## 5 Thrust and Efficiency of a Wing Performing Two Harmonic Motions  
### Thrust Coefficient of a Wing, Performing Two Combined Oscillations  
Assume that the thrust is generated by superposition of two motions \(q_{1}=q_{10}\cos kt\) and \(q_{2}=q_{20}\cos(kt-\psi)\), where \(\psi\) is a phase shift between the contributing motions (e.g., the wing can simultaneously perform heating and pitching oscillations). In this case the period averaged thrust coefficient can be determined by means of the formula  
\[\langle C_{t_{12}}\rangle=\frac{1}{2}q_{10}^{2}\bigg{[}2\langle\overline{C}_{ t_{1}}\rangle+\frac{2}{\kappa^{2}}\langle\overline{C}_{t_{2}}\rangle+\frac{ \cos\psi}{\kappa}\bigg{(}C_{t_{12}}^{q_{1}q_{2}}+k^{2}C_{t_{12}}^{\hat{q}_{1} \hat{q}_{2}}\bigg{)}+\frac{k\sin\psi}{\kappa}\bigg{(}C_{t_{12}}^{q_{1}q_{2}}- C_{t_{12}}^{\hat{q}_{1}q_{2}}\bigg{)}\bigg{]}; \tag{33}\]where \(\langle\overline{C}_{t_{1}}\rangle=\langle C_{t_{1}}\rangle/q_{10}^{2}\), and \(\langle\overline{C}_{t_{2}}\rangle=\langle C_{t_{2}}\rangle/q_{20}^{2}\kappa=q_ {10}/q_{20}\)- ratio of amplitudes of the participating motions; the coefficients \(\langle\overline{C}_{t_{1}}\rangle\) and \(\langle\overline{C}_{t_{2}}\rangle\) are determined with help of the Formula (25), other coefficients are  
\[C_{t}^{q_{1}q_{2}}=C_{s}^{q_{1}q_{2}}+\overline{C}_{x}^{q_{1}q_{2}}; \tag{34}\]  
\[C_{s}^{q_{1}q_{2}}=\frac{h\lambda}{l^{2}}\int\limits_{-l/2}^{l/2}\left(\frac{ \partial\Gamma^{q_{1}}}{\partial v}\frac{\partial\Gamma^{q_{2}}}{\partial v} \right)_{v=0}dz\overline{C}_{x}^{q_{1}q_{2}}=-\frac{\lambda}{4l^{2}}\iint \limits_{S}\left([p]^{\hat{q}_{1}}\frac{\partial f_{q_{2}}}{\partial x}+[p]^{ \hat{q}_{2}}\frac{\partial f_{q_{1}}}{\partial x}\right)dS; \tag{35}\]  
\[C_{t}^{\hat{q}_{1}\hat{q}_{2}}=C_{s}^{\hat{q}_{1}\hat{q}_{2}}+\overline{C}_{x}^ {\hat{q}_{1}\hat{q}_{2}}; \tag{36}\]  
\[C_{s}^{\hat{q}_{1}\hat{q}_{2}}=\frac{h\lambda}{l^{2}}\int\limits_{-l/2}^{l/2} \left(\frac{\partial\Gamma^{\hat{q}_{1}}}{\partial v}\frac{\partial\Gamma^{ \hat{q}_{2}}}{\partial v}\right)_{v=0}dz\overline{C}_{x}^{\hat{q}_{1}\hat{q}_ {2}}=0,C_{s}^{\hat{q}_{1}\hat{q}_{2}}=\frac{h\lambda}{l^{2}}\int\limits_{-l/2}^ {l/2}\left(\frac{\partial\Gamma^{q_{1}}}{\partial v}\frac{\partial\Gamma^{ \hat{q}_{2}}}{\partial v}\right)_{v=0}dz, \tag{37}\]  
\[C_{t}^{q_{1}\hat{q}_{2}}=C_{s}^{q_{1}\hat{q}_{2}}+\overline{C}_{x}^{\hat{q}_{1 }\hat{q}_{2}}\overline{C}_{x}^{\hat{q}_{1}\hat{q}_{2}}=-\frac{\lambda}{4l^{2} }\int\limits_{S}\left[p\right]^{\hat{q}}\frac{\partial f_{q1}}{\partial x}dS \hat{C}_{s}^{\hat{q}_{1}q_{2}}=\frac{h\lambda}{l^{2}}\int\limits_{-l/2}^{l/2} \left(\frac{\partial\Gamma^{\hat{q}_{1}}}{\partial v}\frac{\partial\Gamma^{q _{2}}}{\partial v}\right)_{v=0}dz, \tag{38}\]  
\[C_{t}^{\hat{q}_{1}q_{2}}=C_{s}^{\hat{q}_{1}q_{2}}+\overline{C}_{x}^{\hat{q}_{1 }q_{2}};\overline{C}_{x}^{\hat{q}_{1}q_{2}}=-\frac{\lambda}{4l^{2}}\iint \limits_{S}\left[p\right]^{\hat{q}_{1}}\frac{\partial f_{q_{2}}}{\partial x}dS; \tag{39}\]  
It follows from the expression for thrust generated by combining two different motions (deformations) that for a given Strouhal number there can exist such ratio of their amplitudes \(\kappa=q_{10}/q_{20}\) and phase shift \(\psi\) for which the gain in thrust force as compared to the case of only one motion becomes a maximum. Denoting these quantities as \(\kappa_{opt}\) and \(\psi_{opt}\) we can determine them imposing a condition of the maximum (if it exists) of the period-averaged thrust coefficient.