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Contract No. DE-AC36-08GO28308

A New Structural-Dynamics 

Module for Offshore 

Multimember Substructures 

within the Wind Turbine 

Computer-Aided Engineering 

Tool FAST

Preprint

Huimin Song, Rick Damiani, Amy Robertson, 

and Jason Jonkman

Presented at the 23rd International Ocean, Offshore and Polar 

Engineering Conference – ISOPE 2013

Anchorage, Alaska

June 30 – July 5, 2013

Conference Paper

NREL/CP-5000-58093

August 2013
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A New Structural-Dynamics Module for Offshore Multimember Substructures within the Wind Turbine 

Computer-Aided Engineering Tool FAST

Huimin Song, Rick Damiani, Amy Robertson, Jason Jonkman 

National Renewable Energy Laboratory

Golden, CO, USA

ABSTRACT

FAST, developed by the National Renewable Energy Laboratory, is a 

computer-aided engineering tool for aero-hydro-servo-elastic analysis 

of land-based and offshore wind turbines. This paper discusses the 

development of a new module (SubDyn), which when integrated to 

FAST, will enable loads simulations of offshore wind turbines with 

fixed-bottom, multimember support structures. The main theory and 

strategies for the implementation of the multimember substructure 

dynamics module (SubDyn) within the new FAST modularization 

framework are introduced. SubDyn relies on two main engineering 

schematizations: (1) a linear frame finite-element beam model (LFEB), 

and (2) a dynamics system reduction via Craig-Bampton’s method. To 

test the SubDyn module and preliminarily assess the results against 

results from a commercial finite-element code, we analyzed a jacket 

support structure on its own and as part of an offshore system 

supporting a 5-MW turbine. The preliminary results show good 

agreement between the two codes.

KEY WORDS: Offshore; wind turbine; multimember; support 

structure; beam finite element; dynamic reduction; state space. 

INTRODUCTION

Recently, there has been significant interest in offshore turbine support 

structures. Wind turbines are being placed further offshore because of 

the better wind resource, scarcity of land available for development, 

and reduced visual impact. The United States alone possesses large and 

accessible offshore wind resources that can provide a considerable 

amount of electricity-generating capacity. In fact, the National 

Renewable Energy Laboratory (NREL) estimates that the U.S. offshore 

winds have a gross potential capacity that is four times greater than the 

nation’s present electric capacity (Musial and Ram, 2010). 

Additionally, different geographic locations require different support 

structure designs. In shallow water, where the water depth is less than 

30 meters (m), monopiles and gravity-based substructures that extend 

to the seabed may be used. However, at transitional water depths of 30 

to 60 m, monopiles are no longer economical and may exceed the

manufacturing limit. As a result, fixed-bottom, multimember support 

structures such as jackets and tripods may be used (for water depth that 

is greater than 60 m, floating platforms may be required). This paper 

focuses on the analysis of support structures commonly used in 

transitional water depths. 

Many tools have been developed to analyze offshore wind turbines with 

jacket type support structures. FAST is one such tool. Developed by 

NREL, FAST is a computer-aided engineering (CAE) tool for aerohydro-servo-elastic analysis of land-based and offshore wind turbines.

Previously, FAST only had the capability of examining land-based and 

offshore wind turbines that were placed on either floating hulls or

monopile substructures. But recent modifications made to FAST—

through an updated platform hydrodynamics module (HydroDyn) 

(Song et al., 2012)provide new capabilities to model the 

hydrodynamic loading that is imparted to the offshore multimember 

supports by waves and currents. This paper focuses on further efforts to 

enable the modeling of multimember structures in FAST, specifically, 

the development of structural-dynamics modeling of multimember 

support structures. In particular, we introduce the main theory and 

strategies used to implement SubDyn, the new multimember 

substructure dynamics module within FAST. SubDyn relies on a linear 

frame finite-element beam model (LFEB) that has the option of using a 

dynamics system reduction via the Craig-Bampton method.

LFEB can accommodate different element types, including EulerBernoulli and Timoshenko beam elements (the latter account for shear 

deformation and are better suited to represent low aspect ratio beams

that may be used within frames and to transfer the loads within the 

frame) of either constant or longitudinally tapered cross sections. A 

previous study by Damiani et al. (2013a) has shown that linear theory 

yields acceptable results when analyzing wind turbine substructures. 

The large number of degrees-of-freedom (~103) associated with a 

standard finite-element analysis of a typical multimember structure 

would hamper computational efficiency during wind turbine system 

dynamic simulations. As a result, for this study, we implemented a

Craig-Bampton system reduction to speed up processing time while 

retaining a high level of fidelity in the overall system response. 

