Gyroscopic Coefficients

Let

$$\beta_{r,\,s} =\lambda_s\int_{\mit\Omega} {\mit\Phi}_r'\,{\mit\Phi}_s\,d{\mit\Omega} = -\int_{\mit\Omega} {\mit\Phi}_r'\,D\,{\mit\Phi}_s\,d{\mit\Omega}$$

$$=\int_{\mit\Omega} \left(\frac{\partial {\mit\Phi}_r'}{\partial\t... ...}{\partial\phi}\,\frac{\partial{\mit\Phi}_s}{\partial\phi}\right)d{\mit\Omega},$$ (12.269)

where use has been made of Equations (12.235) and (12.236). It follows from Equations (12.262) and (12.263) that

$$\beta_{r,\,s} = \int_{\mit\Omega}\frac{\cos\theta}{\sin\theta}\left(\frac{\part... ...{\partial\theta}\,\frac{\partial{\mit\Phi}_s}{\partial\phi}\right)d{\mit\Omega}$$

$$\phantom{=}-\int_{\mit\Omega}\frac{1}{\sin\theta}\left(\frac{\par... ...\partial\theta}\,\frac{\partial{\mit\Phi}_s}{\partial\phi}\right)d{\mit\Omega}.$$ (12.270)

However, the second term on the right-hand side of the previous equation integrates to zero with the aid of Equation (12.261). Hence, we are left with

$$\beta_{r,\,s}= -\int_{\mit\Omega}\frac{\cos\theta}{\sin\theta}\le... ...\partial\phi}\,\frac{\partial{\mit\Phi}_s}{\partial\theta}\right)d{\mit\Omega}.$$ (12.271)

Let

$$\beta_{-r,\,s} =\lambda_s\int_{\mit\Omega} {\mit\Phi}_r''\,{\mit\Phi}_s\,d{\mit\Omega} = -\int_{\mit\Omega} {\mit\Phi}_r''\,D\,{\mit\Phi}_s\,d{\mit\Omega}$$

$$=\int_{\mit\Omega} \left(\frac{\partial {\mit\Phi}_r''}{\partial\... ...}{\partial\phi}\,\frac{\partial{\mit\Phi}_s}{\partial\phi}\right)d{\mit\Omega},$$ (12.272)

where use has been made of Equations (12.235) and (12.236). It follows from Equations (12.272) and (12.273) that

$$\beta_{-r,\,s} = -\int_{\mit\Omega}\cos\theta\left(\frac{\partial{\mit\Psi}_r}{\... ...r}{\partial\phi}\,\frac{\partial{\mit\Phi}_s}{\partial\phi}\right)d{\mit\Omega}$$

$$\phantom{=}-\int_{\mit\Omega}\frac{1}{\sin\theta}\left(\frac{\par... ...\partial\theta}\,\frac{\partial{\mit\Phi}_s}{\partial\phi}\right)d{\mit\Omega}.$$ (12.273)

However, the second term on the right-hand side of the previous equation integrates to zero with the aid of Equation (12.271). Hence, we are left with

$$\beta_{-r,\,s}= -\int_{\mit\Omega}\cos\theta\left(\frac{\partial{... ...}{\partial\phi}\,\frac{\partial{\mit\Phi}_s}{\partial\phi}\right)d{\mit\Omega}.$$ (12.274)

Let

$$\beta_{r,\,-s} =\mu_s\int_{\mit\Omega} {\mit\Psi}_r''\,{\mit\Psi}_s\,d{\mit\Omega} = -\int_{\mit\Omega} {\mit\Psi}_r''\,D\,{\mit\Psi}_s\,d{\mit\Omega}$$

$$=\int_{\mit\Omega} \left(\frac{\partial {\mit\Psi}_r''}{\partial\... ...}{\partial\phi}\,\frac{\partial{\mit\Psi}_s}{\partial\phi}\right)d{\mit\Omega},$$ (12.275)

where use has been made of Equations (12.248) and (12.261). It follows from Equations (12.262) and (12.263) that

$$\beta_{r,\,-s} = \int_{\mit\Omega}\cos\theta\left(\frac{\partial{\mit\Phi}_r}{\p... ...r}{\partial\phi}\,\frac{\partial{\mit\Psi}_s}{\partial\phi}\right)d{\mit\Omega}$$

$$\phantom{=}+\int_{\mit\Omega}\frac{1}{\sin\theta}\left(\frac{\par... ...\partial\theta}\,\frac{\partial{\mit\Psi}_s}{\partial\phi}\right)d{\mit\Omega}.$$ (12.276)

However, the second term on the right-hand side of the previous equation integrates to zero with the aid of Equation (12.249). Hence, we are left with

$$\beta_{r,\,-s}= \int_{\mit\Omega}\cos\theta\left(\frac{\partial{\... ...}{\partial\phi}\,\frac{\partial{\mit\Psi}_s}{\partial\phi}\right)d{\mit\Omega}.$$ (12.277)

Finally, let

$$\beta_{-r,\,-s} =\mu_s\int_{\mit\Omega} {\mit\Psi}_r'\,{\mit\Psi}_s\,d{\mit\Omega} = -\int_{\mit\Omega} {\mit\Psi}_r'\,D\,{\mit\Psi}_s\,d{\mit\Omega}$$

$$=\int_{\mit\Omega} \left(\frac{\partial {\mit\Psi}_r'}{\partial\t... ...}{\partial\phi}\,\frac{\partial{\mit\Psi}_s}{\partial\phi}\right)d{\mit\Omega},$$ (12.278)

where use has been made of Equations (12.248) and (12.271). It follows from Equations (12.272) and (12.273) that

$$\beta_{-r,\,-s} = \int_{\mit\Omega}\frac{\cos\theta}{\sin\theta}\left(\frac{\part... ...{\partial\theta}\,\frac{\partial{\mit\Psi}_s}{\partial\phi}\right)d{\mit\Omega}$$

$$\phantom{=}+\int_{\mit\Omega}\frac{1}{\sin\theta}\left(\frac{\par... ...\partial\theta}\,\frac{\partial{\mit\Psi}_s}{\partial\phi}\right)d{\mit\Omega}.$$ (12.279)

However, the second term on the right-hand side of the previous equation integrates to zero with the aid of Equation (12.249). Hence, we are left with

$$\beta_{-r,\,-s}= -\int_{\mit\Omega}\frac{\cos\theta}{\sin\theta}\... ...\partial\phi}\,\frac{\partial{\mit\Psi}_s}{\partial\theta}\right)d{\mit\Omega}.$$ (12.280)

Incidentally, the $\beta_{r,s}$ , $\beta_{-r,s}$ , $\beta_{r,-s}$ , and $\beta_{-r,-s}$ are known collectively as gyroscopic coefficients (Proudman 1916).