# A Three-Mass System

## Model description:

The transfer function of the three-mass-system is much more complex than it is for one dominant elasticity (two-mass-system).

$${G_{\rm mech}}(s) = \underbrace{{\dfrac{1} {T_{ \Sigma} \cdot s}}}_{G_{\rm rs}(s)} \cdot \underbrace{ \dfrac{ a_{7} \cdot s^{4} + a_{6} \cdot s^{3}+a_{5} \cdot s^{2} + a_{4} \cdot s + 1}{a_{3} \cdot s^{4} + a_{2} \cdot s^{3}+ a_{1} \cdot s^{2} + a_{4} \cdot s + 1}} _{G_{\rm nrs}(s)}$$

with

$T_{\Sigma} = T_{\rm M} + T_{{\rm L}1} + T_{{\rm L}2}$

and

\begin{align*} a_{1}&=d_{1}d_{2}T_{{\rm C}1}T_{{\rm C}2}+T_{{\rm L}2}\left(T_{\rm M}+T_{{ \rm L}1}\right) \cdot \frac{T_{C2}}{T_{\Sigma }}+T_{\rm M}\left(T_{{\rm L}1}+T_{{\rm L}2}\right) \cdot \frac{T_{{\rm C}1}}{T_{\Sigma}} \\ a_{2}&=\frac{T_{{\rm C}1}T_{{\rm C}2}}{T_{\Sigma}}\cdot\left(d_{1}T_{{\rm L}2}\left(T_{\rm M}+T_{{\rm L}1}\right)+d_{2}T_{\rm M} \left(T_{{\rm L}1}+T_{{\rm L}2}\right)\right) \\ a_{3}&=\frac{T_{\rm M}T_{{\rm L}1}T_{{\rm L}2}T_{{\rm C}1}T_{{\rm C}2}}{T_{\Sigma}} \\ a_{4}&=d_{1}T_{{\rm C}1}+d_{2}T_{{\rm C}2} \\ a_{5}&=d_{1}d_{2}T_{{\rm C}1}T_{{\rm C}2}+T_{{\rm L}2}T_{{\rm C}2}+\left(T_{{\rm L}1}+T_{{\rm L}2}\right)\cdot T_{{\rm C}1} \\ a_{6}&=\left(\left(d_{1}+d_{2}\right)T_{{\rm L}2}+d_{2}T_{{\rm L}1}\right)\cdot T_{{\rm C}1}T_{{\rm C}2} \\ a_{7}&=T_{{\rm L}1}T_{{\rm L}2}T_{{\rm C}1}T_{{\rm C}2}. \end{align*}

## Publication details:

 Title Application of the Welch-Method for the Identification of Two- and Three-Mass-Systems Publication Type Journal Article Year of Publication 2008 Authors Villwock, S., and Pacas M. Journal IEEE Transactions on Industrial Electronics Volume 55 Issue 1 Start Page 457 Pagination 457-466 Date Published 01/2008 ISSN 0278-0046 Accession Number 9756566 Keywords electric drives, frequency response, identification, machine control, spectral analysis Abstract This paper deals with the measurement of the frequency response of the mechanical part of a drive for the parameter identification of a plant. The system is stimulated by pseudorandom binary signals. The measurement of the frequency response is part of a system identification procedure being carried out during an automatic commissioning of the drive. For the calculation of the frequency response of the mechanics, the Welch-method is applied for spectral analysis. The Welch-method is known from the fields of communications and measurement engineering. This paper addresses the application of this powerful method for the identification of electrical drives. Investigations have pointed out that the pure utilization of conventional identification strategies does not yield satisfying experimental results. Experimental results presented in this paper point out clearly the efficiency and flexibility of the proposed Welch-method. This paper contains many practical aspects and realization details that are important for their implementation on industrial systems. Although in principle, commercial software tools can be utilized for identifying the parameters of the plant, this paper addresses the implementation of the necessary identification algorithms on the embedded control electronics of the drives. The utilization of the Levenberg-Marquardt-algorithm yields excellent results for the identified parameters on the basis of the measured frequency response data. DOI 10.1109/TIE.2007.909753