NREL has recently put considerable effort into improving the overall 

modularity of FAST (Jonkman 2013a). The SubDyn module developed 

and presented in this paper will become a new module that can be 
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at www.nrel.gov/publications.

integrated into the FAST modularization framework through loose or 

tight coupling (including full-system linearization), thereby allowing 

simulations of offshore wind turbine systems in transitional water 

depths under combined wind, wave, and current loading, and including 

hydro-elastic effects. In the new FAST framework, the SubDyn module 

is called by the FAST driver program. In a loosely coupled timeintegration scheme, the FAST driver program transfers data such as 

hydrodynamic forces, substructure responses, loads transmitted to the 

transition piece, and transition piece responses among the SubDyn, 

HydroDyn, and ElastoDyn (a structural (elastic) dynamics module that

models the tower, rotor, nacelle, drivetrain, and the transition piece) 

modules at each time step. Within this scheme, SubDyn tracks its states 

and integrates its equations through its own solver. In a tightly coupled 

time-integration scheme, SubDyn sets up its own equations, but its

states and those of other modules are tracked and integrated by a solver 

that is common to all of the modules.

SubDyn may also be used as a stand-alone module to analyze the 

substructure independently from the rest of the wind turbine system. In 

addition, SubDyn, as a substructure finite-element analysis module, is 

being integrated inside the jacket sizing tool (Damiani and Song, 

2013b) within the NREL wind energy systems engineering initiative. 

In this study, a jacket substructure both on its own and as part of a 5-

MW offshore wind system were analyzed to test SubDyn’s substructure 

integration with the transition piece, tower, and rotor-nacelle/assembly 

(RNA), and to preliminarily assess the results against the commercial

software ANSYS (ANSYS, 2011). For this assessment, the natural 

frequencies and static deflections were computed for the jacket and the 

simplified turbine system. Preliminary verification of the results was 

performed against results from ANSYS models. 

The paper layout is as follows. First we introduce the SubDyn module 

within the FAST modularization framework and its relation with other 

modules. Second, we introduce the SubDyn main theory including the 

LFEB and the dynamic reduction using the Craig-Bampton method. 

The state-space formulations that are to be used in the time-domain 

simulation are then introduced. Finally, two examples are presented to 

show the preliminary results of the SubDyn module.

SUBDYN WITHIN THE FAST MODULARIZATION 

FRAMEWORK 

NREL’s core CAE tool, FAST (Jonkman and Buhl, 2005, Jonkman, 

2013b) is based on advanced engineering models, derived from 

fundamental laws of physics, but with appropriate simplifications and 

assumptions, and supplemented with computational solutions and test 

data (where applicable). 

FAST joins AeroDyn (a rotor aerodynamics module) (Laino and 

Hansen, 2002; Moriarty and Hansen, 2005); HydroDyn (Jonkman, 

2007, 2009; Song et al., 2012) for offshore systems; SubDyn; 

ServoDyn (a control and electrical system (servo) dynamics module); 

and ElastoDyn to enable coupled nonlinear aero-hydro-servo-elastic 

analysis in the time domain. The FAST tool allows users to analyze a 

wide range of wind turbine configurations, including two- or threebladed horizontal-axis rotors, pitch or stall regulation, rigid or teetering 

hubs, upwind or downwind rotors, and lattice or tubular towers. The 

wind turbine can be modeled on land or offshore on fixed-bottom or 

floating substructures.

NREL has recently put considerable effort into improving the overall 

modularity of FAST to enhance its ability to read, implement, and 

maintain source code; increase module sharing and shared code 

development across the wind community; enrich numerical 

performance and robustness; and greatly enhance flexibility and 

expandability to enable further developments of functionality without 

the need to re-code established modules (Jonkman, 2013a). The new 

FAST modularization framework supports module-independent inputs, 

outputs, states, and parameters; states in continuous-time, discrete-time, 

and in constraint form; loose and tight coupling; independent time and 

spatial discretizations; time marching, operating-point determination, 

and linearization; data encapsulation; dynamic allocation; and 

save/retrieve capabilities. One of the benefits of a modular framework 

is that it allows modules to be interchanged, which can be important for 

benchmarking, research, and industrial applications because the 

required model fidelity is reliant on the application.

Fig. 1: SubDyn layout. 

SubDyn is a substructure finite-element module for analyzing 

multimember substructure static and dynamic responses. The 

substructure is considered to be clamped at the seabed and rigidly 

connected to the transition piece at the substructure top nodes (interface 

nodes). Other restraints can be implemented via the input file, with a 

soil-pile interaction module intended for the future. When integrated 

with FAST, loads and responses are transferred between SubDyn, 

HydroDyn, and ElastoDyn via the FAST driver program. At the 

interface nodes, the transition piece displacement, rotation, velocity, 

and acceleration are inputs from the ElastoDyn module and the reaction 

forces at the transition piece are outputs of the SubDyn module. 

SubDyn also outputs the substructure displacements and velocities to 

the HydroDyn module to calculate the hydrodynamic forces that are

inputs for the SubDyn module. In addition, the SubDyn module can 

calculate the member forces, as requested by the user. Figure 1 shows 

the basic layout of the SubDyn module.

The input file defines the substructure geometry, material properties, 

and constraints. Users can define element types; full finite-element 

mode or Craig-Bampton reduction; the number of modes to be retained

in the Craig-Bampton reduction; modal damping coefficients; and the 

number of elements for each member.
3 This report is available at no cost from the 

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SubDyn relies on two main engineering schematizations: (1) a linear 

frame finite-element beam model (LFEB), and (2) a dynamics system 

reduction via Craig-Bampton’s method.

