Herein, to model a steady ride\, -- \,as expected for individual timetrials\, -- \,we assume a constant centre-of-mass speed, while allowing the cadence and power to vary during a lap. Hence, the kinetic energy is constant and the only mechanical energy whose change we need to consider is the increase of gravitational potential energy due to raising the centre of mass upon exiting each curve. The effect of dissipative forces is examined at each point of the lap; the effect of conservative forces is examined as an average. The latter is a small\, -- \,albeit not negligible\, -- \,part of the total power, and its inclusion within a model is a novelty presented herein. It increases the model's empirical adequacy. Following derivations and justifications of expressions that constitute this mathematical model, we present a numerical example. We show that the cadence and power vary slightly during a steady ride. In other words, a constant centre-of-mass speed entails nearly constant cadence and power, as expected for a steady ride and as supported by measurements. Also, we examine changes in the required power due to changes of various quantities, such as air density at a velodrome, laptime and several others, as well as the model sensitivity to input errors. Furthermore, we examine the effects on the required power of slight and gradual changes in speed, which are pertinent to individual time trials.

physics.class-ph↗
arXiv2021

Modelling of cyclist's power to overcome dissipative forces on a velodrome

We model the instantaneous power applied by a cyclist on a velodrome -- for individual pursuits and other individual time trials -- taking into account its straights, circular arcs, and connecting transition curves. The forces opposing the motion are air resistance, rolling resistance, lateral friction and drivetrain resistance. We examine the constant-cadence and constant-power cases, and discuss their results, including an examination of empirical adequacy of the model.

physics.pop-ph↗
arXiv2021

On maximizing VAM for a given power: Slope, cadence, force and gear-ratio considerations

The velocità ascensionale media (VAM) is measurement that quantifies a cyclist's climbing ability. It depends on both the ground speed of a bicycle-cyclist system and the slope of an incline. To maximize the ascent speed, the solution to the brachistochrone problem determines that the optimal curve for the incline is a straight line, which is a hill with a constant slope. The maximum obtainable VAM value by a cyclist increases monotonically with the slope of the incline, but is limited by the maximum sustainable power. These properties -- which are theorems stemming from a standard mathematical model to account for the power required to propel a bicycle -- constitute a mathematical-physics background upon which various strategies for the VAM maximization can be examined in the context of the maximum sustainable power as a function of both the gear ratio and cadence. Recently established records provide an empirical support for these analytical results, which are based on theoretical considerations.

physics.pop-ph↗
arXiv2021

On modelling bicycle power-meter measurements

We combine power-meter measurements with GPS measurements to study the model that accounts for the use of power by a cyclist. The model takes into account the change in elevation and speed along with adverse effects of air, rolling and drivetrain resistance. The focus is on estimating the resistance coefficients using numerical optimization techniques to maintain an agreement between modelled and measured power-meter values, which accounts for the associated uncertainties. The estimation of coefficients is performed for two typical scenarios of road cycling under windless conditions, along a course that is mainly flat as well as a course of near constant inclination. Also, we discuss relations between different combinations of two model parameters, where other quantities are constant, by the implicit function theorem. Using the obtained estimates of resistance coefficients for the two courses, we use the mathematical relations to make inferences on the model and physical conditions. Along with a discussion of results, we provide two appendices. In the first appendix, we illustrate the importance of instantaneous cadence measurements. In the second, we consider the model in constrained optimization using Lagrange multipliers.

physics.pop-ph↗
arXiv2021

On modelling bicycle power for velodromes: Part II Formulation for individual pursuits

We model the instantaneous power on a velodrome--as applied to individual pursuits and other individual time trials--taking into account its straights, circular arcs, and connecting transition curves. The forces opposing the motion are air resistance, rolling resistance, lateral friction and drivetrain resistance. We examine the constant-cadence and constant-power cases, and discuss their results, including an examination of an empirical adequacy of the model. We also examine changes in the kinetic and potential energy.

physics.pop-ph↗
arXiv2020

On modelling bicycle power-meter measurements: Part II. Relations between rates of change of model quantities

Power-meter measurements are used to study a model that accounts for the use of power by a cyclist. The focus is on relations between rates of change of model quantities, such as power and speed, both in the context of partial derivatives, where other quantities are constant, and Lagrange multipliers, where other quantities vary to maintain the imposed constraints.

physics.pop-ph↗
arXiv2020

Selecting velocity models using Bayesian Information Criterion

We present a strategy for selecting the values of elasticity parameters by comparing walk-away vertical seismic profiling data with a multilayered model in the context of Bayesian Information Criterion. We consider $P$-wave traveltimes and assume elliptical velocity dependence. The Bayesian Information Criterion approach requires two steps of optimization. In the first step, we find the signal trajectory and, in the second step, we find media parameters by minimizing the misfit between the model and data.

