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Determinantal method for locally modified structures. Application to the vibration damping of a space launcher

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Abstract

In order to improve the dynamic behaviour of an existing or already designed structure, local modifications can be performed by taking advantage of the relative displacement between two points of the structure. A stiffner, damper or viscoelastic rod may be added and its effect on the initial structure must be assessed. A new formulation is developed, based on the response of the initial structure at the attachment points of the local modification. A determinantal equation results, whose roots are the eigenvalues of the modified structure. The equation is solved numerically with a dedicated algorithm and it is shown that this is faster than performing an eigenvalue problem reanalysis. The method is able to deal with both undamped and damped systems, and can handle several modifications simultaneously. It is applied on the last stage of a space launcher, along with a double modal synthesis method.

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Abbreviations

n :

Total number of degrees of freedom

\({n_i^{(s)}}\) :

Number of internal degrees of freedom of substructure (s)

n j :

Number of junction degrees of freedom

\({n_i^{(s)}}\) :

Number of retained eigenmodes for substructure (s)

n B :

Number of retained branch modes

I :

Set of internal degrees of freedom

J :

Set of junction degrees of freedom

E :

Set of excitation degrees of freedom

η s :

Substructure (s) eigenmode generalized coordinate \({(n_I^{(s)}\times 1)}\)

\({{{\bf {\zeta}}}}\) :

Branch mode generalised coordinate (n B  × 1)

F j :

Junction force (n j  × 1)

q j :

Junction displacement (n j  × 1)

\({{\bf {\Psi}}_{s}}\) :

Matrix of constraint modes of the global structure (n × n j )

X Bj :

Matrix of branch modes (n j  × n B )

\({{\bf {\Psi}}^{(s)}}\) :

Substructure (s) constraint mode \({((n_i^{(s)}+n_j)\times n_j)}\)

\({{\bf {\Psi}}_{i}^{(s)}}\) :

Restriction of constraint modes \({{\bf {\Psi}}^{(s)}}\) on I set \({(n_i^{(s)}\times n_j)}\)

\({{\bf {\Psi}}_{j}}\) :

Restriction of constraint modes \({{\bf {\Psi}}^{(s)}}\) on J set (n j  × n j )

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Authors and Affiliations

  1. Laboratoire de Tribologie et Dynamique des Systèmes, UMR-CNRS 5513, École Centrale de Lyon, 36 Avenue Guy de Collongue, 69134, Ecully Cedex, France

    Denis Brizard, Sébastien Besset & Louis Jézéquel

  2. EADS Astrium, 66, Route de Verneuil, 78133, Les Mureaux, France

    Bernard Troclet

Authors
  1. Denis Brizard
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  2. Sébastien Besset
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  3. Louis Jézéquel
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  4. Bernard Troclet
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Correspondence to Denis Brizard.

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Brizard, D., Besset, S., Jézéquel, L. et al. Determinantal method for locally modified structures. Application to the vibration damping of a space launcher. Comput Mech 50, 631–644 (2012). https://doi.org/10.1007/s00466-012-0695-9

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  • DOI: https://doi.org/10.1007/s00466-012-0695-9

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