diff --git a/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/Model Linearisation.md b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/Model Linearisation.md new file mode 100644 index 0000000..2cea356 --- /dev/null +++ b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/Model Linearisation.md @@ -0,0 +1,308 @@ +# Linear Analysis Background + +The linear analysis calculations reduce the Bladed aeroelastic model to a linear system at each operating point requested by the user. The linear system of equations in state-space form is represented by + +$$ +\begin{array}{r}{\underline{{\dot{\mathbf{x}}}}=\mathbf{A}\underline{{\mathbf{x}}}+\mathbf{B}\underline{{\mathbf{u}}}}\\ {\underline{{\mathbf{y}}}=\mathbf{C}\underline{{\mathbf{x}}}+\mathbf{D}\underline{{\mathbf{u}}}}\end{array} +$$ + +with + +$$ +\underline{{\mathbf{x}}}=\mathbf{x}-\mathbf{x_{0}},\quad\underline{{\mathbf{y}}}=\mathbf{y}-\mathbf{y_{0}},\quad\mathrm{~and~}\quad\underline{{\mathbf{u}}}=\mathbf{u}-\mathbf{u_{0}} +$$ + +where x is a vector of states representing the system, u is the vector of system inputs and $\mathbf{y}$ is the vector of system outputs. The normalised vectors x,y and u are representing the deviation from equilibrium. + +The matrices A, B, C and D represent the linearised relationship between these vectors. This represents a simplification of the full Bladed model which uses a fully non-linear set of equations to calculate the state derivatives and outputs + +$$ +\begin{array}{r}{\dot{\mathbf{x}}=f(t,\mathbf{x},\mathbf{u})}\\ {\mathbf{y}=h(t,\mathbf{x},\mathbf{u}).}\end{array} +$$ + +It is important to note that in order to enable proper linearised wind turbine dynamical systems, the following principles for preparing the model need to be considered: + +· Azimuthal dependency shall be removed which includes wind shear, yaw, rotor imbalance, etc. +· Physical effects that cannot be linearised shall be removed, for instance wind turbulence, stickslip, etc. + +In Bladed, the states fall into two main categories: + +1. Elastodynamic: These are the states that represent the structural modes of the system. Elastodynamic modes are governed by $2^{\mathrm{nd}}$ order equations of motion. Therefore, to be represented in state-space form, each mode is represented by two states - displacement and velocity. This also includes the principal rotor rigid body freedom. + +2. Aerodynamic: These are primarily used to model dynamic stall and dynamic wake. These states are generally $1^{\mathrm{st}}$ order as they are concerned with time-lags. + +With the aerodynamic model in versions 4.7 and earlier, aerodynamic states are not included in model linearization. In the aerodynamic formulation in version 4.8 and later, the user has the option to include the aerodynamic states or not. + +To perform linear analysis, Bladed takes each operating point in turn and finds the steady-state conditions of the turbine (as per the initial conditions in time-domain runs). This means that the rotor is not accelerating and the modal deflections are such that the elastic loads balance the external loading. This defines the values for $\mathbf{x_{0}},\,\mathbf{y_{0}}$ and $\mathbf{u_{0}}$ , the principal equilibrium point about which everything is perturbed. + +For each input or state, Bladed then makes a series of perturbations of increasing magnitude either side of the equilibrium point. The value of the state or input is artificially increased or decreased, the system is solved with these edited values and the state derivatives and outputs are recorded. The number of perturbations and the maximum perturbation magnitude can be defined by the user. + +The elements of the matrices A, B, C and D can then be derived by performing a linear regression of the state derivative against the input or state value at all its perturbed values and its equilibrium value. The gradient of the linear regression gives the value of the element. If the correlation coefficient is less than the minimum correlation coefficient , then the relationship is considered void, and a zero value is given to the element. + +![](images/559b5972f3b586cff78a91c0dfa85bc88bb9a7f9e4c9735de97fbd009af301f1.jpg) + +Figure 1: Example linear regression calculating element ${\bf A}_{7,4}$ with a value of -1.315, with a correlation coefficient of 0.9982. The equilibrium point is shown in green + +Last updated 30-08-2024 + +# Multi-blade Coordinate Transformation + +For linearisation calculations or Campbell diagrams it is recommended to select the multi-blade coordinate transformation, which generates coupled modes referring to the non-rotating coordinate system including the backward and forward whirling modes of the rotor. This is based on theory developed in (Bir, 2008) and (Hansen, 2003). The linearised model is significantly azimuth-dependent, but when transformed to non-rotating coordinates the resulting model of the structural dynamics should be only weakly azimuth-dependent. However, for 2-bladed turbines there is still a strong azimuth dependency. + +# Single mode transformation + +The transformation matrix of displacements of a 3-blade system with azimuths $\psi_{1}$ to $\psi_{3}$ from nonrotating to rotating coordinates is given by + +$$ +\begin{array}{r}{\left[\!\!\begin{array}{l}{q_{1}}\\ {q_{2}}\\ {q_{3}}\end{array}\!\!\right]=\tilde{\mathbf{t}}_{N R\rightarrow R}\left[\!\!\begin{array}{l}{q_{0}}\\ {q_{c}}\\ {q_{s}}\end{array}\!\!\right],}\end{array} +$$ + +with + +$$ +\tilde{\mathbf{t}}_{N R\to R}=\left[\begin{array}{l l l}{1}&{\cos\psi_{1}}&{\sin\psi_{1}}\\ {1}&{\cos\psi_{2}}&{\sin\psi_{2}}\\ {1}&{\cos\psi_{3}}&{\sin\psi_{3}}\end{array}\right]. +$$ + +Multi-blade coordinate transformations are often quoted in the above form, but the primary aim is to go the other way and transform from rotating to non-rotating coordinates. The transformation matrix of displacements of a 3-blade system from rotating to non-rotating coordinates is the inverse of the above matrix given by + +$$ +\mathbf{t}_{R\rightarrow N R}=\frac{1}{3}\left[2\cos\psi_{1}\quad2\cos\psi_{2}\quad2\cos\psi_{3}\right]. +$$ + +Note, that the inverse relation does not hold for the derivatives of this matrix. + +The general transformation matrix for a turbine with an arbitrary number of blades $(n)$ + +$$ +\mathbf{t}_{R\rightarrow N R}=\frac{1}{n}\left[\begin{array}{c c c c c}{1}&{1}&{1}&{\cdots}&{1}\\ {2\cos\psi_{1}}&{2\cos\psi_{2}}&{2\cos\psi_{3}}&{\cdots}&{2\cos\psi_{n}}\\ {2\sin\psi_{1}}&{2\sin\psi_{2}}&{2\sin\psi_{3}}&{\cdots}&{2\sin\psi_{n}}\\ {2\cos j\psi_{1}}&{2\cos j\psi_{2}}&{2\cos j\psi_{3}}&{\cdots}&{2\cos j\psi_{n}}\\ {2\sin j\psi_{1}}&{2\sin j\psi_{2}}&{2\sin j\psi_{3}}&{\cdots}&{2\sin j\psi_{n}}\\ {\vdots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}\\ {1}&{-1}&{1}&{\cdots}&{(-1)^{n}}\end{array}\right], +$$ + +where the last row is the transformation to the differential mode and exists only if there is an even number of blades. For odd bladed turbines, the last row will be a sine cyclic row. The counter $j$ + +goes from 1 to $(n-1)/2$ $^n$ is odd, and from 2 to $(n-2)/2$ ü $^{n}$ is even. + +Dropping the matrix representation the non-rotating coordinates can be calculated as + +$$ +\begin{array}{l}{\displaystyle q_{0}=\frac{1}{n}\sum_{i=1}^{n}q_{i}}\\ {\displaystyle q_{c j}=\frac{2}{n}\sum_{i=1}^{n}q_{i}\cos{\left(j\psi_{i}\right)}}\\ {\displaystyle q_{s j}=\frac{2}{n}\sum_{i=1}^{n}q_{i}\sin{\left(j\psi_{i}\right)}}\\ {\displaystyle q_{d}=\frac{1}{n}\sum_{i=1}^{n}q_{i}(-1)^{n}}\end{array} +$$ + +Returning to the specific case of 3-bladed turbines as an example, the derivative transformation matrices are now calculated. Each azimuth angle $\psi_{i}$ can be expressed in terms of the (assumed constant) rotorspeed $\Omega$ and initial azimuth angle $\Psi_{i}$ as linear relationship + +$$ +\psi_{i}=\Omega t+\Psi_{i}. +$$ + +Taking the time-derivatives of the transformation matrix gives + +$$ +\dot{\bf t}_{R\rightarrow N R}=\frac{\Omega}{3}\left[\!\!\!\begin{array}{c c c}{{0}}&{{0}}&{{0}}\\ {{-2\sin\psi_{1}}}&{{-2\sin\psi_{2}}}&{{-2\sin\psi_{3}}}\\ {{2\cos\psi_{1}}}&{{2\cos\psi_{2}}}&{{2\cos\psi_{3}}}\end{array}\!\!\right] +$$ + +and + +$$ +\ddot{\bf t}_{R\rightarrow N R}=-\frac{\Omega^{2}}{3}\left[\!\!\!\begin{array}{c c c}{{0}}&{{0}}&{{0}}\\ {{2\cos\psi_{1}}}&{{2\cos\psi_{2}}}&{{2\cos\psi_{3}}}\\ {{2\sin\psi_{1}}}&{{2\sin\psi_{2}}}&{{2\sin\psi_{3}}}\end{array}\!\!\right] +$$ + +for the first and second derivatives, respectively. + +# System transformation matrix + +A transformation matrix for the whole state list, including both displacement and velocity states, is required. For the displacement states we have already established in Equation (1) that + +$$ +{\bf q}_{N R}={\bf t}_{R\rightarrow N R}{\bf q}_{R} +$$ + +holds. Taking the time-derivative of Equation (1) gives + +$$ +\dot{\bf q}_{N R}={\bf t}_{R\rightarrow N R}\dot{\bf q}_{R}+\dot{\bf t}_{R\rightarrow N R}{\bf q}_{R} +$$ + +for the velocity states. + +Combining qvR and qnR to a vector of all states (both displacements and velocities) allows us to define a common transformation matrix $\mathbf{T}$ that is of the same dimensions as A. We define + +$$ +\mathbf{T}:=\left[\begin{array}{c c}{\mathbf{\{t}}_{R\rightarrow N R}}&{0}\\ {\vdots}&{\mathbf{t}_{R\rightarrow N R}}\end{array}\right] +$$ + +allowing us to express the combined vector as + +$$ +\begin{array}{r}{\left[\mathbf{q}_{N R}\right]=\left[\mathbf{\dot{t}}_{R\rightarrow N R}\mathbf{\Phi}\;\;\;\;\;\;\;0\mathbf{\Phi}\right]\left[\mathbf{q}_{R}\right].}\end{array} +$$ + +Note that in general the displacement and velocity states are not ordered in this way and a permutation of the system transformation matrix $\mathbf{T}$ will occur. The system transformation matrix is not singular and the inverse can be calculated. + +The derivative of the system transformation matrix is trivially inferred as + +$$ +\dot{\mathbf{T}}=\left[\!\!\begin{array}{c c}{\dot{\mathbf{t}}_{R\rightarrow N R}}&{0}\\ {\ddot{\mathbf{t}}_{R\rightarrow N R}}&{\dot{\mathbf{t}}_{R\rightarrow N R}}\end{array}\!\!\right]. +$$ + +# Transforming the A,B,C,D matrices + +We consider the linear model equations in the rotating frame of reference and define + +$$ +\mathbf{x}_{R}:=\left[\mathbf{q}_{R}\right] +$$ + +to express the principal system as + +$$ +\begin{array}{r}{\dot{\mathbf{x}}_{R}=\mathbf{A}_{R}\mathbf{x}_{R}+\mathbf{B}_{R}\mathbf{u}}\\ {\mathbf{y}=\mathbf{C}_{R}\mathbf{x}_{R}+\mathbf{D}_{R}\mathbf{u}}\end{array} +$$ + +with respect to rotating blade coordinates. + +The transformation of the state vector from rotating to non-rotating coordinates is given as + +$$ +\mathbf{x}_{N R}=\mathbf{T}\mathbf{x}_{R} +$$ + +and its derivative follows as + +$$ +\dot{{\bf x}}_{N R}={\bf T}\dot{\bf x}_{R}+\dot{\bf T}{\bf x}_{R}. +$$ + +By combining Equation (17) with Equation (15) we infer + +$$ +\begin{array}{r l}&{\dot{\mathbf{x}}_{N R}=\mathbf{T}\left(\mathbf{A}_{R}\mathbf{x}_{R}+\mathbf{B}_{R}\mathbf{u}\right)+\dot{\mathbf{T}}\mathbf{x}_{R}}\\ &{\qquad=\Big(\mathbf{T}\mathbf{A}_{R}+\dot{\mathbf{T}}\Big)\mathbf{x}_{R}+\mathbf{T}\mathbf{B}_{R}\mathbf{u}}\\ &{\qquad=\Big(\mathbf{T}\mathbf{A}_{R}+\dot{\mathbf{T}}\Big)\mathbf{T}^{-1}\mathbf{x}_{N R}+\mathbf{T}\mathbf{B}_{R}\mathbf{u}}\end{array} +$$ + +and conclude + +$$ +\begin{array}{r l}&{\mathbf{A}_{N R}=\Big(\mathbf{TA}_{R}+\dot{\mathbf{T}}\Big)\mathbf{T}^{-1}}\\ &{\mathbf{B}_{N R}=\mathbf{TB}_{R}}\end{array} +$$ + +from there. + +Similar transformation in Equation (16) gives + +$$ +\begin{array}{r}{\mathbf{y}=\mathbf{C}_{R}\mathbf{x}_{R}+\mathbf{D}_{R}\mathbf{u}}\\ {=\mathbf{C}_{R}\mathbf{T}^{-1}\mathbf{x}_{N R}+\mathbf{D}_{R}\mathbf{u}}\end{array} +$$ + +for the output y of the linear system. We now define + +$$ +\mathbf{C}_{N R}:=\mathbf{C}_{R}\mathbf{T}^{-1},\quad\mathrm{~and~}\mathbf{D}_{N R}:=\mathbf{D}_{R} +$$ + +for the matrices concerned with the output of the linear model. This completes the derivation of a linear model with respect to a non-rotating frame + +Rotational transformations are exclusively applied to states, which represent the degress of freedom in a mathematical model defined for all blades. These states include blade mode states as well as dynamic stall states, whereas any other individual-blade states such as pitch positions, rates, actuator internal states etc. are not transformed. The matrix T just has unit diagonal elements for rows and columns corresponding to the states and state derivatives which are not transformed. For other rows and columns, the elements of T represent the basic transformation defined above for each group of modes. Note that the elements connecting states and state derivatives also need to be defined by differentiating the equations of the basic transformation, bearing in mind that the derivative of the azimuth angle is equal to the rotor speed (which is assumed constant for this purpose). Model inputs and outputs are not transformed. + +Last updated 26-11-2024 + +# Calculating Coupled Modes + +The Campbell diagram and blade stability analyses are analyses of the matrix A at each specified operating point. Each coupled mode corresponds to an eigenvalue and its eigenvector. Given a (complex) eigenvalue, $\lambda,$ of A, Bladed reports the undamped frequency $\displaystyle(\omega_{n}),$ damped frequency ( $\omega_{d}$ ) and damping ratio $(\zeta)$ according to Figure 1. + +![](images/1bc3025789424b3ce73139d78bc919927c8a5412bd7fc241d6212fc0f2efd0ba.jpg) +Figure 1: Argand Diagram + +The uncoupled mode contributions to each coupled mode are determined by its eigenvector. If the coupled mode has contributions from second-order states (structural states), which are represented by two states in the state vector, then the displacement state is used to determine the contribution. + +In their raw form, these eigenvector contributions represent the relative displacement of each mode and can be used to build up the coupled mode-shape. However, the contributions in the Campbell diagram have been normalised. This is done by modifying the matrix of eigenvectors such that each row and each column have a unit sum. This has the effect of increasing percentage contributions from modes with high mass and stiffness, which contribute very little in displacement but significantly in energy. + +The phase of each contribution, $\phi_{i\,,}$ is determined by the argument of the corresponding complex eigenvector element, $v_{i},$ i.e. + +$$ +\phi_{i}=\arctan\left(\frac{\operatorname{Im}\left(v_{i}\right)}{\operatorname{Re}\left(v_{i}\right)}\right). +$$ + +Last updated 26-11-2024 + +# Naming Coupled Modes + +In cases where coupled modes are computed such as in the Campbell diagram analysis the following sections gives details on the naming. A focus is placed on the behaviour when the multiblade coordinate transform is used. + +# Support structure modes + +For support structure modes, the coupled mode is named after the whole-tower mode that gives the highest contribution. Whole-tower modes are uniquely calculated for the linearisation calculations through a subsequent eigen analysis with fixed-free boundary conditions. This analysis considers the effect of the RNA and any other masses at distal nodes. In case multiple coupled support structure modes share the same whole-tower mode as its prime contributor, then the coupled mode name is made unique by appending letters A,B,C, and so on. + +# Rotor modes rotating frame + +If no MBC transformation is used for the rotor modes, then the following logic applies to naming the coupled rotor modes: + +· If a single blade mode gives $>\!75\%$ contribution to the coupled rotor mode, then the coupled rotor mode is named after that blade mode. In other words, the mode is called "Blade" instead of "Rotor" mode. +· Else, the rotor mode is named after its prime contributor and made unique by appending letters $\mathsf{A},\mathsf{B},\mathsf{C},$ etc. in case multiple coupled rotor modes share the same uncoupled blade mode as prime contributor + +# Rotor modes non-rotating frame + +If an MBC transform is applied then the individual blade modes are transformed to a set of rotor modes. For a three bladed rotor there typically is a collective, cosine-cyclic and sine-cyclic rotor mode. The 1st flapwise modes of all blades will be renamed to rotor 1st flapwise collective, rotor 1st flapwise sine-cyclic and rotor 1st flapwise cosine cyclic. In case the number of blades is even there will be a differential mode as well. + +After the transformation and renaming of the individual blade modes the coupled rotor modes are named. The whirling modes are identified following the logic in the table below. + +Coupled mode name 1st uncoupled mode 2nd uncoupled mode Phase angle $(\phi_{2}-\phi_{1})$ + + +
