Is this project an undergraduate, graduate, or faculty project?
Undergraduate
Project Type
group
Campus
Daytona Beach
Authors' Class Standing
Kelly Wold Aidan Hart Patrick Gilliam
Lead Presenter's Name
Kelly Wold
Lead Presenter's College
DB College of Arts and Sciences
Faculty Mentor Name
Dr. Hemanta Kunwar
Abstract
Numerical Investigation of the Nonlinear Simple Pendulum and the Dependence of Oscillation Period on Initial Angle examines how the oscillation period of a simple pendulum varies with initial angular displacement and evaluates the accuracy of numerical methods in capturing this behavior. In classical treatments, the small-angle approximation simplifies the governing differential equation and predicts a constant period independent of amplitude; however, this assumption breaks down for larger angles, where the system exhibits nonlinear dynamics. The objective of this project is to model the full nonlinear equation of motion and quantify how the period depends on initial conditions. To achieve this, the second-order nonlinear differential equation is solved numerically using Runge-Kutta methods, and oscillation periods are computed across a range of initial angles. These numerical results are compared to the small-angle approximation to determine the range of validity of the linear model. Additional analysis investigates the influence of step size on numerical accuracy, stability, and convergence. Preliminary results indicate that the period increases with larger initial displacements, demonstrating clear deviation from the constant-period prediction of the linear approximation. This project highlights the limitations of analytical simplifications and demonstrates the effectiveness of numerical methods in analyzing nonlinear differential equations, reinforcing their importance in modeling realistic physical systems.
Did this research project receive funding support (Spark, SURF, Research Abroad, Student Internal Grants, Collaborative, Climbing, or Ignite Grants) from the Office of Undergraduate Research?
No
Included in
Dynamics and Dynamical Systems Commons, Numerical Analysis and Computation Commons, Ordinary Differential Equations and Applied Dynamics Commons
Numerical Investigation of the Nonlinear Simple Pendulum and the Dependence of Oscillation Period on Initial Angle
Numerical Investigation of the Nonlinear Simple Pendulum and the Dependence of Oscillation Period on Initial Angle examines how the oscillation period of a simple pendulum varies with initial angular displacement and evaluates the accuracy of numerical methods in capturing this behavior. In classical treatments, the small-angle approximation simplifies the governing differential equation and predicts a constant period independent of amplitude; however, this assumption breaks down for larger angles, where the system exhibits nonlinear dynamics. The objective of this project is to model the full nonlinear equation of motion and quantify how the period depends on initial conditions. To achieve this, the second-order nonlinear differential equation is solved numerically using Runge-Kutta methods, and oscillation periods are computed across a range of initial angles. These numerical results are compared to the small-angle approximation to determine the range of validity of the linear model. Additional analysis investigates the influence of step size on numerical accuracy, stability, and convergence. Preliminary results indicate that the period increases with larger initial displacements, demonstrating clear deviation from the constant-period prediction of the linear approximation. This project highlights the limitations of analytical simplifications and demonstrates the effectiveness of numerical methods in analyzing nonlinear differential equations, reinforcing their importance in modeling realistic physical systems.