1Challenge the future

Impulse Based Substructuring Theory, Improvement and Implementation on real problems

Nazgol HaghighatSupervisor: Prof. Dr. Ir. Daniel J. Rixen

2Challenge the future

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)?

3Challenge the future

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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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

Simpler substructures:

Large and complex structures:

Analysing a structure by studying its subparts

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

By using impulse responses and applying convolution product

Analysing each subsystem individually:• Physical domain• Modal domain• Frequency domain• Time domain

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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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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)?

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IBS general frame work

Obtaining Impulse Response Functions (IRFs):

Decomposing the structure into some subsystems:

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 accelerationCan 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 matrixK linearized stiffness matrixu array of degrees of freedomf external applied forces

Newmark time integration scheme:using finite time difference concept

β , γ Newmark parametersu array of DoFsdt size of the time step

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Numerical models of unit impact

IF impact model:

3 different impact models can be defined:

1. Initial applied velocity (IV) 2. Initial applied force (IF) 3. Applied force at second time step (SF)

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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)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

-4

t(s)

H(1

)LL

(t)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.0160

0.002

0.004

0.006

0.008

0.01

0.012

0.014

t(s)

H(2

)00

(t)

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Does IBS really work?

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016-2

0

2

4

6

8

10

12

14

16x 10

-9

time(s)

Ux (t

)

IBSIF

NI

Dynamic response of the bar system under an excitation described by unit step

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016-8

-6

-4

-2

0

2

4

6

8x 10

-8

time(s)

U x (t)

IBSIF

NI

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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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 IRFs2. Costs in computing convolution product

Suggested solution : Truncating Impulse Response Functions

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Applying Truncation on Different types of IRFs

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

-4

t(s)

H(1

)LL

(t)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.0160

0.002

0.004

0.006

0.008

0.01

0.012

0.014

t(s)

H(2)

00(t)

• Boundary DoFs are fixed

• IRFs are damped with time

Non-floating (sub)structure Floating (sub)structure

• Boundary DoFs are not fixed• IRFs are increased with time

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Truncating IRFs of a non-floating (sub)structure

Multiplying IRFs by a window functionDifferent types of window functions

• Rectangular window • Cosine Window

Effects of applying cosine window function on IRF of a non-floating structure

Amax maximum amplitudeA(t) amplitude at pickα 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)?

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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

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

-8

time(s)

u 11(t)

IBSNI

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

300 302 304 306 308 310 312 314 316 318 3202.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1x 10

-8

time(s)

u 11(t)

IBSNI

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