Forward whirlSine cyclicCosine cyclic> 0.0
Cosine cyclicSine cyclic<0.0
Backward whirlSine cyclicCosine cyclic< 0.0
Cosine cyclicSine cyclic> 0.0
+ +If a coupled mode does not meet the criteria of the whirling modes, then the mode is named after its prime contributor. This is analogous with the naming logic of rotor modes in the rotating frame and support structure modes + +Last updated 04-12-2024 + +# Joining Coupled Modes across Operating Points + +The Campbell diagram displays the frequencies of different coupled displacement modes with respect to the rotor speed together with the most important excitation frequencies given in terms of multiples of the rotor frequency (P). In addition to the frequencies of the coupled modes, the Campbell diagram displays the corresponding damping ratios, which include the effect of structural damping as well as aerodynamic damping. Both characteristics are calculated as described in the article Calculating Coupled Modes and are useful for identifying the critical Operating points that need further analysis. + +# The Joining Process + +Given a set of coupled modes at each operating point, a fundamental step in creating the resulting Campbell diagram is to identify similar modes at adjacent operating points, which allows for joining similar modes across the operating points with line segments, giving the user the impression of continuous change in frequency against rotor speed (or wind speed). This joining process is generally challenging because the coupled modes evolve and change in their contributions between the operating points. + +Similar coupled modes at two adjacent operating points are identified by comparing their complex eigenvectors and frequency in term of the extended modal assurance criterion (MACX) (Vacher, Jacquier, and Buchales, 2010) with frequency weighting. More specifically, the frequency weighted MACX numbers are calculated for all combinations of coupled modes at the two operating points to form a score matrix, which are then used for joining the modes by the GaleShapley algorithm (Gale & Shapley, 1962). A sequence of similar modes at all operating points forms a coupled mode series, which represents a line in the resulting Campbell diagram. + +To ensure that the coupled mode series in the resulting Campbell diagram primarily involves structural dynamics, an initial calculation of coupled modes with only structural states is performed at the first operating point. These structure-only modes are then joined with the coupled modes at the first operating point as described above, which effectively excludes coupled modes that mainly have contributions from aerodynamic states. + +![](images/8f4cdb86718fe4c1b6ab7487c05d0fb3aef859ff9cfbd79295e20fc0c3242153.jpg) + +![](images/f2b29d2c38d71e5383bdd43b5de1d57ddfb37e3e66cd5c15d2e31c832e6614cd.jpg) + +Figure 1: Illustration of the joining process showing how the coupled modes are connected at the operating points. Dots represent coupled modes. Coloured lines represent coupled mode series. + +The resulting joining process is then: + +1. Calculate a set of structure-only modes at the first operating point. These modes will also form the basis of the mode series, representing the lines in the resulting Campbell diagram. +2. Join the structure-only modes with the coupled modes at the first operating point. It is noted that the number of coupled modes is generally larger than the number of mode series, which means that not all coupled modes are included in a mode series. +3. Join the coupled modes that were included in a mode series at the current point (first operating point) with the coupled modes at the next point (second operating point). Repeat until the last operating point is reached. + +# Naming and Exclusion of Coupled Mode Series + +A coupled mode series is named according to the contributions of the structure-only coupled modes at the first operating point only (more details on coupled mode naming can be found here). The contributions and therefore the shape of a coupled mode can change significantly between the range of operating points, and therefore the characteristic of a mode cannot be determined from the name alone. + +A coupled mode series, which includes coupled modes with real eigenvalues (and therefore no oscillatory behaviour) at all operating points, is excluded from the resulting Campbell diagram. This is done because such modes cannot cause resonant behaviour, which is the primary purpose of the Campbell diagram to detect. + +A coupled mode series is also excluded if the coupled mode frequencies at all operating points exceed a user-defined maximum value. + +Last updated 26-11-2024 + +# Align Wind Field with Hub Axis + +Figure 1 illustrates that azimuthal dependency is still invoked in certain cases even though the considerations given in the linear analysis background are followed. This occurs when a tilt angle is considered in the calculations and when the tower flexibility is strongly affecting the rotor orientation. Figure 1 (left) provides an ilustration of the azimuthally independent system, indicating that the solution will be the same regardless of the azimuth angle of the rotor. However, when the tilt angle is considered, Figure 1 (middle), the loads perceived by the upper side of the rotor and the lower side of the rotor vary. This creates an imbalance of the loads similar as adding "a virtual wind shear" on the rotor plane. The situation is worse when the flexibility of the tower affects the rotor orientation strongly as illustrated in Figure 1 (right). Here, one can see that the hub orientation might be tilted even greater which creates a stronger loads imbalance across the rotor. + +![](images/e0baae06c8f1868c90cbef4730f5751d227a99bc3f1a1ded738eb0502ce1c059.jpg) + +Figure 1: llustration of the hub axis orientation relative to the incoming wind direction in the linearisation calculation. Left: rigid wind turbine without tilt angle, middle: rigid wind turbine with tilt angle being considered, right: flexible turbine with tilt angle being considered. + +To avoid the imbalance of the loads due to the tilted hub orientation, Bladed introduces a correction to align the wind field according to the tilted hub axis orientation. The mechanism is clearly illustrated in Figure 2. It can be observed that when the wind field is aligned with the tilted hub axis, the loads experienced by the rotor will be independent of the azimuth angle. This effectively removes the azimuthal dependency of the system due to tilted hub axis orientation. This correction may also be applied for floating wind turbines as illustrated in Figure 3. + +![](images/401c313ff319bf4fd49c69e2882903bdb219b4d519026f6a5e05eed954997a36.jpg) + +Figure 2: Aligning wind field to the tilted hub axis orientation for an onshore wind turbine or a bottom-fixed offshore wind turbine. Left: Azimuthally dependent system, right: azimuthally independent system. + +![](images/cf4934de4feb7e041c7083a932a6b6869c37c1aac65f4e6b35b5209a76ea6651.jpg) +Figure 3: Aligning wind field to the tilted hub axis orientation for a floating wind turbine. Left: Azimuthally dependent system, right: azimuthally independent system. + +The hub axis is determined during the initial conditions routine, where the rotor is not accelerating and the modal deflections are such that the elastic loads balance the external loading. Within the initial conditions routine, iterations are performed and the wind field is adjusted accordingly for every iteration to be parallel to the tilted hub axis. After the the conditions are found, the wind field orientation is fixed to the last found tilted angle in the initial conditions routine. Then, the system is perturbed using the fixed wind field orientation for calculating any of the linearisation type calculations available in Bladed. This process is done for every operating point simulated. + +Last updated 30-08-2024 \ No newline at end of file diff --git a/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/Model Linearisation_origin.pdf b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/Model Linearisation_origin.pdf new file mode 100644 index 0000000..a50f63f Binary files /dev/null and b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/Model Linearisation_origin.pdf differ diff --git a/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/1bc3025789424b3ce73139d78bc919927c8a5412bd7fc241d6212fc0f2efd0ba.jpg b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/1bc3025789424b3ce73139d78bc919927c8a5412bd7fc241d6212fc0f2efd0ba.jpg new file mode 100644 index 0000000..371c44f Binary files /dev/null and b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/1bc3025789424b3ce73139d78bc919927c8a5412bd7fc241d6212fc0f2efd0ba.jpg differ diff --git a/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/3554e94686609b5997a0f2b05764fe7e91db4e99c1bfa08cde43ac6f566f8d58.jpg b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/3554e94686609b5997a0f2b05764fe7e91db4e99c1bfa08cde43ac6f566f8d58.jpg new file mode 100644 index 0000000..8d51c97 Binary files /dev/null and b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/3554e94686609b5997a0f2b05764fe7e91db4e99c1bfa08cde43ac6f566f8d58.jpg differ diff --git a/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/401c313ff319bf4fd49c69e2882903bdb219b4d519026f6a5e05eed954997a36.jpg b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/401c313ff319bf4fd49c69e2882903bdb219b4d519026f6a5e05eed954997a36.jpg new file mode 100644 index 0000000..cb79e54 Binary files /dev/null and b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/401c313ff319bf4fd49c69e2882903bdb219b4d519026f6a5e05eed954997a36.jpg differ diff --git a/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/559b5972f3b586cff78a91c0dfa85bc88bb9a7f9e4c9735de97fbd009af301f1.jpg b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/559b5972f3b586cff78a91c0dfa85bc88bb9a7f9e4c9735de97fbd009af301f1.jpg new file mode 100644 index 0000000..4a61630 Binary files /dev/null and b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/559b5972f3b586cff78a91c0dfa85bc88bb9a7f9e4c9735de97fbd009af301f1.jpg differ diff --git a/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/8f4cdb86718fe4c1b6ab7487c05d0fb3aef859ff9cfbd79295e20fc0c3242153.jpg b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/8f4cdb86718fe4c1b6ab7487c05d0fb3aef859ff9cfbd79295e20fc0c3242153.jpg new file mode 100644 index 0000000..33ca99b Binary files /dev/null and b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/8f4cdb86718fe4c1b6ab7487c05d0fb3aef859ff9cfbd79295e20fc0c3242153.jpg differ diff --git a/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/cf4934de4feb7e041c7083a932a6b6869c37c1aac65f4e6b35b5209a76ea6651.jpg b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/cf4934de4feb7e041c7083a932a6b6869c37c1aac65f4e6b35b5209a76ea6651.jpg new file mode 100644 index 0000000..fdc9ff6 Binary files /dev/null and b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/cf4934de4feb7e041c7083a932a6b6869c37c1aac65f4e6b35b5209a76ea6651.jpg differ diff --git a/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/e0baae06c8f1868c90cbef4730f5751d227a99bc3f1a1ded738eb0502ce1c059.jpg b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/e0baae06c8f1868c90cbef4730f5751d227a99bc3f1a1ded738eb0502ce1c059.jpg new file mode 100644 index 0000000..386818d Binary files /dev/null and b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/e0baae06c8f1868c90cbef4730f5751d227a99bc3f1a1ded738eb0502ce1c059.jpg differ diff --git a/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/f2b29d2c38d71e5383bdd43b5de1d57ddfb37e3e66cd5c15d2e31c832e6614cd.jpg b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/f2b29d2c38d71e5383bdd43b5de1d57ddfb37e3e66cd5c15d2e31c832e6614cd.jpg new file mode 100644 index 0000000..740d537 Binary files /dev/null and b/书籍/力学书籍/BladedTheoryManual/Model Linearisation/auto/images/f2b29d2c38d71e5383bdd43b5de1d57ddfb37e3e66cd5c15d2e31c832e6614cd.jpg differ diff --git a/书籍/力学书籍/BladedTheoryManual/Structural Dynamics/auto/Structural Dynamics.md b/书籍/力学书籍/BladedTheoryManual/Structural Dynamics/auto/Structural Dynamics.md new file mode 100644 index 0000000..1b9fa98 --- /dev/null +++ b/书籍/力学书籍/BladedTheoryManual/Structural Dynamics/auto/Structural Dynamics.md @@ -0,0 +1,1273 @@ +# Introduction to Multibody Dynamics + +In the early days of the industry, wind turbine design was undertaken on the basis of quasi-static aerodynamic calculations, while the effects of structural dynamics were either ignored completely or included through the use of estimated dynamic magnification factors. From the late $1970^{\prime}\varsigma$ research workers began to consider more reliable methods of dynamic analysis, and two basic approaches were considered: finite element representations and modal analysis. + +The traditional use of standard, commercial finite element analysis software packages for solving problems of structural dynamics is challenging in the case of wind turbines. This is because of the presence of rigid body motion of one structural component, i.e.the rotor, with respect to another, i.e. the tower or another support structure type. In principle, the standard finite element method only considers structures in which the deflection occurs about an initial reference position, and for this reason the finite element models that have been developed for wind turbine in the past have been tailored to deal with this problem. + +Dynamic wind turbine models commonly used as the basis of design calculations involve a modal representation of the deformation state. This approach, borrowed from the helicopter industry, has the major advantage that it offers a reliable representation of the dynamics of a wind turbine with relatively few degrees of freedom. Another important advantage is that the structural damping of flexible components can be described conveniently and realistically as modal damping. Obviously the number and type of modal degrees of freedom used to represent the dynamics of a particular wind turbine depends on the configuration and structural properties of the machine. + +In principle, a wind turbine structure may be considered as a collection of a set of interconnected structural components that may undergo large rotations relative to neighbouring components, but also relative to an inertial coordinate system. The natural choice for modelling structural dynamics of said mechanical systems is the multibody dynamics approach that emerged in the late ${}^{1}980^{\prime}\mathbf{s},$ initially for rigid components or bodies (Nikravesh, 1988), but later also for flexible components (Shabana, 1988); (Géradin and Cardona, 2001). The multibody dynamics approach is now used as an integrated part of the design process in the automotive and the aircraft industry, but it has also been used extensively in the space industry. The structural model in Bladed is based on this approach combined with a modal representation of the flexible components like the blades and the tower. This ensures a consistent formulation of the dynamic equations and facilitates the modelling of the turbines based on new structural concepts. +多体动力学导论 + +在行业早期，风力发电机设计是基于准静态气动计算进行的，而结构动力学的影响要么完全被忽略，要么通过使用估计的动态放大系数来包含。从 1970 年代后期开始，研究人员开始考虑更可靠的动态分析方法，并考虑了两种基本方法：有限元表示和模态分析。 + +在风力发电机的情况下，使用标准的、商业化的有限元分析软件包来解决结构动力学问题具有挑战性。这是因为一个结构部件（例如风轮）相对于另一个结构部件（例如塔架或其他支撑结构）存在刚体运动。从原理上讲，标准有限元方法仅考虑相对于初始参考位置发生挠曲的结构，因此过去为风力发电机开发的有限元模型都针对此问题进行了定制。 + +常用的风力发电机动态模型作为设计计算的基础，涉及变形状态的模态表示。这种方法借鉴了直升机行业的经验，具有主要优点，即在相对较少的自由度下提供可靠的风力发电机动力学表示。另一个重要的优点是，可以方便且现实地将柔性部件的结构阻尼描述为模态阻尼。显然，用于表示特定风力发电机动力学的模态自由度的数量和类型取决于机器的配置和结构特性。 + +从原理上讲，风力发电机结构可以被认为是连接的一组结构部件的集合，这些部件可能相对于相邻部件以及相对于惯性坐标系发生大旋转。对所述机械系统进行结构动力学建模的自然选择是多体动力学方法，该方法于 20 世纪 1980 年代后期出现，最初用于刚体部件或刚体（Nikravesh, 1988），后来也用于柔性部件（Shabana, 1988）；(Géradin and Cardona, 2001)。多体动力学方法现在被用作汽车和航空工业设计过程的集成部分，也已广泛应用于航天工业。Bladed 中的结构模型基于这种方法，并结合了柔性部件（如叶片和塔架）的模态表示。这确保了动态方程的一致性，并促进了基于新结构概念的风力发电机建模。 + +## The multibody dynamics approach + +The multibody dynamics approach used for the Bladed structural model was originally proposed by Dr. J.P. Meijaard, Nottingham University, (presently University of Twente, in the Netherlands) under commission of Garrad Hassan and Partners Ltd (Meijaard, 2005), and it was developed particularly for modelling wind turbine structural dynamics. The approach assumes a tree-like structure, which means that structural loops cannot be described outside of flexible components. + +In general, the structural components may be assumed to be rigid, such as yaw and blade bearings, or flexible such as support structures, towers and blades. While rigid components are relatively easy to model, flexible components are more complicated as the motion of a material point of this type of components is generally caused by rigid body motion combined with relative motion due to the deformation. A description of the applied method for modelling flexible components is given in Modelling flexible components. + +The rigid body motion of a component is described in terms of the motion of a set of local component nodes that are characteristic material points, where the motion of the component is assumed to be known. Components can only be interconnected at the nodes, and a connection is imposed by the usual finite element technique by linking the nodes of the components involved in the connection to a global structural node. Due to the assumption of the tree-like structure it is convenient to subdivide the nodes of a component into a proximal node that is linked to a node of its predecessor, and distal nodes that may be linked to nodes of successors. For example a yaw bearing has two nodes, i.e. one proximal node attached to the tower and one distal node attached to the main frame. + +For all components a local Cartesian coordinate system is attached to the proximal node such that the position of origin and the orientation are defined by the position and orientation, respectively, of this node. This local coordinate system is mainly used for flexible components that are described in more details in Modelling flexible components. + +The deformation state of a component is described by generalised strains that represent the degrees of freedom associated with the component. For example a yaw bearing has one generalised strain, which represent the yaw angle. The approach used also allows prescribed strains, which are particularly important in the case of stick-slip friction in bearings, where the bearing may stick if the absolute value of the angular velocity (such as strain rate) approaches zero. It is noted that prescribing a strain element will reduce the effective order of the system of equilibrium equations by one. + +In general, the relative motion of the distal nodes is constrained, for which reason the position and orientation of a distal node are expressed as functions of the position and orientation of the proximal node and the generalised strains. From this fundamental transformation it is straightforward to derive the corresponding transformations for the velocity and the acceleration. The constraints are conveniently expressed in terms of two constraint matrices relating to the nodal velocities and the strain rates. In general, the constraint matrices are time-dependent functions of the position and orientation of the proximal node as well as the strains. + +Components may have mass or may be considered massless. The generalised inertia forces are derived from the principle of virtual work, where the inertia force density is expressed according to D'Alambert's principle. The material velocity and acceleration are derived from a fundamental displacement hypothesis that defines the absolute displacement of all material points of the component as a function of the relative position, the nodal position and orientation and the strain. The result of this analysis shows that the inertia force can be expressed in terms of a mass matrix multiplied by the acceleration vector plus a vector of non-linear inertia forces and stresses, such as centrifugal and Coriolis forces. + +In principle, external loads can only be applied as nodal loads or generalised stresses (equal and opposite loads applied at a generalised freedom). For distributed loads like wind and wave loads the corresponding applied nodal loads and generalised stresses are calculated according to the principle of virtual work. For a yaw bearing the applied generalised stress is simply the moment applied by the yaw actuator. + +Gravity loads are conveniently considered as a distributed applied body force. + +The resulting equilibrium equations of a component are derived by collecting all generalised forces acting on the component. The effect of the distal node(s) constraints is described by Lagrange's methods in terms of internal forces that are expressed by the constraint matrices and a set of yet unknown Lagrange multipliers (Cook, Malkus and Plesha, 1989). A detailed analysis shows that the resulting component equilibrium equations and the transformation for the acceleration may be expressed in matrix form as +叶片动力学多体法，最初由J.P. Meijaard博士（诺丁汉大学，现荷兰图温特大学）受Garrad Hassan and Partners Ltd委托提出（Meijaard, 2005），并特别用于风轮结构动力学建模。该方法假设树状结构，这意味着结构环路不能在柔性构件之外描述。 + +一般来说，结构构件可以假定为刚性的，例如偏航轴承和叶片轴承，或者柔性的，例如支撑结构、塔架和叶片。虽然刚性构件相对容易建模，但柔性构件更为复杂，因为这种类型构件的材料点运动通常是由刚体运动与由于变形引起的相对运动相结合而产生的。柔性构件建模方法的描述见“柔性构件建模”。 + +一个构件的刚体运动由一组局部构件节点（特征材料点）的运动来描述，其中构件的运动被认为是已知的。构件只能在节点处相互连接，并且连接是通过标准的有限元技术实现的，即将参与连接的构件的节点链接到全局结构节点。由于树状结构的假设，将一个构件的节点划分为一个与前驱节点链接的邻近节点和一个可以与后继节点链接的远端节点是方便的。例如，偏航轴承有两个节点，即一个连接到塔架的邻近节点和一个连接到主框架的远端节点。 + +对于所有构件，一个局部笛卡尔坐标系附着在邻近节点上，使得原点的位置和方向由该节点的位置和方向定义。这个局部坐标系主要用于在“柔性构件建模”中更详细描述的柔性构件。 + +一个构件的变形状态由广义应变来描述，这些应变代表与构件相关的自由度。例如，偏航轴承有一个广义应变，代表偏航角度。所用的方法也允许规定应变，这在轴承的粘滑摩擦中尤其重要，因为当绝对角速度（如应变率）接近零时，轴承可能会卡住。需要注意的是，规定一个应变元素会使平衡方程组的有效阶数减少一个。 + +一般来说，远端节点的相对运动受到约束，因此远端节点的位置和方向表示为邻近节点的位置和方向以及广义应变的函数。从这种基本的变换可以很容易地推导出速度和加速度的相应变换。约束可以方便地用两个与节点速度和应变率相关的约束矩阵来表达。一般来说，约束矩阵是邻近节点的位置和方向以及应变的时间相关函数。 + +构件可能具有质量，或者可以被认为是无质量的。广义惯性力是从虚功原理推导出来的，其中惯性力密度根据达朗贝尔原理来表达。材料速度和加速度是从一个基本位移假设推导出来的，该假设定义了构件所有材料点的绝对位移为相对位置、节点位置和方向以及应变的函数。此分析的结果表明，惯性力可以表示为质量矩阵乘以加速度向量，加上一个非线性惯性力和应力向量，例如离心力和科里奥利力。 + +原则上，外力只能施加为节点力或广义应力（施加在广义自由度上的大小相等且方向相反的力）。对于风力和波力等分布载荷，相应的施加的节点力和广义应力是根据虚功原理计算出来的。对于偏航轴承，施加的广义应力只是偏航执行器施加的力矩。 + +重力载荷可以方便地考虑为施加的分布体力。 + +一个构件的平衡方程组是通过收集作用于该构件的所有广义力而推导出来的。远端节点约束的影响由拉格朗日方法描述，用约束矩阵和一组未知的拉格朗日乘子来表达（Cook, Malkus and Plesha, 1989）。详细的分析表明，得到的构件平衡方程和加速度变换可以表示为矩阵形式。 + +$$ +\begin{equation} +\left[ +\begin{array}{ccc} +M_{\mathrm{rr}}^{\mathrm{c}} & M_{\mathrm{r}\epsilon}^{\mathrm{c}} & D_{\mathrm{r}}^{\mathrm{c},T} \\ +M_{\mathrm{r}\epsilon}^{\mathrm{c},T} & M_{\epsilon\epsilon}^{\mathrm{c}} & D_{\epsilon}^{\mathrm{c},T} \\ +D_{\mathrm{r}}^{\mathrm{c}} & D_{\epsilon}^{\mathrm{c}} & 0 +\end{array} +\right] +\left[ +\begin{array}{c} +\dot{\mathbf{v}}^{\mathrm{c}} \\ +\ddot{\boldsymbol{\epsilon}}^{\mathrm{c}} \\ +\boldsymbol{\lambda}^{\mathrm{c}} +\end{array} +\right] ++ +\left[ +\begin{array}{c} +0 \\ +C_{\epsilon\epsilon}^{\mathrm{c}} \\ +0 +\end{array} +\right] +\dot{\boldsymbol{\epsilon}}^{\mathrm{c}} ++ +\left[ +\begin{array}{c} +0 \\ +K_{\epsilon\epsilon}^{\mathrm{c}} \\ +0 +\end{array} +\right] +\boldsymbol{\epsilon}^{\mathrm{c}} += +\left[ +\begin{array}{c} +\mathbf{f}_{\mathrm{a}}^{\mathrm{c}} + \mathbf{f}_0^{\mathrm{c}} \\ +\boldsymbol{\upsigma}_{\mathrm{a}}^{\mathrm{c}} + \boldsymbol{\upsigma}_0^{\mathrm{c}} \\ +\mathbf{a}_2^{\mathrm{c}} +\end{array} +\right] +- +\left[ +\begin{array}{c} +\mathbf{f}_{\mathrm{i}}^{\mathrm{c}} \\ +\boldsymbol{\upsigma}_{\mathrm{i}}^{\mathrm{c}} \\ +0 +\end{array} +\right], +\end{equation} +$$ + +where + +$\mathbf{v}^{\mathrm{c}}$ is a vector of nodal velocities, +$\lambda^{\mathrm{~c~}}$ is a vector of Lagrange multipliers corresponding to the constraints, +$\mathbf{f}_{\mathrm{i}}^{\mathrm{c}}$ and $\boldsymbol{\upsigma}_{\mathrm{i}}^{\mathrm{c}}$ are vectors of non-linear inertia forces and stresses dual to nodal velocities and strain rates, +$\mathbf{f}_{\mathrm{a}}^{\mathrm{c}}$ is a vector of applied nodal forces, +$\upsigma_{\mathrm{a}}^{\mathrm{c}}$ is a vector of applied generalised stresses dual to generalised strain rates, +$\mathbf{f}_{0}^{\mathrm{\,c}}$ is a vector of joint reactions dual to nodal velocities, +$\boldsymbol{\upsigma}_{0}^{\mathrm{c}}$ is a vector of generalised constraint stresses corresponding to prescribed strains, +$\mathbf{a}_{2}^{\mathrm{c}}$ collects the convective terms (quadratic in nodal velocities) in the transformation for the acceleration, +$\mathbf{M}_{\mathrm{rr}}^{\mathrm{c}},\mathbf{M}_{\mathrm{r}\epsilon}^{\mathrm{c}}$ and ${\bf M}_{\epsilon\epsilon}^{\mathrm{c}}$ are the structural mass matrix partitions dual to nodal velocities and strain rates, ${\bf C}_{\epsilon\epsilon}^{\mathrm{c}}$ is the structural damping matrix dual to strain rates, +${\bf K}_{\epsilon\epsilon}^{\mathrm{c}}$ is the structural stiffness matrix dual to strain rates, +$\mathbf{D}_{\mathrm{r}}^{\mathrm{c}}$ and ${\bf D}_{\epsilon}^{\mathrm{c}}$ are the constraint matrix partitions relating to nodal velocities and strain rates. + +Obviously it is not possible to solve this equation due to the unknown joint reaction forces $\mathbf{f}_{0}^{\mathrm{{c}}}$ (section forces), which originates from the connection to other components. In order to solve the system it is necessary to collect all the component equilibrium equations into a global system of the complete structure, which is done using the standard finite element assembly process (Cook,. Malkus and Plesha, 1989). This global system of equations has almost the same form as the component equations and is written in matrix form as +其中 + +$\mathbf{v}^{\mathrm{c}}$ 是节点速度向量， +$\lambda^{\mathrm{~c~}}$ 是对应于约束的拉格朗日乘子向量， +$\mathbf{f}_{\mathrm{i}}^{\mathrm{c}}$ 和 $\boldsymbol{\upsigma}_{\mathrm{i}}^{\mathrm{c}}$ 分别是节点速度和应变率的非线性惯性力和应力向量， +$\mathbf{f}_{\mathrm{a}}^{\mathrm{c}}$ 是施加的节点力向量， +$\upsigma_{\mathrm{a}}^{\mathrm{c}}$ 是对应于广义应变率的施加广义应力向量， +$\mathbf{f}_{0}^{\mathrm{\,c}}$ 是对应于节点速度的节点连接反作用力向量， +$\boldsymbol{\upsigma}_{0}^{\mathrm{c}}$ 是对应于规定应变的广义约束应力向量， +$\mathbf{a}_{2}^{\mathrm{c}}$ 收集了用于加速度变换中的对流项（节点速度的二次项）， +${\bf M}_{\mathrm{rr}}^{\mathrm{c}}$, ${\bf M}_{\mathrm{r}\epsilon}^{\mathrm{c}}$ 和 ${\bf M}_{\epsilon\epsilon}^{\mathrm{c}}$ 分别是对应于节点速度和应变率的结构质量矩阵分区，${\bf C}_{\epsilon\epsilon}^{\mathrm{c}}$ 是对应于应变率的结构阻尼矩阵， +${\bf K}_{\epsilon\epsilon}^{\mathrm{c}}$ 是对应于应变率的结构刚度矩阵， +$\mathbf{D}_{\mathrm{r}}^{\mathrm{c}}$ 和 ${\bf D}_{\epsilon}^{\mathrm{c}}$ 分别是对应于节点速度和应变率的约束矩阵分区。 + +显然，由于未知的节点连接反作用力 $\mathbf{f}_{0}^{\mathrm{{c}}}$ (受力)，无法求解此方程，这些反作用力源于与其他组件的连接。为了求解该系统，必须将所有组件平衡方程收集到一个完整的结构全局系统，这使用标准的有限元组装过程完成（Cook, Malkus 和 Plesha, 1989）。这个全局方程组的形式几乎与组件方程相同，并以矩阵形式写为： + +$$ +\begin{equation} +\left[ +\begin{array}{ccc} +\mathbf{M}_{\mathrm{rr}} & \mathbf{M}_{\mathrm{r}\epsilon} & \mathbf{D}_{\mathrm{r}}^T \\ +\mathbf{M}_{\mathrm{r}\epsilon}^T & \mathbf{M}_{\epsilon\epsilon} & \mathbf{D}_{\epsilon}^T \\ +\mathbf{D}_{\mathrm{r}} & \mathbf{D}_{\epsilon} & \mathbf{0} +\end{array} +\right] +\left[ +\begin{array}{c} +\dot{\mathbf{v}} \\ +\ddot{\boldsymbol{\epsilon}} \\ +\boldsymbol{\lambda} +\end{array} +\right] ++ +\left[ +\begin{array}{c} +\mathbf{0} \\ +\mathbf{C}_{\epsilon\epsilon} \\ +\mathbf{0} +\end{array} +\right] +\dot{\boldsymbol{\epsilon}} ++ +\left[ +\begin{array}{c} +\mathbf{0} \\ +\mathbf{K}_{\epsilon\epsilon} \\ +\mathbf{0} +\end{array} +\right] +\boldsymbol{\epsilon} += +\left[ +\begin{array}{c} +\mathbf{f}_{\mathrm{a}} \\ +\boldsymbol{\sigma}_{\mathrm{a}} + \boldsymbol{\sigma}_0 \\ +\mathbf{a}_2 +\end{array} +\right] +- +\left[ +\begin{array}{c} +\mathbf{f}_{\mathrm{i}} \\ +\boldsymbol{\sigma}_{\mathrm{i}} \\ +\mathbf{0} +\end{array} +\right] +\end{equation} +$$ + +The main difference between component equations and the global system of equations is that the joint reaction forces do not appear in the latter as they have been cancelled out by the assembly process. Consequently the above system can be solved directly with respect to the nodal accelerations $\dot{\mathbf{v}},$ the strain accelerations $\ddot{\epsilon},$ and the Lagrange multipliers $\lambda$. +元件方程组与整体方程组的主要区别在于，后者中没有出现节点反作用力，因为它们已被组装过程抵消。因此，上述系统可以直接求解，得到节点加速度 $\dot{\mathbf{v}}$、应变加速度 $\ddot{\epsilon}$ 和拉格朗日乘子$\lambda$。 + + +Last updated 26-11-2024 + +# Calculation Procedure + +A schematic of the Bladed calculation to evaluate the structural response in the time domain is shown in Figure 1. The numbered steps below map directly to the numbered steps in the diagram. + +1. Modal displacements and velocities ("state values") are known at the start of the time step. 2. Applied loads are calculated based on the external loads and the state values. External loads applied at structural nodes are transformed into applied loads on the modes. + +3. The structural dynamics equation $\mathbf{M}\ddot{\mathbf{q}}+\mathbf{C}\dot{\mathbf{q}}+\mathbf{K}\mathbf{q}=\mathbf{f}$ is solved in modal space to find the state accelerations q, where + +o M, C, and K are the system modal matrices for mass, damping and stiffness respectively 0 q, q, ? and are the state displacements, velocities and accelerations respectively o f is the modal vector of externally applied forces on each state I. The integrator uses the accelerations to find the state values at the next time step. + +In most cases, applied loads depend partially on the nodal positions and velocities which are calculated from the state values. This is convenient as the states values are known at the start of each time step. + +图1显示了Bladed计算方案，用于评估时域内的结构响应。以下编号步骤与图中的编号步骤一一对应。 + +1. 在时间步的开始时，模态位移和速度（“状态值”）已知。 +2. 应用载荷基于外部载荷和状态值进行计算。施加在结构节点上的外部载荷被转换成作用于模态的应用载荷。 + +3. **在模态空间内求解结构动力学方程** $\mathbf{M}\ddot{\mathbf{q}}+\mathbf{C}\dot{\mathbf{q}}+\mathbf{K}\mathbf{q}=\mathbf{f}$，以求得状态加速度 $\ddot{\mathbf{q}}$，其中 + + - $M、C、K$分别是系统模态矩阵，代表质量、阻尼和刚度； + - $\mathbf{q}、\dot{\mathbf{q}}、\ddot{\mathbf{q}}$分别是状态位移、速度和加速度； + - $f$是作用于每个状态的外部力模态向量。 +4. 积分器使用加速度来找到下一个时间步的状态值。 + +在大多数情况下，施加的载荷部分地取决于从状态值计算出的节点位置和速度，这很方便，因为状态值在每个时间步的开始时是已知的。 + +![](images/7c0159c4d4ddffd321638729e7a6b6835f10ff0b86e99d1a65d22686f4c3c716.jpg) + +Figure 1: Schematic of calculation procedure of structural dynamics in time domain simulations Last updated 30-08-2024 + +# Equations of Motion + +Because of the complexity of the coupling of the modal degrees of freedom of the rotating and non-rotating components, the algebraic manipulation involved in the derivation of the equations of motion for a wind turbine is relatively complicated, and the following only gives a brief description of the theory. +由于旋转部件和非旋转部件的模态自由度耦合较为复杂，风轮运动方程推导涉及的代数运算相对繁琐，以下仅对理论进行简要描述。 + +## Degrees of freedom + +Examples of possible degrees of freedom involved in the equations of motion for the structural dynamic model for Bladed are as follows: + +· Blade deflection +· Nacelle yaw +· Tower fore-aft, side-side and torsional deflection (axisymmetric tower model) +· General tower deflection (multi-member tower model): a large number of modes is allowed, including the torsional and axial degrees of freedom. + +In addition, the following dynamics can also be included as required: + +· A sophisticated representation of the power train dynamics. +· A range of different representations of generator and power converter dynamics, including both mechanical models and electrical models which can include network interactions. +· A range of pitch actuator models, from simple passive models to detailed representations of electric servo drives and hydraulic actuators. +· Teeter restraints, passive blade vibration dampers and tower dampers, and yaw system dynamic response. +· Transducer dynamics for control signals. +· All controller dynamics. + +Bladed结构动力学模型运动方程中涉及的自由度示例如下： + +· 叶片挠曲 +· 机舱偏航 +· 塔架前后、侧向和扭转挠曲（塔架轴对称模型） +· 塔架挠曲（多单元塔架模型）：允许大量的模式，包括扭转和轴向自由度。 + +此外，根据需要，还可以包含以下动力学： + +**· 复杂的动力传动系统表示。** +**· 各种发电机和电力转换器动力学表示，包括机械模型和可以包含网络互动的电气模型。** +**· 各种pitch执行器模型，从简单的被动模型到详细的电动伺服驱动器和液压执行器表示。** +**· teeter约束、被动叶片振动阻尼器和塔架阻尼器，以及偏航系统的动态响应。** +**· 控制信号的传感器动力学。** +**· 所有控制器动力学。** +## Formulation of equations of motion + +As described briefly in the multibody dynamics approach the equations of motions of the complete system have been derived using the multibody dynamics approach based on the principle of virtual work. It appears that the solution of the resulting equations is generally difficult to obtain as **the augmented mass matrix including the constraint matrices are generally ill-conditioned**. The system is therefore transformed into a system where the only unknown is the strain accelerations using the so-called constraint elimination method (Géradin and Cardona, 2001).together with the transformation for the velocity given in the form $\mathbf{D}_{\mathbf{r}}^{\mathrm{c}}\mathbf{v}+\mathbf{D}_{\epsilon}^{\mathrm{c}}\dot{\epsilon}=\mathbf{0}.$ The final result of this straightforward transformation can be written in the conventional form + +如同多体动力学方法简要描述的那样，我们基于虚功原理，利用多体动力学方法推导了整个系统的运动方程。 显而易见，**由于包含约束矩阵的增广质量矩阵通常病态**，因此求解得到的方程通常难以获得。 因此，**我们将系统转换成一个仅包含应变加速度的未知量的系统，采用所谓的约束消除法**（Géradin and Cardona, 2001），并结合以如下形式给出的速度变换：$\mathbf{D}_{\mathbf{r}}^{\mathrm{c}}\mathbf{v}+\mathbf{D}_{\epsilon}^{\mathrm{c}}\dot{\epsilon}=\mathbf{0}$。 