## 3 Frequency Design Formula and Approximate Analytical Solution for NTMD
### Nonlinear Design Formula of NTMD  
For Equations (26) and (27), the two equations are simultaneously squared and added together, and trigonometric operations are performed to eliminate the phase term to obtain the following equation:  
\[a_{1}^{2}=\left(-\Omega_{2}^{2}a_{2}-\frac{3}{4}\alpha_{2}\Omega_{2}^{2}a_{2} ^{3}+a_{2}\right)^{2}+\lambda_{2}^{2}a_{2}^{2} \tag{30}\]  
The ratio of the steady-state amplitude of the primary structure and the steady-state amplitude of the NTMD can be set to \(E=a_{1}^{2}/a_{2}^{2}\), and let \(E_{2}=a_{2}^{2}\). Then the above equation can be reduced to the following:  
\[E=\left(-\Omega_{2}^{2}-\frac{3}{4}\alpha_{2}\Omega_{2}^{2}E_{2}+1\right)^{2} +\lambda_{2}^{2} \tag{31}\]  
If we consider \(E\) as a function of the mass damper frequency \(\omega_{2}^{2}\), we can obtain a very small value of the amplitude ratio \(E\) by taking the derivative of \(\omega_{2}^{2}\) and setting it to zero. This allows us to find the value of \(\Omega_{2}^{2}\) corresponding to \(E\) approaching a very small value, denoted as \(\dot{E}=\partial E/\partial\left(\Omega_{2}^{2}\right)=0\). By taking the derivative of both sides of Equation (31) with respect to \(\Omega_{2}^{2}\), we can obtain the following expression:  
\[\left(-\frac{3\alpha_{2}\Omega_{2}^{2}}{4}E\left(\Omega_{2}^{2}\right)+1- \Omega_{2}^{2}\right)\left(-\frac{3\alpha_{2}}{4}E\left(\Omega_{2}^{2}\right) -\frac{3\alpha_{2}\omega_{2}^{2}}{4}\ddot{E}\left(\Omega_{2}^{2}\right)-1 \right)=0 \tag{32}\]  
The conditions under which Equation (32) holds after verification are  
\[-\frac{3\alpha_{2}\Omega_{2}^{2}}{4}E_{2}\left(\Omega_{2}^{2}\right)+1-\Omega_ {2}^{2}=0 \tag{33}\]  
This condition holds because the second bracket in Equation (32), \(\dot{E}_{2}\left(\omega_{2}^{2}\right)\), tends to zero when the energy ratio takes a very small value. For the other two terms of the second bracket of Equation (32), it is impossible to equal zero with respect to the hardened elastic element, so Equation (32) holds provided that the value in the first bracket is zero. When Equation (32) holds, for Equation (29), we have  
\[\sigma\lambda_{2}a_{2}-\lambda_{1}a_{2}\left(\frac{3}{4}\alpha_{2}\Omega_{2}^{ 2}a_{2}^{2}-1+\Omega_{2}^{2}\right)+\alpha_{1}\lambda_{2}a_{2}=f\cos b \tag{34}\]For Equation (28), there are  
\[ \begin{gathered}\sigma a_{2}\Big{(}\Omega_{2}^{2}+\tfrac{3}{4}\alpha_ {2}\Omega_{2}^{2}a_{2}^{2}-1\Big{)}+\lambda_{1}\lambda_{2}a_{2}+\\ \alpha_{1}a_{2}\Big{(}\Omega_{2}^{2}+\tfrac{3}{4}\alpha_{2}\Omega_ {2}^{2}a_{2}^{2}-1\Big{)}+\alpha_{1}a_{2}=f\sin b\end{gathered}  \tag{35}\]  
Let the values in the bracketed terms of Equations (34) and (35) both be zero, then we have:  
\[\sigma\lambda_{2}a_{2}+\alpha_{1}\lambda_{2}a_{2}=f\cos b \tag{36}\]  
\[\lambda_{1}\lambda_{2}a_{2}+\alpha_{1}a_{2}=f\sin b \tag{37}\]  
Adding Equations (36) and (37) squared together while using the trigonometric formula, a simple expression for \(a_{2}^{2}\) can be obtained by eliminating the phase term:  
\[a_{2}^{2}=\frac{f^{2}}{\left(\sigma\lambda_{2}+\alpha_{1}\lambda_{2}\right)^{ 2}+\left(\lambda_{1}\lambda_{2}+\alpha_{1}\right)^{2}} \tag{38}\]  
Substituting Equation (38) into Equation (33) yields  
\[\Omega_{2}=\sqrt{1-\frac{3\alpha_{2}f^{2}}{3\alpha_{2}f^{2}+4\Big{[}\left( \sigma\lambda_{2}+\alpha_{1}\lambda_{2}\right)^{2}+\left(\alpha_{1}+\lambda_ {1}\lambda_{2}\right)^{2}\Big{]}}} \tag{39}\]  
Equation (39) is the nonlinear design equation for the nonlinear tuned mass damper frequency derived in this paper. A similar equation was obtained in reference [19], but the equation obtained in the literature only considers the frequency design for the 1:1 resonance condition, while the Equation (39) obtained in this paper is based on the offset parameter \(\sigma\) and yields an improved frequency design equation that yields the optimal design frequency for the 1:1 resonance and the case near the 1:1 resonance.