physics.geo-ph↗
arXiv2020

On effects of inhomogeneity on anisotropy in Backus average

In general, the Backus average of an inhomogeneous stack of isotropic layers is a transversely isotropic medium. Herein, we examine a relation between this inhomogeneity and the strength of resulting anisotropy, and show that, in general, they are proportional to one another. There is an important case, however, in which the Backus average of isotropic layers results in an isotropic -- as opposed to a transversely isotropic -- medium. We show that it is a consequence of the same rigidity of layers, regardless of their compressibility. Thus, in general, the strength of anisotropy of the Backus average increases with the degree of inhomogeneity among layers, except for the case in which all layers exhibit the same rigidity.

physics.geo-ph↗
arXiv2020

On anisotropy and inhomogeneity parameter estimation using traveltimes

We consider an anisotropic inhomogeneous model to simulate measured vertical-seismic-profile traveltimes. In this model, we assume that velocity increases linearly with depth and anisotropy is the result of elliptical velocity dependence. Using a series of sources in a line to a single receiver, we minimize the least-squares residual between measured and modelled traveltimes to estimate the anisotropy and inhomogeneity parameters of the subsurface. To verify the approach, we construct synthetic traveltime data for a one- and two-layer model and perform the traveltime inversion. We justify the convergence-ensuring specifications and model-parameter restrictions that are intrinsic to the parameter estimation. To determine the approach's practicality, we assess the reliability of results under the influence of noise. From this assessment, we discern the noise threshold for both models and determine that the noise restriction is severely restrictive for increasing model complexity. We estimate parameters, using the same methodology, for a real-data case and conclude with a discussion of results.

physics.geo-ph↗
arXiv2020

On possible issues of Backus average

In this paper, we continue the study of Bos et al. (2018) regarding statistical and numerical considerations of the Backus (1962) product approximation. While the approximation is typically quite good for seismological scenarios, Bos et al. (2018) demonstrate a physical scenario that could, in spite of the stability conditions for isotropic media, lead to an issue within the Backus average. Using the Preliminary Reference Earth Model of Dziewoński and Anderson (1981) and a case study in the upper oceanic crust, we investigate whether this issue is likely to occur in the context of seismology.

physics.geo-ph↗
arXiv2020

On modelling bicycle power for velodromes: Part I: Formulation for individual pursuits

For a moving bicycle, the power can be modelled as a response to the propulsion of the centre of mass of the bicycle-cyclist system. On a velodrome, an accurate modelling of power requires a distinction between the trajectory of the wheels and the trajectory of the centre of mass. We formulate and examine an individual-pursuit model that takes into account the aforementioned distinction. In doing so, we provide details of the invoked physical principles and mathematical derivations, with an emphasis on their limitations. We assume that a velodrome consists of two parallel straights and two semicircular arcs. We neglect the effects of the track inclination along the straights and assume the track inclination along the curves to be constant. For either segment, we consider two distinct black-line speeds. For the latter, the lean-angle expression is derived based on a noninertial frame of the cyclist. Among conclusions quantified by this model is the fact that a constant-cadence approach to an individual pursuit does not minimize the required power.

physics.pop-ph↗
arXiv2019

On relations of anisotropy and linear inhomogeneity using Backus average

The anisotropy of an equivalent medium resulting from the Backus (1962) average is induced by the vertical inhomogeneity among its constituent layers. The velocity field of the constituent isotropic layers increases linearly with depth, which is assumed to be a good seismological description of sedimentary layers Slotnick (1959). We derive an analytical relationship between the anisotropy, characterized by the Thomsen (1986) parameters, and the linear inhomogeneity parameters, which forms a system of three equations for nine unknowns. To obtain well-posedness, we constrain the problem by considering two seismological methods applied to field data. We use the results from the two methods, for a particular region of interest, to assess the validity of the analytical relation.

physics.geo-ph↗
arXiv2019

Guided waves as superposition of body waves

We illustrate properties of guided waves in terms of a superposition of body waves. In particular, we consider the Love and SH waves. Body-wave propagation at postcritical angles--required for a total reflection--results in the speed of the Love wave being between the speeds of the SH waves in the layer and in the halfspace. A finite wavelength of the SH waves--required for constructive interference--results in a limited number of modes of the Love wave. Each mode exhibits a discrete frequency and propagation speed; the fundamental mode has the lowest frequency and the highest speed.

physics.class-ph↗