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Identification and evaluation of uncertainties and errors in dynamic models of casted machine tool components

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Abstract

When calculating the dynamic properties of machine tools by means of the finite-element-analysis, highly correlated component models are needed for the derivation of an accurate model of the assembled machine tool structure. For casted machine tool components this cannot always be ensured due to the significant geometric and material uncertainties inherent in the casting process. In this paper, a stepwise modelling approach for casted machine tool components is proposed that allows the derivation of highly correlated models in a defined frequency range. Based on the assumption of superposition the uncertainties and errors involved in the modelling process are identified and evaluated by means of strain energy considerations. By applying this approach to a casted machine tool column, the initial frequency deviations for the considered first 15 structural modes of up to 15 % from the measured values could be reduced below 3 %. Also deeper insight was gained about the influence of the considered uncertainties and errors on the model correlation, as the deviations could be assigned to their sources.

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Notes

  1. For the derivation of the model, damping is neglected due to the very low loss factor of the material [8]. Hence, the undamped eigenfrequencies are assumed to be equal to the damped ones.

  2. The coupled mass model is a MSC.Nastran-specific intermediate formulation between the lumped and the consistent mass matrix approach [23].

  3. This is due to the fact that a modification of ν has the same effect on the (positive) diagonal and the (negative) off-diagonal terms of the element stiffness matrices [K e ], which almost cancel each other out.

Abbreviations

[K]:

Stiffness matrix

[K e ]:

Element stiffness matrix

[M]:

Mass matrix

[M e ]:

Element mass matrix

{f}:

Excitation force vector

{x}:

Nodal displacement coordinates

{x e }:

Element node positions

{ϕ} r :

r-th eigenvector

{ϕ e } r :

r-th eigenvector at element’s degrees of freedom

ω r :

r-th eigenfrequency in rad s−1

ω r X :

Measured r-th eigenfrequency in rad s−1

ω r A :

Analytical r-th eigenfrequency in rad s−1

f r :

r-th eigenfrequency in Hz

\(\Updelta\omega_{r}\) :

Natural frequency difference for the r-th eigenfrequency in %

U r e :

Modal element strain energy for the r-th eigenvector

u r e :

Modal element strain energy density for the r-th eigenvector

\({\widetilde{U}_{e}}^{k}\) :

Averaged element strain energy ratio for the first k eigenvectors

V e :

Element volume in mm3

f obj :

Objective function

E :

Young’s modulus in N mm−2

ν :

Poisson’s ratio

ρ :

Density in kg m−3

m X :

Measured mass in kg

m A :

Analytical mass in kg

x :

Length in mm

r :

Radius in mm

k :

Integer

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Acknowledgments

This work was supported by the German Research Foundation (DFG) within the research unit FOR-1087 “Damping effects in Machine Tools”.

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Authors and Affiliations

  1. Institute for Machine Tools and Industrial Management, Technische Universitaet Muenchen, Boltzmannstrasse 15, 85748, Garching, Germany

    S. Schwarz, A. Sing & M. F. Zaeh

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  1. S. Schwarz
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  2. A. Sing
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  3. M. F. Zaeh
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Correspondence to S. Schwarz or M. F. Zaeh.

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Schwarz, S., Sing, A. & Zaeh, M.F. Identification and evaluation of uncertainties and errors in dynamic models of casted machine tool components. Prod. Eng. Res. Devel. 8, 175–185 (2014). https://doi.org/10.1007/s11740-013-0506-y

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  • DOI: https://doi.org/10.1007/s11740-013-0506-y

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