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Basic structural dynamics II
Wind loading and structural response - Lecture 11Dr. J.D. Holmes
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Basic structural dynamics II
• 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.
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Basic structural dynamics I
• 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 zeroadmission.edhole.com
Basic structural dynamics I
• Multi-degree of freedom structures – free vibration :
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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Basic structural dynamics I
• Multi-degree of freedom structures – free vibration :
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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Basic structural dynamics I
• 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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Basic structural dynamics II
• 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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Basic structural dynamics II
• Multi-degree of freedom structures – forced vibration
• For forced vibration, external forces pi(t) are applied to each mass i:
(t)px.......kxkxkxkxm 1n1n31321211111
(t)px.......kxkxkxkxm 2n2n32322212122
(t)px.......kxkxkxkxm nnnn3n32n21n1nn
…………………….
• These are coupled differential equations of motionadmission.edhole.com
Basic structural dynamics II
• Multi-degree of freedom structures – forced vibration
• For forced vibration, external forces pi(t) are applied to each mass i:
(t)px.......kxkxkxkxm 1n1n31321211111
(t)px.......kxkxkxkxm 2n2n32322212122
(t)px.......kxkxkxkxm nnnn3n32n21n1nn
…………………….
• These are coupled differential equationsadmission.edhole.com
Basic structural dynamics II
• Multi-degree of freedom structures – forced vibration
• For forced vibration, external forces pi(t) are applied to each mass i:
(t)px.......kxkxkxkxm 1n1n31321211111
(t)px.......kxkxkxkxm 2n2n32322212122
(t)px.......kxkxkxkxm nnnn3n32n21n1nn
…………………….
• These are coupled differential equationsadmission.edhole.com
Basic structural dynamics II
• Multi-degree of freedom structures – forced vibration
• 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 time
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Basic structural dynamics II
• Multi-degree of freedom structures – forced vibration
• 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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Basic structural dynamics II
• Multi-degree of freedom structures – forced vibration
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 3
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Basic structural dynamics II
• Multi-degree of freedom structures – forced vibration
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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Basic structural dynamics II
• Multi-degree of freedom structures – forced vibration
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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Basic structural dynamics II
• Multi-degree of freedom structures – forced vibration
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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Basic structural dynamics II
• Multi-degree of freedom structures – forced vibration
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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Basic structural dynamics II
• Multi-degree of freedom structures – forced vibration
We now have a set of independent uncoupled equations. Each one has the form :
(t)PaKaG jjjjj
p(t)aKaG T
Gen. stiffness
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 usedadmission.edhole.com
Basic structural dynamics II
• Multi-degree of freedom structures – forced vibration
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. force
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Basic structural dynamics II
• Multi-degree of freedom structures – forced vibration
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. coordinate
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Basic structural dynamics II
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)
Basic structural dynamics II
• Cross-wind response of slender towers
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Sinusoidal excitation model :
Equation of motion (jth mode):
Basic structural dynamics II
• Cross-wind response of slender towers
(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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Sinusoidal excitation model :
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
Basic structural dynamics II
Force per unit length of structure =
ψ)tnsin(2 b (z)UCρ2
1j
2a
b = width of tower
• Cross-wind response of slender towers
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Then generalized force in jth mode is :
Pj,max is the amplitude of the sinusoidal generalized force
Basic structural dynamics II
• Cross-wind response of slender towers
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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Basic structural dynamics II
Then, maximum amplitude
• Cross-wind response of slender towers
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)
Basic structural dynamics II
• Cross-wind response of slender towers
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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