Dynamics of Structures - Appunti

Politecnico di Milano

Facoltà di Ingegneria

Corso di laurea in Ingegneria Civile

Course notes

Dynamics of Structures

Student: Lorenzo Sostegni

Academic Year 2021-2022

Contents

1 Introduction 1

1.1 Prototype problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Space discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Assumed mode method . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Lumped mass method . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Dissipation in structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Moving reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Single DoF systems 9

2.1 Undamped systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Free vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Step force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.3 Harmonic force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Damped systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Free vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Harmonic force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.3 Argand diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Frequency domain and Fourier analysis . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Time domain and Duhamel’s integral . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Duality between frequency and time domain . . . . . . . . . . . . . . . . . . 18

2.6 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6.1 Linear acceleration method . . . . . . . . . . . . . . . . . . . . . . . 19

2.7 Moving reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Multi DoF systems 22

3.1 Normal modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 Eigenvalues issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.2 Inverse iteration technique . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Damped systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Internal forces analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Moving reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5 Rayleigh-Ritz method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.6 Frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7 Time domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Random vibration of structures 31

4.1 Spectral density function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Extreme value in a time interval . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3 Combination of two processes . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3.1 Cross properties of two processes . . . . . . . . . . . . . . . . . . . . 36

4.4 Comments on the procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Dynamics of Structures I

5 Wind engineering 39

5.1 Average velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 Morison approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3 Aerodynamic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.3.1 Effect of the section rotation . . . . . . . . . . . . . . . . . . . . . . 43

5.4 Classification of buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.4.1 Tall buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.5 Wind forces in EC1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.5.1 Spectrum of the longitudinal turbulence . . . . . . . . . . . . . . . . 46

5.5.2 Structural factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.6 SDOF model for galloping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.7 Vortex induced vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Measurements 53

7 Seismic engineering 55

7.1 Modelling of ground motion at site . . . . . . . . . . . . . . . . . . . . . . . 57

8 Damping modelling 59

8.1 Hysteretic damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

8.2 Weighted modal damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

8.3 Modal viscous matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

8.4 Rayleigh damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

8.5 Aerodynamic damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

9 Step-by-step procedures 63

9.1 Non-linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Dynamics of Structures II

1

Introduction

Structural dynamics is a quite recent branch of engineering because of the difficulty in

solving a large number of equations, even for the most simple systems. Its main scope is

the study of small oscillations in the neighborhood of stable equilibrium config-

uration (under static loads).

The first problem addressed in this new field of study was machine foundations, which

transmit vibrations to the supporting structure or the soil. In order to prevent the dif-

fusion of vibrations or the safety of structures, we need to understand the response of

systems to forces that are not static.

Figure 1.1: Machine foundations scheme

As we can see from above, the rotor of the machine has, usually, an eccentric mass which

rotating provokes the release of harmonic waves in the form:

F = F cos ωt F = F sin ωt

x y

The foundation, which is very stiff, moves as a rigid body while the ground deforms with

the motion of the machine. In facing the problem, the ground can be modelled as a series

of elastic springs.

Other types of ”simple” problems are hammer foundations and space frames sup-

porting machines. Conceptually, the solution of these problems is not difficult but the

computations, when it is not possible to reduce the problem to a few DoFs, are very com-

plicated.

The main objective of this course is to provide the instruments to face many types of

dynamic loading, such as seismic waves, wind excitation and hydrodynamic forces.

Seismic waves propagate in the ground and transmit their acceleration to structures. The

main aspect of seismic phenomena is the topic of moving constraints: in fact, it is very

difficult to set a problem in which the constraints move. In terms of structural analysis, it

is better to compute forces proportional to the ground acceleration and then apply them

to the structure to study displacements and internal forces generated by the earthquake.

Moreover, differently from other types of loading, the seismic actions are the only ones for

Dynamics of Structures 1

1. Introduction 1.1. Prototype problem

which the engineer accepts several damage to the structure; anyway, safety of people must

be guaranteed. In order to do a more refined analysis, seismic design must take in account

more complicate models such as non-linearity of the problem and plasticity of materials.

In this course the assumption will be always that the problem is linear.

Wind forces are composed by two terms: one static and the other dynamic. When we

analyze wind intensity over time we can observe that generally the mean wind velocity is

around 25 m/s, which causes a static force on our structure, while the fluctuations of this

velocity provoke the dynamic loading. Anyway, wind analysis is often not important in

design since most of the structure are not sensible to such type of load. The only cases

in which wind actions are important are tall buildings and slender bodies (suspended

bridges).

Hydrodynamic forces are present in offshore structures and their treatment is similar to

the one we make for wind.