这种简单变换的最终结果可以写成常规形式。 + + +$$ +\mathbf{M}\dot{\epsilon}+\mathbf{C}\dot{\epsilon}+\mathbf{K}\epsilon={\pmb{\sigma}}. +$$ + +In cases with no prescribed strains it is straightforward to show, that the three system matrices appearing on the left-hand side of the above equation become: +在无给定约束应力的情况下，很容易证明上述方程左侧出现的三个系统矩阵为： +$$ +\mathbf{M}=\mathbf{M}_{\epsilon\epsilon}+\mathbf{D}_{\epsilon\tau}^{T}\mathbf{M}_{\mathrm{rr}}\mathbf{D}_{\epsilon\mathrm{r}}+\mathbf{D}_{\epsilon\mathrm{r}}^{T}\mathbf{M}_{\mathrm{r}\epsilon}+\mathbf{M}_{\mathrm{r}\epsilon}^{T}\mathbf{D}_{\epsilon\mathrm{r}},\quad\mathbf{C}=\mathbf{C}_{\epsilon\epsilon},\quad\mathbf{K}=\mathbf{K}_{\epsilon\epsilon} +$$ + +where $\mathbf{D}_{\epsilon\mathrm{r}}=-\mathbf{D}_{\mathrm{r}}^{-l}\mathbf{D}_{\epsilon}$ is the time-dependent part of the reduction matrix. The right-hand side stress vector of the global system of equations becomes: +其中 $\mathbf{D}_{\epsilon\mathrm{r}}=-\mathbf{D}_{\mathrm{r}}^{-l}\mathbf{D}_{\epsilon}$ 是减缩矩阵随时间变化的项。全局方程组的右侧应力向量变为： +$$ +\begin{array}{r}{\pmb{\sigma}=\mathbf{\sigma}_{\mathbf{a}}+\pmb{\sigma}_{0}-\mathbf{\sigma}_{\mathbf{i}}-\mathbf{M}_{\mathrm{r}\epsilon}^{T}\mathbf{g}_{2}+\mathbf{D}_{\epsilon\mathrm{r}}^{T}\,(\mathbf{f}_{\mathbf{a}}-\mathbf{f}_{\mathbf{i}}-\mathbf{M}_{\mathrm{rr}}\mathbf{g}_{2}),}\end{array} +$$ + + where $\mathbf{g}_{2}=\mathbf{D}_{\epsilon\mathrm{r}}^{-1}\mathbf{a}_{2}$ + +In general, the system mass matrix M is full, due to the coupling of the degrees of freedom, and it contains periodic coefficients because of the time-dependent interaction of the dynamics of the rotor and tower. The system damping and stiffness matrices C and K are generally diagonal and constant. + +Because of their complexity, further details of the equations of motion are not presented here. + +通常情况下，由于自由度之间的耦合，系统质量矩阵 M 是满秩的，并且由于风轮和塔架动力学的时间相关相互作用，它包含周期系数。系统阻尼矩阵 C 和刚度矩阵 K 通常是对称的且为常数。 + +由于其复杂性，此处不详细呈现运动方程的细节。 +## Solution of the equations of motion + +Typically, the equations of motion are solved by time-marching numerical integration of the system of ordinary differential equations using a variable step size, fourth order Runge-Kutta integrator (Kreyszig, 2006). For so-called stiff systems with many high frequency modes (for example wind turbine models with multi-part blades), a fixed step Newmark- $\beta$ integrator (Newmark, 1959) or the Generalised- ${\alpha}$ integrator (Chung, 1993) integrator are recommended to improve simulation performance. + +It is noted that all fixed step integrators in Bladed assume zero structural state accelerations when calculating the geometric stiffness loads. This assumption was done to enable fixed-step integrators to converge within reasonable time step size. In contrast, the implicit Newmark- $\beta$ and Runge-Kutta integrators do not require this simplification in the formulation, which will provide a more accurate solution for dynamic response of long-flexible blades. Therefore, the implicit Newmark- $\cdot\beta$ integrator or Runge-Kutta integrator is recommended over other integrators in Bladed. + +通常，运动方程是通过时间步进数值积分法，使用可变步长、四阶龙格-库塔积分器（Kreyszig, 2006）求解的。对于所谓刚性系统，这类系统具有许多高频模态（例如带有复数叶片的风轮模型），建议使用固定步长的 Newmark-β 积分器（Newmark, 1959）或广义- ${\alpha}$ 积分器（Chung, 1993）积分器，以提高模拟性能。 + +需要注意的是，Bladed 中所有固定步长积分器在计算几何刚度载荷时都假设结构状态加速度为零。这是为了使固定步长积分器能够在合理的步长内收敛。相比之下，隐式 Newmark-β 和龙格-库塔积分器在公式推导中不需要这种简化，从而能为长柔性叶片的动态响应提供更准确的解。因此，在 Bladed 中建议使用隐式 Newmark-β 积分器或龙格-库塔积分器，优于其他积分器。 + +Last updated 15-11-2024 + +# Internal Forces in Flexible Body Components + +A key functionality of the flexible body component is the calculation of internal forces (also known as stress resultants or section forces), which is carried out for a known displacement state at the end of each time step for a dynamic analysis or at the calculated steady state solution for a steady state analysis. A fundamental assumption for the calculation is that the external loads (incl. gravity and inertial loads) are applied to the deflected state of the flexible body component since the second order effects of the external loads have a significant effect on the resulting internal forces. + +The conventional method for calculating internal forces in Bladed is a displacement-based finite element method that calculates the internal forces at the end nodes of a beam element using the element stiffness matrix. + +An equilibrium-based method is available for calculating internal forces, which is derived by equilibrating the internal forces with the external loading. For statically indeterminate flexible bodies complex analysis it is necessary to include stiffness properties, but for statically determinate flexible bodies the calculation can be done without using any stiffness properties. + +For blades, the equilibrium-based method is always used to calculate internal forces. For the tower, either method can be used. + +# Conventional Displacement-Based Finite Element Method + +With the modal model, the deformed shape of the flexible body components like the tower at any instant is a linear combination of the selected mode shape functions. With a reduced number of modes, the resulting deformation may therefore not be accurately predicted, which means that it is not possible to calculate the internal forces directly from the deformations as done by standard finite element technique. + +In order to calculate the internal forces of flexible body components, the deformation at all stations is therefore calculated from a static equilibrium analysis, where the applied force is calculated as the sum of all external forces including the inertial loads. In case that some fundamental degrees of freedom are constrained the system is solved with respect to a reduced set of independent degrees of freedom, and the Lagrange multipliers associated with the constraints are calculated. Finally the internal forces of all beam elements at both ends are calculated from the fundamental equilibrium equation of the beam element. The second order effects of the external loads on the calculated internal forces are accounted for through the geometric stiffness model as described in (Przemieniecki, 1968). + +# Equilibrium-Based Method + +The method employs the multibody_approach used by Bladed for modelling of the complete wind turbine structure on a flexible body component level. With this approach described in (Nim et al.,. 2024), a flexible body is modelled as an assembly of rigid or deformable elements, interconnected at $N$ nodes each including six nodal DOFs, i.e., three translation components and three rotation components. According to the multibody approach, the deformation state of a flexible element is described by $N_{\epsilon}^{\epsilon}$ generalised strains that are collected in the vector $\epsilon^{\mathrm{{e}}}$ . The external element loading including inertial loads is represented by the vector $\mathbf{p}_{\mathrm{r}}^{\mathrm{e}}$ conjugate to nodal motion and the vector $\mathbf{p}_{\epsilon}^{\mathrm{e}}$ conjugate to generalised strains, defining the deformation state of a deformable element with elastic properties defined by the stiffness matrix ${\bf K}_{\epsilon\epsilon}^{\mathrm{e}}$ . The kinematical relations between nodal displacement and the generalised strains motion are described by $N_{\mathrm{c}}^{\mathrm{e}}$ geometric constraint relations in terms of the constraint matrices ${\bf C}_{\mathrm{r}}^{\mathrm{e}}$ and ${\bf C}_{\epsilon}^{\mathrm{e}},$ which associate a set of unknown Lagrange multipliers $\lambda^{\mathrm{~e~}}$ . By application of the principle of virtual work, it appears that the resulting equilibrium equations for an element can be written in a simplified form as + +$$ +\left[\mathbf{C}_{\mathrm{r}}^{\mathrm{e}T}\right]\boldsymbol{\lambda}^{\mathrm{e}}=\left[\begin{array}{l}{\mathbf{p}_{\mathrm{r}}^{\mathrm{e}}+\mathbf{f}_{0}^{\mathrm{e}}}\\ {\mathbf{p}_{\epsilon}^{\mathrm{e}}-\mathbf{K}_{\epsilon\epsilon}^{\mathrm{e}}\epsilon^{\mathrm{e}}}\end{array}\right] +$$ + +Thevector $\mathbf{f}_{0}^{\mathrm{{e}}}$ contains the resulting element internal forces, which originate from the connection to neighbouring elements. This vector cancels out in the assembly process, which means that the resulting equilibrium equations for the flexible body take the form + +$$ +\left[\mathbf{C}_{\mathrm{r}}^{\phantom{\dagger}}T\right]\boldsymbol{\lambda}=\left[\begin{array}{c}{\mathbf{p}_{\mathrm{r}}}\\ {\mathbf{p}_{\epsilon}-\mathbf{K}_{\epsilon\epsilon}\epsilon}\end{array}\right] +$$ + +For statically determinate flexible bodies, the total number of constraints $N_{\mathrm{c}}=\Sigma_{\mathrm{e}}N_{\mathrm{c}}^{\mathrm{e}}$ equals the number of nodal DOFs $6N$ , i.e., $N_{\mathrm{c}}=6N$ implying that $\mathbf{C}_{\mathrm{r}}$ is a square matrix. In this case, the Lagrange multipliers $\lambda$ for the flexible body are calculated from Equation (2) by the linear system $\mathbf{C}_{\mathrm{r}}{}^{T}\hat{\pmb{\lambda}}=\mathbf{p}_{\mathrm{r}}$ with the assumption that the constraint matrix $\mathbf{C}_{\mathrm{r}}$ is invertible. The extracted element Lagrange multipliers $\lambda^{\mathrm{{e}}}$ are then used for calculating the resulting internal forces at the element end nodes by the relationship $\mathbf{f}_{0}^{\mathrm{e}}=\mathbf{C}_{\mathrm{r}}^{\mathrm{e}T}\lambda^{\mathrm{e}}-\mathbf{p}_{\mathrm{r}}^{\mathrm{e}}$ derived by Equation (1). With this approach, the internal forces are determined in terms of the Lagrange multipliers, which represent the internal constraint forces in the element. + +For statically indeterminate flexible bodies, the number of constraints is larger than the number of nodal DOFs, i.e., $N_{\mathrm{c}}>6N$ implying that $\mathbf{C_{r}}$ is a rectangular matrix. Conventionally, the degree of static indeterminacy $D_{\mathrm{s}}$ is used for quantifying the number of redundant forces, which in the present context means that $\ensuremath{D_{\mathrm{s}}}=\ensuremath{N_{\mathrm{c}}}-6\ensuremath{N_{\mathrm{}}}$ In order to calculate the Lagrange multipliers 入 for the flexible body, it is therefore necessary to include both relations in Equation (2). To this end, the geometric constraints are subdivided into a set of $6N$ determinate constraints and the remaining $D_{\mathrm{s}}$ indeterminate constraints. With this subdivision, it appears that Equation (2) can be rewritten as + +$$ +\left[{\bf C}_{\mathrm{rr}}^{{\mathrm{\tiny~\textit{T}}}}\;\;{\bf C}_{\mathrm{er}}^{{\mathrm{\tiny~\textit{T}}}}\right]\left[\boldsymbol{\lambda}_{\mathrm{r}}\right]=\left[\boldsymbol{\mathbf{p}}_{\mathrm{r}}\;\;\right]}\\ {{\bf C}_{r\epsilon}^{{\mathrm{\tiny~\textit{T}}}}\;\;{\bf C}_{\mathrm{e}\epsilon}^{{\mathrm{\tiny~\textit{T}}}}\!\!\!\int\left[\boldsymbol{\lambda}_{\mathrm{e}}\right]=\left[{\bf p}_{\epsilon}-{\bf K}_{\epsilon\epsilon}\epsilon\right]}\end{array} +$$ + +where $\lambda_{\mathrm{r}}$ and $\lambda_{\mathrm{e}}$ include partial permutations of $\lambda$ , while $\mathbf{C}_{\mathrm{rr}}$ and $\mathbf{C}_{\mathrm{r}\epsilon}$ and respectively $\mathbf{C}_{\mathrm{er}}$ and $\mathbf{C}_{\mathrm{e}\epsilon}$ include partial permutations of $\mathbf{C}_{\mathrm{r}}$ and $\mathbf{C}_{\epsilon}$ . In principle, the subdivision of the geometric constraints into determinate and indeterminate partitions corresponds to the introduction of virtual cuts in the structure. The subdivision is done automatically, (Wehage and Haug, 1982), in such a way that the cut structure is statically determinate, which enables the calculation of internal forces by the equilibrium approach. With the application of the constraint elimination method, + +(Géradin and Cardona, 2001), it appears that the Lagrange multipliers $\lambda_{\mathrm{r}}$ associated with the determinate constraints can be eliminated in Equation (3). After a minor rearrangement of terms, it appears that the resulting transformed system can be written in the symmetric form + +$$ +\left[\mathbf{C}_{\epsilon\epsilon}^{*}\quad\mathbf{C}_{\epsilon\epsilon}^{*\ T}\right]\left[\boldsymbol{\epsilon}^{\prime}\right]=\left[\mathbf{p}_{\epsilon}^{*}\right] +$$ + +where $\mathbf{C}_{\mathrm{e}\epsilon}^{*}=\mathbf{C}_{\mathrm{e}\epsilon}+\mathbf{C}_{\mathrm{er}}\mathbf{T}_{\mathrm{r}\epsilon}$ and $\mathbf{p}_{\epsilon}^{*}=\mathbf{p}_{\epsilon}+\mathbf{T}_{\mathrm{r}\epsilon}{}^{T}\mathbf{p}_{\mathrm{r}},$ while ${\bf K}_{\epsilon\epsilon}^{*}={\bf K}_{\epsilon\epsilon}$ is introduced for notational consistency. The transformation matrix $\mathbf{T}_{\mathbf{r}\epsilon}$ relates a variation of the generalised strains to the variation of the nodal DOFs, and it is calculated from the partial permutations $\mathbf{C}_{\mathrm{rr}}$ and $\mathbf{C}_{\mathrm{r}\epsilon}$ assuming that $\mathbf{C}_{\mathrm{rr}}$ is invertible. In principle, the indeterminate constraints $\lambda_{\mathrm{e}}$ together with generalised strains $\epsilon^{\prime}$ can be calculated from the linear system (4), but the actual calculation is done in a more numerically stable way in order to deal with the fact that the coefficient matrix on the left-hand side is ill-conditioned. The determinate Lagrange multipliers $\lambda_{\mathrm{r}}$ corresponding to the indeterminate constraints $\lambda_{\mathrm{e}}$ are then calculated from the equilibrium relation of Equation (3) resulting in the linear system $\mathbf{C}_{\mathrm{rr}}^{\phantom{\dagger}}{}^{T}\lambda_{\mathrm{r}}=\mathbf{p}_{\mathrm{r}}-\mathbf{C}_{\mathrm{er}}{}^{T}\lambda_{\mathrm{e}}$ With known Lagrange multipliers, the resulting internal forces can be calculated by the approach described above and used for statically determinate flexible bodies. + +It is noted that the alternative equilibrium-based method for calculating internal forces is essentially different from the conventional displacement-based finite element method, where the internal forces at the end nodes of a beam element are calculated in terms of the internal elastic forces using the element stiffness matrix. An important difference is that the equilibrium-based method calculates the internal forces at the current modal truncated state during time integration. The number of modes used for describing the modal truncated state can vary significantly depending on the desired accuracy of the analysis. In particular, it is even possible to do analysis without any modes, for flexible bodies that are temporarily assumed rigid, which may be convenient for some initial investigations. + +Last updated 25-10-2024 + +# Introduction to Flexible Components + +In Bladed, the support structure is modelled as a single linear flexible component (body). The blade is also simulated as a flexible component, with the option to be subdivided into multiple linear flexible components in order to accurately model large deflections. + +The fundamental finite element model assumes that the flexible components are linear space beams or more general space frames that comprise assemblies of multiple members modelled by Timoshenko beam elements. As the model is linear, deflections are assumed to be small within eachbody. + +For the blades, the finite element model degrees of freedom can be used directly as the generalised freedoms. For the blades and support structure, modal reduction can be performed to reduce the number of generalised freedoms. Using the modal approach, the deformation is represented by a linear combination of some pre-calculated mode shapes. The scalars of this linear combination are the modal amplitudes(Clough and Penzien, 1993), that represent the generalised strains and hence the degrees of freedom of the component. It is important to note that the mode shape functions are constant in time. + +The applied beam element may be considered as an extension to the standard three-dimensional engineering Timoshenko beam element (Przemieniecki, 1968) with two nodes or stations located at the two ends. This element has twelve fundamental degrees of freedom. These are the three translational and three rotational degrees of freedom at both stations. The deflection at all intermediate points is calculated via interpolation functions that are derived from a set of homogenous equilibrium equations valid for prismatic beam elements. It is important to note that this beam element includes the effect of shear deformation that may be important for some support structures, in particular complicated offshore foundations. The magnitude of the shear deformations relative to bending deformations for an element may be evaluated by the element shear parameter conveniently defined