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Dynamics of Structures 2020-2021

Vibrations : Continuous systems

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Arnaud Deraemaeker

CONTINUOUSSYSTEMS

[email protected]

CONTINUOUS SYSTEMS IN REAL LIFE

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Beam –Bar kinematics

Simple continuous systems : civil engineering

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Simple continuous systems : mechanical engineering

Beam –Bar kinematics

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Cantilever beam

Simply supported beam

Equivalent continuous systems

EQUATIONS OF MOTION FOR BEAMS AND BARS

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Normal forceLongitudinal strain

For bars:

Boundary conditions for bars

- 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)

Equation of motion for bars

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p(x,t) = axial load per unit length on the barA = surface of the sectionr = density

Equilibrium :

Equation of motion for bars

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Bending moment Curvature

Shear force

For beams:

Rotation

Boundary conditions for beams

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

Equation of motion for beams

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

Equilibrium :

Equation of motion for beams

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MODE SHAPES FOR BARS

N degrees of freedom (dofs) system = n eigenfrequencies

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

Eigenfrequencies

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

Characteristic equation: ! The variable is x not t

Mode shapes and eigenfrequencies for bars

General solution

A and B depend on the boundary conditions

Eigenfrequencies

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

Mode shapes


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Mode 2

Mode 3

Mode 1


MODE SHAPES FOR BEAMS

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

Characteristic equation: ! The variable is x not t

Mode shapes and eigenfrequencies for beams

General solution

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

Example : Simply supported beam


Eigenfrequencies Mode shapes

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Example 1: Simply supported beam


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n=1

n=5 n=10


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Example 2: Double cantilever beam


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n=1

n=5 n=10


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Example 3: Cantilever beam


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n=1

n=5

n=2


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https://youtu.be/1Z-d_DlxVSQ

PROJECTION IN THE MODAL BASIS

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https://youtu.be/1Z-d_DlxVSQ



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Orthogonality :

Projection in the modal basis for bars

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

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

Projection in the modal basis

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The solution is the sum of sdof oscillators :

In practice, truncation is needed …


is the last eigenfrequency usedin the truncated sum

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In practice :

The approximation can beimproved using staticcorrection

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

Truncated modal basis

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Orthogonality :

Projection in the modal basis for beams

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Orthogonality conditions


Discrete vs continuous systems

MDOF BARS BEAMS

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For proportional (global) damping models

Rayleigh damping

Loss factor Constant modal damping

Damping models

Modal damping For each mode

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