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
AuthorsVillwock, S., and Pacas M.