Identification and synthesis of components for uncertainty propagation

For automotive structures, built-up of hundreds of components with property spread, knowing the effects of component variability and its propagation through the system assembly is important in order to mitigate noise and vibration problems. To increase the understanding of how the spread propagates into variability in built-up structures, both experimental and computational aspects are considered in this thesis.In the first part of the thesis, methods to identify models from experimental data are developed. Physical insight is often required for accurate experimental models. To this end, two-phase state-space system identification algorithms are developed where physically motivated residual states are included and physically motivated constraints are enforced. The developed identification algorithms are used together with finite element model updating to investigate the variability in dynamical properties between nominally identical components. Furthermore, the accurate and physical experimental models are used in synthesis with the updated finite element models. It is shown that experimental-analytical synthesis of complex and modally dense structures is possible, and that the component variability can be predicted in such assemblies.In the second part of the thesis, methods to reduce the computational cost of variability analysis are developed. An efficient multifidelity interface reduction method is developed for component synthesis. It is also shown that modal truncation augmentation vectors can be computed efficiently from the multifidelity interface reduction basis. Lastly, an efficient uncertainty propagation method is developed, based on a second-order modal model. Utilising several approximations, it is shown that industrial-sized models can be handled with small loss in accuracy compared to a purely Monte Carlo based approach

Model updating of multiple nominally identical car components

A method for estimation of rubber bushing stiffness parameters is presented. Four individual rubber bushings, mounted in a car rear subframe are considered. A traditional model of the bushing elements using a generalised spring model, known as a CBUSH element in Nastran, is compared to a geometrically more realistic approach where the bushing is modelled with solid elements and a linear elastic material model. Each bushing is mass loaded to better reveal the bushing\u27s dynamic behaviour in a lower frequency range of interest. In an initial step, the overall subframe model is updated towards test data.In a second step, the bushing parameters are updated. Three nominally identical components are used to investigate the spread between the identified parameters. The model updating procedure is based on frequency responses and equalised damping. The undamped behaviour at frequencies below 300~Hz are considered. To quantify the parameter uncertainty, with respect to measurement noise for each individual, an uncertainty quantification procedure is proposed, using a linear-in-parameters surrogate model with bootstrapping

State-Space Dynamic Substructuring with the Transmission Simulator Method

Dynamic substructuring is a technique to simplify the analysis of complex structures. The vibrational problems of the constituent substructures are analysed and solved individually and their solutions are then assembled to form the global solution. In experimental dynamic substructuring, at least one of the constituent substructures lis identified experimentally. The coupling interfaces are commonly simplified in such syntheses, which can result in poor prediction quality in many applications. The transmission simulator was introduced to address this problem. Transmission simulators are well-modelled parts attached to the interface of the substructures to be coupled. This allows for distributed interfaces and a relaxation of the coupling conditions by using the transmission simulator\u27s analytical modes as a basis for the coupling equations, at the cost of adding a decoupling step to the substructuring problem. In this paper, the transmission simulator method is translated to the state-space substructuring domain. The methodology is applied to the Society for Experimental Mechanics\u27 substructuring focus group\u27s Ampair A600 test bed in form of experimental-analytical substructuring. The Ampair wind turbine\u27s hub is used as the transmission simulator and is modelled with finite elements while the three blades, individually attached to the real hub, are experimentally identified. The three experimental blade hub systems are then coupled and two finite element hubs decoupled from the system, using the derived method. Finally, this system is compared to a directly measured hub with three blades by means of frequency response functions and modal properties

Prediction of structural dynamic behaviour under uncertainties: With applications to automotive structures

