Exercise 2 - Politecnico di Milano
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Transcript Exercise 2 - Politecnico di Milano
Buckling and harmonic analysis with FEM
E. Tarallo, G. Mastinu
POLITECNICO DI MILANO, Dipartimento di Meccanica
Summary
2
Subjects covered in this tutorial
An introduction to linear perturbation analysis
An introduction to buckling analysis
An introduction to modal analysis (frequency and
complex)
A guided example to evaluate the harmonic response of a
simple structure
Other few exercises (to include in exercises-book)
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Linear perturbation - buckling
3
Linear perturbation means impose a δq around the equilibrium position
A general dynamic system is described fully by the basic equation:
M q Rq K q Q
In a general static problem, Abaqus solves the following equation:
K q Q
The buckling solver is generally used to estimate the critical (bifurcation)
load of “stiff” structures; Abaqus solves the following equation:
K
MN
0
i KMN viMN 0
The buckling analysis includes the effects of preloads (force, moment,
pressure)
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Linear perturbation – modal analysis
4
Starting from general dynamic equation:
M q Rq K q Q
in the “frequency” analysis, Abaqus solves the following equation:
M
M q K q 0
2
MN
K MN MN 0
The “frequency” analysis doesn’t include the effects of loads and damping
Following the “frequency” analysis is possible to perform a “complex”
analysis where the damping (structural and contact effects) is taken into
account.
M
2
MN
iR MN K MN MN 0
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Exercise 1 - buckling
F
5
F
T
Part: 2D beam planar
Material: E=210 GPa, ν=0.3
Section: circular radius 10 mm
Load F: 1 kN
Boundary: bottom U1=U2=0; top U1=0
Problem:
1. Perform buckling analysis with 1 step
2. Add 1 static step with Load T=100 kN and
perform buckling analysis with 2 steps
3. Compare the results btw the analysis
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Exercise 1 – results 1st configuration
1st freq: 1449 Hz
2nd freq: 4852 Hz
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3rd freq: 8504 Hz
Exercise 1 – results 2nd configuration
1st freq: 14.5 Hz
2nd freq: 48.5 Hz
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3rd freq: 85.04 Hz
Exercise 2 – Modal analysis
T
8
Part: 2D beam, L=1000 mm
Section: circular, R=10 mm
Material: E=210 GPa, ν=0.3,
ρ=7800 kg/m3
Boundary: encastre
Analysis: Frequency, Steady-state
dynamic, Dynamic-Implicit
1) Frequency analysis: find first 5 natural frequency
2) Steady-state dynamic: T=-1 kN; frequency range=[1,800] Hz
3) Harmonic response: T=-1000sin(ft) where f=1,100,1000 Hz
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Exercise 2 – definition of frequency and
steady-state steps
Natural Frequencies:
Dynamic Response:
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Exercise 2 – definition of harmonic step
Harmonic Response:
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Exercise 2 – results (1)
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Exercise 2 – results (2)
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Exercise 2 – results (3)
13
1Hz
100Hz
1000Hz
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