1.1 Prototype problem

In this section we will analyze a prototype problem in general terms. The engineering

problem is preceded by the modelling of:

• actions, which is made of:

– ”field modelling”: how to model the fluctuations

– ”interaction modelling”: how the system interacts with the action

• material behaviour (only linear elastic for this course)

• structural behaviour: selecting a structural theory which can simplify reality. This

approximation must be made with attention since the structural theory is load de-

pendent. For instance, beam theory cannot be applied to study local phenomena

such as impacts.

• constraints: in structural dynamics we must be careful about adding constraints. A

chimney could be approximated as a clamped beam and the ground as a rigid clamp.

This type of approximation is wrong when we study the dynamics of this body since

we must be more accurate in representing the real behaviour of the system.

• inertial properties: usually the mass is concentrated (lumped) in the axis of the

element we are considering.

• dissipation forces

Once the modelling is done we linearize the problem which means linearizing:

• actions

• material behaviour: assuming linear elasticity

• structural behaviour: hypothesis of small displacements and oscillations

• dissipation properties: linear viscous damping

Let us imagine to have a beam subjected to a generic load f (x, t). This beam has a

bending stiffness EI(x), a length l and a distributed mass γ(x). If we compute the local

static equilibrium of a piece of this beam of length dx we have:

( dM dT −p(x)

= T =

dx dx ′′

′′ ′′

−M −→

EI(x) v (x) = (x) [EI(x) v (x)] = p(x)

Dynamics of Structures 2

1. Introduction 1.2. Space discretization

How can we turn statics into dynamics?

D’Alembert principle allows us to do this by explicitly putting the time dependence in

every variable. Assuming that mechanical properties do not vary in time we have:

2

∂ v(x, t)

′′

′′

−

EI(x) v (x, t) = f (x, t) γ(x) 2

∂t

′′

′′

γ(x) v̈(x, t) + EI(x) v (x, t) = f (x, t)

We obtained the equation of motion of the beam without dissipation. Since we have a

fourth order derivative in space and a second order derivative in time we will need 4

boundary conditions and 2 initial conditions. For the clamped beam with homogeneous

initial conditions we have:

( ( (

′′

v(0, t) = 0 EI(l) v (l, t) = 0 v(x, 0) = 0

∀t ∀t ∀x

′

′ ′′

v (0, t) = 0 [EI(l) v (l, t)] = 0 v̇(x, 0) = 0

Solving this type of equations is rather difficult and for this reason, in practice, time and

space are discretized and the equations are solved by numerical methods.

1.2 Space discretization

There are two types of space discretization that are widely used by modern computer

codes: the assumed mode method and the lumped mass method. The first method

consists in selecting certain functions Ψ , respecting internal compatibility and geometrical

k

boundary conditions, such that the displacement can be expressed as:

X

v(x, t) = Ψ q (t)

k k

k

Where Ψ is called shape function and q (t) is the lagrangian coordinate. In this way

k k

the unknown v(x, t), that is a continuous function, becomes a discretized set of lagrangian

coordinates.

The conceptual basis of the lumped mass method comes from structural mechanics: know-

ing the stiffness of the element and having concentrated loads applied in the points of our

interest, the displacements can be easily computed. In order to apply this method, it is

important to discretize both forces (static, dynamic, inertia) and the unknown v(x, t).

−→ −→

γ(x) m , m . . . m f (x, t) F (t), F (t) . . . F (t)

1 2 n 1 2 n

How to write the discretized equation of motion?

In this more general case we cannot use local equilibrium since it was violated. In fact,

discretizing systems is like imposing new constraints and making them more stiffer. In

discretized systems, the local equilibrium in the points of the discretization is violated.

One powerful tool we can use is the corresponding of the principle of virtual works in

dynamics: the Hamilton Principle. −

Let us define the lagrangian quantity L = T V , where T is the kinetic energy and V is

the total potential energy. The equation of motion can be derived from:

d ∂L ∂L

− = Q (t) k = 1, . . . , n

k

dt ∂ q̇ ∂q

k k

Where Q (t) is found writing the virtual work performed by dynamic forces:

k n

X

δW = Q (t) δq

j j

j=1

Dynamics of Structures 3

1. Introduction 1.2. Space discretization

1.2.1 Assumed mode method

Now we want to obtain the equation of motion of a generic system applying the assumed

mode method for space discretization. First we write the kinetic energy:

n n

l l

Z Z

1

1 X X

2

γ(x) v̇ (x, t) dx = γ(x)