The automotive industry is moving towards shorter development cycles for new car generations. This means that less expensive prototypes can be built and tested, and that, increasingly, computer models must be used for decision making. Further, the automotive industry is producing thousands of nominally identical cars which are known to exhibit noticeable spread in their vibration characteristics. A car\u27s noise and vibration behaviour is therefore not the same between nominally identical cars. This implies a need for structural dynamic models considering uncertainties for robust decision making. Due to the final products complexity a substructuring approach is considered in this thesis, including experimental and computational methods, where predictive models of components are created, to be assembled for a predictive system response.The first part of this thesis considers the reduction of uncertainties introduced from vibration experiments. A method for sensor placement in vibration experiments is developed, based on the method of effective independence, so that symmetric sensor positions are rejected using system gramians. Further, a measurement system is developed in MATLAB for fast and efficient stepped sine excitation.The second part considers the spread between nominally identical components and the calibration, and an associated parameter uncertainty quantification, of industrial finite element models of said components. Results are reported here for three front and one rear subframe. For model calibration, a model updating procedure is employed that uses a frequency response function based deviation metric and equalised damping. A bootstrapping procedure is subsequently used to quantify parameter uncertainties with respect to the measurement noise. Calibrations are performed for an ensemble of front subframe components. Particular care is taken in the modelling of coupling elements and for the rear subframe the elastic modulus in rubber bushings is estimated using a mass loaded bushing boundary configuration.In the automotive industry high fidelity models are common, with many interface degrees of freedom decreasing the efficiency of component mode synthesis methods. Therefore, a component mode synthesis interface reduction method is developed to speed up the process, using coarse meshes

Identification of physically realistic state-space models for accurate component synthesis

For components that are difficult to model with conventional analytical or numerical tools, experimentally derived state-space models can instead be used in system synthesis. For successful state-space synthesis, a physically realistic model must be identified. For this purpose, a hybrid first- and second-order system description is used here as the basis for identification. In the identification procedure, a physically motivated rigid body rank constraint is imposed together with a reciprocity constraint. The two constraints are enforced during a re-estimation phase of the state-space matrices following after a traditional state-space subspace identification phase. In this paper, two complex and modally dense industrial components are combined into a dynamical system. An experimental model of a car body-in-white structure is identified. The identified subsystem model is coupled with a finite element model of a rear subframe in a system synthesis. The two subsystems are attached through four rubber bushings modelled by finite element procedures. It is shown that the experimental-analytical assembly successfully predicts the reference measured system, with higher accuracy than what could be achieved with a model based solely on finite elements. It is also shown that synthesis with individually calibrated rear subframe models can capture the variability in the coupled system

Data-driven modal surrogate model for frequency response uncertainty propagation

A method is developed for propagation of model parameter uncertainties into frequency response functions based on a modal representation of the equations of motion. Individual local surrogate models of the eigenfrequencies and residue matrix elements for each mode are trained to build a global surrogate model. The computational cost of the global surrogate model is reduced in three steps. First, modes outside the range of interest, necessary to describe the in-band frequency response, are approximated with few residual modes. Secondly, the dimension of the residue matrices for each mode is reduced using principal component analysis. Lastly, multiple surrogate model structures are employed in a mixture. Cheap second-order multivariate polynomial models and more expensive Gaussian process models with different kernels are used to model the modal data. Leave-one-out cross-validation is used for model selection of the local surrogate models. The approximations introduced allow the method to be used for modally dense models at a small computational cost, without sacrificing the global surrogate model\u27s ability to capture mode veering and crossing phenomena. The method is compared to a Monte Carlo based approach and verified on one industrial-sized component and on one assembly of two car components

Physically motivated rank constraint on direct throughput of state-space models

In frequency range vibration testing a few outside band eigenmodes are often included in the system identification to compensate for residual mass and stiffness influences. It has been observed that, in particular, energy conjugate input-output pair transfer functions with strong outside band modes tend to render models with poor fit even after inclusion of mass and stiffness residuals. For such problems the inclusion of another complementary residual term has been found to improve the fit to data. In this paper, modal models identified from acceleration data with a subspace state-space method are considered. The residual mass influence is modelled with a state-space direct throughput while the stiffness and complementary residuals are modelled with extra states. Furthermore, for state-space models on accelerance form it is shown that the direct throughput matrix can be partitioned into a flexible and rigid motion partition. For systems with more inputs and outputs than rigid body modes it is shown that the rigid body motion partition has a bounded rank. The upper bound is equal to the number of rigid body modes. Therefore, for identified models on accelerance form this constraint must be enforced for physical consistency. The proposed method is applied on simulated finite element test data from an automotive component