Nonlinearities come from many aspects, including material 

nonlinearity, axial shortening due to bending, large displacement, and 

so on. The material nonlinearity is not considered here because most of 

the offshore multimember support structures are designed to use steel 

and the maximum stress is intended to be below the yield strength of 

the material. Damiani et al. (2013a) showed that a linear finite-element

method was suitable when analyzing wind turbine substructures, in 

which several wind turbine configurations that varied in base geometry,

load paths, sizes, supported towers, and turbine masses were analyzed 

under extreme loads using nonlinear and linear models. The results

revealed that the nonlinear behavior was mainly caused by the monotower response and had little effect on the multimember support 

structures. Therefore, an LFEB model for the substructure was 

considered appropriate within the FAST modularization framework. 

The dynamic analysis of the wind turbine system requires numerous 

time-step calculations. Yet, the large number of degrees of freedom 

(DOF) associated with a standard finite-element analysis of a typical 

multimember structure could hamper computational efficiency during 

wind turbine system dynamic simulations. Therefore, a Craig-Bampton 

methodology is used to re-characterize the substructure finite-element

model into a reduced DOF model that maintains the fundamental lowfrequency response modes of the structure. In the FAST initialization 

step, SubDyn reduces the large substructure physical degrees of 

freedom (displacements) to a small number of modal DOFs and 

interface (boundary) DOFs, and during each time step, only these DOFs 

and associated velocities need to be calculated.

The state-space formulation sets the equation of motion of the 

substructure system with physical DOFs at the interface and modal 

DOFs representing all interior motions. At each time step, loads are 

exchanged between modules through the driver code; the modal 

responses are calculated inside the SubDyn state-space model; and the 

next time-step responses are calculated by the SubDyn integrator for 

loose coupling and the system integrator for tight coupling.

LINEAR FRAME FINITE-ELEMENT BEAM MODEL

In SubDyn, the LFEB can accommodate different two-node beam 

element types, including Euler-Bernoulli and Timoshenko beam 

elements, either of constant or tapered cross sections. The uniform and 

tapered Euler-Bernoulli beam elements are displacement-based and use

third-order interpolation functions that guarantee the displacement and 

rotation continuity between elements. The uniform Timoshenko beam 

element is derived by introducing the shear deformation into the 

uniform Euler-Bernoulli element, so the displacements are represented 

by third-order interpolation functions as well. For the two-node-tapered 

Timoshenko element, the displacements and rotations are represented 

using first-order interpolation functions, respectively. Therefore, when 

the beam element is thin, the low-order displacement representation 

will result in very stiff behavior (also known as shear locking). To 

minimize the shear-locking effects when using two-node Timoshenko 

tapered elements, we recommend applying reduced integration or a fine 

discretization, or applying three-node- or four-node-tapered elements 

where the displacements are represented by higher order interpolation 

functions. Currently, only a two-node-tapered Timoshenko element is 

provided; three- or four-node-tapered Timoshenko elements will be 

implemented in the future.

Many researchers have presented theories on finite-element formulation 

on tapered Timoshenko beams, such as Greenhill et al. (1985), Hinnant

(1989), and Rao and Gupta (2001). In their research, bending stiffness 

was considered to change only through the section geometry along the 

beam axis. In other words, for a homogeneous, isotropic-tapered beam, 

the bending stiffness EI is simply written as EI(x), where I varies along 

the axis of the beam due to the taper. Many recent studies follow the 

same methodology, such as Abdel-Jaber et al. (2008), Das et al. (2009), 

and De Rosa et al. (2010), where the bending strain energy is simply 

written as EI(x)κ2

/2. In this paper, the tapered beam elements were 

developed following the same methodology and only for circular crosssection beam elements. 

However, Hodges et al. (2008, 2010) pointed out that the bending 

stiffness depends on the taper, which leads to bending and shear 

coupling for a tapered beam with rectangular cross sections. As a result, 

correction factors are required for the bending stiffness when 

computing the strain energy, and the effect of correction factors is 

found to be significant, especially for deep beams. There is no research 

that shows the effect of taper on circular cross sections and further 

research is needed to assess the correction factors for bending stiffness 

associated with circular cross sections. This type of research will be 

considered in a future investigation. 

DYNAMIC SYSTEM REDUCTION

The dynamic analysis of a wind turbine system requires numerous 

time-step calculations. For the multimember structure, the number of 

DOFs can be in the thousands, thereby hampering the computational 

efficiency of wind turbine system dynamic simulations. A typical wind 

turbine system model in FAST has about 20 DOFs. Therefore, a CraigBampton methodology is used to re-characterize the substructure finiteelement model into a reduced DOF model that maintains the 

fundamental low-frequency response modes of the structure. With the 

Craig-Bampton method, the DOFs of the substructure can be reduced to 

about 10 (user defined). This system reduction method was first 

introduced by Hurty (1964) and later expanded by Craig and Bampton 

(1968). 

In SubDyn’s Craig-Bampton reduction, the structure nodes are 

separated into two groups: (1) the boundary nodes that include the 

constrained boundary and interface nodes, and (2) the rest of the nodes. 

