Mechanics Tutorials

These tutorials are designed to showcase the functionality of the sympy.physics.mechanics module through detailed examples and step-by-step instructions. We demonstrate how to model various mechanical systems, derive equations of motion, and solve dynamic problems using Kane’s and Lagrange’s method. Each example includes clear explanations, mathematical formulations, and code to illustrate key concepts.

• Duffing Oscillator with a Pendulum
  • Define Variables
  • Define Kinematics
  • Define Forces
  • Lagrange’s Method
  • References
• A rolling disc
  • A rolling disc, with Kane’s method
  • A rolling disc, with Kane’s method and constraint forces
  • A rolling disc using Lagrange’s Method
• Multi Degree of Freedom Holonomic System
• Nonminimal Coordinates Pendulum
  • Kane’s Method
  • Lagrange’s Method
• A four bar linkage
  • Revealing noncontributing forces
• Linearized Carvallo-Whipple Bicycle Model
  • Declaration of Coordinates & Speeds
  • Declaration of System’s Parameters
  • Kinematics of the Bicycle
  • Inertia and Rigid Bodies
  • Gravitational Loads
  • Nonlinear Equations of Motion
  • Linearized Equations of Motion
  • References
• Atwood Machine Example
  • Define Variables
  • Inertial Frame and Pulley Center
  • Positions and Velocities of the Two Masses
  • Create WrappingCylinder for the Pulley
  • Determine Tangent Points \(T_1\) and \(T_2\)
  • WrappingPathway Over the Cylinder
  • Compute Segment Lengths and Verify Inextensible Rope
  • Define Gravity Forces on Each Mass
  • Collect All Loads for Kane’s Method
  • Kinematic Differential Equation
  • Formulate and Solve via Kane’s Method
  • Numeric Check
  • Conclusion
  • References
• Muscle Wrapping over a Spherical Bone in 3D Space
  • Define Symbols and Import Modules
  • Generalized Coordinates and Physical Parameters
  • Inertial Frame and WrappingSphere Setup
  • Position and Velocity of Contact Point
  • Kinetic Energy Calculation
  • Fixed Points and Tangent Point Determination
  • Total Muscle Path Length Computation
  • Potential Energy and Lagrangian Formulation
  • Equations of Motion Derivation
  • Physical Interpretation of the Equations
  • Conclusion