Tuning of vibration absorbers by an effective modal coupling factor

The simplest family of vibration absorbers comprise inertia absorbers, with an inerter in series with a parallel or series dashpot-spring element, and stiffness absorbers, for which inerter and spring are interchanged. Vibration absorbers are tuned to a structure’s critical resonance, for which scalar structure-absorber equations are derived from an expansion in vibration modes with the absorber in either a free or clamped state. It is demonstrated that the influence from non-targeted residual modes is consistently represented by an effective modal coupling factor (EMCF), evaluated as the relative difference between free and clamped natural frequencies squared. While closely related to the coupling factor for shunted electromechanical transducers, the use of EMCFs determines an entirely new direction for the tuning of mechanical vibration absorbers on a flexible structure. For inertia and stiffness absorbers, the EMCF represents apparent mass or stiffness ratios that also account for the influence from residual modes. Therefore, analytical tuning expressions can be obtained from the derived modal frequency response functions, as demonstrated for a pole-placement method with the EMCF as a new governing absorber parameter. The accuracy of the proposed absorber tuning is validated by a simple numerical example, which shows that analytical tuning expressions can be used directly for vibration damping of flexible structures, provided that the EMCF is included correctly to compensate for the inherent modal interaction.