This guide has been produced by the EURAMET project *Traceable Dynamic Measurement of Mechanical Quantities*. More information about this collaborative project can be found on the project’s web site and the linked EURAMET publishable summary of the project. The aim of the guide is to provide practical information and advice to engineers and technicians in industry who carry out dynamic measurements in the course of their work.

We assume that readers of this guide have a basic understanding of the concept of measurement uncertainty and are familiar with the key general technical terms that arise in understanding measurement and measurement uncertainty. Further information about these topics can be found in the Guide to the expression of uncertainty in measurement (GUM) and the International Vocabulary of Metrology (VIM). A basic introduction to measurement can be found here and a link to a beginner's guide to measurement uncertainty is here.

In the main text of this guide we try to explain concepts in plain language and to avoid as much as possible the introduction of mathematical notation. Nevertheless some mathematical concepts (especially terms associated with signal processing and system identification) are unavoidable and we provide links to resources (textbooks, scientific papers, web pages) that help explain the necessary background.

Any mention of commercial products within these web pages is for information only; it does not imply recommendation or endorsement by the partners in this project.

The views expressed in this guide are those of the authors and of the EMRP IND09 project team.

Please note that when selecting many of the links included here you will be leaving our web pages. We have provided these links to other web sites because they have information that may be useful to you. We do not necessarily endorse the views expressed, or concur with the facts presented on these sites.

The production of this guide was funded by the European Metrology Research Programme (EMRP). The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union.

Preparation of this best practice guide was led by Andy Knott and Trevor Esward of the National Physical Laboratory (UK) with extensive input from members of the EMRP IND09 project team.

We have provided a glossary of technical terms that arise frequently in almost all dynamic measurement applications to assist readers who may be new to this topic.

This best practice guide covers the following topics:

Understanding the performance of measuring systems

Introduction to mathematical modelling of dynamic systems

In a number of application areas national and international standards documents exist that address aspects of dynamic measurement in that application area. An example is provided by those standards that are within the jurisdiction of ISO Technical Committee TC 108 on *Mechanical vibration, shock and condition monitoring. *Our guide is not intended to supersede the advice in the standards produced by bodies such as TC 108 but to provide supplementary information and in particular to identify when methods that have been established in one application area can be applied elsewhere. The guidelines set out here are not intended as a standard and will be superseded when national and international standards are published for the mechanical quantities addressed in the IND09 project.

In November 2012 the International Bureau of Weights and Measures (BIPM) organised an international workshop entitled *Metrology for Dynamic Measurement. *The aim was to bring together experts from the NMIs and industrial users who need traceable, reliable and comparable dynamic measurements, as well as those responsible for writing and applying specification standards and/or environment, energy saving and safety legislation.

The workshop identified the need for:

- a collaboration aimed at characterizing the challenges in the field of dynamic measurements traceable to the SI;
- harmonization of terms and methods so that measurements made by various methodologies can be related to each other, thereby enabling comparability of research work;
- a consistent approach to the estimation of measurement results and their uncertainties; and best practice guides and/or documents which could be adopted as the basis for the measurement aspects of international standardization, regulation and/or legislation.

The material produced by the workshop provides useful background information for readers of this guide. The summary report from the workshop can be found here and individual presentations by those attending the workshop can be found here.

A key contributor to the workshop was André Schäfer of HBM who gave an industrial view of current and future challenges in dynamic measurement. A link to his BIPM talk is here and supporting commentary on his slides is here.

A regular series of workshops on the topic of dynamic measurements is held as part of EURAMET research project 1078. Links to presentations at the various workshops can be found on the project's home page.

The guide is not intended to cover all forms of measurement of time varying quantities but to concentrate on the three mechanical quantities that formed the work tasks of EMRP project IND09, dynamic force, torque and pressure. There are many application areas that we do not address such as dimensional, electrical, flow, optical, mass and acoustical metrology and process control. However these applications share many problems in common with measurements of dynamic force, pressure and torque and knowledge from one application domain may be transferred to other related application domains. For example, the boundary between dynamic pressure for mechanical applications and acoustics (both airborne and underwater) is not well-defined and measurement techniques in these application areas are related, for example, in both applications piezoelectric sensors are employed and the frequencies of interest overlap (especially frequencies for underwater acoustics applications).