as + +$$ +\Omega^{e}=\frac{12E I^{e}}{l^{e^{2}}G A^{e}}, +$$ + +where +$E I^{e}$ is the bending stiffness, +$G A^{e}$ is the corresponding shear stiffness, $l^{e}$ is the element length. + +The beam element supports an arbitrary spatial variation of the stiffness along the beam element, but in the present implementation it is assumed that the bending, torsional, axial and shear stiffness vary linearly. The orientation of the element is defined by the position of the two ends as well as the orientation of the principal axis around the neutral axis (elastic centre). The effect of possible coupling between bending and torsion is included in terms of the position of the shear centre (torsion centre), and a transformation between displacements and forces relating to the shear centre and the neutral axis is included. The resulting stiffness matrix is constant and calculated by numerical integration. For prismatic elements, where the shear centre is located at the neutral axis, the stiffness matrix is identical to the standard engineering Timoshenko beam element (Przemieniecki, 1968). + +An important feature of the derived method is that some fundamental degrees of freedom may be constrained, which is particularly useful in cases where the effect of elongation and/or torsion can be neglected. In order to enable the description of rigid connection the deflection of a beam element may also be constrained completely. The constraints are modelled in terms of a constant constraint matrix together with Lagrange's method (Cook, Malkus and Plesha, 1989). + +Second-order effects of the internal axial forces are included in terms of a geometric stiffness matrix (stress stiffening) that is calculated from the derivatives of the interpolation functions (Clough and Penzien, 1993). For the blades the dominating part of the axial force is caused by centrifugal forces for which reason the term centrifugal stiffness is traditionally used in this case. A similar effect occasionally referred to as geometric destiffening can be observed in the support structure due to gravity loading. + +Further second-order effects of the internal shear forces are accounted for by applying extra external loads based on a method by (Krenk, 2009). This model can particularly enhance the prediction of torsional deflection in blades with a torsional degree of freedom. + +Inertia forces acting on the element and the proximal node are derived as described in the multibody dynamics approach from the fundamental displacement hypothesis using the principle of virtual work. An important feature of the derived method is that the inertia forces are expressed directly in terms of the modal amplitudes, i.e. the strains and corresponding derivatives as originally suggested in (Shabana, 1998). In principle this means that the time for calculating the accelerations scales with the number of mode shapes rather than the number of beam elements of the flexible component. + +Last updated 30-08-2024 + +# Normal and Attachment Modes + +The selection and calculation of mode shape functions follows the idea that was originally suggested by (Craig, 2000) as a modification of the widely used Craig-Bampton method from (Craig, 1968). For both methods the stations are subdivided into boundary stations that may couple to other components and interior stations that do not couple. The boundary stations also represent the component nodes that may link to nodes of other components. In particular the station representing the proximal node is constrained completely in order to exclude rigid body displacement modes. + +With the applied method the modes are generally selected as the union between attachment modesthat may couple to other components and normal modes that may be considered as internal vibration modes. + +![](images/dfb0559f901bb76d71fca50c912bd62e34efba55b69f91f14bad8f5cd5c2161b.jpg) + +Figure 1: llustration of the boundary conditions for attachment and normal modes. + +# Attachment Modes + +The attachment modes are calculated from the component stiffness matrix by a static equilibrium, where the component is fixed at the proximal node and point loads are applied in turn at the distal nodes, as seen in Figure 1. This method yields an attachment mode for every degree of freedom of the distal node of the component (fixed-free boundary condition). + +For the attachment modes of the support structure, the calculation of the mode frequencies include the mass of the rotor and nacelle assembly, and would usually be of lower frequencies than the corresponding normal modes. + +# Normal Modes + +The normal modes are determined directly from the fully assembled finite element mass and stiffness matrices using a generalised eigenvalue problem. This calculation is performed with the component fixed at the proximal node (fixed-free) or at both the proximal and distal nodes (fixedfixed), as illustrated in Figure 1. The type of normal mode (fixed-free vs. fixed-fixed) depends solely on the presence of distal nodes. + +For instance, in the case of a tower, there is always a distal node at the tower top, whereas singlepart blades do not have distal nodes. Multi-part blades have both a proximal node and a distal node for all parts except the last part, which only has a proximal node. Consequently, the last part produces fixed-free normal modes, while the rest produce fixed-fixed normal modes. There might be multiple distal nodes present, such as in dynamic moorings, and in such cases, all the distal nodes are fixed. + +This will produce one normal mode for every degree of freedom in the model, where the number of degrees of freedom is dependent on the boundary condition. + +# Frequencies and Structural Damping + +The frequencies of the attachment modes are calculated by Rayleigh's method (Clough and Penzien, 1993), while the frequencies of the normal modes result from the eigenvalue problem. These frequencies are solely used for describing damping. + +Structural damping is modelled as modal damping (Clough and Penzien, 1993). in terms of a set of damping coefficients (damping ratio) that relate to the mode shape functions. These coefficients are defined as input parameters for the model and may usually be measured directly, for example by exiting a mode and measure the decay of the succeeding oscillation. + +Last updated 10-10-2024 + +# Blade Modes + +The motion of the tapered, twisted and very flexible rotor blades is among the most complex phenomena related to the structural dynamics of a wind turbine. In Bladed, a blade can be represented by one component or several rigidly connected components. Use of several components allows rigorous modelling of large deflections. + +For the single-part blade model, only normal modes with fixed-free boundary conditions are used. This is the classical approach for selecting the vibration modes of a wind turbine blade. In the multi-part blade model, the inner parts use both normal modes with fixed-fixed boundary conditions and attachment modes with fixed-free boundary conditions, while the outermost part uses only normal modes with fixed-free boundary conditions. For more details see normal and attachment modes. + +The naming conventions for blade modes are detailed in the article titled Naming Mode Shapes. + +Each mode is defined in terms of the following parameters: + +Modal frequency, $\omega_{i}$ +Modal damping coefficient, $\xi_{i}$ +Mode shape represented by a vector of the displacement at the stations + +where the subscript $_i$ indicates properties related to the $i^{t h}$ mode. + +For blade with several parts, it is still desirable to calculate and review the coupled eigenmodes for the whole blade. To facilitate this, Bladed performs a subsequent eigen analysis of the blade parts to calculate the modes corresponding to the natural modes of the blade. This is useful both for physical interpretation of the blade mode shapes and for applying damping, as explained in the next section. + +# Specifying blade damping (whole blade damping) + +The blade damping is specified for the natural modes of the whole blade. To allow this, Bladed must calculate the damping on each blade part mode (or generalised finite element freedom) based on the damping of the natural modes of the whole blade. This is done according to theory presented in (Clough and Penzien, 1993) pp240-242. + +Damping should be specified for the coupled modes which would be expected to contribute significantly to the dynamic response of the blade (typically the first \~10 modes). For subsequent higher frequency coupled modes, the damping is assumed to be proportional to modal stiffness, calculated as $\mathbf{C}=a_{1}\mathbf{K}$ . This results in damping that is proportional to modal frequency, so that the responses of higher frequency modes are effectively eliminated by high damping ratios. + +The coefficient $\displaystyle a_{1}$ is defined according to the highest mode for which damping is specified: + +$$ +a_{1}=\frac{2\xi_{c}}{\omega_{c}} +$$ + +where + +· the subscript c refers to highest mode with damping specified, ? $\xi$ refers to the coupled mode damping ratio. + +Damping on these higher frequency coupled modes is given by + +$$ +\xi_{n}=\xi_{c}\left(\frac{\omega_{n}}{\omega_{c}}\right). +$$ + +Once the coupled mode damping values are calculated, the blade part modal frequencies are calculated in Bladed according to (Clough and Penzien, 1993). + +Last updated 11-12-2024 + +# Tower Modes + +The standard axisymmetric tower model in Bladed has one proximal node at the base and one distal node at the top. This implies that the mode shape functions are represented by a combination of attachment modes with fixed-free boundary conditions and normal modes with fixed-fixed boundary conditions. These mode shapes represent the deflection in the fore-aft and side-side directions as well as the torsional deflection and axial elongation. The tower modes are defined in terms of their modal frequency, modal damping and mode shape. + +A multi-member tower uses the same approach, but in this case an arbitrary structure consisting of any number of straight interconnected members is permitted. Since the tower is not assumed to be axisymmetric, the modes are generally three-dimensional and contain all six degrees of freedom at each node, and there may be a foundation at more than one node (proximal nodes). + +The naming conventions for tower modes are detailed in the article titled Naming Mode Shapes. + +Mass and inertia of the nacelle and rotor: For the calculation of the tower support structure modes, the nacelle and rotor are modelled as lumped mass and inertia located at the nacelle centre of gravity and rotor hub, respectively. As the normal modes do not couple to other components it appears that only the frequency of the attachment modes are affected. This means that the mass and inertia of the nacelle and rotor only affect the resulting frequency of the attachment modes. + +# Note on the Difference between Normal and Attachment modes forTowers + +For the support structure, the interpretation of the definition of norma/ modes can sometimes cause confusion. + +The conventional free vibration modes of a support structure include the modes where the top of the tower is free to move with no external forces on it. In Bladed, normal modes only include the modes from the eigen-analysis using the fixed-fixed boundary condition as shown here. However, Bladed also includes the attachment modes which are more realistic, as in reality the tower top will move due to the application of external forcing from the structure above it. + +This means that if the free vibration modes for the tower structure are calculated with only the tower base constrained, they will not match the mode frequencies calculated by Bladed. + +Bladed also calculates coupled vibration modes in the Campbell diagram, shows how the normal and attachment modes combine into coupled vibration modes at a specific operating point. Typically, these coupled modes correspond well to normal modes calculated in other software, with the tower base constrained and the tower top free to move. + +# Note on Comparing Coupled and Uncoupled Modes for Floating Turbines + +For a floating turbine with soft moorings, the deflection shape of the tower when a load is applied to the top in the Campbell diagram calculation does not correspond to the first attachment mode, as the structure is fixed at the modal reference node during the modal analysis. So, the cantilever attachment mode shapes calculated in modal analysis are not seen individually in the Campbell diagram deflections. When a load is applied to the tower top of a floating turbine, the deflection in the tower will take a form that is a combination of various normal and attachment modes, so the first coupled tower mode will be a combination of various uncoupled modes. + +Last updated 11-12-2024 + +# Naming Mode Shapes + +A structural mode shape describes an allowable pattern of translational and rotational displacements of any point of a flexible body. Using FEM, a mode shape is conveniently represented by a column vector of nodal displacements and rotations. In most cases, a mode shape represents a characteristic displacement pattern, such as the flapwise displacement of a blade or the side-to-side displacement of a tower. Naming the mode shapes with industrystandard terms can therefore facilitate easier reference. + +Lower-order mode shapes are often well-defined, with displacements primarily occurring in a single direction. These cases are typically easy to categorise. However, for prebend blades with beam-level cross couplings, the mode shapes typically contain large displacements in various directions. Similarly, for multi-member towers, such as jacket support structures, where the structural members are connected in different directions, determining the primary displacement direction is not straightforward. + +The problem of identifying mode shape names cannot be solved theoretically, as the problem is not rigorously defined. Hence, the presented method is a practical approach, implementing certain conditions that result in meaningful mode shape names in most cases. + +# The Naming Procedure + +This method uses a set of predefined conditions to identify and label mode shapes based on their primary characteristics. The mode shape vector is defined as a column of the mode shape matrix, shown here, where each entry corresponds to a degree of freedom of a node in the proximal frame of the component. + +Example of a normalised mode shape vector containing 2 nodes with 6 degrees of freedom each: + +$$ +\boldsymbol{\psi}_{\mathrm{prox}}=\left[\underbrace{\!\!