Transcript Missie en Visie TU Delft

Impulse Based Substructuring
Theory, Improvement and Implementation on real problems
Nazgol Haghighat
Supervisor: Prof. Dr. Ir. Daniel J. Rixen
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Outlines
• What is IBS?
• Why is IBS important?
• How does IBS work?
• How to apply IBS for longer time simulations?
• Can it be applied on real problems (3D structures)?
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Outlines
• What is IBS?
• Why is IBS important?
• How does IBS work?
• How to apply IBS for longer time simulations?
• Can it be applied on real problems (3D structures)?
Challenge the future
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What is IBS?
• IBS is one of the Dynamic
Substructuring methods
• IBS can be applied to study
performance of a system with time
(Dynamic Analysis)
• IBS is working in time domain and
can be used instead of numerical time
integration (Newmark)
• IBS can be applied only on linear
systems
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Dynamic substructuring
Analysing a structure by studying its subparts
Large and complex structures:
Simpler substructures:
Assembling the substructures:
By considering interface forces (λ)
• They must be in equilibrium
• They must satisfy compatibility condition at interface
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Dynamic substructuring techniques
Analysing each subsystem individually:
•
•
•
•
Physical domain
Modal domain
Frequency domain
Time domain
By using impulse responses and
applying convolution product
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Outlines
• What is IBS?
• Why is IBS important?
• How does IBS work?
• How to apply IBS for longer time simulations?
• Can it be applied on real problems (3D structures)?
Challenge the future
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Why IBS is important?
• Can provide dynamic response of large and complex
structures
• Offers advantages in shock analysis compared to
frequency based substructuring
• Can be implemented easily
• Provides the possibility of analysing a system in the
basic design steps
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Outlines
• What is IBS?
• Why is IBS important?
• How does IBS work?
• How to apply IBS for longer time simulations?
• Can it be applied on real problems (3D structures)?
Challenge the future
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IBS general frame work
Decomposing the structure
into some subsystems:
Obtaining Impulse
Response Functions
(IRFs):
Assembling substructures:
(Assembly Equation)
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Obtaining Impulse Response Functions (IRFs)
1. Applying impact at the input or interface forces positions;
Unit Impact
Can be interpreted as a short vary high acceleration
Can be modelled:
- Experimentally
Hammer impact
- Numerically
Defining special initial conditions in time integration of
motion equation
2. Measuring or computing dynamic response of the system under a
unit impact
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Numerical time integration (Newmark)
Linearized motion equation:
M linearized mass matrix
C linearized damping matrix
K linearized stiffness matrix
u array of degrees of freedom
f
external applied forces
Newmark time integration scheme:
using finite time difference concept
β,γ
u
dt
Newmark parameters
array of DoFs
size of the time step
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Numerical models of unit impact
3 different impact models can be defined:
1.
2.
3.
Initial applied velocity (IV)
Initial applied force (IF)
Applied force at second time step (SF)
IF impact model:
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Discretizing the input force using IF impact
model
Discretizing the input force with IF impact model at
each time step
IBS assembly equation (using IF impact model)
Convolution product :
Compatibility condition :
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Computed IRFs under IF impact model
(Applying Newmark time integration)
-4
1
x 10
0.8
0.6
0.4
LL
H(1)(t)
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.002
0.004
0.006
0.008
t(s)
0.01
0.012
0.014
0.016
0.012
0.014
0.016
0.014
0.012
0.008
00
H(2)(t)
0.01
0.006
0.004
0.002
0
0
0.002
0.004
0.006
0.008
t(s)
0.01
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Does IBS really work?
