Date of Award
Spring 2025
Rights
Access is available to all users
Document Type
Thesis
Degree Name
Master of Science (MS) in Applied Mathematics
Department
Mathematics
Abstract
This thesis presents a force-based mathematical model for simulating dynamic falls in lead sport climbing, with an emphasis on physical realism and empirical validation. The system is modeled as a mass–spring–damper, incorporating gravity, nonlinear rope stiffness, internal damping, and Capstan-style friction at protection points. The goal is to predict peak forces and rope elongation while capturing the complex dynamics of real climbing ropes. Several novel features are introduced. Activation switches ensure forces only engage when the rope is tensioned, preventing premature response. A velocity-sensitive stiffness transition function allows the rope to stiffen smoothly with increasing fall speed, reflecting rate-dependent rope behavior. A hysteresis damping term is included for high fall factors to account for energy loss during rebound. Two modeling approaches are developed. First, a unified force model uses layered physical mechanisms to simulate fall dynamics across different fall factors. This implementation demonstrates the effectiveness of transition stiffening in producing realistic rebound and force attenuation. Second, an interpolation-based model directly fits empirical force–displacement curves from published drop test data, allowing the model to capture nonlinear rope behavior and high-strain dynamics directly from experimental observations. Both approaches successfully capture key aspects of lead fall dynamics, including peak force attenuation, rebound behavior, and rate-dependent rope stiffening. The interpolation model effectively reproduces nonlinear rope response at moderate fall factors, while the unified force model offers greater stability for higher fall conditions. Hysteresis damping improves rebound realism in the unified model but is excluded from the interpolated framework, as empirical data may already reflect these losses.
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Recommended Citation
Evans, Rosemary Christmas, "Mathematical modeling of lead climbing falls" (2025). EWU Masters Thesis Collection. 972.
https://dc.ewu.edu/theses/972
