Response spectra in fixed and rotating coordinates

The linearized equations of motion of a three-bladed wind turbine can be made time-invariant with a multi-blade coordinate transform. This expresses the rotor dynamics in a fixed (non-rotating) coordinate frame. The state equations (3) are implemented in multi-blade coordinates. Sensor inputs associated with each blade – in the present state observer, this is only the blade pitch angle – are transformed to multi-blade coordinates before being fed to the state observer. Solving for response spectra in the fixed coordinate frame is then straightforward. In order to compute the material stresses required for fatigue analysis (Section 4.7), driveshaft and blade load spectra need to be transformed into the rotating frame.

Consider a trio of variablesr^𝐵= [𝑟₁, 𝑟₂, 𝑟₃], with each variable associated with the blade indicated by the subscript.²³ The multi-blade coordinate transform is r^𝜓=T^𝜓_𝐵r^𝐵, where

T^𝜓_𝐵= 1

It will be convenient to write (58) in complex exponential form,

T^𝐵_𝜓 =E₀+Eexp(𝑖Ω𝑡) +E^∗exp(−𝑖Ω𝑡), (60) Let us define the variable r^𝜓 in the following way:

r^𝜓=r^𝜓+ 𝛥r^𝜓+𝛥r̃ ^𝜓. (62) Here r^𝜓 contains the steady-state values, the components of 𝛥r^𝜓 are zero-mean Gaussian processes, and 𝛥r̃ ^𝜓 is periodic. We are given a spectral matrix of 𝛥r^𝜓, S^𝜓(𝑓), or in discrete form S^𝜓(𝑘∆𝑓) = S^𝜓(𝑘). We are also given the steady-stater^𝜓, as well as the Fourier coefficientsc^𝜓_𝑘 describing𝛥r̃ ^𝜓. The goal is to transform these quantities to body (rotating) coordinates. That is, we seek an equivalent form of (62) in body coordinates,

r^𝐵=r^𝐵+ 𝛥r^𝐵+𝛥r̃ ^𝐵, (63) where the zero-mean stochastic term𝛥r^𝐵 is characterized by a spectral matrixS^𝐵(𝑘), and the zero-mean periodic term 𝛥r̃ ^𝐵 by Fourier coefficientsc^𝐵_𝑘.

To develop the appropriate transformations, let’s start in the time domain. The spectra are each the individual Fourier transform of a covariance function; the time-domain counterpart to S^𝜓 is a covariance functionQ^𝜓. By definition,

𝑄^𝜓_𝑗𝑘(𝑡, 𝜏 ) = 𝐸 [(𝑟^𝜓_𝑖(𝑡) − 𝑟^𝜓_𝑖) (𝑟^𝜓_𝑗(𝑡 + 𝜏 ) − 𝑟^𝜓_𝑗)] , (64)

23These could be any of the dynamic variables: displacements, loads, stresses, or whatever.

From (62),

𝑄^𝜓_𝑗𝑘(𝑡, 𝜏 ) = 𝐸 [𝛥𝑟^𝜓_𝑖(𝑡) 𝛥𝑟^𝜓_𝑗(𝑡 + 𝜏 )] + 𝐸 [𝛥𝑟^𝜓_𝑖(𝑡)𝛥𝑟̃ ^𝜓_𝑗(𝑡 + 𝜏 )]

+ 𝐸 [ ̃𝛥𝑟^𝜓_𝑖(𝑡) 𝛥𝑟^𝜓_𝑗(𝑡 + 𝜏 )] + 𝐸 [ ̃𝛥𝑟^𝜓_𝑖(𝑡)𝛥𝑟̃ ^𝜓_𝑗(𝑡 + 𝜏 )] . (65) It is necessary to make a brief digression to specify what is meant by the expected value. In the context of random variables, the fundamental definition of an expected value is

𝐸[𝑥] = ∫

∞

−∞

𝑥 𝜑(𝑥) 𝑑𝑥, (66)

where 𝜑(𝑥) is the probability density over 𝑥. It is useful to picture (66) in discrete space – that is, discretized values of𝑥 – in which case

𝐸[𝑥] =

with𝑝(⋅)the cell probability. Blind application of (66) to a deterministic function simply returns the value of the function: making use of the Law of the Unconscious Statistician,

𝐸[𝑓(𝑥_𝑝)] =

But this is not really what we want. Rather, given that we are working with a stationary stochastic process and repeating periodic signal, the appropriate definition of the expected value is theensemble average over a set of arbitrary times. That is to say, we consider𝑡– the starting time, in the case of the covariance (65) – to be a uniformly-distributed random variable, and then we evaluate the expected value according to (66). In this case,the expected value becomes the time-average. To see this, write

𝐸[𝑓(𝑡)] = lim

By the ergodic theorem, (66) and (69) are the same, for a stationary stochastic process. This means that for both stochastic and periodic quantities, we may define the expected value as the average value over time.

