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

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

3. Equation of Motion (external force) Equation of Motion (base motion)
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. 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.

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

6. 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:

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)
Viscous resistance
Elastic 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

11. Natural circular frequency
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,