introduction to structural dynamics: single-degree-of-freedom (sdof) systems
TRANSCRIPT
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