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Introduction to Structural Dynamics: Single-Degree-of-Freedom (SDOF) Systems

Introduction to Structural Dynamics: Single-Degree-of-Freedom (SDOF) Systems. Geotechnical Engineer’s View of the World. Structural Engineer’s View of the World. Basic Concepts. Degrees of Freedom Newton’s Law Equation of Motion (external force) Equation of Motion (base motion)

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Introduction to Structural Dynamics: Single-Degree-of-Freedom (SDOF) Systems

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  1. Introduction to Structural Dynamics: Single-Degree-of-Freedom (SDOF) Systems

  2. Geotechnical Engineer’s View of the World Structural Engineer’s View of the World

  3. Basic Concepts • Degrees of Freedom • Newton’s Law • Equation of Motion (external force) • Equation of Motion (base motion) • Solutions to Equations of Motion • Free Vibration • Natural Period/Frequency

  4. Lumped mass systems – masses can be assumed to be concentrated at specific locations, and to be connected by massless elements such as springs. Very useful for buildings where most of mass is at (or attached to) floors. Degrees of Freedom The number of variables required to describe the motion of the masses is the number of degrees of freedom of the system Continuous systems – infinite number of degrees of freedom

  5. Degrees of Freedom Single-degree-of-freedom (SDOF) systems Vertical translation Horizontal translation Horizontal translation Rotation

  6. According to Newton’s Law: If the mass is constant: Newton’s Law Consider a particle with mass, m, moving in one dimension subjected to an external load, F(t). The particle has: m F(t)

  7. Equation of Motion (external load) Mass Dashpot External load Spring Dashpot force External load Spring force From Newton’s Law, F = mü Q(t) - fD - fS = mü

  8. Equation of Motion (external load) Elastic resistance Viscous resistance

  9. Equation of Motion (base motion) Newton’s law is expressed in terms of absolute velocity and acceleration, üt(t). The spring and dashpot forces depend on the relative motion, u(t).

  10. Solutions to Equation of Motion • Four common cases • Free vibration: Q(t) = 0 • Undamped: c = 0 • Damped: c≠ 0 • Forced vibration: Q(t) ≠ 0 • Undamped: c = 0 • Damped: c ≠0

  11. Solutions to Equation of Motion Undamped Free Vibration Solution: where Natural circular frequency How do we get a and b? From initial conditions

  12. Solutions to Equation of Motion Undamped Free Vibration Assume initial displacement (at t = 0) is uo. Then,

  13. Solutions to Equation of Motion Assume initial velocity (at t = 0) is uo. Then,

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