\left[u_{x_{1}}\quad u_{y_{1}}\quad u_{z_{1}}\quad\theta_{x_{1}}\quad\theta_{y_{1}}\quad\theta_{z_{1}}\quad u_{x_{2}}\quad u_{y_{2}}\quad u_{z_{2}}\quad\theta_{x_{2}}\quad\theta_{y_{2}}\quad\theta_{z_{2}}\right]}_{\mathrm{Node}\;1}\right]^{T} +$$ + +where the first 6 entries correspond to the translational and rotational degrees of freedom for the first node, while the next 6 entries correspond to those of the second node. By examining the values in the vector, one can observe that the first entry is the largest. This would indicate a fore-aft or flapwise mode, depending on whether it pertains to a tower or a blade. However, this approach is not suitable, as the values of the translational and rotational degrees of freedom use differing units. Therefore, a more sophisticated method is necessary. + +The procedure for naming mode shapes in Bladed involves a series of checks to determine the primary characteristic of the mode shape. + +# 1. Determine if the Mode is Longitudinal $(u_{z})$ + +Evaluate the following condition: + +$$ +u_{z m a x}>K\cdot u_{x y_{m a x}} +$$ + +where + +$$ +\begin{array}{r l}&{u_{z\operatorname*{max}}=\operatorname*{max}\big(u_{z_{1}}\quad u_{z_{2}}\quad\cdot\cdot\cdot\quad u_{z_{n}}\big),}\\ &{u_{x y_{m a x}}=\operatorname*{max}\left(\sqrt{u_{x_{1}}^{2}+u_{y_{1}}^{2}}\quad\sqrt{u_{x_{2}}^{2}+u_{y_{2}}^{2}}\quad\cdot\cdot\cdot\quad\sqrt{u_{x_{n}}^{2}+u_{y_{n}}^{2}}\right),}\end{array} +$$ + +$\kappa$ is a constant calibration factor. + +The left-hand side of the condition is the maximum longitudinal displacement in the mode shape vector, and the right-hand side is the maximum displacement magnitude in the $\times$ and y directions compared across all nodes. + +# 2. Determine if the Mode is Torsional $(\theta_{z})$ + +If the mode is not axial, evaluate if it is a torsional mode: + +$$ +\theta_{z m a x}>K\cdot\theta_{x y_{m a x}} +$$ + +where + +$$ +\begin{array}{r l}&{\theta_{z_{\operatorname*{max}}}=\operatorname*{max}\big(\theta_{z_{1}}\quad\theta_{z_{2}}\quad\cdot\cdot\cdot\quad\theta_{z_{n}}\big),}\\ &{\theta_{x y_{m a x}}=\operatorname*{max}\bigg(\sqrt{\theta_{x_{1}}^{2}+\theta_{y_{1}}^{2}}\quad\sqrt{\theta_{x_{2}}^{2}+\theta_{y_{2}}^{2}}\quad\cdot\cdot\cdot\quad\sqrt{\theta_{x_{n}}^{2}+\theta_{y_{n}}^{2}}\bigg)}\end{array} +$$ + +The left-hand side of the condition is the maximum torsional displacement in the mode shape vector, and the right-hand side is the maximum rotation (bending) magnitude in the x and y directions compared across all nodes. + +Additionally, the following condition must also be true: + +$$ +\frac{1}{2}\frac{L}{G_{r a t i o}}\theta_{z m a x}>K\cdot u_{x y_{m a x}} +$$ + +The left-hand side of the condition is the approximate tangential displacement resulting from the torsional rotation. This assumes a small angle approximation and uses the constant $G_{r a t i o}=15,$ which is the typical ratio between the length and the cross-section of a wind turbine structure (both tower and blades). $L$ is a characteristic length: for whole-blade modes, it is the length of the blade; for blade part modes, it is the part length; and for towers, it is the tower height. + +# 3. Determine if the Mode is Transverse $(u_{x},u_{y})$ + +If the mode is neither axial nor torsional, it must be a mode in either the $u_{x}$ Or $u_{y}$ direction (transverse). This is determined by evaluating the largest $\times$ or y displacement at the maximum displacement magnitude, Urymax' + +# List of Mode Shape Names + +Table 1: Translation of mode types to blade or tower mode names. Note that the axes refer to the proximal frame of the component. + +Mode Type + +Blade + +Tower + +Longitudinal $\left(u_{z}\right)$ + +Vertical + +Tower + +Torsional $(\theta_{z})$ + +Torsional + +Torsional + +Transverse $(u_{x},u_{y})$ + +Flapwise, Edgewise + +Fore-aft, Side-side + +Last updated 11-12-2024 + +# Modal Analysis Output + +# Generalised mass and stiffness properties + +During modal analysis, Bladed calculates the "generalised mass matrix" and "generalised stiffness matrix" by transforming the flexible component finite element mass/stiffness matrices using the normalised mode shape matrix $\Psi$ as shown below. The purpose of this is to transform the finite element mass and stiffness matrices into modal space (often referred to as the generalised coordinates). + +$$ +\begin{array}{r}{\mathbf{M}_{g e}=\boldsymbol{\Psi}^{T}\mathbf{M}_{f e}\boldsymbol{\Psi}}\\ {\mathbf{K}_{g e}=\boldsymbol{\Psi}^{T}\mathbf{K}_{f e}\boldsymbol{\Psi}}\end{array} +$$ + +where + +· $\mathbf{M}_{f e}$ and $\mathbf{K}_{f e}$ are the finite element mass and stiffness matrices for the complete flexible component. They are described in terms of $6N_{n}$ -by- $6N_{n}$ square matrices, where $N_{n}$ is the total number of nodes in the flexible component. +· $\mathbf{M}_{g e}$ and ${\bf K}_{g e}$ are the modal mass and stiffness matrices. They are described in terms of $N_{m}$ -by- $.N_{m}$ square matrices, where $N_{m}$ is the total number of modes specified for the flexible component $\Psi$ is the mode shape matrix, which is used to transform between the finite element and modal domain. Consequently, the mode shape matrix has the dimensions below, with one mode shape defined on each column. + +The diagonal elements of this generalised mass and stiffness matrices are then reported by Bladed for each mode. + +![](images/3a667c36c91f37aaa8c16230d06a2749a9e91ee4a0b09280cbd05651745286fe.jpg) + +# Natural frequency calculation + +Using the Rayleigh principle (Clough and Penzien, 1993), the mode natural frequencies can be calculated as + +$$ +\omega_{i}^{2}=\;\frac{K_{g e,i}}{M_{g e,i}}, +$$ + +where $K_{g e,i}$ and $M_{g e,i}$ are the terms on the leading diagonal of the generalised stiffness and mass matrices. + +# Note on the "effective modal mass" + +There is also a quantity called the "effective modal mass" associated with each mode shape, which is not reported by Bladed. This quantity can represent (for example) the part of the total mass in each mode shape that responds to a specified unit displacement (for example a unit translation of the whole component in a certain direction). The "effective modal mass" for a particular mode, i, can be expressed as: + +$$ +M_{\mathrm{eff},i}=\frac{l_{i}^{2}}{M_{g e,i}}, +$$ + +where + +$$ +\mathbf{l}=\mathbf{\rho}\left[\begin{array}{c}{l_{1}}\\ {l_{2}}\\ {\cdot}\\ {\cdot}\\ {l_{\mathbf{N}_{m}}}\end{array}\right]=\,\Psi^{T}\mathbf{M}_{f e}\mathbf{r} +$$ + +and r is the displacement vector of each degree of freedom when the component is subject to a unit displacement. + +It is possible to add up the "effective modal mass" for each mode and compare that to the total component mass to evaluate the contribution of each mode to a certain displacement field. This summation of "effective modal mass" is sometimes used to help choose the number of modes required for a simulation. + +However, it is not clear in general which set of displacements should be used to calculate the "effective modal mass". For example, a unit displacement of the whole component is appropriate for (say) simulating an earthquake, but does not give information in general as to the significance of a mode when subject to arbitrary forcing and displacement. For this reason, the "effective modal mass" is not reported by Bladed. + +Last updated 11-12-2024 + +# Geometric Stiffness Models + +Geometric stiffening models account for changes in structural response due to structural deflection from the reference (not deflected) state. Bladed provides models that include contributions from element axial and shear internal forces. + +# Geometric stiffness due to element axial forces + +Traditional geometric stiffness models account for the effect of element internal axial forces on structural stiffness. This is illustrated schematically in Figure 1. Centrifugal loads in the structural elements lead to a restoring load that tends to increase the stiffness of the blade. A linear finite element model for an initially straight blade is illustrated on the left side of Figure 1. A centrifugal force applied to the blade in its deflected position does not cause a bending moment along the blade. This is normal for linear finite element (FE) models as deflections are assumed to be small. On the right side of the diagram, the effect of geometric stiffness is illustrated. As the centrifugal load is applied in the deflected blade position, a bending moment is generated in the blade. This extra bending moment can change the blade flapwise and edgewise frequencies. + +![](images/c48399e0e79b7a2b47ed45609a405baddc76883869eb435b58fbf5c77367a5bd.jpg) +Figure 1: Element axial forces causing bending moments in a blade. + +Geometric stiffness forces in the axial direction are responsible for the well-known "centrifugal stiffening" effect, where blade vibrational frequencies increase with rotor speed. + +# Geometric stiffness due to element shear forces + +There are also geometric stiffening forces associated with element internal shear forces. Figure 2 illustrates how torsion moments can be generated in the blade by application of shear forces to the blade in its displaced position. On the right side of the diagram, as the drag or lift load is applied in the deflected blade position, a torsional moment is generated in the blade. This extra torsional moment can affect the blade torsional dynamics. + +![](images/ebe52afdc3940fe0f0ddc81bcb38324d4f988c9469b26583a9d63c32b1082029.jpg) +Figure 2: Element shear forces causing torsion moments in a blade + +When evaluating the geometric stiffness effect of shear forces, it is important to account for the change in orientation of the torsion axis of the blade elements due to deflection. This is illustrated in Figure 3, where the difference in internal torsional load between the "reference" and deflected coordinate system is shown. Whether this affect is included depends on whether IgnoreAxesorientationDifferencesForshear is set true or false . For more details see the theory section about Translation and orientation offset between neutral and shear axes. + +![](images/9c4d734bb0d937679f80dd7bd1d6e2e0f39ba3f179fcb63197dc4653444ffe10.jpg) +Figure 3: Element shear forces causing torsion moments in a blade + +Last updated 13-12-2024 + +# Bend-Twist Coupling Relationships in Beam Elements + +This article describes how bend-twist coupling effects can be accounted for in Bladed beam elements. The built-in Bladed model of bend-twist coupling due to shear centre offset from the neutral axis are described. Additionally, the specification of user-defined bend-twist coupling terms is discussed. + +# Co-incident shear and neutral axes + +If the elastic centre and shear centre coincide, the constitutive relationship between strain and internal load for a beam element can be expressed as a diagonal matrix as shown below. Note that this equation is formulated in the local element coordinate system (i.e. it is rotated according to blade structural twist, prebend and sweep). + +![](images/6a3c44ac0ff44832599cf6e68b2c38da0375f6ead341f2470f6cec1bb9ba4305.jpg) + +The $6\!\times\!6$ constitutive matrix is referred to in this document as $\bar{\bar{\mathbf{C}}},$ where the double over-bar denotes the local element coordinate system. + +It is also noted that this equation may easily be transformed to the local principal coordinate system if the principal axis direction is constant for successive beam elements in the blade. In this case, the principal x-axis equals the element z-axis and the principal y-axis equals the element yaxis in the opposite direction, which implies that the principal z-axis equals the element x-axis (neutral axis). Further details of the relation between the local principal coordinate system and the element coordinate system may be found in the article about Blade Local Element Axes System. + +# Translational offset between neutral and shear axes + +It is possible to define a translational offset between the neutral axis and the shear centre within the blade section, as illustrated in Figure 1. + +![](images/2232c8006b1981075f53ab84f9ac8e8216e9b12e69cab2faf4743d7390036300.jpg) +Figure 1: Shear centre offset from neutral axis + +The translational offset between shear and neutral axes is taken into account using the following calculation, which transforms the shear properties onto the neutral axis position. + +![](images/9fc72b2fea82719d5e0d97c537f08630b11f1aa83453593ef611fb2df3a700ec.jpg) + +where + +$$ +\mathbf{Y_{S}}=\begin{array}{l l l}{\left[\begin{array}{l l l}{0}&{0}&{0}\\ {-z_{c s}}&{0}&{0}\\ {y_{c s}}&{0}&{0}\end{array}\right]}&{\qquad\quad\mathbf{I}\;=\;\left[\begin{array}{l l l}{1}&{0}&{0}\\ {0}&{1}&{0}\\ {0}&{0}&{1}\end{array}\right]}\end{array} +$$ + +Entries $y_{c s}$ and $z_{c s}$ define the position of the shear centre relative to the neutral axis. + +Expanding the above expression gives the following constitutive relationship around the neutral axis. The effect of shear centre offset is to introduce additional coupling between shear strain and torsional moment, and between bending strain and shear force. + +$$ +\left[\!\!{\begin{array}{c}{F_{x}}\\ {F_{y}}\\ {F_{z}}\\ {M_{x}}\\ {M_{y}}\\ {M_{z}}\end{array}}\right]=\left[\!\!{\begin{array}{c c c c c c c c}{\mathbf{\big[}E A}&&&&{\big|}&&&\\ {0}&{G A_{y}}&&{\big|}&&{\mathbf{sym}}&\\ {0}&{0}&{G A_{z}}&{\big|}&&&\\ {-}&{-}&{-}&{-}&{-}&{-}&{-}\\ {0}&{-z_{c s}G A_{y}}&{y_{c s}G A_{z}}&{\big|}&{G I_{x}}&&\\ {0}&{0}&{0}&{\big|}&{0}&{E I_{y}}&\\ {0}&{0}&{0}&{\big|}&{0}&{0}&{E I_{z}}\end{array}}\!\!\right]\left[\!{\begin{array}{c}{\gamma_{x}}\\ {\gamma_{y}}\\ {\gamma_{z}}\\ {\kappa_{x}}\\ {\kappa_{y}}\\ {\kappa_{z}}\end{array}}\right] +$$ + +where, $G I_{x}=G I_{x}^{\mathrm{~*~}}+\;G A_{y}z_{c s}^{2}+\;G A_{z}y_{c s}^{2}$ and $G{I_{x}}^{*}$ is the torsional stiffness defined around the shear (torsional) axis. + +# Translation and orientation offset between neutral and shear axes + +Optional by selecting "ignore blade shear centre axis orientation transformation" in Additional Items. + +In general, the elastic and shear axes are not parallel, so it can be important to take account of the orientation difference between them. The orientation difference between the shear axis and the elastic axis is illustrated by the $\theta$ terms in Figure 2. + +![](images/5dadf76cb98bcff1f0577218862c9577ce405f9aac0dc87622dcbfda2bcfded4.jpg) +Figure 2: Orientation difference between shear and elastic axes + +The combined translational and orientation offset between shear and neutral axes is taken into account using the following calculation, which transforms the shear properties onto the neutral axis position. + +![](images/680f8193c8469e09860cf395332fffdb1655a1d252d0b84377df481c7a5d1a09.jpg) + +where + +$$ +\mathbf{Y_{S}}=\begin{array}{c c c}{\left[\begin{array}{c c c}{0}&{0}&{0}\\ {-z_{c s}}&{0}&{0}\\ {y_{c s}}&{0}&{0}\end{array}\right]}&{\qquad\quad\mathbf{R_{s}}=\frac{1}{L^{e}}\begin{array}{c c c}{\left[\begin{array}{c c c}{L_{s}^{e}}&{0}&{0}\\ {-\Delta y_{c s}}&{L^{e}}&{0}\\ {-\Delta z_{c s}}&{0}&{L^{e}}\end{array}\right]}\end{array} +$$ + +Entries $\Delta y_{c s}$ and $\Delta z_{c s}$ describe the change in position of the shear centre within the beam element, in order to describe the shear axis orientation. + +The effect of shear centre translation and orientation offset is to introduce additional coupling between bending and torsional moments, resulting in the following constitutive relationship around the neutral axis. + +$$ +\left[\begin{array}{c}{F_{x}}\\ {F_{y}}\\ {F_{z}}\\ {M_{x}}\\ {M_{y}}\\ {M_{z}}\end{array}\right]=\begin{array}{c c c c c c c}{\displaystyle\left[\begin{array}{c c c c c c c}{\scriptstyle E A}&&&&&{\mid}&&&\\ {0}&{\scriptstyle G A_{y}}&&&{\mid}&&&{\mathbf{sym}}&\\ {0}&{0}&{\scriptstyle G A_{z}}&&{\mid}&&&\\ {-}&{-}&{-}&{-}&{-}&{-}&{-}&{-}\\ {0}&{-z_{c s}G A_{y}}&{y_{c s}G A_{z}}&{\mid}&{\scriptstyle G I_{x}}&&\\ {0}&{\scriptstyle0}&&{0}&{\mid}&{-\frac{\Delta y_{c s}E I_{y}}{L e}}&{\scriptstyle E I_{y}}&\\ {0}&{0}&{0}&{\mid}&{-\frac{\Delta z_{c s}E I_{z}}{L e}}&{0}&{\scriptstyle E I_{z}}\end{array}\right]\left[\begin{array}{c}{\gamma_{x}}\\ {\gamma_{y}}\\ {\gamma_{z}}\\ {\kappa_{x}}\\ {\kappa_{y}}\\ {\kappa_{z}}\end{array}\right], +$$ + +where + +$$ +G I_{x}=\bigg(\frac{L^{e}}{L_{s}^{e}}\bigg)^{2}G I_{x}^{*}+G A_{y}z_{c s}^{2}+G A_{z}y_{c s}^{2}+\frac{\Delta y_{c s}^{2}E I_{y}+\Delta z_{c s}^{2}E I_{z}}{L^{e^{2}}} +$$ + +and $G{I_{x}}^{*}$ is the torsional stiffness defined around the shear (torsional) axis. This transformation results in extra bend-twist off-diagonal coupling terms, as well as a change to the torsional stiffness around the neutral axis. + +# User-defined bend-twist coupling + +The user can directly add extra off-diagonal terms to the constitutive matrix as shown + +$$ +\begin{array}{r}{\bar{\bar{\mathbf{C}}}=\left[\begin{array}{c c c c c c c}{E A}&&&{|}&&&\\ {0}&{G A_{y}}&&{|}&&{\mathrm{sym}}&\\ {0}&{0}&{G A_{z}}&{|}&&&\\ {-}&{-}&{-}&{-}&{-}&{-}&{-}\\ {0}&{0}&{0}&{|}&{G I_{x}^{*}}&&\\ {0}&{0}&{0}&{|}&{C_{x y}}&{E I_{y}}&\\ {0}&{0}&{0}&{|}&{C_{x z}}&{C_{y z}}&{E I_{z}}\end{array}\right]}\end{array} +$$ + +The transformations described in previous sections based on shear axis position relative to neutral axis are also applied, resulting in the following relationship by selecting the option "ignore blade shear centre axis orientation transformation" in Additional Items. + +$$ +{\left[\begin{array}{l}{F_{x}}\\ {F_{y}}\\ {F_{z}}\\ {M_{x}}\\ {M_{y}}\\ {M_{z}}\end{array}\right]}\,=\,{\left[\begin{array}{l l l l l l l l}{\mathbf{I}}&{\mathbf{0}}&{}&{}&{}&{}&{}&{}&{}\\ {\mathbf{I}}&{\mathbf{0}}&{}&{}&{}&{}&{}&{}\\ {\mathbf{I}}&{\mathbf{0}}&{}&{}&{}&{}&{}&{}\\ {\mathbf{T}}&{-}&{-}&{-}&{-}&{-}&{-}&{}&{-}\\ {\mathbf{0}}&{}&{\mathbf{0}}&{\mathbf{0}}&{}&{|}&{G\mathbf{I}_{x}^{*}}&{}&{}\\ {\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{|}&{C_{x y}}&{E I_{y}}&{}&{}\\ {\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}&{|}&{C_{x z}}&{C_{y z}}&{E I_{z}}\end{array}\right]}{\left[\begin{array}{l l}{\mathbf{I}}&{\mathbf{Y}}\\ {\mathbf{Y}}\\ {\mathbf{0}}&{\mathbf{I}}\end{array}\right]}\,\Biggr[{\boldsymbol{Y}}\cdot\mathbf{Y}\left[{\begin{array}{l}{\mathbf{X}}\\ {\mathbf{Y}}\\ {\mathbf{X}}\\ {\mathbf{H}}\\ {\mathbf{X}}\end{array}}\right]}\end{array}\right]} +$$ + +The following transformation is used as default without enabling the option. + +
Fc Fy Fz I Mx [Ys T My MzEA 0 0Yy
GAy 0GAsym[I
0 一 RsT 0一一一 一Ys 0 RsY2(9)
00GI* cKc
000CryEIyKy
000CxzEIz C yzKz.
