Question 2.7: The power transmission system from a diesel engine to a prop...

The complete set of free-body diagrams for this system is shown in Fig. 2.17.We develop the system analysis by beginning at the left-hand  end and writing the describing equation for each element and any necessary connecting point  equations until the system has been completely described.
For the engine (moving parts and flywheel lumped together into an ideal inertia in which friction is ignored),

\frac{d\Omega _1}{dt}=\left(\frac{1}{J_e} \right)(T_e-T_c) .            (2.51)

For the fluid coupling (negligible inertia),

T_c=C_c(\Omega _1^2-\Omega _2^2).                        (2.52)

At the junction between the fluid coupling and the driveshaft,

T_c=T_K.            (2.53)

For the driveshaft (ideal rotational spring – negligible friction and inertia),

\frac{dT_K}{dt}=K(\Omega _2-\Omega _3).                 (2.54)

For the propeller (ideal inertia – negligible friction),

\frac{d\Omega _3}{dt}=\left(\frac{1}{J_P} \right)(T_k-T_w).               (2.55)

Equations (2.51)–(2.55) constitute a necessary and sufficient set of five equations for this system containing five unknowns:\Omega _1,\Omega _2,T_c,T_K, and\Omega _3
Note that additional dampers to ground at points (1), (2), and (3) would be required if bearing friction at these points were not negligible. Rearranging Eq. (2.52) into the form \Omega _2=f_2(T_c,\Omega _1)yields

\Omega _2^2=\Omega _1^2-\frac{T_c}{C_c}                 (2.52a)

or

\Omega _2=SSR\left(\Omega _1^2-\frac{T_c}{C_c} \right),           (2.52b)

where SSR denotes “signed square root,” e.g.,

SSR(X)=\frac{X}{\sqrt{\left|X\right| } } .

Combining Eqs. (2.51) and (2.53), we have

\frac{d\Omega _1}{dt}=\left(\frac{1}{J_e} \right) (T_e-T_k);                (2.56)

and combining Eqs. (2.52b), (2.53), and (2.54) yields

\frac{dT_K}{dt}=K\left[SSR\left(\Omega _1^2-\frac{T_K}{C_c} \right)-\Omega _3 \right] .               (2.57)

Equations (2.55), (2.56), and (2.57) constitute a necessary and sufficient set of equations for this system, containing the three unknown variables\Omega _3,T_K, and\Omega _1 Because of the nonlinearity in the fluid coupling, it is not possible to combine these equations algebraically into a single input–output differential equation. However, in their present form they are a complete set of state variable equations ready to be integrated numerically on a computer (see Chap. 5).