Dynamic reduction strategies to extend modal analysis approach at higher frequencies

Abstract

Modal analysis on huge finite element models requires a numerical simplification in terms of total number of degrees of freedom (d.o.f.'s). The above aim is generally achieved by choosing some master d.o.f.'s and condensing the structure matrices on those d.o.f.'s. Static condensation (i.e. stiffness matrix reduction) is theoretically an errorless operation; on the other hand, mass condensation can only be approximate in dynamic applications. In order to accomplish matrix condensation, the technique mainly used is the so-called Guyan reduction. The present paper outlines the limitations of the technique, introducing some significant improvements. These are related to the inertia conservation properties of the reduced mass matrix and the condensed mass matrix assembly by means of fictitious and appropriate stiffness connections that are different from those obtained by the stiffness model. The effectiveness of the modified approach is demonstrated with respect to the modal analysis results obtained by Guyan approach, through three different test cases. ᭧

A Reduction of the Equations of the Linear Stationary Vibrating Systems Without Limitations of Dissipation Using the Method of Modal Truncation

“…The obtained mathematical model Eqs. (11) and (14) of the system under investigation in its state variables is based on the application of variables for defining the state of the system provided in the Eq. (4).…”

“…This version differs from the above-mentioned one (see the Eqs. (11) and (14)) in a considerably simpler structure of the differential equations; in addition, it provides an opportunity of a major reduction of these equations using the method of modal truncation.…”

“…Other methods of elimination of high natural frequencies are obtained on the relevant reduction of the equations describing the vibrating system [6][7][8][9][10][11][12][13][14]. Among those methods, the method of modal truncation where the initial system of equations is divided to a number of independent equations is widely used; each of such independent equations describes modal vibrations of one natural frequency and then the equations corresponding to high natural frequencies are rejected (see, for example [10][11][12][13][14]). It is considered that the errors appearing in this case would be permissible, if the frequencies of the rejected modal vibrations are at least 1.5-2 times higher than the natural or resonance frequencies the investigator takes an interest in [15].…”

Free vibration analysis of a von Koch beam

“…'s of their ends, results are accurate for mode I, f 1;n =ð0:3f s;n Þ % 0:4 (the relative error is less than 3‰), and acceptable for mode II, f 2;n =ð0:3f s;n Þ % 1:5 (the relative error being nearly 2%). For higher modes, frequency values are overestimated by more than 50%: more sophisticated techniques (see (Savini and Vivo, 2007) and related references) or Guyan's reduction with respect to a larger number of degrees of freedom must hence be considered. Values reported in Table 3 corresponding to mode III are computed by choosing as masters the 15 d.o.f.…”

Fast approximation of synthesized frequency response functions with automated multi-level substructuring (AMLS)

“…Computation times are rising steeply, whereas time-tomarket should be reduced. Solving these FE models for natural frequencies and eigenvectors in an acceptable amount of time is only possible with high performance computers [4,5]. Therefore, there has been an increasing need for new and innovative solution methods and much research has focused on different ways of increasing the applicable frequency range of deterministic methods like the fast multi-pole boundary element method (MPBEM) [6,7] and the wave based method (WMB) [4,8].…”