Introduction to Structural Dynamics: Single-Degree-of-Freedom (SDOF) Systems.
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Transcript of Introduction to Structural Dynamics: Single-Degree-of-Freedom (SDOF) Systems.
![Page 1: Introduction to Structural Dynamics: Single-Degree-of-Freedom (SDOF) Systems.](https://reader033.fdocuments.us/reader033/viewer/2022061419/56649da15503460f94a8d72d/html5/thumbnails/1.jpg)
Introduction to Structural Dynamics:Introduction to Structural Dynamics:
Single-Degree-of-Freedom (SDOF) SystemsSingle-Degree-of-Freedom (SDOF) Systems
![Page 2: Introduction to Structural Dynamics: Single-Degree-of-Freedom (SDOF) Systems.](https://reader033.fdocuments.us/reader033/viewer/2022061419/56649da15503460f94a8d72d/html5/thumbnails/2.jpg)
Geotechnical Engineer’sGeotechnical Engineer’sView of the WorldView of the World
Structural Engineer’sStructural Engineer’sView of the WorldView of the World
![Page 3: Introduction to Structural Dynamics: Single-Degree-of-Freedom (SDOF) Systems.](https://reader033.fdocuments.us/reader033/viewer/2022061419/56649da15503460f94a8d72d/html5/thumbnails/3.jpg)
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
![Page 4: Introduction to Structural Dynamics: Single-Degree-of-Freedom (SDOF) Systems.](https://reader033.fdocuments.us/reader033/viewer/2022061419/56649da15503460f94a8d72d/html5/thumbnails/4.jpg)
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.
![Page 5: Introduction to Structural Dynamics: Single-Degree-of-Freedom (SDOF) Systems.](https://reader033.fdocuments.us/reader033/viewer/2022061419/56649da15503460f94a8d72d/html5/thumbnails/5.jpg)
Degrees of FreedomDegrees of Freedom
Single-degree-of-freedom (SDOF) systems
Vertical translation Horizontal translation Horizontal translation Rotation
![Page 6: Introduction to Structural Dynamics: Single-Degree-of-Freedom (SDOF) Systems.](https://reader033.fdocuments.us/reader033/viewer/2022061419/56649da15503460f94a8d72d/html5/thumbnails/6.jpg)
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)
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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
![Page 8: Introduction to Structural Dynamics: Single-Degree-of-Freedom (SDOF) Systems.](https://reader033.fdocuments.us/reader033/viewer/2022061419/56649da15503460f94a8d72d/html5/thumbnails/8.jpg)
Equation of Motion (external load)Equation of Motion (external load)
Elastic resistanceElastic resistanceViscous resistanceViscous resistance
)(tQkuucum
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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)(
)(
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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
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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
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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
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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