Introduction

I.1. Experiments for solid and structural mechanics

The modern mechanics of solids and structures relies heavily on the numerical solution of a mechanical problem. Since the early 1970s, the Finite Element Method was widely used for very complex cases. In the present day, a Computer-Aided Design software which generally integrates a small calculation module predicts the behavior of complete mechanical systems, something impossible as few as twenty years ago. The training of a mechanical technician or engineer today largely incorporates this tool, sometimes abandoning the practical work altogether. However, the numerical calculation only responds, in a more or less accurate manner, to an inevitably idealised mathematical problem. It is therefore necessary to validate the simplifying assumptions introduced in the modeling. Furthermore, the values used in the calculation should be well-known (structural damping, binding strength, or boundary conditions). This all requires experimental work which is sometimes difficult, even in the case of a relatively simple behavior that can be easily modelled. Firstly, numerical codes have to be fed with experimental data. For example, the current development of elaborate composite parts requires characterization of the anisotropic stiffness tensor (9 parameters), whereas the contemporary practice reflects only the properties of the plate (4 to 6 parameters) where one dimension is negligible in the face of others. Furthermore, the boundary conditions, either restraint or contact, are often subject to strong assumptions that an experimental approach can improve, by defining a recessed stiffness, for example.

But mechanical design is based on various functioning patterns of the proposed device. There is of course normal functioning, very often under static loading, but also a dynamic functioning linked to possible shocks, abrupt load changes (e.g. emergency stop), or a challenging external environment with variations in temperature or humidity. Moreover, any mechanical device must guarantee a certain lifetime. In a conventional design approach, it is possible to size the apparatus by numerical method for some cases and then experimentally test the prototype with the objective of validation, whereas other cases will only be studied experimentally.

Finally, even though Mechanics is an ancient discipline, the formalism is sometimes lacking. It is then necessary to return to the basic approach of experimental science and conduct experiments for understanding. These situations beyond the mathematical formalism are very common in everyday life: in the study of interaction between two solids in contact - tribology - friction and wear are beyond the scope of intrinsic material properties and modeling of infinitesimal elements, as is usually done in mechanical modeling. More recently, mechanicians were interested in the mechanics of powders, where the material studied is neither a liquid nor a solid. The recent interest in biomechanics also raises the question of the nature of the medium studied; the skin, for example, could be considered as a linear elastic material, or hyper-elastic, anisotropic, viscoelastic, poro-elastic. Therefore, presently, a well-conducted experimental study is the only reasonable approach to this category of problems.

These different types of experiments rely on common concepts such as data processing, choice of sensors, or experimental modelling. However, the strategies are quite different, depending on whether we can or cannot rely on a reliable formalism. The three following examples will illustrate the experimental approaches for different purposes, directly related to the degree of knowledge of a system.

I.1.1. Study of a bicycle wheel; an example of a complete structural validation

This work was conducted as part of a technology transfer from a university lab to an SME, in the form of a doctoral thesis [MOU 98]. The objective was to provide the company with a software to assist the designing of bicycle wheels. In particular, the software should be able, via a Finite Element analysis, to recognize and analyze the natural modes of a wheel.

The program was written in MATLAB© using a graphical interface and numerical analysis facilities. This solution enables the SME not to invest human and financial resources in a generic finite element software; the developed application can be used by the technicians of the research department without any special knowledge of the calculation method.

From the mechanical point of view, the numerical modelling is as follows:

• - the spoke beams are highly slender structures with negligible flexural rigidity and compression. Their behavior has a geometric nonlinearity. So we have: [I.1]
• - given the number of spokes and considering the thickness of the rim relative to its diameter, it is approximated as a simple beam element (not a curve). The section of the rim is complex, such that the beam element is a strong approximation required to maintain a reasonable calculation time;
• - the hub is considered infinitely rigid;
• - the connections are assumed to be perfect; the point of application of stress of the spokes is shifted with respect to the torsion center of the rim.

The main elements of the research method of eigenvalues and eigenvectors are:

• - a search for solutions to the dynamic equation in a pseudo-modal base which enables a reduced calculation time;
• - numerical method of resolution of the nonlinear behavior of the wheel is the incremental Newton-Raphson method. The change of state is divided into n steps, for which the stiffness matrix is updated at each step; the total change is the sum of individual changes.

The software developed is used to find the static behavior, frequency response, and the time response of a bicycle wheel with defined assembly. This software has been validated by an experimental approach, particularly for the frequency response. The assembly is reproduced in Figure I.1.

Figure I.1. Assembly for frequency analysis of a bicycle wheel (according to [MOU 98])

The wheel is mounted on flexible supports simulating free-free boundary conditions. An accelerometer is placed on the upper side of the hub. The excitation takes place on one side of the rim. This excitation requires movement off the periodic plan.

Just as the digital model is questionable due to various assumptions and required approximations, a test like this is only an approximation of the real situation. This is an experimental model, simplifying the structure, the boundary conditions and the load. The experimental model also offers only a few measuring points, based on a priori judgment of the designers of this model, which gives a limited view of the examined physical reality. Finally, the modifications of the experimental model in relation to the physical reality it explores leads to distortion of the obtained solution.

In this specific case, an accelerometer weight sensor is generally likely to alter the natural modes of the wheel. Likewise, the positioning of the accelerometer may also affect the observation of certain occurrences. Therefore, an accelerometer placed at the node of a mode does not allow its identification.

It may be noted, according to these rules, that the choice of positioning is especially important: in an infinitely rigid zone, the accelerometer does not change the stiffness matrix. With regard to assumed or calculated modes, it can be predicted that the accelerometer will be sensitive to different degrees. For example, Figure I.2 shows the "2F-plan" mode which is barely visible and the "umbrella" mode that should be easily identifiable.

Figure I.2. Examples of vibration modes of a bicycle wheel (according to [MOU 98])

Figure I.3. Numerical and experimental response of a bicycle wheel (according to [MOU 98])

Comparison of numerical and experimental approaches gives the results shown in Figure I.3. The first resonance, which corresponds to the "2F off-plan" mode shows a very good theory/experiment correlation. In contrast, the frequencies corresponding to other modes differ more and more, until the error reaches 15%. Even if the prediction model works well, this variance is a representative of many modal analyses: the approximations are manifested especially when the frequency is high.

On the other hand, the theoretical and experimental values of the transmittances are somewhat similar. But these values, which are directly related to damping (structural damping, spoke connections), show the acuteness of the natural frequency to be taken into account: with zero damping, the structure will break; with a critical damping , the natural frequency will be in noise.

This example shows that a mechanical analysis cannot be conducted without the three traditional pillars of science: a well-established mechanical model, a predictive tool using numerical analysis, and experimental tool for validation. From this point of view, the numerical model and the experimental model are both based on a set of assumptions that are generally not the same. A discussion between the numerical and experimental approach of both these sets of assumptions is required to ensure proper understanding of what is being studied.

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