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Basic structural dynamics I

Wind loading and structural response - Lecture 10

Dr. J.D. Holmes

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Basic structural dynamics I

• Topics :

• e!ision o" single degree-o" "reedom !i#ration t$eory

• esponse to sinusoidal e%citation

Refs. : R.W. Clough and J. Penzien ‘Dynamics of Structures’ 1!"

  R.R. Craig ‘Structural Dynamics’ 1#1

J.D. $olmes ‘Wind %oading of Structures’ &''1

• &ulti-degree o" "reedom structures ' Lect. 11

• esponse to random e%citation

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Basic structural dynamics I

(c −

)*uation of free +i,ration :

)(am-le : masss-ringdam-er system :

k

c m

(

'/((c(m   =++  

mass 0 acceleration s-ring force 2 dam-er force

- kx

- c(dx/dt)

=(m

  /(.

e*uation of motion

• (ingle degree o" "reedom system :

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Basic structural dynamics I

• (ingle degree o" "reedom system : 

)*uation of free +i,ration :

)(am-le : masss-ringdam-er system :

Ratio of dam-ing to critical c3cc : 

k

c

 m(

'/((c(m   =++  

m/ &

c=ζ 

often e(-ressed as a -ercentage

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Basic structural dynamics I

• (ingle degree o" "reedom system :

Dam-er remo+ed :

k

 m(4t5

)*uation of motion :   '/((m   =+

(m/(   −=

is an e*ui+alent static force 4‘inertial’ force5(m −

6ndam-ed natural fre*uency :π 

ω 

&m

/ 

&7

1n 11   ==

Period of +i,ration8 9 :11

1&9

n==

ω 

π 

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Basic structural dynamics I

• (ingle degree o" "reedom system : 

nitial dis-lacement ;o

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Basic structural dynamics I

• (ingle degree o" "reedom system :

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Basic structural dynamics I

• (ingle degree o" "reedom system :

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Basic structural dynamics I

• (ingle degree o" "reedom system : 

Res-onse to sinusoidal e(citation :

)*uation of motion :tsin

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Basic structural dynamics I

• (ingle degree o" "reedom system :

+ritical damping ratio '

damping controls amplitude at

resonance

0.1

1.0

10.0

100.0

0 1 ,

n/n1

      H      (     n      )

ζ/0.01

ζ/0.0

ζ/0.1ζ/0.,

ζ/0.

@t n3n1 1.'8 $4n15 13&ζ 9hen8 &/?<

('

ma(  ≅

Dynamic am-lification factor8 $4n5

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Basic structural dynamics II

• esponse to random e%citation : 

Consider an a--lied force =ith s-ectral density S

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Basic structural dynamics II

• esponse to random e%citation :

S-ecial case constant force s-ectral density S

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nd o" LectureJo$n Holmes

,,-0-234 JHolmes5lsu.edu