Picking up Eq. (65) where we left off, it is clear that the two terms in the middle are zero. They are independent, so

𝐸 [𝛥𝑟_𝑖^𝜓(𝑡)𝛥𝑟̃ _𝑗^𝜓(𝑡 + 𝜏 )] = 𝐸 [𝛥𝑟^𝜓_𝑖(𝑡)] 𝐸 [ ̃𝛥𝑟^𝜓_𝑗(𝑡 + 𝜏 )] = 0, (70) and similarly for the other term. Then,

𝑄^𝜓_𝑗𝑘(𝑡, 𝜏 ) = 𝐸 [𝛥𝑟^𝜓_𝑖(𝑡) 𝛥𝑟^𝜓_𝑗(𝑡 + 𝜏 )] + 𝐸 [ ̃𝛥𝑟^𝜓_𝑖(𝑡)𝛥𝑟̃ ^𝜓_𝑗(𝑡 + 𝜏 )] . (71) We are therefore free to consider the stochastic and periodic processes separately. Should we later wish to form the combined covariance or spectra, these can be obtained by superposition.

Let us simplify the terminology by making the time𝑡implicit. So, for instance, we will write (71) as

𝑄^𝜓_𝑗𝑘(𝜏 ) = 𝐸 [𝛥𝑟^𝜓_𝑖(0) 𝛥𝑟^𝜓_𝑗(𝜏 )] + 𝐸 [ ̃𝛥𝑟^𝜓_𝑖(0)𝛥𝑟̃ ^𝜓_𝑗(𝜏 )] . (72) The development that follows is best accomplished in discrete time, and discrete frequencies, since this is how it is programmed. Working with discrete increments also allows us to avoid integrals,

limits, and delta functions, and the operations are easier to visualize. Let 𝜏 = 𝑝∆𝜏 and 𝑓 = 𝑘∆𝑓.

The discrete form of the complex exponential is then

exp(𝑖𝜔𝜏 ) =exp(𝑖2𝜋𝑘∆𝑓 𝑝∆𝜏 ) =exp(𝑖2𝜋𝑘∆𝑓 𝑝

where𝑤 does not necessarily need to be an integer.

Now, we want to arrive at the Fourier representation of 𝛥r̃ ^𝐵,

̃𝛥r^𝐵(𝑝) =

along with the covariance in body coordinates,

Q^𝐵(𝑝) = 𝐸 [(r^𝐵(0) −r^𝐵) (r^𝐵(𝑝∆𝜏 ) −r^𝐵)^𝑇] . (76) since, even if there is some energy in the random process at the rotor frequency, the phase is undeter-mined. As for the periodic part𝛥r̃ ^𝜓, it can be expanded in a Fourier series akin to (75),

̃𝛥r^𝜓(𝑝) =

and The transform from MBC to body coordinates can be written

r^𝐵+ 𝛥r^𝐵+𝛥r̃ ^𝐵 =T^𝐵_𝜓(r^𝜓+ 𝛥r^𝜓+𝛥r̃ ^𝜓) . (87) In light of (71) and (86), it is clear that we can separate the stochastic part from the periodic part, like

𝛥r^𝐵=T^𝐵_𝜓𝛥r^𝜓 (88)

and

r^𝐵+𝛥r̃ ^𝐵=T^𝐵_𝜓(r^𝜓+𝛥r̃ ^𝜓) . (89) In other words, the stochastic signal remains stochastic: it influences neither the mean nor the periodic signal.

The spectra can be obtained directly from the Fourier coefficients. Denoting as S̃^𝐵 the periodic part of the spectral matrix,

̃S^𝐵= 1

∆𝑓 c^𝐵 (c^𝐵)^∗𝑇. (92) As for the stochastic part, the covariance transforms as

Q^𝐵(𝑝) =T^𝐵_𝜓(0)Q^𝜓(𝑝)T^𝐵,𝑇_𝜓 (𝑝). (93) The spectral matrixS^𝐵(𝑘) is the (discrete) Fourier transform of (93). This is

S^𝐵(𝑘) =∆𝜏

This can be written as the sum of three terms, Note how part of the energy at the frequency𝑓 in rotating coordinates comes from𝑓 ±Ω frequencies in multi-blade coordinates.

Obtaining stress spectra at points in the driveshaft requires a similar transformation into the rotating frame. The transform from nacelle to driveshaft coordinates involves a rotation about the 𝑧^𝑛 = 𝑧^𝑑 axis by the rotor azimuth angle Ψ, which is identical to Ψ₁ from above. The transform can be written in the same form as (60),

T^𝑑_𝑛 =E₀+Eexp(𝑖Ω𝑡) +E^∗exp(−𝑖Ω𝑡), (100)

The remainder of the derivation, (62) through (99), is unchanged.