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Introduction to Mass-Spring Systems

Introduction to Mass-Spring Systems. Graphics & Media Lab.@SNU. Mass-Spring Systems. Particles (dimensionless mass points) interconnected with springs. Introduction to Elastic Springs. Spring characteristics Stiffness constant: k Initial spring length: l Current spring length: x

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Introduction to Mass-Spring Systems

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  1. Introduction toMass-Spring Systems Graphics & Media Lab.@SNU

  2. Mass-Spring Systems • Particles (dimensionless mass points) interconnected with springs

  3. Introduction to Elastic Springs • Spring characteristics • Stiffness constant: k • Initial spring length: l • Current spring length: x • Hooke’s law: F = k(x - l)

  4. Forces acting on the Mass Points • Internal force • External force • Gravity, etc. • Resulting force

  5. Static Equilibrium • System of mass points • Static equilibrium

  6. Equation for the Dynamic Movements • Equation of motion • Newton’s 2nd law • Equation of motion with damping

  7. From Static Eq. to Dynamic Eq. • Static equilibrium • Equation of motion with damping

  8. Mass-Spring Dynamics • Equation of motion for mass point i at time t • 2nd order differential equation • f is used for acceleration and damping • Without damping term, simply f = ma mass position damping coefficient acceleration damping spring + external

  9. Numerical Integration • 2nd order ODE  two coupled 1st order ODEs velocity acceleration

  10. Damping Revisited • Point damping • Damping force in opposite direction to the velocity • Damped spring • Damping force proportional to the velocity of the spring

  11. much more resistant in direction than in Topology and Stability • Stability with respect to deformation stable but not general not stable Requires well-chosen stable topologies!

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