1. Basic structural dynamics II
Wind loading and structural response - Lecture 11
Dr. J.D. Holmes
Basic structural dynamics II

2. Basic structural dynamics II
Topics :
multi-degree-of freedom structures - free vibration
multi-degree-of freedom structures - forced vibration
response of a tower to vortex shedding forces

3. Basic structural dynamics I
Multi-degree of freedom structures - :
Consider a structure consisting of many masses connected together by elements of known stiffnesses m1 m2 m3 mn x1 xn x3 x2
The masses can move independently with displacements x1, x2 etc.

4. Basic structural dynamics I
Multi-degree of freedom structures – free vibration :
Each mass has an equation of motion
For free vibration:
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 zero

5. Basic structural dynamics I
Multi-degree of freedom structures – free vibration :
In matrix form :
Assuming harmonic motion : {x }= {X}sin(t+)
This is an eigenvalue problem for the matrix [k]-1[m]

6. 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 (2nj); {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

7. Basic structural dynamics I
Mode shapes - : m1 m2 m3 mn
Mode 3
Mode 2 m1 m2 m3 mn
Mode 1 m1 m3 mn m2
Number of modes, frequencies = number of masses = degrees of freedom

8. 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

9. Basic structural dynamics II
Multi-degree of freedom structures – forced vibration
For forced vibration, external forces pi(t) are applied to each mass i:
…………………….
These are coupled differential equations of motion

10. Basic structural dynamics II
Multi-degree of freedom structures – forced vibration
For forced vibration, external forces pi(t) are applied to each mass i:
…………………….
These are coupled differential equations

11. Basic structural dynamics II
Multi-degree of freedom structures – forced vibration
For forced vibration, external forces pi(t) are applied to each mass i:
…………………….
These are coupled differential equations

12. Basic structural dynamics II
Multi-degree of freedom structures – forced vibration
In matrix form :
Mass matrix [m] is diagonal
Stiffness matrix [k] is symmetric
{p(t)} is a vector of external forces –
each element is a function of time

13. 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

14. Basic structural dynamics II
Multi-degree of freedom structures – forced vibration
Assume that the response of each mass can be written as:
for i = 1, 2, 3…….n
ij is the mode shape coordinate representing the position of the ith mass in the jth mode. It depends on position, not time
aj(t) is the generalized coordinate representing the variation of the response in mode j with time. It depends on time, not position
+ a2(t) 
i2
Mode 2 mi
xi(t)
i1
= a1(t) 
Mode 1
+ a3(t) 
i3
Mode 3

15. 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 :

16. 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 :
Gj is the generalized mass in the jth mode
The matrix [K] is also diagonal, with the jth term equal to :

17. 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

18. 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 :
Gen. mass
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

19. 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 :
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 used

20. 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 :
Gen. force
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

21. 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 :
Gen. coordinate
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

22. Basic structural dynamics II
Cross-wind response of slender towers
f(t)
Cross-wind force is approximately sinusoidal in low turbulence conditions

23. Basic structural dynamics II
Cross-wind response of slender towers
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)

24. Basic structural dynamics II
Cross-wind response of slender towers
Sinusoidal excitation model :
Equation of motion (jth mode):
j(z) is mode shape
Pj(t) is the ‘generalized’ or effective force =
Gj is the ‘generalized’ or effective mass =

25. Basic structural dynamics II
Cross-wind response of slender towers
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
Force per unit length of structure =
Cl = cross-wind (lift) force coefficient
b = width of tower

26. Basic structural dynamics II
Cross-wind response of slender towers
Then generalized force in jth mode is :
Pj,max is the amplitude of the sinusoidal generalized force

27. Basic structural dynamics II
Cross-wind response of slender towers
Then, maximum amplitude
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)

28. Basic structural dynamics II
Cross-wind response of slender towers
Maximum deflection at top of structure
(Section in ‘Wind Loading of Structures’)
where j is the critical damping ratio for the jth mode, equal to
Strouhal Number for vortex shedding ze = effective height ( 2h/3)
(Scruton Number or mass-damping parameter) m = average mass/unit height

29. End of Lecture John Holmes 225-405-3789 JHolmes@lsu.edu