Let's look into this digital Model Order Reduction and at the same time it's parameterization, more in the sense of a flow chart. First of all, on top, we are starting with an overall system modeling. It might be a finite element model of computation, fluid dynamics model, a coupling of both and so forth. Any proper matrix modeling method can be used. Then we have to consider the projections, the model reduction and its parameterization. For finding such a projections there are different methods. It's just previously mentioned. Departmentalization means either Taylor's pension or interpolation technique and so forth. Once these reduced model, and these reduced parametrized model has been established, the simulations can be done on this reduced model. It's not only the simulations, but essential is and keep in mind for practical applications in the real world development processes is, one has to consider the iteration steps on modifying the parameters. So one has to repeat it again and again, but on the contents of reduced model, doing all these steps. We have to know, is it good enough? Can it well represents the behavior of the system? Can it well represents the behavior of the system even after modifications of parameter? So we have to assess the reduced and parametrized model. Often used technique is to look into relevant transfer functions, especially in computational and structure interaction problems. The transfer functions of loads to responses, the transfer functions of actuations to the the system responses and system outputs are looked into, with and without model reduction. Here in the center of this u-curve, we see such a typical transfer function over the frequency of the system. Originally it calculated with blue curve, which is hidden behind the red curve, if you wish so, which represents a large-scale models, where 1,400 so-called states or modes or Eigen frequencies or vibrations are determined. On the reduced model, only 26 states or Eigen frequencies and modes are used, and you see that such a transfer function and the representation of the significantly reduced model and the original larger scale model can be assessed by a transfer function. At least it tells us set up to 20-30 hertz, these reduced model works very well. If we are looking into this assessment, we have to as always, when one is looking into assessments, to establish assessment criteria. Which means in this case, we are looking into the coverage frequency band and the covered modes. Covered means into relevant frequency band and relevant modes, effective masses are the residual modes which are not considered, and formal control point of view, of course, controllability and observability plays an important role. Because we are talking about parameter variation and parameterized reduced order model, we also have to look to which extends is parameterized Reduced Order Model valid for these different parameter changes? In order to better understand what we mean with this, gust load behavior or this aero-elastic behavior and the turbulence. You see here the performance and the behavior of simulation model. Namely, when we start this model, we see that this aircraft and this finite element model of this aircraft is our elastic model of the aircraft experience a rigid body deformation, which also causes or may cause some discomfort of the passenger. But also we see at the same time, the wings of the aircraft are vibrating and get an elastic deformation in the vibration, dynamic deformation, which in this case is relatively small, but there are other cases where these vibrations might become large. Now to make things more concrete, we are looking into series of applications in aerospace systems. Not only from a model reduction and motor condensation and parameterization point of view, but also I will address some points of the physical and technical background of these applications. Let's start again with this is aircraft model and aircraft simulation model and behavior model, where we are looking into different lift load distributions. As mentioned in the introduction, we have three different lift load distributors. So first one is the so-called stationary one. When we neglect the fuse large influence, it's elliptic lift distribution or weathervane. Assuming that the aircraft is flying into turbulences and into a gust, the wing might talk and the angle of attack might increase, especially at the very edges of the wing. This means that the lift is increased at the outer part of the wing. The overall lift is increased, which causes not so comfortable behavior of the passengers, but at the same time also causes an increase of the so-called bending loads and bending forces at the wing root, which are shown here in these red diagram. So in order to reduce this, this gust load defect, what is looking into such a load control technique, which are different ways to achieve this, namely the so-called passive way via aero-elastic tailoring. The wing is designed in such a way, and the parameters of the wing are determined in such a way that due to the natural pressure distribution over the wing, the wing is deformed in such a way where this negative effect of gust loads is minimized and reduced. Alternatively or additionally, one can do this also by active means, by properly controlled use of flaps, where these flaps are used to enforce these proper bending torsion behavior to get a good overall least distribution which minimizes and reduces the bending moments at wing root and also reduces the not so comfortable behavior for the passenger. From this behavior, we see that these strong interactions of aerodynamics and vibration has to be avoided. So these flutter cases, as they're called, have to be avoided not only in order to avoid large deformations, but also to avoid such large forces, such as structures, ring and in this case also the bridge starts to break. So it's a modeling of this behavior results into so-called Aero-servo-elastic simulation models. Aero means aerodynamic, servo means control and activation, and elastic means evaporation, deformation behavior of the aircraft. This means that we first of all have to establish finite element simulation model for structural dynamics, which you see here for an aircraft are so-called half model. On the left hand upper side see differential equations in time into left center side, and also including rigid body modes on the lower part. The upper part of this aircraft shows an elastic eigenmode vibration frequency of the ring together with the smaller amplitude in the fuselage assuming this structural dynamics behavior to be validly represented by these models, we have to also take into account the aerodynamic behavior. It might be either steady behavior or especially in gust, it's an unsteady behavior. So then takes usually the doublet-lattice method or any other computation fluid dynamics analysis technique and these too large molecules have to be combined and coupled together, and in order then to carry out the parametric studies, one does not want to treat these investigations on these two coupled large-scale models, but on the reduced model and the parameterized reduced model like the determination of the eigenvalues, which are sketched on the lower right-hand side, whether they are the real parts, whether they are the imaginary parts of these eigenvalues. So what does active dynamic load alleviation mean technically for the aircraft? This can be seen in this figure and this is sketch for this plane to embody configuration where we assume set is planted to embody aircraft is excited dynamically. Why are aerodynamic disturbances? These disturbances are measured by sensors in front of the aircraft, for example, by certain radar techniques or pressure sensors, but the response is also measured by acceleration sensors in the wing, which gives for the feedback loop of this overall control loop, the sensor inputs of this behavior of the wing. By applying send a feed for combined, feed-forward, feedback control loop, means set the overall response, the dynamic response of serving can be drastically reduced. This is shown here in this transfer function. Where typical displacement at the wingtip is shown versus the frequency of this wing, the resonance or high frequency of this wing and we see here is this blue curve where no active action is taken, no flaps are used to counteract the disturbances and the influences of the gust. When the control loop is applied, which triggers a flap such that this influence is reduced. This is shown in the red curve and we see that practically the overall response now is only half of that of the uncontrolled behavior. The next application is taken from Space Engineering. We are looking into the vibration control and the vibration design of very flexible solar arrays. I'm saying very flexible because these solar arrays as one can see here on the left-hand side of a satellite in orbit, of course, it's not a view in the solar array on the ground in a test or in the test laboratory. Since solar arrays are quite large and floppy structures, so the eigen or resonance frequencies are also very low, but they might be even smaller than a tenth of a hertz. So in case such solar arrays start vibrating, they might introduce disturbances on the satellite, which one wants to avoid to the most possible extent because it might cause small tumbling movements of the overall satellite, which influences its overall performance and behavior. We were looking into more detail, and we are looking into more detail of such a solar array panel. Namely, you see here on the right-hand side, such as singles solar array sandwich panel, which we want to control in its vibration behavior both in orbit and during launch and for investigation, we first started with the overall or full-scale model and we simulated the behavior of and the influence of time parameters of such a sandwich panel named as Young's modulus, thickness distribution of the sandwich panel, and also possibly positioning of non-structured buses. The original simulation model at more than 11,000 degrees of freedom and the drastically reduced model, which we were looking into and handling together with parameter studies and together with interfacing this controlled behavior, only has finally had 51 degrees of freedom. What we did, we're looking into different modes, vibrational modes also, among others to determine proper positioning of actuators to control vibration and control always means you always have a proper positioning of sensors in order to guarantee good observability of the behavior of this plate.
