Application of quantifier elimination to inverse free vibration problems for inhomogeneous beams and bars

• Κύριος συγγραφέας: Ioakimidis, Nikolaos
• Άλλοι συγγραφείς: Ιωακειμίδης, Νικόλαος
• Μορφή: Technical Report
• Γλώσσα: English
• Έκδοση: 2018
• Θέματα: Inverse problems; Vibration problems; Free vibrations; Harmonic vibrations; Inhomogeneous beams; Clamped beams; Clamped–simply-supported beams; Linear mass density; Flexural rigidity; Deflection; Mode shape; Inhomogeneous bars; Longitudinal rigidity; Elastic restraints; Inequality constraints; Positivity conditions; Computer algebra; Symbolic computations; Cylindrical algebraic decomposition; Universal quantifier; Quantified formulae; Quantifier elimination; Quantifier-free formulae; Mathematica; Αντίστροφα προβλήματα; Προβλήματα ταλαντώσεων; Ελεύθερες ταλαντώσεις; Αρμονικές ταλαντώσεις; Μη ομογενείς δοκοί; Πακτωμένες δοκοί; Δοκοί με πάκτωση–άρθρωση; Γραμμική πυκνότητα μάζας; Καμπτική δυσκαμψία; Βέλος κάμψεως; Ιδιομορφή; Μη ομογενείς ράβδοι; Δυστένεια; Ελαστικοί περιορισμοί; Ανισοτικοί περιορισμοί; Συνθήκες θετικότητας; Υπολογιστική άλγεβρα; Συμβολικοί υπολογισμοί; Κυλινδρική αλγεβρική ανάλυση; Καθολικός ποσοδείκτης; Τύποι με ποσοδείκτες; Απαλοιφή ποσοδεικτών; Τύποι χωρίς ποσοδείκτες

Inverse free vibration problems for inhomogeneous beams under various boundary conditions were extensively studied by Elishakoff and his collaborators during the last two decades. In these problems, the linear mass density of the beam is assumed to have a polynomial form known in advance. Moreover, the mode shape of the beam is also assumed to have a simple polynomial form known in advance and, evidently, satisfying the four boundary conditions at the ends of the beam. Then, on the basis of the related ordinary differential equation, it is possible to determine the unknown flexural rigidity of the beam, which, naturally, should also have a polynomial form. Obviously, the linear mass density is selected to be a continuously positive function, but the same should also happen for the initially unknown flexural rigidity. The latter positivity requirement is the subject of the present results. This positivity is assured by determining the related necessary and sufficient positivity conditions on the whole vibrating beam by using the modern computational method of quantifier elimination, which is mainly based on the Collins cylindrical algebraic decomposition algorithm. Here the implementation of quantifier elimination in the computer algebra system Mathematica is used as the computational tool for the derivation of the present conditions, which leads to the elimination of the universal quantifier in the positivity condition and constitutes the equivalent quantifier-free formula. At first, the simple inverse vibration problem of a clamped beam is studied with respect to the aforementioned positivity requirement. Next, the inverse vibration problem of a beam clamped at one end and simply-supported at the other end is also studied. The resulting positivity conditions for the flexural rigidity of the beam are rather simple only for one or two parameters in the linear mass density of the beam, but they become sufficiently complicated for three parameters. An inverse problem of free axial vibrations of inhomogeneous bars is also studied in brief. The present computational approach constitutes a simple, efficient and mathematically rigorous way for the derivation of positivity conditions in inverse free vibration problems for the flexural/longitudinal rigidities of beams/bars. On the other hand, it constitutes an extension of previous recent quantifier elimination results concerning the related inverse buckling problem, where the same computational approach, that of quantifier elimination, was also successfully used.