Basic Structural - Wind Engineering - Lecture Slides, Slides of Environmental Law and Policy

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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

Basic structural dynamics I

• Multi-degree of freedom structures – free vibration :
  • Each mass has an equation of motion

For free vibration:

m 1 x 1 k 11 x 1 k 12 x 2 k 13 x 3 .......k1nxn  0

m 2 x 2 k 21 x 1 k 22 x 2 k 23 x 3 .......k2nxn  0

m (^) n xn kn1x 1 kn2x 2 kn3x 3 .......knnxn  0

mass m 1 :

mass m 2 :

mass mn:

Note coupling terms (e.g. terms in x 2 , x 3 etc. in first equation) stiffness terms k 12 , k 13 etc. are not necessarily equal to zero

Basic structural dynamics I

• Multi-degree of freedom structures – free vibration :

In matrix form : Assuming harmonic motion : {x }= {X}sin(t+)

m   x  k x   0

ω^2 m  X  k  X

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

 k  1 m  X (1/ω^2 ) X

Basic structural dynamics I

• Mode shapes - :

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

Mode 2

m 1

m 2

m 3

mn

m 1

m 2

m 3

mn

Mode 1 Mode 3

m 1

m 3

mn

m 2

• Multi-degree of freedom structures – forced vibration
  • For forced vibration, external forces pi(t) are applied to each mass i:

m 1

m 2

m 3

mn

x 1

xn

x 3 x 2

Pn

P 3 P 2 P 1

• Multi-degree of freedom structures – forced vibration
  • For forced vibration, external forces pi(t) are applied to each mass i:

m 1 x 1 k 11 x 1 k 12 x 2 k 13 x 3 .......k1nxn p 1 (t)

m 2 x 2 k 21 x 1 k 22 x 2 k 23 x 3 .......k2nxn p 2 (t)

m (^) n xn kn1x 1 kn2x 2 kn3x 3 .......knnxn pn(t)

• These are coupled differential equations

• Multi-degree of freedom structures – forced vibration
  • For forced vibration, external forces pi(t) are applied to each mass i:

m 1 x 1 k 11 x 1 k 12 x 2 k 13 x 3 .......k1nxn p 1 (t)

m 2 x 2 k 21 x 1 k 22 x 2 k 23 x 3 .......k2nxn p 2 (t)

m (^) n xn kn1x 1 kn2x 2 kn3x 3 .......knnxn pn(t)

• These are coupled differential equations

• 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

• 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

j 1

xi (t) ij.aj(t)

i = a 1 (t) 

Mode 1

• a 2 (t)  (^) i

Mode 2

• a 3 (t) 

i

Mode 3

• 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

G   a K  a    T p(t) 

2 ij

n

i 1

Gj mi 



j

2 j

2 ij

n

i 1

i

2

K j  ωj m  ω G





• 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

G   a K  a    T p(t) 

P (t)   p(t)  .pi(t)

n

i 1

ij

T

j j 



   

• Multi-degree of freedom structures – forced vibration

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

G j aj Kjaj  Pj(t)

G   a K  a    T p(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 used Docsity.com

• 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

G j aj Kjaj  Pj(t)

G   a K  a    T p(t) 

Gen. force