This work proposes an augmented extended Kalman filter based state-input estimator for mechanical systems defined by implicit equations of motion which is then applied to estimate the six wheel center loads and the strain field on a vehicle suspension test rig.Implicit equations of motion typically arise in the definition of flexible multibody models and also in their time resolution, because implicit time-discretization schemes are normally employed to obtain a stable solution. The presented methodology can be applied to such case and analytical expressions are derived for the necessary linearizations, providing the means for a computationally efficient estimation procedure.The six wheel center loads and the strain field on a vehicle suspension system are valuable quantities during the vehicle design phase (e.g. for durability analysis), hence they are often directly measured during elaborate full vehicle testing campaigns. This work demonstrates that a flexible multibody model representation allows to accurately reconstruct the time domain signals of the six loads and of the full strain field, starting from a minimal set of six measured strains, hence providing an appealing alternative to direct measurement methods. The experimental validation on the suspension test rig shows that all estimated quantities can be accurately reconstructed, given that the system simulation model incorporates an adequate level of accuracy.

A novel nonlinear parametric model order reduction technique for the solution of contact problems in flexible multibody dynamics is presented. These problems are characterized by significant variations in the location and size of the contact area and typically require high-dimensional finite element models having multiple inputs and outputs to be solved. The presented technique draws from the fields of nonlinear and parametric model reduction to construct a reduced-order model whose dimensions are insensitive to the dimensions of the full-order model. The solution of interest is approximated in a lower-dimensional subspace spanned by a constant set of eigenvectors augmented with a parameter-dependent set of global contact shapes. The latter represent deformation patterns of the interacting bodies obtained from a series of static contact analyses. The set of global contact shapes is parameterized with respect to the system configuration and therefore continuously varies in time. An energy-consistent formulation is assured by explicitly taking into account the dynamic parameter variability in the derivation of the equations of motion. The performance of the novel technique is demonstrated by simulating a dynamic gear contact problem and comparing results against traditional model reduction techniques as well as commercial nonlinear finite element software. Copyright NLPMOR METHOD FOR EFFICIENT GEAR SIMULATIONS 1163 meshes. These approaches separate the bulk deformation of the interacting components from the nonlinear local displacement field at the contact surfaces, computing the former using linear FE on a coarse mesh and obtaining the latter from classical contact theory. Most notably, Vijayakar [3] combines traditional FE with the Boussinesq solution for a point-load acting on an infinite halfspace, whereas Andersson and Vedmar [2] compute the local deformation due to Hertzian contact pressure using a formula derived by Weber and Banaschek [4]. Although these approaches require significantly less degrees of freedom as compared to traditional FE-based contact simulations, the procedure of matching the analytical and FE solutions at a certain depth below the contact surface is computationally involved, resulting in relatively high costs per time increment [5]. Moreover, an energy-consistent adaptation of these approaches to the formalism of flexible multibody dynamics (FMBS) has yet to be pursued. Finally, the question as to whether an infinite half-space serves as a good approximation of the actual contact zone is highly case-specific.The use of conventional FE to represent flexible bodies in multibody simulations can be rendered practical with the aid of model order reduction (MOR) techniques. These techniques originated from the fields of control theory [6] and structural dynamics [7] and were primarily designed for reducing the sizes of, respectively, first-order and second-order linear systems. Using the floating frame of reference formulation [8], the linear elastic deformation of a flexible body is separate...

The potential of the Augmented Kalman Filter algorithm is tested in this paper for joint state-input estimation in structural dynamics field. In view of inverse load identification, the filter is compared with the Transfer Path Analysis Matrix Inversion technique, commonly used for industrial applications. An existing Optimal Sensor Placement strategy for Kalman Filter is adopted and validated on real experimental data. The advantages of the proposed methods, through strain measurements information, are identified in the effort needed for data-acquisition and data-processing. The effectiveness of the filter and the quality of the results are demonstrated in this paper for an industrial test-case, such as a rear twistbeam suspension.

