03Lect11StructDynII

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Basic structural dynamics II

Wind loading and structural response - Lecture 11Dr. J.D. Holmes


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Topics : multi-degree-of freedom structures - free vibration

response of a tower to vortex shedding forces multi-degree-of freedom structures - forced vibration


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Basic structural dynamics I

Multi-degree of freedom structures - : Consider a structure consisting of many masses connected

together by elements of known stiffnesses

x1

xn

x3

x2

m 1

m 2

m 3

m n

The masses can move independently with displacements x 1, x2 etc.


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Multi-degree of freedom structures free vibration : Each mass has an equation of motionFor free vibration:

0x.......k xk xk xk xm n1n31321211111 !

0x.......k xk xk xk xm n2n32322212122 !

0x.......k xk xk xk xm nnn3n32n21n1nn ! .

mass m 1:

mass m 2:

mass m n:

Note coupling terms (e.g. terms in x 2, x3 etc. in first equation)

stiffness terms k12

, k13

etc. are not necessarily equal to zero


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Multi-degree of freedom structures free vibration :In matrix form :

Assuming harmonic motion : {x }= {X}sin( [ t+J )

? A _ a ? A _a _a0xk xm !

? A _ a ? A _ aXk Xm2 !

This is an eigenvalue problem for the matrix [k] -1[m]

? A ? A _ a _ aX)(1/Xmk 21 !


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Multi-degree of freedom structures free vibration :

There are n eigenvalues, P j and n sets of eigenvectors { J j}

for j=1, 2, 3 n

Then, for each j :

[ j is the circular frequency (2 Tn j); {J j} is the mode shape for mode j.

They satisfy the equation :

The mode shape can be scaled arbitrarily - multiplying both sides of the

equation by a constant will not affect the equality

? A ? A _ a _ a _ a j2 j j j j1 )(1/mk J J J !!

? A _ a ? A _ a j j2 j k m J J !


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Mode shapes - :

Number of modes, frequencies = number of masses = degrees of freedom

Mode 2

m 1

m 2

m3

m n

m 1

m2

m3

m n

Mode 3Mode 1

m1

m3

mn

m2


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Multi-degree of freedom structures forced vibration For forced vibration, external forces p i(t) are applied

to each mass i:

m 1

m 2

m 3

mn

x1

xn

x3

x2

P n

P 3P 2

P 1


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to each mass i:

(t) px.......k xk xk xk xm 1n1n31321211111 !

(t) px.......k xk xk xk xm 2n2n32322212122 !

(t) px.......k xk xk xk xm nnnn3n32n21n1nn !

.

These are coupled differential equations of motion


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to each mass i:

(t) px.......k xk xk xk xm 1n1n31321211111 !

(t) px.......k xk xk xk xm 2n2n32322212122 !


.

These are coupled differential equations


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to each mass i:

(t) px.......k xk xk xk xm 1n1n31321211111 !

(t) px.......k xk xk xk xm 2n2n32322212122 !


.

These are coupled differential equations


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Multi-degree of freedom structures forced vibration In matrix form :

Mass matrix [m] is diagonal? A _ a ? A _ a _ a p(t)xk xm !

S tiffness matrix [k] is symmetric

{p(t)} is a vector of external forces each element is a function of time


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Multi-degree of freedom structures forced vibrationM odal analysis is a convenient method of solution of the forced vibration problem when the elements of

the stiffness matrix are constant i.e.the structure islinear The coupled equations of motion are transformed

into a set of uncoupled equations

Each uncoupled equation is analogous to the equationof motion for a single d-o-f system, and can be solvedin the same way


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Multi-degree of freedom structures forced vibration

for i = 1, 2, 3 .n

m ixi(t)

a j(t) is the g eneralized coordinate representing the variation of theresponse in mode j with time. It depends on time , not position

Assume that the response of each mass can be written as:

J ij is the mode shape coordinate representing the position of theith mass in the jth mode. It depends on position , not time

!

