Nonlinear dynamic P‐delta interaction between TMD and the frame structure under proportional internal resonances

Summary The effect of axial loads on the dynamic interaction of a shear frame and tuned mass dampers (TMDs) is evaluated under harmonic and seismic lateral loads. Taking into account the nonlinear geometric stiffness created by the small and large TMD gravity load on the frame columns, the nonlinear differential equations of motion of coupled TMD and frame are derived. Then, the interaction between the small and large TMD and the frame is studied, and the equations of motion are investigated. Nonlinear P‐delta effects couple the TMD and frame responses. By using the method of multiple scales, the asymptotic solution of the nonlinear differential equations is derived, and the effect of two‐to‐one internal resonance, called proportional resonance, is examined. System stability under near‐internal resonance is investigated by plotting the modal amplitude responses vs. variations of excitation frequency and amplitude. The jump and saturation phenomena can be observed in the plots. A parametric study is conducted on the effect of variations of the quantities, e.g., damping ratios of the structure and TMD, the mass ratio, and the axial load ratio, on the response amplitude of the structure and TMD. The time and frequency responses of two linear and nonlinear dynamics for the TMD–structure interaction under harmonic and seismic excitation are also studied and compared.