Perspective and Design Aspects ... (practical issues such as installation and design are affected by depth) ... Serviceability limit state (SLS, safe operation).
Dynamics and Vibration of Offshore Energy Structures
Dr Madjid Karimirad
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Outline MARINTEK Introduction Offshore Energy Structures Perspective and Design Aspects Why dynamics/vibration? Examples and case studies Dynamic analyses Aero-hydro-servo-elastic Damping (positive and negative) Wave-induced and Wave-wind-induced Integrated coupled dynamics Conclusions
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Dr Madjid Karimirad
Offshore Engineering: exciting and challenging business
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Dr Madjid Karimirad
Oil and Gas, Maritime and Ocean industries Hydrodynamic, Structural dynamics, Fluid and Structural Mechanics
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Dr Madjid Karimirad
Expertise in offshore technology Laboratory testing
Normal operation and extreme loads Global motion responses, structural responses, fatigue loading on component /structures Mooring and cable system, Marine operations, installation, support vessels and access systems
Numerical simulations and developments Logistics support analysis and maintenance strategies Coupled dynamic simulations
Estimation of loads, motions and power generation Mooring-system optimization Sea keeping and vessel performance Weather-sensitive marine operations
Full scale measurements (structural loads and motions) Dr Madjid Karimirad
Introduction Offshore renewable energy structures, offshore wind, ocean energy (wave energy converters and tidal turbines), have a phenomenal development.
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Dr Madjid Karimirad
Offshore Renewable Energy Structures Wind turbines (WT) Wave energy converters (WEC) Tidal turbines Hybrid devices
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Dr Madjid Karimirad
Deep Offshore
Onshore: • visual and noise impacts • best sites are already in use Offshore: • cannot be seen or heard • offshore wind is steadier/stronger (more electricity) Good-potential in moderate and deep water:
floating offshore structures
Similar to what experienced in the offshore oil and gas technology.
• Cost-effective solution: (cost of fixed turbines increases with water depth) (practical issues such as installation and design are affected by depth) • Conflicts with tourism, military and naval forces, sailing and shipping are less
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Dr Madjid Karimirad
Perspective and Design
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Dr Madjid Karimirad
Why dynamics and vibration? • • • • • • • • • •
Aerodynamic Hydrodynamic Extreme loads Wave-wind-induced Structural dynamics Marine operation Slamming, Ringing, Springing and Whipping Sloshing ….
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Dr Madjid Karimirad
Challenges
Offshore Energy Structures are Aero-Hydro-Servo-Elastic Multi-Body System
Coupling is important: Aerodynamic and Hydrodynamic damping and excitation forces are highly affected by each other through relative motions.
Influencing 1) Power Production 2) Structural integrity FLS, ULS, ALS and SLS Motion responses Structural dynamic response Fault conditions 3) Functionality 4) Several design load cases
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Dr Madjid Karimirad
Modeling and theory Hydrodynamics
Aerodynamics
Structure
DLL interface for mooring system action, Full modeling of mooring systems
Control Aero-hydro-servo-elastic
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Dr Madjid Karimirad
