## Appendix A  
The mathematic proofs of Equation (4) are shown in this appendix section. The energy dissipation of ECD under a cycle of harmonic excitation is  
\[\begin{split} E_{\text{ECD}}&=4\int_{0}^{t_{0}} \frac{2F_{max}}{\frac{\alpha}{\alpha_{T}}+\frac{\alpha_{T}}{\alpha}}\text{d} t=4\int_{0}^{\frac{t_{0}}{2\alpha_{T}}}\frac{2v_{T}F_{max}}{(v_{0}v_{0}( \alpha_{T}))^{2}}\text{d}t\\ &=8v_{cr}F_{max}\int_{0}^{\frac{t_{0}}{2\alpha_{T}}}\frac{1}{ \frac{v_{T}^{2}}{(v_{0}v_{0}(\alpha_{T}))^{2}}}\text{d}t\end{split} \tag{1}\]The indefinite integral in Equation (A1) can be rewritten as  
\[\int\frac{1}{1+\frac{v_{cr}^{2}}{\left(u_{0}\omega\cos\left(at\right)\right)^{2}}} \mathrm{d}t=\int\frac{1}{1+\frac{v_{cr}^{2}\left(\sec\left(at\right)\right)^{2} }{\left(u_{0}\omega\cos\left(at\right)\right)^{2}}}\mathrm{d}t\] (A2)  
Substitute \(s=\omega t\) and \(\mathrm{ds}\,=\omega\mathrm{d}t\) into Equation (A2), it becomes  
\[\int\frac{1}{1+\frac{v_{cr}^{2}}{\left(u_{0}\omega\cos\left(at \right)\right)^{2}}}\mathrm{d}t =\frac{1}{\omega}\int\frac{1}{\frac{v_{cr}^{2}\left(u_{0}^{2} \omega^{2}\right)}{\frac{v_{cr}^{2}\left(u_{0}^{2}\omega^{2}\right)}{u_{0}^{2} \omega^{2}}+\sec^{2}\left(s\right)}}\mathrm{d}s\] (A3) \[=\frac{1}{\omega}\int\frac{\sec^{2}\left(s\right)}{\left(1+\tan^ {2}\left(s\right)\right)\left(1+\frac{v_{cr}^{2}\left(t+\tan^{2}\left(s\right) \right)}{u_{0}^{2}\omega^{2}}\right)}\mathrm{d}s\]  
Substitute \(p=\tan\left(s\right)\) and \(\mathrm{d}p=\sec^{2}\left(s\right)\)ds into Equation (A3), it becomes  
\[\int\frac{1}{1+\frac{v_{cr}^{2}}{\left(u_{0}\omega\cos\left(at \right)\right)^{2}}}\mathrm{d}t =\frac{1}{\omega}\int\frac{1}{\frac{v_{cr}^{2}\left(u_{0}^{2} \omega^{2}\right)}{\frac{v_{cr}^{2}\left(u_{0}^{2}\omega^{2}\right)}{u_{0}^{2} \omega^{2}}+1}}\mathrm{d}p\] (A4) \[=\frac{1}{\omega}\int\frac{1}{\left(\frac{v_{cr}^{2}}{1+1}-\frac{ v_{cr}^{2}}{v_{cr}^{2}p^{2}+v_{cr}^{2}+u_{0}^{2}\omega^{2}}\right)}\mathrm{d}p\] \[=-\frac{v_{cr}^{2}}{\omega^{2}}\int\frac{1}{\frac{v_{cr}^{2}p^{2} +v_{cr}^{2}+u_{0}^{2}\omega^{2}}\mathrm{d}p}+\frac{1}{\omega}\int\frac{1}{p^{ 2}+1}\mathrm{d}p\] \[=-\frac{v_{cr}^{2}}{\omega^{2}}\int\frac{1}{\left(v_{cr}^{2}+u_{0 }^{2}\omega^{2}\right)}\int\frac{1}{\frac{v_{cr}^{2}p^{2}}{v_{cr}^{2}+u_{0}^{2} \omega^{2}}+1}\mathrm{d}p+\frac{1}{\omega}\int\frac{1}{p^{2}+1}\mathrm{d}p\] \[=-\frac{v_{cr}^{2}}{\omega^{2}\left(v_{cr}^{2}+u_{0}^{2}\omega^{ 2}\right)}\int\frac{1}{\frac{v_{cr}^{2}p^{2}}{v_{cr}^{2}+u_{0}^{2}\omega^{2}} +1}\mathrm{d}p+\frac{1}{\omega}\int\frac{1}{p^{2}+1}\mathrm{d}p\]  
Substitute \(v=\frac{v_{cr}p}{\sqrt{v_{cr}^{2}+u_{0}^{2}\omega^{2}}}\) and \(\mathrm{d}v=\frac{v_{cr}}{\sqrt{v_{cr}^{2}+u_{0}^{2}\omega^{2}}}\mathrm{d}p\) into Equation (A4), it becomes  
\[\int\frac{1}{1+\frac{v_{cr}^{2}}{\left(u_{0}\omega\cos\left(at \right)\right)^{2}}}\mathrm{d}t =-\frac{v_{cr}}{\omega\sqrt{v_{cr}^{2}+u_{0}^{2}\omega^{2}}} \int\frac{1}{p^{2}+1}\mathrm{d}v+\frac{1}{\omega}\int\frac{1}{p^{2}+1} \mathrm{d}p\] (A5) \[=\frac{\tan^{-1}\left(p\right)}{\omega}-\frac{v_{cr}\tan^{-1} \left(v\right)}{\omega\sqrt{v_{cr}^{2}+u_{0}^{2}\omega^{2}}}=t-\frac{v_{cr} \tan^{-1}\left(\frac{v_{cr}\tan\left(at\right)}{\sqrt{v_{cr}^{2}+u_{0}^{2} \omega^{2}}}\right)}{\omega\sqrt{v_{cr}^{2}+u_{0}^{2}\omega^{2}}}\]  
According to Equations (A1) and (A5)  
\[E_{\mathrm{ECD}} =8v_{cr}F_{\mathrm{max}}\int_{0}^{\frac{\pi}{2\omega}}\frac{1}{1+ \frac{v_{cr}^{2}}{\left(u_{0}\omega\cos\left(at\right)\right)^{2}}}\mathrm{d}t\] (A6) \[=8v_{cr}F_{\mathrm{max}}(\lim_{t\rightarrow\frac{\pi}{2\omega}}(t- \frac{v_{cr}\tan^{-1}\left(\frac{v_{cr}\tan\left(at\right)}{\sqrt{v_{cr}^{2}+u _{0}^{2}\omega^{2}}}\right)}{\omega\sqrt{v_{cr}^{2}+u_{0}^{2}\omega^{2}}})-0)\] \[=8v_{cr}F_{\mathrm{max}}(\frac{\pi}{2\omega}-\frac{\pi v_{cr}}{2 \omega}\frac{\pi v_{cr}}{2\omega}\frac{\pi v_{cr}}{2\omega})\] \[=\frac{4v_{cr}F_{\mathrm{min}}\pi}{\omega}(1-\frac{v_{cr}}{\sqrt{v_{ cr}^{2}+u_{0}^{2}\omega^{2}}})\]