Test-Driven Development of a Substructuring Technique for the Analysis of Electromagnetic Finite Periodic Structures

In this paper, we follow the Test-Driven Development (TDD) paradigm in the development of an in-house code to allow for the finite element analysis of finite periodic type electromagnetic structures (e.g., antenna arrays, metamaterials, and several relevant electromagnetic problems). We use unit and integration tests, system tests (using the Method of Manufactured Solutions—MMS), and application tests (smoke, performance, and validation tests) to increase the reliability of the code and to shorten its development cycle. We apply substructuring techniques based on the definition of a unit cell to benefit from the repeatability of the problem and speed up the computations. Specifically, we propose an approach to model the problem using only one type of Schur complement which has advantages concerning other substructuring techniques.

Analysis of the Vibrations of Operators’ Seats in Agricultural Machinery Using Dynamic Substructuring

The vibrations produced by an agricultural machine are transmitted to the seat of the tractor operator and must comply with the limitations imposed by international and national regulations. An agricultural machine is generally composed of a tractor that can be linked to different machines required to perform a large number of agricultural tasks. In this paper, substructuring techniques are proposed to investigate the dynamics of the agricultural machine and to evaluate the resulting vibration exposure to the tractor driver in different configurations of the machine. These techniques allow one to couple reduced-order models or experimental models of the component subsystems to obtain the response of the whole system. In the results, the vibration exposure of the tractor operator is evaluated for different configurations of the agricultural machine, by observing the frequency response function (inertance and transmissibility) and the transient response to a given excitation. In conclusion, these techniques allow one to investigate a large number of different configurations and a wide range of operating conditions with a light computational burden and without asking the manufacturers to share sensitive design details.

Friction-induced vibrations in the framework of dynamic substructuring

In complex vibrating systems, contact and friction forces can produce a dynamic response of the system (friction-induced vibrations). They can arise when different parts of the system move one with respect to the other generating friction force at the contact interface. Component mode synthesis and more in general substructuring techniques represent a useful and widespread tool to investigate the dynamic behavior of complex systems, but classical techniques require that the component subsystems and the coupling conditions (compatibility of displacements and equilibrium of forces) are time invariant. In this paper, a substructuring method is proposed that, besides accounting for the macroscopic sliding between substructures, is able to consider also the local vibrations of the contact points and the geometric nonlinearity due to the elastic deformation, by updating the coupling conditions accordingly. This allows to obtain a more reliable model of the contact interaction and to analyze friction-induced vibrations. Therefore, the models of the component substructures are time invariant, while the coupling conditions become time dependent and a priori unknown. The method is applied to the study of a finite element model of two bodies in frictional contact, and the analysis is aimed to the validation of the proposed method for the study of dynamic instabilities due to mode coupling.

Reference Conditions and Substructuring Techniques in Flexible Multibody System Dynamics

The floating frame of reference (FFR) formulation is widely used in multibody system (MBS) simulations for the deformation analysis. Nonetheless, the use of elastic degrees-of-freedom (DOF) in the deformation analysis can increase significantly the problem dimension. For this reason, modal reduction techniques have been proposed in order to define a proper set of assumed body deformation modes. Crucial to the proper definition of these modes when the finite element (FE) FFR formulation is used is the concept of the reference conditions, which define the nature of the deformable body coordinate system. Substructuring techniques, such as the Craig–Bampton (CB) method, on the other hand, have been proposed for developing efficient models using an assembly of their lower order substructure models. In this study, the appropriateness and generality of using the CB method in MBS algorithms are discussed. It is shown that, when a set of reference conditions are not applied, the CB transformation leads to the free–free deformation modes. Because a square CB transformation is equivalent to a similarity transformation that does not alter the problem to be solved, the motivation of using the CB method in MBS codes to improve the solution is examined. This paper demonstrates that free–free deformation modes cannot be used in all applications, shedding light on the importance of the concept of the FE/FFR reference conditions. It is demonstrated numerically that a unique model resonance frequency is achieved using different modes associated with different reference conditions if the shapes are similar.

The influence of a human hand-arm system on the vibrational dynamic behaviour of a compliant mechanical structure

The aim of this study is to provide an approach to predicting human influence on a compliant mechanical structure using a substructuring technique. Substructuring techniques allow us to obtain detailed information on the vibrational behaviour of an assembly of structures by characterization of each structure separately. In this manuscript, a hand-arm system is coupled with a vibrating structure using a substructuring technique. A lightweight and compliant vibrating beam is used to demonstrate the concept. To demonstrate the feasibility of accurately predicting the hand-arm systems’ influence on the beam, we selected one position and tested it using four push forces. The characteristics of the hand-arm system for each configuration were coupled with the dynamic characteristics of the beam only over a frequency range of [5; 300] Hz. For each of the four configurations, the coupling predicts the influence of the hand on the vibrational behaviour of the beam. Reliable predictions were obtained for the vibrational behaviour of the assembly. The results indicate that the substructuring approach predicted the vibrational behaviour of the hand-arm-beam assembly with less than 3% error.

Eliminating Boundary Nodes in Substructuring of Large Structures

Substructuring techniques have been widely used in model reduction of large structures. In these methods a large structure is partitioned into several components and reduced components are built. Boundary degrees-of-freedom (DoF) at the interfaces between components are used to assemble the reduced components and to form a reduced model of the original structure. In the current substructuring methods the boundary DoF or a transformation of these DoF remain in the reduced model. In this paper a methodology is suggested that could eliminate the boundary DoF from the reduced model which in turn leads to having even a smaller reduced model. This method which uses a different partitioning of the DoF of the structure is illustrated for a two-component structure. An example on a simple structure shows how the method can be implemented. The results show that the same level of accuracy compared to a standard substructuring can be obtained with fewer number of DoF in the reduced model.