Limit states and load cases Ultimate limit state (ULS, extreme loads) Serviceability limit state (SLS, safe operation) Accidental limit state (ALS, fault conditions) Fatigue limit state (FLS, fatigue damage) Power production Power production plus fault condition Start-up Normal shutdown Emergency shutdown Parked Parked plus fault condition Transport, assembly, maintenance and repair
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Dr Madjid Karimirad
Structural dynamics and motion responses Rigid body motions
Elastic deflections Eigen modes
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Dr Madjid Karimirad
Dynamics of mooring and foundation Mooring system
Foundation: soil-pile
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Dr Madjid Karimirad
Dynamics/vibration of mechanical components Pitch and yaw mechanisms
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Dr Madjid Karimirad
Dynamics /vibration of mechanical systems
Wavestar: attenuator WEC
Flow induced vibration (FIV) Hydraulics system and cooling unit Water hammer (fluid hammer) Multiphase flow
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Dr Madjid Karimirad
Dynamics /vibration of components Mechanical components, Drivetrain, Gear box, low/high speed shafts, mechanical brake
Fatigue and wear
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Dr Madjid Karimirad
VIV and VIM Vortex induced vibration (VIV), elastic response Fluid structure interaction (FSI) Hydroelasticity
Slender structures: Risers Power cables and umbilica Mooring lines
Vortex induced motion (VIM), Rigid body motions
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Dr Madjid Karimirad
Dynamic interaction Hybrid platforms Coupled wave and wind Hydrodynamic coupling Shared mooring and anchoring Synergy
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Dr Madjid Karimirad
Model testing considering dynamics/vibration
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Dr Madjid Karimirad
Dynamic response analyses Single Degree of Freedom (SDOF)
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Single Degree of Freedom (SDOF) An idealized, one degree of freedom spring-mass-damper system. Equation of motion: balancing the inertia, stiffness, damping and external forces, Newton laws Lagrange equations: “Lagrangian” L is defined (𝐿 = 𝐸𝐾 − 𝐸𝑃 ).
Lagrange equations for conservative systems: 𝑑 𝜕𝜕 𝑑𝑑 𝜕𝑞̇
−
𝜕𝜕 𝜕𝜕
=𝑄
q is noting to generalized motion and Q is indicating generalized forces. For SDOF, the kinetic, potential energies and generalized forces are:
𝐸𝐾 (𝑥, 𝑥̇ ) = 0.5𝑚𝑥̇ 2 𝐸𝑃 (𝑥) = 0.5𝑘𝑥 2 Lagrange equation for such system: 𝑄 = 𝑓 − 𝑐𝑥̇ 𝑑 𝜕𝐸𝐾 𝑑𝑑 𝜕𝑥̇
−
𝜕𝐸𝐾 𝜕𝜕
+
𝑚𝑥̈ 𝑡 + 𝑐𝑥̇ 𝑡 + 𝑘𝑘 𝑡 = 𝑓 𝑡 ,
𝜕𝐸𝑃 𝜕𝜕
+ 𝑐𝑥̇ − 𝑓 = 0
𝑥 0 = 𝑥0 𝑎𝑎𝑎 𝑥̇ 0 = 𝑣0 23
Dr Madjid Karimirad
Free Vibration Solution has two parts: a free response and a forced response. For free vibrating, external force is zero (𝑓 𝑡 = 0). Assuming response as 𝑥 𝑡 = 𝑋𝑒 𝜇𝜇 :
(𝑚𝜇2 + 𝑐𝑐 + 𝑘)𝑋𝑒 𝜇𝜇 = 0
Non-trivial solution is 𝑚𝜇 2 + 𝑐𝑐 + 𝑘 = 0, has two roots:
𝜇1,2 = −
𝑐 2𝑚
𝑐 2 𝑘 ) − 2𝑚 𝑚
± (
Initial displacement and velocity are used to determine coefficients 𝑋1 𝑎𝑎𝑎 𝑋2 for solution of second order ordinary differential equation (ODE) corresponding to 𝜇1 𝑎𝑎𝑎 𝜇2 . Dynamic response, 𝜇1 𝑎𝑎𝑎 𝜇2 are strongly affected by system damping (𝑐). Four cases are possible depending to damping magnitude.
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Dr Madjid Karimirad
Decay tests
𝜔𝑛 =
Under-damped system, 0 < 𝑐 < cc
𝑘/𝑚
𝜔𝑛
𝑐𝑐 = 2 𝑚𝑚
𝜁 = 𝑐/𝑐𝑐
𝜁 2 − 1 is “damped natural frequency”, (𝜔𝑑 )
𝑣 +𝜁𝜔𝑛 𝑥0 Using initial displacement and velocity: 𝑥 𝑡 = 𝑒 −𝜁𝜔𝑛 𝑡 ( 0 𝑠𝑠𝑠𝜔𝑑 𝑡 + 𝑥0 𝑐𝑐𝑐𝜔𝑑 𝑡) 𝜔𝑑
Decay test is performed to find natural frequency and damping (initial velocity is zero).
