The article introduces the nonlinear negative stiffness base isolators (NNBI). These isolators are used to reduce the dynamic responses of single degrees of freedom systems (SDOF). Lagrange’s equation is applied to derive the nonlinear governing equations of motion of the isolated SDOF system. The statistical linearization method is applied to linearise each nonlinear element of highly nonlinear governing equations of motion of the SDOF system isolated by NNBI. The mathematical formulas for the optimal design parameters for NNBI applied to systems with single degrees of freedom are derived using the \(H_2\) and \(H_{\infty }\) optimisation approach. A parametric study is performed using these optimal design parameters. According to that, higher isolator mass ratio values lead to lower frequency ratios. In contrast, it increases at higher increasing length ratio values. Higher values of the isolator mass ratio result in lower damping ratios. Additionally, it reduces with increasing length ratio values. To determine the superior dynamic response reduction capacity (%) of NNBI, the dynamic responses of the NNBI-controlled SDOF systems are compared with the traditional base isolators (TBI)-controlled SDOF systems. Transfer function forms and harmonic balance method are applied to determine the nonlinear dynamic responses of the isolated SDOF systems. According to the transfer function formation results, \(H_{2}\) optimized NNBI and linearised NNBI’s dynamic response reduction capacities are 45.81 % and 45.79 % superior to \(H_{2}\) optimized BI. Whereas, \(H_{\infty }\) optimized NNBI and linearised NNBI’s dynamic response reduction capacities are 66.19 % and 66.17 % superior to \(H_{\infty }\) optimized BI. According to the harmonic balance method results, \(H_{2}\) -optimized NNBI is 65.07 % superior to \(H_{2}\) -optimized BI. \(H_{\infty }\) -optimized NNBI is 10.67 % superior to \(H_{\infty }\) -optimized BI. The suggested technique is further validated by time-domain analysis utilising actual earthquake recordings and random excitations that exhibit power spectra often used in seismic analysis, taking into account the Clough-Penzien power spectrum. The research found that the dynamic response reduction capability of \(H_2\) optimised NNBI is 35.94% higher than that of BI. The dynamic response reduction capability of optimised NNBI is 21.20% higher than that of BI. The analytical study is mathematically accurate and applicable to practical applications.