SUMMARYBody flexibility is of ever-increasing importance in multibody simulation. The current state-of-the-art simulation techniques typically have difficulties with systems in which flexible bodies can be loaded in many degrees of freedom (DOFs). However, in many applications, loading is possible in many DOFs but only few are loaded simultaneously at any given moment, such that at any moment only a low-dimensional part of the reduced body flexibility description contributes to the solution. Static Modes Switching, the methodology proposed in this paper, exploits this by judiciously choosing the body flexibility mode set and at any moment only including those modes that contribute to the solution. Static Modes Switching does not improve simulation accuracy; however, it can significantly reduce simulation times. In a numerical experiment, results using Static Modes Switching match results for a conventional model using the same mode set. The approximation errors are negligible compared to the accuracy loss encountered when using a mode set without compensation for the quasi-static response of the high-frequency dynamics, while both simulation results are obtained at a comparable computational cost. Static Modes Switching numerically introduces discontinuities when removing a mode from the mode set. This is overcome by time integration schemes exhibiting high-frequency numerical damping.

In view of the tendency towards ever lighter and more powerful machines and even shorter design cycles, it becomes essential to have virtual prototyping tools that allow for fast and reliable numerical simulations. Current state-of-the-art structural dynamics and flexible multibody simulation techniques usually involve the solution of matrix systems with thousands to millions of variables. Model order reduction schemes are used to keep computational effort affordable at the expense of a minimal loss of accuracy. These techniques typically face difficulties with systems in which flexible bodies can be loaded in many degrees of freedom and rarely allow for accurate local stress and strain evaluation. The present work proposes to address both issues for the particular but frequent case of moving loads or boundary conditions. This behavior is found in most of the contact problems and systems that include sliding components for which many loading or boundary locations are possible but only a few of them are active at a certain moment in time. The proposed scheme exploits a reduction vector space that continuously varies in time by means of a parametric definition of the external load position. Contrary to the majority of the parametric model order reduction schemes that allow mainly for quasi-static parametric variations, the proposed approach can be used efficiently for time simulation of dynamically varying parametric models. This is achieved by considering the implicit time dependency of the reduction vector space using Galerkin projections or alternatively by direct substitution of the reduced kinetic energy, potential energy and generalized forces in the Lagrange equations. It is shown, by developing a consistent mathematical framework, that the price to pay for the very compact reduction space obtained is the evaluation of some extra terms in the equations of motion. Numerical examples are used to assess the accuracy of the proposed method. Results show the potential of this strategy with particular focus on displacement and stress fields and furthermore highlight its real-time potential. Moreover the developed framework together with the numerical results allow for a deeper physical understanding of the complex phenomena related to this category of time varying multiple-input/multiple output systems.

Contact modeling is an active research area in the field of multibody dynamics. Despite the important research effort, two main challenging issues, namely accuracy and speed, are not yet jointly solved. One main issue remains the lack of model order reduction schemes capable to efficiently treat systems where multiple, a priori unknown, input-output locations are present. This work first analyzes the importance of including the necessary residual attachment modes by numerical simulation of two gears meshing in an ad-hoc flexible multibody model. Given the large number of residual attachment modes needed, the methodology named static modes switching is extended and successfully applied to improve efficiency. The method proposes an on line selection of residual attachment modes for accurate local deformation prediction. The applicability to impact problems is discussed through numerical experiments and the automatic selection strategy is based purely on geometrical information. Results show that the method can be applied to gear meshing simulation, obtaining a high level of accuracy while preserving computational efficiency. Comparisons are made between modally reduced models, full nonlinear finite element and the proposed strategy

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An analytical model that allows for a computationally efficient analysis of face-milled spiral bevel gears, is presented. The methodology builds on the consideration that the mating tooth flanks are designed to transmit motion in a nearly conjugate manner. A multibody approach to tooth contact analysis is proposed that assumes contact between rigid surfaces. By taking advantage of the action surfaces for each flank pair, contact is detected in a computationally efficient and accurate way. An analytical load distribution model is used to translate the detected penetration into resulting contact forces, under the assumption that the flank penetration matches the deformation of the teeth if they were flexible. To account for the global tooth deformation Tredgold's approximation in combination with a set of expressions based on beam theory are utilized, while the local contact deformation is modeled based on Hertz theory. The methodology is validated against nonlinear finite element simulations. A comparison in terms of transmission error, contact patterns and contact pressure is provided. Contrary to FE simulations the proposed methodology requires significantly less computational effort, allowing further extension to optimization or system analysis problems.