1
Vibration Theory

This chapter is an introduction to the theory of vibrations, a term commonly used to define the study of representation models, which are used to describe the behavior of systems in a vibration regime. The scope of this theory is relatively wide and still is in constant development, mainly within the framework of dynamic analysis of structures (modal analysis or other ones), which expands beyond the framework of the course and for which there are already very good books.

Here, we will strive to develop the traditional approach of systems at 1 and 2 degree of freedom (DOF) with coupling in a free or forced regime that allows a good introduction to other, more sophisticated, techniques, and especially allows the comprehension and optimization of methods for vibration testing enforced by normative documents and carried out in testing laboratories.

1.1. Problem

1.1.1. Justification of tests in a mechanical environment

Any industrial system or equipment is subject during its lifetime to the attack of environments that may degrade its performance with the appearance of different failures. The now classic concepts of quality and reliability are therefore used to define its availability involved in all phases of its design. It seems natural and even necessary to minimize as much as possible the causes of these failures.

Therefore, the specifications of the product base are generally guaranteed by a quality assurance plan by acquiring the conviction that the specifications are kept, and this belief is maintained in time through a priori and a posteriori actions (measurements, tests, preventive maintenance, etc.).

Yet, the development of a part of this belief must be done from environmental tests. Here, the need to simulate experimentally is thus clear and thus through the appropriate tests, the constraints of operating the most representative of real conditions that the material meets during its service use. In particular in the case of equipment with strong evolution of design, and/or conditions of use, and/or technologies (new components and/or report processes, and/or materials).

For questions related to reproducibility, accessibility and cost, we will be limited here to laboratory tests, except in special cases.

Thus, we generally define several categories of tests according to the usual chronology:

• - design phase: feasibility and validation of tests.
• - development phase: model tests, qualification, formula, accelerated tests.
• - production phase: batch testing and debugging.
• - storage phase: storage testing, ageing.
• - end of lifetime phase: withdrawal of service.

These tests are subject to the choice of the successful normative documents, whose development can be done through a variety of methods. Through the analysis and synthesis of data of the severity levels of the real environment, if we perfectly know the conditions of use, this results in the concept of "tailorization" (the latter being typically used when the stakes are high as well as the risks of an under- or overvaluation and if the material does not pass the bar of standard tests), or failing that by the adaptation of "standard" levels of severity usually found in the common normative documents edited by (non-exhaustive) organizations such as:

• - at a national level:
  • - the AFNOR (Association Française de NORmalisation, French Standardization Association), NF standards.
  • - the UTE (Union Technique de l'Electricité, Technical Union of Electricity).
  • - the CCT (Comité de Coordination des Télécommunications, Coordinating Committee of Telecommunications), CCTU standards.
  • - the Ministry of Defense: formerly GAMEG13, AIR and SEFT standards, etc.

• - at a European level:
  • - the CENELEC (Comité Européen de Normalisation ELECtronique, European Committee of Electronic Normalization), CECC and EN standards and currently the STANAG 4370 within the NATO area. See, in particular, "Best Practices" by EDSTAR (https://edstar.eda.europa.eu/best-practice-recommendations).

• - at an international level:
  • institutional:
    • - ISO (International Standard Organization).
    • - IEC (Comité Electrotechnique International, International Electro-technical Committee).
    • - UIC (Union Internationale des Chemins de Fer, International Union of Railways).
  • sectorial:
    • - DO 160 or EUROCAE ED 14.
    • - ANSI.

• - at the foreign military standards level:
  • national level:
    • - the DEF STAN 0035 (United Kingdom).
    • - the MIL STD 810 (United States).
    • - the STANAG 4370 (NATO area).
    • - BV Standards (Germany), etc.

• international scale:
  • - MIL STD 810 (because it is considered a reference in all regions of the world).

1.1.2. Quality of environmental tests

• - RG Aero 00011 "quality of environmental tests".
• - Re Aero 601-11 "quality provisions in mechanical environment testing".
• - Re Aero 601-12 "quality provisions in climatic environment-specific tests".

1.1.3. Generating sets of vibrations

• - NF E90-200: Generating sets of vibration - chart intended for the layout of characteristics.
• - NF E90-210: Electrodynamic test facilities used for the generation of vibrations - methods of description of features.
• - NF E90-220: Servohydraulic vibration tests resources - methods of description of features.
• - NF E90-230: Auxiliary tables for vibration generators - methods of description of features.

1.1.4. "Shock and vibration" terminology

• - ISO 2041: Vibration and shock - vocabulary.

1.1.5. Testing methods

For the non-specialized civilian field within a brand of activity, the following are the main methods of the NF or EN or CEI:

• - NF/EN/CEI 60068-2-1 to 2-53 (see detailed list in Appendix I).
• - the defense sector now refers to, for the NATO area, the methods of the STANAG 4370-AECTP 300 (climatic tests) and 400 (mechanical tests). For the international perimeter outside of NATO, these are the methods of MIL STD 810 complemented by those of the DEF STAN 0035 that apply;
• - the civil aviation sector is represented by the DO160 methods.

  In addition, for some specific areas, we can quote:

• - NF EN 61373: Railway applications - rolling material - shock and vibration tests.

1.1.6. Uncertainty in measurement

• - Guide for the expression of the uncertainty in measurement.
• - ISO 5725-1 to 6: Exactitude (accuracy and fidelity) of the results and methods of measurement.
• - FD ISO/TR 22971: Exactitude (accuracy and fidelity) of the results and methods of measurement. Practical guidelines for the use of ISO 5725-2 for the design, implementation and statistical analysis of the results of repeatability and interlaboratory reproducibility.
• - NF ISO 21748: Guidelines relating to the use of repeatability, reproducibility and accuracy estimates in the evaluation of uncertainty in measurements.

  ISO 21748: provides the guidelines in order to:

• - evaluate the measurement uncertainties from data obtained during interlaboratory diffusivity tests conducted in accordance to ISO 5725-2.
• - compare the results of an interlaboratory test with the uncertainty of a measurement obtained by applying formal principles of uncertainty propagation.

ISO 5725-3 provides additional models for the measurement of intermediate fidelity. However, although the same general approach can be applied to the use of these extended models, the assessment of uncertainty from these models is not covered in this international standard.

ISO 21748 is applicable in all areas of measurement and testing requiring the determination of an uncertainty associated with a result. It does not describe the use of repeatability data in the absence of reproducibility data.

ISO 21748 assumes that the recognized non-negligible systematic effects are corrected, either by applying a digital correction in the context of the measurement method, or by searching for and eliminating the origin of these effects.

ISO 21748 recommendations are tentative above all else. It is recognized that, even if they constitute a valid method of uncertainty evaluation, in many aspects, other appropriate methods may also be adopted. In general, it is understood that the references made in ISO 21748 toward results, methods and measurement processes, also apply to results, methods and testing processes.

1.1.7. Interlaboratory comparison and proficiency testing

• - NF EN ISO/CEI 17043: Conformity assessment - general requirements regarding proficiency tests.
• - NF EN DIS 13528: Statistical methods for use in proficiency testing by interlaboratory comparisons.

1.1.8. Metrology management

• - NF EN ISO 10012: Systems of measurement management - requirements for processes and measuring equipment.

The following concepts are also defined:

• - environmental agent: One of the physical, chemical, biological, etc., phenomena that comprises the environment which is...