This paper focuses on the mathematical modelling required to support the development of new primary standard systems for traceable calibration of dynamic pressure sensors. We address two fundamentally different approaches to realising primary standards, specifically the shock tube method and the drop-weight method. Focusing on the shock tube method, the paper presents first results of system identification and discusses future experimental work that is required to improve the mathematical and statistical models. We use simulations to identify differences between the shock tube and drop-weight methods, to investigate sources of uncertainty in the system identification process and to assist experimentalists in designing the required measuring systems. We demonstrate the identification method on experimental results and draw conclusions.

Analysis of dynamic measurements is of growing importance in metrology as an increasing number of applications requires the determination of measurands showing a time-dependence. Often linear time-invariant (LTI) systems are appropriate for modelling the relation between the available measurement data and the required time-dependent values of the measurand. Estimation of the measurand is then carried out by deconvolution.This paper is a tutorial about the application of digital deconvolution filters to reconstruct a time-variable measurand from the measurement signal of a LTI measurement apparatus. The goal of the paper is to make metrologists aware of the potentialities of digital signal processing in such cases. A range of techniques is available for the construction of a digital deconvolution filter. Here we compare various approaches for a form of dynamic model that is relevant to many metrological applications and we discuss the consequences for these approaches of the different ways in which information about the LTI system may be expressed. We consider specifically the methods of minimum-phase all pass decomposition, asynchronous time reversal using the exact inverse filter and the construction of stable infinite impulse response and finite impulse response approximate inverse filters by a least squares approach in the frequency domain. The methods are compared qualitatively by assessing their numerical complexity and quantitatively in terms of their performance for a simulated measurement task.Taking into account numerical complexity and underlying assumptions of the methods, we conclude that when a continuous model of the LTI system is available, or when the starting point is a set of measurements of the frequency response of a system, application of least squares in the frequency domain for the construction of an approximate inverse filter is to be preferred. On the other hand, asynchronous time reversal filtering using the exact inverse filter appears superior when a discrete model of the LTI system is available and when causality of the deconvolution filter is not an issue.

A method for the primary calibration of hydrophones in the frequency range up to 60 MHz is described. The current National Physical Laboratory (NPL) primary standard method of calibrating ultrasonic hydrophones from 500 kHz to 20 MHz is based on optical interferometry. The acoustic field produced by a transducer is detected by an acoustically transparent but optically reflecting pellicle. Optical interferometric measurements of pellicle displacement at discrete frequencies in tone-burst fields are converted to acoustic pressure, and the hydrophone for calibration is substituted at the same point, allowing sensitivity in volts per pascal to be obtained directly. For calibrations up to 60 MHz, the interferometer is capable of measuring the displacement of the pellicle as a function of frequency in a harmonically rich nonlinear field up to and including the 12th harmonic of the shocked field generated by a 5 MHz focusing transducer, allowing hydrophones to be calibrated by substitution in the same field. Sources of uncertainty in the new method have been investigated. Best combined random and systematic uncertainties at the 95% confidence level for the new method are 7% at 20 MHz, 11% at 40 MHz, and 16% at 60 MHz.

A laser interferometer designed to measure acoustic displacements at megahertz frequencies, which has been the basis of a primary standard for the calibration of ultrasonic hydrophones for over ten years, is described. The interferometer is of the Michelson type and is designed to measure the acoustic particle displacement by sensing the movement of a thin plastic membrane placed in the field of an ultrasonic transducer. The acoustic pressure is derived from the measurement of displacement and the hydrophone is calibrated by substituting it for the pellicle. Various sources of uncertainty are described, including acoustooptic corrections, the frequency response of the interferometer, the acoustic properties of the thin membrane, and the lack of ideal plane-wave conditions. Highest calibration accuracy is achievable for membrane hydrophones, with a relative standard uncertainty, for a confidence level of 95 %, of 0.040 at 0.5 MHz, 0.035 from 1 MHz to 7 MHz, 0.046 at 20 MHz and increasing to 0.250 at 60 MHz.The dissemination of the primary standard calibration method, which uses membrane hydrophones as secondary standards, is also described. These hydrophones are shown to have predictable performance properties and long-term stability, making them ideal secondary standards and choice as gold-standard reference devices worldwide.

