Residualisation

As described in Section 4.5, the largest permissible time-step for stable time marching of the system (3.23) or (3.55) using the RK4 scheme is determined by the period of the highest frequency mode in the linear system (a discussion of the stability bound- aries of various ODE time marching schemes is found in textbooks such as [139]). As will be demonstrated by simulations in Section 6.3, the structural model requires the inclusion of high-frequency modes in order to conserve system momentum. As a result the permissible time-step becomes very small compared to the typical timescale of the low-frequency dynamics, increasing the computational cost required to compute a given length of time in the simulation.

The problem of stiffness in ODEs has been extensively studied in the literature on numerical methods [139]. For displacement-based formulations in particular, damping is added to the Newmark method [1]. The time-stepping scheme will thus be able to march at a larger timestep than the highest eigenvalue permits without becoming unstable, while retaining good accuracy for low-frequency responses that are typically of interest. However the method relies on using the structure of the generalised equations of motion which is difficult to translate into the intrinsic modal formulation where a different set of primary variables is used. As described previously, other integration

schemes that include numerical damping are either too limited in their applicability or caused convergence problems in the implicit iterations.

In the aeroelastic system in which we are interested in this work however, we are primarily interested in the low-frequency behaviour. The use of a beam-type description of a slender structure becomes less accurate at wavelengths comparable to the typical section size of the structure, furthermore the strength of aerodynamic damping in an aeroelastic system increases with frequency and will remove any high frequency dynam- ics very quickly. Thus we are only interested in the inclusion of high-frequency modes as a nonlinear correction term to the low-frequency modes in the structural model. This philosophy is the same as the residualisation method used in linear model reduction [124] where rather than reducing the size of a system by simply truncating parts of the dy- namics altogether (balanced truncation), the truncated dynamics are instead solved as algebraic equations and coupled back to the retained parts of the dynamics, which also ensure that the solution to the static problem is preserved. In linear model reduction, the truncation method is normally used to remove low-frequency modes, whereas resid- ualisation is used for high-frequency ones as is the case here. There exists a difference between the balanced residualisation technique and the method used currently, where balanced residualisation operates on linear systems, the current method seeks to retain the coupling in nonlinear dynamics.

We will now describe the algebraic procedure by which the residualisation is carried out in this work. First, we take the structural modal equations (3.28) as the starting point of this analysis. It is thus implied that a finite number of natural modes of the system are used resulting in a diagonal WD matrix that is diagonal by definition. If this

is not the case it will be straightforward to orthogonalise the modes until this is achieved, using the method in Section 4.3.2. The aerodynamic and flight dynamic couplings are not considered part of the residualisation process and are computed after residualisation is performed. We now define a pair of modes φ_1,j and φ_2,j as being high-frequency when their associated eigenvalue, ωj, is higher than a specified cut-off frequency, ωC.

Otherwise it is regarded as being low-frequency. We now split q into two parts containing the low- and high-frequency modes, q_Land q_H, respectively, that is, q =h q⊤

L q⊤H

i⊤

. Using this, (3.28) can be split into

˙qL= WLqL+ ΓL(q)q + ηL, (4.56a)

˙qH = WHqH+ ΓH(q)q + ηH (4.56b)

set of high-frequency harmonic oscillators, operates at a very different frequency to the quadratic part of the equation, which contains contributions from geometric nonlinear couplings, and to the external forcing. Therefore if only the low-frequency dynamics are of interest, the system can be approximated by regarding a time-averaged qH that reacts

instantaneously to excitations from the ΓH(q)q and ηH terms. This approximation

effectively removes the high-frequency dynamics from the q_H states and converts (4.56) into a set of differential-algebraic equations (DAE), or

q_{H = −WH}−1(ΓH(q)q + ηH), (4.57a)

˙qL= WLqL+ ΓL(q)q + ηL, (4.57b)

with q = h q⊤_L q⊤_H i

⊤

. This set of equations is marched by first iterating the value of q_H using the currently known values of system states q_L with the first equation of (4.57), then computing the value of ˙q_L using the second equation to obtain the time derivatives for time marching. The iteration of qH can be carried out as

q_{H,k+1 = −WH}−1(ΓH(qk)qk+ ηH) (4.58)

which converges quickly provided the eigenvalues in WH are large. More elaborate

iteration schemes such as Newton-Raphson can also be used for faster convergence but were not necessary in the results shown in this work.

In practice, the residualisation method was able to significantly speed up the compu- tation by providing an increase in maximum stable timestep, a result that a truncation of high-frequency also achieves. However residualisation of high-frequency components pro- vides the benefit of being much more accurate in predicting the low-frequency response of the structure than a truncation can achieve. This is because it retains coupling in- teractions from high-frequency modes that occur at a far slower rate than their natural frequencies. The effectiveness of this method will be demonstrated in Section 6.3 and compared against full mode simulation using (3.28).