Comprehensive design strategy of the NSD–TMD controlled system with closed-form frequency-amplitude solutions

Abstract

A negative stiffness device (NSD) utilizing pre-compressed helical springs has been integrated with diverse vibration control techniques. Recently, the incorporation of NSD has been proposed to enhance the efficacy of a tuned mass damper (TMD) controlled system. This study introduces two design strategies for the NSD–TMD system, which functions either as a TMD with an adjustable operating bandwidth or as a pure non-linear energy sink. First, the non-linear force of NSD was expressed in a Taylor series approximation form incorporating a negative linear stiffness and a positive cubic stiffness term. Second, the equations of motion were formulated, and closed-form solutions were derived using the complexification–averaging method. The accuracy of the Taylor series approximation was substantiated within a specified range through a comparison of force–displacement curves. Third, the accuracy of the dynamic closed-form solutions was affirmed by comparing the amplitude–frequency curves with the numerical simulations. For a case study, an actual timber pedestrian bridge with decaying natural frequency was selected, and the two design strategies were examined individually. The vibration control performance of the TMD and NSD–TMD was also comparatively analyzed. The assessment focused on the peak force transmissibility value and the effective operating frequency bandwidth. Results highlighted a notable enhancement in operational performance, particularly for primary structures exhibiting diminishing modal frequencies. In a TMD-controlled system, the efficacy of vibration control deteriorates and may even amplify the dynamic response when a modal frequency shift occurs in the primary structure. Conversely, both design strategies of NSD–TMD can achieve lower transmissibility and broaden the operating bandwidth.