The constrained boundary nodes are the nodes that are constrained at 

the seabed, the interface nodes are the nodes connected to the transition 

piece, and the remaining nodes are labeled as interior nodes.

The fundamental assumption of the Craig-Bampton method is that the 

contribution to the displacement of the interior nodes can be 

approximated by only a subset of the interior generalized DOFs. This 

reduced basis is chosen as the first few eigenvectors that are arranged 

by increasing eigenfrequencies. 

The Craig-Bampton method is used to obtain the reduced mass, 

damping, and stiffness matrices of the substructure. Then, in loose 

coupling, the current time-step response of the substructure can be 

obtained through the processing of the instantaneous loads by

HydroDyn and the transition piece motion by ElastoDyn, including the 

known SubDyn states (these states are discussed in the next section). 

The next time-step modal DOFs can be obtained using an integrator 

within SubDyn. In tight coupling, the response will be obtained by an 

integrator that is common to all modules. 
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The derivation of the system reduction is as follows. After the LFEB

assembly, the system equation of motion can be partitioned as follows: 

RR RL R R RR RL RR RL R R Rg

LR LL L L LR LL LR LL L L Lg

M M CC KKU U U F F

M M CC KKU U U F F

            +             ++ =

            + 

 

  (1.0) 

Where MC K , , and are mass, damping, and stiffness matrices, 

respectively; and U F and are nodal DOFs and applied forces, 

respectively. A derivative with respect to time is represented by a dot, 

so that U is the derivative of U . Note that the applied forces include 

the hydrodynamic forces from HydroDyn ( ) , F F R L and gravity forces

( ) , F F Rg Lg , which are considered as static forces that are lumped at each 

node. The subscript R stands for the restrained DOFs, which include 

the constrained DOFs at the seabed and the interface DOFs connected 

to the transition piece. The subscript L stands for all of the interior 

DOFs. Typically, the term CLL is the only nonzero term in the damping 

matrix. 

By using the Craig-Bampton transformation, the interior DOFs are 

transformed from physical DOFs to modal DOFs: 

( ) ( )

( )

2

0 0 0

0 2 0

BB Bm R R BB R

mB m m m m m

T

R Rg R L Lg

T

m L Lg

M M U U K U

M I q q q

FF FF

F F

ζ

                  + +

           Ω Ω

  + +Φ +   =  

  Φ +  

 

 

 (1.1) 

where qm are the reduced modal DOFs according to the m number of 

modes retained (provided by the user). Ωm is the diagonal matrix 

containing the frequencies corresponding to the m lowest interior 

modes Φm . The diagonal matrix that contains the modal damping 

coefficients corresponding to each retained internal mode is ζ . And 

ΦR represents constrained modes. 

The internal modes, ΦL , are obtained by solving the eigenvalue

problem: 

2 K M LL L Φ= Φ ω LL L (1.2) 

and ΦL is selected so that T Φ Φ= L LL L M I (mass normalized), where I

is the identity matrix. Φm is the truncated set of ΦL (keeping m of the 

total modes), and Ωm contains the corresponding eigenfrequencies. 

The restrained modes, ΦR , are calculated as: 

1

R LL LR K K− Φ =− (1.3) 

The matrix partitions in Eq. (1.1) are calculated as follows: 

T T

BB RR RL R R LR R LL R

T T

mB m LR m LL R

T

Bm mB

BB RR RL R

MMM M M

M MM

M M

KKK

= + Φ +Φ +Φ Φ

=Φ +Φ Φ

=

= +Φ

 (1.4) 

The substructure interface nodes are rigidly connected to the transition 

piece, which is presently considered as a rigid block, so it is convenient 

to use the transition piece DOFs (one node with 6 DOFs at the 

reference point of the transition piece) in place of the interface DOFs. 

The interface DOFs, UR , and the transition piece DOFs are related as 

follows: 

U TU R I TP = (1.5) 

where TI is a ( ) 6 6 ⋅ × NIN matrix, NIN is the number of interface 

nodes, and UTP is the 6 DOFs of the rigid transition piece. Matrix TI

can be written as follows: 

( )

1 1

1 1

1 1

100 0

010 0

001 0

000 1 0 0

000 0 1 0

000 0 0 1

, 1,2, , 100 0

010 0

001 0

000 1 0 0

000 0 1 0

000 0 0 1

I

i i

i i

i i

Z Y

Z X

Y X

T i NIN

Z Y

Z X

Y X

  ∆ −∆

  −∆ ∆

∆ −∆

= = ∆ −∆

−∆ ∆

∆ −∆

 

    

   

 (1.6) 

where 

i INi TP

i INi TP

i INi TP

XX X

YY Y

ZZ Z

∆= −

∆= −

∆= −

(1.7) 

where ( XYZ INi INi INi , , ) is the coordinate of th i interface node and 

( XYZ TP TP TP , , ) is the coordinate of the transition piece reference point

within the global coordinate system. 

