Anwendung balkenförmiger schwingungstilger für biegeschwingungssysteme

Abstract

Tuned viscoelastic dampers, comprising a mass connected through a viscoelastic link to a vibrating structure, have been considered from many points of view. For effective operation, the resonant frequency of such a damper must be made somewhat lower than the undamped resonant frequency of the structure. If the resonant frequency of the structure is low, some difficulty is encountered in constructing a good tuned damper system. In such a damper system the viscoelastic materials must be both soft and strong enough to give the necessary tuning frequencies without the damper being too heavy. For this reason a damper, based on a different principle for lateral vibrating structures is described in this paper. The main vibrating system is a clamped-free beam. The damper is also a clamped-free beam which is accomodated in an opening of the main system. The damper beam can vibrate freely in this opening and creates a new degree of freedom in the total system. The different equations for the lateral vibrations of the main and of the damper beam can be formulated according to the classical Bernoulli - Euler theory. For theoretical analysis, the system has been simplified. The main beam has partially constant rectangular sections. The auxiliary beam has a constant rectangular section. The total system has four different parts with constant sections. Structural damping of both beams has also been considered by a complex modulus of elasticity. For vibrations in a normal mode, the Euler - Bernoulli equation can be formulated as an ordinary differential equation. A general solution for each beam section is a sum of trigonometric and hyperbolic functions. The coefficients in the general expressions can be determined from the 16 boundary conditions of the beam damper system. The equations for the forced vibrations of the main beam, which can be excited by a prescribed motion at some point in the system or by a harmonic force are written in matrix notation: |A| · |X| = |B| The components of the matrix |A| are functions of the dimensionless frequency parameter and the geometric and the dynamic properties of the system. The matrix |B| contains the exciting functions, and the matrix |X| contains the unknown coefficients of the general solutions. For the solution of matrix |X| the Wilkinson method is used. The computed examples show, for a given beam-damper system, the amplitude of the vibration of the main beam as a function of frequency; the effect of tuning on the natural frequencies of the main beam; the effect of damping on the amplitudes of the main beam; the effect of the mass ratio MDamper/MBeam on the natural frequencies and resonant amplitudes of the main system; and the effect of the attaching point of the damper on the natural frequencies and resonant amplitudes of the main beam.