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

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

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Publication details: 

TitleApplication of the Welch-Method for the Identification of Two- and Three-Mass-Systems
Publication TypeJournal Article
Year of Publication2008
AuthorsVillwock, S., and Pacas M.
JournalIEEE Transactions on Industrial Electronics
Volume55
Issue1
Start Page457
Pagination457-466
Date Published01/2008
ISSN0278-0046
Accession Number9756566
Keywordselectric drives, frequency response, identification, machine control, spectral analysis
AbstractThis 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.
DOI10.1109/TIE.2007.909753