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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

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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

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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.

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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.

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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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