In terms of the transition piece DOFs, the system equation of motion 

(1.1) after the boundary constraints are applied (the DOFs of the nodes 

that are restrained at the seabed are removed from the equation of 

motion) becomes: 

2

0 0 0

0 2 0

BB Bm TP TP BI B TP TP

mB m m m m m m

M M U U K U F

M I q q ζ q F

                        + +=             Ω  Ω      

     

    (1.8) 

with

( )

( )

T

BB I BB

T

Bm I Bm

T

mB Bm

T

BB I BB I

T TT

TP TP I Rg I R L Lg

T

m m L Lg

M TM T

M TM

M M

K TK T

F F TF T F F

F FF

=

=

=

=

= + +Φ +

=Φ +





 







 (1.9) 

where the overhead bar stands for the matrices/vectors after the 

boundary conditions are applied, and T F TF TP I R = is the transition piece

reaction force. Equation (1.8) is the equation of motion of the 

substructure after the Craig-Bampton reduction. The total DOFs of the 

substructure are reduced from ( 6× total number of nodes) to ( ) 6 + m . 
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During the FAST initialization, the driver program calls the SubDyn 

initialization where the parameter matrices , , , MMMK BB mB Bm BB    , 

, , Φ Φm RI T , the arrays FRg and FLg , and the internal frequency matrix 

Ωm are calculated. The substructure responses at each time step can 

then be obtained by using the state-space formulation discussed below. 

STATE-SPACE FORMULATION

The substructure responses at each time step are represented by statespace formulations when integrated within the FAST framework. The 

state-space formulation of substructure dynamic systems can be 

developed in terms of inputs, outputs, states, and parameters. The 

notations highlighted here are consistent with those used in Jonkman 

(2013a). Inputs (identified by u) are a set of values supplied to SubDyn, 

and, along with the states, are needed to calculate future states and the 

system’s output. Outputs (y) are a set of values calculated by and 

returned from SubDyn that depend on the states, inputs, and/or 

parameters through output equations (with functions Y). States are a set 

of internal values of SubDyn that are influenced by inputs and used to 

calculate future state values and the output. In SubDyn, only continuous 

states are considered. Continuous states (x) are states that are 

differentiable in time and characterized by continuous time differential 

equations (with functions X). Parameters (p) are a set of internal system 

values that are independent of the states and inputs. Furthermore, 

parameters can be fully defined at initialization and characterize a 

system’s state equations and output equations.

In SubDyn, the inputs are defined as: 

1

2

3

4

TP

TP

TP

L

u U

u U u u U

u F

 

    = =  

 

   



 (1.10) 

where FL are the hydrodynamic forces on every interior node of the 

substructure from HydroDyn; UTP , UTP

 , and UTP are transition piece 

deflections (6 DOFs), velocities, and accelerations, respectively. 

In first-order form, the states are defined as: 

1

2

m

m

x q x x q

    = =        (1.11) 

From the system equation of motion, the state equation can be written 

as a standard linear system state equation: 

X Ax Bu F =++ X (1.12) 

where 

2

0 000 0 0 , , 2 00 T X T

m m mB m m Lg

I

AB F

ζ M F

       =    = =      −Ω − Ω − Φ   Φ  

In SubDyn, the outputs to the ElastoDyn module are the reaction forces 

at the transition piece FTP : 

1 TP y F = − (1.13) 

The output equation for 1 y can be found as: 

−= + + Y Cx Du F 11 1 Y (1.14) 

where 

( )

2

1

1

2

0

Bm m Bm m

T TT

BB BB Bm mB Bm m I R

TT T

Y Bm m Lg I Rg R Lg

CM M

D K M MM M T

F M F TF F

=− Ω − Ω   ζ  

=   − Φ − Φ  

= Φ − +Φ

 

   



The outputs to HydroDyn and other modules are the deflections and 

velocities of the substructure: 

2

R

L

R

L

U

U

y

U

U

 

    =  

 

   





(1.15) 

The output equation for 2 y can be written as: 

Y Cx Du 22 2 = + (1.16) 

where 

2

2

0 0

0

0 0

0

0 00

0 00

0 00

0 00

m

m

I

R

I

R

C

T

T

D

T

T

 

  Φ  =  

    Φ

 

  Φ =

  Φ

SubDyn can also calculate member forces by starting from the forces 

computed at the nodes of the elements that are contained in the member 

as: 

 Element Inertia load: 

 Element Static load: 

e ee

I

e ee

S

F MU

F KU

=

=



 (1.17) 

where e K and e M are element stiffness and mass matrices, 

respectively. And Ue and Ue  are element nodal deflections and 

accelerations respectively, which can be obtained from Eq. (1.15) and 

Eq. (1.18): 

R

L

U Cx Du F

U

      =++

   



 (1.18) 

where 

2

0 0

2

0 0 0

0 0

0

mm m m

I

T

R I m mB m m

T

m m Lg

C

T

D

T M

F

F

ζ

  =     −Φ Ω − Φ Ω

  =     Φ −Φ Φ Φ

    =     Φ Φ  



There is no good way to quantify the damping forces for each element, 

so the element damping forces are not calculated.
6 This report is available at no cost from the 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

At time t = 0 , u(0) and x(0) are specified as initial conditions. During 

each subsequent time step, the inputs and states are known values, 

where inputs u t( ) are data coming from the ElastoDyn module and 

HydroDyn module, and states x t( ) are calculated from previous timestep integration. All of the parameter matrices are calculated in the

SubDyn module initiation. With known u t( ) and x t( ) , x t ( ) can be 

calculated using the state equation xt X uxt () ( , ,) = , and the outputs 

1 y t( ) and 2 y t( ) can be calculated using the output functions Y uxt 1 ( ) , ,

and ( ) 2 Y uxt , , . The element forces can also be calculated using Eq. 

