1. Nazgol Haghighat Supervisor: Prof. Dr. Ir. Daniel J. Rixen
Impulse Based Substructuring Theory, Improvement and Implementation on real problems
Nazgol Haghighat
Supervisor: Prof. Dr. Ir. Daniel J. Rixen

2. 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)?

3. 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)?

4. 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

5. 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

6. Dynamic substructuring techniques
Analysing each subsystem individually:
Physical domain
Modal domain
Frequency domain
Time domain
By using impulse responses and applying convolution product

7. 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)?

8. 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

9. 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)?

10. IBS general frame work Decomposing the structure into some subsystems:
Obtaining Impulse Response Functions (IRFs):
Assembling substructures: (Assembly Equation)

11. Obtaining Impulse Response Functions (IRFs)
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
Measuring or computing dynamic response of the system under a unit impact

12. 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 Newmark parameters
u array of DoFs
dt size of the time step

13. Numerical models of unit impact
3 different impact models can be defined:
Initial applied velocity (IV)
Initial applied force (IF)
Applied force at second time step (SF)
IF impact model:

14. 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 :

15. Computed IRFs under IF impact model (Applying Newmark time integration)

16. Dynamic response of the bar system under a periodic excitation
Does IBS really work?
Dynamic response of the bar system under an excitation described by unit step
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)

17. 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

18. 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)?

19. Is IBS efficient for long time simulations?
Costs in computing IRFs
Costs in computing convolution product
Suggested solution : Truncating Impulse Response Functions

20. Applying Truncation on Different types of IRFs
Non-floating (sub)structure
Floating (sub)structure
Boundary DoFs are fixed
IRFs are damped with time
Boundary DoFs are not fixed
IRFs are increased with time

21. 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

22. 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

23. 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)?

24. 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

25. Truncated IBS solving procedure
Computing the IRFs (jacket structure and wind mill)
Removing rigid body impulse response form the IRFs (windmill)
Windowing the vibrational IRFs
Computing convolution product using truncated IRFs
Adding rigid body responses (wind mill)
Computing interface forces

26. Results
Dynamic response of DoF(1) at the interface due to a unit step exciting DoF(1) of the interface (truncation at t=130 (s))

27. Results
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))

28. Conclusion and future work
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

30. Rigid body response analytically