Logarithmic decrement method is used to calculate damping ratios as a function of two succeeding response amplitudes (𝑋𝑖 > 𝑋𝑖+1 for a system with positive-damping). 𝜁=
𝑐 𝑐𝑐
= 𝛿/ 𝜋 2 + 𝛿 2 𝑋𝑖
𝛿 = ln(𝑋
𝑖+1
)
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Dr Madjid Karimirad
Forced vibration When structures are subjected to loads, responses compose: steady state and transient.
Transient response decays with decay frequency (𝜔𝑑 ) while steady state response oscillates with external load frequency (𝜔). Assume Harmonic force, 𝑓 𝑡 = 𝐹𝐹𝐹𝐹𝐹𝐹, forcing-frequency (𝜔). After several cycles, system responds only at external forcing-frequency. Harmonic steady-state response can be assumed as: 𝑥 𝑡 = 𝐴𝐴𝐴𝐴𝐴𝐴 + 𝐵𝐵𝐵𝐵𝐵𝐵. 𝑘 − 𝑚𝜔2 𝐴= 𝐹 (𝑘 − 𝑚𝜔 2 )2 +(𝑐𝑐)2
&
𝐵=
𝑐𝑐 𝐹 (𝑘 − 𝑚𝜔 2 )2 +(𝑐𝑐)2
x t = Acosωt + Bsinωt = Xcos ωt + φ in which the amplitude of motion (X) is A2 + B 2 . Phase between applied force and response is: φ =
−B A
=−
cω k−mω2
Response always lags the external forcing (𝜑 is negative). 26
Dr Madjid Karimirad
Dynamic amplification factor (DAF)
Assuming response and external force as 𝑥 𝑡 = 𝑋𝑒 𝑖𝑖𝑖 and 𝑓 𝑡 = 𝐹𝑒 𝑖𝑖𝑖 , and re-derive: F
Static displacement (Xst ) is , hence: k 𝑋 1 = 𝑋𝑠𝑠 1−𝜛2 +𝑖 2𝜁𝜁
𝑋 𝐹
, 𝜛=
𝜔 𝜔𝑛
=
1
𝑘−𝑚𝜔2 +𝑖(𝑐𝑐)
Ratio of dynamic response amplitude and static amplitude is “dynamic amplification factor”, DAF=
𝑋 𝑋𝑠𝑠
.
by which static displacement responses are amplified for dynamic external forcing.
Damping ratio for mechanical systems (lightly damped) is around 2%. Offshore bottom-fixed structures such as jackets: the damping ratio of 5% is practical. Wind turbine rotor aerodynamic-damping is around 10%. 27
Dr Madjid Karimirad
Multi Degree of Freedom Systems
Dynamic response equation
Dynamic time domain analysis
R I ( r , r , t ) + R D ( r , r, t ) + R S ( r , t ) = R E ( r , r, t )
Inertia
Damping
Internal structural reaction
External
Eigen-Value Analysis N-degrees of freedom system • Lumped mass method • Finite element method Natural Periods of Floating Structures
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Dr Madjid Karimirad
Hydrodynamics 5000 Fsurge Fsway Fheave
4000 3000
Hydro-force (kN)
2000
Radiation and Diffraction
1000 0 -1000 -2000 -3000 -4000 -5000 500
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x 10
550
600
650
700
750 time (sec)
800
850
900
1000
6
Fsurge Fsway Fheave
16 14 12
2
spec Hydro-force (kN /rad/sec)
950
10 8 6 4 2 0 0
0.2
0.4
0.6
0.8
1 1.2 frequency (rad/sec)
1.4
1.6
1.8
2
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Dr Madjid Karimirad
Aerodynamics
500
400
Aero-force (kN)
300
200
Fheave Fsway Fsurge
100
0
-100 500
550
600
650
700
750 time (sec)
800
850
900
950
1000
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Dr Madjid Karimirad
Aerodynamic and hydrodynamic damping
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Dr Madjid Karimirad
Control Variable-speed generator-torque controller and PI collective blade pitch-controller Acts based on the relative wind velocity Operating wind turbine, active control 800 700 Maximum power
Thrust (kN)
600
Constant power
500 400 300 200 100 0
0
5
10
15
20
25
Relative wind speed (m/sec)
Instabilities: very large motions with a wide range of frequencies
Servo-induced negative damping?