(1.17). The next time-step states xt t ( ) + ∆ can be obtained by

integration: 

 

[ ] Integrate ut xt xt ( ), ( ), ( ) ( )  → + ∆ x t t (1.19) 

For loose coupling, each module will have its own integrator, whereas

for tight coupling, the states from all the modules will be integrated 

simultaneously using an integrator that is common to all modules. 

Currently, the state-space formulation is not implemented, but it will be 

integrated within the FAST modularization framework in the near 

future. 

REACTION CALCULATION

The reactions at the base of the structure are the member forces at the 

base nodes. These are usually provided in member local reference 

frames. Additionally, the user may request an overall reaction R�⃗ (six 

forces and moments per Eq. (1.20)) lumped at the center of the 

structure (tower centerline) and mudline, i.e., at the reference point 

(0,0,-Wdpth) in the global reference frame, with Wdpth denoting the water 

depth. R�⃗ is a six element array that can be calculated in matrix form as 

follows: 

R�⃗ = �

Fx

⋮

Mz

� = TReact FReact (1.20) 

where FReact is a (6Nreact) array containing the forces and moments at 

the Nreact boundary nodes in global coordinate frame, and TReact is a (6 

x 6Nreact) matrix, as follows: 

TReact =

⎣

⎢

⎢

⎢

⎢

⎡ 1 0 0

0 1 0

0 0 1

0 0 0

0 0 0

0 0 0

⋯

1 0 0

0 1 0

0 0 1

0 0 0

0 0 0

0 0 0

0 −∆z1 ∆y1

∆z1 0 −∆x1

−∆y1 ∆x1 0

1 0 0

0 1 0

0 0 1

⋯

0 −∆zNreact ∆yNreact

∆zNreact 0 −∆xNreact

−∆yNreact ∆xNreact 0

1 0 0

0 1 0

0 0 1⎦

⎥

⎥

⎥

⎥

⎤

(1.21) 

where ∆xi, ∆yi, and ∆zi (i=1..Nreact) are coordinates of the boundary 

nodes with respect to the reference point. 

EXAMPLES AND RESULTS

Case 1: Modal Analysis of a Jacket Substructure

Fig. 2: Offshore Code Comparison Collaboration Continuation (OC4) 

jacket. 

A jacket designed to support the NREL 5-megawatt (MW) reference 

turbine, as used in the International Energy Agency (IEA) Wind Task 

30 Offshore Code Comparison Collaboration Continuation (OC4) 

research project, was analyzed using SubDyn. This support structure 

was originally designed by Vemula et al. (2010) for the UpWind 

Project. The design was adopted for the OC4 project at a water depth of 

50 m (as shown in Fig. 2), and the structure was cantilevered at the mud 

line. The interface nodes were rigidly connected to the transition piece,

which was modeled in this study as a massless point without inertia. 

The main geometric parameters of the jacket are provided in Table 1, 

but more details can be found in Vorpahl et al. (2011). 

Table 1. OC4 jacket parameters

Parameter Value

Leg outer diameter (OD) [m] 1.2

Leg thickness [m]

0.05 (up to the first bay)-

0.035−0.04 (transition 

piece)

Brace OD [m] 0.8

Brace thickness [m] 0.02

Pile OD [m] 2.082

Pile thickness [m] 0.6

The jacket support structure was modeled with SubDyn using full 

finite-element and Craig-Bampton’s method (C-B), with a number of 

internal modes, as well as with ANSYS. Two-node, nontapered 

Timoshenko beam elements were used in all of the SubDyn models. 

The steel density was artificially augmented to 8,500 kg/m3

, namely to 

account for secondary steel not directly modeled. In the ANSYS model, 

each member was discretized into 10 beam elements (the number of 

elements was proven to not be a factor of the analysis results), and the 

top of the jacket was modeled as a rigid region using rigid constraints. 

For the SubDyn models, the connection to the transition piece was 

modeled using the transformation matrix to reduce the interface nodes 

to the transition piece reference point, as shown in Eq. (1.6). In the 

SubDyn full finite-element model, members were discretized into two 
7 This report is available at no cost from the 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

elements each, resulting in 1,014 DOFs. Multiple Craig-Bampton (C-B) 

reductions were performed within SubDyn to determine the sensitivity 

of the structural frequencies to the number of internal modes chosen. 

The SubDyn C-B calculations used 4, 8, and 12 internal modes for a

total of 10, 14, and 18 DOFs, respectively. 

Fig. 3: Frequencies of the OC4 jacket using ANSYS and SubDyn. 

