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Chapter 12 Structural Dynamics
1
Chapter 12
Structural Dynamics
12.1 Basics of Structural Dynamics
12.2 Step-by-Step: Lifting Fork
12.3 Step-by-Step: Two-Story Building
12.4 More Exercise: Ball and Rod
12.5 More Exercise: Guitar String
12.6 Review
Chapter 12 Structural Dynamics
Section 12.1 Basics of Structural Dynamics
2
Section 12.1
Basics of Structural Dynamics
•
Viscous Damping
Key Concepts
•
Material Damping
•
Coulomb Friction
•
•
Lumped Mass Model
Modal Analysis
•
•
Single Degree of Freedom Model
Harmonic Response Analysis
•
•
Undamped Free Vibration
Transient Structural Analysis
•
•
Damped Free Vibration
Explicit Dynamics
•
•
Damping Coefficient
Response Spectrum Analysis
•
•
Damping Mechanisms
Random Vibration Analysis
Chapter 12 Structural Dynamics
Section 12.1 Basics of Structural Dynamics
3
Lumped Mass Model: The Two-Story Building
[5] Total
bending stiffness
of the second-
floor's beams
and columns.
[4] Total bending
stiffness of the
first-floor's beams
and columns.
[3] Total mass
lumped at the roof
floor.
[2] Total
mass lumped at
the first floor.
m
1
[1] A two-degrees-of-
freedom model for finding
the lateral displacements
of the two-story building.
m
2
k
1
k
2
c
1
c
2
[7] Energy dissipating
mechanism of the
second floor.
[6] Energy dissipating
mechanism of the first
floor.
 Chapter 12 Structural Dynamics
Section 12.1 Basics of Structural Dynamics
4
Single Degree of Freedom Model
x
∑
=
ma
p
−
kx
−
cx
=
m

x
m

x
+
cx
+
kx
=
p
m
k
p
c
•
We will use this single-degree-of-freedom lumped mass model to
explain some basic behavior of dynamic response.
•
The results can be conceptually extended to general multiple-
degrees-of-freedom cases.
 Chapter 12 Structural Dynamics
Section 12.1 Basics of Structural Dynamics
5
Undamped Free Vibration
If no external forces exist, the equation for the
one-degree-of-freedom system becomes
T
=
2
π
ω
m

x
+
cx
+
kx
=
0
t
)
If the damping is negligible, then the equation
becomes
m

x
+
kx
=
0
The
(
)
x
=
A
sin
ω
t
+
B
time (t)
k
m
f
=
ω
Natural frequency:
(rad/s) or
2
π
(Hz)
ω=
T
=
1
Natural period:
f
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