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Ameliorating negative damping controller frequency should be less than the natural frequency floating wind turbine
Tuning just affects the over-rated wind speed without affecting the below-rated wind speed
Dynamic system: the rigid body rotation of the rotor coupled to the aerodynamics and PI-control of the blade pitch angle.
Very important for: •Fatigue limit state (FLS) •Power production •Drive train loads
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Wave-induced vs. Wave-Wind-induced
Using the same irregular waves mean of responses are primarily wind-induced, standard deviations of responses are primarily wave-induced
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Integrated analysis- Turbulent wind
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Power
2
x 10
6
spec powerR (kW2/rad/sec)
Wave (head sea), Wind (head sea) Wave (quarter sea), Wind (head sea) Wave (beam sea), Wind (head sea)
Wind-induced
1.8 1.6
Slowly-varying responses
1.4 1.2 1 Wave-induced
0.8
3P rotor harmonics
0.6 0.4 0.2 0.5
1
1.5
2 2.5 frequency (rad/sec)
3
3.5
4
4.5
4 2
3.5 3 2.5 2 1.5
1P rotor harmonic
edgewise flapwise
1 relative balde tip elastic deflections (m)
spec blade-tip-deflections (m2/rad/sec)
0 0
1 0.5 0 0
0 Wind-induced, Slowly-varying motions -1 -2 2P rotor harmonic
-3
edgewise flapwise
Wave-induced
-4 500
600
550
0.5
1
650
1.5
700
750 time (sec)
850
800
2 2.5 frequency (rad/sec)
3
3P rotor harmonic
900
3.5
950
4
1000
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Dr Madjid Karimirad
Dynamics/vibration analysis for EPCI projects Engineering and Design Procurement and Construction Transpiration and Installation Marine operations Operation and Maintenance …
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Dr Madjid Karimirad
Fatigue Damage Assessment: A Stochastic Dynamic Analysis
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Real-Time Hybrid Model Testing • Mass distribution • Numerical model • Actuators M
Motions, and loads
Froude scaling
Full-scale
Model-scale
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Dr Madjid Karimirad
Conclusions
For functionality and structural integrity, the dynamics, vibration and structural responses are needed. Reliable design requires accurate calculation of dynamic loads/responses.
Offshore structures are complicated, respect to dependency of loads and load-effects, dynamics and vibration; i.e. Aero-Hydro-Servo-Elastic Multi-Body Systems. Dynamic analysis is introduced; starting with a SDOF system. Equation of motion and its characteristics are explained. Aerodynamics, Hydrodynamics and Servo actions are briefly discussed. Damping (both positive and "negative") are illustrated. Wave-induced, Wind-Wave-induced and Integrated time-domain dynamic and vibration analyses are introduced. Dynamic response analysis is the base for design of offshore structures; and, normally, limit states analyses are based on combinations of individual dynamic analysis, i.e. FLS (Fatigue Limit State) is based on accumulated damages. This shows importance of performing correct dynamic and vibration analyses. Model testing and engineering aspects of dynamic/vibration analyses are highlighted. 40
Dr Madjid Karimirad
Thanks for your attention
Norsk Marinteknisk Forskningsinstitutt
