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Dynamics of Structures 2017-2018 5. Continuous Systems
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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
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Dynamics of Structures 2017-2018 5. Continuous Systems
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Continuous systems in real life
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4Beam – Bar kinematics
Simple continuous systems
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More complicated systems
Dulles airport, Washington D.C.
building S, Solbosch, ULB
3D kinematics
Response of a continuous system
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Dynamics of Structures 2017-2018 5. Continuous Systems
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Vibrations of continuous systems
N degrees of freedom (dofs) system = n eigenfrequencies
Continuous system = infinite number of DOFs = infinite number of eigenfrequencies
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Cantilever beam
Simply supported beam
Vibrations of continuous systems : equivalent continuous systems
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Boundary conditions for beams and bars
Normal force
Longitudinal strain
For bars:
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Bending moment
Curvature
Shear force
Boundary conditions for beams and bars
For beams:
Rotation
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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
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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
Bar in traction-compression : mode shapes and eigenfrequencies
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Bar in traction-compression : mode shapes and eigenfrequencies
Mode 2
Mode 3
Mode1
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Bar in traction-compression : orthogonality conditions
Property :
Proof:
Mode i :Mode j :
=
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Bar in traction-compression : orthogonality conditions
Mass orthogonality
Stiffness orthogonality
Define
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Bar in traction-compression : projection of the solution in the modal basis
Modal basis
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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
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Equations of motion : particular solution
Modal basis
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Equations of motion : modal basis solution
The solution is the sum of sdof oscillators :
In practice, truncation is needed …
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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
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Discrete versus continuous systems
Orthogonality conditions
Projection in the modal basis
Response to harmonic excitation
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Equivalent continuous systems : beam in bending
- transversal displacement of the neutral axis y(x,t)
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Equivalent continuous systems : beam in bending
p(x,t) = load per unit length on the beamA = surface of the section = density
Equilibrium :
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Beam in bending : mode shapes and eigenfrequencies
General solution (p(x,t)=0) :
Characteristic equation:
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General solution
Example 1: Simply supported beam
Beam in bending : mode shapes and eigenfrequencies
A, B, C and D depend on the boundary conditions
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Beam in bending : mode shapes and eigenfrequencies
Eigenfrequencies :
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Beam in bending : mode shapes and eigenfrequencies
Mode shapes :
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Beam in bending : mode shapes and eigenfrequencies
n=1
n=5 n=10
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General solution
Example 2: Double cantilever beam
Beam in bending : mode shapes and eigenfrequencies
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Beam in bending : mode shapes and eigenfrequencies
Transcendental equation, no analytical solution
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Beam in bending : mode shapes and eigenfrequencies
Roots of the equation : with
Eigenfrequencies :
Asymptotic approach :
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Beam in bending : mode shapes and eigenfrequencies
Mode shapes :
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Beam in bending : mode shapes and eigenfrequencies
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Beam in bending : mode shapes and eigenfrequencies
n=1
n=5 n=10
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Beam in bending : mode shapes and eigenfrequencies
Simple cantilever beam
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Beam in bending : mode shapes and eigenfrequencies
Transcendental equation, no analytical solution
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Beam in bending : mode shapes and eigenfrequencies
Roots of the equation : with
Eigenfrequencies :
Next roots:
First two roots:
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Beam in bending : mode shapes and eigenfrequencies
Mode shapes :
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Beam in bending : mode shapes and eigenfrequencies
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Beam in bending : mode shapes and eigenfrequencies
n=1
n=5
n=2
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Beam in bending : mode shapes and eigenfrequencies
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Beam in bending : orthogonality of the modeshapes
Property :
Proof:
Mode iMode j
=
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Beam in bending : orthogonality of the modeshapes
Mass orthogonality
Stiffness orthogonality
Define
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Beam in bending : projection on the modal basis
Modal basis
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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
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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
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Example of a finite element model of a bridge
(http://fynitesolutions.com)
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Example of a finite element model of a bridge
[http://www.scielo.br]
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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
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Tuned mass damper attached to continuous systems
Finite element model of structure without DVA
Projection on modal basis
Around frequency :
Single mode approximation:
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Tuned mass damper attached to continuous systems
Valid only around where the TMD is active
Equivalent sdof system at point d excited by fd
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Tuned mass damper attached to continuous systems