24.09.2014
$As part of the European Metrology Research Programme EMRP, methods for the dynamic calibration of torque transducers have been developed within the scope of the joint research project IND09 "Traceable Dynamic Measurement of Mechanical Quantities" [1]. The research objective is the development of procedures to determine the dynamic behaviour of torque transducers by means of a model-based calibration.
The model parameters that describe the dynamic properties of the transducers are to be identified on the basis of measurement data. Dynamic measurements are performed with a dynamic torque measuring device which has been specially developed for this purpose (see Figure 1).
Figure 1: Dynamic torque measuring device with torque transducer.
For the determination of the model parameters of the torque transducer, the model of the transducer has to be extended to include the measuring device. Torque transducers are always mechanically coupled to the environment at both ends.
Therefore, the coupled environment may influence the behaviour of the transducer and must, as here in the case of the calibration, also be taken into account. The different model components of the measuring device and of the transducer are shown in Figure 2.
Figure 2: Model with assigned components of the measuring device (blue) and transducer (orange). The determined angular positions \(\varphi_{\text{i}}\) are shown on the left.
The model of transducer and measuring device consists of a series arrangement of coupled mass moments of inertia \(J\) which are connected by torsional spring-damper elements (\(c\), \(d\)).
For an identification of the unknown parameters of the torque transducer it is indispensable that the model properties of the measuring device are known. Therefore, these parameters had been determined before with auxiliary devices specially developed for this purpose [2, 3].The model of transducer and measuring device is linear and time invariant. Mathematically, it is described as an inhomogeneous differential equation system, and the following is valid:
$$ J\ddot{\varphi}+D \dot{\varphi}+C \varphi=M\text{.}$$
In this equation,\(J\) denotes the matrix of mass moments of inertia, \(C\) the stiffness matrix, \(D\) the damping matrix , \(\varphi\), \(\dot{\varphi}\), \(\ddot{\varphi}\) refers to the vectors of angle, angular speed and angular acceleration, respectively, and \(M\) describes the forced oscillations of the rotational exciter.
For the identification of the model parameters, two transfer functions are determined which are calculated from the measurement data. These transfer functions describe the dynamic behaviour of the upper and of the lower part of the measuring device. The differential equation system allows the respective model parameters to be calculated by means of the transfer functions.
The angular positions above and below the transducer \(\varphi_{\text{H}}\), \(\varphi_{\text{B}}\) cannot be directly measured. However, they can be described by means of a proportionality factor \(\rho\), assuming the proportionality of the transducer signal \(u_{\text{DUT}}(t)\) to the torsion\(\varphi_{\text{H}}\) - \(\varphi_{\text{B}}\).
$$ u_{\text{DUT}}(t)=\rho\cdot(\varphi_{\text{H}}(t)-\varphi_{\text{B}}(t))=\rho\cdot\triangle_{\text{HB}}(t)\text{.}$$
Figure 3: Transfer functions for the model parameter identification.
The measurement results \(u_{\text{DUT}}(t)\), \(\varphi_{\text{M}}\), \(\varphi_{\text{E}}\) allow the model parameters to be identified by non-linear approximation.
To analyse the sensitivity of the two transfer functions to changes of the transducer parameters, two typical transducers were simulated. It turned out that – depending on the transducer – some parameters may be difficult to determine if they hardly influence the transfer function. Conversely, this means also that the parameter changes do not have any influence worth mentioning on the dynamic behaviour of the measuring device.
The procedure is described in detail in [4].
Literature:
[1] C. Bartoli et al., "Traceable Dynamic Measurement of Mechanical Quantities: Objectives and First Results of this European Project" in International Journal of Metrology and Quality Engineering; 3, 127–135 (2012)
DOI: 10.1051/ijmqe/2012020
[2] L. Klaus, Th. Bruns, M. Kobusch, "Modelling of a Dynamic Torque Calibration Device and Determination of Model Parameters" in ACTA IMEKO Vol. 3, No. 2 (2014), pp. 14-18, online at:
http://acta.imeko.org/index.php/acta-imeko/article/view/IMEKO-ACTA-03%20%282014%29-02-05/253
[3] L. Klaus, M. Kobusch, "Experimental Method for the Non-Contact Measurement of Rotational Damping" in Proc. of Joint IMEKO International TC3, TC5 and TC22 Conference, 2014, Cape Town, South Africa, online at:
http://www.imeko.org/publications/tc22-2014/IMEKO-TC3-TC22-2014-003.pdf
[4] L. Klaus, B. Arendacká, M. Kobusch, Th. Bruns, "Model Parameter Identification from Measurement Data for Dynamic Torque Calibration" in Proc. of Joint IMEKO International TC3, TC5 and TC22 Conference, 2014, Cape Town, South Africa, online at:
http://www.imeko.org/publications/tc3-2014/IMEKO-TC3-2014-018.pdf
Contact persons:
Leonard Klaus, Department 1.7, WG 1.73, e-mail: leonard.klaus(at)ptb.de
Michael Kobusch, Department 1.7, WG 1.73, e-mail: michael.kobusch(at)ptb.de