Dynamics of structures - Personal...

Dynamics of Structures 2017-2018 5. Continuous Systems

1

1

5.Continuous systems

Dynamics of structures

Arnaud Deraemaeker ([email protected])

*Examples of continuous systemssimple systemsmore complicated systems

*Response of continuous systemsbar in traction-compressionbeam in bendingmore complicated systems

*TMD applied to continuous systems

Outline of the chapter

2


2

Continuous systems in real life

3

4Beam – Bar kinematics

Simple continuous systems


3

5

More complicated systems

Dulles airport, Washington D.C.

building S, Solbosch, ULB

3D kinematics

Response of a continuous system

6


4

Vibrations of continuous systems

N degrees of freedom (dofs) system = n eigenfrequencies

Continuous system = infinite number of DOFs = infinite number of eigenfrequencies

8

Cantilever beam

Simply supported beam

Vibrations of continuous systems : equivalent continuous systems


5

9

Boundary conditions for beams and bars

Normal force

Longitudinal strain

For bars:

10

Bending moment

Curvature

Shear force

Boundary conditions for beams and bars

For beams:

Rotation


6

Equivalent continuous systems : bar in traction-compression

- can be seen as an infinite number of small mass-spring systems in series-> Infinite number of eigenfrequencies and mode shapes

- axial displacement u(x,t)

p(x,t) = load per unit length on the beamA = surface of the section = density

Equilibrium :

Equivalent continuous systems : bar in traction-compression


7

Bar in traction-compression : mode shapes and eigenfrequencies

General solution (p(x,t)=0) :

Characteristic equation:

! The variable is x not t

General solution

A and B depend on the boundary conditions

Eigenfrequencies

Example : Bar fixed at x=0 and x=L

Mode shapes



8


Mode 2

Mode 3

Mode1

16

Bar in traction-compression : orthogonality conditions

Property :

Proof:

Mode i :Mode j :

=


9

17

Bar in traction-compression : orthogonality conditions

Mass orthogonality

Stiffness orthogonality

Define

18

Bar in traction-compression : projection of the solution in the modal basis

Modal basis


10

19

The solution can be obtained by solving an infinite set of independent equations of the type

This equation corresponds to the equation of motion of a sdof system with

*mass (modal mass)

*stiffness

*angular eigenfrequency

*excitation (modal excitation)

Bar in traction-compression : projection of the solution in the modal basis

20

Equations of motion : particular solution

Modal basis


11

21

Equations of motion : modal basis solution

The solution is the sum of sdof oscillators :

In practice, truncation is needed …

22

Truncated modal basis

In practice :

Approximation can beimproved using staticcorrection (not detailed here)

Rules for truncation : - depends on the frequency band of excitation- depends on the frequency band of interest for the response

is the last eigenfrequency usedin the truncated sum


12

23

Discrete versus continuous systems

Orthogonality conditions

Projection in the modal basis

Response to harmonic excitation

24

Equivalent continuous systems : beam in bending

- transversal displacement of the neutral axis y(x,t)


13

25

Equivalent continuous systems : beam in bending

p(x,t) = load per unit length on the beamA = surface of the section = density

Equilibrium :

26

Beam in bending : mode shapes and eigenfrequencies

General solution (p(x,t)=0) :

Characteristic equation:


14

27

General solution

Example 1: Simply supported beam


A, B, C and D depend on the boundary conditions

28


Eigenfrequencies :


15

29


Mode shapes :

30


n=1

n=5 n=10


16

31

General solution

Example 2: Double cantilever beam


32


Transcendental equation, no analytical solution


17

33


Roots of the equation : with

Eigenfrequencies :

Asymptotic approach :

34


Mode shapes :


18

35


36


n=1

n=5 n=10


19

37


Simple cantilever beam

38


Transcendental equation, no analytical solution


20

39


Roots of the equation : with

Eigenfrequencies :

Next roots:

First two roots:

40


Mode shapes :


21

41


42


n=1

n=5

n=2


22

43


44

Beam in bending : orthogonality of the modeshapes

Property :

Proof:

Mode iMode j

=


23

45

Beam in bending : orthogonality of the modeshapes

Mass orthogonality

Stiffness orthogonality

Define

46

Beam in bending : projection on the modal basis

Modal basis


24

47

More complex structures: the finite element approach

3n dofs system = 3n mode shapes and eigenfrequencies

are the shape functions

is the number of nodes in the finiteelement mesh -> 3n degrees of freedom

48

Computation of the response

Projection on the modal basis (computed based on K and M)

•In practice, the number of dofs in the finite element model is dictated by the details of the geometry and for large models, it is not possible to compute all the modeshapes.•In addition, the number of modes in the frequency band of interest is usually quite low (i.e 10 to 50 modes)

Important reduction when projecting on the modal basis


25

49

Example of a finite element model of a bridge

(http://fynitesolutions.com)

50

Example of a finite element model of a bridge

[http://www.scielo.br]


26

51

Modeling of damping

•Damping introduced at the level of the matrices :

-The damping matrix can be constructed using physical constitutive laws including dissipative effects (viscoelasticity, friction …). This is rarely done because in practice, a lot of the damping comes from the joints which are usually represented by a simplified model …

•Modal damping

Damping of each mode is usually introduced :- after identification based on experimental data- in a forfaitary manner, knowing the type of structure studied

is often used as a first guess for lightly damped structures

-Rayleigh damping is often used to build the damping matrix, but is gives a very poor fit to experimental data (only two parameters, no physical meaning)

TMD applied to continuous structures

52


27

53

Tuned mass damper attached to continuous systems

Finite element model of structure without DVA

Projection on modal basis

Around frequency :

Single mode approximation:

54


Valid only around where the TMD is active

Equivalent sdof system at point d excited by fd


28

55


Dynamics of structures - Personal...

Documents

Transcript of Dynamics of structures - Personal...