Fig. 3 shows the lowest 12 natural frequencies of the OC4 jacket, as

calculated by ANSYS, SubDyn full finite-element, and SubDyn C-B, 

with the different levels of modal reduction discussed above. Fig. 3

shows that the ANSYS and SubDyn full finite-element results matched

very well. The slight difference in the 11th mode is caused by the 

different methods used by the two codes to treat the rigid connection at 

the interface nodes. By using SubDyn C-B, good results were achieved 

at low frequencies by using just a few internal modes, which will 

greatly accelerate the subsequent simulation time. The SubDyn results 

show that using only four internal modes provided fairly accurate 

approximations of the four lowest modes of the jacket structure. 

Increasing the number of internal modes improves the higher-frequency 

predictions. Because higher-frequency modes contribute little to the 

dynamic response to low-frequency forcing, a user can select the 

number of internal modes based on the forcing frequency range to 

reduce the number of substructure DOFs.

Case 2: Modal Analysis and Static Response of a Wind Turbine 

System

The support structure in Case 1 was then incorporated into a full 

offshore wind system by placing the NREL 5-MW turbine and tower

atop the OC4 jacket substructure (see also Fig. 4). This system was then 

studied to test SubDyn’s ability to accurately represent the 

eigenfrequencies and static response of a turbine system. Static 

analyses were performed under two rotor thrust loads [2,000 and 4,000 

(kN) in magnitude] applied at the assumed RNA center of mass (CM) 

and directed along the horizontal x-axis. Additionally, the influence of 

the RNA self-weight and the jacket self-weight were considered in 

SubDyn. The tower, transition piece, RNA mass, and trust parameters 

are listed in Table 2.

Table 2. Tower geometric, inertial, and thrust parameters

Parameter Value

Deck height above mean sea 

level (MSL) [m] 16 

Transition-piece length [m] 4

Tower-base OD [m] 5.6

Tower-base thickness [m] 0.032

Tower-top OD [m] 4

Tower-top thickness [m] 0.03

Tower length [m] 68

RNA vertical offset [m] 2.34

Turbine hub height [m] 90

RNA mass [ton] 350

Jacket mass [ton] 581+(427 pile/grout)

Transition piece mass [ton] 666

Tower mass [ton] 230

Thrust [kN] 2,000 and 4,000

In the ANSYS model, five pipe elements were utilized per jacket 

member, and 20 tapered beam elements were used for the tower.

According to the analysis results, the number of elements proved to not 

be a factor. For the SubDyn full finite-element model, we used 80 

elements for the tower and 258 elements for the remainder of the jacket 

structure. For the SubDyn C-B models, we used a driver code to call 

SubDyn to run a full finite-element analysis of the tower, RNA, and 

transition piece, and then SubDyn was called to run a C-B reduction 

analysis of the jacket substructure. The matrices were subsequently 

assembled by the driver code to obtain results of the whole system. 

Eighty elements were used for the tower and various numbers of jacket 

internal modes were used for different cases. The tower, RNA, and 

transition piece models were assembled with the jacket substructure 

model at the interface, which was the transition piece reference point. 

For all of the models, the RNA was considered as a point mass 

(rotational inertia was ignored) and located at the RNA center of mass. 

In all cases, the transition piece was considered as a concentrated mass 

with inertia that was located at the transition piece reference point. The 

maximum relative error in calculated mass between the ANSYS and 

SubDyn models was less than 0.06%. The added mass was not included 

in the current models, but in the final FAST framework, the added mass 

will be transmitted to SubDyn at each time step from HydroDyn.
8 

This report is available at no cost from the 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

Fig. 4: OC4 wind turbine system and transition piece models. 

Table 3. Natural frequencies of the OC4 wind turbine system [Hz]

Mode 

No. 

ANSYS

Full 

FEM

SubDyn 

Full 

FEM CB: 12 CB: 8 CB: 4 CB: 1

1 0.3190 0.3190 0.3190 0.3190 0.3190 0.3190

2 0.3190 0.3190 0.3190 0.3190 0.3190 0.3190

3 1.1936 1.1944 1.1945 1.1945 1.1947 1.1947

4 1.1936 1.1944 1.1945 1.1945 1.1947 1.1951

5 3.4425 3.3738 3.3803 3.3803 3.4211 3.4211

6 4.1253 4.1288 4.1321 4.1322 4.1375 4.1377

7 4.1253 4.1292 4.1323 4.1325 4.1378 4.1862

8 5.3391 5.3417 5.3480 5.3497 5.3523 5.3523

9 6.6953 6.6991 6.7020 6.7027 6.7088 6.7096

10 6.6953 6.7016 6.7038 6.7034 6.7103 9.5861

The natural frequencies of the OC4 wind turbine system are listed in 

Table 3; Fig. 5 shows the first 6 eigenfrequencies in graphical format. 

The results are from the different modelsthe ANSYS full FEM, the 

SubDyn full FEM, and the SubDyn C-Bwith substructure reduction 

using 1, 4, 8, and 12 internal modes.

From these results, all of the models agree almost equally. However, a

slight variance was noted in the fifth mode between the ANSYS and 

SubDyn models, which was due to the different ways ANSYS and 

SubDyn modeled the transition piece. With only one internal mode of 

the substructure, good approximation of the first nine modes of the 

entire system was achieved. As a result, the total DOFs were reduced 

by ~1,000. A comparison of Fig. 3 and Fig. 5 shows that the lowest 

modes are mostly associated with the tower, transition piece, and RNA

(for the whole system). The number of retained substructure modes did 

not significantly affect the low-frequency modes of the entire structure. 