The national metrology institutes and their partners participating in EUROMET Project 463 developed finite element methods (FEM) for calculation of the pressure distortion coefficients, including their uncertainties, of pressure balances operated at pressures up to 1 GPa and applied them to a PTB 1 GPa piston–cylinder assembly. The methods use axisymmetric models developed and analysed on the basis of the experimental data including the elastic properties of the piston–cylinder materials, pressure-dependent density and viscosity of the pressure-transmitting fluid, dimensions of the piston and cylinder and the piston–cylinder clearance as well as the conditions at the piston–cylinder boundaries. Results such as pressure distributions and radial distortions along the piston–cylinder engagement length, pressure distortion coefficients and their uncertainties as well as piston fall rates dependent on pressure are presented for the free deformation (FD) and the controlled-clearance operating modes of the assembly. The theoretical results are verified by comparing them with the distortion coefficients determined by an experimental method and with the jacket pressure distortion coefficients. The participants' results demonstrate good agreement of the distortion coefficients up to 1 GPa but rather large differences in the uncertainties of the distortion coefficients as well as in the pressure distributions, gap profiles and piston fall rate at maximum pressure. The FEM distortion coefficients obtained for the real piston–cylinder gap profile are in good agreement with the coefficients determined by the experimental method; the FEM values obtained for the ideal gap agree well with the distortion coefficients furnished by the simplified theory. For the real gap model, the uncertainty of the gap geometry is the main uncertainty source. The total standard uncertainties of the controlled-clearance distortion coefficient obtained by different methods lie between (0.078 and 0.17) × 10−6 MPa−1 at 400 MPa and between (0.04 and 0.098) × 10−6 MPa−1 at 1 GPa.

Abstract. Nine european national metrology institutes (NMIs) are collaborating in a new project funded by the european metrology research programme (EMRP) to establish traceable dynamic measurement of the mechanical quantities force, pressure, and torque. The aim of this joint research project (JRP) is to develop appropriate calibration methods, mathematical models, and uncertainty evaluation. The duration of the project is 3 years for a global amount of e3.6 million. It began in September 2011.

This paper is concerned with bringing together the topics of uncertainty evaluation using a Monte Carlo method, distributed computing for data parallel applications and pseudo-random number generation. A study of a measurement system to estimate the absolute thermodynamic temperatures of two high-temperature blackbodies by measuring the ratios of their spectral radiances is used to illustrate the application of these topics. The uncertainties associated with the estimates of the temperatures are evaluated and used to inform the experimental realization of the system. The difficulties associated with determining model sensitivity coefficients, and demonstrating whether a linearization of the model is adequate, are avoided by using a Monte Carlo method as an approach to uncertainty evaluation. A distributed computing system is used to undertake the Monte Carlo calculation because the computational effort required to evaluate the measurement model can be significant. In order to ensure that the results provided by a Monte Carlo method implemented on a distributed computing system are reliable, consideration is given to the approach to generating pseudo-random numbers, which constitutes a key component of the Monte Carlo procedure.

Results for pressure distortion coefficients (λ) obtained by five national metrology institutes with finite element methods (FEM) for PTB and LNE 1 GPa piston-cylinder units are presented. For the PTB unit they demonstrate good agreement of the FEM and the experimental distortion coefficients but rather large differences in the uncertainties, pressure distributions, gap profiles and piston fall rates. The reasons for differences between the theoretical real-gap and experimental λ obtained for the LNE unit require further clarification. The uncertainty of the gap geometry is identified as the main uncertainty source.