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.
","abstract_html":"<p>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. </p>","abstract_has_math":false,"creators":["Evans, Rosemary Christmas"],"institution":null,"degree_name":"Master of Science (MS) in Applied Mathematics","degree_level":"Thesis","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-01-01T08:00:00Z","date_published":"2025-01-01T08:00:00Z","updated_at":"2026-07-24T02:13:07Z","subjects":["Dynamic Systems","Non-linear Dynamics"],"languages":[],"rights":["Access is available to all users"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://dc.ewu.edu/theses/972","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Evans, Rosemary Christmas"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science (MS) in Applied Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Dynamic Systems","Non-linear Dynamics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["Access is available to all users"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://dc.ewu.edu/theses/972"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Mathematical modeling of lead climbing falls"]}]}],"canonical_facts":{"dc:creator":["Evans, Rosemary Christmas"],"dc:description.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.
"],"dc:identifier":["https://dc.ewu.edu/theses/972"],"dc:rights":["Access is available to all users"],"dc:subject":["Dynamic Systems","Non-linear Dynamics"],"dc:title":["Mathematical modeling of lead climbing falls"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Science (MS) in Applied Mathematics"]},"updated_at":"2026-07-24T02:13:07Z"}