-9
16
-8
x 10
8
x 10
IBSIF
14
IBSIF
6
NI
12
NI
4
10
U (t)
x
x
U (t)
2
8
6
0
-2
4
2
-4
0
-6
-2
0
0.002
0.004
0.006
0.008
time(s)
0.01
0.012
0.014
Dynamic response of the bar system under
an excitation described by unit step
0.016
-8
0
0.002
0.004
0.006
0.008
time(s)
0.01
0.012
0.014
Dynamic response of the bar system under a
periodic excitation
Results of IBS (IF impact model) are exactly equal to results
of Newmark time integration (constant average acceleration)
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0.016
Advantage and disadvantage of IBS
The IRFs (of substructure) can be computed once
and being used several times in different analysis
Convolution product can be applied only for
the same length of time of IRFs
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Outlines
• What is IBS?
• Why is IBS important?
• How does IBS work?
• How to apply IBS for longer time simulations?
• Can it be applied on real problems (3D structures)?
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Is IBS efficient for long time simulations?
1. Costs in computing IRFs
2. Costs in computing convolution product
Suggested solution : Truncating Impulse Response Functions
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Applying Truncation on Different types of
IRFs
Non-floating (sub)structure
Floating (sub)structure
-4
1
x 10
0.014
0.8
0.012
0.6
0.01
0.4
0.008
00
H(2)(t)
LL
H(1)(t)
0.2
0
0.006
-0.2
0.004
-0.4
0.002
-0.6
0
-0.8
• Boundary DoFs are fixed
• IRFs are damped with
time
-1
0
0.002
0.004
0.006
0.008
t(s)
0.01
0.012
0.014
0.016
0
0.002
0.004
0.006
0.008
t(s)
0.01
• Boundary DoFs are not fixed
• IRFs are increased with time
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0.012
0.014
0.016
Truncating IRFs of a non-floating (sub)structure
Multiplying IRFs by a window function
Different types of window functions
• Rectangular window
• Cosine Window
Effects of applying cosine window
function on IRF of a non-floating structure
Amax maximum amplitude
A(t) amplitude at pick
α
design variable
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Truncating IRFs of a floating (sub)structure
IRFfloating system
IRFvibrational mode
IRFpure rigid body mode
IRFpure rigid body mode :
Can be obtained analytically (null Space of stiffness matrix)
IRFvibrational mode = IRFfloating system - IRFpure rigid body mode
IRFvibrational mode : Can be truncated by applying window function
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Outlines
• What is IBS?
• Why is IBS important?
• How does IBS work?
• How to apply IBS for longer time simulations?
• Can it be applied on real problems (3D structures)?
Challenge the future
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Applying IBS on a 3D structure
Problem Definition:
• Offshore wind turbine (3D)
• 2 substructures: Jacket structure and
wind mill (tower +RNA)
• External force: unit step at the interface
• Applied method: Truncated IBS
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Truncated IBS solving procedure
1. Computing the IRFs (jacket structure and wind mill)
2. Removing rigid body impulse response form the IRFs
(windmill)
3. Windowing the vibrational IRFs
4. Computing convolution product using truncated IRFs
5. Adding rigid body responses (wind mill)
6. Computing interface forces
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Results
-8
5
x 10
IBS
NI
4.5
4
3.5
11
u (t)
3
2.5
2
1.5
1
0.5
0
0
50
100
150
200
250
300
time(s)
Dynamic response of DoF(1) at the interface due to a unit
step exciting DoF(1) of the interface (truncation at t=130 (s))
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Results
-8
3.1
x 10
IBS
NI
3
2.9
11
u (t)
2.8
2.7
2.6
2.5
2.4
2.3
300
302
304
306
308
310
time(s)
312
314
316
318
320
Dynamic response of DoF(1) at the interface due to a unit step
exciting DoF(1) of the interface (zoomed at the last 20(s))
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Conclusion and future work
Conclusion :
• Results of IBS (IF impact model) are exactly equal to results of
Newmark time integration (average constant velocity)
• Truncation (cosine window) is a reliable solution for reducing
computation cost of IBS
Future work :
• Studying results of applying IBS on more 3D structures
• Trying more types of window functions
• Improve IBS to cover also non-linear problems
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Rigid body response analytically
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