+ +It is noted that the transformation of these equations to the local principal coordinate system as described in Co-incident shear and neutral axes will introduce a change of the sign of the offdiagonal terms that relate to bending about the element y-axis. + +Last updated 13-12-2024 + +# Support Structure Superelement + +For offshore turbines, the jacket support structure is sometimes modelled as a superelement. The superelement can be included as a component in the multibody framework in a similar way to the other flexible components. + +# Model Creation + +The finite element model of the jacket must be converted into reduced form and exported from a offshore modelling code for example Sesam, SACS or ROSA. The Bladed superelement feature has been designed with (Craig-Bampton, 2000), reduction for the superelement in mind. CraigBampton is the preferred method as an accurate dynamic response of the jacket can be retained in the superelement. Other reduction methods could in principle be used but are not considered in this document. + +Craig-Bampton modes for the superelement consist of constraint and norma/ mode shapes. An important concept is division of the jacket nodes into the boundary node and interior nodes. The boundary node is the node where the superelement connects to the tower base. The interior nodes are all of the other nodes in the jacket. + +Constraint modes describe the displacement of the interior jacket nodes when six unit displacements (3 translational and 3 rotational) are applied at the superelement interface node (the boundary node). These mode shapes provide the static response of the superelement and allow the superelement interface node to move. + +For the normal modes, the jacket structure is constrained at the base and the interface node and the eigenmodes are calculated. As the jacket is constrained at both ends, these are also referred to as interior modes (they include motion of the interior nodes only). The normal modes enhance the dynamic response of the superelement. The union of these two sets of modes can provide an accurate dynamic model of the jacket and motion of the interface node. + +The outputs of the superelement creation process are a mass matrix and stiffness matrix for the reduced model. A wave load time history for each mode is also output. A reduction basis $[R]$ based on the constraint and normal mode shapes, is used to reduce the jacket finite element matricesasfollows + +$$ +\begin{array}{r l}&{\overline{{\mathbf{M}}}=\mathbf{R}^{T}\mathbf{M}_{F E}\mathbf{R}}\\ &{\overline{{\mathbf{K}}}=\mathbf{R}^{T}\mathbf{K}_{F E}\mathbf{R}}\\ &{\overline{{\mathbf{f}}}=\mathbf{R}^{T}\left[\mathbf{f}_{i}\right]}\end{array} +$$ + +where: + +$\mathbf{R}$ is the reduction basis that transforms the jacket FE model into a superelement model +$\mathbf{M}_{F E}$ is the finite element mass matrix +$\bf{K}_{F E}$ is the finite element stiffness matrix + +$\overline{{\bf{M}}}$ is the superelement mass matrix $\overline{{\mathbf{K}}}$ is the superelement stiffness matrix $\mathbf{f}_{b}$ is the external applied wave load at the boundary node $\mathbf{f}_{i}$ is the external applied wave load at the interior nodes · f is the superelement external applied wave load + +It is important to note that the first six degrees of freedom in the superelement model correspond to displacement of the interface node (boundary node). The remaining degrees of freedom correspond to the interior normal modes. Sufficient normal modes should be included to obtain an accurate dynamic response. The reduced equations of motion for the superelement are expressed asfollows + +$$ +\overline{{\mathbf{M}}}\left[\stackrel{\cdot\cdot}{\mathbf{\ddot{x}}_{b}}\right]+\overline{{\mathbf{K}}}\left[\stackrel{\mathbf{x}_{b}}{\mathbf{\Delta}}\right]=\overline{{\mathbf{f}}} +$$ + +where + +$\mathbf{x}_{b}$ are the six boundary DoF displacements (equivalent to $>$ constraint mode amplitudes). + +$\pmb{\eta}_{i}$ is the amplitude of each normal mode (istands for interior) + +The method applied to complete modal reduction for the superelement method in Bladed is referred to as the Craig-Bampton method. + +# Superelement Mode Shape Matrix + +The reduced matrices $\overline{{\bf M}}$ and $\overline{{\mathbf{K}}}$ include allthe necessary properties to describe the dynamics of the superelement in Bladed. However, it is also necessary for Bladed to know how the interface node in the Bladed structural definition couples to the superelement modal displacements. In other words, how this interface node moves when each superelement degree of freedom is activated. + +By default, a mode shape matrix for the superelement is assumed by Bladed as shown below. The 6 boundary degrees of freedom are assumed to correspond to translational and rotational motion in the Bladed global coordinate system. The normal mode amplitudes do not cause motion of the interface node. However, note that the normal modes do contribute to the dynamic response of the superelement due to off diagonal terms in the superelement mass and stiffness matrices. + +
262 Cb3 Minterface x disp 1 0 0 0 0 0 0 0 0 0 Cb4 Cinterface y disp 0 1 0 0 0 0 0 0 0 0 b5 Cinterface z disp 0 0 1 0 0 0 0 0 0 0 66 0 0 0 1 0 0 0 0 0 Cinterface x rot 0 Mi 0 0 0 0 1 0 0 0 0 0 Minterface x rot Mi2 10 interface x rot 0 0 0 0 1 0 0 0 0 Mi3 Mi4
+ +In the case that the superelement has been generated in a coordinate system different from that used in Bladed, an additional transformation is required. + +# Superelement Damping Definition + +The integrated and superelement approaches use a different structural mode basis, as illustrated in Figure 1. In the integrated approach, modes cover the entire support structure, whereas in the superelement approach, separate mode shapes are defined for the superelement and tower modes. + +![](images/a15e7271bf976fe310fb2fb588a0f4d9199bb59242f6033df204d5ac7087422c.jpg) +Figure 1: Modal basis in integrated and superelement approaches + +In Bladed, damping is defined on the modal degrees of freedom. This leads to the question how can equivalent damping be defined in the integrated and superelement approaches? + +In this section, the methods and challenges for damping definition are discussed for both integrated and superelement approaches. A method is then described that overcomes these dificulties and allows equivalent damping to be defined on integrated and superelement approaches. + +# Structural Damping for Integrated Models + +For an integrated model, typically modal damping ratios are defined on the uncoupled Craig Bampton vibration support structure modes. + +The modes are used to calculate modal stiffness and mass matrices for the support structure. The structure of the modal mass and stiffness matrices are shown in Equations (6) and (7). The stiffness matrix is diagonal but the mass matrix is not. + +$$ +\mathbf{M}_{\mathrm{SS}}=\left[\begin{array}{l l l l}{\mathbf{M}_{\mathrm{SS11}}}&{\mathbf{M}_{\mathrm{SS12}}}&{\mathbf{M}_{\mathrm{SS13}}}&{\cdot\cdot\cdot}\\ {\mathbf{M}_{\mathrm{SS21}}}&{\mathbf{M}_{\mathrm{SS22}}}&{\mathbf{M}_{\mathrm{SS23}}}&{}\\ {\mathbf{M}_{\mathrm{SS31}}}&{\mathbf{M}_{\mathrm{SS32}}}&{\mathbf{M}_{\mathrm{SS33}}}&{}\\ {\vdots}&{}&{}&{\cdot\cdot}\end{array}\right] +$$ + +$$ +\mathbf{K}_{\mathrm{SS}}=\left[\begin{array}{c c c c}{\mathbf{K}_{\mathrm{SS11}}}&{0}&{0}&{\cdots}\\ {0}&{\mathbf{K}_{\mathrm{SS22}}}&{0}&{}\\ {0}&{0}&{\mathbf{K}_{\mathrm{SS33}}}&{}\\ {\vdots}&{}&{}&{\ddots}\end{array}\right] +$$ + +where subscript Ss is short for support structure. + +To calculate the frequency of each mode, the RNA (rotor inertia assembly) inertia associated with each attachment mode must also be taken into account. For each attachment mode, a quantity $\mathbf{M}_{\mathrm{i,rotor}}$ is defined as follows + +$$ +\mathbf{M}_{\mathrm{i,rotor}}=\Psi^{T}\mathbf{M}_{\mathrm{RNA}}\boldsymbol{\Psi} +$$ + +where ${\bf M}_{\mathrm{RNA}}$ is the $6\!\times\!6$ RNA inertia matrix and $\Psi$ is the mode shape matrix for the tower top node Only. + +The modal angular frequency $\omega_{i}$ is then calculated as + +$$ +\omega_{i}=\sqrt{\frac{K_{\mathrm{ii}}}{M_{\mathrm{ii}}+M_{\mathrm{i,rotor}}}} +$$ + +where ${M}_{i,r o t o r}$ is non-zero for each attachment mode and zero for the normal modes. + +The support structure modal damping matrix is the calculated as shown below + +$$ +\mathbf{C}_{\mathrm{SS}}=2\left[\begin{array}{c c c}{\frac{\zeta_{1}\mathbf{K}_{\mathrm{SS11}}}{\omega_{1}}}&{0}&{\cdots}\\ {0}&{\frac{\zeta_{2}\mathbf{K}_{\mathrm{SS22}}}{\omega_{2}}}&{}\\ {\vdots}&{}&{\ddots}\end{array}\right] +$$ + +where + +$\zeta_{i}$ are the modal damping ratios, +$\omega_{i}$ are modal angular frequencies (rad/s), +$\mathbf{K}_{\mathrm{SSii}}$ are the diagonal terms of the modal stiffness matrix. + +# Structural Damping for Superelement Models + +Bladed allows damping to be defined separately on the tower and superelement, which each have their own mode shapes and damping definitions. + +For the superelement, typically Rayleigh damping parameters are available based on the superelement mass and stiffness matrices. + +$$ +\mathbf{C}_{\mathrm{SE}}=a_{0}\mathbf{M}_{\mathrm{SE}}+a_{1}\mathbf{K}_{\mathrm{SE}} +$$ + +The tower modal damping matrix will take a similar form to that for the integrated approach. + +$$ +\mathbf{C}_{\mathrm{T}}=2\left[\begin{array}{c c c}{\frac{\zeta_{1}\mathbf{K}_{\mathrm{T11}}}{\omega_{1}}}&{0}&{\cdots}\\ {0}&{\frac{\zeta_{2}\mathbf{K}_{\mathrm{T22}}}{\omega_{2}}}&{}\\ {\vdots}&{}&{\ddots}\\ {\vdots}&{}&{\ddots}\end{array}\right] +$$ + +These matrices are simply combined to give a support structure damping matrix for the following format. + +$$ +\mathbf{C}_{\mathrm{SS}}=2\left[\begin{array}{l l}{\mathbf{C}_{\mathrm{SE}}}&{\mathbf{O}}\\ {\mathbf{O}}&{\mathbf{C}_{\mathrm{T}}}\end{array}\right] +$$ + +where subscript SE is short for superelement and the subscript $\mathbf{T}$ is short for tower. The matrix O contains elements of zeroes. + +There are two potential problems with this approach. + +Firstly, the "cross terms" between the tower and superelement components are zero. This means that it is not possible with this approach to specify damping in a Bladed superelement model that is exactly equivalent to damping values that are specified on whole support structure modes. + +Secondly, the supplied superelement damping matrix does not typically account for the effect of the inertia of the tower and RNA. + +# Unified Damping for Superelement and Integrated Methods + +It has been explained that integrated model damping and superelement model damping are not equivalent, for the following reasons: + +1. Damping is specified on a different set of modes for the superelement and integrated approaches. +2. The superelement damping method does not include cross terms between the tower and superelement damping. + +3. The superelement damping approach does not take into account the inertia of the tower and RNA when calculating the superelement damping. + +To unify the damping for the superelement and integrated approaches, it is proposed to define the damping on a set of modes that is common to both approaches such as the coupled vibrational modes for the support structure. If the structural properties of the superelement are defined in a valid way, then the support structure coupled modes will be very similar to the integrated case. + +# Theory Basis + +The aim is to specify damping on the support structure coupled modes, and then transform this damping onto the actual degrees of freedom for the tower and superelement which are used in the simulation. This method is based on theory presented in (Clough, 1993). + +Consider the partitions of the system mass and stiffness matrices relating to the support structure degrees of freedom. For the superelement approach, the uncoupled support structure mass and stiffness matrices have the form shown in Equations (14) and (15). + +$$ +\begin{array}{r}{\mathbf{M}_{\mathrm{SS,uncoupled}}=\left[\begin{array}{l l}{\mathbf{M}_{\mathrm{SE}}}&{\mathbf{O}}\\ {\mathbf{O}}&{\mathbf{M}_{\mathrm{T}}}\end{array}\right]}\\ {\mathbf{K}_{\mathrm{SS,uncoupled}}=\left[\mathbf{K}_{\mathrm{SE}}\quad\mathbf{O}\right]}\end{array} +$$ + +For the integrated approach, the same matrices have a more simple form based on the form shown in Equation (6) and (7) + +$$ +\begin{array}{r l}&{\mathbf{M}_{\mathrm{SS,uncoupled}}=\mathbf{M}_{\mathrm{SS}},}\\ &{\mathbf{K}_{\mathrm{SS,uncoupled}}=\mathbf{K}_{\mathrm{SS}}.}\end{array} +$$ + +Note that $\mathbf{M}_{\mathrm{SS}}$ and ${{\bf{M}}_{T}}$ include the influence of the rigid RNA inertia on each tower attachment mode. For the purpose of the damping calculation, $\mathbf{M}_{\mathrm{SE}}$ is edited to include the influence of the tower and the rigid RNA on each constraint mode, using a method equivalent to the formulation in Equation (8). + +The coupled mode shapes for the support structure are found by solving the structural eigen problem using the uncoupled mode shape matrices. The number of coupled mode shapes for the support structure is equal to the sum of the number of tower and superelement modes. + +$$ +\mathbf{K}_{\mathrm{SS,\,uncoupled}}\psi_{i}=\omega_{i}^{2}\mathbf{M}_{\mathrm{SS,\,uncoupled}}\psi_{i} +$$ + +where: + +$\psi_{i}$ are the mode shape vectors, $\omega_{i}$ are modal angular frequencies rad/s. + +The assembled square mode shape matrix $\Psi$ , where each column holds an individual mode shape $\psi_{i},$ describes coupled mode shapes in terms of the contributions from the uncoupled mode + +shapes. The mode shape matrices are used to transform uncoupled properties to coupled properties + +$$ +\begin{array}{r l}&{{\bf K}_{\mathrm{coupled}}=\Psi^{T}{\bf K}_{\mathrm{uncoupled}}\Psi,}\\ &{{\bf M}_{\mathrm{coupled}}=\Psi^{T}{\bf M}_{\mathrm{uncoupled}}\Psi.}\end{array} +$$ + +Damping is specified on the coupled modes using either proportional + +$$ +\mathbf{C}_{\mathrm{coupled}}=a_{0}\mathbf{M}_{\mathrm{coupled}}+a_{1}\mathbf{K}_{\mathrm{coupled}} +$$ + +or modal damping + +$$ +\mathbf{C}_{\mathrm{coupled}}=2\left[\begin{array}{c c c c}{\frac{\zeta_{1}\mathbf{K}_{\mathrm{coupled11}}}{\omega_{1}}}&{0}&{\cdots}\\ {0}&{\frac{\zeta_{2}\mathbf{K}_{\mathrm{coupled22}}}{\omega_{2}}}&{}\\ {\vdots}&{}&{\ddots}\\ {\vdots}&{}&{\ddots}\end{array}\right] +$$ + +where $\scriptstyle a_{0}$ and $\boldsymbol{a}_{1}$ are proportionality constants, $\zeta_{i}$ are modal damping ratios, and $\omega_{i}$ are modal angular frequencies rad/s. + +This damping on the coupled modes is then transformed back onto the uncoupled modes for use in the simulation + +$$ +\mathbf{C}_{\mathrm{SS}}=\big[\boldsymbol{\Psi}^{T}\big]^{-1}\mathbf{C}_{\mathrm{coupled}}\boldsymbol{\Psi}^{-1} +$$ + +$C_{\mathrm{SS}}$ is a full matrix that has coupling terms between the tower and superelement components (for a superelement model), or coupling between the support structure modes (for an integrated model). The matrix $C_{S S}$ also includes the influence of the RNA inertia. + +# Results Comparison + +The effect of specifying damping on the support structure coupled modes is demonstrated in Figure 2. + +The coupled mode damping ratios for an integrated and superelement model are compared, using a jacket support structure from the study (Bladed-Sesam Verification, 2019) with the RNA included. The support structure coupled mode frequencies are derived by running a Campbell diagram calculation with rigid RNA. The target damping ratio is $0.5\%$ on the first two modes, and $1.0\%$ on the second two modes. + +The orange bars show the damping ratios when damping is defined separately on the tower and superelement modes. The grey and blue bars show the damping ratios for integrated and superelement approaches with damping defined on the support structure coupled modes. It is seen that defining the damping on support structure coupled modes results in equivalent damping for the two approaches up to 5Hz. + +Definition of damping on the uncoupled superelement and tower modes results in incorrect damping on the first two modes. The uncoupled mode damping can be tuned to give the desired + +damping ratio for the first few coupled modes. + +![](images/57e60dea54fc6349af0b858ecb15f203cfaa6bc28ce5f28bfbb49be913e4307f.jpg) + +Figure 2: Support structure coupled mode damping ratios for integrated and superelement models, including effect of specifying damping on the support structure coupled modes + +# Specifying Superelement Damping + +In Bladed, it is recommended to specify damping on the support structure coupled modes using a unified damping.approach. However, this functionality is not available in Bladed 4.8, so for this version a different method must be used to achieve the target damping for the support structure modes when using a superelement method. + +The uncoupled tower modal damping and superelement damping specified by the user are used directly in the calculation, in the form of Equation (13). This simple approach means that it is not straightforward to achieve equivalent damping for an integrated structure and superelement structure demonstrated by a comparison of damping between the different modelling approaches. + +The target damping ratios for the first few support structure coupled modes can be achieved by tuning the damping on the superelement and/or tower. A possible strategy for such a tuning exercise is: + +· Define target damping values for the desired support structure coupled modes · For each support structure coupled mode where the damping needs to be adjusted, examine the contributions of the uncoupled modes using a Campbell diagram calculation for a parked turbine and with a rigid rotor + +· Raise or lower the damping on the uncoupled modes that are important in the coupled mode in order to adjust towards the target damping ratio + +This method typically requires some iteration to achieve the desired damping on the first few support structure coupled modes. It can be necessary to adjust both the superelement and tower damping values. + +Example results of this tuning exercise are illustrated in Figure 3, using the same model for the comparison of damping study. The blue and grey bars across all coupled mode frequencies are compared. It is observed that, while it was possible to achieve the target damping on the first two support structure coupled modes, higher modes, such as at 3.51Hz do not achieve the target damping ratio. + +![](images/6a5dd588e420c40358f0c7050dc80f863482889b95a2d70bb9ad46e8a2cfc989.jpg) +Figure 3: Support structure coupled mode damping ratios for integrated and superelement models, including effect of tuning the damping for the superelement model. + +# Craig-Bampton Reduction Basis + +The aim of the reduction is to reduce the number of degrees of freedom in the jacket structure and export the resulting matrices in a compact form ready for import to Bladed for dynamic analysis. + +To perform the Craig-Bampton reduction, the finite element model equations are expressed with the boundary and interior nodes in matrix partitions + +$$ +\begin{array}{r l}{\left[\mathbf{M}_{b b}\right.}&{\mathbf{M}_{i b}\right]\left[\ddot{\mathbf{x}}_{b}\right]+\left[\mathbf{K}_{b b}\right.}&{\mathbf{K}_{i b}\right]\left[\mathbf{x}_{b}\right]=\left[\mathbf{f}_{b}\right]}\\ {\left.\mathbf{M}_{b i}\right.}&{\mathbf{M}_{i i}\right]\left[\ddot{\mathbf{x}}_{i}\right]+\left[\mathbf{K}_{b i}\right.