Basic Structural Dynamics - Wind Engineering - Lecture Slides, Slides of Environmental Law and Policy

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• Topics :
• Revision of single degree-of freedom vibration theory
• Response to sinusoidal excitation

Refs. : R.W. Clough and J. Penzien ‘Dynamics of Structures’ 1975

R.R. Craig ‘Structural Dynamics’ 1981

J.D. Holmes ‘Wind Loading of Structures’ 2001

• Multi-degree of freedom structures – Lect. 11
  • Response to random excitation

• Single degree of freedom system :

Equation of free vibration :

Example : mass-spring-damper system :

Ratio of damping to critical c/cc :

k

c

m

x

mx  cx kx  0

2 mk

c  

often expressed as a percentage

• Single degree of freedom system :

Damper removed :

k

m

x(t)

Equation of motion : mx kx  0

kx mx

mx ^ is an equivalent static force (‘inertial’ force)

Undamped natural frequency :





m 2

k

n

1

Period of vibration, T :

1 1

T

n

• Single degree of freedom system :

Free vibration following an initial displacement :

-0.

-0.

-0.

-0.

0

1

0 1 2 3 4 5

time/T

amplitude

• Single degree of freedom system :

Free vibration following an initial displacement :

-0.

-0.

-0.

-0.

0

1

0 1 2 3 4 5

time/T

amplitude

t

C e

 1 



• Single degree of freedom system :

Critical damping ratio – damping controls amplitude at

resonance

0 1 2 3 4

n/n 1

H(n)

=0.2^ =0.

At n/n 1 =1.0, H(n 1 ) = 1/2 Then, 2kζ

F

x

0 max 

Dynamic amplification factor, H(n)

Basic structural dynamics II

• Response to random excitation :

Consider an applied force with spectral density SF(n) :

σ S (n)dn (1/k). H(n) .SF(n)dn

2

0

2 x

0

2 x (^)  

 

k

c

F(t)^ m

Spectral density of displacement :

|H(n)|^2 is the square of the dynamic amplification factor (mechanical admittance)

S (n) (1/k) .H(n) .SF(n)

2 2 x 

Variance of displacement :

see Lecture 5