!n

1 j jiji (t).a(t)x J

J i1= a1(t) v

Mode 1

+ a2(t) v J i2

Mode 2

+ a3(t) vJ i3

Mode 3


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In matrix form :

[J ] is a matrix in which the mode shapes are written ascolumns([J ]T is a matrix in which the mode shapes are written as

rows)Differentiating with respect to time twice :

_ a ? A _ aa(t)x J !

_ a ? A _ a(t)ax J !


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Multi-degree of freedom structures forced vibrationBy substitution, the original equations of motion reduceto:

The matrix [G] is diagonal, with the jth term equalto :

The matrix [K] is also diagonal, with the jth term equal to :

G j is the g eneralized mass in the jth

mode

? A _a ? A _a ? A_ a p(t)aK aG TJ !

2ij

n

1i

i j m J !

!

j2 j

2ij

n

1i

i2 j j m !!

!

J


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The right hand side is a single column, with the jthterm equal to :

P j(t) is the g eneralized force in the jth mode


_ a _ a (t). p p(t)(t)P in

1iij

T j j !!! J J


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We now have a set of independent uncoupledequations. Each one has the form :

Th is is t he same in form as t he equation of motion of asin g le d.o.f. system, and t he same solutions for a j (t)

can be used

(t)Paa j j j j j !


G en. mass


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(t)Paa j j j j j !


G en. stiffness


can be used


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can be used

(t)Paa j j j j j !


G en.force


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can be used

(t)Paa j j j j j !


G en.coordinate


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B asic structur al dynamics II

f(t)

Cross-wind response of slender towers

Cross-wind force isapproximatelysinusoidal in lowturbulence conditions


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S inusoidal exci tation model :

Equation of motion ( jth mode):



(t)Paa j j j j j !

J j(z) is mode sha pe

P j(t) is the gener alized or effe ctive f or ce =

G j is the gener alized or effe ctive mass = h

0

2 j dz(z)m(z) J

h

0j dz(z)t)f (z, J


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S inusoidal exci tation model :

A pplied f or ce is assumed to be sinusoidal with a f r equency

equal to the vortex shedd ing f r equency, ns

Maxim um am plitude occ ur s at r esonanc e when ns=n j

CP = cr oss-win d (lif t) f or ce coeff icient


F or ce per unit length of structur e = )tnsin (2 b(z)UC

2

1 j

2a

TN

b = width of tower



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Then gener alized f or ce in jth mode is :

P j,max is the am plitude of the sinusoidal gener alized f or ce



)tnsin (2P jmax j,! T

!!h

0 j

2

ja

h

0 j jdz(z) (z))tnsin (2 bC

2

1dz(z)t)f(z,(t) J J N

!h

0 j

2a dz(z) (z) bC2

1 J N


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Then, maxim um am plitude


j j2

j2

max j,

j j

max j,max

Gn8

P

2

Pa !!

N ote analog y with singl e d.o.f system r esult (Lectur e 10)

S u bstituting f or P j,max :

Then, maxim um def lection on structur e at height, z,

(S lide 14 - consi der ing only 1st mode con tr i bution)

max jmax (z) .a(z)x J !

j j2

j2

z2

z1j

2a

max Gn8

dz(z)(z) bC21

a

!

J N


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Maxim um def lection at to p of structur e

(S ection 11.5.1 in Wind Loading of S tructur es)

wher e ^ j is the cr itical dam ping r atio f or the jth mode, equal to j j j

K G

C

2

)(z

bn

)(z bn

Ste

j

e

s !!

(Scruton Number or mass- dam ping par ameter)m = aver age mass/ unit height

Strouhal Number f or vortex shedd ingze = effe ctive height (} 2h/3)



2a

j

b

m4

Sc

^ T !

!!L

0

2 j

2

h

0 j

2 j j

2

h

0 j

2a

max

dz(z)StSc4

dz(z)C

StG16

dz(z) bC

b

(h)x

J

J J NN


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End of LectureJohn Holmes

225-405-3789 [email protected]

03Lect11StructDynII

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