Therefore, users may reliably limit the number of substructure modes 

and still obtain high-fidelity results for the whole system.

Fig. 5: Natural frequencies of the OC4 wind turbine system. 

Table 4 and Table 5 show the x-direction deflections (see Fig. 4)

calculated at the RNA center of mass and tower base, respectively, with

thrusts applied at the RNA center of mass in the x-direction, which is 

the downwind direction. The thrust magnitude was arbitrarily varied 

between 2,000 kN and 4,000 kN, or approximately between the 

maximum expected thrust and a value that would cause yield in a 

limited region of the support structure. These results show that the 

outputs of SubDyn and ANSYS are very close, with small differences 

caused by the different ways that ANSYS and SubDyn model the 

transition piece (see Fig. 4). The number of internal modes used in the 
9 

This report is available at no cost from the 

National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.

SubDyn C-B model did not affect the deflection at the interface node or 

the static response of the tower and RNA. This outcome is to be

expected because the Craig-Bampton method returns the exact static 

solutions at interface nodes via the restrained modes. Therefore, for the 

static analysis, the use of internal modes is not needed to achieve the 

correct solution for the entire turbine system.

Table 4. X-direction deflections at the RNA center of mass

Thrust

at 

RNA 

[kN]

ANSYS

[m]

SubDyn [m]

Full 

FEM C-B:12 C-B: 8 C-B: 4 C-B: 1

2000 1.2089 1.2084 1.2084 1.2084 1.2084 1.2084 

4000 2.4178 2.4168 2.4168 2.4168 2.4168 2.4168 

Table 5. X-direction deflections at the tower base

Thrust 

at 

RNA 

[kN]

ANSYS 

[m]

SubDyn [m]

Full 

FEM C-B: 12 C-B: 8 C-B: 4 C-B: 1

2000 0.1375 0.1375 0.1375 0.1375 0.1375 0.1375 

4000 0.2750 0.2750 0.2750 0.2750 0.2750 0.2750 

CONCLUSIONS

This paper discussed recent upgrades made to FAST to enable loads 

simulations of offshore wind turbines with fixed-bottom, multimember 

support structures (e.g., jackets and tripods, which are commonly used 

in transitional-depth waters).

In particular, the main theory and strategies for implementing the 

multimember substructure structural-dynamics module (SubDyn) 

within the new FAST modularization framework were introduced. 

SubDyn relies on two main engineering schematizations: (1) a linear 

frame finite-element beam model (LFEB), and (2) a dynamics system 

reduction via Craig-Bampton’s method. The state-space formulation of 

SubDyn, which is important for time-domain simulations in the FAST 

framework, was also introduced.

The SubDyn LFEB can accommodate different element types, 

including Euler-Bernoulli and Timoshenko beam elements, either 

constant cross section or tapered. In this study, a Craig-Bampton 

system reduction was implemented to accelerate processing time while 

retaining a high level of fidelity in the overall system response. In 

general, the state-space formulation of the substructure equation of 

motion simplifies the time-domain calculation. 

SubDyn can be used as a stand-alone module to perform modal analysis 

and analyze static response of multimember wind turbine substructures. 

In the future, SubDyn will be integrated into the FAST modularization 

framework through both loose and tight coupling and its full 

capabilities will therefore be available in a future release of FAST. 

An example multimember support structure, the Offshore Code 

Comparison Collaboration Continuation (OC4) jacket, was analyzed 

using SubDyn and ANSYS. For all of the models, the modal analysis 

results were in agreement. Using only a few internal modes, the 

SubDyn C-B models also revealed good results, which greatly reduces 

the number of degrees of freedom of the substructure. The second 

example showed the modal analysis and static response of an offshore 

system consisting of a National Renewable Energy Laboratory 5-

megawatt wind turbine atop the OC4 substructure. The natural 

frequencies and the static deflectionsunder two different turbine rotor 

thrust levelsat the RNA center of mass and tower base, were 

calculated using SubDyn and ANSYS. The results revealed that the 

number of internal modes of the substructure had little effect on the 

lower-frequency modes of the entire wind turbine system. Therefore, 

limiting the number of internal modes will lead to relatively highfidelity results, because the overall system dynamics are generally 

dominated by the low-frequency response. The static deflection at the 

RNA center of mass and tower base showed that the SubDyn results 

agreed with the ANSYS results, and the number of internal modes had 

no effect on the static response at the interface node, which was 

collocated at the transition piece reference point in this case (which had 

no effect on the tower and RNA either). 

Based on the two example cases, the preliminary SubDyn results 

agreed with the ANSYS model outputs, thereby indicating that

SubDyn, as a substructure module or a stand-alone code, may be 

considered as a reliable tool when analyzing multimember support 

structures for wind turbine systems. 

ACKNOWLEDGMENTS

This work was performed at the National Renewable Energy 

Laboratory, in support of the U.S. Department of Energy (under 

contract #DEAC36-08GO28308). 

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