This guide does not provide detailed advice on digital signal processing (DSP) and digital filter design. We concentrate specifically on those mathematical topics that are directly related to data analysis and uncertainty evaluation and to the correction of time series data for measurement system effects (e.g., deconvolution). However we provide links to information on DSP and related topics for readers who require more detailed information.

For practical purposes a dynamic measurement can be regarded as a measurement where the physical quantity being measured varies with time and where this variation may have a significant effect on the measurement result. This is in contrast to a static measurement in which the value of the quantity of interest stays constant. A measurement may also be quasi-static, i.e., the quantity varies with time but this variation has no effect on the measurement result. In this guide we provide advice on how to handle dynamic measurements as defined in the first sentence of this paragraph, that is, for cases in which it is inappropriate to treat a measurement as static or quasi-static.

Many industrial applications that require the measurement of quantities such as force, torque and pressure are dynamic but owing to the lack of commonly accepted procedures or documentary standards for dynamic calibration, transducers and sensors are often calibrated by static methods. However, mechanical sensors exhibit distinctive dynamic behaviour that shows an increasing deviation from static sensitivity characteristics as frequency increases. The lack of standards for dynamic calibration also applies to the electrical conditioning components of the measurement chain.

Lack of traceability has prevented industry from making reliable measurements in a range of applications with consequences for safety, quality and efficiency of designed systems. Our project has attempted to fill that gap and establish within several European National Measurement Institutes (NMIs) the primary and secondary standards and associated calibration methods to support dynamic measurements of force, torque and pressure.

The need for a dynamic calibration of a sensor or a measuring system arises when the speed of the response of the sensor (or system) is too slow to capture accurately the time history of the measurand. An alternative (and equivalent) way of expressing this is to say that the bandwidth of the measuring system is insufficient to respond fully to the frequency content of the signal being measured.

There are two ways of describing and therefore analysing a dynamic system. The first considers the time domain behaviour of the measurand and the time domain response of the measuring system, and the second considers the frequency domain behaviour of the measuring system. In the time domain the key concept is the impulse response. This is the time-domain output of a system arising from a short impulse-like input signal. An ideal impulse would exist for an infinitely short period of time and have infinitely large amplitude. Such an input signal is unachievable in practical applications so impulse responses are typically measured using a short pulse or a step signal so that the output must be regarded as an approximation to the system's impulse response. If the impulse response of a system is known (perhaps as a result of a calibration or a system identification process), the convolution of the input signal with the system's impulse response predicts the time domain output of the system.

The second approach recognises that it is often more effective and more convenient to represent the system response in the frequency domain, which is referred to as the system's transfer function and is the Laplace Transform of the impulse response. The mathematical operations of convolution and deconvolution are easier to perform in the frequency domain using transfer functions. A calibration of a sensor or measuring system employed in dynamic applications typically aims to derive, by means of a system identification process, the impulse response or transfer function of the system, given a known input and a measured output.

If calibration information about a measuring system is known (in the form a transfer function or an impulse response), this can be used to estimate the input signal by correcting the measured output by means of a deconvolution process.

If a system of interest can be defined as linear and time-invariant, its behaviour is described fully by its impulse response. A__ linear time-invariant__ (LTI) system exhibits the behaviour described by the following characteristics.

Assume that a time domain input _{} to a system produces an output _{}, that an input _{} produces an output _{}, and that _{} produces the output _{} Such a system is linear. Now consider a system in which the input is _{} and the output is _{} and that an input _{} produces an output _{}. Such a system is time invariant because the output is not dependent on the time at which the input is applied. A system that exhibits both characteristics outlined above is a linear time-invariant system.

The section of this guide on mathematical modelling of dynamic systems contains more information about impulse responses, transfer functions, convolution and LTI systems. An introduction can be found on the web site of the Physikalisch-Technische Bundesanstalt (PTB), Germany, at Analysis of dynamic measurements.