T = Ψ (x) Ψ (x) q̇ (t) q̇ (t) dx =

j j

k k

2 2

0 0 j=1 k=1

n n n n

l

Z 1

1 X X X X

γ(x) Ψ (x) Ψ (x) dx q̇ (t) q̇ (t) =

= m q̇ (t) q̇ (t)

j j j

k k jk k

2 2

0

j=1 j=1

k=1 k=1

Since j, k = 1, . . . , n we have that coefficients m belong to a square matrix that is also

jk

symmetric (m = m ), called consistent mass matrix. Therefore:

jk kj 1 T

T = q̇ m q̇

2

The vector q is called configuration vector. In obtaining this result we made two as-

sumptions: mass distribution is constant over time, as well as shape functions; constraints

are rigidly fixed. The total potential energy is: n n

l l

Z Z

1 1 X X ′′ ′′

′′2

(E) (x) q (t) q (t) dx =

Ψ (x) Ψ

V = V = EI(x) v (x, t) dx = EI(x) j k

j k

2 2

0 0 j=1 k=1

n n

n n

l

Z

1 1

X X

X X

′′ ′′

= EI(x) Ψ (x) Ψ (x) dx q (t) q (t) = k q (t) q (t)

j j

k jk k

j k

2 2

0

j=1 j=1

k=1 k=1

As for the mass matrix, coefficients k belong to a square symmetric matrix called stiff-

jk

ness matrix. 1 T

V = q k q

2

The vector of generalized forces is obtained in the following:

n

l l

Z Z X

δW = f (x, t) δv(x, t) dx = f (x, t) Ψ (x) δq dx =

j j

0 0 j=1

n l l

Z Z

X −→

= f (x, t) Ψ (x) dx δq Q (t) = f (x, t) Ψ (x) dx

j j j j

0 0

j=1

Since we understood that kinetic energy depends only on q̇ and TPE on q we can write:

k k

∂T ∂L ∂V

∂L −

= and =

∂ q̇ ∂ q̇ ∂q ∂q

k k k k

Computing the derivatives we find:

" #

T

∂ q̇ ∂ q̇

∂T 1 T Tk

= m q̇ + q̇ m = e m q̇

∂ q̇ 2 ∂ q̇ ∂ q̇

k k k

" #

T

∂q ∂q

∂V 1 T Tk

= k q + q k = e k q

∂q 2 ∂q ∂q

k k k

The equation of motion is:

Tk Tk Tk −→

e m q̈ + e k q = Q = e Q m q̈ + k q = Q

k

Dynamics of Structures 4

1. Introduction 1.2. Space discretization

1.2.2 Lumped mass method

In this method we consider the forces lumped in selected points; knowing the flexibility

coefficients associated to every displacement we are able to compute the value of the

lagrangian coordinates by the following equation:

q = ηQ

Where η is the flexibility matrix. The kinetic energy of the system is:

n n

1 1 1

X X

2 2 T

T = m v (x, t) = m q̇ (t) = q̇ m q̇

j j

j j

2 2 2

j=1 j=1

The total potential energy can be written following the Clayperon theorem:

1 1 1 1

T

−1 −1

(E) T T T

V = V = q

Q q = η q = q η q = q k q

2 2 2 2

In the end we obtain the same equation of motion:

m q̈ + k q = Q

The advantage of this method is the fact that the mass matrix is diagonal, meaning that

numerical algorithms have a greater performance using this type of matrix.

Is stiffness matrix positive definite?

The answer is yes and it can be demonstrated by considering the Taylor’s expansion of

C

the total potential energy around the stable configuration .

0

V = V + V + V + . . .

0 1 2

The first term represents the potential energy of the applied static loads, which do not

contribute in the equation of motion since the derivative for constant terms is null. The

second term is null for the stationary property of total potential energy. The third term

is always greater than 0 and for this reason the matrix k is positive definite. Greater

derivatives are not considered since we are under the small oscillations hypothesis.

Anyway there are cases in which the condition of positive definiteness is not respected.

Static loads applied to the structure could enter in the equation of motion in cases in

which they provoke instability. Let us imagine a clamped beam in which a static axial

load P is applied (compression). This beam is subjected to a dynamic distributed load

f (x, t) that makes the beam oscillating. The oscillations provoke an axial displacement

for which a work is done by the concentrated load. The potential energy associated to

this work is: l

Z

1 2

′

(P ) (P )

−W −P · −P ·

V = = u = v (x, t) dx

2 0

Because: ′2 l

l l

Z Z Z

1

v 2

′ ′

− − − dx = v (x, t) dx

u = l cos v dx = l 1 2 2

0 0 0

As we can see the potential energy associated to the static load is not linear and for this

reason falls in the second order variation of TPE. If we apply the assumed mode method

we have: n n n n

l

Z

1 1 (G)

X

X X X