An Inverse Method to Assign Receptances by Using Classical Vibration Absorbers

The classical mass-spring absorber can be thought of as a very simple structural modification. In this paper the inverse problem of assigning receptances to a dynamic system by using one or more simple mass-spring absorbers is considered. The absorber parameters can be determined using selected receptances from the original system. When more than one absorber is applied, the equations that determine the modified-system receptances may be rearranged to form a set of nonlinear algebraic equations, multivariate polynomials in the absorber parameters. The desired absorber parameters are determined by the roots of these equations, which may be found by Newton's method or by using Gröbner bases. Realistic solutions require that positive values be found for the mass and stiffness of each of the added absorbers. Consequently there may be no acceptable solution, a unique solution, or there may be finitely many acceptable solutions. Whenever two receptances are to be assigned at a single frequency by a single absorber, then the available absorber mass and stiffness values are severely restricted. However, arbitrary assignment is possible for many other problems, except for the condition of positiveness on the masses and stiffnesses.