Introduction to Structural Dynamics:Introduction to Structural Dynamics:

Single-Degree-of-Freedom (SDOF) SystemsSingle-Degree-of-Freedom (SDOF) Systems

Geotechnical Engineer’sGeotechnical Engineer’sView of the WorldView of the World

Structural Engineer’sStructural Engineer’sView of the WorldView of the World

Basic ConceptsBasic 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/FrequencyNatural Period/Frequency

Degrees of FreedomDegrees 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 FreedomDegrees of Freedom

Single-degree-of-freedom (SDOF) systems

Vertical translation Horizontal translation Horizontal translation Rotation

Newton’s LawNewton’s Law

uvelocity uonaccelerati

uposition

Consider a particle with mass, m, moving in one dimension subjected to an external load, F(t). The particle has:

According to Newton’s Law:

)(tFumdt

d

If the mass is constant:

)(tFumudt

dmum

dt

d

m

F(t)

Equation of Motion (external load)Equation of Motion (external load)

MassDashpot

SpringExternal load

External loadDashpot force

Spring force

From Newton’s Law, F = mü

Q(t) - fD - fS = mü

Equation of Motion (external load)Equation of Motion (external load)

Elastic resistanceElastic resistanceViscous resistanceViscous resistance

)(tQkuucum

Equation of Motion (base motion)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).

b

b

b

t

umkuucum

kuucuum

kuucuum

kuucum

0)(

)(

Solutions to Equation of MotionSolutions 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

)(tQkuucum

Solutions to Equation of MotionSolutions to Equation of Motion

Undamped Free Vibration

0 kuum

Solution:

tbtatu oo cossin)(

where

m

ko Natural circular frequencyNatural circular frequency

How do we get a and b? From initial conditions

Solutions to Equation of MotionSolutions to Equation of Motion

Undamped Free Vibration

tbtatu oo cossin)(

Assume initial displacement (at t = 0) is uo. Then,

bu

bau

bau

o

o

ooo

)1()0(

)0(cos)0(sin

Solutions to Equation of MotionSolutions to Equation of Motion

Assume initial velocity (at t = 0) is uo. Then,

o

o

oo

ooo

ooooo

oooo

ua

au

bau

bau

tbtau

)0()1(

)0(sin)0(cos

sincos