Comparison of Hilbert Transform and Sine Fit Approaches for the Determination of Damping Parameters

For the analysis of rotational damping measurements in the time domain, two different identification procedures were compared. The first procedure that was investigated incorporates a Hilbert transform of the data, which enables an analysis by means of a linear regression calculation. The second approach is a direct nonlinear approximation of a damped sine function to the measurement data. The two approaches were compared using both simulated data and measured data. The results of the comparison are presented below.

For the determination of damping parameters, either forced vibrations can be analysed in the frequency domain, or free, decaying oscillations can be analysed in the time domain. For technical reasons, the rotational damping properties of the components of a dynamic torque calibration device were determined by analyzing exponentially decaying oscillations [1]. The occurring oscillations are decaying sinusoids with the magnitude  $A$, the angular frequency $\omega$, the phase angle  $\varphi$ and the decay rate $\delta$.

$$ x(t)=A \cdot e^{-\delta t} \cdot \sin (\omega t + \varphi)$$

For the measurement of the damping properties, both  $\omega$, and $\delta$ need to be identified.

To determine the damping parameters from measurement data which had been obtained by experiments, two different approaches were compared.

1) Hilbert-transform

The acquired measurement time series data  $x(t)$  can be considered as the real part of a complex analytical signal  $\underline{x}(t)$, giving

$$\underline{x}(t)=x(t)+\tilde{x}(t)\text{.}$$

The Hilbert transform of the real part (i.e. of the measurement data) is the imaginary part $\tilde{x}(t)$ and corresponds to a convolution in the time domain [2]

$$\mathcal{H}(x(t))=\tilde{x}(t)=x(t)*\frac{1}{\pi t}\text{.}$$

Thus, with the Hilbert transform, the envelope of the signal  $A$ can be calculated, giving

  $$ A(t)=\sqrt{x^2(t)+\tilde{x}{^2}(t)}$$

and the instantaneous phase angle can be calculated by means of the arctan function, giving

$$ \varphi (t)=\tan^{-1} \left( \frac{\tilde{x}(t)}{x(t)}\right)\text{.} $$

Based on the envelope in logarithmic scale, the decay rate can be derived using a linear regression. The same applies to the angular frequency with the linear relation from time and phase angle

$$ \varphi (t)=\omega\cdot t\text{.}$$

2) Sine fit

The acquired damped sine signal can also be approximated directly by such a function with an additional offset parameter  $B$ .

$$x(t)=A \cdot e^{-\delta t}\cdot \sin (\omega t + \varphi)+B\text{.}$$

In contrast to the approach using the Hilbert transform, the sine function is nonlinear in its parameters. Therefore, iterative nonlinear regression algorithms need to be applied.

In the first step, both approaches were compared with simulated data. For this purpose, damped sine oscillations with properties similar to the real measurements were generated and superposed with random noise. Both approaches were applied to the simulated data and compared regarding the deviations to the parameters used for the data generation.

Both procedures proved to be suitable for the determination of the parameters; therefore, in the next step, both of them were applied to measured data, and the results were compared. For this purpose, the squared sum of errors (SSE) was calculated based on the residuals and compared. A typical result for a determination based on the Hilbert transform is depicted in Fig. 1.

Figure 1: Measured data (blue) and approximated magnitude of the envelope and of the instantaneous phase (magenta). Depicted in red is the data used for the analysis.

Comparing the SSEs, it becomes apparent that the direct approximation of the damped sine is advantageous: the SSEs were predominantly smaller. Besides, with the Hilbert transform, it is necessary to define a data range manually for the analysis in order to avoid influences from the measurement noise.

Detailed information about the comparison can be found in [2].

References:

[1] Leonard Klaus, Michael 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, 2014, http://www.imeko.org/publications/tc22-2014/IMEKO-TC3-TC22-2014-003.pdf

[2] Leonard Klaus, “Comparison of Hilbert Transform and Sine Fit Approaches for the Determination of Damping Parameters”, Proc. of XXI IMEKO World Congress 2015, Prague, Czech Republic, 2015, http://www.imeko.org/publications/wc-2015/IMEKO-WC-2015-TC15-323.pdf

Contact persons:

Leonard Klaus, FB 1. 7, AG 1. 73, E-Mail: leonard.klaus(at)ptb.de
Michael Kobusch, FB 1. 7, AG 1. 73, E-Mail: michael.kobusch(at)ptb.de