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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

m1

m2

m3

mn

The masses can move independently with displacements x1, x2 etc.School.edhole.com



• Multi-degree of freedom structures – free vibration :

• Each mass has an equation of motion

For free vibration:

0x.......kxkxkxkxm n1n31321211111

0x.......kxkxkxkxm n2n32322212122

0x.......kxkxkxkxm nnn3n32n21n1nn …………………….

mass m1:

mass m2:

mass mn:

Note coupling terms (e.g. terms in x2, x3 etc. in first equation)

stiffness terms k12, k13 etc. are not necessarily equal to zeroSchool.edhole.com




In matrix form :

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

0xkxm

XkXmω2

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

X)(1/ωXmk 21

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

for j=1, 2, 3 ……n

Then, for each j :

j is the circular frequency (2nj); {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

j2jjjj

1 )(1/ωλmk

jj2j kmω

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• Mode shapes - :

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

Mode 2

m1

m2

m3

mn

m1

m2

m3

mn

Mode 3Mode 1

m1

m3

mn

m2

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• Multi-degree of freedom structures – forced vibration

• For forced vibration, external forces pi(t) are applied to each mass i:

m1

m2

m3

mn

x1

xn

x3

x2

Pn

P3

P2

P1

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(t)px.......kxkxkxkxm 1n1n31321211111

(t)px.......kxkxkxkxm 2n2n32322212122

(t)px.......kxkxkxkxm nnnn3n32n21n1nn

…………………….

• These are coupled differential equations of motionSchool.edhole.com





(t)px.......kxkxkxkxm 1n1n31321211111

(t)px.......kxkxkxkxm 2n2n32322212122

(t)px.......kxkxkxkxm nnnn3n32n21n1nn

…………………….

• These are coupled differential equationsSchool.edhole.com




• In matrix form :

Mass matrix [m] is diagonal

p(t)xkxm

Stiffness matrix [k] is symmetric

{p(t)} is a vector of external forces –

each element is a function of timeSchool.edhole.com




• Modal 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 is linear

The coupled equations of motion are transformed into a set of uncoupled equations

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

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for i = 1, 2, 3…….n

mi

xi(t)

aj(t) is the generalized coordinate representing the variation of the response in mode j with time. It depends on time, not position

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

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

n

1jjiji (t).a(t)x

i1

= a1(t)

Mode 1

+ a2(t) i2

Mode 2

+ a3(t) i3

Mode 3School.edhole.com




In matrix form :

[] is a matrix in which the mode shapes are written as columns

([]T is a matrix in which the mode shapes are written as rows)

Differentiating with respect to time twice :

a(t)x

(t)ax

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By substitution, the original equations of motion reduce to:

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

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

Gj is the generalized mass in the jth

mode

p(t)aKaG T

2ij

n

1iij mG

j2j

2ij

n

1ii

2jj GωmωK

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

Pj(t) is the generalized force in the jth mode

p(t)aKaG T

(t).pp(t)(t)P i

n

1iij

Tjj

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

This is the same in form as the equation of motion of a single d.o.f. system, and the same solutions for aj(t) can be used

(t)PaKaG jjjjj

p(t)aKaG T

Gen. mass

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(t)PaKaG jjjjj

p(t)aKaG T

Gen. stiffness


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(t)PaKaG jjjjj

p(t)aKaG T

Gen. force

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(t)PaKaG jjjjj

p(t)aKaG T

Gen. coordinate

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f(t)

• Cross-wind response of slender towers

Cross-wind force is approximately sinusoidal in low turbulence conditions

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Sinusoidal excitation model :

Assumptions :

• sinusoidal cross-wind force variation with time

• full correlation of forces over the height

• constant amplitude of fluctuating force coefficient

‘Deterministic’ model - not random

Sinusoidal excitation leads to sinusoidal response (deflection)



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Equation of motion (jth mode):



(t)PaKaG jjjjj

j(z) is mode shape

Pj(t) is the ‘generalized’ or effective force =

Gj is the ‘generalized’ or effective mass = h

0

2j dz(z) m(z)

h

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

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Applied force is assumed to be sinusoidal with a frequency equal to the vortex shedding frequency, ns

Maximum amplitude occurs at resonance when ns=nj

C = cross-wind (lift) force coefficient


Force per unit length of structure =

ψ)tnsin(2 b (z)UCρ2

1j

2a

b = width of tower


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Then generalized force in jth mode is :

Pj,max is the amplitude of the sinusoidal generalized force



ψ)tnsin(2P jmaxj,

h

0j

2ja

h

0jj dz(z) (z)Uψ)tn sin(2π bCρ

2

1dz(z) t)f(z,(t)P

h

0j

2a dz(z) (z)UbCρ

2

1

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Then, maximum amplitude


jj2

j2

maxj,

jj

maxj,max

ζGn8π

P

ζ2K

Pa

Note analogy with single d.o.f system result (Lecture 10)

Substituting for Pj,max :

Then, maximum deflection on structure at height, z,

(Slide 14 - considering only 1st mode contribution)

maxjmax (z).a(z)x

jj2

j2

z2

z1j

2a

maxζGn8π

dz(z) (z)UbCρ2

1

a

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Maximum deflection at top of structure

(Section 11.5.1 in ‘Wind Loading of Structures’)

where j is the critical damping ratio for the jth mode, equal to jj

j

KG

C

2

)(zU

bn

)(zU

bnSt

e

j

e

s

(Scruton Number or mass-damping parameter) m = average mass/unit height

Strouhal Number for vortex shedding ze = effective height ( 2h/3)



2a

j

bρ

m4Sc

L

0

2j

2

h

0j

2jj

2

h

0j

2a

max

dz(z)StSc4π

dz(z)C

StζG16π

dz(z)bCρ

b

(h)x

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End of Lecture John Holmes

225-405-3789 [email protected]

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mn x n

k1n x n

nn x n

forced vibration

p1 tm2 x

n eigenvalues

mode j

mass matrix