}&{\mathbf{K}_{i i}\right]\left[\mathbf{x}_{i}\right]=\left[\mathbf{f}_{i}\right]}\end{array} +$$ + +wheresubscript $j$ refers to interior nodes, and $^{b}$ for boundary nodes. All nodes are interior nodes except for the superelement interface node, which is at the boundary of the superelement. Note that damping is excluded at this stage and can be added directly to the reduced model. + +The motion of the boundary nodes will be described by constraint modes, which correspond to unit displacements and rotations of the boundary node. The boundary node is retained in the reduction method. + +To perform the reduction, it is desirable to express the motion of the interior nodes in terms of the static and dynamic response. + +$$ +\mathbf{x}_{i}=\mathbf{x}_{i,\mathrm{stat}}+\mathbf{x}_{i,\mathrm{dyn}} +$$ + +The static response of the interior nodes is described by constraint modes. Constraint modes describe the displacement of the interior nodes based on the displacement of the boundary nodes. + +$$ +\begin{array}{r l}&{\mathbf{x}_{i,\mathrm{stat}}=-\mathbf{K}_{i i}^{-1}\mathbf{K}_{i b}\mathbf{x}_{b}}\\ &{\mathbf{\mu}=\boldsymbol{\Psi}_{i b}\mathbf{x}_{b}}\end{array} +$$ + +$\Psi_{i b}$ is the matrix of constraint modes. + +The dynamic response of the interior nodes is captured by normal modes calculated by solving the eigen problem for the interior nodes + +$$ +\left(\mathbf{K}_{i i}-\omega_{j}^{2}\mathbf{M}_{i i}\right)\phi_{i j}=0 +$$ + +A chosen number of normal modes m are retained, and collected in a matrix of normal modes + +$$ +\Phi_{i}=[\phi_{i1},\phi_{i2},\ldots,\phi_{i m}] +$$ + +The dynamic response can then be expressed by the normal mode shapes and modal amplitudes, $\pmb{\eta}_{i}$ + +$$ +\mathbf{x}_{i,\mathrm{dyn}}=\Phi_{i}\pmb{\eta}_{i} +$$ + +The deflection of interior nodes can therefore be expressed as + +$$ +\mathbf{x}_{i}=\Psi_{i b}\mathbf{x}_{b}+\Phi_{i}\pmb{\eta}_{i} +$$ + +Expanding into matrix form, to include boundary and interior nodes + +$$ +\begin{array}{r}{\left[\mathbf{x}_{b}\right]=\left[\phantom{-}\mathbf{x}_{b}\right.}\\ {\left.\mathbf{x}_{i}\right]=\left[\phantom{-}\mathbf{\Psi}_{i b}\mathbf{x}_{b}+\Phi_{i}\eta_{i}\right]}\\ {=\left[\begin{array}{l l}{\mathbf{I}}&{\mathbf{O}}\\ {\mathbf{\Psi}_{i b}}&{\Phi_{i}\displaystyle\right]\left[\phantom{-}\mathbf{\Psi}_{m_{i}}^{\mathbf{x}_{b}}\right]}\\ {\phantom{-}=\mathbf{R}\left[\phantom{-}\mathbf{\Psi}_{\eta_{i}}^{\mathbf{x}_{b}}\right]}\end{array} +$$ + +The matrix R is called the reduction basis and reduces the interior degrees of freedom from 6 x number of interior nodes to the number of normal modes $m$ + +This response of the original finite element system is expressed as + +$$ +\mathbf{M}\left[{\ddot{\mathbf{x}}}_{b}\right]+\mathbf{K}\left[{\mathbf{x}}_{b}\right]=\left[{\mathbf{f}}_{b}\right] +$$ + +The superelement system is defined by substituting in the reduction matrix from (34) + +$$ +\mathbf{M}\mathbf{R}\left[\ddot{\mathbf{x}}_{b}\right]+\mathbf{K}\mathbf{R}\left[\mathbf{x}_{b}\right]=\left[\mathbf{f}_{i}\right] +$$ + +Pre-multiply by $\mathbf{R}^{T}$ leads to the reduced equations of motion expressed as + +$$ +\overline{{\mathbf{M}}}\left[\stackrel{\cdot\cdot}{\mathbf{\ddot{x}}_{b}}\right]+\overline{{\mathbf{K}}}\left[\stackrel{\mathbf{x}_{b}}{\mathbf{\Delta}}\right]=\overline{{\mathbf{f}}} +$$ + +where $\overline{{\mathbf{M}}},\overline{{\mathbf{K}}}$ and f are the reduced mass matrix, stiffness matrix and force vector respectively and are defined as follows + +$$ +\begin{array}{r}{\boldsymbol{\overline{{\mathbf{M}}}}=\boldsymbol{\mathbf{R}}^{T}\boldsymbol{\mathbf{M}}\boldsymbol{\mathbf{R}}}\\ {\boldsymbol{\overline{{\mathbf{K}}}}=\boldsymbol{\mathbf{R}}^{T}\boldsymbol{\mathbf{K}}\boldsymbol{\mathbf{R}}}\\ {\boldsymbol{\overline{{\mathbf{f}}}}=\boldsymbol{\mathbf{R}}^{T}\left[\begin{array}{l}{\mathbf{f}_{b}}\\ {\mathbf{f}_{i}}\end{array}\right]}\end{array} +$$ + +Last updated 11-10-2024 + +# Modelling Foundation Supports + +This section describes how linear and non-linear foundation supports are taken into account in the model. A dynamic model for non-linear foundation loading is described in detail. + +# Linear and non-linear foundations + +Linear foundations can be defined at structural nodes and have constant stiffness. The stiffness and mass is included directly in the support structure matrices in order to be included in the dynamic solution. + +Non-linear foundations can be defined as a lookup table between the foundation node displacement and the reaction load from the foundation. These lookup tables are often referred to as "p-y curves" which refers to the lateral stiffness definition of a foundation, although lookup tables in all 6 nodal degrees of freedom can be defined. Non-linear curves result in a variable foundation stiffness (the p-y curve gradient), as illustrated in Figure 1. + +Only a constant stiffness value can be included in the support structure stiffness matrix. Therefore, the initial slope of the curve (labelled $K_{01}$ in the figure) is included in the support structure stiffness matrix, and the non-linear part is included as an applied load. + +![](images/0fe3cc9ef4f678fc3e6e81d7a7c24f45f33b741fdb41c1851702e584d6b36263.jpg) +Figure 1: Example of a non-linear p-y curve + +# Dynamic model for non-linear foundations + +For non-linear foundation p-y curves, support structure modal deflections are often not accurate enough to be used for looking up the foundation load in the p-y curve. This is because the nonlinear nature of the p-y curve means that small differences in displacement can result in large + +differences in foundation stiffness. Consequently, the stiffness of the foundation and the reaction force may not be accurately predicted if modal displacements are used for the foundation load lookup. It is therefore desirable to use the finite element (FE) model displacements directly which more accurately determine the foundation deflections for lookup in the p-y curve. + +The FE model displacements are found by solving: $\mathbf{f}=\mathbf{K}\mathbf{x},$ to find x for the support structure, where + +f is the vector of applied loads $=$ (externalloads $^+$ inertial loads) on each node of the support structure, +K is the FE stiffness matrix for the support structure. This is constant. +x is the vector of support structure nodal displacements + +The vector of FE displacements, x, can then be used as lookup in the non-linear p-y curve. + +Unfortunately, f is not known at the start of each time step, as it includes the effect of inertial forces, which depend on the system accelerations. Therefore, x isn't known at the start of each time step either, so Bladed must use x from the previous time step when calculating the foundation applied loads. + +An algorithm schematic is presented in Figure 2 which shows the calculation carried out in Bladed when using the dynamic non-linear foundation load feature. The number of each step maps directly to the numbers in the flow diagrams depicted in Figure 2. Note that steps 1-4 are described in calculation procedure. + +5. The modal accelerations are used to calculate the acceleration of each node, and therefore the inertial loading at each node can be calculated. + +6. The equation $\mathbf{f}=\mathbf{K}\mathbf{x}$ is solved. The applied loads include contributions from inertia and external loading. $\mathbf{\deltaK}$ is the support structure stiffness matrix. The FE deflections x are found. + +7. The FE deflections x are used to lookup the foundation forces in the p-y curves. + +8. On the following time step, the foundation applied loads are used when evaluating the structural dynamics in step 3. + +![](images/e86c80eca7a683f34cee3f6e156e90f1e6df57716c088fe7c679401aea824625.jpg) + +Figure 2: Schematic of calculation procedure of structural dynamics in time domain simulations, including dynamic non-linear p-y curve forces (shown with blue arrows) + +Last updated 30-08-2024 + +# Modelling Moorings for Floating Turbines Catenary mooring model: Auto-population + +This section describes the theory behind the auto-population functionality for the "Manually defined" sub-type of catenary moorings. + +Catenary mooring lines in Bladed are represented by non-linear applied loads calculated from a combined stiffness, damping and inertia matrix. For a single section mooring line, the stiffness matrix can be auto-populated from a set of line properties as described in this section. + +The classical catenary equations required to describe the statics of a single mooring line are given in many references, as for example in (Faltinsen, 1990). In this section, only a brief summary is mentioned. + +Consider a cable which is partially suspended and partially lying on the seabed as sketched in Figure 1. The cable might be slack to reduce anchor loads and an initial pretension might be induced through use of winches, to pull on the cables to give a desired cable configuration. In such arrangements, the principal loads in the line are the self weight $(w)$ , and the analysis can be based on the equations for elastic catenary lines. + +Referring to Figure 1, the base catenary equations for a mooring line can be defined for an element in the mooring line $(d s)$ + +$$ +\begin{array}{l}{\displaystyle{d T-\rho g A d z=\left[w\sin\varphi-F_{h_{t}}\left(1+\frac{T}{A E}\right)\right]d s}}\\ {\displaystyle{\qquad}}\\ {\displaystyle{T d\varphi-\rho g A z d\varphi=\left[w\cos\varphi+F_{h_{n}}\left(1+\frac{T}{A E}\right)\right]d s}}\end{array} +$$ + +where we assume + +· a horizontal seabed and a simplified geometrical description with the mooring line all contained in the same plane $(x_{1},x_{3})$ +· a negligible he bending stifness, which can be a good approximation for chains and lines with a large curvature radius. +We denote the mean hydrodynamic forces per unit length in the normal and tangential direction +acting in the line element by $F_{h_{n}}$ and $F_{h_{t}}.$ +as well as the weight per unit length of the line in the water by $_w$ + +We further use + +$A$ as the line sectional area, $\boldsymbol{E}$ as the elastic stiffness (Young modulus) $T$ as the tension in the line element + +Explicit solutions of the above equations are only found for simple cases, and some simplified solutions are given in the literature. An example of such solutions is obtained for an inelastic cable, assumed to be homogeneous with constant weight $(w)$ (per unit length), and with large associated stiffness $\boldsymbol{E}$ (so the elasticity in the line is neglected). The hydrodynamic mean loads are also assumed to be zero ( $\'{}F_{h_{t}}=F_{h_{n}}=0\'{}$ ). The derivation of such equations can be found in [Faltinsen, 1990]. + +![](images/b87594e5187abf411991fc4114dd56e1a01ec809462818858dc30401f818aa34.jpg) +Figure 1: Schematic drawing of a common arrangement of a mooring line that is lying on the seabed (Faltinsen, 1990). + +For the special case of an inelastic and homogeneous mooring line, composed with only one segment, and for which the mean hydrodynamic loads are zero, the classic inelastic catenary equations are given by: + +$$ +\begin{array}{l}{\displaystyle s=\frac{T_{x}}{w}\sinh\left(\frac{w}{T_{x}}x\right)}\\ {\displaystyle\,z=\frac{T_{x}}{w}\left[\cosh\left(\frac{w}{T_{x}}x\right)-1\right]}\\ {\displaystyle T_{z}=w s}\\ {\displaystyle T=\sqrt{T_{x}^{2}+T_{z}^{2}}}\end{array} +$$ + +where + +$_x$ and $_z$ are the coordinates of the mooring line, +$\pmb{s}$ is the length of the mooring line from the origin to a certain point $\left({{\bf{x}},}{\bf{Z}}\right)$ in the line, $\boldsymbol{w}$ is the weight (force) per unit length of the mooring line, +$T_{x}$ is the horizontal tension in the mooring line (constant), +$T_{z}$ is the vertical tension in the mooring line at a certain point $(x,z)$ in the line, $_T$ is the resultant tension of the mooring line at a point $(x,z)$ in the line. + +Note that these equations (3, 4, 5, 6) are defined in a plane $(x,z)$ and are associated with a coordinate system with the origin located at the point of contact between the mooring line and the sea floor. The $_x$ axis should point towards the attachment point, and the $_z$ axis towards the water free surface, see Figure 2. + +The relationship between the horizontal tension in the mooring line and the horizontal displacement of the attachment point is found by solving the following equation (see Figure 2): + +$$ +X-x_{a}=L-s_{a}, +$$ + +where + +$X$ is the distance from the anchor point to the attachment point (a), $\boldsymbol{x}_{a}$ is the horizontal coordinate of the attachment point, $Z_{a}$ is the vertical distance to the attachment point, $\pmb{\mathscr{s}}_{a}$ is the length of the line from the origin of the coordinate system, and $L$ is the total length of the mooring line. + +![](images/ec1ca994ee9b5db1f792011ca82495579b22e4f333ec6bb982cbe578857655e8.jpg) +Figure 2: Schematic drawing of a float moored with a mooring line. + +Squaring the expressions for $s_{a},$ and taking into account the hyperbolic Pythagorean identity, $\cosh^{2}x-\sinh^{2}x=1,$ the value of $\pmb{\mathscr{s}}_{a}$ is givenby + +$$ +s_{a}=\sqrt{z_{a}^{2}+2z_{a}\frac{T_{x}}{w}}\;. +$$ + +The value for $\scriptstyle{x_{a}}$ can be obtained from + +$$ +x_{a}=\frac{T_{x}}{w}\mathrm{cosh}^{-1}\left(1+\frac{z_{a}}{T_{x}}w\right)\!. +$$ + +The relationship between the horizontal tension at the attachment point $(T_{x})$ and its displacement $(X)$ is obtained by solving + +$$ +L-Z_{a}\sqrt{1+2\frac{T_{x}}{w Z_{a}}}+\frac{T_{x}}{w}\mathrm{cosh}^{-1}\left(1+\frac{Z_{a}w}{T_{x}}\right)-X=0. +$$ + +This equation provides the basic relationship to be considered in a procedure for populating a stifness lookup table based on the inelastic catenary equations for mooring lines composed with a single segment. + +Each entry in the lookup table is to represent the horizontal and vertical displacement of the attachment point $(\Delta X,\Delta Z)$ of the mooring line to an initial position $(X_{0},Z_{0})$ + +$$ +X=X_{o}+\Delta X +$$ + +$$ +Z_{a}=Z=Z_{a}0+\Delta Z +$$ + +The horizontal tension at the attachment point is obtained as a function of the horizontal and vertical displacement of the attachment point, i.e. + +$$ +T_{x}=f(\Delta X,\,\Delta Z). +$$ + +Note that this equation is non-linear and transcendental ( ${\cal T}_{x}$ is in and out of $\mathrm{cosh^{-1}}$ ).Such type of equations can be solved using an iterative method, such as Newton-Raphson or bisection. + +The value for the vertical tension $(T_{z})$ at the attachment point is then given by + +$$ +T_{z}=w\ s_{a} +$$ + +and the total line tension at the attachment point can be obtained by + +$$ +T=\sqrt{T_{x}^{2}+T_{z}^{2}}. +$$ + +# Dynamic mooring model + +The dynamic mooring model, unlike other models, represents a mooring line in an explicit and fully-coupled manner. Normally the multibody dynamics (MB) model includes just the turbine, with moorings treated as just one of many load sources treated separately from the model. Such sources only interact with the MB model by means of applied loads at nodes. In the case of dynamic moorings, the MB model is extended to include the mooring lines themselves. Hydrostatic and hydrodynamic loads are calculated and applied to mooring elements (segments) in the same way as for support structure elements. Hydrodynamic loads are based on Morison's equation, which is appropriate for the slender elements involved. The Morison drag term can be applied axially as well as perpendicular to the segment, with separate drag coefficient values defined for each of the two cases. In the MB tree structure, the proximal ends of the mooring lines are attached to the fairleads and the distal ends terminate at the anchors, so in MB terms they are not connected to anything. + +Anchor and seabed forces are calculated from line kinematics, based on simple spring-damper models and are applied as point forces at the node points of the line. A line is discretised as a series of cylindrical elements with an optional single degree of freedom (i.e. axial extension). These elements are connected to each other by universal joints with optional user-defined damping at the joint axes. Inline floats or clump weights, where present, are modelled as simple vertical forces, modified where necessary in the case where a weight makes contact with the + +seabed or a float pierces the sea surface. In the former case, a simple spring-damper model of applied seabed force is used, just as for mooring segments. In the latter case, the buoyancy force is assumed to vary linearly with the submerged proportion of the float's height. No hydrodynamic forces on weights or floats are modelled. + +Last updated 06-09-2024 + +# Structural loads + +The structural loads at the multibody nodes, e.g. the blade root, the power train and the tower base, are calculated as the joint reactions by the component equilibrium equation described in the multibody dynamics approach. The inertial loads are calculated by integration of the mass properties and the total acceleration vector at each station according to the principle of virtual work. The total acceleration vector includes modal, centrifugal, Coriolis and gravitational components. Second-order effects of axial forces due to structural deflections are taken into account, i.e. the contribution to bending moments caused by the applied axial forces acting on the deflected structure. For example, the contribution to tower base bending moment caused by the weight of the nacelle takes into account the true position of the nacelle centre of gravity including the deflection of the tower top. + +Last updated 30-08-2024 \ No newline at end of file diff --git a/书籍/力学书籍/BladedTheoryManual/Structural Dynamics/auto/Structural Dynamics_origin.pdf b/书籍/力学书籍/BladedTheoryManual/Structural Dynamics/auto/Structural Dynamics_origin.pdf new file mode 100644 index 0000000..89b2003 Binary files /dev/null and b/书籍/力学书籍/BladedTheoryManual/Structural Dynamics/auto/Structural Dynamics_origin.pdf differ diff --git a/书籍/力学书籍/BladedTheoryManual/Structural Dynamics/auto/images/0fe3cc9ef4f678fc3e6e81d7a7c24f45f33b741fdb41c1851702e584d6b36263.jpg b/书籍/力学书籍/BladedTheoryManual/Structural Dynamics/auto/images/0fe3cc9ef4f678fc3e6e81d7a7c24f45f33b741fdb41c1851702e584d6b36263.jpg new file mode 100644 index 0000000..0bced5b Binary files /dev/null and b/书籍/力学书籍/BladedTheoryManual/Structural 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带3.5气动与fast3.5对比 相同\n- Herowind 带4.0气动与fast4.0对比 相同\n- Herowind 带hrl气动与fast对比 需气动支持15MW\n\nP1 Bladed交流问题汇总","x":-700,"y":134,"width":440,"height":560}, - {"id":"c18d25521d773705","type":"text","text":"# 计划\n这周要做的3~5件重要的事情，这些事情能有效推进实现OKR。\n\nP1 必须做。P2 应该做\n\n\nP1 柔性部件 叶片、塔架主动力惯性力算法 主线\n- 变形体动力学 简略看看ing\n- 柔性梁弯曲变形振动学习，主线 \n\t- 广义质量 刚度矩阵及含义\n- 如何静力学求解\n\t- 基于本构方程 读孟的论文\n\t- normal mode shape 能否使用？\n\nP1 模型线性化原理\nP1 结果对比\n- Bladed与FAST之间的对比 去掉角度\n\nP1 如何优雅的存储、输出结果。\nP1 推进气动、控制、多体、水动 耦合计算\nP2 yaw 自由度再bug确认 已知原理了\n\nP1 模型线性化调研\n \n","x":-594,"y":-693,"width":450,"height":347}, + {"id":"c18d25521d773705","type":"text","text":"# 计划\n这周要做的3~5件重要的事情，这些事情能有效推进实现OKR。\n\nP1 必须做。P2 应该做\n\n\nP1 柔性部件 叶片、塔架主动力惯性力算法 主线\n- 变形体动力学 简略看看ing\n- 柔性梁弯曲变形振动学习，主线 \n\t- 广义质量 刚度矩阵及含义\n- 如何静力学求解\n\t- 基于本构方程 读孟的论文\n\t- normal mode shape 能否使用？\n\nP1 模型线性化原理\nP1 结果对比\n- Bladed与FAST之间的对比 去掉角度 不能直接比较\n- 坐标系转换\n\nP2 如何优雅的存储、输出结果。\nP1 推进气动、控制、多体、水动 耦合计算\nP2 yaw 自由度再bug确认 已知原理了\n\n \n","x":-594,"y":-693,"width":450,"height":347}, {"id":"30cb7486dc4e224c","type":"text","text":"# 8月已完成\n\n","x":260,"y":134,"width":440,"height":560} ], "edges":[]