Derivation of feasibility conditions in classical vibration problems by using the method of quantifier elimination

• Κύριος συγγραφέας: Ioakimidis, Nikolaos
• Άλλοι συγγραφείς: Ιωακειμίδης, Νικόλαος
• Μορφή: Technical Report
• Γλώσσα: English
• Έκδοση: 2018
• Θέματα: Vibration problems; Design problems; Shear buildings; Natural frequencies; Natural modes; Vibration modes; Rayleigh quotient; Frequency equation; Upper bounds; Lower bounds; Maximization; Minimization; Symbolic computations; Universal quantifier; Existential quantifier; Quantified formulae; Quantifier elimination; Quantifier-free formulae; Computer algebra systems; Mathematica; Προβλήματα ταλαντώσεων; Προβλήματα σχεδιασμού; Διατμητικά κτίρια; Ιδιοσυχνότητες; Ιδιομορφές; Ιδιομορφές ταλαντώσεων; Πηλίκο του Rayleigh; Εξίσωση συχνοτήτων; Άνω φράγματα; Κάτω φράγματα; Μεγιστοποίηση; Ελαχιστοποίηση; Συμβολικοί υπολογισμοί; Καθολικός ποσοδείκτης; Υπαρξιακός ποσοδείκτης; Τύποι με ποσοδείκτες; Απαλοιφή ποσοδεικτών; Τύποι χωρίς ποσοδείκτες; Συστήματα υπολογιστικής άλγεβρας

The modern computational method of quantifier elimination, which leads to QFFs (quantifier-free formulae) after the elimination of the quantifier(s) involved and the related quantified variables, is applied to classical vibration problems here with respect to the natural frequencies of vibration and by using the efficient implementation of quantifier elimination in the popular computer algebra system Mathematica. These vibration problems concern: (i) The natural frequencies of a uniform five-storey shear building. In this problem, by using the frequency equation the natural frequencies of this building are determined (here in dimensionless form) as well as related upper and lower bounds. (ii) The related problem based on the Rayleigh quotient, which is minimized or maximized here by using again the method of quantifier elimination. This minimization/maximization permits the determination of the smallest/largest natural frequency, respectively, of the same five-storey shear building. (iii) The design problem for a three-storey shear building in such a way that its first (smallest) natural frequency is not less than a predefined value. The related QFFs may include a root of a polynomial or, better, be expressed without using such a root, but only with the use of two or three inequalities including the reduced stiffnesses of the three floors of the building and, evidently, the predefined lower bound of its first natural frequency as well. The above three applications show that quantifier elimination constitutes a powerful and, simultaneously, practically useful computational tool in classical vibration problems related to the natural frequencies of vibration although computational restrictions are present. In the present vibration problems, these restrictions mainly concern the total number of variables, which should not be more than four to six.