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

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)?

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)?

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

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

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

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)?

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

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)?

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

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

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

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:

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 :

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

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)

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

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)?

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

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

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

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

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)?

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

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

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))

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))

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

Rigid body response analytically