Introduction to Structural Dynamics SingleDegreeofFreedom SDOF Systems Structural

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

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

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

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 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 Single-degree-of-freedom (SDOF) systems Vertical translation Horizontal translation Rotation

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) According to Newton’s Law: If the mass is constant:

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

Equation of Motion (external load) Viscous resistance Elastic resistance

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).

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

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

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

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