Summary
A homogeneous beam supported by a nonlinear cubic rotational spring and excited by a prescribed harmonic translational motion was analysed by the harmonic balance method. Harmonic and sub-harmonic solutions were determined as functions of frequency, excitation amplitude and material damping. The results were verified against numerical time integrations of the governing nonlinear differential equations. The numerical method was based on a component mode synthesis technique, using free normal modes and inertia relief attachment modes. It was found that the predictions by the harmonic balance method were in excellent agreement with the numerical solution in a frequency interval covering the three first cantilevered eigenfrequencies of the beam. The only exceptions were two narrow frequency intervals in which the numerical solution showed quasiperiodic oscillations.
Übersicht
Ein homogener Balken, der mit einer kubischen Drehsteifigkeit eingespannt und von einer harmonischen vorgegebenen Bewegung angeregt ist, wurde mit der Methode der harmonischen Balance untersucht. Harmonische und subharmonische Lösungen wurden als Funktion der Frequenz, der anregenden Amplitude und der Werkstoffdämpfung berechnet. Die Ergebnisse wurden mit numerischen Zeitintegrationen der nichtlinearen Differentialgleichungen verglichen. Die numerische Methode war auf einer Methode der dynamischen Substrukturtechnik aufgebaut, wobei freie Schwingungsformen mit statischer Kondensation verwendet wurden. Es zeigte sich, daß in einem Frequenzband, welches die drei ersten Eigenfrequenzen eines einseitig eingespannten Balkens überdeckte, die Vorhersagen der harmonischen Balance in sehr guter Übereinstimmung mit der numerischen Lösung waren. Die einzigen Ausnahmen waren zwei kleine Frequenzbänder, in denen die numerische Lösung quasiperiodische Schwingungen zeigte.
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Gudmundson, P. On the accuracy of the harmonic balance method concerning vibrations of beams with nonlinear supports. Ing. arch 59, 333–344 (1989). https://doi.org/10.1007/BF00534063
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DOI: https://doi.org/10.1007/BF00534063